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Reflection: What resonated with you the most after viewing this trailer?

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Monthly letter writing campaign - please see communication(s) from Lia.

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This method calls for students working in groups to create problems, point out common errors, find solutions, and then explain the process.

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A Collaborative Approach to Mistake Analysis. 

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A Collaborative Approach to Mistake Analysis. 

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This method calls for students working in groups to create problems, point out common errors, find solutions, and then explain the process.

Courtesy of Emma Chiappetta

"Construct viable arguments and critique the reasoning of others." The Common Core asserts, "Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and-if there is a flaw in an argument-explain what it is."

"Attend to precision. Mathematically proficient students try to communicate precisely to others."

"Look for and express regularity in repeated reasoning. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results."

Many math teachers use mistake analysis as an instructional strategy for helping students to think metacognitively and to develop a deeper understanding of critical procedures. The approach that I will share takes this strategy to the next level by getting students up out of their seats, working collaboratively, and thinking critically about common misconceptions. The activity requires students to not only identify an error but also reflect on their own learning to generate their own ideas of what the common mistakes are and then explain the thinking of their peers.

As you read through the procedure, think about how students are deeply engaging with each of these practices.

Here's how it works. Randomly separate the students into groups of three, and assign each group to a board. Each iteration of the exercise has three rounds.

3-Round Mistake Analysis Exercise

Round one: Each group of students is prompted to generate a problem related to the current unit of study in which one of their peers might make a mistake. They write the problem and solve it incorrectly, intentionally performing the common error that they identified but without noting where the error occurs in the solution.

Round two: Each group rotates to the next board, so that they are looking at a problem and an incorrect solution generated by the previous group. Using a different color marker, they must identify the error in the solution and solve the problem correctly.

Round three: Each group rotates again so that now they are looking at a problem, an incorrect solution, and a correct solution, none of which were written by the group currently standing at that board. They verbally explain to the rest of the class the mistake that was made by the first group and the correct solution provided by the second group.

Benefits for Students

There are a slew of benefits to this approach. My favorite is that students must do several different types of thinking.

Generating an example of a common misconception requires a great deal of metacognition on their part. Identifying an error is an exercise in precision and critical evaluation of a process. Next, they flex their problem-solving skills. Finally, they need to construct an argument and explain someone else's solution. Thinking about the same content from all of these perspectives and modalities encourages a much deeper understanding of the material.

Another aspect of this protocol that I love is that every student has an entry point. Struggling students can reflect on a mistake they have made on a previous assignment, while high-flying students can think critically about a subtle mistake that might stump their classmates.

This collaborative approach to mistake analysis is a fabulous way to facilitate teamwork in a math classroom that, again, allows students at all levels to participate. Even the students struggling the most will be able to talk about what mistakes they have made and what types of problems are most challenging to them. Adding an additional requirement, that only one student may wield the writing utensil during each round, also forces everyone to participate and communicate.

Finally, the multifaceted stages of the exercise boost energy levels in the room. The buzz in the room is palpable as students move from one board to the next, brainstorm, debate, and change their thinking modality. Plus, I've always been a strong believer that we think better when we are standing up and writing big!

This strategy can be used at any stage in the learning process: near the beginning of a unit to nip misconceptions in the bud, in the middle when students are in the thick of formative assessment, or at the end as a review exercise. While I've found success using it in the math classroom, I imagine it could be adapted for any discipline. For example, English teachers might have students generate sentences with subtle but common grammatical errors. The possibilities are endless.

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A Collaborative Approach to Mistake Analysis

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This method calls for students working in groups to create problems, point out common errors, find solutions, and then explain the process.

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As schools nationwide contend with declining math scores, some districts are dedicating more time to the practice of foundational skills.

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Should More Time Be Spent Learning Math Facts? 

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Should More Time Be Spent Learning Math Facts? 

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It's the end of math class at Forest Hills Elementary School in Sidman, Pennsylvania, and after a short break, fourth-grade teacher Dawn McCall's students are getting down to business-with an additional 20 minutes of math work.

McCall's students know the routine: They pair off with a partner-usually a higher-performing student and a lower-performing one-and work through short sets of problems together that target weak or missing skills, like multidigit subtraction or multiplication. One student solves the problem while describing their thinking aloud; the other student offers support and highlights mistakes. Then they switch roles.

This is SpringMath, a short and intensive math intervention that McCall's school is implementing for all students, regardless of their achievement levels. The targeted daily practice is designed to strengthen foundational skills, and it's rapidly putting grade-level math mastery within reach for a lot more students, according to McCall. "To succeed at fractions and decimals, you have to know your multiplication and division facts," McCall said. "If you know those multiplication and division facts, things like reducing fractions is going to be a lot easier. It's that little extra bit of practice every day."

Since Forest Hills Elementary began implementing the math intervention at the beginning of 2018-slowly adding a few grades at a time, beginning with fourth through sixth grade-student test scores for some participating groups have steadily improved, according to assistant superintendent Dr. Rob Dill. In 2022, 56 percent of fourth graders at the school scored proficient or advanced on the Pennsylvania System of School Assessment state test; by fifth grade, 76 percent of those same students scored proficient or above. Meanwhile, math achievement across the nation is headed in the opposite direction. Only 36 percent of fourth graders scored proficient in math on the 2022 National Assessment of Educational Progress (NAEP) after a historic five-point drop following pandemic school closures. For the decade before that, scores had hovered around only 40 percent proficiency. Eighth-grade achievement dropped even more-down eight points from 2019, with only 26 percent of students scoring proficient; 40 percent scored below basic.

Now, a growing number of researchers, school leaders, and educators are calling attention to research-based solutions to combat slippage in math, including the crucial role that adequate practice time plays in student performance. "They are considered dirty words, but drill and practice, and explicit instruction on how to procedurally solve math problems, are evidence-based strategies that work," said SUNY University at Albany math researcher Ben Solomon. There's often a misconception that rote math practice is dull and uninteresting, but "think about how adults thought phonics was boring, but kids actually feel awesome when they can be successful at learning to read," said University of Winnipeg professor of mathematics Anna Stokke. "It's the same with math. Teaching foundational math skills does not need to be mindless and boring. Adults who have already mastered a topic don't view things with the same sense of excitement that kids do. Kids get excited about math when they feel successful, regardless of the topic."

The Fluency Question

A key missing ingredient in higher math achievement is math fluency, especially in the basic operations of addition, subtraction, multiplication, and division. Research shows that fluency in these foundational skills is critical to math performance and achievement, but experts and educators say there's a disconnect between how much practice students need to achieve fluency-especially students who struggle with math-and how much they often get in classrooms. The National Mathematics Advisory Panel includes the importance of practice in basic operations among its top findings and recommendations-"Computational proficiency with whole number operations is dependent on sufficient and appropriate practice to develop automatic recall," the authors write-yet schools and teachers say that, for a variety of reasons, students aren't getting enough practice to solidify foundational concepts.

Researchers like Solomon say that memorizing multiplication tables, timed practice with feedback, and the dreaded worksheet of problems actually matter to long-term math achievement because they help students build the kind of domain knowledge in foundational skills needed to perform more complex math-and that some kinds of math practice are better than others. Evidence shows that procedural and conceptual understanding go hand in hand and are mutually reinforcing, but experts say that in many classrooms an overcorrection has been made that puts too much focus on conceptual understanding at the expense of procedure. "Conceptual understanding really has been overemphasized to the point that students don't have procedural skills," Stokke said.

Lack of procedural skill often keeps students from ever reaching the creative, open-ended side of math. "We need to give them the basic tools and then play with them in many different ways so they can explore the beauty of math," said cognitive scientist Daniel Ansari, professor and Canada research chair at the University of Western Ontario in London, Ontario. "But first they need the basic building blocks."

Obstacles to practice time

Math practice gets shortchanged for a constellation of reasons-an overstuffed curriculum and the drive to cover an overwhelming number of standards; test-prep pressure; and instructional philosophies that consider repetitive practice, sometimes negatively referred to as "rote" or "drill and kill," to be uniformly harmful and to be avoided at all costs.

When Lee Williams, mom to two middle school students in Williamson County, Tennessee, checks in on their homework, she said, the most notable thing is that they're moving through math at lightning speed-a common refrain among parents interviewed for this story. Both kids are tackling complex concepts in math class, yet they haven't mastered the fundamentals of arithmetic that make learning more challenging topics possible. "They move to a new topic every week in elementary and don't ever master the basic functions," she said. "They start with algebraic concepts before they even have multiplication and division facts at the ready-in third grade."

In classrooms, teachers say there's just not enough time for extra practice. Pressured to cover every state standard with equal emphasis, they often can't spend extra time on fundamentals like memorizing multiplication tables, or even on the standards themselves. "It's a huge problem," said Sarah Powell, associate professor at the University of Texas at Austin, who researches math instruction and trains teachers. Some districts go so far as to put out a calendar dictating what every teacher should be working on every week of the year-regardless of how far along (or behind) their students are in the curriculum, Powell noted. "We have trapped ourselves with having so many expectations; at each grade level there are these standards, and you are supposed to cover these 30 things and not let anything go," she said.

But when it comes to standards, it's important to recognize that they're not all of equal importance, said Phil Daro, who helped write the Common Core math standards. Some of the fundamentals, such as single-digit addition, subtraction, multiplication, and division for the youngest students, are critical to spend ample time teaching and have kids practice. "Students need to know foundational math facts fluently," Daro said. "Practice needs to focus on the foundational skills. If you fill that practice bucket with more and more stuff, that will dilute the amount of time and effort available for practicing the real foundation."

Adding to the problem, researchers say that popular math curricula, and even some teacher training programs, de-emphasize practice's crucial role in math learning. "Many of the newer textbooks don't provide enough practice problems," said Doug Rohrer, a professor and math education researcher at the University of South Florida.

Where does practice belong?

A few years ago, SpringMath founder Amanda VanDerHeyden overheard a tutoring session where a student was trying to divide 148 by 3. Instead of doing long division, the student skip-counted by 3s all the way to 148, which didn't work out evenly. After several more attempts, the student finally arrived at the correct solution. "The student has the horsepower, he conceptually understands," she thought to herself at the time. "But he has been deprived of the most efficient way to get there."

Like learning any skill, after students learn a particular concept like long division, in order to consolidate that learning and reproduce it in the most efficient way possible, they have to practice until it becomes second nature. But because learning isn't a linear process, students should regularly be moving between foundational skills and application, with teachers frequently planning tasks that encourage retrieval of that material in a variety of ways. In most math classrooms, says VanDerHeyden, students are required to stop working on a skill too soon, cutting short valuable opportunities for repetition and limiting their ability to lock the learning into long-term memory.

SpringMath is based on a learning framework called the instructional hierarchy, a framework for how students acquire new skills that includes four distinct stages, including acquisition, fluency, generalization, and adaptation. According to this model (and there are a variety of learning frameworks), teaching looks different in the acquisition stage when students are brand-new to a concept, compared with when students can take a skill like long division and adapt it to other things, like complex story problems. A deeper awareness of how learning works, said University of Florida school psychology assistant professor Kathrin Maki, makes it easier to use techniques like inquiry learning at optimal times-but not when kids haven't yet had enough practice to adapt new skills to new problems.

"Inquiry learning isn't necessarily bad," said Maki. "It's that those techniques are only helpful when kids have basic, prerequisite skills. Something we struggle with in math instruction is trying to implement those [inquiry-based] techniques too soon. Kids who are in the acquisition and proficiency stages, those are kids who need a lot of repeated practice and explicit instruction."

The goal of practice is to move foundational skills into long-term memory so they become quick and automatic. Since math constantly builds new skills on top of existing ones, it's "relentlessly hierarchical," said Maki, and once foundational skills like addition and multiplication facts are locked into long-term memory, learning new, more complex math skills like multiplying fractions gets easier; working memory doesn't have to work so hard. "Long-term memory is effectively limitless; that's our superpower," said researcher and teacher Greg Ashman. "Things that are extremely hard to do when you're a novice become easy when you've got lots of stuff in your long-term memory."

Best practice for practice

Improving math fluency doesn't necessarily require endless hours of drilling, research shows. Short but frequent bursts, using a variety of retrieval practice strategies-think flash cards and brain dumps, for example, that get students to identify what they know about a concept, thus strengthening long-term retention-are more effective than longer, less frequent practice. "Teachers tend to focus on getting info into students' heads," said cognitive scientist Pooja Agarwal, coauthor of Powerful Teaching: Unleash the Science of Learning. "But a big component of learning is getting the information out of our heads." Frequent low-stakes quizzes, entry and exit tickets, and some digital games, like FactFreaks, can offer effective retrieval practice.

Retrieval strategies like spaced practice, where students revisit concepts over a period of time with breaks in between, and interleaved practice, where mixing up different types of math problems helps students remember information better than when they work through long blocks of one type of problem, are strongly backed by the science of learning. Interleaving is so effective, University of Florida researcher Rohrer said, because students must decide which problem-solving strategy to use before solving. "In math textbooks, students don't have a chance to choose a strategy because a typical math assignment is devoted to a single concept," he said. "Which means that they usually know the strategy for each problem before they even read it." Interleaving, however, "gives students a chance to learn what they need to know."

In the Forest Hills school district in Pennsylvania, assistant superintendent Dill said a series of evidence-based reforms, including a new curriculum, have rewired how the district teaches math from kindergarten to high school-but the importance of the extra daily practice can't be understated. "Kids who are now fourth- and fifth-graders, they've been doing this for a few years," Dill said of the intervention. "And their skill development is far better than we've ever seen before." And in McCall's classroom, she's seeing overall math confidence blossom. "They know they can do it," she said. "They've learned those math facts."

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Should More Time Be Spent Learning Math Facts?

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As schools nationwide contend with declining math scores, some districts are dedicating more time to the practice of foundational skills.

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Consider these ideas for giving students meaningful praise that will encourage their learning.

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The Power of Compliments in the Math Classroom 

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The Power of Compliments in the Math Classroom 

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Consider these ideas for giving students meaningful praise that will encourage their learning.

Dobrila Vignjevic / iStock

When you see a student who is off task or having disruptive behavior, it can really draw your attention. Our brains are naturally wired to search for the negative or things not going right, especially in the math classroom, where the content can be so black and white, right or wrong.

There's a lot of research about how specific positive praise can decrease unwanted behavior. It might help even more to rewire our thinking to say that we can use specific positive praise, or compliments, to increase desired behaviors.

The recommended praise-to-reprimand ratio is 4:1, with about six praise statements for every 15 minutes. This may seem like a lot, but once you adjust your mindset to see the positive, you'll have access to unlimited opportunities, but these expected behaviors have to be carried out in a targeted way in order to have the most impact both academically and behaviorally.

4 Traits of Motivating Compliments

1. Compliments must be authentic and unexpected. This is one of the largest components that can dictate whether this effort is successful. Students can tell if compliments are disingenuous or an effort to control their behavior.Pulling up next to a child and observing them as they work for a moment before asking, "Can I give you a compliment?" is a great way to catch them off guard in the best of ways. Then once they are ready to receive it, make eye contact and point to where they've done what you noticed.

2. Compliments must be specificand observable. They must name the student or group and exactly what they were doing well. If compliments are vague, it can send an unclear message about which behavior you would like to be repeated. Instead of saying, "Good job on this problem," it will be easier for students to repeat what you want with a statement like, "Strong mathematicians label their representations with context like you did here on your strip diagram, when you labeled the trucks and that it was pumpkins in each truck."

3. Compliments must be timely. Feedback is always the most impactful when it occurs exactly when the behavior occurs. This allows the student, and any others nearby, to obtain true clarity. Here's what this could look like: "Sadie, you just read the problem again when your comprehension was unclear. That's a great strategy instead of guessing. You were being a self-aware mathematician!" If feedback is always given after work has been turned in, we can miss out on witnessing great behaviors that happen during the solving process.

4. Compliments must emphasize process, not ability. It's very important that the compliments be specific to a behavior that students can replicate. For example, praising a student's grit or thoughtfulness and how it has positively impacted their work is so much more powerful than saying someone is "good at math."

How to Increase the Value of a Compliment

In order to increase the validity of the intended change, you can also add on a value potential statement. This is a statement after the compliment that explains why it's important to continue the behavior.

Give compliments on abstract skills. If the student is persevering on a difficult math problem, you can say, "Wow, James.I see you're drawing a representation to help you understand the context. When you use representations, it helps to make relationships and patterns visible in complex problems. That's what good mathematicians do."

You can also give compliments on classroom management skills that impact the learning. This might look like, "Thank you so much, Dhanavi, for putting all of the pieces back where they go in the math station. That's so thoughtful as a member of our class. When you do this, the next group can get started right away and will have more time to learn."

Another type of compliment could be on specific content-related skills that you want students to continue. "Talli, your number line is drawn so accurately with equal spaces, labels, and appropriate intervals. This will help you be more efficient and accurate, especially next year when you use them to find equivalent fractions."

As you deliver these compliments, you shouldn't be afraid to show your excitement. This can be called "Making Students Famous." We want to celebrate students for what they're doing well; especially in a math classroom, where students may have anxiety or reservations about their ability, compliments can even create a positive physical reaction. You'll notice that students will be taken aback and even sit up straighter as they receive your feedback. Why is that? In a post on Individual Matters, psychologist Dr. Katrina Katen notes that "praise triggers the brain's release of dopamine: a 'feel good' neurotransmitter that is strongly associated with pleasure, reward, and motivation."

Getting Started

When you take this back to your classroom to try out, start with one math assignment or part of the math workshop. List a math behavior you want to focus on. As your students are working, call them by name, and celebrate that specific behavior and the value it will hold for them when they continue to do it.

Until it becomes natural to you, you can prepare using the following example.

Focus: Productive struggle

I can see that you are thinking so hard. You used the Three Reads routine and made notes for the important parts. That's what good mathematicians do! When you do this, it helps you to process difficult problems to find a way to get started.

I see that you made a mistake here and crossed it out before you found another way to solve. Sometimes mathematicians have to try ways that don't work to get to a strategy that does. When you aren't afraid to make mistakes, it helps you to take risks and build grit to accomplish great things. I admire that quality!

Wow, [student name]! I noticed you got a little overwhelmed by this problem, but you got up and took a walk, then came back and got right to work. When you do that, it sends oxygen to your brain and helps you think more clearly. That's a great strategy for when you get overwhelmed. I use that strategy sometimes too!

Behaviors to look for: Students making mistakes and trying a new way, students using resources around the room to get started, students not giving up

It can take a few tries to make this feel natural, but it's easy to buy into after you do it just once and see the reaction on one of your students' faces. Next time you're tempted to call out a reprimand, just scan the room for someone who is doing the desired behavior well, and give them the positive feedback they deserve. You will enhance your classroom culture in no time.

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The Power of Compliments in the Math Classroom

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To effectively harness the power of praising students in math class, consider these simple strategies for giving meaningful compliments.

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A monthly after-school activity guides students to have fun exploring science, technology, engineering, and math.

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A monthly after-school activity guides students to have fun exploring science, technology, engineering, and math.

Two years ago my school started a program called STEM Events to provide our students with opportunities to see how the concepts they study in separate classes like chemistry and geometry are really integrated in real-life situations.

The program highlights the connections among the various disciplines of STEM (science, technology, engineering, and math-and sometimes we add the arts, for STEAM). It also encourages our students-all girls-to explore STEM-related concepts in a noncompetitive and fun way and to develop a passion for these concepts and disciplines. Women are underrepresented in the STEM fields, so this is particularly worthwhile for our school.

The STEM subjects are important because they provide a basic quantitative description of the world around us-they pervade every aspect of our lives. And the U.S. Department of Commerce has projected that employment in STEM occupations would increase by 8.9 percent from 2014 to 2024, while other occupations are expected to grow by 6.4 percent-and STEM degree holders have a higher average income than workers with other degrees, even when they work in non-STEM careers.

The more we expose our students to the STEM disciplines in both structured (classrooms, competitions, lectures) and unstructured (clubs, summer camps, after-school activities) environments, the more familiar they become with these concepts, activities, and challenges, which is what led us to create our program.

STEM Events: Informal and Fun

STEM Events is a series of informal gatherings held once a month at the end of the school day for 60 to 75 minutes. Each one is led by a small team of teachers who want to highlight the interdisciplinary nature of what we study in class and what we do in everyday life. These teachers aren't all teachers of STEM disciplines, although generally at least one of the teachers in each team is a STEM expert. We invite all faculty members to participate and share their interests and hobbies, as there is always a science or math connection.

One of our French teachers, for example, has a passion for origami, which inspired a STEM event called "The Math and Art of Origami." It introduced students to the idea that the ancient art of paper folding was not only alive and well but also very useful when applied to sciences like biology (heart stents) and astrophysics (space telescopes) because it inspires novel thinking in situations where something has to be packed small and then become bigger at the delivery point. There are also myriad connections between origami and math, some of which were explored in a session at the National Council of Teachers of Mathematics conference in April in Washington, DC.

In other STEM events, students created binary bracelets (using this lesson plan for the main activity, which followed some unplugged computational thinking); built a snow village complete with LED lights; and programmed and 3D-printed cookie cutters and baked cookies with them.

Other popular events included "Patterns in Weaving," organized and led by an art teacher who had plenty of ideas and resources; "Math Gatherings," which involved going on a campus-wide math treasure hunt; and a friendly, low-stakes Sprouts tournament with our brother school. ( Sprouts is a strategy game created by two mathematicians at Cambridge University in the early 1960s.)

What has worked well for us: interesting, engaging pursuits that students don't naturally associate with the STEM fields, to emphasize the reach and diversity of such disciplines; hands-on, fun activities that catch the students' attention and bring home the importance of looking at a challenge from many perspectives in order to understand and appreciate it in depth; and informal, multilayered conversations that encourage making connections, looking for explanations, sharing ideas, collaborating, identifying and understanding the challenges, and problem solving together. This is not a time for worksheets.

"The girls loved being creative, exploring code, and making jewelry during this STEM activity!" is one of the many positive comments we've heard. The popularity of this program explains its endurance despite the pressure of an already jam-packed school day and overscheduled afternoons-I consider it a real victory since initiatives often die off after a few tries. Our students love spending time with their friends and doing something really different from what they do in school and outside of school.

The flexibility in terms of schedule and content of this program is one of its strengths. As its main goal is to spark and sustain curiosity about the STEM disciplines and there is no special curriculum to cover, our teams have been able to set their own schedule according to their own constraints. More importantly, each team can create its own content based on the passions of the teachers and the students. Plus, when many teachers participate, the ideas multiply and the workload decreases.

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A monthly after-school activity guides students to have fun exploring science, technology, engineering, and math.

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Although Google is sunsetting Jamboard, there are plenty of other online whiteboards out there for teachers. Here are some to try.

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Although Google is sunsetting Jamboard, there are plenty of other online whiteboards out there for teachers. Here are some to try.

Westend61 on Offset / Shutterstock

"Google Jamboard is winding down."This is the message that Google sent to educators, giving them until December 31, 2024, to make some decisions about how and where they want to migrate their Jam data. This certainly is a shock to educators who have become comfortable and confident in Jamboard integration with staff and students. There are a number of options for replacing Jamboard, which can feel overwhelming. How can educators know what will work best for schools, teachers, and students? To make this decision, it's important to consider what programs you have access to at your school site, the age of your students, and the activities that you previously used Jamboard to support.

Google-Recommended Whiteboard Apps

Google has recommended the following whiteboard applications: FigJam by Figma, Lucidspark by Lucid, and Miro. All of these can sync with Google Meet, which is useful for remote and hybrid teaching.

FigJam by Figma is a free online whiteboard with a wide set of collaboration and creative options. FigJam has internal timers, voting options, templates, stickers, widgets, plug-ins, and more. Unlike Jamboard, FigJam allows embedded video, media, and other links. Figma and FigJam have become popular with educators. There is also a button in FigJam that allows you to import your downloaded Jamboard PDFs, and it separates or extracts out each board in your Jamboard automatically. When you create a Figma for Education account, you will also gain access to Figma that allows for realistic prototyping that may benefit MakerSpace, graphic design, engineering, computer science, and other classes.

Lucidspark by Lucid provides many of the same options as found in Jamboard with additional commenting features, voting, and collaborative options. The platform Lucidspark is free for educators and presents a library of templates and seamless integration options with many other digital platforms. Lucidspark allows the educator to support and moderate discussions using their collaborator controls.

Templates in the library are curated by educators, for educators, and there are lessons posted to support these. Lucidspark lets you import your PDF Jamboards, and when you import, you can navigate through the PDF pages. You can also download a picture file of one Jamboard Slide and upload that into Lucidspark if you prefer to see a separate, individual Jamboard Slide.

Miro is another option that is designed for collaborative whiteboard with options for diagramming, data visualization, and feedback. Miro is built to integrate with a variety of applications that may be utilized at your school. You can drag your Jamboard PDFs to a Miro Board and then will need to navigate to the page you want to view or extract the pages to separate them. Educators may individually sign up and have up to 100 students. There is a cost for integration at the institution level.

These three options are the ones that Google has suggested as replacements. It's important for you as a teacher to first check if your school or district has made a decision about the replacement that may work best. These three options have some overlap. They all will let you and your students collaborate. Figma has a look that often appeals to students with a variety of templates and widgets, Lucidspark has an educational focus on diagramming, and Miro prides itself on app integration.

Other Options to Consider

If you're looking beyond the three suggestions above, here are some tools that might feel familiar to students and be particularly useful for younger learners.

Kamihas collaborative learning and whiteboarding tools, accessibility features, integrations, and some specific tools built for teachers and students. One of the biggest differences is that Kami provides increased security options, including allowing the teacher to turn off particular features for students. Students can add annotations and can only alter their own text. They cannot move or delete the work added by another student. The teacher can move and delete something from a student. Kami has a free version and upgrade options.

Canva is another option that may work really well for schools, teachers, and students. Many schools may already have students and staff using Canva in a variety of creative capacities, so it might not feel just like one more thing. Digital whiteboards in Canva help you capture ideas and give you infinite space to collaborate. Use a template to brainstorm or plan out a project and stay in Canva to create the project itself.

There are two ways you can also import PDFs into Canva. The first step is to download your Jamboard as a PDF. If you want your project to still look like a Jamboard with separate slides, navigate to Projects and click Add New, Upload, then click on the file. Each board of the Jamboard imports as a Canva Slide. You can search for sticky notes in the elements and add as needed. If you want to add the Jamboards to a whiteboard, click the whiteboard option in the main menu in Canva, click a blank whiteboard or another template, and then import the PDF, and all of the boards can be brought into one collaborative space.

Finally, if you want to remain in a Google workspace with buttons that may already be familiar to you, Google Slides is an option. One of the reasons why educators appreciated Jamboard was the simplified tool set for collaboration, and this is available in Slides as well.

If you want to use Slides, you will need to save each frame as an image. Then you can put each image on a slide. If you want to mimic the look of a Jamboard, you need to create the shapes that replicate sticky notes, add text, and draw lines and use scribble features.

Although it is a challenge that Jamboard is sunsetting, the good news is that there are many options available to educators. The questions are, what will integrate best with the tools already used in the classroom, what will increase creative output and brainstorming, and what is appropriate for the age of students in the classroom?

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Although Google is sunsetting Jamboard, there are plenty of other online whiteboards out there for teachers. Here are some to try.

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The Top 10 Education Podcasts for 2024  

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The Top 10 Education Podcasts for 2024  

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Here is a list of the top 10 podcast for teachers in 2024. Some of these podcasts will look familiar! I like to give you guys as much *updated* content as possible!

*Please submit 1 reflection per podcast / per episode. Please include the specific Podcast title in your reflection form (e.g. The Harvard EdCast Podcast - Episode: What it Takes to Change a School ).*

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Discover the top education podcasts popular right now. The guide below was put together by Horizon’s team of experts who found some of the most fun, informative and insightful educational podcasts out there today.

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4 Ways to Build a Culture of Professional Learning 

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4 Ways to Build a Culture of Professional Learning 

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4 Ways to Build a Culture of Professional Learning

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When school leaders encourage teachers to improve their craft through better professional development, the entire community benefits.

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Kickstarting Collaboration With a Lockbox Challenge 

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Kickstarting Collaboration With a Lockbox Challenge 

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"With the pandemic, kids have been staring at screens for a couple of years now," says Susan Bistransin, of Prince George's County Public Schools. "And they relish being able to get away from that. I think part of what they lost-not just learning loss during the pandemic-was socialization. Working in teams is something that doesn't come naturally to most kids anymore."

Bistransin and other teachers at Parkdale High School in Riverdale, Maryland, have found that high school students are more engaged in off-screen activities, particularly ones where kids get to work together. In Oscar Peña's financial literacy class, kids collaborate on a regular basis. Eleventh grader Iraima enjoys the fact that she gets to talk with other classmates and share opinions. "I've definitely built a lot of friendships throughout the class."

In this fun lesson, kids race against the clock to test their knowledge of what they've learned about online banking. Like escape rooms, this kind of activity is built around a story or scenario, and it works well for reviewing information at the end of a unit or before an assessment.

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Kickstarting Collaboration With a Lockbox Challenge

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When teens work together to solve puzzles, they practice much-needed social skills-and engage more deeply with the content.

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4 Ways to Make Classroom Participation More Inclusive 

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4 Ways to Make Classroom Participation More Inclusive 

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4 Ways to Make Classroom Participation More Inclusive

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When students feel seen and heard, engagement goes up. These simple strategies help teachers make classroom discussions more equitable for all.

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The Role of Coaching in Developing Playful Lessons 

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The Role of Coaching in Developing Playful Lessons 

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Many teachers want their classroom to be a place where learning is more joyful, but with the demands of content standards, time pressures, and classroom management, that can sometimes feel out of reach. In Denmark, some schools have solved this challenge by adopting an instructional coaching model: Educators trained in playful learning collaborate and consult with teachers regularly to help infuse socialization, activity, creativity, and meaning into their lessons. At Vorbasse School in Billund, when playful learning guide Maria Astrup Bendixen worked with math teacher Nicolaj Bech, they transformed a statistics lesson into a more connected and memorable experience for his middle school students.

To learn more about the research behind the practices seen in the video, check out the links below.

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The Role of Coaching in Developing Playful Lessons

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Giving teachers time to collaborate on making classroom activities more active, social, and meaningful can result in higher student engagement-and deeper understanding of academic content.

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7 Zones Every Middle School Classroom Should Have

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Setting up and organizing a classroom environment with more intention can help maximize the space for student engagement—and learning.

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Creating Opportunities for All Teachers to Grow 

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Creating Opportunities for All Teachers to Grow 

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At P.S. 249, in Brooklyn, New York, educators have leaned into an innovative professional development model called "lab sites" to help each other improve their teaching practice. Once a week, during PD time, teams of teachers get together to explore specific teaching strategies or activities they need help with. Teachers hone their skills by working with a small group of students who volunteer to stay after school, while their grade-level colleagues observe. Having students be part of the teachers' experience allows them real-time teaching and learning interactions, while the students get extra academic support. After the students leave, the group provides feedback to the teacher who was being observed on what went well and what still needs work. Teachers implement what they've learned in their classrooms later in the week and return the following week to let everyone know how it went.

The teachers at P.S. 249 were first introduced to lab sites in a Teachers College workshop about seven years ago, and they loved working with their peers and the students so much, they adapted it and made it part of their regular professional development. The low-stakes practice builds on the strengths of all teachers and gives new teachers, especially, an opportunity to grow and learn from their peers in a safe, supportive environment.

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Creating Opportunities for All Teachers to Grow

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When grade-level teams of teachers work together on skills and strategies to help them improve their practice every week, everyone benefits.

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3 Ways for Schools to Engage Families of Older Students 

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3 Ways for Schools to Engage Families of Older Students 

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From building more effective communication habits to creating a more welcoming back-to-school night to considering setting up a home visit system, there are many ways that secondary school administrators can work to improve family engagement, even as students get older. The best of these approaches will welcome caregivers of all backgrounds to partner with the school, increase community involvement, and ultimately boost student achievement.

To learn more about the benefits of the strategies covered in this video and others, explore these resources cited in the video:

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3 Ways for Schools to Engage Families of Older Students

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Though sustaining a strong connection between school and home can become more difficult in middle and high school, it's worth the extra effort to support long-term student success.

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Cross-grade buddy programs are well-regarded for building positive school culture. With the right planning, they can boost academic and social skills as well-for both older and younger students.

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Bringing Math and Physics to Life With a Model Car Derby 

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Bringing Math and Physics to Life With a Model Car Derby 

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Every year at Casco Bay High School in Portland, Maine, teachers stage a model car race as the culminating event for an integrated math and physics project that helps students connect to content in a hands-on way. Teams of students design, test, and tinker with several car designs before the big day in an attempt to build the fastest-and the slowest-model cars. And on race day, an atmosphere of competition and celebration builds excitement for their demonstrations of learning. Only one team wins, but everyone walks away with a better grasp on higher-level STEM concepts.

To learn more about the research behind the practices seen in the video, check out the links below.

The American Academy of Pediatrics' report on how play enhances development in young children (2018)

Kayleigh Skene, Christine M. O'Farrelly, Elizabeth M. Byrne, Natalie Kirby, Eloise C. Stevens, and Paul G. Ramchandani's meta-analysis and review on whether guidance during play can enhance children's learning and development in educational contexts (2022)

Rachel Parker, Bo Stjerne Thomsen, and Amy Berry's article on learning through play at school (2022)

Rosangela Bando, Emma Näslund-Hadley and Paul Gertler's study "Effect of Inquiry and Problem Based Pedagogy on Learning: Evidence from 10 Field Experiments in Four Countries" (2019)

Lu Zhang and Yan Ma's "A study on the impact of project-based learning on student learning effects: a meta-analysis study" (2023)

Cheng-Huan Chen and Yong-Cih Yang's study "Revisiting the effects of project-based learning on students' academic achievement: A meta-analysis investigating moderators" (2019)

Linda Darling-Hammond, Lisa Flook, Channa Cook-Harvey, Brigid Barron and David Osher's article "Implications for educational practice of the science of learning and development" (2019)

Helena M. Thuneberg, Hannu S. Salmi and Franz X. Bogner's research on the link between cognitive learning and hands-on inquiry in math (2018)

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Bringing Math and Physics to Life With a Model Car Derby

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Giving students a design challenge-and a compelling reason to delve into complex engineering-makes them more engaged, motivated, and committed to deepening their own understanding.

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Boosting Math Skills With Daily Practice 

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Boosting Math Skills With Daily Practice 

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Boosting Math Skills With Daily Practice

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Adding an extra period of math every day gives students an opportunity to review what they’ve learned and strengthen their mathematical thinking.

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3 Activities That Integrate SEL Into Elementary Math 

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3 Activities That Integrate SEL Into Elementary Math 

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Building social and emotional learning (SEL) into elementary math lessons not only is possible but can benefit students' emotional awareness as well as their mathematical thinking and skills. Plus, giving students meaningful opportunities to build empathy and connect their own emotions (and the feelings of those around them) with class content helps them relate to math concepts in new ways.

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3 Activities That Integrate SEL Into Elementary Math

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When social and emotional learning is woven into core math curriculum, elementary students can grow in their math skills, social awareness, and emotional literacy-all at the same time.

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In middle and high school, incorporating mood-boosting, brain-stimulating attention breaks into lessons can head off student inattention, boredom, and fatigue at the pass. To find many more brain breaks designed especially for older students, read Paige Tutt's article for Edutopia titled " 17 Brain Breaks Tailored for High Schoolers." To learn more about the neuroscience behind brain breaks, see Dr. Judy Willis's article " Using Brain Breaks to Restore Students' Focus" or Ethan R. Buch, Leonardo Claudino, Romain Quentin, Marlene Bönstrup, and Leonardo G. Cohen's 2021 research on how rest intervals strengthen the consolidation of skills.

To learn more about each of the breaks shown in the video, visit these sources:

Think Outside the Box activity from Tina Centineo, via X (formerly Twitter)

Jigsaw Jumble activity from "8 Easy, Awesome Brain Break Games for High School Students," by Jenn Breisacher, Student-Centered World

Sequence of Exercises and Calming Break from "54 Educational Brain Breaks Your Students Will Love," by Stephanie Sanders, We Are Teachers

Balloon Toss from AwesomelyTeacher, via X

Rock, Paper, Scissors Tournament from Barre Unified Union School District

Mysterious Image Game from Peter Embleton, via X

Tower of Terror Cup-Stacking Game from Jonathan Alsheimer, via X

Collaborative Class Story from Dora Hartsell, via X

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Students are still learning, even during brief breaks, which serve to help them decompress, refocus, and process new information.

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4 Ways to Help Teens Strengthen Social and Emotional Literacy 

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4 Ways to Help Teens Strengthen Social and Emotional Literacy 

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While much energy is invested in helping young children begin to identify and articulate their emotions, regulate themselves, and learn social skills, that focus tends to taper off as students get older. But social and emotional skills are just as critical in middle and high school, as students learn how to navigate more complex situations and relationships, confront higher expectations that bring more anxiety and stress, and grapple with bullying and social pressures. From gratitude journaling to mindfulness exercises, encouraging positive inner voice to connecting SEL to curriculum, these four strategies will make space for teens to keep building the lifelong skills that will help them throughout school and beyond.

To learn more about the research cited in the video, check out the links below.

Christina Cipriano, Michael J. Strambler, Lauren H. Naples, Cheyeon Ha, Megan Kirk, Miranda Wood, Kaveri Sehgal, Almut K. Zieher, Abigail Eveleigh, Michael McCarthy, Melissa Funaro, Annett Ponnock, Jason C. Chow, and Joseph Durlak's meta-analysis of evidence for social and emotional learning in universal school-based SEL interventions (2023)

Lan Nguyen Chaplin, Deborah Roedder John, Aric Rindfleisch, and Jeffrey J. Froh's studies on the impact of gratitude on adolescent materialism and generosity (2018)

Carolyn MacCann, Yixin Jiang, Luke E. R. Brown, Kit S. Double, Micaela Bucich, and Amirali Minbashian's meta-analysis on how emotional intelligence predicts academic performance (2020)

Sander Thomaes, Iris Charlotte Tjaarda, Eddie Brummelman, and Constantine Sedikides's research on how self-talk can benefit the mathematics performance of children with negative competence beliefs (2019)

Róisín M. Flanagan and Jennifer E. Symonds's literature review on children's self-talk in naturalistic classroom settings in middle childhood (2022)

Mary L. Phan, Tyler L. Renshaw, Julie Caramanico, Jeffrey M. Greeson, Elizabeth MacKenzie, Zabryna Atkinson-Diaz, Natalie Doppelt, Hungtzu Tai, David S. Mandell, and Heather J. Nuske's review of outcomes and evidence quality in research on mindfulness-based school interventions (2022)

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4 Ways to Help Teens Strengthen Social and Emotional Literacy

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When middle and high school students are given the chance to build social and emotional skills during the school day, they get better at self-regulation and grow more socially aware-both in and out of the classroom.

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Deepening Academic Understanding With Tangible Experiences 

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Deepening Academic Understanding With Tangible Experiences 

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Books offer the opportunity to step into someone else's world, and middle school ELA teacher Jennifer Montgomery's seventh-grade class did just that during their book study of Linda Sue Park's A Long Walk to Water. In a Socratic seminar, students discussed how the author builds empathy for the characters, whose lives are playing out during two different time periods in Sudan. But to bring the book's themes to life for her classroom in Eminence, Kentucky, Montgomery had her middle school students do their own walk-traveling a mile while carrying a jug filled with water. In the discussion that followed, students shared their own reflections and insights, emerging from this exploration with a stronger sense of empathy-and a solid experience of perspective-taking that will stick with them for years to come.

To learn more about the research behind the practices seen in the video, check out the links below.

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Deepening Academic Understanding With Tangible Experiences

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By providing middle school students a chance to walk in someone else's shoes, teachers can help them develop empathy-and a stronger connection to content.

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Building Problem-Solving Skills Through 'Speed Dating' 

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Building Problem-Solving Skills Through 'Speed Dating' 

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When English language arts teacher Cathleen Beachboard was trying to come up with new ways of encouraging her students at Fauquier High School in Warrenton, Virginia, to be solution-oriented, she noticed that while kids often get stuck on their own problems, they are quick to find solutions to other people's. Her husband jokingly said, "Wouldn't it be cool if you could just speed-date problems?" And the problem speed-dating activity was born. Beachboard has used this quick and simple strategy in her classes ever since.

"Today we were working on argumentative writing. But you could use this in any scenario, in any classroom, for any stuck point that the kids are having with any piece of content," she says.

In addition to hearing multiple ideas for solutions to their problems, kids get to know each other-and learn to see each other as experts in the room.

"Problem speed dating ultimately teaches them that as a society or as a person, you're not alone," Beachboard says. "And reaching out to others is important because that's where the magic happens. It happens when we use each other and we use the strength that every person has."

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Building Problem-Solving Skills Through 'Speed Dating'

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When students solve each other's problems, they gain confidence in their own skills and witness the power of collaboration in real time.

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To say that students are engaged in Nicole Goepper's French class at Fauquier High School in Warrenton, Virginia, is an understatement. Her lessons are peppered with games, songs, and even dancing, which keeps students on their toes. Her whiteboard relay strategy is especially captivating-and it can be used to assess any concept that can be assigned as a task that students complete by working together.

In Goepper's class, she most frequently uses it for verb conjugation. She calls out a verb, counts down from three, "and it's off to the races!" Each student on the team completes one subject-verb portion of the task and passes to their adjacent teammate, until all subjects are conjugated.

Says Goepper, "If somebody gets stuck, they've got their team to help them, but they're still the one that's actually writing it. Once the task goes all the way around and is completed, they stick the whiteboard on their head and say, "Fini!" And by sticking it on their head, the other teams can't see what they wrote. Also, when it's on their head, they can't change answers. No cheating in this fast-paced game!

Goepper awards points for each round: "First place, they get three points. Second place gets two points. And then if any team just has it correct, they get a point. If they did get it wrong initially, sorry, no points, but if they fix it and show me, OK, you get a point.They are encouraged to go back and fix mistakes."

At the end of all the rounds, the team with the most points gets paper euros, which they save up throughout the year and use for prizes.

The benefits of this strategy are hard to number. In addition to being a fun informal assessment, Goepper adds, "it's an easy prep for the teacher. It can be done impromptu. If you start seeing that something needs to be reviewed, it's easy to go into. Because they're in teams, there's that interaction because they want to win. And by having the pressure from your teammates wanting to win, you get a high level of achievement. It might be messy handwriting, but that's OK. "

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The team competition is fierce in this informal assessment activity, in which students have to work together to win.

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Exploring Writing, Social Studies, and Math in a Map Challenge

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Exploring Writing, Social Studies, and Math in a Map Challenge

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Each fifth grader in Andrea Ketcham's class in Eminence, Kentucky, has been charged with creating a world and asking someone else to navigate it. In this interdisciplinary project, students use a range of skills-from spatial awareness and mapmaking to communication skills and coding. To touch on English language arts standards, students use descriptive language to write about their imaginary land. Next, to align with social studies standards, they're tasked with drawing a map depicting it, complete with key features like grid points and a compass rose. When their maps are complete, the students come up with questions that challenge a classmate to identify characteristics of their land and program a Sphero BOLT robot to follow directions and get to a specific destination. By the end, students have mastered content from multiple subject areas-and stayed engaged and excited through the whole process.

To learn more about the research behind the practices seen in the video, check out the resources below.

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Exploring Writing, Social Studies, and Math in a Map Challenge

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Encouraging students to bring their creativity and imagination to challenging content makes it more accessible-and helps them learn it more deeply.

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Jump-Starting Academic Learning With Movement and Dance

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Jump-Starting Academic Learning With Movement and Dance

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French teacher Nicole Goepper is famous-for dancing with her students in the school courtyard. But even inside the walls of her northern Virginia classroom at Fauquier High School, kids are on their feet.

"I truly believe that there is something in our brains that connect words with movement," says Goepper, who references research showing how movement cements recall and boosts mood, helping to improve the well-being of students. Goepper saw this firsthand when she faced postpandemic declines in student mental health.

Goepper incorporates movement every chance she gets-from pairing new vocabulary with gestures, to acting out French accent marks, to the pièce de résistance: going outside to dance.

"Usually we dance in the middle of class. I use it as a reward for something we've done in class [kids get to pick which French pop songs they want to dance to], and I almost use it as a transition."

The benefits? "It's a brain break; it gets them moving, gets their hearts pumping, and that blood is going up into their brain. And it's like a primer, getting them ready to go [back inside to] learn more stuff."

But it's also a cultural connection, "which is a huge piece of learning and understanding of world language. When I introduce any new song to them, there's always, you know, who is singing it? Who is this person? A little backstory of the artist. And then just the general meaning of what's behind the song." Students learn about the song's relevance to French and French diaspora culture.

Finally, although it's a well-established fact that movement benefits younger students, Goepper believes it's just as beneficial for high school students. "There are kids who in the beginning don't want to put themselves out there," admits Goepper. "But then they realize that it's fun, and they feel left out if they're not joining in with us. I win over 99 percent of them."

Christian Andrä, Brian Mathias, Anika Schwager, Manuela Macedonia, and Katharina von Kriegstein's research on how learning foreign language vocabulary with gestures and pictures enhances memory for school children (2020)

Lawrence Shapiro and Steven Stolz's comprehensive review of studies on embodied cognition (2019)

Panagiotis Kosmas, Andri Ioannou, and Panayiotis Zaphiris's research on embodied learning and its effects on children's memory and language skills (2019)

Spyridoula Vazou, Panagiota Gavrilou, Evangelia Mamalaki, Anna Papanastasiou, and Nefeli Sioumala's study on the integration of physical activity and its influence on academic motivation (2012)

Christopher R. Madan and Anthony Singhal's article about using actions to enhance memory (2012)

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Jump-Starting Academic Learning With Movement and Dance

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The benefits of movement in the classroom aren't limited to younger students. Pairing new words and concepts with gestures or dance moves locks in understanding-and active brain breaks prime students to learn even more.

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Making Meaning of High School With Self-Reflection

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Making Meaning of High School With Self-Reflection

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In many high schools, only the top few highest-performing students get the chance to give a speech sharing a reflection on their school journey. But at Casco Bay High School, in Portland, Maine, every graduating senior experiences this opportunity. Through a series of introspective writing exercises, students craft a speech to deliver to an audience of classmates, family, and the greater school community that helps them make meaning of their four years of high school and look to the future with confidence. The result is a powerful, joyful event-and an established rite of passage that the school's younger students look forward to.

Says high school senior Kaya, "Now that it's done and I performed it, [I know] I wouldn't have been able to be successful without being pushed. It's the genuine care and love that the teachers have because they want you to succeed. They know your greatness and potential."

English language arts teacher Susan McCray, who supports and guides the students as they write their "final word" narratives, believes the experience is transformative for the students. "Every single kid is heard," she says. "And cherished, in that moment. Right? Loved, in that moment. Because that's what happens up there."

To learn more about the research behind the practices seen in the video, check out the links below.

Jerusha Conner, Michael Posner, and Bright Nsowaa's study on the relationship between student voice and student engagement in urban high schools (2022)

Kristina Zeiser, Carrie Scholz, and Victoria Cirks's report on how implementing and measuring student-centered learning practices can maximize student agency (2018)

Janneke van de Pol, Monique Volman, Frans Oort, and Jos Beishuizen's study on the effects of scaffolding in the classroom and its relation to student achievement, task effort, and appreciation of support (2015)

Cheryl J. Travers, Dominique Morisano, and Edwin A. Locke's qualitative study of self-reflection, growth goals, and academic outcomes (2014)

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Making Meaning of High School With Self-Reflection

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Building a tradition where high school seniors share a personal narrative has helped this school community honor each and every student experience.

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Boosting Student Belonging With the Jigsaw Technique

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Boosting Student Belonging With the Jigsaw Technique

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The "jigsaw classroom" is a cooperative learning approach originally designed by psychologist Elliot Aronson to build up empathy in recently desegregated classrooms. In the jigsaw system, each student receives one small part of a bigger lesson-then, in small groups, students have to pool their knowledge together and teach each other what they learned. "Aronson compared it to gift-giving," says social psychologist Geoffrey Cohen. "When I give a gift, I feel a sense of attachment to the person I'm giving it to-and vice versa."

Cohen walks us through the classic jigsaw research from the 1970s and lays out how the strategy can help the modern-day teacher looking to boost empathy, collaboration, and a sense of belonging in their classroom. To learn more from Daniel Leonard's interview with Cohen for Edutopia, read " The Science of Belonging and Connection."

To read the original research, along with several newer studies on the approach, check out the links below:

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Boosting Student Belonging With the Jigsaw Technique

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This classic group learning strategy can help students communicate, collaborate, and empathize with one another-fighting back against a growing sense of disconnection in schools.

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In Jennifer Montgomery's seventh-grade English class in Eminence, Kentucky, there are no students zoning out during class discussions. That's because she expects the students in the outer circle-the ones who are traditionally supposed to observe and actively listen to the students in the inner circle-to look for opportunities to jump in when they have something to add to the conversation. She reserves a single empty seat in the inner circle for these contributors, labeled "the hot seat," and it's made all the difference for bumping up participation in her Socratic seminars. "It propels the discussion," she says. "Sometimes it turns the discussion in a different direction. And it keeps them on their toes."

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Providing the active listeners in the outer ring of a Socratic circle a way to jump in with their burning questions and comments keeps all students engaged.

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Sustaining a Schoolwide Culture of Playful Learning

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Sustaining a Schoolwide Culture of Playful Learning

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In Eminence, Kentucky, a small rural district has dedicated their entire school to reveling in the joy and excitement of learning. From makerspaces to interactive hallways to a slide that sends students from the top level to the ground, they have created a space that captures the imagination-and has become the pride of their economically challenged community. But the approach goes well beyond the physical building. Teachers are supported in professional development sessions that encourage them to inject a sense of "surprise and delight" into their lessons, as well as in guiding students through meaningful projects that connect to their community. So school becomes a place where academic rigor is accomplished in an active, collaborative, and engaging way.

To learn more about the research behind the practices seen in the video, check out the links below.

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Sustaining a Schoolwide Culture of Playful Learning

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Educators in this small town prioritized innovation, creativity, and wonder in every aspect of their school, and the message to students was clear-dream big, because anything is possible.

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How to Welcome Newcomers in the Classroom

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How to Welcome Newcomers in the Classroom

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Making Physical Education More Well-Rounded

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Making Physical Education More Well-Rounded

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PE classes have come a long way, points out New Jersey wellness teacher Suzanne Horsley, whose job title reflects her district's dedication to making the class about more than just physical exercise. When she was in school, "we tended to the movement side of things, but we didn't necessarily attend to the emotional or the social."

Today, in Horsley's class, kids still get exposed to the standard fare of physical skills, but they also have ample opportunity to practice things like cooperation, team-building, perseverance, and kindness. All of her classes incorporate personal goal-setting, Socratic seminars, and time for personal and whole group reflection. One special feature is daily journal time, which supports goal-setting and self-awareness (not to mention writing practice). Even the youngest of students keep a journal-Horsley encourages her kindergarten students to draw their goals and accomplishments.

Horsley goes out of her way to create a space where all students feel comfortable trying out new physical skills. "Wellness is a class that we learn in front of each other," she says. "It's not like any other subject area. So if we don't set up a safe space for learning, we're not going to have everybody involved in the learning process."

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Making Physical Education More Well-Rounded

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Adding practices like reflective journaling, Socratic seminars, and team-building games to gym class is a perfect way for kids to learn and practice key SEL skills.

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Research shows that taking a more playful approach in the classroom could benefit students. But what does it mean to bring learning and play together, for students of all ages? For the Making Learning More Playful video series, we traveled to six schools in four countries to discover some of the characteristics of playful learning. Along the way, we saw a lot of great examples: students engaged in meaningful, hands-on projects; dancing and moving during active, energetic lessons; learning in the outdoors; collaborating to do great iterative work, driven by curiosity and passion; and even playful assessment. The common thread was clear-when more play is intentionally built into teaching and learning, students are enthusiastic and focused, agency and ownership soars, and the classroom environment is a joyful one.

To learn more about the research behind the practices seen in the video, check out the links below.

The American Academy of Pediatrics' report on how play enhances development in young children (2018)

Kayleigh Skene, Christine M. O'Farrelly, Elizabeth M. Byrne, Natalie Kirby, Eloise C. Stevens, and Paul G. Ramchandani's meta-analysis and review on whether guidance during play can enhance children's learning and development in educational contexts (2022)

Rachel Parker, Bo Stjerne Thomsen, and Amy Berry's article on learning through play at school (2022)

Jennifer M. Zosh, Kathy Hirsh-Pasek, Emily J. Hopkins, Hanne Jensen, Claire Liu, Dave Neale, S. Lynneth Solis, and David Whitebread's literature review on redefining play as a spectrum (2018)

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By bringing playful approaches into the classroom, teachers can engage more students at a deeper level, improve learning environments, and boost motivation.

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Required Content Reflection (for each AmeriCorps hours opportunity)

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Required Content Reflection (for each AmeriCorps hours opportunity)

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Please submit a reflection regarding the specific content that you have just completed...

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The effect of letter writing on a long-term care resident with loneliness.

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Abstract

Older adults living in long term care (LTC) facilities may experience increased isolation and loneliness. This was compounded with the Coronavirus Disease 2019 (COVID-19) pandemic. Loneliness and isolation increase the risk for physical, psychological, and social decline. This case report discusses the effect of a letter writing initiative on feelings of loneliness and connection in a long-term care resident. Personalized care is the standard in the long-term care setting and letter writing between two people can contribute to personalized meaningful care as evidenced by the example provided. Further research is needed to explore social connection and loneliness and methods to combat these issues with a personalized approach for different populations within LTC facilities.

Keywords: Loneliness, Connection, Letters, Personalized care

Introduction

Loneliness and isolation in the older adult population is a growing concern.¹,² Long-term care residents may experience increased isolation and loneliness.³,⁴ With social distancing parameters and visitor restriction in long term care facilities (LTCFs) due to Coronavirus Disease 2019 (COVID-19), the risk for loneliness and isolation intensified.⁵ There are numerous well documented physical and psychological sequalae to loneliness and isolation such as cognitive and functional decline, poor health outcomes, increased rates of hospitalizations, and increased mortality.⁶,⁷ Growing evidence suggests engagement in meaningful activities can improve quality of life, mental health, and reduce complications such as frailty.⁸,⁹ Strategies to combat loneliness and promote connection in LTCFs have historically involved outside groups providing activities, music, and other events. Group gatherings within facilities were planned and encouraged by activity directors. Due to restrictions these types of activities were curtailed or modified during the pandemic. Evidence-based meaningful strategies to engage the older adult in the long-term care setting were needed even before the pandemic¹⁰ and continue as we progress to the endemic stage.

Person Centered Care is considered a priority in in long term care.¹¹,¹² Studies suggest person-centered care offers an evidence-based approach wherein health care workers can incorporate patient preferences, feelings, culture, and life history into patient interactions.¹³

One emphasis of person-centered care is nurturing relationships.¹⁴,¹⁵ Supporting relationship building can promote connection. Data suggest decreased feelings of social connection is linked to mental health issues including loneliness, cognitive decline, and depression.⁸

Case report

An 82-year-old female resident of a long-term care facility participated in a letter writing initiative. The initiative received Institutional Review Board Approval at the local university. The resident had lived in the facility for 5 years and was living in a room alone stating she had "problems finding a roommate because none of her roommates could talk to her." Her family lived out of state and was only able to visit a couple of times a year. This was during the COVID-19 pandemic and visitors and outside activities were significantly curtailed. Once consent was obtained and prior to letters being sent, a pre intervention UCLA Loneliness Scale was completed.

The UCLA Loneliness Scale is an evidence based reliable and valid tool for use in the older adult. 16, 17, 18, 19 Items within the scale have also been noted to suggest an association between loneliness and social connection.¹⁷ The UCLA Loneliness Scale comprises 20 separate questions ranked from 0 to 3 ranging from "I never feel this way", "I sometimes feel this way", "I rarely feel this way" to "I often feel this way". Higher scores suggest a higher degree of loneliness. There have been several revisions of the scale as shorter versions (Fetzer, n.d.). Although the revised scales are shorter, specific questions on the original scale related to connection were of interest to the investigator and the original UCLA Loneliness Scale was utilized. For this resident, the pre intervention scale indicated a score of 33 out of a possible 60 points with a score greater than 30 indicating a moderate to high level of loneliness.¹⁸

A nursing student was assigned to the resident sent letters weekly for 10 weeks. After an introductory letter, the student also sent artwork, crafts, and other creative correspondence. With the introductory letter the student sent a self-addressed stamped envelope encouraging the resident to write back. The resident elected to send 4 letters to the student during the period of the intervention. After the 10-week period of letter correspondence, the UCLA loneliness scale was completed, and the resident scored 12 out of a possible 60 points. Of particular interest were three individual items within the scale.¹⁸ Table 1 details the question and resident responses.

Table 1

Item

Pre-Intervention

Post Intervention

"I am unhappy doing so many things alone"

I often feel this way

I rarely feel this way

"I lack companionship"

I often feel this way

I rarely feel this way

"I feel left out"

I often feel this way

I rarely feel this way

In a post interview the resident shared several insights regarding the letter writing initiative. She stated "So many do not have loved ones and to hear from young people is amazing. I enjoyed it immensely. I hope for this to continue even after the pandemic is over". When asked about her favorite parts of the letter writing she indicated "getting mail" and our shared interests such as "faith, pets, and some television shows." She also stated in a post interview the letters provided something to talk to the other residents who were receiving letters about, and she felt more "connected" to some of the other residents. This is an ongoing project. Anecdotal data from the facility and verbal statements suggest there was a potential clinical significance to the project for this and other residents.

Discussion

This initiative allowed older adults to connect without advanced technology and to potentially correspond in a way that had meaning to them in their young and middle adult years. Letter writing is a reflective, tangible activity taking time and purposefulness to send letters. Letter writing may represent a potential method to allow resident choice and promote person centered care. Although there are a growing number of older adults who are technologically adept, there remains a significant number of older adults (specifically those 75 years and older, less affluent, or with lower levels of educational attainment) who are unfamiliar or uncomfortable with technology.²⁰,²¹ Older adults may lack access to technology or the skills necessary to utilize technology.²¹ Older adults may also face physical challenges making it difficult to manipulate devices as well as have a lack of confidence in using technological devices. Studies evaluating the impact of different types of social engagement and the benefit for different individuals are warranted. Further study of the 3 questions from the scale with significant change in response merit further investigation.

The letters prompted connection among shared interests. Further research is needed to identify the presence and consequences of social isolation and loneliness among long term care residents.⁴,²² Loneliness is multi-faceted. Distinct dimensions of loneliness and feelings of connection need to be explored among residents in LTCFs.²⁰,²³

Conclusion

The case study presented points to the potential benefits to both participants with his low technological initiative. It is not cost prohibitive and is easily implemented. Although the letter writers were nursing students this type of project could be readily adapted by numerous types of individuals or groups within the community. Personalized care is the standard in the long-term care setting and letter writing between two people can contribute to personalized meaningful care as evidenced by the case study discussed. Further research is needed to explore social connection and loneliness and methods to combat these issues with a personalized approach for different populations within LTC facilities.

References

6. Kasper J., Freedman V. Johns Hopkins University School of Public Health; 2022. National Health and Aging Trends Study User Guide: Rounds 1-9 Beta Release.[ https://www.nhats.org 2020. Available at. Accessed October 21. Google Scholar]

8. Bethell J., Aelick K., Babineau J., et al. Social connection in long-term care homes: a scoping review of published research on the mental health impacts and potential strategies during COVID-19. J Am Med Dir Assoc. 2021;22(2):228-237. [[ PMC free article] [ PubMed] Google Scholar]

10. Brimelow R.E., Wollin J.A. Loneliness in old age: interventions to curb loneliness in long-term care facilities. Act Adapt Aging. 2017;4:301. [ Google Scholar]

15. Soklaridis S., Ravitz P., Adler Nevo G., Lieff S. Relationship-centred care in health: a 20-year scoping review. Patient Exp J. 2016;3(1):130-145. [ Google Scholar]

16. Hughes M., Waite L., Hawkley L., Cacioppo J. A short scale for measuring loneliness in large surveys. Res Aging. 2004;26(6):655-672. [[ PMC free article] [ PubMed] Google Scholar]

21. Hunsaker A., Hargittai E. A review of internet use among older adults. New Media Soc. 2018;20(10):3937-3954. [ Google Scholar]

Articles from Geriatric Nursing (New York, N.y.) are provided here courtesy of Elsevier

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The effect of letter writing on a long-term care resident with loneliness

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Older adults living in long term care (LTC) facilities may experience increased isolation and loneliness. This was compounded with the Coronavirus Disease 2019 (COVID-19) pandemic. Loneliness and isolation increase the risk for physical, psychological, ...

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BELOVED COMMUNITY

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Today, we commemorate the 51st anniversary of Dr. Martin Luther King, Jr.'s death: his assassination by James Earl Ray in Memphis, Tennessee, where King was preparing to march on behalf of sanitation workers on strike. In marking this tragic anniversary, we also celebrate the U.S. Civil Rights Movement King did so much to advance. Indeed, most 21st-century Americans see King as a national hero; we regard his writings, speeches, and sermons as key documents of the expanding American experiment, and we honor his memory through monuments, parks, schools, a holiday, and the names of our boulevards.

As much as King has become an American champion and even a legend, we should also-and especially on this day-recall the international scope of his vision: how King's notion of "the beloved community" grew to encompass all the people of the world.

King refined his philosophy of nonviolent resistance to racial segregation, racist brutality, and injustice over many years, but even in the beginning, he saw the international dimensions of the civil rights struggle. In July of 1956, he gave a speech to the Home Mission Societies of Christian Friends in Green Lake, Wisconsin, while in the midst of the Montgomery bus boycott; the state statute requiring racial segregation on intrastate buses had just been overturned in district court, but the boycott continued and the case would not be decided by the U. S. Supreme Court until November of that year. With his feet in Wisconsin and his heart in Alabama, King nevertheless brought his mind and the minds of his listeners to global horizons: "Now this determination on the part of the Negro to struggle and to struggle, until segregation and discrimination have passed away, springs from the same longing for human dignity that motivates oppressed peoples all over the world. This is not only a nation in transition, but this is a world in transition." In a later version of the speech, King said that actions like "non-cooperation and boycotts are not ends within themselves... the end is redemption and reconciliation. The aftermath of non-violence is the creation of a beloved community."

King drew on a variety of religious and philosophical sources as he developed the basic tenets of this philosophy. The non-aggression principle of , as popularized by Mahatma Gandhi, is the best known of these sources, as outlined in King's 1959 Sermon on Gandhi. He also benefited from the ideas of the Fellowship of Reconciliation, an international interfaith group founded in 1914 in an attempt to prevent World War I; the U.S. chapter was attended by Josiah Royce, a theologian and philosopher who first articulated the idea of the beloved community. (Royce received one of the first doctorate degrees granted by Johns Hopkins in 1878. We have some of his early writings in the University Archives.) Over the years, King's description of the ideals and practices of nonviolent resistance grew more specific, as he and his brother and sister activists sought to implement it in their arduous and dangerous work.

In April 1957, when he gave his sermon on " The Birth of a New Nation," King's beloved community was explicitly global. In this speech, celebrating Ghana's successful fight for independence from British colonialism, he connected that struggle to the Indian independence movement and the U.S. struggle against segregation-and he invoked again the goal of a beloved community that leaves no one out. King's view of the beloved community also clearly embraces migrants to and from all nations. In a 1966 note to Cesar Chavez, leading farmworkers who were largely undocumented, King said, "Our separate struggles are really one-a struggle for freedom, for dignity and for humanity." And in his 1963 " Letter from Birmingham Jail," a response to local clergy who objected to "outsiders coming in," he issued perhaps his most eloquent defense of the beloved community as a global phenomenon: "Injustice anywhere is a threat to justice everywhere. We are caught in an inescapable network of mutuality, tied in a single garment of destiny. Whatever affects one directly, affects all indirectly." His murder interrupted the launch of the Poor People's Campaign, a multi-pronged crusade against U.S. poverty through policies to provide jobs, unemployment insurance, a fair minimum wage, and improved opportunities for education that was organized by over fifty multiracial organizations. While its immediate goals were national, its premise was transnational-a recognition that poverty at home was linked to economic and political power dynamics that created wealth and poverty everywhere.

In his last Sunday sermon, preached at the National Cathedral in Washington, D.C., King described the "great revolution" of his time in terms that are extraordinarily relevant to the networked, global dimensions of our lives in 2018. What was taking place in 1968, as he saw it, was "in a sense... a triple revolution; that is, a technological revolution, with the impact of automation and cybernation; then there is a revolution of weaponry, with the emergence of atomic and nuclear weapon of warfare; then there is a human rights revolution, with the freedom explosion that is taking place all over the world." He closed the sermon by reminding his listeners not to despair on the hard road to freedom, because "however dark it is, however deep the angry feelings are, and however violent the explosions," what we are capable of making is "a beautiful symphony of brotherhood."

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Dr. Martin Luther King, Jr.'s Beloved Community

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Today, we commemorate the 51st anniversary of Dr. Martin Luther King, Jr.'s death: his assassination by James Earl Ray in Memphis, Tennessee, where King was preparing to march on behalf of sanitation workers on strike. In marking this tragic anniversary, we also celebrate the U.S. Civil Rights Movement King did so much to advance.

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What You Should Know About Martin Luther King, Jr. Day of Service:

The history behind MLK Day, and how you can celebrate it!

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What You Should Know About Martin Luther King, Jr. Day of Service

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The history behind MLK Day, and how you can celebrate it.

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2024 MLK Day of Service: Fellow Reflection

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2024 MLK Day of Service: Fellow Reflection

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Please submit a reflection regarding your 2024 MLK Day of Service experience!

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Learn one teacher's strategies to return from break well-rested and recharged.

In my teaching experience, winter break arrives at the moment when both the students and I need it most. As we are nearing winter break, I look forward to some time to relax and renew my energy before school resumes in January. Some years I realized a day or two before the end of my time off that I wasn't any more rested than I had been in the days leading up to break. Over the years there are some things that I have found to help me have a truly revitalizing winter break.

Before break, I designate days on my calendar when I won't check email or do anything work-related. Allocating days when I won't engage in work helps me to be intentional about spending time with family as well as spending mindful time alone. Around a week before winter break, write down all work-related to-do items. Check off as many as possible before break. Listing what needs to get accomplished helps me stay organized. Even better, if I succeed in checking off all of the items, I return from the break with a clean slate.

A few days before break, plan a lesson for the first day back. Write out all ideas in detail and gather any materials needed to implement the lesson. Having a day planned for my return allows me to enjoy the break without worrying about the first day back.

The winter break can be a busy time filled with shopping, cooking, and chores, which means running the risk of arriving at the last day of break and realizing that I haven't taken any time for myself. To make sure the break isn't only full of responsibilities, I make a list of fun things I hope to do over break and then schedule them on a calendar. I include items such as outdoor activities, leisurely walking around a bookstore, taking exercise classes, baking, traveling, or watching a movie. Designating time for these activities means they are more likely to happen.

Nurture Relationships

Over the years, some of my best friends have started out as colleagues. I have found that having those supportive relationships at work helps reduce stress throughout the year. Over long breaks, I like to make plans with a colleague or group of colleagues to hang out and enjoy each other's company. We commit to not talking about work during that time together. Building relationships with colleagues outside of the work environment can reveal what is important to them other than their work.

During the school year, my schedule is often very busy, and I don't have a lot of time to connect with friends and extended family. The winter break is an opportunity to visit friends and family outside of school.

I read throughout the year, but it is typically work-related. During the break, I forgo reading young adult literature or nonfiction and instead find a relaxing book purely for pleasure. Over winter break, choose one "beach read" and take some time to read for fun. I also spend some time casually reading online for inspiration. I scroll Twitter and Instagram and read some of my favorite education bloggers for inspiration. This allows me to gather some new ideas for the second half of the year but doesn't feel like work.

Reflect and Set a Goal

Try at least one new technology tool with students before the end of the year.

Ask students to name some of their favorite books they have read this year, and choose a few to read myself.

Send at least one positive note home to each child's parent before the end of the school year.

Collaborate with colleagues in another subject to add interdisiplinary elements to a current assignment.

Incorporate new texts based on student interest and needs.

Find opportunities to make changes to assignments in order to provide more student choice.

Attend more events with students outside of school, such as sports, concerts, or plays.

I have never been much for New Year's resolutions, but I find that making one professional goal for the second half of the school year is inspiring without being stressful. When making a goal, I often begin by reviewing student feedback from first-semester student surveys. From these surveys, I may learn that students feel we need to spend more time on a certain aspect of the curriculum, students enjoy being read aloud to, or certain students feel distracted by the organization of the desk. I combine that feedback with ideas from the social media.

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Learn one teacher's strategies to return from break well-rested and recharged.

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Curiosity is a powerful tool that can be harnessed to guide middle school students to come up with solutions to real problems.

Inquiry-based learning provides students with the opportunity to explore real-world connections and problems through curious exploration and empowerment. Imagine sitting in a classroom as a seventh grader and being asked the question "How might we educate our community on their civic duty?"

Such a simple question, loaded with content to investigate, areas to explore, empathy to be considered, and agency to be born. But, how does this question create agency? Why does empathy need to be considered?

Student agency is when students become active agents in the process of learning and engage with the world with sustained, courageous curiosity.

With this in mind, it seems only natural that inquiry-based learning would lead to student agency. However, there's a trick to making sure that student agency is at the core of your designed inquiry-based learning prompt. That trick is empathy.

Ground Inquiry-Based Learning in Empathy

Empathy is found all around us. There's a reason we laugh when others laugh or smile when others smile. Emotions are contagious by design, but emotion is rooted in a cognitive idea. Whether it's the idea that we should be happy when friends succeed or that we should be energized to make a change that our community came together for, empathy can be a key factor in organized success. In turn, your inquiry-based learning question should be rooted in empathy.

Empathize with your students. Give them a voice in formulating their own question that is rooted in a topic or concept you're studying. You can also give students the task to empathize with other stakeholders to solve a problem they may see in their community. Through the use of empathy interviews, students can deduce a problem, team up cohesively to create a solution, and feel the empowerment of empathy in driving their learning.

Interviews CAN UNLOCK Empathy

In the question shared earlier, "How might we educate our community on their civic duty?" did you catch the "we" aspect of the question? The strategic wording creates a sense of collectivism that surrounds the problem students are being tasked with and allows for natural agency to occur. Furthermore, the question isn't designed for a simple solution but rather pushes a cognitive lift through exploration and creation. This question was given to seventh-grade students at my school.

Geared toward societal awareness in their social studies class, students were tasked with conducting a minimum of six empathy interviews with their guardians, school staff members, and other community members whom they encountered or had access to. The results of the empathy interviews were synthesized, and specific domains of interest were created to help categorize the learning outcomes. At this point, the facilitator (we prefer the language facilitator because it adds an additional layer of agency as the adult becomes the facilitator of student learning, rather than the teacher of content) drafted the question, starting with the stem "How might we..."

How might we educate our community on their civic duty?

How might we catalyze positive local and state perceptions of our community?

How might we promote healthy living through agricultural awareness and activity?

How might we promote equity and inclusion in our community?

The following are some sample inquiry-based questions from students' last empathy interviews:

Students then organically formed teams to tackle the question of their choosing. Some teams were made up of three students, other teams were four, and some teams were partnerships of two. Teams then chose their inquiry-based learning prompt and began the process of diving in and creating solutions. Questions may be used by more than one team, as the solutions may be vastly different than their peers', thus promoting additional agency in their product. Students then present to an authentic audience made up of their peers, stakeholders, and some of the same people with whom they conducted their empathy interviews.

This just adds another layer of authenticity to students' learning experience and creates agency and empowerment because they're able to share their findings with those who care the most about the issue.

Getting Good Questions Inspires Actionable Solutions

The questions drafted from the empathy interviews are designed to be more broad than confined, so that students have a wide area to explore without the confines of a singular, unactionable topic. When students conduct their empathy interviews, they are asked to home in on a specific aspect of their classroom studies and engage with their interviewees on that concept. Students are given the same parameters that the facilitator is given. These parameters state the following about the final question:

1. It has clear and direct real-world applications.

2. It is relevant to stakeholders' passions and interests.

3. It develops critical consciousness.

4. It promotes student agency and empowerment.

Naming, unpacking, and owning these tenets for a "How might we..." question with students creates for them a sense of ownership in their own learning by understanding the process, reason, and outcomes of their targeted solution-based design opportunity. Starting with a topic or concept with real-world application allows students to engage in inquiry that is aligned with established learning standards enhanced with relevancy.

Recognizing interests and passions through empathy allows students to develop the critical consciousness necessary to become global citizens. Agency and empowerment promotes the self-efficacy needed to be agents of change in an ever- evolving world.

After several years of this focus on inquiry-based learning (in classrooms and at a schoolwide level), our students have an increased sense of self. During our end-of-year reflection, when students highlight their biggest takeaways from their learning, they overwhelmingly highlight their excitement to express themselves creatively without repercussion and share their fulfillment in being heard, valued, and respected by their peers and teachers.

As evidenced through their continued passion around their own learning, assessment data, social and emotional learning data, and shared student dialogue, it's clear that they feel empowered to be agents of change, independent in their thoughts, collaborative in their discourse, and confident in their abilities to be problem solvers.

Ideas and inspiration for empathy-driven questions are everywhere, and it's important for teachers to keep an ear out for that inspiration in any given context. Keeping a continuous open dialogue with your students about their individual interests and passions is a great way to unearth ideas that could be great opportunities to have students become agents of their own learning.

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Using Inquiry-Based Learning to Encourage Student Agency

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Curiosity is a powerful tool that can be harnessed to guide middle school students to come up with solutions to real problems.

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Not all students feel comfortable participating verbally in class discussions, and these tools invite everyone to contribute.

Alexey Yaremenko / iStock

As teachers, we can ensure that "all means all" by introducing, modeling, and providing opportunities for students to use hand signals or gestures in the classroom. Gestures are a tool that can support equitable classroom participation by increasing engagement and allowing us to read the room to assess who understands a concept and tailor instruction accordingly.

When teachers ensure that all students are responding, we show that we value every student and have high expectations for every student to contribute. Gestures can prove helpful during independent, partner, small group, and whole group instruction. Below, I offer ways to implement them.

Using Gestures with Large Groups

When used in whole group situations, hand motions enable every student to respond at once. For example, "Show me with your face and body how the character was feeling at this point"or "Show me with your thumb how well you understood that strategy: thumbs up for 'I understand completely' and sideways for 'I'm getting there.'"

Using gestures is great for all and necessary for some, including quieter students, students with anxiety, multilingual students, and students who communicate without speaking. A student may be hesitant or unable to share with the whole group using words, and a hand motion allows them to express their thinking.

Assessment and Engagement

When I hold all students accountable for participating, energy and engagement increase. I can take a quick assessment of understanding. For example, "Show me with your fingers how many sounds you heard in that word." I reteach or scaffold as needed, either in the moment or during conferring or instruction at a later time.

Interactive read-alouds are an excellent time to use gestures. With younger kids, I have students act out the story with their hands, to retell it. Students can turn and talk briefly and use a variety of gestures depending on the question; for example, "Tell your partner the two different perspectives" might allow them to hold one hand flat, palm up for the first perspective and to raise the other hand for the second perspective.

Small Groups

When partners and small groups use gestures, they are more likely to bounce ideas off each other. For example, when using the American Sign Language (ASL) "agree" motion, one student can encourage their partner to continue and then build upon what their partner said, using the " I have something to add " gesture.

To assess partnerships and small group dialogues, I glance around the room to see which gestures are being used and get a quick idea of how the conversations are going; I can visit students who need coaching or a check in.

I have also seen students use hand motions when working independently to remind themselves to add detail (place one fist on top of the other), focus (hold two hands parallel with palms facing), or try a different strategy or word (start with palm flat then flip it over), showing how these moves become internalized and enhance learning.

Some of My Most-Used Hand Motions

Two of my favorite signals are . A sense of connection builds when classmates and/or the teacher agrees with a student. Teaching the disagreegesture and modeling how to respectfully disagree deepens conversations as we consider different perspectives.

When a listener uses the elaborate gesture, the speaker feels heard and is encouraged to say more or clarify. The speaker responds with more precise language, an important communication skill. By modeling and asking students to paraphrase, we teach one of the most important listening skills. In order to paraphrase, a listener needs to deeply listen and rephrase to summarize what has been said.

When teaching sound articulation, teachers ask students to point to where sounds are articulated to help students distinguish phonemes, for example: "say /th/ and point to your tongue between your teeth or feel the vibration in your throat. " One of my colleagues teaches or asks her kindergarten students to make a gesture for important high-frequency words; they love to show me their gestures as they warm up by reading the words.

When I introduce sentence stems, I emphasize certain words. For example, when using "I think this strategy is the best because ___," students use a flat hand extending out from their bodies with the word because. Recently, I saw a teacher use a connect signal with students to show how facts in nonfiction reading connect. As soon as she introduced it, students honed in on the instruction.

Supporting Classroom Culture

In addition to ensuring that all students have the tools to communicate and respond, we can use hand signals to keep instruction flowing. A student using the ASL signal for bathroom allows a teacher to know they are heading to the bathroom without having to stop instruction. Similarly, if a student needs to reset and reflect on classroom expectations, the teacher can give an inconspicuous letter R signal.

Start simple with gestures and introduce them with a sense of playfulness and a clear expectation that all will participate. Then, over time, go deeper and layer on complexity. Be responsive to what students need next. This will increase a sense of novelty and new learning. Using hand motions is such a powerful teaching move, a huge equity tool, and a way to increase engagement.

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Making Classroom Participation More Equitable With Hand Gestures

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Not all students feel comfortable participating verbally in class discussions, and these tools invite everyone to contribute.

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K-12 District Set Out to Learn How Our Teachers Were Using Tech

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K-12 District Set Out to Learn How Our Teachers Were Using Tech

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Technology Solutions That Drive Education

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So, when a 2022 JAMA Pediatrics study found that during the COVID-19 pandemic, screen time among children ages 3 to 18 went up 59 percent globally, our district took notice. We also considered a 2019 longitudinal study of Australian children reported in the International Journal of Behavioral Nutrition and Physical Activity. The study notes that among the following categories of screen time - social, educational, passive, interactive and other - educational screen time provided the most benefit.

Curious as to how much screen time students across our district were experiencing and how they were using technology to learn, our district leaders decided to take a closer look. To get a sense of what was happening in the classroom, our instructional team facilitated a series of "learning walks." What we discovered was eye-opening.

RELATED: Check out the research on screen time and school-aged kids.

We Designed a Student-Centered Tool to Observe Ed Tech Use

We started the process by creating a student-centered observation tool to help us reflect during the walks. The tool implemented the four C's of 21st century learning: creativity, communication, collaboration and critical thinking. It also included Ruben Puentedura's SAMR ( substitution, augmentation, modification and redefinition) model of technology integration, and some popular blended-learning models such as station rotation, whole-group rotation, flipped classroom and playlist. In addition, we paid close attention to the program or tool that teachers and students used with particular technologies.

We then traveled to all 16 of our campuses (10 elementary schools, three middle schools and three high schools) to see blended learning in action. For each school, we formed teams of six to 10 people, which included district- and school-level staff. We spent an hour observing up to six classrooms, followed by a half hour to debrief.
DIG DEEPER: Learn how technology supports personalized learning in K-12.
The Truth About Higher Student Engagement

After completing all 16 learning walks, district and school leadership gathered at a monthly meeting where they identified similarities and differences among their observations as well as any valuable takeaways.

We found higher levels of student engagement when teachers used gamification tools; when teachers provided student voice and choice to display student mastery; and when they prioritized communication, collaboration, creativity and critical thinking.

Some school leaders noticed large amounts of substitution and augmentation in student learning experiences. There was more evidence of students using technology for communication and collaboration rather than for critical thinking or creativity. There were also some instances where the technology use could have been more purposeful and engaging instead of serving as a babysitter.
DISCOVER: Here's how technology enables collaboration on K-12 projects.
So, our group equipped each principal with a lesson plan containing an appropriate topic and lesson structure requiring them to choose how they would integrate technology using the models upon which we based the rubric. They then shared their creations with small groups of their peers, asking further questions and navigating the ensuing discussion.

Targeted Professional Development Is Key to Ed Tech Integration

While the learning walk process was not perfect, I think it showed our district and school-level administrators the importance of meaningful, engaging technology integration within the classrooms. It also revealed the importance of planning for technology integration by identifying the tools, the structure for integration and the purpose behind the technology use.

Each school came to its own conclusions on how next to proceed. Some decided to conduct more learning walks for teachers to learn from each other, and others promoted teacher-led professional development. On the district level, we provided targeted technology-focused PD for teachers, which focused on tech integration, equipping teachers with ideas about what could be and not mandating any one approach.

So far, we've received positive feedback on the introduction of new tools, the reiteration of what integration could look like and reminders on student ownership in the process.

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What Happened When Our K-12 District Set Out to Learn How Our Teachers Were Using Tech

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After comparing what we saw in the classroom to 21st century ed tech best practices, we found room for improvement.

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Technology Solutions That Drive Education

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Now that many students have devices and access to technology, educators and school leaders must work to narrow the digital divide through equity of use. If students aren't exposed to technology and taught how to use it, they will fall behind their peers.

"Educators should make sure logging in is a really easy, smooth process," Highfill says. "Once I get everyone logged in, the No. 1 thing I have to get students to learn how to do is share their screen."

This not only helps her work through problems with students, she says, but also helps students take a more active role in their learning. Students will find new ways to achieve a goal or manipulate a technology and can show the class - and the teacher - how they've accomplished it by sharing their screen. "You empower them and put them in the teaching role," Highfill adds.

What Are the Benefits of Technology for Students?

Through technology, schools can support all students. There are roughly 60 grade school students and nearly 250 high school students enrolled at Pleasanton Virtual Academy. "I'm so excited our district put in that investment," Highfill says. "We're a public school virtual academy. They invested in a quality virtual academy to meet the needs of all students."

Even students who are learning in an in-person environment are using technology in their daily lives. Integrating it into the classroom gives them an opportunity to learn to use tech in a meaningful way.
Build the themes of digital citizenship into instruction and business planning.
"If you have the skills and know how to research and find information and discern whether that information is true or not, that's going to help you not only in school with your schoolwork, but also with life in general," Lim says.

"I watch the kids, and they're very addicted to their devices," says Highfill. "So, it's my new teaching point: How can you take a digital diet, and how can you identify when tech is not doing good things for you?"

Highfill says that anytime there's a fear about introducing technology to the classroom, educators should use that. "We have to teach students how to take care of themselves if they're going to use technology," she says.

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Technology in the Classroom & The Benefits for K-12 Schools

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Adoption & introduction of educational technology in the classroom is good for both students & educators. Learn how it works, why it's important & the benefits.

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Ongoing Service Opportunities: Monthly Community Calendar Listings

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Sales from Thriftsmart’s store support 4 other nonprofits including Mercy Health Clinic, The Belize Project, African Leadership, and New Hope Academy. Volunteers are needed to help sort donations, color code and organize clothing on the sales floor, and organize shoes at the Thriftsmart store on Nolensville Road. If you have donations of clothing or shoes, Thriftsmart certainly welcomes your donation. All volunteers under the age of 18 are required to bring a signed Volunteer Release Form with them. Dress Code: Please wear clothing that is appropriate for a “family” oriented store. Shoes must be closed toe and no heel for your safety. Make sure to ‘SIGN IN’ in order to receive credit for coming. This project does not qualify for COURT-ORDERED COMMUNITY SERVICE HOURS. If you are needing to do community service hours please call our store number at (615)-833-8200 to get ahold of the volunteer coordinator, or go in person to speak to them. Thank you.

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Institute of Educational Technology, The Open University, Milton Keynes, United Kingdom

There is a widespread notion that educational systems should empower learners with skills and competences to cope with a constantly changing landscape. Reference is often made to skills such as critical thinking, problem solving, collaborative skills, innovation, digital literacy, and adaptability. What is negotiable is how best to achieve the development of those skills, in particular which teaching and learning approaches are suitable for facilitating or enabling complex skills development. In this paper, we build on our previous work of exploring new forms of pedagogy for an interactive world, as documented in our Innovating Pedagogy report series. We present a set of innovative pedagogical approaches that have the potential to guide teaching and transform learning. An integrated framework has been developed to select pedagogies for inclusion in this paper, consisting of the following five dimensions: (a) relevance to effective educational theories, (b) research evidence about the effectiveness of the proposed pedagogies, (c) relation to the development of twenty-first century skills, (d) innovative aspects of pedagogy, and (e) level of adoption in educational practice. The selected pedagogies, namely formative analytics, teachback, place-based learning, learning with drones, learning with robots, and citizen inquiry are either attached to specific technological developments, or they have emerged due to an advanced understanding of the science of learning. Each one is presented in terms of the five dimensions of the framework.

Introduction

In its vision for the future of education in 2030, the Organization for Economic Co-operation and Development ( OECD, 2018) views essential learner qualities as the acquisition of skills to embrace complex challenges and the development of the person as a whole, valuing common prosperity, sustainability and wellbeing. Wellbeing is perceived as "inclusive growth" related to equitable access to " quality of life, including health, civic engagement, social connections, education, security, life satisfaction and the environment" (p. 4). To achieve this vision, a varied set of skills and competences is needed, that would allow learners to act as "change agents" who can achieve positive impact on their surroundings by developing empathy and anticipating the consequences of their actions.

Several frameworks have been produced over the years detailing specific skills and competences for the citizens of the future (e.g., Trilling and Fadel, 2009; OECD, 2015, 2018; Council of the European Union, 2018). These frameworks refer to skills such as critical thinking, problem solving, team work, communication and negotiation skills; and competences related to literacy, multilingualism, STEM, digital, personal, social, and "learning to learn" competences, citizenship, entrepreneurship, and cultural awareness ( Trilling and Fadel, 2009; Council of the European Union, 2018). In a similar line of thinking, in the OECD Learning Framework 2030 ( OECD, 2018) cognitive, health and socio-emotional foundations are stressed, including literacy, numeracy, digital literacy and data numeracy, physical and mental health, morals, and ethics.

The question we are asked to answer is whether the education vision of the future, or the development of the skills needed to cope with an ever-changing society, has been met, or can be met. The short answer is not yet. For example, the Programme for International Student Assessment (PISA) has been ranking educational systems based on 15-year-old students' performance on tests about reading, mathematics and science every 3 years in more than 90 countries. In the latest published report ( OECD, 2015), Japan, Estonia, Finland, and Canada are the four highest performing OECD countries in science. This means that students from these countries on average can " creatively and autonomously apply their knowledge and skills to a wide variety of situations, including unfamiliar ones" ( OECD, 2016a, p.2). Yet about 20% of students across participating countries are shown to perform below the baseline in science and proficiency in reading ( OECD, 2016b). Those most at risk are socio-economically disadvantaged students, who are almost three times more likely than their peers not to meet the given baselines. These outcomes are quite alarming; they stress the need for evidence-based, effective, and innovative teaching and learning approaches that can result in not only improved learning outcomes but also greater student wellbeing. Overall, an increasing focus on memorization and testing has been observed in education, including early years, that leaves no space for active exploration and playful learning ( Mitchell, 2018), and threatens the wellbeing and socioemotional growth of learners. There is an increased evidence-base that shows that although teachers would like to implement more active, innovative forms of education to meet the diverse learning needs of their students, due to a myriad of constraints teachers often resort to more traditional, conservative approaches to teaching and learning ( Ebert-May et al., 2011; Herodotou et al., 2019).

In this paper, we propose that the distance between educational vision and current teaching practice can be bridged through the adoption and use of appropriate pedagogy that has been tested and proven to contribute to the development of the person as a whole. Evidence of impact becomes a central component of the teaching practice; what works and for whom in terms of learning and development can provide guidelines to teaching practitioners as to how to modify or update their teaching in order to achieve desirable learning outcomes. Educational institutions may have already adopted innovations in educational technology equipment (such as mobile devices), yet this change has not necessarily been accompanied by respective changes in the practice of teaching and learning. Enduring transformations can be brought about through "pedagogy," that is improvements in "the theory and practice of teaching, learning, and assessment" and not the mere introduction of technology in classrooms ( Sharples, 2019). PISA analysis of the impact of Information Communication Technology (ICT) on reading, mathematics, and science in countries heavily invested in educational technology showed mixed effects and "no appreciable improvements" ( OECD, 2015, p.3).

The aim of this study is to review and present a set of innovative, evidence-based pedagogical approaches that have the potential to guide teaching practitioners and transform learning processes and outcomes. The selected pedagogies draw from the successful Innovating Pedagogy report series ( https://iet.open.ac.uk/innovating-pedagogy), produced by The Open University UK (OU) in collaboration with other centers of research in teaching and learning, that explore innovative forms of teaching, learning and assessment. Since 2012, the OU has produced seven Innovating Pedagogy reports with SRI international (USA), National Institute of Education (Singapore), Learning In a NetworKed Society (Israel), and the Center for the Science of Learning & Technology (Norway). For each report, teams of researchers shared ideas, proposed innovations, read research papers and blogs, and commented on each other's draft contributions in an organic manner ( Sharples et al., 2012, 2013, 2014, 2015, 2016; Ferguson et al., 2017, 2019). Starting from an initial list of potential promising educational innovations that may already be in currency but not yet reached a critical mass of influence on education, these lists were critically and collaboratively examined, and reduced to 9-11 main topics identified as having the potential to provoke major shifts in educational practice.

After seven years of gathering a total of 70 innovative pedagogies, in this paper seven academics from the OU, authors of the various Innovating Pedagogy reports, critically reflected on which of these approaches have the strongest evidence and/or potential to transform learning processes and outcomes to meet the future educational skills and competences described by OECD and others. Based upon five criteria and extensive discussions, we selected six approaches that we believe have the most evidence and/or potential for future education:

* Formative analytics,

* Teachback,

* Place-based learning,

* Learning with robots,

* Learning with drones,

* Citizen inquiry.

Formative analytics is defined as "supporting the learner to reflect on what is learned, what can be improved, which goals can be achieved, and how to move forward" ( Sharples et al., 2016, p.32). Teachback is a means for two or more people to demonstrate that they are progressing toward a shared understanding of a complex topic. Place-based learning derives learning opportunities from local community settings, which help students connect abstract concepts from the classroom and textbooks with practical challenges encountered in their own localities. Learning with robots could help teachers to free up time on simple, repetitive tasks, and provide scaffolding to learners. Learning with drones is being used to support fieldwork by enhancing students' capability to explore outdoor physical environments. Finally, citizen inquiry describes ways that members of the public can learn by initiating or joining shared inquiry-led scientific investigations.

Devising a Framework for Selection: The Role of Evidence

* Relevance to effective educational theories: the first criterion refers to whether the proposed pedagogy relates to specific educational theories that have shown to be effective in terms of improving learning.

* Research evidence about the effectiveness of the proposed pedagogies: the second criterion refers to actual studies testing the proposed pedagogy and their outcomes.

* Relation to the development of twenty-first century skills: the third criterion refers to whether the pedagogy can contribute to the development of the twenty-first century skills or the education vision of 2030 (as described in the introduction section).

* Innovative aspects of pedagogy: the fourth criterion details what is innovative or new in relation to the proposed pedagogy.

* Level of adoption in educational practice: the last criterion brings in evidence about the current level of adoption in education, in an effort to identify gaps in our knowledge and propose future directions of research.

A major component of the proposed framework is effectiveness, or the generation of evidence of impact. The definition of what constitutes evidence varies ( Ferguson and Clow, 2017; Batty et al., 2019), and this often relates to the quality or strength of evidence presented. The Strength of Evidence pyramid by John and McNeal (2017) (see Figure 1) categorizes different types of evidence based on their strength, ranging from expert opinions as the least strong type of evidence to meta-analysis or synthesis as the strongest or most reliable form of evidence. While the bottom of the pyramid refers to "practitioners' wisdom about teaching and learning," the next two levels refer to peer-reviewed and published primary sources of evidence, both qualitative and quantitative. They are mostly case-studies, based on either the example of a single institution, or a cross-institutional analysis involving multiple courses or institutions. The top two levels involve careful consideration of existing resources of evidence and inclusion in a synthesis or meta-analysis. For example, variations of this pyramid in medical studies present Randomized Control Trials (RCTs) at the second top level of the pyramid, indicating the value of this approach for gaining less biased quality evidence.

Another approach proposed by the innovation foundation Nesta presents evidence on a scale of 1 to 5, showcasing the level of confidence with the impact of an intervention ( Puttick and Ludlow, 2012). Level 1 studies describe logically, coherently and convincingly what has been done and why it matters, while level 5 studies produce manuals ensuring consistent replication of a study. The evidence becomes stronger when studies prove causality (e.g., through experimental approaches) and can be replicated successfully. While these frameworks are useful for assessing the quality or strength of evidence, they do not make any reference to how the purpose of a study can define which type of evidence to collect. Different types of evidence could effectively address different purposes; depending on the objective of a given study a different type of evidence could be used ( Batty et al., 2019). For example, the UK government-funded research work by the Educational Endowment Foundation (EEF) is using RCTs, instead of for example expert opinions, as the purpose of their studies is to capture the impact of certain interventions nationally across schools in the UK.

Education, as opposed to other disciplines such as medicine and agriculture, has been less concerned with evaluating different pedagogical approaches and determining their impact on learning outcomes. The argument often made is the difficulty in evaluating learning processes, especially through experimental methodologies, due to variability in teaching conditions across classrooms and between different practitioners, that may inhibit any comparisons and valid conclusions. In particular, RCTs have been sparse and often criticized as not explaining any impact (or absence of impact) on learning, a limitation that could be overcome by combining RCT outcomes with qualitative methodologies ( Herodotou et al., 2017a). Mixed-methods evaluations could identify how faithfully an intervention is applied to different learning contexts or for example, the degree to which teachers have been engaged with it. An alternative approach is Design-Based Research (DBR); this is a form of action-based research where a problem in the educational process is identified, solutions informed by existing literature are proposed, and iterative cycles of testing and refinement take place in order to identify what works in practice in order to improve the solution. DBR often results in guidelines or theory development (e.g., Anderson and Shattuck, 2012).

An evidence-based mindset in education has been recently popularized through the EEF. Their development of the teaching and learning toolkit provides an overview of existing evidence about certain approaches to improving teaching and learning, summarized in terms of impact on attainment, cost and the supporting strength of evidence. Amongst the most effective teaching approaches are the provision of feedback, development of metacognition and self-regulation, homework for secondary students, and mastery learning ( https://educationendowmentfoundation.org.uk). Similarly, the National Center for Education and Evaluation (NCEE) in the US conducts large-scale evaluations of education programs with funds from the government. Amongst the interventions with the highest effectiveness ratings are phonological awareness training, reading recovery, and dialogic reading ( https://ies.ed.gov/ncee/).

The importance of evidence generation is also evident in the explicit focus of Higher Education institutions in understanding and increasing educational effectiveness as a means to: tackle inequalities and promote educational justice (see Durham University Evidence Center for Education; DECE), provide high quality education for independent and lifelong learners (Learning and Teaching strategy, Imperial College London), develop criticality and deepen learning (London Center for Leadership in Learning, UCL Institute of Education), and improve student retention and performance in online and distance settings [Institute of Educational Technology (IET) OU].

The generation of evidence can help identify or debunk possible myths in education and distinguish between practitioners' beliefs about what works in their practice as opposed to research evidence emerging from systematically assessing a specific teaching approach. A characteristic example is the "Learning Styles" myth and the assumption that teachers should identify and accommodate each learner's special way of learning such as visual, auditory and kinesthetic. While there is no consistent evidence that considering learning styles can improve learning outcomes (e.g., Rohrer and Pashler, 2010; Kirschner and van Merriënboer, 2013; Newton and Miah, 2017), many teachers believe in learning styles and make efforts in organizing their teaching around them ( Newton and Miah, 2017). In the same study, one third of participants stated that they would continue to use learning styles in their practice despite being presented with negative evidence. This suggests that we are rather in the early days of transforming the practice of education and in particular, developing a shared evidence-based mindset across researchers and practitioners.

In order to critically review the 70 innovative pedagogies from the seven Innovating Pedagogy reports, over a period of 2 months the seven authors critically evaluated academic and gray literature that was published after the respective reports were launched. In line with the five criteria defined above, each author contributed in a dynamic Google sheet what evidence was available for promising approaches. Based upon the initial list of 70, a short-list of 10 approaches was pre-selected. These were further fine-tuned to the final six approaches identified for this study based upon the emerging evidence of impact available as well as potential opportunity for future educational innovation. The emerging evidence and impact of the six approaches were peer-reviewed by the authoring team after contributions had been anonymized, and the lead author assigned the final categorizations.

In the next section, we present each of the proposed pedagogies in relation to how they meet the framework criteria, in an effort to understand what we know about their effectiveness, what evidence exist showcasing impact on learning, how each pedagogy accommodates the vision of the twenty-first century skills development, innovation aspects and current levels of adoption in educational practice.

Selected Pedagogies

Formative Analytics

Relevance to Effective Educational Theories

As indicated by the Innovating Pedagogy 2016 report ( Sharples et al., 2016, p.32), "formative analytics are focused on supporting the learner to reflect on what is learned, what can be improved, which goals can be achieved, and how to move forward." In contrast to most analytics approaches that focus on analytics of learning, formative analytics aims to support analytics for learning, for a learner to reach his or her goals through "smart" analytics, such as visualizations of potential learning paths or personalized feedback. For example, these formative analytics might help learners to effectively self-regulate their learning. Zimmerman (2000) defined self-regulation as "self-generated thoughts, feelings and actions that are planned and cyclically adapted to the attainment of personal learning goals." Students have a range of choices and options when they are learning in blended or online environments as to when, what, how, and with whom to study, with minimal guidance from teachers. Therefore, "appropriate" Self-Regulated Learning (SRL) strategies are needed for achieving individual learning goals ( Hadwin et al., 2011; Trevors et al., 2016).

With the arrival of fine-grained log-data and the emergence of learning analytics there are potentially more, and perhaps new, opportunities to map how to support students with different SRL ( Winne, 2017). With trace data on students' affect (e.g., emotional expression in text, self-reported dispositions), behavior (e.g., engagement, time on task, clicks), and cognition (e.g., how to work through a task, mastery of task, problem-solving techniques), researchers and teachers are able to potentially test and critically examine pedagogical theories like SRL theories on a micro as well as macro-level ( Panadero et al., 2016; D'Mello et al., 2017).

Research Evidence About the Effectiveness of the Proposed Pedagogies

There is an emergence of literature that uses formative analytics to support SRL and to understand how students are setting goals and solve computer-based tasks ( Azevedo et al., 2013; Winne, 2017). For example, using the software tool nStudy ( Winne, 2017) recently showed that trace data from students in forms of notes, bookmarks, or quotes can be used to understand the cycles of self-regulation. In a study of 285 students learning French in a business context, using log-file data ( Gelan et al., 2018) found that engaged and self-regulated students outperformed students who were "behind" in their study. In an introductory mathematics course amongst 922 students, Tempelaar et al. (2015) showed that a combination of self-reported learning dispositions from students in conjunction with log-data of actual engagement in mathematics tasks provide effective formative analytics feedback to students. Recently, Fincham et al. (2018) found that formative analytics could actively encourage 1,138 engineering learners to critically reflect upon one of their eight adopted learning strategies, and where needed adjust it.

Relation to the Development of Twenty-First Century Skills

Innovative Aspects of Pedagogy

By using fine-grained data and reporting this directly back to students in the form of feedback or dashboards, the educational practice is substantially influenced, and subsequently innovated. In particular, instead of waiting for feedback from a teacher at the end of an assessment task, students can receive formative analytics on demand (when they want to), or ask for the formative analytics that link to their own self-regulation strategies. This is a radical departure from more traditional pedagogies that either place the teacher at the center, or expect students to be fully responsible for their SRL.

Level of Adoption in Educational Practice

Teachback

Relevance to Effective Educational Theories

The method of Teachback, and the name, were originally devised by the educational technologist Gordon Pask (1976), as a means for two or more people to demonstrate that they are progressing toward a shared understanding of a complex topic. It starts with an expert, teacher, or more knowledgeable student explaining their knowledge of a topic to another person who has less understanding. Next, the less knowledgeable student attempts to teach back what they have learned to the more knowledgeable person. If that is successful, the one with more knowledge might then explain the topic in more detail. If the less knowledgeable person has difficulty in teaching back, the person with more expertise tries to explain in a clearer or different way. The less knowledgeable person teaches it again until they both agree.

A classroom teachback session could consist of pairs of students taking turns to teach back to each other a series of topics set by the teacher. For example, a science class might be learning the topic of "eclipses." The teacher splits the class into pairs and asks one student in each pair to explain to the other what they know about "eclipse of the sun." Next, the class receives instruction about eclipses from the teacher, or a video explanation. Then, the second student in the pair teaches back what they have just learned. The first student asks questions to clarify such as, "What do you mean by that?" If either student is unsure, or the two disagree, then they can ask the teacher. The students may also jointly write a short explanation, or draw a diagram of the eclipse, to explain what they have learned.

The method is based on the educational theory of "radical constructivism" (e.g., von Glaserfeld, 1987) which sees knowledge as an adaptive process, allowing people to cope in the world of experience by building consensus through mutually understood language. It is a cybernetic theory, not a cognitive one, in which structured conversation and feedback among individuals create a system that "comes to know" by creating areas of mutual understanding.

Research Evidence About the Effectiveness of the Proposed Pedagogies

Some doctors and healthcare professionals have adopted teachback in their conversations with patients to make sure they understand instructions on how to take medication and manage their care. In a study by Negarandeh et al. (2013) with 43 diabetic patients, a nurse conducted one 20-min teachback session for each patient, each week over 3 weeks. A control group ( N = 40) spent similar times with the nurse, but received standard consultations. The nurse asked questions such as "When you get home, your partner will ask you what the nurse said. What will you tell them?" Six weeks after the last session, those patients who learned through teachback knew significantly more about how to care for their diabetes than the control group patients. Indeed, a systematic review study of 12 published articles covering teachback for patients showed positive outcomes on a variety of measures, though not all were statistically significant ( Ha Dinh et al., 2016).

Relation to the Development of Twenty-First Century Skills

Teachback has strong relevance in a world of social and conversational media, with "fake news" competing for attention alongside verified facts and robust knowledge. How can a student "come to know" a new topic, especially one that is controversial. Teachback can be a means to develop the skills of questioning knowledge, seeking understanding, and striving for agreement.

Innovative Aspects of Pedagogy

The conversational partner in Teachback could be an online tutor or fellow student, or an Artificial Intelligence (AI) system that provides a "teachable agent". With a teachable agent, the student attempts to teach a recently-learned topic to the computer and can see a dynamic map of the concepts that the computer agent has "learned" ( www.teachableagents.org/). The computer could then attempt to teachback the knowledge. Alternatively, AI techniques can enhance human teachback by offering support and resources for a productive conversation, for example to search for information or clarify the meaning of a term.

Rudman (2002) demonstrated a computer-based variation on teachback. In this study, one person learned the topic of herbal remedies from a book and became the teacher. A second person then attempted to learn about the same topic by holding a phone conversation with the more-knowledgeable teacher. The phone conversation between the two people was continually monitored by an AI program that detected keywords in the spoken dialogue. Whenever the AI program recognized a keyword or phrase in the conversation (such as the name of a medicinal herb, or its properties), it displayed useful information on the screen of the learner, but not the teacher. Giving helpful feedback to the learner balanced the conversation, so that both could hold a more constructive discussion.

Level of Adoption in Educational Practice

The method has seen some adoption into medical practice ( https://bit.ly/2Xr9qY5). It has also been tested at small scale for science education ( Gutierrez, 2003). Reciprocal teaching has been adopted in some schools for teaching of reading comprehension ( Oczuks, 2003).

Placed-Based Learning

Place-based learning derives learning opportunities from local community settings. These help students to connect abstract concepts from the classroom and textbooks with practical challenges encountered in their own localities. "Place" can refer to learning about physical localities, but also the social and cultural layers embedded within neighborhoods; and engaging with communities and environments as well as observing them. It can be applied as much to arts and humanities focused learning as science-based learning. Place-based learning can encompass service learning, where students, and teachers solve local community problems, and through place-based learning acquire and learn a range of skills ( Sobel, 2004). Mobile and networked technologies have opened up new possibilities for constructing and sharing knowledge, and reaching out to different stakeholders. Learning can take place while mobile, enabling communication across students and teachers, and beyond the field site. The physical and social aspects of the environment can be enhanced or augmented by digital layers to enable a richer experience, and greater access to resources and expertise.

Relevance to Effective Educational Theories

Place-based learning draws upon experiential models of learning (e.g., Kolb, 1984), where active engagement with a situation and resulting experiences are reflected upon to help conceptualize learning, which in turn may trigger further explorations or experimentation. It may be structured as problem-based learning. Unplanned or unintentional learning outcomes may occur as a result of engagements, so place-based learning also draws on incidental learning (e.g., Kerka, 2000). Place-based learning declares that a more "authentic" and meaningful learning experience can happen in relevant environments, aligning with situated cognition, that states that knowledge is situated within physical, social and cultural contexts ( Brown et al., 1989). Learning episodes are often encountered with and through other people, a form of socio-cultural learning (e.g., Vygotsky, 1978). Networked technologies can enhance what experiences may be possible, and through the connections that might be made, recently articulated as connectivism (e.g., Siemens, 2005; Ito et al., 2013).

Research Evidence About the Effectiveness of the Proposed Pedagogies

Place-based learning draws on a range of pedagogies, and in part derives its authority from research into their efficacy (e.g., experiential learning, situated learning, problem-based learning). For example, in a study of 400 US high school students Ernst and Monroe (2004) found that environment-based teaching both significantly improved students' critical thinking skills, and also their disposition toward critical thinking. Research has shown that learning is very effective if carried out in "contexts familiar to students' everyday lives" ( Bretz, 2001, p.1112). In another study, Linnemanstons and Jordan (2017) found that educators perceived students to display greater engagement and understanding of concepts when learning through experiential approaches in a specific place. Semken and Freeman (2008) trialed a method to test whether "sense of place" could be measured as learning outcome when students are taught through place-based science activities. Using a set of psychometric surveys tested on a cohort of 31 students, they "observed significant gains in student place attachment and place meaning" (p.1042). In an analysis of 23 studies exploring indigenous education in Canada, Madden (2015) showed that place-based education can play an effective role in decolonizing curriculum, fostering understandings of shared histories between indigenous and non-indigenous learners in Canada. Context-aware systems that are triggered by place can provide location relevant learning resources ( Kukulska-Hulme et al., 2015), enhancing the ecology of tools available for place-based learning. However, prompts to action from digital devices might also be seen as culturally inappropriate in informal, community based learning where educational activities and their deployment needs to be considered with sensitivity ( Gaved and Peasgood, 2017).

Relation to the Development of Twenty-First Century Skills

Critical thinking and problem solving are central to this experiential-based approach to learning. Contextually based, place-based learning requires creativity and innovation by participants to manage and respond to often unexpected circumstances with unexpected learning opportunities and outcomes likely to arise. As an often social form of learning, communication and collaboration are key skills developed, with a need to show sensitivity to local circumstances. An ability to learn the skills to manage social and cross-cultural interactivity will be central for a range of subject areas taught through place-based learning, such as language learning or human geography. Increasingly, place-based learning is enhanced or augmented by mobile and networked technologies, so digital literacy skills need to be acquired to take full advantage of the tools now available.

Innovative Aspects of Pedagogy

Place-based learning re-associates learning with local contexts, at a time when educators are under pressure to fit into national curricula and a globalized world. It seeks to re-establish students with a sense of place, and recognize the opportunities of learning in and from local community settings, using neighborhoods as the specific context for experiential and problem-based learning. It can provide a mechanism for decolonializing curriculum, recognizing that specific spaces can be understood to have different meanings to different groups of people, and allowing diverse voices to be represented. Digital and networked technologies extend the potential for group and individual learning, reaching out and sharing knowledge with a wider range of stakeholders, enabling flexibility in learning, and a greater scale of interactions. Networked tools enable access to global resources, and learning beyond the internet, with smartphones and tablets (increasingly owned by the learners themselves) as well as other digital tools linked together for gathering, analyzing and reflection on data and interactions. Context and location aware technologies can trigger learning resources on personal devices, and augment physical spaces: augmented reality tools can dynamically overlay data layers and context sensitive virtual information ( Klopfer and Squire, 2008; Wu et al., 2013).

Level of Adoption in Educational Practice

Place-based learning could be said to pre-date formal classroom based learning in the traditional sense of work based learning (e.g., apprenticeships), or informal learning (e.g., informal language learning). Aspects of place-based learning have a long heritage, such as environmental education and learning though overcoming neighborhood challenge, with the focus on taking account of learning opportunities "beyond the schoolhouse gate" ( Theobald and Curtiss, 2000). Place-based learning aligns with current pedagogical interests in education that is "multidisciplinary, experiential, and aligned with cultural and ecological sustainability" ( Webber and Miller, 2016, p.1067).

Learning With Robots

Relevance to Effective Educational Theories

Learning through interaction and then reflecting upon the outcomes of these interactions prompted Papert (1980) to develop the Logo Turtles. It can be argued that these turtles were one of the first robots to be used in schools whose theoretical premises were grounded within a Constructivist approach to learning. Constructivism translates into a pedagogy where students actively engage in experimental endeavors often based within real-world problem solving undertakings. This was how the first turtles were used to assist children to understand basic mathematical concepts. Logo turtles have morphed into wheeled robots in current Japanese classrooms where 11- and 12-year olds learn how to program them and then compete in teams to create the code needed to guide their robots safely through an obstacle course. This latter approach encourages children to "Think and Learn Together with Information and Communication Technology" as discussed by Dawes and Wegerif (2004). Vygotsky's theoretical influence is then foregrounded in this particular pedagogical context, where his sociocultural theory recognizes and emphasizes the role of language within any social interaction to prompt cognitive development.

Research Evidence About the Effectiveness of the Proposed Pedagogies

Relation to the Development of Twenty-First Century Skills

Teaching a robot to undertake a task through specific instructions mimics the way human teachers behave with pupils when they impart a rule set or heuristics to the pupils using a variety of rhetoric techniques in reaction to the learner's latest attempt at completing a given task. This modus operandi has been well documented by Jerome Bruner and colleagues and has been termed as "scaffolding" ( Wood et al., 1979). This latter example illustrates a growing recognition of the expanding communicative and expressive potential found through working with robots and encouraging teamwork and collaboration.

Innovative Aspects of Pedagogy

The robot can undertake a number of roles, with different levels of involvement in the learning task. Some of the examples mentioned above demonstrate the robot taking on a more passive role ( Mubin et al., 2013). This is when it can be used to teach programming, such as moving the robot on a physical route with many obstacles. Robots can also act as peers and learn together with the student or act as a teacher itself. The "interactive cat" (iCat) developed by Philips Research is an example of a robotic teacher helping language learning. It has a mechanical rendered cat face and can express emotion. This was an important feature with respect to social supportiveness, an important attribute belonging to human tutors. Research showed that social supportive behavior exhibited by the robot tutor had a positive effect on students' learning. The supportive behaviors exhibited by iCat tutor were non-verbal behavior, such as smiling, attention building, empathy, and communicativeness.

Level of Adoption in Educational Practice

Interest in learning with robots in the classroom and beyond is growing but purchasing expensive equipment which will require technical support can prevent adoption. There are also ethical issues that need to be addressed since "conversations" with embodied robots that can support both learning and new forms of assessment must all sustain equity within an ethical framework. As yet these have not been agreed within the AI community.

Learning With Drones

Relevance to Effective Educational Theories

Outdoor fieldwork is a long-standing student-centered pedagogy across a range of disciplines, which is increasingly supported by information technology ( Thomas and Munge, 2015, 2017). Within this tradition, drone-based learning, a recent innovation, is being used to support fieldwork by enhancing students' capability to explore outdoor physical environments. When students engage in outdoor learning experiences, reflect on those experiences, conceptualize their learning and experiment with new actions, they are engaging in experiential learning ( Kolb, 1984). The combination of human senses with the multimedia capabilities of a drone (image and video capture) means that the learning experience can be rich and multimodal. Another key aspect is that learning takes place through research, scientific data collection and analysis; drones are typically used to assist with data collection from different perspectives and in places that can be difficult to access. In the sphere of informal and leisure learning in places such as nature reserves and cultural heritage sites ( Staiff, 2016), drone-based exploration is based on discovery and is a way to make the visitor experience more attractive.

Research Evidence About the Effectiveness of the Proposed Pedagogies

There is not yet much research evidence on drone-based learning, but there are some case studies, teachers' accounts based on observations of their students, and pedagogically-informed suggestions for how drones may be applied to educational problems and the development of students' knowledge and practical skills. For example, a case study conducted in Malaysia with postgraduate students taking a MOOC ( Zakaria et al., 2018) was concerned with students working on a video creation task using drones, in the context of problem-based learning about local issues. The data analysis showed how active the students had been during a task which involved video shooting and editing/production. In the US, it was reported that a teacher introduced drones to a class of elementary students with autism in order to enhance their engagement and according to the teacher the results were "encouraging" since the students stayed on task better and were more involved with learning ( Joch, 2018). In the context of education in Australia, Sattar et al. (2017) give suggestions for using drones to develop many kinds of skills, competences and understanding in various disciplines, also emphasizing the learners' active engagement.

Relation to the Development of Twenty-First Century Skills

Sattar et al. (2017) argue that using drone technology will prepare and equip students with the technical skills and expertise which will be in demand in future, enhance their problem-solving skills and help them cope with future technical and professional requirements; students can be challenged to develop skills in problem-solving, analysis, creativity and critical thinking. Other ideas put forward in the literature suggest that drone-based learning can stimulate curiosity to see things that are hidden from view, give experience in learning through research and analyzing data, and it can help with visual literacies including collecting visual data and interpreting visual clues. Another observation is that drone-based learning can raise issues of privacy and ethics, stimulating discussion of how such technologies should be used responsibly when learning outside the classroom.

Innovative Aspects of Pedagogy

Drones enable learners to undertake previously impossible actions on field trips, such as looking inside inaccessible places or inspecting a landscape from several different perspectives. There is opportunity for rich exploration of physical objects and spaces. Drone-based learning can be a way to integrate skills and literacies, particularly orientation and motor skills with digital literacy. It is also a new way to integrate studies with real world experiences, showing students how professionals including land surveyors, news reporters, police officers and many others use drones in their work. Furthermore, it has been proposed as an assistive technology, enabling learners who are not mobile to gain remote access to sites they would not be able to visit ( Mangina et al., 2016).

Level of Adoption in Educational Practice

Accounts of adoption into educational practice suggest that early adopters with an interest in technology have been the first to experiment with drones. There are more accounts of adoption in community settings, professional practice settings and informal learning than in formal education at present. For example, Hodgson et al. (2018) describe how ecologists use drones to monitor wildlife populations and changes in vegetation. Drones can be used to capture images of an area from different angles, enabling communities to collect evidence of environmental problems such as pollution and deforestation. They are used after earthquakes and hurricanes, to assess the damage caused by these disasters, to locate victims, to help deliver aid, and to enhance understanding of assistance needs ( Sandvik and Lohne, 2014). They also enable remote monitoring of illegal trade without having to confront criminals.

Citizen Inquiry

Citizen science is an increasingly popular activity that has the potential to support growth and development in learning science. Active participation by the public in scientific research encourages this. This is due to its potential to educate the public-including young people-and to support the development of skills needed for the workplace, and contribute to findings of real science research. An experience that allows people to become familiar with the work of scientists and learn to make their own science has potential for learning. Citizen science activities can take place online on platforms such as Zooniverse, which hosts some of the largest internet-based citizen science projects or nQuire ( nQuire.org.uk), which scaffolds a wide range of inquiries, or can be offline in a local area (e.g., a bioblitz). In addition, mobile and networked technologies have opened up new possibilities for these investigations (see e.g., Curtis, 2018).

Relevance to Effective Educational Theories

Most current citizen science initiatives engage the general public in some way. For example, they may be in the role of volunteers, often non-expert individuals, in projects generated by scientists such as species recognition and counting. In these types of collaboration the public contributes to data collection and analysis tasks such as observation and measurement. The key theory which underpins this work is that of inquiry learning. " Inquiry-based learning is a powerful generalized method for coming to understand the natural and social world through a process of guided investigation" ( Sharples et al., 2013, p.38). It has been described as a powerful way to encourage learning by encouraging learners to use higher-order thinking skills during the conduct of inquiries and to make connections with their world knowledge.

Inquiry learning is a pedagogy with a long pedigree. First proposed by Dewey as learning through experience it came to the fore in the discovery learning movement of the sixties. Indeed, the term citizen inquiry has been coined which " fuses the creative knowledge building of inquiry learning with the mass collaborative participation exemplified by citizen science, changing the consumer relationship that most people have with research to one of active engagement" ( Sharples et al., 2013, p.38).

Researchers using this citizen inquiry paradigm have described how it " shifts the emphasis of scientific inquiry from scientists to the general public, by having non-professionals (of any age and level of experience) determine their own research agenda and devise their own science investigations underpinned by a model of scientific inquiry. It makes extensive use of web 2.0 and mobile technologies to facilitate massive participation of the public of any age, including youngsters, in collective, online inquiry-based activities" ( Herodotou et al., 2017b). This shift offers more opportunities for learning in these settings.

Research Evidence About the Effectiveness of the Proposed Pedagogies

Research has shown that learning can be developed in citizen science projects. Herodotou et al. (2018) citing a review by Bonney et al. (2009) have found that systematic involvement in citizen science projects produces learning outcomes in a number of ways, including increasing accuracy and degree of self-correction of observations. A number of studies have examined the learning which takes place during the use of iSpot (see Scanlon et al., 2014; Silvertown et al., 2015). Preliminary results showed that novice users can reach a fairly sophisticated understanding of identification over time ( Scanlon et al., 2014). Also, Aristeidou et al. (2017, p 252) examined citizen science activities on nQuire, and reported that some participants perceived learning as a reason for feeling satisfied with their engagement, with comments such as "insight into some topics" and "new information."

Through an online survey, Edwards et al. (2017) reported that citizen science participants of the UK Wetland Bird Survey and the Nest Record Scheme had learned on various dimensions. This was found to be related in part to their prior levels of education. Overall, there is a growing number of studies investigating the relationship between citizen science and learning with some positive indications that projects can be designed to encourage learning (Further studies on learning from citizen science are also discussed by Ballard et al., 2017, and Boakes et al., 2016).

Relation to the Development of Twenty-First Century Skills

The skills required by citizens in the twenty-first century are those derived from citizen science projects. They " need the skills and knowledge to solve problems, evaluate evidence, and make sense of complex information from various sources." ( Ferguson et al., 2017, p.12). As noted by OECD (2015) a significant skill students need to develop is learn to " think like a scientist." This is perceived as an essential skill across professions and not only the science-related ones. In particular, STEM education and jobs are no longer viewed as options for the few or for the "gifted." " Engagement with STEM can develop critical thinking, teamwork skills, and civic engagement. It can also help people cope with the demands of daily life. Enabling learners to experience how science is made can enhance their content knowledge in science, develop scientific skills and contribute to their personal growth. It can also increase their understanding of what it means to be a scientist" ( Ferguson et al., 2017, p.12).

Innovative Aspects of Pedagogy

One of the innovations of this approach is that it enables potentially any citizen to engage and understand scientific activities that are often locked behind the walls of experimental laboratories. Thinking scientifically should not be restricted to scientists; it should be a competency that citizens develop in order to engage critically and reflect on their surroundings. Such skills will enable critical understanding of public debates such as fake news and more active citizenship. Technologically, the development of these skills can be supported by platforms such as nQuire, the vision of which is to scaffold the process of scientific research and facilitate development of relevant skills amongst citizens.

Level of Adoption in Educational Practice

Citizen science activities are mainly found in informal learning settings, with rather limited adoption to formal education. " For example, the Natural History Museum in London offers citizen science projects that anyone can join as an enjoyable way to interact with nature. Earthworm Watch is one such project that runs every spring and autumn in the UK. It is an outdoor activity that asks people to measure soil properties and record earthworms in their garden or in a local green space. Access to museums such as the Natural History Museum is free of charge allowing all people, no matter what their background, to interact with such activities and meet others with similar interests." ( Ferguson et al., 2017, p.13) At the moment, adoption is dependent on individual educators rather than a policy. Two Open University examples are the incorporation of the iSpot platform into a range of courses from short courses such as Neighborhood Nature to MOOCs such as An introduction to Ecology on the FutureLearn platform. In recent years there are more accounts of citizen science projects within school settings (see e.g., Doyle et al., 2018; Saunders et al., 2018; Schuttler et al., 2018).

Discussion

In this paper, we discussed six innovative approaches to teaching and learning that originated from seven Innovating Pedagogy reports ( Sharples et al., 2012, 2013, 2014, 2015, 2016; Ferguson et al., 2017, 2019), drafted between 2012 and 2019 by leading academics in Educational Technology at the OU and institutions in the US, Singapore, Israel, and Norway. Based upon an extensive peer-review by seven OU authors, evidence and impact of six promising innovative approaches were gathered, namely formative analytics, teachback, place-based learning, learning with robots, learning with drones, and citizen inquiry. For these six approaches there is strong or emerging evidence that they can effectively contribute to the development of skills and competences such as critical thinking, problem-solving, digital literacy, thinking like a scientist, group work, and affective development.

The maturity of each pedagogy in terms of evidence generation varies with some pedagogies such as learning with drones being less mature and others such as formative analytics being more advanced. In Table 1, we used the evidence classifications in Figures 1, 2 to provide our own assessment of the overall quality of evidence (strength of evidence and level of confidence (scale 1-5) based on NESTA's standards of evidence shown in Figure 2) for each pedagogy, as a means to identify gaps in current knowledge and direct future research efforts.

Table 1. Future directions of selected pedagogies.

Figure 2. Standards of evidence by Nesta.

The proposed pedagogies have great potential in terms of reducing the distance between aspirations or vision for the future of education and current educational practice. This is evident in their relevance to effective educational theories including experiential learning, inquiry learning, discovery learning, and self-regulated learning, all of which are interactive and engaging ways of learning. Also, the review of existing evidence showcases their potential to support learning processes and desirable learning outcomes in both the cognitive and emotional domain. Yet, this list of pedagogies is not exhaustive; additional pedagogies that could potentially meet the selection criteria-and which can be found in the Innovating Pedagogy report series-are for example, playful learning emphasizing the need for play, exploration and learning through failure, virtual studios stressing learning flexibility through arts and design, and dynamic assessment during which assessors support learners in identifying and overcoming learning difficulties.

Conclusions

In this paper we presented six approaches to teaching and learning and stressed the importance of evidence in transforming the educational practice. We devised and applied an integrated framework for selection that could be used by both researchers and educators (teachers, pre-service teachers, educational policy makers etc.) as an assessment tool for reflecting on and assessing specific pedagogical approaches, either currently in practice or intended to be used in education in the future. Our framework goes beyond existing frameworks that focus primarily on the development of skills and competences for the future, by situating such development within the context of effective educational theories, evidence from research studies, innovative aspects of the pedagogy, and its adoption in educational practice. We made the case that learning is a science and that the testing of learning interventions and teaching approaches before applying these to practice should be a requirement for improving learning outcomes and meeting the expectations of an ever-changing society. We wish this work to spark further dialogue between researchers and practitioners and signal the necessity for evidence-based professional development that will inform and enhance the teaching practice.

Author Contributions

CH: introduction, discussion, confusion sections, revision of manuscript. MS: teachback. MG: place-based learning. BR: formative analytics. ES: citizen inquiry. AK-H: learning with drones. DW: learning with robots.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

References

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Innovative Pedagogies of the Future: An Evidence-Based Selection

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The calendar below makes it easy to find opportunities and events that are scheduled for a specific day and time. Use the basic search below to find opportunities by date, keyword, and more. The advanced search tool at the bottom of this page gives you even more options to refine your results.

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1. Easy EdTech Podcast

Hosted by Monica Burns, this podcast for teachers provides clear and practical advice for both new and seasoned instructors around implementing EdTech in the classroom. Burns' approach provides many examples and tangible strategies you can use in your next lesson plan.

2. The 10-Minute Teacher Podcast

This is the first podcast I subscribed to, and I keep returning to it for inspiration and ideas to share with my students. Host Vicki Davis covers everything from augmented reality to escape rooms and other innovative techniques to try in the classroom.

I especially enjoy the guest interviews in which actual teachers share their ideas and strategies.

3. Beginning Teacher Talk

Although this podcast hosted by Dr. Lori Friesen has "beginning teacher" in the title, all teachers can benefit from tuning in. In this podcast for teachers, Dr. Friesen offers useful advice on typical first-year teacher struggles.

Topics include classroom management, working with parents, and developing creative techniques to use in the classroom.

4. The Burned-In Teacher Podcast

Amber Harper turns teacher burnout on its head through her podcast for teachers. In this series, Harper shares strategies with teachers on the verge of burnout, supports them through their burnout, and encourages them to grow through this experience.

5. EduMagic: Future Teacher Podcast

Shameless plug alert! I host the EduMagic podcast, which provides strategies for aspiring educators from kindergarten through college. My approach gives future teachers the confidence to try new things in their field placements, classrooms, and student teaching experiences, combining theory and practice.

6. This Teacher Life

This podcast for teachers is hosted by one of my favorite Instagram teachers, Monica Genta. Genta's approach to teaching and learning is not only practical but fun for educators and students. She offers a wealth of inspiring ideas that teachers can put into practice right away.

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In traditional math classrooms-and in classrooms where challenging, unfamiliar work is often assigned, more generally-the progression " I do, we do, you do" often becomes the default approach to learning, according to researcher and professor of mathematics at Simon Fraser University, Peter Liljedahl.

It makes sense in many cases, particularly when difficult concepts need to be addressed in time windows that are compressed by bell schedules, holiday breaks, andsummer vacation.But used too frequently, this approach, Liljedahl recently told the Cult of Pedagogy, inhibits higher order thinking and results in students who "mimic" teachers. Students who engage in too much rote work miss out on some of the challenging, sometimes confusing work that builds up their self esteem to face difficult thinking tasks in the future.

"By and large students spend most of their class time not thinking, at least not in ways we know they need to think in order to be successful in mathematics." Liljedahl explains. "If they're not thinking, they're not learning."

Liljedahl, the author of Building Thinking Classrooms in Mathematics, Grades K-12: 14 Teaching Practices for Enhancing Learning advocates for "Thinking Classrooms", which offers a different take on how classroom work is organized, how tasks are assigned, and how students learn and work together. His conclusions are based on more than a decade of research, experimentation, and collaboration with over 400 K-12 teachers.

In a 2017 article for Edutopia, Liljedahl clarified that a "non-thinking" classroom is "predicated on an assumption that the students either could not or would not think." When presented with difficult problems, students in these classrooms have a hard time pushing through to find their own solutions, he argued, and often wait for teachers to step in to do the heavy lifting for them.

To stimulate independent thinking, Liljedahl says, reorganize some of your classroom approaches: Start with hard puzzles and problems that push kids to their limits; confront the fundamental passivity of classroom seating; and use highly structured group activities to promote discussion, peer review, and iterative thinking.

Starting With 'Thinking Tasks'

Instead of starting a lesson with direct instruction, give students novel "thinking tasks" they can work on, ideally in groups. Liljedahl describes these tasks as problem solving activities and mental puzzles that, early on in the school year, should be "highly engaging, non-curricular tasks" to motivate students and get them in the mindset of challenging themselves. As the school year progresses and students become more accustomed to this mode of working and thinking, the activities and challenges can be replaced with tasks directly related to the curriculum.

Liljedahl points out that the tasks should be carefully sequenced so they get incrementally more challenging. "The goal of thinking classrooms is not to get students to think about engaging with non-curricular tasks day in and day out-that turns out to be rather easy," he told Cult of Pedagogy. "Rather, the goal is to get more of your students thinking, and thinking for longer periods of time, within the context of curriculum, which leads to longer and deeper learning."

Liljedahl has a long list of sample "thinking tasks" to look through. They include challenging dice-related problems, such as: Imagine a typical 6-sided die, and notice that the sum of opposite faces is always seven. The one is across from the six, the two is across from the five, and so on. Now imagine that you were making your own six-sided die that did not have this restriction. How many different dice could be made?

Use Standing, Randomized Group Work

Central to Liljedahl's approach is student collaboration and group work. Instead of grouping students by ability, or allowing them to choose their own groups to work in, his research has shown him students work more effectively-and are more likely to contribute-in randomized groups. According to Liljedahl, interviews with students show that randomized groups "break down social barriers within the room, increase knowledge mobility, reduce stress, and increase enthusiasm for mathematics."

At Design39Campus, a K-8 school in San Diego, eighth grade math teacher Kyle Asmus puts Liljedahl's approach into practice by using a random group generator (such as this one from Classtools or this one from Keamk) to ensure that different kids work together across multiple exercises.

To ensure that all students feel included, Asmus assigns group roles: The scribe writes down possible solutions; the speaker communicates the group's thinking to the broader classroom; the inquirer asks the teacher questions; and the manager makes sure the rest of the group stays on track.

Having students stand while they engage in this collaborative, messy thinking is yet another way to engage them, according to Liljedahl: It makes them much less likely to withdraw from the work, or assume that others will handle it. "It turns out that when students are sitting, they feel anonymous," Liljedahl told Cult of Pedagogy. "And when students feel anonymous, they disengage."

Work on Non-Permanent, Vertical Surfaces

In a thinking classroom, students put notebooks away and participate in group work while standing at vertical non-permanent surfaces such as whiteboards, blackboards, or windows-surfaces that Liljedahl believes promote more risk-taking.

According to Liljedahl, his experimentation with students shows that when comparing a group working on a whiteboard versus a group working on flip chart paper, the group working on whiteboards start working within 20 seconds.

"They'll start making notations on the board. They'll try anything and everything because they feel like they can just erase it if it's wrong," he told Cult of Pedagogy. Meanwhile, students working on chart paper take upwards of three minutes to make a single notation, because, Liljedahl said, they often wait until what they write is perfect-"and that hesitation leads to a lower form of thinking."

Design39 students also practice Liljedahl's "vertical surfaces" learning technique, which Asmus says promotes "high quality collaboration" and facilitates "questions and rich conversations" among students. According to Asmus, when students are presented with challenging tasks and work vertically, the time for students to get to a task is not only quicker, but the time they spend on a task is longer.

The large surfaces spread out around the room also allow students to see the work their peers are doing in other groups, and build off of each other's understanding. "It makes it easy for the whole class to see what everybody else is doing so we can be inspired by each other's ideas," said Iniyaa, a student in Asmus' class.

Answer the Right Questions

As students work vertically in groups, teachers can easily see how they're progressing and bounce around the room. Questions will undoubtedly arise, but Liljedahl says teachers should avoid answering questions asked for the purpose of reducing student effort and getting to an answer more quickly-such as "is this right?" Instead, they should prioritize addressing student questions that will lead to further independent thinking.

In her elementary school classroom in Brooklyn, teacher Tori Filler says that instead of rushing to provide hints and solutions to the hard parts of a lesson, she often asks them to evaluate what's tough about it, and encourages them to sort it out on their own before stepping in to help.

If most of the class is struggling, ask more questions and turn the struggle into a productive discussion for everyone. Asking questions like "What makes this hard?" or "What have we tried?" gets students to think metacognitively and develop the skills to push through challenging work on their own.

Evaluate What You Value

To succeed in a thinking classroom, students need to develop skills like perseverance, academic courage, collaboration, and curiosity, among others. But according to Liljedahl, if we want students to develop these competencies, then we should find ways to evaluate them on it.

"What we choose to evaluate tells students what we value, and, in turn, students begin to value it as well," he writes.

He argues for a mix of both formative and summative assessment in math classrooms that focus less on end products and student ranking, and more on the work leading up to those end products and collaboration between groups to get there.

The educators at Bite Sized Learning, for example, suggest evaluating students on the work they produce, but also on how well they persevere and make an effort in response to challenges, how well they set individual goals and monitor progress toward achieving them, and how well they share information and resources with group members to solve problems and make decisions.

Liljedahl's own formative assessments focus on informing students "about where they are and where they're going in their learning." This can take the form of observations, check-for-understanding questions, or even unmarked quizzes. Summative assessments, meanwhile, "should focus more on the processes of learning than on the products, and should include the evaluation of both group and individual work," Liljedahl writes.

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How to Turn Your Math Classroom Into a 'Thinking Classroom'

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The researcher Peter Liljedahl evangelizes for practices that prioritize and stimulate more hard thinking in classrooms.

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Dixie Ross has taught every level of math offered in Texas public high schools and trained hundreds of AP calculus teachers in summer institutes. Over 40 years, she's developed strong views on what's wrong with American math education, but one problem has rankled her since she first walked into a classroom: unequal access to higher-level courses.

Too many students are held back from advanced math that could provide direct pathways into college level math and STEM jobs, said Ross, a former presidential awardee for her teaching. What irks her most is that decisions about who gets tracked into or out of these higher-level courses are too often based on a student's race.

"There are kids who can be successful in math, but the opportunities are not there for them," Ross told me, in an eye-opening conversation that came in response to a survey The Hechinger Report sent to our readers last month. "I wish I had some magic bullet solution but haven't found it yet. And I have been looking for four decades. "

Ross was among more than 465 Hechinger Report readers who responded to our survey, with thoughtful feedback that is already informing our coverage of America's math crisis. We welcome hearing from readers as we visit classrooms and campuses, digging into questions about what kind of math should be taught at what age, and how best to boost lagging performance, close racial achievement gaps and help students catch up after the pandemic.

"There are a lot of holes and gaps from distance learning. The math content got shrunk down and the fluency just wasn't there. It's heartbreaking."
Giavanni Coleman, math teacher in Haywood, California

Several people pointed to gaps in availability of courses in STEM classes, which should not come as a surprise: Two out of five Black and Latino students surveyed for a recent joint report from the Education Trust and Equal Opportunity Schools said they have a passion for studying science, technology and engineering and want to go to college, yet only three percent were enrolled in AP STEM classes.

The issue of learning loss and recent NAEP test score declines - the largest ever recorded - also loomed large in survey responses, highlighting the devastating impact the pandemic had on students and families, in particular in schools that serve large numbers of Black and Latino students.

So did the issue of U.S. student performance compared with other countries: Our 15-year-olds rank behind 30 countries and one region on one international test, while our fourth graders trail 14 countries on another. So it makes sense that some teachers who answered the survey want to know how high-performing countries are teaching math, along with what cultural barriers might be in the way. "Are there schools that replicate best practices of countries like Japan and Finland and demonstrate better outcomes?" one educator asked.
Related: Plunging NAEP scores make clear the long and difficult road ahead to pandemic recovery
Survey results also confirmed there is a lot of anxiety about math. Some of it arises from recent test scores showing dismal middle school performance: Students who started middle school early in the pandemic lost more ground in math than any other group and are still struggling.

Fears that teachers are insufficiently trained in math and that poor math skills harm America's competitiveness and weaken our ability to fill critical jobs came up often in our survey. So did worries that high schools are placing too much emphasis on calculus and not enough on practical skills like data analysis and statistics for an increasingly high-tech world.

Several readers noted that families need more support than ever in overcoming their own math fears, along with additional tools and strategies for playfully supporting and supplementing their children's math knowledge. That means challenging age-old assumptions that some people simply aren't good at math.

And some teachers had specific ideas about what must change in math education: Giavanni Coleman, a 20-year veteran who teaches fifth- and sixth-grade math in Hayward, California, told us that schools must build a stronger foundation in math early on, and wants to see more investment in teacher training and early childhood math to help infuse a love of numbers at a young age.

"It takes time, and money, and human capital and training," Coleman told me in a follow-up conversation.

Coleman was also among the many teachers worried about pandemic learning loss. "There are a lot of holes and gaps from distance learning," she said. "The math content got shrunk down and the fluency just wasn't there. It's heartbreaking."

Here are a few top themes that concerned our readers:

Reducing anxiety or fear of math among students and helping them to understand why it matters.

Highlighting the importance of basic arithmetic (addition, subtraction, multiplication, division) in math teaching and focusing on establishing strong foundations before advancing to more conceptual lessons.

Learning in more detail about what effective math instruction looks like for preschoolers and kindergarteners

Understanding why subjects like algebra and calculus have become so important in college admissions and whether statistics or data science should matter more, along with how curricula are chosen and which work best

Tracing how math instruction has changed throughout history and ensuring that math lessons aren't outdated

Analyzing how math instruction and student performance changed after the introduction of the Common Core standards

Related: After common core a mysterious spike in failure rate among New York High School students
We also discovered common themes that concerned particular groups.

Parents were most likely to mention concerns about math curricula, math anxiety and their hope that math instruction would place greater emphasis on problem-solving instead of memorization and repetition.

Respondents from higher education were also most likely to mention reducing anxiety or fear of math among their students, along with the hope they can learn to both love math and understand why it matters to their careers.

And all groups worry that there aren't enough sufficiently qualified and experienced math teachers, in part due to low pay and poor working conditions.

Teacher Ross believes in recruiting great math students to become math teachers and wants to put all students on track to take advanced math unless they opt out of it. They should then be required to take any classes they fail until they pass, she thinks.

"Are there schools that replicate best practices of countries like Japan and Finland and demonstrate better outcomes?"
Educator who replied to Hechinger's survey

"We need to make sure kids understand that their decision to take or not take certain math classes will largely determine the economic opportunities that will be available to them," she said.

The survey results will be enormously helpful, but one of the most important ways of improving math came from a student I contacted after speaking with Ross. Carla Edith Brayton was an undocumented immigrant from Mexico living in Texas when she landed in Ross's math classes. She worked after school and nights at a local McDonald's, and while she'd always been good at math and loved the subject, she often fell asleep in class and felt discouraged.

Ross never allowed her to give up and pushed her to apply for scholarships and attend college. Brayton is now 29, a civil engineer and mother of two, the first in her family to attend college - she graduated from the University of Texas at San Antonio in 2017 - and to own a home. She loves her job and said none of her success would have happened if Ross had not encouraged her.

"Someone simply took the time to notice and believed in me. That's what changed my life," Brayton told me, noting that she has found a way to pay it forward by speaking at school career days, describing her background and the higher-level math classes she might otherwise have been shut out of.

"Education is the key for all people," she said. "It certainly was for me."
This story about math education was produced by The Hechinger Report, a nonprofit, independent news organization focused on inequality and innovation in education. The Gates Foundation has recently begun a major funding effort for math education projects around the country, and is among the many funders of The Hechinger Report. Sign up for our weekly newsletters.

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COLUMN: How can we improve math education in America? Help us count the ways

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Among the top themes were unequal access to higher-level math courses, how best to close racial achievement gaps, helping students catch up after the pandemic, assessing why American students lag behind in math.

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Playing games can be a great way to practice or review skills related to algebraic expressions.

For some students, learning algebraic expressions feels like drudgery, but it doesn't have to be that way. These three activities will get your students moving, talking, and thinking deeply about algebraic expressions, all while having fun. They're also a great way to practice or review skills that have already been introduced and explicitly taught.

3 Ways to Have Fun With Algebraic Expressions

1. Vocabulary Four Corners. Many students are overwhelmed by the prospect of translating a word problem into an algebraic expression. To help students feel confident recognizing keywords and appropriately converting them into numerical operations, you can introduce students to Vocabulary Four Corners.

In this version of Four Corners, you invite all students to move around the room based on a given vocabulary word. First, designate each corner of your classroom as an operation-addition, subtraction, multiplication, or division. Next, have all students stand behind their desks. Then, present students with a key vocabulary word and instruct them to move to the corresponding corner of the room. Once students have selected a corner, you can give them time to discuss their choice in groups and then call on a few students to share their thinking with the class.

This activity is a great way to help students review keywords while getting active in the classroom, moving around, and having a chance to talk in groups. If you want to make the activity more challenging, you can ask students to explain their thinking by making up sample story problems that use the given keyword.

2. How'd You Do It? This next activity is a great way to help students recognize the multitude of possibilities when working with algebraic expressions. There are a few different ways to implement this activity, depending on the amount of time you want to spend, but the basic structure is this: You present students with an algebraic term, and they come up with as many different algebraic expressions as possible that would simplify to your given term.

One way to do this is with a gallery walk. First, you need to post chart paper around the classroom and put one term on the top of each piece of paper. Next, split students into small groups and assign each group a piece of chart paper. Then, give students a set amount of time to spend writing their algebraic expressions on the chart paper before moving on to the next piece of paper as a group.

Once every group has had a chance to work on each piece of paper, you can then give students additional time to walk around the room and look at all of the expressions. Students may notice patterns in the types of numbers or operations their classmates used, or they may identify expressions that do not simplify to the given number. Students should share these thoughts within their small groups and then with the class at large.

If you don't want to use a gallery-walk approach, this same activity can work with small groups seated around the classroom. You can provide students with a term and a given amount of time to work together to come up with as many expressions as possible. You can then have groups share their expressions. If you want to make the activity more of a competition, you can even assign points for the numbers of unique expressions each group finds.

This activity helps students get creative and recognize how many possibilities exist when creating equivalent expressions. Depending on how difficult you want to make the activity, you can set additional parameters: Students must use negative numbers, students must use at least three operations, students must use division, etc.

An example of what students might come up with follows.

3. Equivalent Expression Stories. This last activity is another great way to have students practice recognizing and creating equivalent expressions. For this activity to work, you'll need to split students up into groups and, ideally, have them seated in columns or rows.

The first student in each column/row will be given a piece of paper with an algebraic expression on it. They will then write an equivalent expression underneath it and fold the top of the paper down. Then, they will pass the paper back to the next student. The second student won't be able to see the original expression, only the one written by the first student.

The second student will then repeat the process: Write an equivalent expression, fold the paper, pass it back. Once the paper reaches the last student, you will collect each group's work and show each equivalent expression story to the class.

Students can then discuss how each expression changed as it was passed, whether each expression remained equivalent to the original one, and any patterns they noticed. Similarly to the "How'd You Do It?" activity, you can scale up the difficulty by setting parameters around what students should use in their expressions.

A sample expression story might look like this:

Original expression: 5 x + 3 y

There are plenty of ways to personalize math activities to best fit the needs and interests of your students, but these three activities provide a jumping-off point to get your students engaged and excited about algebraic expressions.

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3 Fun Activities to Teach Algebraic Expressions

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Using fun activities to teach algebra gives kids a chance to review lessons on algebraic expressions in collaboration with their classmates.

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Transformation of the mathematics classroom with the internet...

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Transformation of the mathematics classroom with the internet...

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1 Introduction

The central theme for this special issue is the evolution and transformation of the classroom with the growing integration of the internet and interactive digital devices into mathematics teaching and mathematics teacher education. At the time of finalising this special issue, the entire globe is in the middle of the COVID-19 pandemic. Because of the extreme relevance of online and blended learning programmes at the current time, an editorial on the timeliness of this issue will be included in this special issue. Since the 1970s technology has changed mathematics education and it will certainly be a major factor in how education in the future differs from education today. Educators realise that we need to rethink the entire model of education and redesign it so that it is more student-centred. This means adopting new technologies, but it also means giving up on certain attitudes about what constitutes educational success. These new technologies also seriously influence the nature of mathematics, e.g., application of procedures is becoming less important and new ways of validation (and practising in general) in mathematics are being developed. However, it is beyond the scope of this special issue to explore this matter in depth.

With the rapid development of new digital technology, specific characteristics of these developments are continuously changing. After the baby boomers and the Generation X groups, new micro-generations evolved in cycles of about four years, giving birth to new concerns, new motivations and new challenges in all aspects of their lives (Morin 2016). These new micro-generations are as follows:

The Echo-Boomers or first digital natives Gen Y, born between 1989 and 1994; with the establishment of the Web, they grew up with the new technology.

For the net generation, often called Gen Z, born between 1994 and 1998, the internet is an essential part of their lives. They are sometimes called the hyper-connected 'selfie' generation and are attached to their smart phones as if they have become extensions of their personalities.

The Post-Millennial Generation Z or young mobile generation, born between 2002 and 2006, who did their learning through social networks and mobile technology.

The youngest group of students, born between 2006 and 2010, is named Gen Z-Silent Generation, who have been connected since birth.

Although the micro-generations differ, there are strong common characteristics-they multitask and want information quickly using visuals, sounds, and colour from multiple multimedia sources in a novel or useful way (Dineva et al. 2019). Our current students have grown up in a digital world of computers, internet, and social online media such as Instagram, Facebook, Twitter, Google and other social networks (Jukes et al. 2010). They learn by interacting with other individuals online. Many students have a blog on the Web and a profile on the internet (Curtis 2009). They like to be active and collaborate using the latest technology and visualisation opportunities. Today's students prefer seeking their own information rather than being presented with it (Morin 2016), they prefer on-demand access to knowledge, disseminated over the internet, and absorb knowledge rapidly across different channels. Using networks to share and create new knowledge, they are in frequent contact with their friends.

In a special issue of ZDM in 2012 (Borba and Llinares 2012), the emergent field regarding online mathematics teacher education showed key topics such as communities and networks of teachers in online environments; sustainability of these communities; knowledge-building practices in technology-mediated work group interactions; and online interactions among teachers. These topics also gave rise to new theoretical developments. The exponential development of interactive digital devices, and the use of the internet in teaching mathematics and online learning environment since the last special issue, support the need to frame some of these initiatives developing in the world (Borba et al. 2013; Borba et al. 2016; Trouche et al. 2013).

The development of digital technology use in mathematics education has been taking place in distinct phases (Borba et al. 2016). In recent years, the development of the internet has introduced a relationship revolution-communication has changed dramatically (Borba et al. 2016; Engelbrecht and Harding 2005a, b; Van de Sande 2011). This development brings us the possibility of two-way communication via the internet, and enhanced opportunities for collaborative learning. It also brings us the personalisation of the internet through personal devices. The role of social media is becoming increasingly important, moving the education process from a 'push' to a student-centred 'pull' approach in which the students become an integral part of many facets of the process (Martinovic et al. 2013). Furthermore, the introduction of digital resources and tools questions the efficacy of current teacher practices and traditional classrooms (Drijvers et al. 2013; Gueudet and Pepin 2020). Social aspects of the internet become more and more relevant and notions such as 'humans-with-media' emphasise that if media are changed, the entire knowledge-acquiring process may change (Borba et al. 2018). Moreover, as humans develop and construct new media, these media seem to transform and 'construct' a new human.

In this development process, the classroom, as we know it, may change entirely from a physical area with defined boundaries to a virtual environment including various components that will probably be determined by the student rather than only by the teacher. Mobile technology, personal learning environments, digital learning objects and other artefacts are 'stretching' the classroom, transforming the classroom, to the extent that it can hardly be recognised as such. Currently it seems clear that digital technology is 'deconstructing' the notion of the classroom. Flipped classrooms change the notion of what is in and outside of the classroom, and also change the roles of students and teachers. There is a profusion of online resources (e.g., widgets, videos), designed with respect to specific mathematical content, which transforms the presentation of content and allows students access to solving mathematical tasks and sharing their mathematical explorations. Our students also have to evaluate the quality of the knowledge disseminated over the internet-they need to be able to select valid resources.

On the other hand, the development of mobile digital technologies, such as forums, wikis, Twitter, Instagram and Facebook is allowing the new generation of different kinds of learning opportunities, supported in new social interaction spaces. These interactive technologies allow students to collaborate with their peers when they use multimedia and the internet, allowing new social ways of knowledge construction (Goos and Geiger 2012; Llinares and Olivero 2008). Finally, different blends are being forged between face-to-face education and online distance education.

In this paper, following mainly Borba et al. ( 2016) and other surveys and descriptions of the state of the art (Borba et al. 2013; Silverman and Hoyos 2018), we aim to support the building of the domain blended mathematics education. By addressing three main strands, the different papers in this special issue provide particular views and describe steps in each of the following areas:

Principles of design: How mathematics educators enact the principles of design in Massive Open Online Courses (MOOCs) and blended approaches to designing professional development opportunities and mathematics teaching contexts.

Social interaction and construction of knowledge: How technologies in online contexts support social interaction among participants as a medium to support mathematical knowledge construction and teaching competences.

Tools and resources: Different meanings associated with the idea of online resources and how their use is conceptualised in different mathematics teaching contexts, given the emergence of new online mathematics resources and ways of teaching.

While the papers in this issue address some of the topics in these domains, this collection is far from exploring all the different paths of digital technology in the mathematics classroom. We conclude the paper by identifying some trends and issues to be developed in the future.

Before we go any further, we should mention that, as we were finalising this paper, it became clear that the COVID-19 crisis was more than 'a little cold' as some prominent politicians claimed. The COVID-19 pandemic has 'paused' the world, and it has paused many mathematics education activities (e.g.. ICME 14, PME 44). In a certain way it also paused our paper. It seemed odd to write about digital technology, without connecting it to the current situation, which strongly impacts the world-wide use of digital technology in teaching. On the one hand, the worldwide lockdown increases the rate of change of using digital technology in education but at the same time it creates an awareness of the need to feel connected to each other. Rather than rewriting the paper, we added a section at the end in which we 'blend' the formal conclusions of the paper with questions and doubts that arose from the 'agency' of COVID-19, or that were catalysed by it.

2 How mathematics teacher educators enact the principles of design in blended approaches and MOOCs

Scaling up of professional development is concerned with how the programme is implemented across multiple instructors and settings and about the participation effects, such as course participants' changes in practice. Some initiatives focused on identifying factors that contribute to the sustainability and scaling up of professional development (Goos et al. 2018), recognising that examining their impact is complex, and lacking solid frameworks to explain learning in open online settings (Joksimovic et al. 2018). Digital technologies allow new approaches to design, implement and analyse large scale professional development and determine the influence on teacher practice and student achievement (Bell et al. 2010). For example, the internet allows development of new approaches through which it is possible to support the sustainability and scaling up of professional development focused on the introduction of dynamic technologies in using Web-based toolkits (Clark-Wilson and Hoyles 2019). The growing activities of designing MOOCs and using blended learning approaches are defining new issues related to how design principles can be generated.

2.1 MOOCs

Borba et al. ( 2016) mentioned MOOCs as a relevant theme, as follows:

The potential of MOOCs to disrupt the institutional and hierarchical nature of traditional education, offering students opportunities to access courses without prerequisites, without fees (unless they require a record of course completion), and the potential of MOOCs to affect access to and the quality of mathematics education is not well understood. (p. 606)

Right now, the reflection on the design, implementation and sustainability of MOOCs in mathematics education, generates theoretically informed perspectives and how these issues can inform decision making.

The design of MOOCs in different cultural contexts places emphasis on key aspects such as the possibility of sharing materials and ideas, and implicates methodologies of mathematics teaching and the different theoretical frameworks used to support their designs. In this special issue, the two papers focused on MOOCs present different conceptual perspectives to support the principles of design. The common feature in both papers is how the principles of design were theoretically informed and how the conceptual frames influenced the implementation of MOOCs.

Taranto and Arzarello ( 2020) report on a conceptual framework that guided the design of their MOOC and its use in interpreting the dynamics characterising teacher education. The conceptual framework presents the necessary hybridisation of three different theoretical approaches for mathematics teacher education through MOOCs, namely, the meta-didactical transposition model, the instrumentation/ instrumentalisation process and the network of knowledge of connectivism. Taranto and Arzarello argue that the conceptual framework allows the identification of specific features of the new learning environments defined by the MOOCs-how the interactions between participants change their knowledge and beliefs.

Hollebrands and Lee ( 2020) adopted a different conceptual approach to report on the effects of design principles of MOOCs. They examined how design principles were enacted in the development of the MOOC and how these influenced the engagement of participants. The principles considered were self-directed learning, learning from multiple voices, job-connected learning and peer-supported learning. From these principles and using the Interconnected Model of Professional Growth (Clarke and Hollingsworth 2002), Hollebrands and Lee considered the ways in which elements of the external domain influenced the personal domain of participants. The conceptual framework used emphasises the teacher change occurring through the process of enacting and reflecting on practice. In particular, they studied how the personal domain of participants in the MOOCs-knowledge, beliefs, and attitudes-interacts, as participants engaged with and reflected upon elements from the external domain, namely, readings, tasks, and frameworks, that define the MOOCs. How design principles are enacted in the design of MOOCs and how they support the learning opportunities for participants became a point for reflection.

A common feature, in the paper of Taranto and Arzarello, and that of Hollebrands and Lee, is their reflection on how the design principles adopted by designers of MOOCs influence participants' personal and social aspects. However, the ideas that support the design principles differ. On the one hand, Hollebrands and Lee use research-based practices used in face-to-face mathematics teacher education and design principles for the online environment to frame their decisions. On the other hand, Taranto and Arzarello hybridise three theoretical perspectives on mathematics teaching and learning to create a new conceptual framework for the purpose of making decisions as designers, researchers and teacher educators.

Other issues are related to how designers consider time constraints to strengthen the impact of the interventions. Three of these needs can be identified. Firstly, an issue is how theoretical perspectives can help us to improve our understanding of the processes of design and implementation of MOOCs and blended initiatives. Secondly, we should identify factors contributing to learning and sustainability in the MOOCs and blended approaches-factors influencing teacher participation in these initiatives. Finally, we have to evaluate different MOOCs and blended learning approaches across the educational settings to provide information about the role played by different contextual and cultural variables.

2.2 Blended approaches

Blended learning provides students with both delivery options, whatever their current stage of development may be (Chaney 2016; Vasileiou 2009). By integrating online learning into the system, blended learning expands the learning environment into the virtual world where traditional limitations are removed. Through the online component, differentiation between student needs becomes easier and combines with the social aspect of the actual classroom to create a strong learning system.

A blended learning system includes the important face-to-face interaction that Vygotsky considered to be vital and thus provides all of the benefits of the social aspects of learning (Ting and Chao 2013). Blended learning recognises the need for peer interaction and practical application in order to bring learning to maturity. Blended learning engages students since it offers them an opportunity to develop their own opinions, consider new ideas in collaboration with other students online, and try out their own ideas in a relatively anonymous environment (Holley and Oliver 2010).

Three papers in this special issue describe blended approaches to professional development for mathematics teachers as a reply to contextual needs, namely, improving primary and middle teachers' mathematics knowledge for teaching, focusing on 'out-of-field' mathematics teachers, and supporting the introduction of technology in mathematics teaching. They describe professional development initiatives from Chile, Ireland, and South Africa. One other paper describes a blended approach for first year engineering mathematics in Australia.

Martinez, Guíñez, Zamora, Bustos and Rodríguez ( 2020) describe a blended learning professional development programme for primary and middle school teachers aimed at developing mathematical knowledge for teaching. The ways in which face-to-face workshops and virtual activities are integrated illustrates the use of design principles by an instructional model characterised by a learning sequence and a construction model. The construction model is formed by four types of activities: activation, analysis, institutionalisation and practice. The authors underline that the design principles allow participants to have a high level of autonomous self-directed online learning and have learning opportunities to unpack and analyse elementary mathematics from a teaching perspective. Three general principles are used to design and analyse the implementation of this programme: the constructivist view of learning; contextualised problem-based learning, and that the teaching of mathematics requires specific knowledge. These principles are operationalised during the design, for example by means of the design of online activities around contexts operating as a general frame to problematise various aspects of the content. This feature allows generated learning situations in which tasks help to develop different aspects of mathematics knowledge for teaching.

Goos, O'Donoghue, Ní Ríordáin, Faulkner, Hall and O'Meara ( 2020) focus their initiative on secondary teachers that have to teach mathematics with no formal training or education in the field- out-of-field teachers. They analyse the design principles underpinning the development and delivery of a blended learning programme of professional development for out-of-field teachers of secondary school mathematics in Ireland. Three theoretical frameworks informed their analysis of the blended learning design. The first framework examines definitions, dimensions, and rationales for blended learning, the second framework characterises out-of-field teaching as a boundary-crossing event, and the third concerns effective teacher professional development using structural and core features. They integrated these frameworks within a blended learning context, attempting to find out how such an environment contributes to effective professional learning for out-of-field mathematics teachers. They found that engagement is long in duration and intensive among participants, but also learnt that epistemic considerations must be addressed when professional mathematicians engage with school mathematics teachers in curriculum development. For example, the nature of blended learning deserved more attention because a better appreciation could lead to better learning opportunities and outcomes for teachers.

A relatively new development in the professional development of teachers, with a strong foundation in mathematics education, is called lesson study. It is based on teachers collaborating to design lessons. This collaboration, however, can be challenging for isolated teachers who cannot communicate face-to-face on a regular basis. Joubert, Callaghan and Engelbrecht ( 2020) conducted a study in which they presented a blended mode course in the use of technology in teaching to practising teachers from different subject fields, including mathematics. The purpose of this course was to develop teachers' knowledge and skills in the use and integration of mobile technology in their teaching. In their study they investigated how lesson study can be adapted into a blended format to support isolated teachers who cannot meet face-to-face on a regular basis, using an Learning Management System (LMS) to communicate. They identified aspects that should be incorporated into a blended lesson study process to support isolated teachers in teaching with technology. A framework with three dimensions emerged, supported by aspects relating to teachers' collaboration, to the instructional design, and to the iterative improvement process.

Quinn and Aarão ( 2020) addressed the issue of what the contributions of online and face-to-face experiences should be in an ideal first year engineering mathematics blended learning experience. Developments trialled included online quizzes for changing attitudes and teaching foundational concepts, online lecture options in addition to face-to-face lectures, the adoption of what they called board tutorials (a technique that supports peer-to-peer learning and improves engagement), and learning outcomes supported by a problem-solving approach for more complex engineering modelling problems and online interactive problems. The individual student could determine the amount of blending. They found online quizzes to be a useful tool that, coupled with supporting material, can level the playing field in relation to the assumed knowledge for students transitioning to university. In addition, the use of online quizzes of foundational knowledge efficiently replaced a myriad of potential time consuming tutor-student conversations, such as diagnosis, feedback, independent work and the re-assessment, which would be impractical in a face-to-face environment. Quinn and Aarão also used board tutorials, changing the passive tutorials from 'mini-lectures', to experiences that both students and teachers valued highly. Quinn and Aarão argue that successful implementation of blended learning in first year mathematics occurs when one automates as many of the routine conversations between teacher and student as possible, and reserve what tutor-student interaction time you have for high-impact face-to-face learning activities such as board tutorials and for supporting project work.

The papers in this special issue that are mentioned in this section and centre on the design principles of scaling up professional development initiatives, show the needs that should be addressed by research. These needs focus on how to understand teacher change as a consequence of participating in MOOCS and blended learning initiatives.

3 Social networking and construction of knowledge

Social constructivist learning theory, as proposed and developed by Vygotsky, has long been seen to improve student engagement and learning, and many studies support this theory (Grady et al. 2012; Schmidt 2013). In their research on instructional technology, Pepin et al. ( 2017) found that many studies on the topic are predominantly framed by socio-cultural theories underlining the role of discourse in learning. The development of information and communication technologies has undergirded the emergence of new forms of discourse and has the potential to change social relations and the ways through which we come to understand the development of knowledge (Llinares and Olivero 2008; Llinares and Valls 2010; Clay et al. 2012). Interaction in online contexts allows us to consider the links between the processes of meaning construction and of participation underpinning learning. The links between construction of meaning and participation is scaffolded by social artefacts or tools, such as online collaboration, mind mapping or sharing narratives in online forums to discuss relevant aspects in mathematics teaching. Sharing interaction spaces, such as those that facilitate asynchronous1 online discussion, creates opportunities for participants to reorganise their knowledge in the course of the social interaction. In this sense, the affordance of new media helps participants to communicate knowledge in multimodal ways generating different ways of discourse.

Different theoretical perspectives about learning and knowledge have been used to understand the links between interaction in online contexts and the construction of knowledge to conceptualise technology-mediated interaction (Clay et al. 2012; Goos and Geiger 2012; Llinares and Valls 2010). Recently, new perspectives have emerged that consider how newly introduced media reorganise human thinking, favouring connections and group discussion (Borba et al. 2018). These approaches underline that media act and interact in knowing when participants interact collaboratively (in blended or online contexts). It is assumed that the nature of the interface between participants affects the potential for knowledge building (Borba and Llinares 2012). The notion of knowledge construction in collaborative settings assumes that the nature of participation and content of discourse is related to how the process of construction of knowledge is developed. One aspect here is the different forms of discourse that the participants adopt, and how participants create points of focus around a nexus that organises the negotiation of meanings. The media inserted in the specific online context can provide different ways of articulating the discourse for the participants to notice, represent, interpret or use theoretical elements. Two papers in this special issue focus their attention on different ways of articulating the discourse in collaborative contexts, namely, the construction of collaborative mind maps, and sharing and discussing narratives about teaching in an online forum. These two different types of discourse generated in collaborative online contexts focus the participants' attention on meanings, combining participation and reification processes. In these initiatives, several aspects of mathematics knowledge for teaching emerge, and mathematically significant pedagogical opportunities for mathematics teaching that builds on student thinking.

Cendros-Araujo and Gadanidis ( 2020) report on how the use of collaborative and multimodal technologies supports the construction of knowledge. The specific context is an undergraduate blended course in a primary teacher education programme in Canada, including the use of different tools for online collaborative mind mapping activities. The theoretical frame underlines the notion that new media reorganise human thinking. In this case mind maps are understood as ways of visualising mathematics education knowledge that is collectively constructed, and the different technological tools are used as means to support and organise topics, creating visual connections and inserting other different semiotic representations such as videos or images. The analysis of artefacts created by the prospective teachers (mind maps) and the flux of interaction during the construction process enabled Cendros-Araujo and Gadanidis to report on a grounded theory of knowledge building through mind mapping. They describe how pre-service mathematics teachers construct knowledge when they interact through online collaborative mind mapping and how the different technical characteristics of the tools impact on the reorganisation of knowledge by the different means of discourse. In this case the technological tools in a collaborative context allow new ways of knowing to be shown through a new type of discourse-such as integrating the visual and text, and combining graphical, narrative and symbolic realisations.

Fernandez, Llinares and Rojas ( 2020) report on the development of prospective secondary mathematics teachers' noticing as a consequence of sharing narratives of their own teaching in an asynchronous forum with other colleagues and university tutors. In this case, narratives of his/her own teaching are the artefacts that allow a prospective mathematics teacher to reify what is noticed on each occasion. The context is an online distance teacher education programme in Costa Rica aimed at developing the competence of prospective teachers to identify mathematically significant pedagogical opportunities (MOSTs) in order to build on student thinking. In this intervention, integration of thinking and doing is evident when prospective teachers reason about the teaching events in order to decide how to act. This paper shows two features of how online social interaction influences the development of teaching competence, such as noticing. The first feature is defined by the role of writing and sharing narratives in an online forum, and the second is defined by the role of feedback as a way of interactive collaboration. In this case, the development of noticing was related to aspects of discourse (communication processes) such as identifying MOSTs, providing more details of students' thinking and providing explicit reasons behind the prospective teachers' actions.

Both papers can be considered to be instances of how an online context, supporting the participants' interaction, helps to rearrange their own activity and, at the same time, as a way in which participants think and share their knowledge using certain technologies. These ideas are supported by the description of stages of knowledge building through mind mapping-introducing a topic, building a concept, and making sense of the whole picture-and by the ways in which the MOSTs are taken advantage of, and by the reasons given for the prospective teachers' actions. These ideas conform to the approach called humans-with-media, developed by Borba and Villarreal ( 2005). Furthermore, collaborative construction of mind mapping and sharing narratives of their own teaching can be seen as resources for thinking and communicating knowledge when participants are attempting to build new knowledge. The papers underline three relevant features, as follows. Firstly, how the media supporting social interaction can deploy several semiotic possibilities, giving forms to different types of discourse. Secondly, how sharing and co-creating tools generate the context in which participants can compare and share their ideas and justify and evaluate their arguments. Finally, the role played by cognitive scaffolding such as feedback from others and prompts to carry out the activities. These features help us to understand a little more about the relationships between social interaction in online environments and the knowledge construction and development of teaching competence.

4 Resources, tools and new learning environments: changing the relationships between mathematical knowledge, learners and teachers

In this special issue, several types of tools are analysed and discussed as they concern mathematics teaching and mathematics teacher education, showing new ways of thinking but also defining how we frame our activities as mathematics educators. Tools mediate the actions of mathematics educators, allowing us to design new learning environments, but this new context also determines new ways of thinking. There are several papers that focus on how using different types of tools determines changes in the relationships between mathematical knowledge and learners and teachers. Furthermore, the development of the new digital tools (or new uses of old tools) has generated the necessity of exploring innovative uses of digital technology.

The papers in Sect. 4.1 show three different approaches to digital tools: describing innovative practices; making explicit principles of design, and developing a new teaching stage. Innovative practices include using arts and digital technology to create virtual instruments with music software (Scucuglia) or the production of videos as a resource to produce meaning and to change the dynamics in classrooms, reported on by Oechsler and Borba. Another innovative practice is how students in mathematics courses can benefit by seeking assistance on the internet for help in solving their mathematical problems (Sanchez-Aguilar and Esparza Puga).

A second approach, as explored by the papers in Sect. 4.2, is to consider how new digital tools are framed by conceptual perspectives. In particular these perspectives influence the design of assessment tasks for computer based assessment to assess students' mathematical learning (Yerushalmy and Olsher), or they influence the production of tools such as video as resources to support teachers' professional learning, in which existing resources support teachers' professional learning (Bennison, Goos and Geiger). Finally, they illustrate how designing a new organisation of teaching supported by online access to video, shows that the typical lecture and homework elements of a course can be reversed in a flipped classroom environment (Voigt, Fredriksen and Rasmussen).

4.1 Tools, new ways of thinking and new practices

The new ways of using different digital tools portray a range of different teaching scenarios. The papers in this special issue consider the different meanings associated with the design and implementation of resources, tools and learning contexts, and changes in mathematics learning perspectives and the digital tools. The meanings associated with the notion of design and implementation of resources, tools and learning contexts, change mathematics learning perspectives; and the digital tools presented in different papers in this special issue show new ways of interaction between students, teachers and mathematical knowledge, defining new practices. Traditional tools, such as videos and assessment tasks, are used differently and define new practices. Furthermore, the accessibility of huge sources of information provided by the internet, generates new needs, such as how to determine criteria in looking for help to solve problems, or explicit conceptual frameworks to develop online resources to support teacher professional learning or to assess students' mathematical learning. There is also an increased need for students to think critically about the wealth of new opportunities associated with all the various resources with which they can interact, and how they can be empowered to develop this critical attitude towards evaluating new resources.

Scucuglia ( 2020) reports on the integrated use of the arts and digital technology in mathematics education creating pedagogic scenarios. In this case, the creation of a virtual instrument with music software is used for developing new ways of mathematics teaching. This approach generates new teaching scenarios that provide aesthetic mathematical experiences. The theoretical frame takes into account that mathematical knowledge is not produced by humans alone but by humans-with-media, in this case, analysing the pre-service teachers' mathematical experience when they are engaged in musical production. This new learning scenario underlines the link between music and mathematics education. The focus on the link between mathematics and music using digital tools opens possibilities for considering the hypothetical potential of thinking-with-media and music in terms of representation, patterning and algorithms, as aspects of computational thinking.

Oechsler and Borba ( 2020) investigated how the creation of videos with mathematical content, by the students themselves, may contribute to the process of changing the classroom, and how this activity can become a teaching and learning tool. They ground their discussion in social semiotics, a theory that considers the context of production and the negotiations between actors, to analyse how the production of videos in the classroom can help in the communication of mathematical knowledge and in the change of the dynamics in the classroom. They found that video production provided a classroom dynamic in which students could become protagonists of the teaching and learning process, with teachers mediating this activity. They argue that video production is a different way to express mathematics, and it is particularly well-suited to expressing what students have understood. Using videos, a new kind of mathematics can emerge in the classroom, integrating its traditional symbolic language with other modes, such as language, gesture, image and music. In production of the videos, students showed their understanding of the content, and through this activity, the students themselves became aware of their difficulties and sought ways to overcome them. In this sense, video production assists in encouraging students' discussion and reflection about content and its exposition to produce meaning and promotes a change in the dynamics of the classroom, breaking the barrier between the classroom and the outside world. This activity is seen as a new, emergent facet of education, in which the student searches for content outside the classroom.

Sánchez Aguilar and Puga ( 2020) used monitoring software to observe how a group of students use the internet to solve a mathematical task, whose mathematical underpinnings they are not completely familiar with, to produce a characterisation of the help-seeking behaviours that students display. They made use of self-reports, complemented with the analysis of students' solving of mathematics tasks, supported by the use of monitoring software. They found that students manifest instrumental help-seeking behaviours mostly associated with the procedural items of the task. A general pattern of behaviour, manifested by the students who participated in the study, was dominated by the use of search engines and keywords to identify sources of mathematical help. They observed how the internet and its resources were fundamental for some students to seek for help, and that, with this help, students were able to successfully solve a mathematical task on a topic partially unknown to them, which could be interpreted as a positive development of a self-regulated learning strategy. However, some students only used appropriate keywords to perform the help-seeking process-so students could find ways to solve mathematical tasks in which there was no need to exert a priori reasoning about the structure of the task or the nature of the mathematical situation at stake, before deciding on specific algorithmic procedures.

A common issue, considered by all three papers in this section, is the use of digital tools to emphasise a purposeful, jointly undertaken activity (creating virtual instruments with music software to underline mathematical thinking, videos with mathematical content, and using the internet to look for help). In these cases, the use of different types of tools shows how to create spaces for multiple voices in the learning environments, underlining the technological mediation played by the digital tools. These new uses of tools define new practices in mathematics education.

4.2 Developing new tools

The design of digital tools has generated the necessity to make the conceptual framing underpinning their development more explicit. The development of new tools strengthens the growth of this field, linking the quality of available resources and learning. This is the focus in papers in this issue by Bennison, Goos and Geiger, and by Yerushalmy and Olscher. Bennison and colleagues report on a research-informed instructional design approach to developing an online resource to support teacher professional learning. This approach provides a framework to determine the utility of existing resources and identify new needs, and also a conceptual framework to inform the development of new sets of videos. The specific focus is how video resources targeting specific teacher learning needs can be designed and made available via open access online.

Yerushalmy and Olscher report on a study concerning a special kind of task for assessing students' reasoning skills when establishing the validity of geometry statements about the similarity of triangles. By creating examples, students have to verify claims that argue for conjunction or disjunction of given relations. They focus on characterising the properties of the conjunction/disjunction design for automatically assessing conceptions related to examples generated by the learner with interactive diagrams. Their analysis shows that the Seeing the Entire Picture (STEP) environment, which supports interactive example eliciting tasks, and the design principles of conjunction and disjunction of geometric relations, enabled them to assess the students' exploration of the logic of universal claims, characterise successful and partial answers, and differentiate between students. By analysing the student-generated example spaces, they explored the opportunities of the environment and the specific task design pattern to automatically provide feedback and assess students' mathematical skills based on logical relations between examples and universal statements.

The papers of Bennison, Goos and Geiger ( 2020), and Yerushalmy and Olsher ( 2020) show how mathematics educators should take into account the conceptual frame underpinning the design of new digital tools (to provide opportunities for teacher professional learning and to asses students' mathematical learning). Explicit principles of design to develop tools (and determine ways of using them), is a growing field in mathematics teacher education (Van Es et al. 2020) and teaching mathematics (Leung and Baccaglini-Frank 2017).

4.3 New ways of teaching: flipped classrooms

The idea of a flipped classroom, where students watch earlier prepared lectures outside the classroom, has recently become quite popular in blended learning (Schmidt 2013).

In this pedagogical model, students completed preparatory activities (e.g., readings and reflections) before class and then participated in collaborative activities in class. (Crouch and Mazur 2001).

The flipped classroom approach in mathematics education is related to increasing in-class time for task/practice, the possibility of integrating new knowledge with existing beliefs and real-time feedback. But challenges are also reported, such as students' unfamiliarity with flipped learning, and significant start-up effort on the part of instructors (Lo et al. 2017).

5 Implications for future directions

In 2016, Borba et al. ( 2016) identified five trends of development in e-learning in mathematics education that need to be addressed:

1. 1.
The relationship between students and mathematics created by student access to mobile technologies disrupts the traditional flow of mathematics knowledge from teacher to student, and that is not well understood from a research perspective.

2. 2.
The role of MOOCs, disrupting the institutional and hierarchical nature of traditional education is not well understood.

3. 3.
The availability of online mathematics learning resources means that many students now turn to these resources before they consult a teacher or a textbook, raising questions about how the resources are facilitated to foster conceptual understanding.

4. 4.
Current technologies, such as social media, provide extensive collaborative and social networking affordances. This raises questions about the design and use of LMSs and personal learning environments.

5. 5.
Teachers are still uncertain about the amount and nature of blending in blended learning courses and how to employ a flipped classroom model to make the classroom a place for extension and elaboration rather than direct instruction.

Although some of the issues in these trends are being addressed, the questions are still very much open. From the surveyed literature, it is becoming increasingly clear that more empirical evidence is needed to determine the effectiveness of online or blended instruction in classrooms at all levels of mathematics education and how the use of digital tools determines new practices.

The nature of mathematics is also changing. With computing devices that can do the procedural mathematics faster and more accurately than humans, there is a shift from seeing mathematics as an application of procedures to an emphasis on creative problem solving (Devlin 2011).

Along with the trends mentioned above, we identify four domains that help us to see changes caused by the use of the internet: the changing classrooms; new ways of thinking and human-with-media; collaboration in online contexts, and hyper-personalisation of learning.

5.1 The changing classroom

Singh ( 2018) related a story of a mathematics teacher who used to give out "insanely hard" mathematics problems. He encouraged students to get help from any teacher in or outside the school, but excluding himself. His main objective was not really to get students to arrive at the correct answer or solution-he wanted to initiate mathematics conversations outside the classroom. He wanted people talking about mathematics. His ideas are central to the new approach of expanding the classroom outside the boundaries that we are used to.

The physical classroom, as we know it, is changing. Menninger ( 2011) encouraged educators to seriously consider and discuss the changes needed in the classroom. He wanted educators to regard teaching as a work of art, in that.

...a total work of art serves not only as an all encompassing intellectual, emotional, and spiritual experience, but also as a means of conceptualizing the act of teaching itself; instruction should inspire while it informs, just as art informs as it inspires. (p. 97)

Many current initiatives take a technology push approach in which learning content is pushed onto a group of students in a closed environment in a one-size-fits-all, centralised, static, top-down, and knowledge-push as in models of traditional learning. Researchers who feel a shift towards a more open student-pull model for learning are needed-a shift towards a more personalised, social, open, dynamic, and knowledge-pull model (Borba et al. 2016). Chatti et al. ( 2010) suggested the 3P learning model, consisting of three core components, namely, personalisation, participation, and knowledge-pull, as a new approach for addressing the growing complexity and constant change in knowledge that is required for the new generation.

The hyper-connected students of today live in a world of instant interpersonal communication and unlimited access to information and educational resources (Christen 2009). To an extent, this networked world, and the powerful learning tools it offers, has started to penetrate the typical classroom. Schools should take full advantage of the twenty-first century learning technologies. We are entering an educational transformation that aligns the learning with the learners themselves and the employment that awaits them after they leave school.

We may think that few people still believe that 'the role of the teacher is to transmit information'. However, many teachers-with-blackboard, or teachers-with-power-point are still considering themselves as information transmitters. Almost parallel to this issue, at the end of last century there was a heated debate about distance versus face-to-face education. As many researchers predicted, it now looks as if a blended, changing classroom is taking shape. Whether one calls it face-to-face, blended or distance, it almost always is a blended experience, in which on the one hand, students who are in a traditional classroom use internet and mobile technology constantly, and in which on the other hand, almost every distance education course has some face-to-face components.

As argued by Souto and Borba ( 2018), the internet has become the main source of transmitting information, changing traditional classrooms. The role of teachers, and of the community (face-to-face and virtual) is to build knowledge and propose new problems that have not already been solved on the internet. Together with these approaches, we still have (and it seems necessary) to have traditional instruction in many instances. Different ways of communicating mathematical ideas transform the production of knowledge, and it seems likely that the classroom (as we currently know it) will not fit in future education systems. The papers in this issue are instances of how the meaning of the classroom is changing in mathematics teaching as well as mathematics teacher education.

5.2 New ways of thinking and humans-with-media

In Vygotsky's theory, social interaction with others plays the primary role in the construction of learning-interaction with adults/teachers but also with peers. This interplay enables the student to move to higher levels of understanding and achievement (Blatchford et al. 2003).

The traditional online learning environment was first viewed with some skepticism and expected to be less effective at developing higher cognitive thinking processes than traditional classroom learning (Chaney 2016; Cicconi 2014). However, with the currently available features, blended learning incorporates the social aspect more fully through the presence of many other students, teachers and online resources. Chaney ( 2016) suggested that in blended learning the combination of a human teacher with online resources provides an effective way for students to construct learning socially (Chaney 2016). Researchers suggested that online blended learning fits well with Vygotsky's concept of a zone of proximal development despite challenges that arise (Cicconi 2014; Deulen 2013).

There is a growing relationship between humans and media, as originally proposed more than twenty years ago. Some authors have claimed that artefacts shape the human mind, but Borba and Villarreal ( 2005) documented that things happen the other way around as well: humans shape technology beyond the design of tools and of digital tools. Besides, technology is seen as having agency. Digital technology is saturated with humanity in its design and in its conception, and humans are impregnated by technology, and in particular digital technology.

Souto and Borba ( 2018) showed examples of how the third generation of activity theory may be transformed by the notion of humans-with-media as agents of production of knowledge. From the theoretical perspective of human capital and activity theory (Souto and Borba 2018), artefacts, community, and subjects are separate vertices of the multiple triangle diagram that illustrates activity in a social approach. Souto and Borba ( 2018) show examples of the internet being community and subject. The notion of humans-with-media, which has as one of three pillars the first generation of activity theory, with the notion of reorganisation of thinking (Tikhomirov 1981), may now transform the current, and more thorough third generation of activity theory.

5.3 Collaboration in online contexts

Collaboration in learning is becoming increasingly possible and popular in new teaching and learning contexts from the internet, and takes place in different formats. The concept of personal learning environments (PLE) was introduced through the work of, e.g., Attwell ( 2007), Chatti et al. ( 2010) and Wild et al. ( 2010). PLEs are systems that enable students to take control of their own learning, setting their own learning objectives, and managing their own learning content to achieve these learning objectives (Borba et al. 2016). A PLE can consist of subsystems, such as a desktop application or some web-based service integrating formal and informal learning, using social networks, and could include collaboration possibilities, such as small groups, to connect a range of resources and systems in an individual space.

PLEs differ from learning management systems (LMS) in that the LMS is course-wide (or institution-wide), while a PLE is individual. When students do not have control over what is taught but do have control over what is learnt (Tu et al. 2012), they create a PLE, a collection of all tools they use for learning, thereby enabling a student-controlled integration of myriad learning tools and services into a personalised space (Bidarra and Araújo 2013). The idea of a personal learning network (PLN) is related to the concept of a PLE. PLNs extend the PLE framework to include an informal learning network of people with whom to connect for the specific purpose of learning (Borba et al. 2016). In a PLN there is an understanding among participants that they are connecting for the purpose of active learning (Lalonde 2012). Although these environments are not commonly used in mathematics education yet, they are well-known concepts in other disciplines, such as computer science. It is envisaged that, in the foreseeable future, an increasing number of students and institutions will embark on this route of collaborating online to support their learning.

The issues regarding collaboration in online contexts in the papers of this issue underline some of the latter features, including, in particular, how the media supporting social interaction deploy several forms of semiotic possibilities. Also, they underline how features of participants in interactions, such as justifying their positions and evaluating their arguments, are intermingled with the cognitive scaffolding that is more difficult to identify in traditional teaching settings.

5.4 Hyper-personalisation of learning

Students no longer recite their lessons in chorus as they did years ago, but we are still far from a really personalised educational system (Paludan 2006). Despite our awareness of a disparity between students, the normal practice is to lump learners together by date of birth. Today, unique, personal characteristics such as creativity, a sense of humour, and special competencies are recognised as important in education (Paludan 2006). Learning can be enhanced when the instructional process accommodates the various learning styles of students (Lin et al. 2017).

New teaching contexts, with the use of the internet, offer every student a personalised approach to learning where they control their own pacing and where they can see themselves as successful students (Staker 2011). In fact, some authors are of the opinion that students should be allowed to choose their own learning pathways (Chaney 2017).

[They] are able to select learning formats to fit their changing needs...It is not the role of the teacher to prescribe the nature of the blend. (George-Walker and Keeffe 2010, p. 12)

The idea of hyper-personalisation has become quite popular in internet marketing. Adaptive hypermedia aims to enhance the functionality of hyperlink-based systems by making the user interaction process personalisable (Brusilovsky et al. 1998a). Adaptive hypermedia is an alternative to the traditional 'one-size-fits-all' approach in the development of hypermedia systems, in that they build a model of the goals, preferences and knowledge of each individual user and this model is used throughout the interaction with the user in order to adapt to the needs of that particular user (Brusilovsky 1996; Kurilovas 2016). In a learning situation, a student in an adaptive educational hypermedia system could be given a presentation that is adapted specifically to his or her knowledge of the topic (Hothi et al. 2000) and the most relevant links to proceed further will be suggested (Brusilovsky et al. 1998b; Kavcic 2004).

Mohan ( 2013) predicted that the future of all student education-from kindergarten to post-graduate level-will be hyper-personalised. Each student will focus on having their own teacher, their own curriculum and their own books or other resources. In such an environment, the teacher, using adaptive hypermedia, will increasingly become a person who understands the unique needs of each student. Not all students in a class will be at the same level. Some might surge ahead in mathematics while others in literature or art. It will help students to excel in something, rather than be ordinary at everything (Mohan 2013).

5.5 Panic-gogy

The arrival of the COVID-19 pandemic in 2020 forced universities and schools to move to online teaching instead of the traditional face-to-face approach, and this is likely to continue for the indefinite future (Han 2020). This situation makes online and blended learning an essential topic in the teaching of mathematics and other disciplines. If nothing else, the worldwide lockdown has given us a glimpse into the future-a future of remote workspaces, online interactions and digital service delivery. More and more people may be working or studying from home using online educational and meeting platforms, replacing existing processes with digital equivalents, ramping up measurement instruments on websites, interaction on social media, and marketing activities. E-learning is becoming popular worldwide. As a result of the pandemic, blended and online learning has developed from important to essential.

Teachers all over the world have a tongue-in-cheek name for what everybody is forced to do now: Panic-gogy-for panic + pedagogy (Kamanetz 2020).

Panic-gogy means understanding students' practical resources and problems, including availability of devices and the internet, family responsibilities, students sent home who need to find a new place to live, and financial constraints. But it also means how teachers are going to move into this environment with their teaching approaches.

Many teachers do not have the same experience of online instruction as they have of face-to-face teaching and all of a sudden there are many 'experts' giving advice on how an online approach should be employed. Teachers encounter new problems and feel somewhat isolated and uncomfortable in the environment. Teachers are uncertain about the level of students' commitment to learning.

One does get the feeling that, as a result of the COVID-19 crisis, in many instances, institutions move over to using technology without really making use of the existing available research on the topic. Granted, COVID-19 has forced institutions into using technology without the luxury of time to consider research or best practice-they are running crisis management to survive. Moreover, some instititutions use this opportunity to profit from the situation, saving on their face-to-face activities.

Most of us agree that teaching should be student-centred: "Teaching should not be based on what knowledge the professors can impart, but instead on what students need" (Han 2020). But when one suddenly has to transfer to an entirely new teaching environment, many teachers tend to just convert their traditional courses to an online platform. We have labelled that a "domestication of a new media": one does not take advantage of the agency of a new medium and simply uses practices from teaching with the old medium (Borba and Villarreal 2005).

I think the first thing is, we are not building online courses or converting your face to face courses to online learning. Really, what we're doing is we are trying to extend a sense of care to our students and trying to build a community that's going to be able to work together to get through the learning challenges that we have. .... so if people think that in three to five days they're going to rejigger their course and build some super amazing online platform, that's probably unlikely to happen. (DeRosa in Kamanetz 2020).

In many countries, there are also digital divides and social inequalities that have to be taken into account. There are many students (and teachers) who do not have access to technology and connectivity.

In the COVID-19 context where social distancing is encouraged, it will be essential to pay even more attention to the human connection and to find ways to ensure that human interaction is continued.

This special issue shows parts of this transformation that are happening in regular education, pre-service and in-service teacher education.

6 Conclusions

It is generally accepted that we do not need a formal 'curriculum' to explicitly teach a young child to speak. An environment permeated by orality seems a friendly one for a small child to try a word, to repeat, and to interiorise and create or construct new sentences.

The classic philosopher Pierre Levy (Levy 1993) discussed the different levels of orality. In current terminology, we can connect his ideas to the multimodal discourse (Bezemer et al. 2016) that is generated by digital videos, LMSs, and most discourses that are products of collectives of humans-with-media, including computers. Borba and Villarreal ( 2005) claimed that expressions made by collectives that include computers, may be part of an orality at a next level.

In this phase of technological development, communication that involves different computer platforms, e.g. LMS and social media, is a combination of icons, videos, regular writing, orality, images, graphs and video-clips that seems to bring a new consequence to mathematics education.

Traditionally, education used to be very much teacher driven. However, over the last decades, the focus has been changing from a situation of students passively absorbing information from an educator who is teaching by writing on the blackboard-sometimes referred to as pushing knowledge-to a more student driven approach, where students take control of the learning process-referred to as a pull process (Bassendowski and Petrucka 2013). This approach encourages students to select and transform information, discover principles, make hypotheses and decisions beyond the given information (Jung and Latchem 2009). Students can become involved in the design and development as well as the delivery of curricula (Lightner et al. 2007). An environment that supports the development of communities and collaborative discussion opportunities can assist students to comprehend and synthesise information, as independent and critical thinkers (Jansen et al. 2011).

In the traditional push approach the idea is that only the educator has legitimate knowledge and this knowledge is being transmitted one-way to students. A pull model involves the interest and commitment of students to create communities of trust, knowledge sharing, cooperation, and collaboration (Bassendowski and Petrucka 2013) as some papers in this special issue show. Pull approaches are characterised by constructivist and connectivist models-some of which are created by students, as in socially shared environments (Willems 2009). In these models students demand, request, and even create the particular products or information that they need. Students become active participants in their learning by working alongside the educators in both traditional and online settings (Ahn and Class 2011; Willems 2009).

A message throughout our entire paper is that the official 'curriculum' is currently playing a role that is overemphasised. Our modern students want a bigger say in how they are taught and what they are taught. Comparable to a child learning to speak, they want to decide on what mathematics they learn and how, in a pull approach, rather than a curriculum that is pushed onto them by the educational system. We may be at the beginning of a transformation of the classroom. Will mathematics change in the same way that it changed with the appearance of 'paper and pencil'? However, although the papers in this issue push the field forward, there are still many issues to address.

A wide array of media and technology is available to create new hybrid forms of teaching. The integration of technology enables educators to create learning experiences that actively and meaningfully pull students into course content. This technology may form thinking collectives (Levy 1993) with teachers that can break the walls of the regular 'cubic' classroom that is associated with lecturing.

Questions arise considering the new social interactions, the design of new teaching settings and about the new ways of thinking in the use of digital tools. Some of these questions are as follows:

What technology should students use to support their own learning as well as collaboratively the learning from other students?

How do we provide ways for students to evaluate and reflect critically about the learning and the resources with which they interact?

How can mathematics educators develop research-based principles of design regarding new teaching contexts that digital tools provide?

How can social media tools be combined with the best practices in teaching and contribute effectively to student engagement?

Papers in general, and survey papers in particular, age rapidly when the theme is digital technology (and mathematics education). New trends in mathematics education, and as we emphasise in this paper, changes in the (mathematics) classroom itself and the speed of digital technology may accelerate this 'deterioration' of the paper. With the COVID-19 crisis this may be an even bigger problem. Once this paper is published, or when it reaches a given reader, the COVID-19 problem may have been resolved with an appropriate vaccine and/or treatment for those infected, or we may still be at the stage we are now, with many airports, borders and frontiers closed, or we may be in a 'cloudy scenario', with some activities being resumed, but with everyone using masks and avoiding shaking hands and bodily contact.

The question is, what has this to do with mathematics education and digital technology? Besides the impact on conferences and on the transforming of mathematics classrooms, we may have to ask broader questions:

Digital technology intensified travelling and our way of living, so it is also partly responsible for the present crisis. Is it possible that the use of digital technology can generate a similar crisis in mathematics education?

Conversely, if the crisis lasts for a long period, would digital technologies be able to provide alternative ways to implement mathematics education?

There is not much research on online mathematics education for young children, but if the crisis lasts for a long time, are we going to implement it without sufficient research? If the current crisis is over soon, are we going to develop research on mathematics education for a possible 'COVID-2X' crisis?

In this paper, among others, we have anthropomorphised media, talking about agency. The notion of humans-with-media as the collective that produces knowledge, may synthesise it, as we discussed in this paper. The COVID-19 virus (SARS-CoV-2) is a non-living being: can we talk about the impact (agency) of COVID-19 on mathematics education and on the world?

These questions are too broad for this paper, and along with other questions will be the theme of other papers. We hope to be there to write and to read them!

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Authors and Affiliations

1. University of Pretoria, Pretoria, South Africa

Johann Engelbrecht

2. Universidad de Alicante, Alicante, Spain

Salvador Llinares

3. UNESP, Rio Claro, Brazil

Marcelo C. Borba

Corresponding author

Correspondence to Johann Engelbrecht.

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Special issue of ZDM Mathematics Education (2020, Nº 5 September) " Online mathematics education and e-learning ".

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Engelbrecht, J., Llinares, S. & Borba, M.C. Transformation of the mathematics classroom with the internet. ZDM Mathematics Education 52, 825-841 (2020). https://doi.org/10.1007/s11858-020-01176-4
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Transformation of the mathematics classroom with the internet - ZDM - Mathematics Education

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Growing use of the internet in educational contexts has been prominent in recent years. In this survey paper, we describe how the internet is transforming the mathematics classroom and mathematics teacher education. We use as references several reviews of use of the internet in mathematics education settings made in recent years to determine how the field has evolved.

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Integration of computational thinking in K-12 mathematics education: a systematic review on CT-based mathematics instruction and student learning

Abstract

There has been substantial research undertaken on the integration of computational thinking (CT) in K-12 mathematics education in recent years, particularly since 2018 when relevant systematic reviews were conducted on the topic. Many empirical studies in this area have yet to elaborate clearly and explicitly on how CT may support mathematics learning, or otherwise, in CT-based mathematics activities. Addressing this research gap, we conducted a systematic review on the integration of CT in K-12 mathematics education with a focus on CT-based mathematics instruction and students learning under such instruction. The Web of Science database was searched for in terms of studies published from 2006 to 2021, from which 24 articles were selected to provide illustrations of CT-based mathematics instruction and related student learning, and they were further analyzed according to education levels and contexts, programming tools, learning outcomes in CT and mathematics, and the mutual relationship between CT and mathematics learning. Among the results, this review found that geometrized programming and student-centered instructional approaches were facilitators of productive learning in CT and mathematics. Moreover, CT-based mathematics learning entails an interactive and cyclical process of reasoning mathematically and reasoning computationally, which can occur when: (1) applying mathematics to construct CT artefacts; (2) applying mathematics to anticipate and interpret CT outputs; and (3) generating new mathematical knowledge in parallel with the development of CT. The findings contribute to an in-depth understanding of what, and how, CT-based mathematics instruction impacts student learning in K-12 contexts.

Introduction

In the last two decades, considerable literature has grown around the theme of computational thinking (CT), namely, "the thought processes involved in formulating problems and their solutions, so that the solutions are represented in a form that can be effectively carried out by an information-processing agent" (Wing, 2011, p. 20). First to coin the term, Wing ( 2006, 2008) emphasized that CT is a fundamental skill required in daily life and a kind of analytical thinking that shares close connections with mathematical thinking, engineering and scientific practices. Research on CT has received increasing attention, especially regarding its integration into K-12 educational contexts (e.g., Feldhausen et al., 2018; Jocius et al., 2021; Sırakaya et al., 2020; Swaid, 2015). One kind of such integration that has obvious connection, yet little is known about, is CT's integration in mathematics instruction and its associated learning outcomes (e.g., Barr & Stephenson, 2011; Sneider et al., 2014; Weintrop et al., 2016).

Using CT and programming tools to learn mathematics can be traced back to Papert's Logo programming ( 1980) and further developed by Weintrop et al. ( 2016), who illustrate the resemblance of CT practices and those of science and mathematics practices, i.e., data handling, modelling and simulation, computational problem-solving, and systems thinking. With CT-based mathematics instructional approaches becoming more widely adopted in K-12 contexts in recent years (e.g., Miller, 2019; Pei et al., 2018; Shumway et al., 2021), a number of design-based research have been conducted to improve the conceptual basis and practice of a "computationally enhanced mathematics education" (Ng & Cui, 2021, p. 848). Related studies have also advanced the instructional design, including the programming tasks to facilitate students' mathematics learning. These studies encompass Kotsopoulos et al.'s ( 2017) pedagogical framework for CT activities, designing for unplugged, tinkering, making, remixing in K-12 mathematical problem-solving in the domains of combinatorics (e.g., De Chenne & Lockwood, 2022), algebra (Bråting & Kilhamn, 2021), number theory and mathematical modelling (e.g., Ng & Cui, 2021) and geometry (e.g., Miller, 2019). Significantly, these studies evidence the reciprocal relationship between computer science and mathematical concepts, in the sense that the application of CT deepens mathematical disciplinary knowledge, while the context of CT-based mathematics learning improves one's CT skills (Pei et al., 2018). However, as argued by Lockwood and De Chenne ( 2020), while programming seems to be effective in learning certain mathematical topics, it cannot be concluded that it would be superior to paper-and-pencil, nor that all domains of mathematics are generally suitable for integration with computing. In addition, students were found to experience various challenges due to potential differences between the two types of thinking (i.e., computational and mathematical) when solving mathematical problems in programming contexts (Cui & Ng, 2021). Thus, there remains a lack of consensus in the interrelationship and components of overlap between CT and mathematics, especially surrounding the K-12 education contexts in which strong emphasis has been placed on CT development in age-appropriate and interdisciplinary ways (Hong Kong Curriculum Development Council, 2020; International Society for Technology in Education, 2016; National Research Council, 2013). An important question to be further explored is to clarify how computing can be effectively used to foster K-12 mathematics thinking and learning (Hickmott et al., 2018), and vice versa.

While existing reviews have explored the multi-faceted linkage of CT and K-12 education (Barcelos et al., 2018; Grover & Pea, 2013; Hickmott et al., 2018; Hsu et al., 2018; Tang et al., 2020), a scarcity of reviews has discussed the linkage or interplay between CT and K-12 mathematics thinking and learning (Shumway et al., 2021). The most recent and relevant review was conducted by Hickmott et al. ( 2018), which was a scoping review of how CT has been applied in K-12 mathematics classrooms broadly. The categories of analyses were mainly descriptive, including implicit or explicit connections to domains of mathematics, activity approaches, and evidence of impact on student learning (by quantitative methods of studies). Hickmott et al. ( 2018) highlighted the need for "studies that explicitly linked the learning of mathematics concepts with computational thinking were uncommon in the reviewed literature" (p. 65). In response, this paper aims to provide an in-depth examination of extant studies of K-12 practices that explicitly link CT with mathematics thinking and learning. In this way, we contribute toward understanding and synthesizing the interplay between CT and mathematics learning as evident in empirical studies, especially given the vast number of studies that have been published since Hickmott et al.'s ( 2018) review.

In this paper, we report on a systematic review with the dual purpose of reviewing broadly the characteristics of CT-based mathematics instruction and student learning under such environment, and then provide an in-depth analysis and discussion of the interplay between CT and mathematics learning outcomes from extant empirical studies. To do so, we conducted a review of studies that detail both teaching and learning processes in CT-based mathematics activities. It addresses the research gap regarding the impact of CT-based mathematics instruction on student learning, and also more clearly and explicitly identify the interplay between CT and mathematics learning. To achieve our dual foci, we pose the following four research questions (RQs):

1. 1.
In what educational contexts is CT integrated in K-12 mathematics education?

2. 2.
What tools and instructional approaches are used in CT-based mathematics activities?

3. 3.
What theoretical constructs or conceptual frameworks have been used, and what students' learning outcomes are characterized in CT-based mathematics activities?

4. 4.
What kinds of interplay between CT and mathematics thinking and learning are observed?

Methodology

We used the PRISMA systematic review method (Page et al., 2021a, b) to review the impact of CT-based mathematics instruction on student learning. In what follows, we describe the search process, selection process and method of data analysis.

Search process

The Web of Science (WOS) database was used as the source for literature search. We used "computational thinking" AND "math*" AND ("school*" OR "educat*") AND ("primary" OR "elementary" OR "secondary" OR "middle" OR "high" OR "K-12") as the search terms to search in "topic", which included searching in titles, abstracts, author keywords, and keywords plus. In consistent with other systematic reviews on CT (Barcelos et al., 2018; Hsu et al., 2018), we limited our search to the publication date starting from 2006, when Jeannette Wing first introduced the term CT; as such, the date range of the publications searched was between 2006-01-01 and the date we performed the search, i.e., 2021-11-15. With WOS automatically removing duplicate articles, we obtained 290 articles in total for further screening.

Selection process

The overall goal of the selection process was to improve relevance of the searched articles as pertaining to the goals of the current review. Our selection process consisted of three stages, and the procedures are summarized in Fig. 1. In the following, we discuss each stage in detail.

At Stage 1, non-English articles and non-journal articles (e.g., Conference Papers/Abstract/Unpublished papers) were removed to ensure language accessibility and quality of research of the literature pool, resulting in 164 remaining papers. We excluded unpublished articles because of the need for peer review to ensure quality. Consistent with similar review (Tang et al., 2020), our decision to exclude conference papers and abstract was based on the following reasons: (a) according to Bradford's Law (Testa, 2009), the majority of noteworthy scientific findings have been disclosed in journal articles. Conference papers in this situation would present findings that were similar to those in journal publications. (b) Our systematic review aimed to examine detailed teaching process or student learning process. In general, conference papers did not report such information due to the limited space.

In the next two stages, we developed screening criteria as aligned with our research purpose to investigate the integration of CT in K-12 mathematics education. Although we included keywords of "K-12", "CT", and "math" in the search terms, there was no guarantee that the search records were relevant to those terms, as we only limited the search in titles, abstracts, and keywords. Therefore, at Stage 2, we further reviewed the title, abstract, and full text (when necessary) holistically to eliminate studies that were not conducted in K-12 contexts (e.g., at collegiate levels) or were not focused on CT and/or mathematics education. For example, some articles only mentioned the term "STEM (Science, Technology, Engineer and Mathematics)", "mathematics", or "CT" in the abstracts, but the primary focus of the studies was neither CT-related nor mathematics education-related, i.e., these articles are not about the teaching, learning, and affect issues of CT and mathematics. After the screening of this stage, we removed 111 articles, and the remaining 53 articles entered the full-text screening at the next stage.

At Stage 3, we aimed to screen for those studies that provide adequate descriptions of the teaching and learning processes to shed light on the relationship between CT-based mathematics instruction and student learning outcomes in CT and mathematics. First, the articles must report on empirical studies, so that the quality of student learning with reference to the instructional design can be empirically investigated. As such, we excluded theoretical papers, reviews, and articles that described provisional activities without implementation. Second, we identified studies that analyzed student learning processes and outcomes upon engaging in some form of CT-based mathematics instruction. This means we excluded studies that did not offer insight into features of student learning, such as research focusing only on teachers' learning (e.g., Bouck et al., 2021), and articles that only described the process of implementation or design of a course, task, or learning program, without explicitly reporting on student learning (e.g., Matsumoto & Cao, 2017). Regarding the third and final criterion, we continued to determine whether the article explicitly described any features of the CT-based mathematics instructional process (e.g., tools, tasks, instructional approaches); this inclusion criterion directly relates to our research questions. The first author and the second author independently conducted screening against these criteria. For discrepant screening results, the authors discussed each one until mutual agreement. This screening stage was concluded with 24 articles that constituted the final collection of review articles.

Data analysis

Although our research questions seem to be more inclined to retain the studies using qualitative research methods, the numerical results (Fig. 2, left) does not show significant difference in the use of the three research methods (qualitative at 42%, quantitative at 33%, mix method at 25%). We further categorized the data collection methods (Fig. 2, right) and found that observation and test are the two most frequent ways for assessing and having access to student learning.

Upon completion of the screening process, the first and the third author initially proposed a set of codes which addressed the RQs upon their reading of the 24 articles. For example, in response to answer RQ1 (educational context), the deductive codes included education level (grade), educational setting (school education or after-school activity), and disciplinary context (multidisciplinary or CT-based mathematics context). Likewise, in response to RQ2 and RQ3, the authors prescribed codes that addressed the programming tools and instructional approach adopted, as well as learning outcomes (associated with CT or mathematics).

To arrive at the second- and third-level codes, the first and third author performed constant comparative analysis (Strauss & Corbin, 1990) to identify and refine the codes that adequately describe the methods and results highlighted in the empirical studies until saturation of categories. Specifically, the authors independently coded six of the articles (or 25% of the selected articles) and discussed the coding results to (1) operationalize the meaning of each code; and (2) delete, add, and synthesize coding items that were helpful to answer the RQs. This process yielded the second- and third-level codes, and a finalized coding scheme with code definitions as presented in Table 1. In the case of RQ4, the authors used constant comparative analysis (Strauss & Corbin, 1990) to obtain new findings regarding the interplay between CT and mathematics. Open coding was first used to create initial codes by extracting information from the literature that was pertinent to student learning. Then, the authors practiced axial coding and selective coding to create a core category to connect them and achieved the final codes.

Upon completion of the coding scheme, the authors coded another eight articles independently, and the inter-rater agreement exceeded 90%. At last, the first author coded all the remaining articles, and when encountering uncertain coding, all authors discussed until agreement. We report on the coding results in the next section.

Result

In this section, we first describe the educational context, where CT-based mathematics teaching and learning activities are implemented (section " Educational context of CT-based mathematics instruction"); this includes reporting on the education level, educational setting, and disciplinary context of the reviewed studies. Then, we discuss the pedagogical aspects of the articles (section " Pedagogical aspects of CT-based mathematics instruction"), which include the tools and instructional approaches used during CT-based mathematics instruction. Finally, we review students' learning outcomes in relation to CT and mathematics (section " Interplay between CT and mathematics learning outcomes"), both in terms of the diverse constructs used to characterize student learning, as well as the interplay between CT and mathematics learning outcomes as evident in the reviewed literature. We discuss the results along with our further insights in each sub-section.

Educational context of CT-based mathematics instruction

To begin, we use Fig. 3 to summarize the literature count regarding the education level in which the studies are conducted. Notably, more researchers have carried out studies in primary educational contexts, followed by lower secondary education. This may be related to researchers' attention to the importance of students' early exposure to CT (Jurado et al., 2020; Sáez-López et al., 2019) alongside the emergence of child-friendly programming tools for easy CT experimentation during a critical cognitive developmental period for CT learning (Rodríguez-Martínez et al., 2020). Among the 24 reviewed articles, five included research participants across education levels; these studies highlight the feasibility of conducting CT-based mathematics activities to students with varying knowledge bases and skill levels. The cross-grade activities used among these studies are afforded by the spiral mathematics curricula across K-12 education, such as the recurrent arithmetic topics in kindergarten and lower primary education (Sung et al., 2017), the relevant transition from arithmetic to algebraic thinking in the middle school grades (Ng & Cui, 2021), and the interrelated 2D geometry topics across secondary education (Wilkerson-Jerde, 2014). In addition, these studies commonly provided an open-ended constructionist learning environment which involved the students creating CT artefacts with little direct instruction (Papert & Harel, 1991), during which the students could explore and develop non-prescribed learning outcomes in both mathematics and CT. As argued by Muñoz et al. ( 2020) and Ng et al. ( 2021), CT-based mathematics activities are suitable for cross-grade teaching and learning, because they support non-traditional curricula learning, with the former study showing that mathematics learning could take on a range of domains with little prior mathematics knowledge necessary, and the latter study concluding that the developmental sequence of learning mathematics maybe altered by the use of computational tools. In particular, Ng et al. ( 2021) highlights the conceptual connection between skip counting and geometric sequences, which takes roughly 7 years to learn in traditional curricula; but in the Scratch programming environment, it is as straightforward as replacing the symbol for operation. In summary, we synthesize from the reviewed studies the potential for CT-based mathematics instruction to achieve cross-grades mathematics learning. At the same time, they also stimulate questions regarding the constraints of disciplinary mathematics learning within a highly structured curricula which is predominantly intended for using paper-and-pencil as the medium of learning. As such, we also draw attention to the constraints of research and practice in CT-based mathematics instruction given the current curricula structure.

As part of educational context, we analyzed the educational settings to understand whether CT-based mathematics instruction has been applied in formal or informal education, with the former being compulsory and the latter being voluntary. We found that 17 reviewed studies were conducted within a formal schooling environment, and 7 studies were conducted outside of formal education settings (of which 4 were in primary education and 3 were in secondary education). In particular, the 7 studies were contextualized in after-school programs or summer camps, which are also considered as non-formal education, since they are intentional and structured when compared to spontaneous and unstructured activities in daily informal settings (Eshach, 2007). We also investigated the disciplinary context of CT-based mathematics instructions to examine whether a single or multiple disciplines were involved in the learning activities. We found that a minority of studies ( n = 7) aimed to develop or support students' CT and mathematics learning in an interdisciplinary or multidisciplinary way (e.g., STEM, STEAM, Math + Science). For example, the study of Gilchrist et al. ( 2021) was situated in a pandemic awareness STEM outreach curriculum focused on medicine, mathematics and technology, where students developed an epidemic outbreak simulator using the Python programming with CoCalc. In another study, students were asked to use KIBO robots to solve STEAM-related problems in designing and engineering a real-life artefact (Jurado et al., 2020).

The remaining 17 articles reported on students' CT and mathematics learning in the context of solving CT-based mathematical problems (see more discussion in section " Interplay between CT and mathematics learning outcomes"). For example, Wilerson-Jerde (2014) explored how secondary students create and analyze fractal structure with Categorizer, while Psycharis and Kallia ( 2017) posed a statistical problem for students to solve with MATLAB programs. At the primary education level, students used Scratch to learn geometric figures by drawing squares and spirolaterals (Miller, 2019). It is evident that most practices of CT-based mathematics instruction are situated in mathematical (disciplinary) problem-solving contexts rather than in STEM (multidisciplinary) contexts. The lack of studies which address multidisciplinary learning in CT-based mathematics instructions is likely associated with our earlier findings that most studies were conducted in formal education context. Moreover, it has been well-documented in literature that STEM integration is difficult to practice, especially when the focus of STEM integration is on achieving mathematics learning (English, 2016). Together, the findings surrounding educational levels and settings prove that CT-related research in both STEM and mathematics education is at a beginning point, and that more empirical research is needed to inform effective integration of CT in formal K-12 curricula context.

Pedagogical aspects of CT-based mathematics instruction

In this section, we focus on the pedagogical tools and approaches that have been developed and utilized to support CT-based mathematics instruction and student learning.

Tools

While CT is generally related to computer science and programming tools, CT learning activities are not limited to plugged activities (implemented with the use of computers). Our review shows that unplugged CT activities (implemented without the use of computer) are often carried out by researchers as precursory activities for plugged CT activities, such that students would experience some target CT concepts and practices in advance, and this is conducive to students' adaptation to programming (plugged) activities (Bouck & Yadav, 2020; Sung et al., 2017). In Bouck and Yadav's ( 2020) study, for example, an unplugged game of moving plastic cups on a surface was used to help students gain a better appreciation for what it is algorithms and debugging before students began coding to move a robot. In terms of plugged activities, we identified 18 different tools and platforms used in CT-based mathematics activities across the literature, with some being conventional tools for programming and technology instruction (e.g., Python, Scratch, Robot Dash), and some being more commonly used in mathematical practices (e.g., Sketchpad, GeoGebra, MATLAB). Based on the programming platform interface, we divide these tools into two categories: screen-based programming and tangible programming (Table 2).

Under screen-based programming, by which users program within a screen (e.g., computer screen), we identified three sub-categories of programming languages: text-based programming, block-based programming, and geometrized programming. We define text-based programming tools as those requiring strict, precise, and complex syntactic rules to code, thereby often requiring a keyboard to input text (Horn & Jacob, 2006), such as Processing, Python, Matlab, and MaLT2 (Fig. 4a). In block-based programming environment (e.g., Scratch, ViMAP), students can drag and drop the visual script blocks on the screen with a mouse or their hands to program. These script blocks constitute more natural and intuitive ways to operate a program while providing useful visual cues on how and where to use program commands (Fig. 4b). Unique to CT-based mathematics instruction is a geometrized programming language, which, as Sinclair and Patterson ( 2018) proposed, is a third screen-based programming language whose elements are geometric objects rather than texts or blocks. One typical example is the Geometer's Sketchpad, and another example is Lattice Land (see Pei et al., 2018), whose interface enables users to interact directly with dots to construct segments and polygons and investigate lattice geometry as a form of geometrized programming (Fig. 4 c).

a Interface of text-based programming tool MaLT2 (Grizioti & Kynigos, 2021, p. 10); b a block-based programming tool mBot (Sáez-López et al., 2019, p. 1414); c segments and a polygon drawn in a geometrized programming tool Lattice Land (Pei et al., 2018, p. 78); d a Bee-bot robot, a form of tangible programming used in Muñoz et al., ( 2020, p. 7)

Tangible programming requires coders to use hardware objects with physical user interface to build programs, as opposed to "write" or "drag-and-drop" on the computer screen. With tangible programming, young coders can create procedural syntax to perform physical actions such as moving between two locations by scanning a code sequence ( KIBO Robot), pressing command buttons on the machine ( Bee-bot Robot; Fig. 4d), entering commands with an external remote ( Botley), and placing the desired tiles on the programming board ( Cubetto).

Although extant studies have considered the cognitive demand required on the programming tool in age-appropriate ways during CT-based mathematics activities, few researchers have explicitly communicated the affordances of the programming tool in supporting the target mathematics learning content. In response, we aim to generate further understanding from the reviewed empirical studies on this question. First, the reviewed studies suggest that the physicality of tangible programming is supportive to the learning of number, measurement, geometric shapes and spatial concepts and skills (Jurado et al., 2020; Muñoz et al., 2020; Shumway et al., 2021). Sáez-López et al. ( 2019) further recommended the use of block-based programming and robots for learning number concepts in CT-based mathematics activities, saying "there are many advantages to teaching computational concepts, coordinates, values and integer numbers as motivation for the student to learn how to operate the robot. The immediate feedback and response of the robot when operating with numbers is a powerful and highly motivational tool for students" (p. 1422). Conversely, text-based programming requires higher level of abstraction; these features make it suitable to deal with more advanced number topics, such as statistics (Gilchrist et al., 2021; Psycharis & Kallia, 2017), number sequence (Chan et al., 2021), and algebraic relationships (Kaufmann & Stenseth, 2021). We note that although the spreadsheet (e.g., Excel) is not generally defined as a CT tool, it executes some statistical functions and output upon inputting text-based formulae, and thus, it was considered as a text-based CT tool in both studies of Chan et al. ( 2021) and Valovičová et al. ( 2020).

Uniquely in a geometrized programming environment, the grammar and rules for execution are the properties of geometric objects; only when the properties of geometric objects are satisfied can the correct structure of geometric figure be constructed. Studies which adopted geometrized programming would take a "point" (in a Euclidean sense) as the most basic programming element or input, from which one can generate a more complex output, such as constructing a segment by joining two points or further formed a quadrilateral by constructing four segments. As Sinclair and Patterson ( 2018) described, applying actions on geometric objects (e.g., dragging a point) is like executing a program, and the actual behavior of the objects provides feedback that helps the coder to debug or proceed to program. In summary, geometrized programming tool as a form of computing tool is especially congruent to the learning of geometry. If future development in CT tools can be specialized to be strongly associated with a particular form or learning content at the tool level, it will open a new venue of educators to support mathematics learning in CT contexts.

Instructional approaches

Regarding CT-based mathematics instruction, we identified several common instructional approaches, and we further sub-categorized them by task structure (i.e., the organization of learning tasks and learning processes): problem-based learning (Gilchrist et al., 2021; Kaufmann & Stenseth, 2021; Ng & Cui, 2021), inquiry-based learning (Gilchrist et al., 2021; Pei et al., 2018), and project-based learning (Sinclair & Patterson, 2018). While reviewing the studies, we also note other learning modes, such as embodied learning (Sung et al., 2017), game-based learning (Grizioti & Kynigos, 2021), and pair- or group-learning (Echeverría et al., 2019; Grizioti & Kynigos, 2021; Kaufmann & Stenseth, 2021; Muñoz et al., 2020). In the present review, we focus on analyzing the characteristics of problem-based learning, project-based learning, and inquiry-based learning, since they are more dominant in the literature.

In general, the studies adopting one of these task structures share the common characteristic of implementing student-centered approaches, thereby positioning students as active mathematics learners, computational thinkers and problem solvers. They are also constructivist- or constructionist-oriented, revealing the epistemological underpinning the movement of CT for mathematics. While sharing in common the goal of constructionist or "learning-by-making" (Papert & Harel, 1991, p. 1), we note that the kinds of "making" vary among the three task structures: "making" stands for creating a programmable solution in problem-based learning, constructing a CT artefact in project-based learning, and formulating complex explanations through explorations in inquiry-based learning. In the following, we first provide an example of how each task structure manifest under CT-based mathematics activities to illuminate patterns within and across them, followed by discussing their implications for advancing the integration of CT and mathematics in K-12 contexts.

To begin, our review points to the high compatibility of problem-based learning in CT-based mathematics activities, given the central connection that CT is a mode of problem solving and a kind of analytical thinking in common with mathematical problem solving and reasoning (Baldwin et al., 2013; Wing, 2008). In this context, the main objective is to apply CT and mathematical knowledge and skills-as they are developed-during the course of problem solving. An example is provided in Kaufmann and Stenseth ( 2021), who utilized problem-based learning by first introducing the computational tool, then providing a default program as the starting problem for the students. Without providing any other scaffoldings, students grouped in pairs were asked to propose solutions to rectify the coding, so that the wheel can rotate at a constant speed and the size of the wheel can vary without affecting how the wheel rotates. This approach facilitates "autonomy and active learning experiences, to promote uncertainty and active methodologies centred on the student" (Sáez-López et al., 2019, p. 1406).

Different from problem-based learning, the goal of project-based learning is to produce CT artefacts with some project specifications; typically, the students encounter multiple problems related to CT or mathematics and need to make clear plans and reasonable decisions within their knowledge to satisfy the requirements of the project. Sinclair and Patterson ( 2018) illustrated a study adopting project-based learning, where a teacher first shared some examples of machines created with Sketchpad, led the class to brainstorm similar machines they could make, and provided instructions for the "Machine Project" with specific features (e.g., it should have animated features and involve trigonometry in some way). After students made a project proposal, they worked on their projects independently over the next 2 months and finally created a working machine with Sketchpad (see an example in Fig. 5).

Akin to the other two task structures, inquiry-based learning unfolds with an open question with no obvious solutions in CT-based mathematics context. Uniquely, this form of learning often led to new insights or discovery about mathematics by students upon being guided by a series of exploration and teacher scaffolding (Savery, 2015). This approach was adopted by Pei et al. ( 2018), who designed an activity in Lattice Land and asked students to explore the area of triangles. The teacher scaffolded students' inquiry-based learning by generating different shapes of triangles, so that students can move the vertices of triangles to identify relationships between triangular area and perimeter.

Interplay between CT and mathematics learning outcomes

When implementing CT-based mathematics activities, one assumption educators have been adopting is that the integration of computer science and mathematics education can generate affordances that a sole focus on one may not have. Naturally, a central question researchers have been inquiring is: how do students' CT and mathematics learning interact when they are involved in CT-based mathematics instruction, and what are the consequences of such interactions? While an intuitive and straightforward answer to this question is that the influences are bi-directional and beneficial for student learning, our analysis reveals non-trivial results-that there exist complex interplays and even mismatches when coordinating computational and mathematical modes of thinking. In the following sub-sections, we first review the theoretical constructs and conceptual frameworks used to characterize students' computational concepts and practices as associated with their CT-based mathematical activities. Then, we discuss the ways in which mathematics is used and learned in such contexts.

Computational concepts and practices associated with CT-based mathematical activities

Since the introduction of the notion of CT, researchers have put forward different views on the definition and components of CT (e.g., Brennan & Resnick, 2012; Hoyles & Noss, 2015; Weintrop et al., 2016). Among the 24 reviewed articles, 11 articles specified the CT definitions or frameworks used, as summarized in Table 3. Among these frameworks, Brennan and Resnick ( 2012) proposed one of the most influential framings of CT for K-12 education that consists of three dimensions: computational concepts, computational practices, and computational perspectives. We focus our following discussion on the first two dimensions as they are addressed more often in the literature.

Computational concepts

We use Table 4 to summarize the CT concepts studied by prior researchers. In addition to providing the definitions commonly accepted in computer science and adopted in the reviewed literature, we offer our interpretations of how each computational concept is operationalized and illustrated in CT-based mathematics context. In doing so, we set out to create a theoretical tool for mathematics education researchers to integrate CT in their research and teaching, and more importantly, to continually enrich and refine the definitions in future research.

One prevalent CT concept mentioned in the literature is variable. In working with CT-based mathematics tasks, students often need to define a variable and establish its initial value, and then this variable should take on subsequent values when the program is run. Besides representing a quantity, the use of variables may also be extended to represent an algebraic expression (e.g., result of operating with two or more variables; Ng & Cui, 2021), or geometric location (e.g., locus of a point having a variable location; Sinclair & Patterson, 2018) which could be stored, retrieved and operated on. The observed range of manifestation of variables provides the empirical bases for future research to learn and expand on the diverse ways taken up by the use of variables in CT-based mathematics activities.

In addition, we highlight two other important CT concepts, conditionals and loops. A conditional is a statement needed to be tested in a program, upon which further actions would take place. A student's use of conditionals often requires their knowledge of number properties (e.g., remainder of a division is non-zero; Ng & Cui, 2021), algebraic properties (e.g., when a parametric equation determines the amount of rotation of the wheels; Ng & Cui, 2021; Sinclair & Patterson, 2018) or geometric properties (e.g., when a draggable point determines how much a given shape is scaled by; Sinclair & Patterson, 2018). In particular, Sinclair and Patterson ( 2018) noted the frequent applications of conditionals in conjunction with the signum function, sgn( x) = {− 1, 0, 1} to create conditions in a geometrized programming context, which "worked as a checkpoint of sorts, operating as an if-then statement in a block of computer code" (p. 68). A loop is a code structure that enables a repetition of a certain mathematical procedure. In the case of Miller's ( 2019) study, the students formulated mathematical procedures to draw a square using Scratch. This involved considering what actions (e.g., moving and turning) were repeated, how they were repeated (e.g., constant distance and angle), and how many times to repeat (e.g., 4 times). Furthermore, Sinclair and Patterson ( 2018) conceptualized the concept of loops as being closely related to the notions of iteration and infinity. They considered a triangle construction by joining the midpoints of the line segments which formed the triangle in the previous iteration. This process of constructing line segments between the midpoints (of the previous iteration) and generating new midpoints (of the current iteration) would yield an infinite number of triangles whose area would approach zero. Other researchers have considered the productive relationship between iterations and infinity in CT-based mathematical activities (Arnon et al., 2014; Weller et al., 2003); however, limited mathematical topics have been studied regarding the interplay between the concepts of infinity and loops in K-12 contexts. We suggest that there is high potential for CT-based mathematics tasks to take up these concepts in topics, such as sequences and series with infinite terms as well as fractal geometry.

Computational practices

First, we identify systems thinking to be one of the CT practices at the first level. Systems thinking entails analyzing the problem systematically by investigating a complex system as a whole, understanding the relationships within a system, thinking in levels, communicating information about a system, defining systems and managing complexity (Weintrop et al., 2016). We further categorize systems thinking into three sub-practices: decomposition, abstraction, and algorithmic thinking.

Decomposition refers to breaking down a complex problem into many small problems; it can also be applied in a mathematical sense of "decomposing numbers, shapes, or problems in mathematics" by "considering the part-whole relationships" (Shumway et al., 2021, p. 4). As illustrated in Pei et al.'s ( 2018) study, a student named Alice attempted to solve a problem of finding the area of an irregular polygon by breaking it down into sub-areas of triangles and rectangles. Besides decomposing a complex problem into smaller, manageable sub-problems, abstraction in the sense of abstracting key information and identifying patterns of the problem can also facilitate a solution process. A typical example of this practice is the reasoning process called pattern recognition, that is, looking for similarities between and within problems, e.g., to "see patterns, recognis[e] the unit of repeat, deduce the pattern and abstract the general structure" (Miller, 2019, p. 924). Pei and colleague's ( 2018) study demonstrated pattern recognition between problems; they described a group of students abstracting a general mathematical formula from a set of data, so that they could calculate the area of any lattice polygon regardless of its shape or size.

If decomposition and abstraction serve to simplify a complex problem at the thinking level, then algorithmic thinking serves to enact the simplification at the practice level through designing a concise algorithm to solve the problem. The reviewed studies commonly defined algorithmic thinking as the systematic means of articulating a problem's solution by well-defined rules, with logical, ordered, and discrete steps to be executed by a processing agent (Kaufmann & Stenseth, 2021; Ng & Cui, 2021; Valovičová et al., 2020). An important feature of algorithmic thinking is automation, which offers efficiency in problem solving by introducing automatic steps to be taken (Valovičová et al., 2020). After completing the algorithmic design for each sub-problem, it is necessary to combine these small parts to form a more complex algorithm, which is called modularizing (Ng & Cui, 2021).

While systems thinking has mainly to do with problem solving in general, its relevance to mathematical content is not particularly apparent. Our review identifies another first-level CT practice, data practices, with stronger mathematical connections, since they relate to collecting, creating, manipulating, analyzing, and visualizing data (Weintrop et al., 2016). Under data practices, we further characterized two inter-related practices, simulation and modelling (Dickes et al., 2020; Gilchrist et al., 2021). Often, CT-based mathematical problems require one to construct a model, represent the data from an experiment (e.g., simulating the spread of a disease; Gilchrist et al., 2021; modeling the calculation of a footprint problem; Dickes et al., 2020), and predict the future trend of the simulated problem (e.g., address problems related to disease transmission through the generated computational model; Gilchrist et al., 2021; generated and discussed the modeling graphs, made predictions related to footprint size using ViMAP's grapher; Dickes et al., 2020). Besides, data practices can serve the purpose of learning a mathematical concept during inquiry-based learning. In Pei et al.'s ( 2018) study, students used microworld to explore and derive the Pick's Theorem through gathering data, and recording, sorting, and examining the data set to look for and generalize relationships. This study was an example in which "students used computational tools to facilitate these data-driven computational thinking practices" (p. 85).

The remaining two categories of practices (i.e., Brennan & Resnick's [ 2012] reusing and remixing, testing and debugging) are specific to computational problem solving, in the sense that mathematical problem solving is necessarily situated in the programming environment and tool use. Reusing and remixing involve the coders building on and extending some existing codes and ideas to create a new or more complex program (Ng & Cui, 2021). Testing and debugging are necessary to ensure the solution can solve the tasks successfully (Cui & Ng, 2021). Testing often occurs before debugging, which is to "ensure that things work" (Ng & Cui, 2021, p. 6). If not, then debugging will follow to recognize, locate and fix errors or mistakes when they arise (Bouck & Yadav, 2020; Ng & Cui, 2021).

To summarize, our contribution is toward a systematic organization of the set of computational practices evident in the literature. It is worth noting that these practices can occur simultaneously to facilitate problem solving; meanwhile, they are the very CT learning outcomes co-developed with mathematics learning outcomes during the course of problem solving. An illustrative example was given by Sinclair and Patterson ( 2018), who discussed a variety of other CT concepts and practices evident when students constructed a complex machine in Sketchpad (see an example in Fig. 5). First, they decomposed the machine into components and actions into smaller movements and animations ( decomposition). When creating small animations, they needed to understand the sequence of the steps to link them together into a complete operation ( algorithmic thinking). To create these decomposed steps, students also needed to transform real phenomena (e.g., to create a parabolic trajectory) into mathematical objects (e.g., trigonometric or quadratic curve) ( abstraction and pattern recognition). There were also multiple CT concepts involved in the activity: algebraic or geometric parameters that determine the characteristics of a machine ( variable), the use of functions as conditions to control certain motions ( conditionals), and the concept of subroutines and loop were used to operate programs controlled by certain buttons and to repeat certain actions. The vast examples provided in this section show that CT concepts and practices can fruitfully support mathematical meaning-making and representations of mathematical relationships when situated in the task of constructing a meaningful CT artefact. Thus, mathematical explorations can serve as a site for students to engage in CT in explicit ways, where the mathematical operations change what appear on the interface.

Mathematical learning outcomes demonstrated in computational contexts

Besides the aforementioned CT learning outcomes, the reviewed literature also conveys a range of mathematical ideas covered in CT-based mathematics instruction (see a summary in Table 5). Several studies have also researched the associated mathematical processes, such as problem-solving (Gilchrist et al., 2021; Psycharis & Kallia, 2017), algebraic thinking (Ng & Cui, 2021), modeling (Araya, 2021; Sinclair & Patterson, 2018), mathematical reasoning (Kaufmann & Stenseth, 2021), and generalization and pattern recognition (Miller, 2019; Wilerson-Jerde's, 2014). To further demonstrate how students' mathematical reasoning facilitates and emerges from computational contexts, we focus our discussion on three aspects. First, we discuss how students apply their mathematical knowledge to construct CT artefacts, including the challenges they may experience during the application. Next, we discuss how they draw upon their mathematics to anticipate and interpret CT outputs and how these processes are tied to their debugging practices. Then, we discuss the reciprocal influence in the sense that students' engagement with CT-math integrated tasks are generative to growth in their mathematical knowledge, and we identify ways through which such learning occurs.

Applying mathematical knowledge to construct CT artefacts

Numerous researchers have stressed the important role of mathematical knowledge in solving CT-based mathematical tasks, and in particular, in constructing computer programs to produce certain mathematical outcomes (e.g., Bouck & Yadav, 2020; Dickes et al., 2020; Grizioti & Kynigos, 2021; Miller, 2019). Influenced by Papert's ( 1980) Logo programming, several studies involved students constructing computer programs to generate a certain geometric shape (e.g., right triangles, squares, rectangles, parallelograms, trapezoids, and rhombuses; Bouck & Yadav, 2020; Miller, 2019). In doing so, the students necessarily drew on their knowledge of the constituting elements of the shapes (e.g., number of sides and angle size) and the relationship of those elements (e.g., parallelism or perpendicularity). A similar task design has been extended to the context of atypical shapes. For example, in Miller's ( 2019) study, a group of second graders (aged 7-8 years) were asked to use Scratch to draw a spirolateral (see Fig. 7a). The students were able to identify the structure of the spirolateral which pertained growing and repeating patterns (Fig. 7b, c), and then, they translated their observed structure into Scratch codes (Fig. 7 d).

a Spirolateral shown to students, b, c students' conceived structure of the spirolateral, and d sample Scratch codes (Miller, 2019, pp. 923-924)

In algebraic contexts, a study was conducted by Dickes et al. ( 2020), in which a group of third graders used the command "repeat" to model the relationship between total distance and step size (Total Distance = No. of Repeats × Step size); this reflected that the students had constructed a multiplicative relationship between the quantities involved. As another example, Ng and Cui ( 2021) designed the Doubling Machine problem in which a bank account balance increased by a doubling amount of deposit in each subsequent week (i.e., $3, $9, $21, $45...). One pair of students observed that the next number always equals to the previous number times two plus three (i.e., \({t}_{n}= 2{t}_{n}+ 3\)) and translated this relationship into codes "[set button_value to][button_value * 2]" followed by "[set button_value to][button_value + 3]." In each of these examples, mathematics was used in an applied manner, and the students translated their conceived mathematical relationships and structures to computer languages and programs.

Despite the supporting role of mathematics in the construction of CT artefacts, there have been studies focused on the challenges students experience during their enactment of mathematics in computational contexts. As argued by Ng and colleagues (2021), these challenges were due to the mismatches or contradictions between CT and mathematical thinking (Cui & Ng, 2021; Ng et al., 2021; Sinclair & Patterson, 2018). For example, displaying a result requires an additional code during programming (e.g., calculating 5 + 3 = 8 and then displaying 8), whereas in mathematics, carrying out an operation automatically leads to a result (e.g., calculating 5 + 3 = 8 is a single operation; Cui & Ng, 2021; Ng et al., 2021). In addition, Jackiw and Finzer ( 1993) explained how debugging a sketch in dynamic geometry environments might differ from debugging in traditional programming environments. That is, the former requires "reasoning from the manifestation of a bug backwards...towards its cause" (ibid, p. 301), while the latter requires identifying the manifestation of a bug, locating the source of a bug, and re-executing the program circuitously.

The notions of variables and the way they are operated on also differ in mathematics and computing contexts. For example, in mathematics, the use of a variable in an equation already implies that the variable can take on any values, but in programming environments, one needs to write a line of codes to tell the computer to increase the variable by one (Cui & Ng, 2021). Another issue relates to the use of equal signs. In mathematics, one way of interpreting an equation is that two entities on both sides are equal (e.g., \(y=x*2\)), but in programming, it means "performing some repetitive actions on a single quantity" (e.g., Set [variable] to [variable] *2 or [variable] = [variable]*2) (Cui & Ng, 2021, p. 19). As a result, students often encountered difficulties translating their conceived mathematical relationship from a problem situation into its programming equivalent.

To conclude, we highlight the take-away message by these empirical findings: despite the fact that mathematical knowledge plays a supportive role in constructing CT artefacts, the differences between CT and mathematical thinking may constrain students from understanding or practicing specific CT or mathematical concepts. As suggested by many scholars, these discrepancies might lead to problems in programming even if the students understand the problem situation and the mathematical relationships behind. Therefore, it is crucial for mathematics education researchers and mathematics educators to address these discrepancies to improve the integration of CT into mathematics learning.

Applying mathematical knowledge to anticipate and interpret CT artefacts

During programming, students may apply their mathematical knowledge to anticipate and interpret CT artefacts beyond making a workable program. We further classify four forms of such application based on the literature: (1) making sense of outputs, (2) anticipating outputs, (3) mediating debugging, and (4) code modeling.

Making sense of outputs

One form of such application is applying one's mathematical knowledge to make sense of the outputs. In Wilerson-Jerde's ( 2014) study, a group of middle graders used a fractal construction tool (Fig. 8) to construct a collection of fractals. In doing so, the students interpreted why the produced fractals looked the way they were based on the creation rules (e.g., square-size, rotation-angle, reflection-axis). In a prior lesson, the students had already learned about fractals, including different types of fractals and how they could be produced. Therefore, the aim of the current activity was in supporting students' reflection upon features of fractals and in making sense of the rules that produce them in a CT context.

Another study by Pei et al. ( 2018) reported on a student making connections between her exploration within the Lattice Land environment and the triangle area formula she had previously learned. By interacting with Lattice Land, the student could see how moving a vertex along the line parallel to the base would change the shape of the triangle yet keep the area fixed. At first, the student was surprised by this observation, but then she recalled the area formula for a triangle (\(Area=\frac{1}{2}\times Base \times Height\)), which explained the conservation of area in the geometrized CT environment. Similar to the previous theme, the reasoning processes entailed by these two examples involve students' enactment of existing mathematical knowledge; what distinguish them from the prior theme is that the students were building novel connections between their mathematical knowledge and the CT outputs. Therefore, mathematics is not merely used in an applied manner, and the students deepen and reorganize their mathematical knowledge as they recontextualize it in a CT environment.

Anticipating outputs

Studies observing this theme relate to the way students draw on their mathematical knowledge to anticipate the outputs of a computer program. The students are aware of the mathematics needed to evaluate the program and they compare its outputs to the intended goal. In Shumway et al.'s ( 2021) study, a group of kindergarten students (aged 5-6 years) engaged in an activity with the Cubetto robot coding toy, and they were asked to create programs to instruct the robot to move between two destinations. The robot's movement provided students instant feedback about the accuracy of the program in terms of the desired movement. When the robot moved some steps short of or over the goal, the students would revise the program by adding or removing some codes. This evidences how the students' knowledge of space, measurement, and number operations facilitated their anticipation of the robot movement and the associated debugging actions. As Shumway et al. ( 2021) described, students can "[think] one step ahead of the robot in anticipation of evaluating the accuracy of their sequence of codes in the program" (p. 20).

Dickes et al. ( 2020) offered another example of students realizing their program was incorrect due to their outputs being inconsistent with their mathematical calculations. Specifically, the students noticed that the total distance represented in their graph was 240 units, while the total distance calculated using the rate equation was 300 units (15 step size \(\times\) 20 number of steps = 300 units of total distance). They subsequently revisited the program and corrected the programming error to obtain a desired result. Similar to the previous example, the students' mathematical thinking guided their evaluation of the computer program and motivated their debugging.

Mediating debugging practices

We also identify examples, where mathematical reasoning not only motivated students' debugging practices but also guided and mediated the debugging practices themselves. In a study by Kaufmann and Stenseth ( 2021), a group of students in Grade 8 and Grade 9 (age 13 and 14) were asked to troubleshoot a program, so that a wheel would rotate along a line, such that its diameter hits the target markers on the line. The students initially used intuition and trial-and-error to change their codes without any mathematical basis; eventually, they attended to the circumference and its relationship to other quantities to debug the program successfully. In this example, mathematics was not applied to construct the program per se, but it was used for locating and fixing the bugs. Kaufmann and Stenseth ( 2021) further proposed a model to characterize how mathematics mediates the complex process of troubleshooting; as they shown in Fig. 9, the use of mathematics is intertwined within the iterative process of testing, observing and analyzing, making hypothesis, and changing program.

Code modeling

Code modeling refers to students interpreting the mathematics underlying a sequence of codes, sometimes during the process of debugging. Ng and Cui ( 2021) described a pair of students who debugged a program for generating a prime number detector. They modeled the outcomes of two set of codes, with one having the conditions ('remainder = 0' and 'remainder > 0') within a repeat loop and one outside the loop, and they realized that they should have placed the conditions within the loop to check the remainder before (as opposed to after) carrying out the next division. Different from the previous examples, this instance is illustrative of students using their mathematical knowledge to model the behavior of a given set of codes. Code modeling allowed them to realize that the codes behaved in a way against the intended mathematical operations, and hence to revise the program accordingly.

Summary

Examples in these four themes commonly involve students applying their mathematical knowledge to anticipate or interpret CT artefacts, but there are nuances between them. First, using mathematics to make sense of CT outputs can engender novel mathematical connections that were not present previously. In terms of the other three themes which relate directly to debugging practices, students can (1) recognize a buggy program by comparing the outputs with the anticipated outcomes; (2) fix the bugs by reconsidering the mathematical relationships relevant to the situation; or (3) fix the bugs by modeling the mathematical behavior underlying the program. We consider it significant to tease out and illustrate these nuances across the literature, since they shed light on the foundational role of mathematics for CT as well as the insights into how mathematics is used in explicit but different ways in CT contexts.

Generating new mathematical knowledge in parallel with CT development

First, students can construct mathematical ideas and relationships by reflecting on CT outputs. For example, in Wilerson-Jerde's ( 2014) study, the students were asked to use the Categorizer tool to create categories that reflected certain features about fractals (e.g., density, "crispness" or "fuzziness", twisting structure, branching structure, sponging structure, and area reduction). By observing a collection of fractals produced by the computer, the students were offered opportunities to "explore important mathematical properties of those structures, and offered more ways to construct fractals with particular mathematical properties" (p. 118). In another study by Pei et al. ( 2018), students collected data from a number of polygons in Lattice Land and derived the Pick's Theorem1 by reasoning with the data and generalizing pattern from the data. In each of these examples, students made use of the affordances of computational tools to create numerous examples and outcomes that they could reflect upon and generalize from. The rich mathematical conversations around those CT outputs facilitated the students' constructions of deep mathematical ideas and relationships.

Another avenue through which new mathematical ideas are constructed is reflecting on the programming processes and code features. In Miller's ( 2019) study, a group of students created codes for making a square and deduced a generalized rule for calculating the perimeter of a square by identifying the repeating patterns from the codes (i.e., perimeter of a square equals four times its length). The author concluded that these primary school students "were forming generalizations when writing computer codes" (ibid, p. 925), suggesting that teaching mathematics through coding may provide opportunities for students to "see the mathematics in a deeper way and build a more connected understanding" (ibid, p. 923). As another example, Pei et al. ( 2018) described a classroom discussion centered on calculating the areas of triangles whose side lengths were not whole numbers in the Lattice Land environment (Fig. 10). Using the "draw segment" tool, the students came up with different ways to "box" the triangles and calculate their areas. These novel methods were only possible when students used the construction function of the programming tool and reflected on the construction processes and products.

Third, CT-based mathematics instruction motivates a new form of mathematical knowledge that is represented by computational languages and unique to CT contexts. For example, Ng and Cui ( 2021) discussed a group of elementary students who had no prior knowledge of algebraic expressions and variables, but they could use iterations like '[set total to] [total + money]' and '[set money to] [money *2]' to model a geometric series. It was noteworthy that these students could make use of commands and codes to model mathematical relationships whose algebraic forms were not yet accessible to them in paper-and-pencil context. Therefore, the authors concluded that mathematical content can be "purposefully synergized in a programming context" (ibid, p. 859) to make advanced and historically challenging ideas accessible to younger children and "can serve as a bridge for children to advance their arithmetic thinking toward building coherent and meaningful learning of more advanced algebra in later years" (ibid, p. 858). Shumway et al. ( 2021) made a similar claim by showing evidence of kindergarten students expressing their emerging knowledge of spatial orientation, spatial movement, measurement, and numbers by imitating a robot toy's motion, suggesting their mathematical knowledge is embedded in their programming activities. The author also argued that those activities afforded students coordinating interconnected domains of mathematical knowledge (i.e., simultaneously using their spatial reasoning and knowledge of numbers) and constructing dynamic perspective of relevant concepts, which went beyond the kindergarten standards.

In summary, this group of literature provides rich evidence that confirms the affordances of CT-math integration in K-12 education. Such integration supports students' abstraction of a variety of mathematical ideas (e.g., geometric properties, formulas, theorems). Through interacting with the CT environment, students are offered opportunities to explore a variety of mathematical contents through reflecting upon the CT construction processes and products. Furthermore, the CT-math integration opens up new possibilities of learning mathematics, ultimately transforming traditional views of the nature of mathematics and mathematics learning. As the literature showed, the integration increases accessibility of mathematical ideas to young kids and allows more mathematical connections made. It also contributes to the emergence of a new form of mathematical knowledge and reasoning mode that bear characteristics of computing (e.g., syntax representations of algebraic relationships), which leave seeds for students' future mathematical abstraction and development.

Discussion and future research directions

CT is the thought process entailing abstracting, algorithmic thinking, and computational modeling in combination with the use of programming tools to solve a series of complex problems (Wing, 2006). When engaging CT-based mathematical activities, students not only produce CT artefacts and programmable solutions in a physical sense, but also construct meanings for various CT and mathematical concepts. Despite positive findings about the impact of CT-based mathematical activities on student learning, consensus has not been reached regarding how to integrate the two fields in coherent and meaningful ways. This systematic review has clarified, with concrete examples, the topics, education levels, pedagogical arrangements including learning resources, and the degree of integration among other STEM disciplines in which recent studies have investigated in relation to CT-based mathematics learning. Together, these studies have shown that no matter which descriptor of learning context is concerned, CT should not be introduced as an add-on to mathematics learning, but considerations for research and practice is warranted at a conceptual level to fully understand and maximize the ways CT may support mathematics learning.

Our study sets out to review the impact of CT-based mathematics instruction on student learning through four aspects: educational context, pedagogical approach, theoretical construct related to CT and mathematics learning outcomes, and the interplay between CT and mathematics in CT-based mathematical activities. From each of these aspects, our review suggests several affordances of CT which can foster and enrich mathematics learning: first, we observe that programming activities afford non-prescribed mathematics learning in unstructured ways and with little prior knowledge necessary. Second, problem-based learning and project-based learning are particularly compatible with CT-based mathematics instruction. Third, CT integration in mathematics education has proven to be supportive in the content of algebra and geometry across education levels. Finally, CT can serve as both the means (when applying mathematical knowledge to anticipate or interpret CT artefacts) and goals (when applying mathematical knowledge to construct CT artefacts) of learning in CT-based mathematics activities. That is, CT-based mathematical activities afforded students to develop and apply their CT alongside their mathematical knowledge. In particular, we evidenced students' CT development by their ability to use diverse commands, flexibility in sequencing codes, and efficiency in debugging (Dickes et al., 2020; Kaufmann & Stenseth, 2021; Ng & Cui, 2021); simultaneously as the students' CT concepts and practices emerged, the CT-based mathematics activities also became progressively more mathematically grounded and rigorous.

Altogether, these characteristics have important implications to how we organize mathematics learning activities in computationally enhanced ways, and if taken up, can be highly transformative in the way we structure the mathematics curricula and instruction. Wilkerson-Jerde ( 2014) posits that, "computational ideas provide new and powerful ways of thinking about math and science phenomena" (p. 101); specifically, we highlight the re-orientation of sequencing of mathematical topics as well as the pedagogical aspects of mathematics instructions that are made possible in CT environments (Ng et al., forthcoming). In addition, we draw attention to the mutual support and affordance of CT and mathematical activities, and in turn, highlight the potential of designing CT instruction around mathematical inquiry for co-developing CT and mathematical understandings (Pei et al., 2018).

One way forward [...] is through the creation of blended curricula that infuse computational thinking practices into topics from other domains. In doing so, the destination domain can serve as a context for developing and using computational thinking practices while also presenting learners with new, computationally facilitated approaches for exploring disciplinary ideas. (p. 76)

Besides, we conclude that research on the following areas is currently lacking:

1. 1.
CT-based mathematics instructions taking place in formal and interdisciplinary education settings. This also implies that teachers' professional development concerning the emergent competency of CT-based mathematics thinking is indispensable.

2. 2.
design experiments within geometrized programming environments, and more generally, the role of different computing tools in supporting mathematics learning;

3. 3.
empirical studies highlighting the different ways in which CT and mathematics learning co-develop, including the potential of integrating CT in other domains of mathematics (e.g., calculus and statistics), and what can be learned from CT-based mathematics instructions in these contexts, both in terms of how CT support mathematics learning, and vice versa.

In conclusion, this study lays the groundwork for future research into more coherent and effective integration of CT in K-12 mathematics education. We encourage more research on the integration of child-friendly programming instructional designs and tools for adoption in primary education or lower to begin integration of CT concepts early in mathematics teaching and learning. Regarding undergraduate mathematics education, we suggest that, due to the nature of the mathematical content, the integration of CT in this context seems to be more natural and common than K-12 education. Different than K-12 mathematics teaching that often rely on paper-and-pencil, it is not uncommon to find an undergraduate mathematics class that includes CT components. However, despite the predominate CT-math integration at the collegiate level, there has been few research that provides fine-grained analysis of the interplay between CT and mathematics learning, although recent work by Lockwood, Sand, and colleagues have paid explicit attention to this research line (Lockwood, 2022; Lockwood & De Chenne, 2020, 2021; Sand et al., 2022). In addition, more interestingly, these scholars' findings echo the types of interplay we have identified in the current review, and future research can continue to gain insights into how the interactions between CT and mathematical thinking may manifest differently or similarly in both contexts (Lockwood & Mørken, 2021; Lockwood, et al., 2019).

Notably, although most studies provide empirical evidence in the mutual relationship between CT and mathematics learning outcomes, authors of two articles conveyed that students' CT engagement did not support their mathematical learning (Chan et al., 2021; Psycharis & Kallia, 2017). Given the contradictory claims of these two studies with other studies reviewed, there remains future work in understanding how to maximize the effective role of CT in students' mathematical learning and vice versa. Specifically, to offer an explanation for these discrepant results, researchers need to not only focus on students' learning outcomes and their correlations with CT engagement, but also focus on the qualitative nuances in the students' learning processes and identify and compare the essential features of the tasks and instructional approaches that generate the discrepant learning outcomes.

Availability of data and materials

This is a systematic review of prior literature. All articles we reviewed here are accessible in Web of Science database.

Notes

The Pick's Theorem states that suppose that a polygon has integer coordinates for all of its vertices. Let I be the number of integer points that are interior to the polygon, and let B be the number of integer points on its boundary (including vertices as well as points along the sides of the polygon). Then, the area A of the polygon is: \(A=0.5B+I-1\).

Abbreviations

CT:

Computational thinking

PRISMA:

Preferred Reporting Items for Systematic Reviews and Meta-Analyses

STEM:

Science, Technology, Engineering and Mathematics

STEAM:

Science, Technology, Engineering, Arts and Mathematics

WOS:

Web of Science

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Acknowledgements

This paper was fully supported by the Hong Kong Research Grants Committee, General Research Fund (Ref No. 14603720).

Funding

This study is supported by the Hong Kong Research Grants Committee, General Research Fund (Ref No. 14603720).

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Authors and Affiliations

1. Department of Curriculum and Instruction, Faculty of Education, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

Huiyan Ye, Oi-Lam Ng & Ching Sing Chai

2. Faculty of Education, The University of Hong Kong, Pokfulam, Hong Kong

Biyao Liang

Contributions

All authors contributed to the design and implementation of this study. Literature searches with agreed-upon search terms were completed by the first author. Then, the first, second and third author collaboratively worked on the process of literature selection, coding, and themes identification. The first author and second author conducted the analysis and interpretation of the reviewed literature with help of the third and fourth author. Findings were discussed and agreed by all authors. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Oi-Lam Ng.

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Ye, H., Liang, B., Ng, OL. et al. Integration of computational thinking in K-12 mathematics education: a systematic review on CT-based mathematics instruction and student learning. IJ STEM Ed 10, 3 (2023). https://doi.org/10.1186/s40594-023-00396-w
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Integration of computational thinking in K-12 mathematics education: a systematic review on CT-based mathematics instruction and student learning - International Journal of STEM Education

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There has been substantial research undertaken on the integration of computational thinking (CT) in K-12 mathematics education in recent years, particularly since 2018 when relevant systematic reviews were conducted on the topic. Many empirical studies in this area have yet to elaborate clearly and explicitly on how CT may support mathematics learning, or otherwise, in CT-based mathematics activities.

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Career challenges like reorganizations, layoffs, and a lack of personal fulfillment can feel like big bumps in the road, but instead of seeing them as obstacles, start to look for the opportunities. Personal career development is critical for resilience in the face of uncertainty and change. The authors, who train over 100,000 people a year in career development, have identified four common challenges that get in the way of people's growth. They categorize them as when, who, what, and where challenges. Here's how you can think and act creatively to overcome these challenges and continually invest in your career development.

Change and uncertainty are now a given in our careers. As a result, people are re-evaluating what they want from their work. For many, the traditional career ladder has been replaced by "squiggly" careers - non-linear career moves where progression goes beyond promotion and allows people to develop in different directions. In the context of constant change, personal career development is now a must-have rather than a nice-to-have.

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What's Holding Back Your Career Development?

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Career challenges like reorganizations, layoffs, and a lack of personal fulfillment can feel like big bumps in the road, but instead of seeing them as obstacles, start to look for the opportunities. Personal career development is critical for resilience in the face of uncertainty and change.

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Introduction to Khanmigo | Khan for Educators (US) | Khan Academy

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Introduction to Khanmigo | Khan for Educators (US) | Khan Academy

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Introduction to Khanmigo | Khan for Educators (US) | Khan Academy

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AI for education | Life skills | Khan Academy

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AI for education | Life skills | Khan Academy

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Let Khan Academy and our partners-including Code.org, Common Sense Education, aiEDU, and Professors Ethan and Lilach Mollick-be your trusted partners as we navigate this exciting learning journey together! What is generative AI and how does it work? What are some best practices for teaching and learning with generative AI? This course-in-progress answers those questions and many more. Welcome!

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AI for education | Life skills | Khan Academy

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Khan Academy believes that AI and Large Language Models hold the potential to transform education. By providing teachers and students resources and tools to design joyful, personalized, interactive, engaging learning experiences that drive deep learning in the months and years ahead, we hope to become a trusted partner in this learning journey.

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Khan Academy’s AI Tool for the Classroom: Teacher + Student Edition

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Students aren't consumers of facts. They are active creators of knowledge. Schools aren't just brick-and-mortar structures -- they're centers of lifelong learning. And, most important, teaching is recognized as one of the most challenging and respected career choices, absolutely vital to the social, cultural, and economic health of our nation.

Today, the seeds of such a dramatic transformation in education are being planted. Prompted by massive revolutions in knowledge, information technology, and public demand for better learning, schools nationwide are slowly but surely restructuring themselves.

Leading the way are thousands of teachers who are rethinking every part of their jobs -- their relationship with students, colleagues, and the community; the tools and techniques they employ; their rights and responsibilities; the form and content of curriculum; what standards to set and how to assess whether they are being met; their preparation as teachers and their ongoing professional development; and the very structure of the schools in which they work. In short, teachers are reinventing themselves and their occupation to better serve schools and students.

New Relationships and Practices

Traditionally, teaching was a combination of information-dispensing, custodial child care and sorting out academically inclined students from others. The underlying model for schools was an education factory in which adults, paid hourly or daily wages, kept like-aged youngsters sitting still for standardized lessons and tests.

Teachers were told what, when, and how to teach. They were required to educate every student in exactly the same way and were not held responsible when many failed to learn. They were expected to teach using the same methods as past generations, and any deviation from traditional practices was discouraged by supervisors or prohibited by myriad education laws and regulations. Thus, many teachers simply stood in front of the class and delivered the same lessons year after year, growing gray and weary of not being allowed to change what they were doing.

Many teachers today, however, are encouraged to adapt and adopt new practices that acknowledge both the art and science of learning. They understand that the essence of education is a close relationship between a knowledgeable, caring adult and a secure, motivated child. They grasp that their most important role is to get to know each student as an individual in order to comprehend his or her unique needs, learning style, social and cultural background, interests, and abilities.

This attention to personal qualities is all the more important as America continues to become the most pluralistic nation on Earth. Teachers have to be committed to relating to youngsters of many cultures, including those young people who, with traditional teaching, might have dropped out -- or have been forced out -- of the education system.

Their job is to counsel students as they grow and mature -- helping them integrate their social, emotional, and intellectual growth -- so the union of these sometimes separate dimensions yields the abilities to seek, understand, and use knowledge; to make better decisions in their personal lives; and to value contributing to society.

They must be prepared and permitted to intervene at any time and in any way to make sure learning occurs. Rather than see themselves solely as masters of subject matter such as history, math, or science, teachers increasingly understand that they must also inspire a love of learning.

In practice, this new relationship between teachers and students takes the form of a different concept of instruction. Tuning in to how students really learn prompts many teachers to reject teaching that is primarily lecture based in favor of instruction that challenges students to take an active role in learning.

They no longer see their primary role as being the king or queen of the classroom, a benevolent dictator deciding what's best for the powerless underlings in their care. They've found they accomplish more if they adopt the role of educational guides, facilitators, and co-learners.

The most respected teachers have discovered how to make students passionate participants in the instructional process by providing project-based, participatory, educational adventures. They know that in order to get students to truly take responsibility for their own education, the curriculum must relate to their lives, learning activities must engage their natural curiosity, and assessments must measure real accomplishments and be an integral part of learning.

Students work harder when teachers give them a role in determining the form and content of their schooling -- helping them create their own learning plans and deciding the ways in which they will demonstrate that they have, in fact, learned what they agreed to learn.

The day-to-day job of a teacher, rather than broadcasting content, is becoming one of designing and guiding students through engaging learning opportunities. An educator's most important responsibility is to search out and construct meaningful educational experiences that allow students to solve real-world problems and show they have learned the big ideas, powerful skills, and habits of mind and heart that meet agreed-on educational standards. The result is that the abstract, inert knowledge that students used to memorize from dusty textbooks comes alive as they participate in the creation and extension of new knowledge.

New Tools and Environments

One of the most powerful forces changing teachers' and students' roles in education is new technology. The old model of instruction was predicated on information scarcity. Teachers and their books were information oracles, spreading knowledge to a population with few other ways to get it.

But today's world is awash in information from a multitude of print and electronic sources. The fundamental job of teaching is no longer to distribute facts but to help children learn how to use them by developing their abilities to think critically, solve problems, make informed judgments, and create knowledge that benefits both the students and society. Freed from the responsibility of being primary information providers, teachers have more time to spend working one-on-one or with small groups of students.

Recasting the relationship between students and teachers demands that the structure of school changes as well. Though it is still the norm in many places to isolate teachers in cinderblock rooms with age-graded pupils who rotate through classes every hour throughout a semester -- or every year, in the case of elementary school -- this paradigm is being abandoned in more and more schools that want to give teachers the time, space, and support to do their jobs.

Extended instructional periods and school days, as well as reorganized yearly schedules, are all being tried as ways to avoid chopping learning into often arbitrary chunks based on limited time. Also, rather than inflexibly group students in grades by age, many schools feature mixed-aged classes in which students spend two or more years with the same teachers.

In addition, ability groups, from which those judged less talented can rarely break free, are being challenged by a recognition that current standardized tests do not measure many abilities or take into account the different ways people learn best.

One of the most important innovations in instructional organization is team teaching, in which two or more educators share responsibility for a group of students. This means that an individual teacher no longer has to be all things to all students. This approach allows teachers to apply their strengths, interests, skills, and abilities to the greatest effect, knowing that children won't suffer from their weaknesses, because there's someone with a different set of abilities to back them up.

To truly professionalize teaching, in fact, we need to further differentiate the roles a teacher might fill. Just as a good law firm has a mix of associates, junior partners, and senior partners, schools should have a greater mix of teachers who have appropriate levels of responsibility based on their abilities and experience levels. Also, just as much of a lawyer's work occurs outside the courtroom, so, too, should we recognize that much of a teacher's work is done outside the classroom.

New Professional Responsibilities

Aside from rethinking their primary responsibility as directors of student learning, teachers are also taking on other roles in schools and in their profession. They are working with colleagues, family members, politicians, academics, community members, employers, and others to set clear and obtainable standards for the knowledge, skills, and values we should expect America's children to acquire. They are participating in day-to-day decision making in schools, working side-by-side to set priorities, and dealing with organizational problems that affect their students' learning.

Many teachers also spend time researching various questions of educational effectiveness that expand the understanding of the dynamics of learning. And more teachers are spending time mentoring new members of their profession, making sure that education school graduates are truly ready for the complex challenges of today's classrooms.

Reinventing the role of teachers inside and outside the classroom can result in significantly better schools and better-educated students. But though the roots of such improvement are taking hold in today's schools, they need continued nurturing to grow and truly transform America's learning landscape. The rest of us -- politicians and parents, superintendents and school board members, employers and education school faculty -- must also be willing to rethink our roles in education to give teachers the support, freedom, and trust they need to do the essential job of educating our children.

Judith Taack Lanier is a distinguished professor of education at Michigan State University.

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Redefining the Role of the Teacher: It's a Multifaceted Profession

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A closer look at what being an educator really means.

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Happy 30 years, AmeriCorps!

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AmeriCorps 30th Anniversary: A Movement, Not a Moment

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How can we relate this discussion to our work as a fellow?

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Earvin "Magic" Johnson: Understand Your Customers and Over-Deliver

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The password is: consider

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Teach Like a Champion 3.0: 63 Techniques that Put Students on the Path to College

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Teachers can take steps to create a positive classroom culture before school even begins that will pay off throughout the year.

Your classroom can be a place where students (and you!) enjoy being and learning. Building community while creating a safe learning environment helps keep students engaged in learning with each other. Here are a few ideas on fostering positive relationships that have worked well for me.

Before Students Arrive

Learn as much as you can about your students ahead of time. Have a system that allows you quick access to educational plans along with a place to log communication, and write notes about what helps particular students find success.

Open the door of communication. Send a note to students and parents welcoming them to your class, giving any necessary information, and sharing your excitement for the year. Keep it short and straightforward. This also means that your first communication home is positive!

Label your classroom. Write your name, classroom number, and subject next to the door, and post it in the front of the room. I found a Bitmoji template online and used it to create a poster with pictures about my interests, which sparked conversations with students.

Create a seating system. I have found that having a seating chart on the first day of school eases students' anxiety about where to sit and allows me to start to learn names right away. Having numbers on the desks creates a way to easily group students, assign roles, and allow for students to move around the room, to get supplies and/or work in small groups.

First Day of School

Greet your students at the door. Smiling is highly encouraged.

Introduce yourself. Share pictures and go over any essential information about your class and expectations.

Play a name game. Students must learn each other's names to build community and have everyone feel seen in the classroom. Whatever game you pick, remember to have clear instructions visible and to model the game while students are all still seated.

One idea is to put students in small groups (again, the numbered seating system can be helpful) and have each group stand in a circle, tossing a soft object around. They need to list everyone who has had the object and call out the name of the next person before tossing it to them. The game continues until everyone has been named. If groups finish quickly, you can add another object to toss or have students add their favorite animal to their name. Eventually, you can start combining groups to learn more names.

First Week of School

Balance community-building activities with subject matter. This keeps the focus on creating an engaged, positive learning environment.

Have a brief conversation with each student. This creates a positive connection and helps set the tone for the year. Use their survey from the first day as a conversation starter. Check in on how students are doing and acknowledge any education plans in place or ways you can support them.

Remember to take your cues from students and never force communication. Let them know you care about their success, are here for them, and are happy they're in your class. This is paramount for having a strong base when/if behaviors come up throughout the year. (For more information about this strategy, check out " The Power of a 45-Second Investment in Relationship Building.")

Establish norms and class expectations. As a classroom community, think through what you want the learning environment to look and feel like. There are many different ways to do this, and whatever you come up with, be sure to clearly post, reinforce, and revisit these norms and expectations.

Be proactive. Teach and review procedures, guidelines, and expectations for every activity. It may feel redundant, but giving simple, straightforward instructions (verbal and written) can alleviate disruptive behaviors. If everyone knows what's going on, what to expect, and what to do, students feel safer, are more engaged, and become positive contributing members to the classroom community.

Play 4 Corners. Use labeled corners (1-4) in your classroom to give students four options to choose from, discuss, and share out. To organize this, I use a Google slideshow with a diagram of the corners and choices with a timer embedded in the slide.

For example, I ask students to choose a season they enjoy (1: Winter; 2: Spring; 3: Summer; 4: Autumn), go to that corner, find an elbow partner (or group of 3), and ask each person to give their name and share their reason. After one minute of discussion, I pick a student from each corner to share what they said or heard (or they can pick a volunteer from their group to speak).

Play "How Well Do You Know Your Teacher?." This could be in the form of a Kahoot, a Jeopardy-type game, a worksheet, or any game you can think of. It's fun for students to get to know who you are as a person, and the more interesting, school-appropriate facts they know, the easier it is to connect with a wider range of students over shared interests.

One suggestion for this and other activities is to consistently use random groupings. This helps students learn each other's names more quickly and get comfortable working with others.

Use a people match activity. Using a people match activity gets students out of their seats, talking to each other and making connections over similarities. Encourage students to have conversations with each other while doing this activity. Students can keep the paper to remember what they had in common.

After the excitement of the first weeks of school winds down, remember that it's never too late for students to learn names and make connections with each other. Building in brief icebreaker activities will keep students engaged in learning and growing. The purpose of all these activities is to humanize each member of the classroom community (including you), which makes the overall learning experience safer, kinder, and more positively engaging.

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Building Community Before the First Day of Middle School and Beyond

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Teachers can take steps to create a positive classroom culture before school even begins that will pay off throughout the year.

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Three Steps to Mastering Multiplication Facts (1)

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The American Dream and Public Schools Introduction (Hoschild, 2003) (1)

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Student Social Class and Teacher Expectations

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Reading for their Life- (Re) Building the Textual Lineages of African American Males (Michael W. Smith)

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Ladson-Billings (2006) From the achievement gap to the education debt

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Equality and justice for all Examining Race in Education Scholarship (Brayboy et. al, 2007)

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Engaging African American Males in Reading (Alfred W. Tatum)

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Enabling Texts- Texts that Matter (Alfred W. Tatum)

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Copy of Omi & Winant (1994) Racial formation in the United States [Chapter 4]

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Copyright © 2006 by Arthur Benjamin and Michael Shermer

All rights reserved.
Published in the United States by Three Rivers Press, an imprint of the Crown
Publishing Group, a division of Random House, Inc., New York.
www.crownpublishing.com

Originally published in different form as Mathemagics by Lowell House,
Los Angeles, in 1993.

Three Rivers Press and the Tugboat design are registered trademarks of Random House, Inc.

Library of Congress Cataloging-in-Publication Data

Benjamin, Arthur.
Secrets of mental math : the mathemagician's guide to lightning calculation and amazing math tricks / Arthur Benjamin and Michael Shermer.- 1st ed.
p. cm.
Includes bibliographical references.
1. Mental arithmetic-Study and teaching. 2. Magic tricks in mathematics education. 3. Mental calculators. I. Shermer, Michael. II. Title.
QA111.B44 2006
510-dc22

2005037289

eISBN: 978-0-307-34746-6

v3.1_r1

I dedicate this book to my wife, Deena, and daughters, Laurel and Ariel.

My dedication is to my wife, Kim, for being my most trusted confidante and personal counselor.

-Michael Shermer
Acknowledgments
The authors wish to thank Steve Ross and Katie McHugh at Random House for their support of this book. Special thanks to Natalya St. Clair for typesetting the initial draft, which was partly supported by a grant from the Mellon Foundation.

Arthur Benjamin especially wants to acknowledge those who inspired him to become both a mathematician and a magician-cognitive psychologist William G. Chase, magicians Paul Gertner and James Randi, and mathematicians Alan J. Goldman and Edward R. Scheinerman. Finally, thanks to all of my colleagues and students at Harvey Mudd College, and to my wife, Deena, and daughters, Laurel and Ariel, for constant inspiration.
Contents
Cover

Title Page

Copyright

Dedication

Acknowledgments

Foreword by Bill Nye (the Science Guy®)

Foreword by James Randi

Prologue by Michael Shermer

Introduction by Arthur Benjamin

Answers

Bibliography

About the Author
Foreword by Bill Nye (the Science Guy®)
I like to think about the first humans, the people who came up with the idea to count things. They must have noticed right away that figuring on your fingertips works great. Perhaps Og (a typical ancient cave guy) or one of his pals or associates said, "There are one, two, three, four, five of us here, so we need five pieces of fruit." Later, "Hey, look," someone must have said (or grunted), "you can count the number of people at the campfire, the number of birds on a tree, stones in a row, logs for a fire, or grapes in a bunch, just with your fingers." It was a great start. It's probably also how you came to first know numbers.

You've probably heard that math is the language of science, or the language of Nature is mathematics. Well, it's true. The more we understand the universe, the more we discover its mathematical connections. Flowers have spirals that line up with a special sequence of numbers (called Fibonacci numbers) that you can understand and generate yourself. Seashells form in perfect mathematical curves (logarithmic spirals) that come from a chemical balance. Star clusters tug on one another in a mathematical dance that we can observe and understand from millions and even billions of kilometers away.

We have spent centuries discovering the mathematical nature of Nature. With each discovery, someone had to go through the math and make sure the numbers were right. Well, Secrets of Mental Math can help you handle all kinds of numbers. You'll get comfortable with calculations in a way that will let you know some of Nature's numerical secrets, and who knows where that might take you?

As you get to know numbers, the answer really is at your fingertips. That's not a joke, because that's where it all begins. Almost everyone has ten fingers, so our system of mathematics started with 1 and went to 10. In fact, we call both our numbers and our fingers "digits." Coincidence? Hardly. Pretty soon, though, our ancestors ran out of fingers. The same thing has probably happened to you. But we can't just ignore those big numbers and (this is a joke) throw up our hands.

We need numbers-they're part of our lives every day, and in ways we typically don't even notice. Think about a conversation you had with a friend. To call, you needed a phone number, and the time you spent on the phone was measured in numbers of hours and minutes. Every date in history, including an important one like your birthday, is reckoned with numbers. We even use numbers to represent ideas that have nothing to do with counting. What's your 20? (I.e., Where are you? From the old police "10" codes, like 10-4 for "yes.") What's the 411 on that gal? (I.e., What's her background; is she dating anyone? From the number for telephone information.) People describe one another in numbers representing height and weight. And, of course, we all like to know how much money we have or how much something costs in numbers: dollars, pesos, yuan, rupees, krona, euros, or yen. Additionally (another joke), this book has a time-saving section on remembering numbers-and large numbers of numbers.

If, for some reason, you're not crazy about math, read a little further. Of course I, as the Science Guy, hope you do like math. Well, actually, I hope you love math. But no matter how you feel about math, hatred or love, I'd bet that you often find yourself just wanting to know the answer right away, without having to write down everything carefully and work slowly and diligently-or without even having to stop and grab a calculator. You want the answer, as we say, "as if by magic." It turns out that you can solve or work many, many math problems almost magically. This book will show you how.

What makes any kind of magic so intriguing and fun is that the audience seldom knows how the trick is performed. "How did she do that ...?" "I don't know, but it's cool." If you have an audience, the tricks and shortcuts in Secrets of Mental Math are a lot like magic. The audience seldom knows how a trick is performed; they just appreciate it. Notice, though, that in magic, it's hardly worth doing if no one is watching. But with Secrets, knowing how it works doesn't subtract from the fun (or pun). When arithmetic is easy, you don't get bogged down in the calculating; you can concentrate on the wonderful nature of numbers. After all, math runs the universe.

Dr. Benjamin got into this business of lightning-fast calculating just for fun. We have to figure he impressed his teachers and classmates. Magicians might make some in their audience think that they have supernatural powers. Mathemagicians, at first, give the impression that they're geniuses. Getting people to notice what you're doing is an old part of sharing ideas. If they're impressed, they'll probably listen to what you have to say. So try some "mathemagics." You may impress your friends, all right. But you'll also find yourself performing just for yourself. You'll find you're able to do problems that you didn't think you could. You'll be impressed ... with yourself.

Now, counting on your fingers is one thing (one finger's worth). But have you ever found yourself counting out loud or whispering or making other sounds while you calculate? It almost always makes math easier. The problem, though, is that other people think you're a little odd ... not even (more math humor). Well, in Secrets of Mental Math, Dr. Benjamin helps you learn to use that "out-loud" feature of the way your brain works to do math problems more easily, faster, and more accurately (which is surprising), all while your brain is thinking away-almost as if you're thinking out loud.

You'll learn to move through math problems the same way we read in English, left to right. You'll learn to handle big problems fast with good guesses, actually great guesses, within a percent or so. You will learn to do arithmetic fast; that way you can spend your time thinking about what the numbers mean. Og wondered, "Do we have enough fruit for each person sitting around the fire? If not, there might be trouble." Now you might wonder, "Is there enough space on this computer to keep track of my music files ... or my bank account? If not, there might be trouble."

There's more to Secrets than just figuring. You can learn to take a day, month, and year, then compute what day of the week it was or will be. It's fantastic, almost magical, to be able to tell someone what day of the week she or he was born. But, it's really something to be able to figure that the United States had its first big Fourth of July on a Thursday in 1776. April 15, 1912, the day the Titanic sank, was a Monday. The first human to walk on the moon set foot there on July 20, 1969, a Sunday. You'll probably never forget that the United States was attacked by terrorists on September 11, 2001. With Secrets of Mental Math, you'll always be able to show it was a Tuesday.

There are relationships in Nature that numbers describe better than any other way we know. There are simple numbers that you can count on your hands: one, two, three, and on up. But there are also an infinite number of numbers in between. There are fractions. There are numbers that never end. They get as big as you want and so small that they're hard to imagine. You can know them. With Secrets of Mental Math, you can have even these in-between numbers come so quickly to your mind that you'll have a bit more space in your brain to think about why our world works this way. One way or another, this book will help you see that in Nature, it all adds up.
Foreword by James Randi
Mathematics is a wonderful, elegant, and exceedingly useful language. It has its own vocabulary and syntax, its own verbs, nouns, and modifiers, and its own dialects and patois. It is used brilliantly by some, poorly by others. Some of us fear to pursue its more esoteric uses, while a few of us wield it like a sword to attack and conquer income tax forms or masses of data that resist the less courageous. This book does not guarantee to turn you into a Leibniz, or put you on stage as a Professor Algebra, but it will, I hope, bring you a new, exciting, and even entertaining view of what can be done with that wonderful invention-numbers.

We all think we know enough about arithmetic to get by, and we certainly feel no guilt about resorting to the handy pocket calculator that has become so much a part of our lives. But, just as photography may blind us to the beauty of a Vermeer painting, or an electronic keyboard may make us forget the magnificence of a Horowitz sonata, too much reliance on technology can deny us the pleasures that you will find in these pages.

I remember the delight I experienced as a child when I was shown that I could multiply by 25 merely by adding two 0s to my number and dividing by 4. Casting out 9s to check multiplication came next, and when I found out about cross-multiplying I was hooked and became, for a short while, a generally unbearable math nut. Immunizations against such afflictions are not available. You have to recover all by yourself. Beware!

This is a fun book. You wouldn't have it in your hands right now if you didn't have some interest either in improving your math skills or in satisfying a curiosity about this fascinating subject. As with all such instruction books, you may retain and use only a certain percentage of the varied tricks and methods described here, but that alone will make it worth the investment of your time.

I know both the authors rather well. Art Benjamin is not only one of those whiz kids we used to groan about in school but also has been known to tread the boards at the Magic Castle in Hollywood, performing demonstrations of his skill, and on one occasion he traveled to Tokyo, Japan, to pit his math skills against a lady savant on live television. Michael Shermer, with his specialized knowledge of science, has an excellent overview of practical applications of math as it is used in the real world.

If this is your first exposure to this kind of good math stuff, I envy you. You'll discover, as you come upon each delicious new way to attack numbers, that you missed something in school. Mathematics, particularly arithmetic, is a powerful and dependable tool for day-to-day use that enables us to handle our complicated lives with more assurance and accuracy. Let Art and Michael show you how to round a few of the corners and cut through some of the traffic. Remember these words of Dr. Samuel Johnson, an eminently practical soul in all respects: "Arithemetical inquiries give entertainment in solitude by the practice, and reputation in public by the effect."

Above all, enjoy the book. Let it entertain you, and have fun with it. That, with the occasional good deed, a slice of pizza (no anchovies!), and a selection of good friends is about all you can ask of life. Well, almost all. Maybe a Ferrari ...
Prologue by Michael Shermer
My good friend Dr. Arthur Benjamin, mathematics professor at Harvey Mudd College in Claremont, California, takes the stage to a round of applause at the Magic Castle, a celebrated magic club in Hollywood, where he is about to perform "mathemagics," or what he calls the art of rapid mental calculation. Art appears nothing like a mathematics professor from a prestigious college. Astonishingly quick-witted, he looks at home with the rest of the young magicians playing at the Castle-which he is.

What makes Art so special is that he can play in front of any group, including professional mathematicians and magicians, because he can do something that almost no one else can. Art Benjamin can add, subtract, multiply, and divide numbers in his head faster than most people can with a calculator. He can square two-digit, three-digit, and four-digit numbers, as well as find square roots and cube roots, without writing anything down on paper. And he can teach you how to perform your own mathematical magic.

Traditionally, magicians refuse to disclose how they perform their tricks. If they did, everyone would know how they are done and the mystery and fascination of magic would be lost. But Art wants to get people excited about math. And he knows that one of the best ways to do so is to let you and other readers in on his secrets of "math genius." With these skills, almost anyone can do what Art Benjamin does every time he gets on stage to perform his magic.

This particular night at the Magic Castle, Art begins by asking if anyone in the audience has a calculator. A group of engineers raise their hands and join Art on the stage. Offering to test their calculators to make sure they work, Art asks a member of the audience to call out a two-digit number. "Fifty-seven," shouts one. Art points to another who yells out, "Twenty-three."

Directing his attention to those on stage, Art tells them: "Multiply 57 by 23 on the calculator and make sure you get 1311 or the calculators are not working correctly." Art waits patiently while the volunteers finish inputting the numbers. As each participant indicates his calculator reads 1311, the audience lets out a collective gasp. The amazing Art has beaten the calculators at their own game!

Art next informs the audience that he will square four two-digit numbers faster than his button-pushers on stage can square them on their calculators. The audience asks him to square the numbers 24, 38, 67, and 97. Then, in large, bold writing for everyone to see, Art writes: 576, 1444, 4489, 9409. Art turns to his engineer volunteers, each of whom is computing a two-digit square, and asks them to call out their answers. Their response triggers gasps and then applause from the audience: "576, 1444, 4489, 9409." The woman next to me sits with her mouth open in amazement.

Art then offers to square three-digit numbers without even writing down the answer. "Five hundred and seventy-two," a gentleman calls out. Art's reply comes less than a second later: "572 squared is 327,184." He immediately points to another member of the audience, who yells, "389," followed by Art's unblinking response: "389 squared will give you 151,321." Someone else blurts out, "262." "That'll give you 68,644." Sensing he delayed just an instant on that last one, he promises to make up for it on the next number. The challenge comes-991. With no pause, Art squares the number, "982,081." Several more three-digit numbers are given and Art responds perfectly. Members of the audience shake their heads in disbelief.

With the audience in the palm of his hand, Art now declares that he will attempt to square a four-digit number. A woman calls out, "1,036," and Art instantly responds, "That's 1,073,296." The audience laughs and Art explains, "No, no, that's much too easy a number. I'm not supposed to beat the calculators on these. Let's try another one." A man offers a challenging 2,843. Pausing briefly between digits, Art responds: "Let's see, the square of that should be 8 million ... 82 thousand ... 649." He is right, of course, and the audience roars their approval, as loudly as they did for the previous magician who sawed a woman in half and made a dog disappear.

It is the same everywhere Art Benjamin goes, whether it is a high school auditorium, a college classroom, a professional conference, the Magic Castle, or a television studio. Professor Benjamin has performed his special brand of magic live all over the country and on numerous television talk shows. He has been the subject of investigation by a cognitive psychologist at Carnegie Mellon University and is featured in a scholarly book by Steven Smith called The Great Mental Calculators: The Psychology, Methods, and Lives of Calculating Prodigies, Past and Present. Art was born in Cleveland on March 19, 1961 (which he calculates was a Sunday, a skill he will teach you in Chapter 9). A hyperactive child, Art drove his teachers mad with his classroom antics, which included correcting the mathematical mistakes they occasionally made. Throughout this book when teaching you his mathematical secrets, Art recalls when and where he learned these skills, so I will save the fascinating stories for him to tell you.

Art Benjamin is an extraordinary individual with an extraordinary program to teach you rapid mental calculation. I offer these claims without hesitation, and ask only that you remember this does not come from a couple of guys promising miracles if you will only call our 800 hotline. Art and I are both credentialed in the most conservative of academic professions-Art in mathematics and I, myself, in the history of science-and we would never risk career embarrassment (or worse) by making such powerful claims if they were not true. To put it simply, this stuff works, and virtually everyone can do it because this art of "math genius" is a learned skill. So you can look forward to improving your math skills, impressing your friends, enhancing your memory, and, most of all, having fun!
Introduction
Ever since I was a child, I have loved playing with numbers, and in this book I hope to share my passion with you. I have always found numbers to have a certain magical appeal and spent countless hours entertaining myself and others with their beautiful properties. As a teenager, I performed as a magician, and subsequently combined my loves of math and magic into a full-length show, called Mathemagics, where I would demonstrate and explain the secrets of rapid mental calculation to audiences of all ages.

Since earning my PhD, I have taught mathematics at Harvey Mudd College, and I still enjoy sharing the joy of numbers with children and adults throughout the world. In this book, I will share all of my secrets for doing math in your head, quickly and easily. (I realize that magicians are not supposed to reveal their secrets, but mathemagicians have a different code of ethics. Mathematics should be awe inspiring, not mysterious.)

What will you learn from this book? You will learn to do math in your head faster than you ever thought possible. After a little practice, you will dramatically improve your memory for numbers. You will learn feats of mind that will impress your friends, colleagues, and teachers. But you will also learn to view math as an activity that can actually be fun.

Too often, math is taught as a set of rigid rules, leaving little room for creative thinking. But as you will learn from Secrets, there are often several ways to solve the same problem. Large problems can be broken down into smaller, more manageable components. We look for special features to make our problems easier to solve. These strike me as being valuable life lessons that we can use in approaching all kinds of problems, mathematical and otherwise.

"But isn't math talent something that you are born with?" I get this question all the time. Many people are convinced that lightning calculators are prodigiously gifted. Maybe I was born with some curiosity about how things work, whether it be a math problem or a magic trick. But I am convinced, based on many years of teaching experience, that rapid math is a skill that anyone can learn. And like any worthwhile skill, it takes practice and dedication if you wish to become an expert. But to achieve these results efficiently, it is important that you practice the right way. Let me show you the way!

Mathemagically,

Dr. Arthur Benjamin

Claremont, California
Chapter 0 Quick Tricks:
Easy (and Impressive) Calculations
In the pages that follow, you will learn to do math in your head faster than you ever thought possible. After practicing the methods in this book for just a little while, your ability to work with numbers will increase dramatically. With even more practice, you will be able to perform many calculations faster than someone using a calculator. But in this chapter, my goal is to teach you some easy yet impressive calculations you can learn to do immediately. We'll save some of the more serious stuff for later.
INSTANT MULTIPLICATION
Let's begin with one of my favorite feats of mental math-how to multiply, in your head, any two-digit number by eleven. It's very easy once you know the secret. Consider the problem:

32×11

To solve this problem, simply add the digits, 3 + 2 = 5, put the 5 between the 3 and the 2, and there is your answer:

352

What could be easier? Now you try:

53 × 11

Since 5 + 3 = 8, your answer is simply

583

One more. Without looking at the answer or writing anything down, what is

81×11?

Did you get 891? Congratulations!

Now before you get too excited, I have shown you only half of what you need to know. Suppose the problem is

85 × 11

Although 8 + 5 = 13, the answer is NOT 8135!

As before, the 3 goes in between the numbers, but the 1 needs to be added to the 8 to get the correct answer:

935

Think of the problem this way:

Here is another example. Try 57 × 11.

Since 5 + 7 = 12, the answer is

Okay, now it's your turn. As fast as you can, what is

77 × 11?

If you got the answer 847, then give yourself a pat on the back. You are on your way to becoming a mathemagician.

Now, I know from experience that if you tell a friend or teacher that you can multiply, in your head, any two-digit number by eleven, it won't be long before they ask you to do 99 × 11. Let's do that one now, so we are ready for it.

Since 9 + 9 = 18, the answer is:

Okay, take a moment to practice your new skill a few times, then start showing off. You will be amazed at the reaction you get. (Whether or not you decide to reveal the secret is up to you!)

Welcome back. At this point, you probably have a few questions, such as:

"Can we use this method for multiplying three-digit numbers (or larger) by eleven?"

Absolutely. For instance, for the problem 314 × 11, the answer still begins with 3 and ends with 4. Since 3 + 1 = 4, and 1 + 4 = 5, the answer is 3454. But we'll save larger problems like this for later.

More practically, you are probably saying to yourself,

"Well, this is fine for multiplying by elevens, but what about larger numbers? How do I multiply numbers by twelve, or thirteen, or thirty-six?"

My answer to that is, Patience! That's what the rest of the book is all about. In Chapters 2, 3, 6, and 8, you will learn methods for multiplying together just about any two numbers. Better still, you don't have to memorize special rules for every number. Just a handful of techniques is all that it takes to multiply numbers in your head, quickly and easily.
SQUARING AND MORE
Here is another quick trick.

As you probably know, the square of a number is a number multiplied by itself. For example, the square of 7 is 7 × 7 = 49. Later, I will teach you a simple method that will enable you to easily calculate the square of any two-digit or three-digit (or higher) number. That method is especially simple when the number ends in 5, so let's do that trick now.

To square a two-digit number that ends in 5, you need to remember only two things.

1. The answer begins by multiplying the first digit by the next higher digit.

2. The answer ends in 25.

For example, to square the number 35, we simply multiply the first digit (3) by the next higher digit (4), then attach 25. Since 3 × 4 = 12, the answer is 1225. Therefore, 35 × 35 = 1225. Our steps can be illustrated this way:

How about the square of 85? Since 8 × 9 = 72, we immediately get 85 × 85 = 7225.

We can use a similar trick when multiplying two-digit numbers with the same first digit, and second digits that sum to 10. The answer begins the same way that it did before (the first digit multiplied by the next higher digit), followed by the product of the second digits. For example, let's try 83 × 87. (Both numbers begin with 8, and the last digits sum to 3 + 7 = 10.) Since 8 × 9 = 72, and 3 × 7 = 21, the answer is 7221.

Similarly, 84 × 86 = 7224.

Now it's your turn. Try

26 × 24

How does the answer begin? With 2 × 3 = 6. How does it end? With 6 × 4 = 24. Thus 26 × 24 = 624.

Remember that to use this method, the first digits have to be the same, and the last digits must sum to 10. Thus, we can use this method to instantly determine that

31 × 39 = 1209
32 × 38 = 1216
33 × 37 = 1221
34 × 36 = 1224
35 × 35 = 1225

You may ask,

"What if the last digits do not sum to ten? Can we use this method to multiply twenty-two and twenty-three?"

Well, not yet. But in Chapter 8, I will show you an easy way to do problems like this using the close-together method. (For 22 × 23, you would do 20 × 25 plus 2 × 3, to get 500 + 6 = 506, but I'm getting ahead of myself!) Not only will you learn how to use these methods, but you will understand why these methods work, too.

"Are there any tricks for doing mental addition and subtraction?"

Definitely, and that is what the next chapter is all about. If I were forced to summarize my method in three words, I would say, "Left to right." Here is a sneak preview.

Consider the subtraction problem

Most people would not like to do this problem in their head (or even on paper!), but let's simplify it. Instead of subtracting 587, subtract 600. Since 1200 − 600 = 600, we have that

But we have subtracted 13 too much. (We will explain how to quickly determine the 13 in Chapter 1.) Thus, our painful-looking subtraction problem becomes the easy addition problem

which is not too hard to calculate in your head (especially from left to right). Thus, 1241 − 587 = 654.

Using a little bit of mathematical magic, described in Chapter 9, you will be able to instantly compute the sum of the ten numbers below.

Although I won't reveal the magical secret right now, here is a hint. The answer, 935, has appeared elsewhere in this chapter. More tricks for doing math on paper will be found in Chapter 6. Furthermore, you will be able to quickly give the quotient of the last two numbers:

359 ÷ 222 = 1.61 (first three digits)

We will have much more to say about division (including decimals and fractions) in Chapter 4.
MORE PRACTICAL TIPS
Here's a quick tip for calculating tips. Suppose your bill at a restaurant came to $42, and you wanted to leave a 15% tip. First we calculate 10% of $42, which is $4.20. If we cut that number in half, we get $2.10, which is 5% of the bill. Adding these numbers together gives us $6.30, which is exactly 15% of the bill. We will discuss strategies for calculating sales tax, discounts, compound interest, and other practical items in Chapter 5, along with strategies that you can use for quick mental estimation when an exact answer is not required.
IMPROVE YOUR MEMORY
In Chapter 7, you will learn a useful technique for memorizing numbers. This will be handy in and out of the classroom. Using an easy-to-learn system for turning numbers into words, you will be able to quickly and easily memorize any numbers: dates, phone numbers, whatever you want.

Speaking of dates, how would you like to be able to figure out the day of the week of any date? You can use this to figure out birth dates, historical dates, future appointments, and so on. I will show you this in more detail later, but here is a simple way to figure out the day of January 1 for any year in the twenty-first century. First familiarize yourself with the following table.

For instance, let's determine the day of the week of January 1, 2030. Take the last two digits of the year, and consider it to be your bill at a restaurant. (In this case, your bill would be $30.) Now add a 25% tip, but keep the change. (You can compute this by cutting the bill in half twice, and ignoring any change. Half of $30 is $15. Then half of $15 is $7.50. Keeping the change results in a $7 tip.) Hence your bill plus tip amounts to $37. To figure out the day of the week, subtract the biggest multiple of 7 (0, 7, 14, 21, 28, 35, 42, 49, ...) from your total, and that will tell you the day of the week. In this case, 37 − 35 = 2, and so January 1, 2030, will occur on 2's day, namely Tuesday:

How about January 1, 2043:

Exception: If the year is a leap year, remove $1 from your tip, then proceed as before. For example, for January 1, 2032, a 25% tip of $32 would be $8. Removing one dollar gives a total of 32 + 7 = 39. Subtracting the largest multiple of 7 gives us 39 − 35 = 4. So January 1, 2032, will be on 4's day, namely Thursday. For more details that will allow you to compute the day of the week of any date in history, see Chapter 9. (In fact, it's perfectly okay to read that chapter first!)

I know what you are wondering now:

"Why didn't they teach this to us in school?"

I'm afraid that there are some questions that even I cannot answer. Are you ready to learn more magical math? Well, what are we waiting for? Let's go!
Chapter 1 A Little Give and Take:
Mental Addition and Subtraction
For as long as I can remember, I have always found it easier to add and subtract numbers from left to right instead of from right to left. By adding and subtracting numbers this way, I found that I could call out the answers to math problems in class well before my classmates put down their pencils. And I didn't even need a pencil!

In this chapter you will learn the left-to-right method of doing mental addition and subtraction for most numbers that you encounter on a daily basis. These mental skills are not only important for doing the tricks in this book but are also indispensable in school, at work, or any time you use numbers. Soon you will be able to retire your calculator and use the full capacity of your mind as you add and subtract two-digit, three-digit, and even four-digit numbers with lightning speed.
LEFT-TO-RIGHT ADDITION
Most of us are taught to do math on paper from right to left. And that's fine for doing math on paper. But if you want to do math in your head (even faster than you can on paper) there are many good reasons why it is better to work from left to right. After all, you read numbers from left to right, you pronounce numbers from left to right, and so it's just more natural to think about (and calculate) numbers from left to right. When you compute the answer from right to left (as you probably do on paper), you generate the answer backward. That's what makes it so hard to do math in your head. Also, if you want to estimate your answer, it's more important to know that your answer is "a little over 1200" than to know that your answer "ends in 8." Thus, by working from left to right, you begin with the most significant digits of your problem. If you are used to working from right to left on paper, it may seem unnatural to work with numbers from left to right. But with practice you will find that it is the most natural and efficient way to do mental calculations.

With the first set of problems-two-digit addition-the left-to-right method may not seem so advantageous. But be patient. If you stick with me, you will see that the only easy way to solve three-digit and larger addition problems, all subtraction problems, and most definitely all multiplication and division problems is from left to right. The sooner you get accustomed to computing this way, the better.
Two-Digit Addition
Our assumption in this chapter is that you know how to add and subtract one-digit numbers. We will begin with two-digit addition, something I suspect you can already do fairly well in your head. The following exercises are good practice, however, because the two-digit addition skills that you acquire here will be needed for larger addition problems, as well as virtually all multiplication problems in later chapters. It also illustrates a fundamental principle of mental arithmetic-namely, to simplify your problem by breaking it into smaller, more manageable parts. This is the key to virtually every method you will learn in this book. To paraphrase an old saying, there are three components to success-simplify, simplify, simplify.

The easiest two-digit addition problems are those that do not require you to carry any numbers, when the first digits sum to 9 or below and the last digits sum to 9 or below. For example:

To solve 47 + 32, first add 30, then add 2. After adding 30, you have the simpler problem 77 + 2, which equals 79. We illustrate this as follows:

The above diagram is simply a way of representing the mental processes involved in arriving at an answer using our method. While you need to be able to read and understand such diagrams as you work your way through this book, our method does not require you to write down anything yourself.

Now let's try a calculation that requires you to carry a number:

Adding from left to right, you can simplify the problem by adding 67 + 20 = 87; then 87 + 8 = 95.

Now try one on your own, mentally calculating from left to right, and then check below to see how we did it:

How was that? You added 84 + 50 = 134 and added 134 + 7 = 141.

If carrying numbers trips you up a bit, don't worry about it. This is probably the first time you have ever made a systematic attempt at mental calculation, and if you're like most people, it will take you time to get used to it. With practice, however, you will begin to see and hear these numbers in your mind, and carrying numbers when you add will come automatically. Try another problem for practice, again computing it in your mind first, then checking how we did it:

You should have added 68 + 40 = 108, and then 108 + 5 = 113, the final answer. Was that easier? If you would like to try your hand at more two-digit addition problems, check out the set of exercises below. (The answers and computations are at the end of the book.)
Three-Digit Addition
The strategy for adding three-digit numbers is the same as for adding two-digit numbers: you add from left to right. After each step, you arrive at a new (and simpler) addition problem. Let's try the following:

Starting with 538, we add 300, then add 20, then add 7. After adding 300 (538 + 300 = 838), the problem becomes 838 + 27. After adding 20 (838 + 20 = 858), the problem simplifies to 858 + 7 = 865. This thought process can be diagrammed as follows:

All mental addition problems can be done by this method. The goal is to keep simplifying the problem until you are just adding a one-digit number. Notice that 538 + 327 requires you to hold on to six digits in your head, whereas 838 + 27 and 858 + 7 require only five and four digits, respectively. As you simplify the problem, the problem gets easier!

Try the following addition problem in your mind before looking to see how we did it:

Did you reduce and simplify the problem by adding left to right? After adding the hundreds (623 + 100 = 723), you were left with 723 + 59. Next you should have added the tens (723 + 50 = 773), simplifying the problem to 773 + 9, which you then summed to get 782. Diagrammed, the problem looks like this:

When I do these problems mentally, I do not try to see the numbers in my mind-I try to hear them. I hear the problem 623 + 159 as six hundred twenty-three plus one hundred fifty-nine; by emphasizing the word hundred to myself, I know where to begin adding. Six plus one equals seven, so my next problem is seven hundred and twenty-three plus fifty-nine, and so on. When first doing these problems, practice them out loud. Reinforcing yourself verbally will help you learn the mental method much more quickly.

Three-digit addition problems really do not get much harder than the following:

Now look to see how we did it:

At each step I hear (not see) a "new" addition problem. In my mind the problem sounds like this:

Your mind-talk may not sound exactly like mine (indeed, you might "see" the numbers instead of "hear" them), but whatever it is you say or visualize to yourself, the point is to reinforce the numbers along the way so that you don't forget where you are and have to start the addition problem over again.

Let's try another one for practice:

Do it in your mind first, then check our computation below:

This addition problem is a little more difficult than the last one since it requires you to carry numbers in all three steps. However, with this particular problem you have the option of using an alternative method. I am sure you will agree that it is a lot easier to add 500 to 759 than it is to add 496, so try adding 500 and then subtracting the difference:

So far, you have consistently broken up the second number in any problem to add to the first. It really does not matter which number you choose to break up, but it is good to be consistent. That way, your mind will never have to waste time deciding which way to go. If the second number happens to be a lot simpler than the first, I sometimes switch them around, as in the following example:

Let's finish up by adding three-digit to four-digit numbers. Since most human memory can hold only about seven or eight digits at a time, this is about as large a problem as you can handle without resorting to artificial memory devices, like fingers, calculators, or the mnemonics taught in Chapter 7. In many addition problems that arise in practice, especially within multiplication problems, one or both of the numbers will end in 0, so we shall emphasize those types of problems. We begin with an easy one:

Since 27 hundred + 5 hundred is 32 hundred, we simply attach the 67 to get 32 hundred and 67, or 3267. The process is the same for the following problems:

Because 40 + 18 = 58, the first answer is 3258. For the second problem, since 40 + 72 exceeds 100, you know the answer will be 33 hundred and something. Because 40 + 72 = 112, the answer is 3312.

These problems are easy because the (nonzero) digits overlap in only one place, and hence can be solved in a single step. Where digits overlap in two places, you require two steps. For instance:

This problem requires two steps, as diagrammed the following way:

Practice the following three-digit addition exercises, and then add some (pun intended!) of your own if you like until you are comfortable doing them mentally without having to look down at the page. (Answers can be found in the back of the book.)

Carl Friedrich Gauss: Mathematical Prodigy

A prodigy is a highly talented child, usually called precocious or gifted, and almost always ahead of his peers. The German mathematician Carl Friedrich Gauss (1777-1855) was one such child. He often boasted that he could calculate before he could speak. By the ripe old age of three, before he had been taught any arithmetic, he corrected his father's payroll by declaring "the reckoning is wrong." A further check of the numbers proved young Carl correct.

As a ten-year-old student, Gauss was presented the following mathematical problem: What is the sum of numbers from I to 100? While his fellow students were frantically calculating with paper and pencil, Gauss immediately envisioned that if he spread out the numbers I through 50 from left to right, and the numbers 51 to 100 from right to left directly below the 1-50 numbers, each combination would add up to 101 (I + 100, 2 + 99, 3 + 98, ...). Since there were fifty sums, the answer would be 101 × 50 = 5050.TO the astonishment of everyone, including the teacher, young Carl got the answer not only ahead of everyone else, but computed it entirely in his mind. He wrote out the answer on his slate, and flung it on the teacher's desk with a defiant "There it lies." The teacher was so impressed that he invested his own money to purchase the best available textbook on arithmetic and gave it to Gauss, stating, "He is beyond me, I can teach him nothing more."

Indeed, Gauss became the mathematics teacher of others, and eventually went on to become one of the greatest mathematicians in history, his theories still used today in the service of science. Gauss's desire to better understand Nature through the language of mathematics was summed up in his motto, taken from Shakespeare's King Lear (substituting "laws" for "law"): "Thou, nature, art my goddess; to thy laws/My services are bound."
LEFT-TO-RIGHT SUBTRACTION
For most of us, it is easier to add than to subtract. But if you continue to compute from left to right and to break down problems into simpler components, subtraction can become almost as easy as addition.
Two-Digit Subtraction
When subtracting two-digit numbers, your goal is to simplify the problem so that you are reduced to subtracting (or adding) a one-digit number. Let's begin with a very simple subtraction problem:

After each step, you arrive at a new and easier subtraction problem. Here, we first subtract 20 (86 − 20 = 66) then we subtract 5 to reach the simpler subtraction problem 66-5 for your final answer of 61. The problem can be diagrammed this way:

Of course, subtraction problems are considerably easier when there is no borrowing (which occurs when a larger digit is being subtracted from a smaller one). But the good news is that "hard" subtraction problems can usually be turned into "easy" addition problems. For example:

There are two different ways to solve this problem mentally:

1. First subtract 20, then subtract 9:

But for this problem, I would prefer the following strategy:

2. First subtract 30, then add back 1:

Here is the rule for deciding which method to use: If a two-digit subtraction problem would require borrowing, then round the second number up (to a multiple of ten). Subtract the rounded number, then add back the difference.

For example, the problem 54 − 28 would require borrowing (since 8 is greater than 4), so round 28 up to 30, compute 54 − 30 = 24, then add back 2 to get 26 as your final answer:

Now try your hand (or head) at 81 − 37. Since 7 is greater than 1, we round 37 up to 40, subtract it from 81 (81 − 40 = 41), then add back the difference of 3 to arrive at the final answer:

With just a little bit of practice, you will become comfortable working subtraction problems both ways. Just use the rule above to decide which method will work best.
Three-Digit Subtraction
Now let's try a three-digit subtraction problem:

This particular problem does not require you to borrow any numbers (since every digit of the second number is less than the digit above it), so you should not find it too hard. Simply subtract one digit at a time, simplifying as you go.

Now let's look at a three-digit subtraction problem that requires you to borrow a number:

At first glance this probably looks like a pretty tough problem, but if you first subtract 747 − 600 = 147, then add back 2, you reach your final answer of 147 + 2 = 149.

Now try one yourself:

Did you first subtract 700 from 853? If so, did you get 853 − 700 = 153? Since you subtracted by 8 too much, did you add back 8 to reach 161, the final answer?

Now, I admit we have been making life easier for you by subtracting numbers that were close to a multiple of 100. (Did you notice?) But what about other problems, like:

If you subtract one digit at a time, simplifying as you go, your sequence will look like this:

What happens if you round up to 500?

Subtracting 500 is easy: 725 − 500 = 225. But you have subtracted too much. The trick is to figure out exactly how much too much.

At first glance, the answer is far from obvious. To find it, you need to know how far 468 is from 500. The answer can be found by using "complements," a nifty technique that will make many three-digit subtraction problems a lot easier to do.
Using Complements (You're Welcome!)
Quick, how far from 100 are each of these numbers?

57 68 49 21 79

Here are the answers:

Notice that for each pair of numbers that add to 100, the first digits (on the left) add to 9 and the last digits (on the right) add to 10. We say that 43 is the complement of 57, 32 is the complement of 68, and so on.

Now you find the complements of these two-digit numbers:

37 59 93 44 08

To find the complement of 37, first figure out what you need to add to 3 in order to get 9. (The answer is 6.) Then figure out what you need to add to 7 to get 10. (The answer is 3.) Hence, 63 is the complement of 37.

The other complements are 41, 7, 56, 92. Notice that, like everything else you do as a mathemagician, the complements are determined from left to right. As we have seen, the first digits add to 9, and the second digits add to 10. (An exception occurs in numbers ending in 0-e.g., 30 + 70 = 100-but those complements are simple!)

What do complements have to do with mental subtraction? Well, they allow you to convert difficult subtraction problems into straightforward addition problems. Let's consider the last subtraction problem that gave us some trouble:

To begin, you subtracted 500 instead of 468 to arrive at 225 (725 − 500 = 225). But then, having subtracted too much, you needed to figure out how much to add back. Using complements gives you the answer in a flash. How far is 468 from 500? The same distance as 68 is from 100. If you find the complement of 68 the way we have shown you, you will arrive at 32. Add 32 to 225, and you will arrive at 257, your final answer.

Try another three-digit subtraction problem:

To compute this mentally, subtract 300 from 821 to arrive at 521, then add back the complement of 59, which is 41, to arrive at 562, our final answer. The procedure looks like this:

Here is another problem for you to try:

Check your answer and the procedure for solving the problem below:

Subtracting a three-digit number from a four-digit number is not much harder, as the next example illustrates:

By rounding up, you subtract 600 from 1246, leaving 646, then add back the complement of 79, which is 21. Your final answer is 646 + 21 = 667.

Try the three-digit subtraction exercises below, and then create more of your own for additional (or should that be subtractional?) practice.
Chapter 2 Products of a Misspent Youth:
Basic Multiplication
I probably spent too much time of my childhood thinking about faster and faster ways to perform mental multiplication; I was diagnosed as hyperactive and my parents were told that I had a short attention span and probably would not be successful in school. (Fortunately, my parents ignored that advice. I was also lucky to have some incredibly patient teachers in my first few years of school.) It might have been my short attention span that motivated me to develop quick ways to do arithmetic. I don't think I had the patience to carry out math problems with pencil and paper. Once you have mastered the techniques described in this chapter, you won't want to rely on pencil and paper again, either.

In this chapter you will learn how to multiply in your head one-digit numbers by two-digit numbers and three-digit numbers. You will also learn a phenomenally fast way to square two-digit numbers. Even friends with calculators won't be able to keep up with you. Believe me, virtually everyone will be dumbfounded by the fact that such problems can not only be done mentally, but can be computed so quickly. I sometimes wonder whether we were not cheated in school; these methods are so simple once you learn them.

There is one small prerequisite for mastering the skills in this chapter-you need to know the multiplication tables through ten. In fact, to really make headway, you need to know your multiplication tables backward and forward. For those of you who need to shake the cobwebs loose, consult the multiplication chart below. Once you've got your tables down, you are ready to begin.
Multiplication Table of Numbers 1-10 2-BY-1 MULTIPLICATION PROBLEMS
If you worked your way through Chapter 1, you got into the habit of adding and subtracting from left to right. You will do virtually all the calculations in this chapter from left to right as well. This is undoubtedly the opposite of what you learned in school. But you'll soon see how much easier it is to think from left to right than from right to left. (For one thing, you can start to say your answer aloud before you have finished the calculation. That way you seem to be calculating even faster than you are!)

Let's tackle our first problem:

First, multiply 40 × 7 = 280. (Note that 40 × 7 is just like 4 × 7, with a friendly zero attached.) Next, multiply 2 × 7= 14. Then add 280 plus 14 (left to right, of course) to arrive at 294, the correct answer. We illustrate this procedure below.

We have omitted diagramming the mental addition of 280 + 14, since you learned how to do that computation in the last chapter. At first you will need to look down at the problem while doing the calculation. With practice you will be able to forgo this step and compute the whole thing in your mind.

Let's try another example:

Your first step is to break down the problem into small multiplication tasks that you can perform mentally with ease. Since 48 = 40 + 8, multiply 40 × 4 = 160, then add 8 × 4 = 32. The answer is 192. (Note: If you are wondering why this process works, see the Why These Tricks Work section at the end of the chapter.)

Here are two more mental multiplication problems that you should be able to solve fairly quickly. First calculate 62 × 3. Then do 71 × 9. Try doing them in your head before looking at how we did it.

These two examples are especially simple because the numbers being added essentially do not overlap at all. When doing 180 + 6, you can practically hear the answer: One hundred eighty ... six! Another especially easy type of mental multiplication problem involves numbers that begin with five. When the five is multiplied by an even digit, the first product will be a multiple of 100, which makes the resulting addition problem a snap.

Try your hand at the following problem:

Notice how much easier this problem is to do from left to right. It takes far less time to calculate "400 plus 35" mentally than it does to apply the pencil-and-paper method of "putting down the 5 and carrying the 3."

The following two problems are a little harder.

As usual, we break these problems down into easier problems. For the one on the left, multiply 30 × 9 plus 8 × 9, giving you 270 + 72. The addition problem is slightly harder because it involves carrying a number. Here 270 + 70 + 2 = 340 + 2 = 342.

With practice, you will become more adept at juggling problems like these in your head, and those that require you to carry numbers will be almost as easy as those that don't.
Rounding Up
You saw in the last chapter how useful rounding up can be when it comes to subtraction. The same goes for multiplication, especially when you are multiplying numbers that end in eight or nine.

Let's take the problem of 69 × 6, illustrated below. On the left we have calculated it the usual way, by adding 360 + 54. On the right, however, we have rounded 69 up to 70, and subtracted 420 − 6, which you might find easier to do.

The following example also shows how much easier rounding up can be:

The subtraction method works especially well for numbers that are just one or two digits away from a multiple of 10. It does not work so well when you need to round up more than two digits because the subtraction portion of the problem gets difficult. As it is, you may prefer to stick with the addition method. Personally, for problems of this size, I use only the addition method because in the time spent deciding which method to use, I could have already done the calculation!

So that you can perfect your technique, I strongly recommend practicing more 2-by-1 multiplication problems. Below are twenty problems for you to tackle. I have supplied you with the answers in the back, including a breakdown of each component of the multiplication. If, after you've worked out these problems, you would like to practice more, make up your own. Calculate mentally, then check your answer with a calculator. Once you feel confident that you can perform these problems rapidly in your head, you are ready to move to the next level of mental calculation.
3-BY-1 MULTIPLICATION PROBLEMS
Now that you know how to do 2-by-1 multiplication problems in your head, you will find that multiplying three digits by a single digit is not much more difficult. You can get started with the following 3-by-1 problem (which is really just a 2-by-1 problem in disguise):

Was that easy for you? (If this problem gave you trouble, you might want to review the addition material in Chapter 1.) Let's try another 3-by-1 problem similar to the one you just did, except we have replaced the 0 with a 6 so you have another step to perform:

In this case, you simply add the product of 6 × 7, which you already know to be 42, to the first sum of 2240. Since you do not need to carry any numbers, it is easy to add 42 to 2240 to arrive at the total of 2282.

In solving this and other 3-by-1 multiplication problems, the difficult part may be holding in memory the first sum (in this case, 2240) while doing the next multiplication problem (in this case, 6 × 7). There is no magic secret to remembering that first number, but with practice I guarantee you will improve your concentration, and holding on to numbers while performing other functions will get easier.

Let's try another problem:

Even if the numbers are large, the process is just as simple. For example:

When first solving these problems, you may have to glance down at the page as you go along to remind yourself what the original problem is. This is okay at first. But try to break the habit so that eventually you are holding the problem entirely in memory.

In the last section on 2-by-1 multiplication problems, we saw that problems involving numbers that begin with five are sometimes especially easy to solve. The same is true for 3-by-1 problems:

Notice that whenever the first product is a multiple of 1000, the resulting addition problem is no problem at all. This is because you do not have to carry any numbers and the thousands digit does not change. If you were solving the problem above in front of someone else, you would be able to say your first product-"three thousand ..."-out loud with complete confidence that a carried number would not change it to 4000. (As an added bonus, by quickly saying the first digit, it gives the illusion that you computed the entire answer immediately!) Even if you are practicing alone, saying your first product out loud frees up some memory space while you work on the remaining 2-by-1 problem, which you can say out loud as well-in this case, "... three hundred seventy-eight."

Try the same approach in solving the next problem, where the multiplier is a 5:

Because the first two digits of the three-digit number are even, you can say the answer as you calculate it without having to add anything! Don't you wish all multiplication problems were this easy?

Let's escalate the challenge by trying a couple of problems that require some carrying.

In the next two problems you need to carry a number at the end of the problem instead of at the beginning:

The first part of each of these problems is easy enough to compute mentally. The difficult part comes in holding the preliminary answer in your head while computing the final answer. In the case of the first problem, it is easy to add 5400 + 360 = 5760, but you may have to repeat 5760 to yourself several times while you multiply 8 × 9 = 72. Then add 5760 + 72. Sometimes at this stage I will start to say my answer aloud before finishing. Because I know I will have to carry when I add 60 + 72, I know that 5700 will become 5800, so I say "fifty-eight hundred and ..." Then I pause to compute 60 + 72 = 132. Because I have already carried, I say only the last two digits, "... thirty-two!" And there is the answer: 5832.

The next two problems require you to carry two numbers each, so they may take you longer than those you have already done. But with practice you will get faster:

When you are first tackling these problems, repeat the answers to each part out loud as you compute the rest. In the first problem, for example, start by saying, "Twenty-eight hundred plus five hundred sixty" a couple of times out loud to reinforce the two numbers in memory while you add them together. Repeat the answer-"thirty-three hundred sixty"-several times while you multiply 9 × 7= 63. Then repeat "thirty-three hundred sixty plus sixty-three" aloud until you compute the final answer of 3423. If you are thinking fast enough to recognize that adding 60 + 63 will require you to carry a 1, you can begin to give the final answer a split second before you know it-"thirty-four hundred and ... twenty-three!"

Let's end this section on 3-by-1 multiplication problems with some special problems you can do in a flash because they require one addition step instead of two:

In general, if the product of the last two digits of the first number and the multiplier is known to you without having to calculate it (for instance, you may know that 25 × 8 = 200 automatically since 8 quarters equals $2.00), you will get to the final answer much more quickly. For instance, if you know without calculating that 75 × 4 = 300, then it is easy to compute 975 × 4:

To reinforce what you have just learned, solve the following 3-by-1 multiplication problems in your head; then check your computations and answers with ours (in the back of the book). I can assure you from experience that doing mental calculations is just like riding a bicycle or typing. It might seem impossible at first, but once you've mastered it, you will never forget how to do it.
BE THERE OR B2: SQUARING TWO-DIGIT NUMBERS
Squaring numbers in your head (multiplying a number by itself) is one of the easiest yet most impressive feats of mental calculation you can do. I can still recall where I was when I discovered how to do it. I was thirteen, sitting on a bus on the way to visit my father at work in downtown Cleveland. It was a trip I made often, so my mind began to wander. I'm not sure why, but I began thinking about the numbers that add up to 20, and I wondered, how large could the product of two such numbers get?

I started in the middle with 10 × 10 (or 102), the product of which is 100. Next, I multiplied 9 × 11 = 99, 8 × 12 = 96, 7 × 13 = 91, 6 × 14 = 84, 5 × 15 = 75, 4 × 16 = 64, and so on. I noticed that the products were getting smaller, and their difference from 100 was 1, 4, 9, 16, 25, 36, ... - or 12, 22, 32, 42, 52, 62, ... (see table below).

I found this pattern astonishing. Next I tried numbers that add to 26 and got similar results. First I worked out 132 = 169, then computed 12 × 14 = 168, 11 × 15 = 165, 10 × 16 = 160, 9 × 17 = 153, and so on. Just as before, the distances these products were from 169 was 12, 22, 32, 42, and so on (see table below).

There is actually a simple algebraic explanation for this phenomenon (see Why These Tricks Work, this page). At the time, I didn't know my algebra well enough to prove that this pattern would always occur, but I experimented with enough examples to become convinced of it.

Then I realized that this pattern could help me square numbers more easily. Suppose I wanted to square the number 13. Instead of multiplying 13 × 13,

why not get an approximate answer by using two numbers that are easier to multiply but also add to 26? I chose 10 × 16 = 160. To get the final answer, I just added 32 = 9 (since 10 and 16 are each 3 away from 13). Thus, 132 = 160 + 9 = 169. Neat!

This method is diagrammed as follows:

Now let's see how this works for another square:

To square 41, subtract 1 to obtain 40 and add 1 to obtain 42. Next multiply 40 × 42. Don't panic! This is simply a 2-by-1 multiplication problem (specifically, 4 × 42) in disguise. Since 4 × 42 = 168, 40 × 42 = 1680. Almost done! All you have to add is the square of 1 (the number by which you went up and down from 41), giving you 1680 + 1 = 1681.

Can squaring a two-digit number be this easy? Yes, with this method and a little practice, it can. And it works whether you initially round down or round up. For example, let's examine 772, working it out both by rounding up and by rounding down:

or

In this instance the advantage of rounding up is that you are virtually done as soon as you have completed the multiplication problem because it is simple to add 9 to a number ending in 0!

In fact, for all two-digit squares, I always round up or down to the nearest multiple of 10. So if the number to be squared ends in 6, 7, 8, or 9, round up, and if the number to be squared ends in 1, 2, 3, or 4, round down. (If the number ends in 5, you do both!) With this strategy you will add only the numbers 1, 4, 9, 16, or 25 to your first calculation.

Let's try another problem. Calculate 562 in your head before looking at how we did it, below:

Squaring numbers that end in 5 is even easier. Since you will always round up and down by 5, the numbers to be multiplied will both be multiples of 10. Hence, the multiplication and the addition are especially simple. We have worked out 852 and 352, below:

As you saw in Chapter 0, when you are squaring a number that ends in 5, rounding up and down allows you to blurt out the first part of the answer immediately and then finish it with 25. For example, if you want to compute 752, rounding up to 80 and down to 70 will give you "Fifty-six hundred and ... twenty-five!"

For numbers ending in 5, you should have no trouble beating someone with a calculator, and with a little practice with the other squares, it won't be long before you can beat the calculator with any two-digit square number. Even large numbers are not to be feared. You can ask someone to give you a really big two-digit number, something in the high 90s, and it will sound as though you've chosen an impossible problem to compute. But, in fact, these are even easier because they allow you to round up to 100.

Let's say your audience gives you 962. Try it yourself, and then check how we did it.

Wasn't that easy? You should have rounded up by 4 to 100 and down by 4 to 92, and then multiplied 100 × 92 to get 9200. At this point you can say out loud, "Ninety-two hundred," and then finish up with "sixteen" and enjoy the applause!

Computer the following:

Zerah Colburn: Entertaining Calculations

One of the first lightning calculators to capitalize on his talent was Zerah Colburn (1804-1839), an American farmer's son from Vermont who learned the multiplication tables to 100 before he could even read or write. By the age of six, young Zerah's father took him on the road, where his performances generated enough capital to send him to school in Paris and London. By age eight he was internationally famous, performing lightning calculations in England, and was described in the Annual Register as "the most singular phenomenon in the history of the human mind that perhaps ever existed." No less than Michael Faraday and Samuel Morse admired him.

No matter where he went, Colburn met all challengers with speed and precision. He tells us in his autobiography of one set of problems he was given in New Hampshire in June 1811: "How many days and hours since the Christian Era commenced, 1811 years ago? Answered in twenty seconds: 661,015 days, 15,864,360 hours. How many seconds in eleven years? Answered in four seconds; 346,896,000." Colburn used the same techniques described in this book to compute entirely in his head problems given to him. For example, he would factor large numbers into smaller numbers and then multiply: Colburn once multiplied 21,734 × 543 by factoring 543 into 181 × 3. He then multiplied 21,734 × 181 to arrive at 3,933,854, and finally multiplied that figure by 3, for a total of 11,801,562.

As is often the case with lightning calculators, interest in Colburn's amazing skills diminished with time, and by the age of twenty he had returned to America and become a Methodist preacher. He died at a youthful thirty-five. In summarizing his skills as a lightning calculator, and the advantage such an ability affords, Colburn reflected, "True, the method ... requires a much larger number of figures than the common Rule, but it will be remembered that pen, ink and paper cost Zerah very little when engaged in a sum."
WHY THESE TRICKS WORK
This section is presented for teachers, students, math buffs, and anyone curious as to why our methods work. Some people may find the theory as interesting as the application. Fortunately, you need not understand why our methods work in order to understand how to apply them. All magic tricks have a rational explanation behind them, and mathemagical tricks are no different. It is here that the mathemagician reveals his deepest secrets!

In this chapter on multiplication problems, the distributive law is what allows us to break down problems into their component parts. The distributive law states that for any numbers a, b, and c:

(b + c) × a = (b × a) + (c × a)

That is, the outside term, a, is distributed, or separately applied, to each of the inside terms, b and c. For example, in our first mental multiplication problem of 42 × 7, we arrived at the answer by treating 42 as 40 + 2, then distributing the 7 as follows:

42 × 7 = (40 + 2) × 7 = (40 × 7) + (2 × 7) = 280 + 14 = 294

You may wonder why the distributive law works in the first place. To understand it intuitively, imagine having 7 bags, each containing 42 coins, 40 of which are gold and 2 of which are silver. How many coins do you have altogether? There are two ways to arrive at the answer. In the first place, by the very definition of multiplication, there are 42 × 7 coins. On the other hand, there are 40 × 7 gold coins and 2 × 7 silver coins. Hence, we have (40 × 7) + (2 × 7) coins altogether. By answering our question two ways, we have 42 × 7 = (40 × 7) + (2 × 7). Notice that the numbers 7, 40, and 2 could be replaced by any numbers (a, b, or c) and the same logic would apply. That's why the distributive law works!

Using similar reasoning with gold, silver, and copper coins we can derive:

(b + c + d) × a = (b × a) + (c × a) + (d × a)

Hence, to do the problem 326 × 7, we break up 326 as 300 + 20 + 6, then distribute the 7, as follows: 326 × 7 = (300 + 20 + 6) × 7 = (300 × 7) + (20 × 7) + (6 × 7), which we then add up to get our answer.

As for squaring, the following algebra justifies my method. For any numbers A and d

A2 = (A + d) × (A - d) + d2

Here, A is the number being squared; d can be any number, but I choose it to be the distance from A to the nearest multiple of 10. Hence, for 772, I set d = 3 and our formula tells us that 772 = (77 + 3) × (77 − 3) + 32 = (80 × 74) + 9 = 5929. The following algebraic relationship also works to explain my squaring method:

(z + d)2 = z2 + 2zd + d2 = z(z + 2d) + d2

Hence, to square 41, we set z = 40 and d = 1 to get:

412 = (40 + 1)2 = 40 × (40 + 2) + 12 = 1681

Similarly,

(z − d)2 = z(z − 2d) + d2

To find 772 when z = 80 and d = 3,

772 = (80 − 3)2 = 80 × (80 − 6) + 32 = 80 × 74 + 9 = 5929
Chapter 3 New and Improved Products:
Intermediate Multiplication
Mathemagics really gets exciting when you perform in front of an audience. I experienced my first public performance in eighth grade, at the fairly advanced age of thirteen. Many mathemagicians begin even earlier. Zerah Colburn (1804-1839), for example, reportedly could do lightning calculations before he could read or write, and he was entertaining audiences by the age of six! When I was thirteen, my algebra teacher did a problem on the board for which the answer was 1082. Not content to stop there, I blurted out, "108 squared is simply 11,664!"

The teacher did the calculation on the board and arrived at the same answer. Looking a bit startled, she said, "Yes, that's right. How did you do it?" So I told her, "I went down 8 to 100 and up 8 to 116. I then multiplied 116 × 100, which is 11,600, and just added the square of 8, to get 11,664."

She had never seen that method before. I was thrilled. Thoughts of "Benjamin's Theorem" popped into my head. I actually believed I had discovered something new. When I finally ran across this method a few years later in a book by Martin Gardner on recreational math, Mathematical Carnival (1965), it ruined my day! Still, the fact that I had discovered it for myself was very exciting to me.

You, too, can impress your friends (or teachers) with some fairly amazing mental multiplication. At the end of the last chapter you learned how to multiply a two-digit number by itself. In this chapter you will learn how to multiply two different two-digit numbers, a challenging yet more creative task. You will then try your hand-or, more accurately, your brain-at three-digit squares. You do not have to know how to do 2-by-2 multiplication problems to tackle three-digit squares, so you can learn either skill first.
2-BY-2 MULTIPLICATION PROBLEMS
When squaring two-digit numbers, the method is always the same. When multiplying two-digit numbers, however, you can use lots of different methods to arrive at the same answer. For me, this is where the fun begins.

The first method you will learn is the "addition method," which can be used to solve all 2-by-2 multiplication problems.
The Addition Method
To use the addition method to multiply any two two-digit numbers, all you need to do is perform two 2-by-1 multiplication problems and add the results together. For example:

Here you break up 42 into 40 and 2, two numbers that are easy to multiply. Then you multiply 40 × 46, which is just 4 × 46 with a 0 attached, or 1840. Then you multiply 2 × 46 = 92. Finally, you add 1840 + 92 = 1932, as diagrammed above.

Here's another way to do the same problem:

The catch here is that multiplying 6 × 42 is harder to do than multiplying 2 × 46, as in the first problem. Moreover, adding 1680 + 252 is more difficult than adding 1840 + 92. So how do you decide which number to break up? I try to choose the number that will produce the easier addition problem. In most cases-but not all-you will want to break up the number with the smaller last digit because it usually produces a smaller second number for you to add.

Now try your hand at the following problems:

The last problem illustrates why numbers that end in 1 are especially attractive to break up. If both numbers end in the same digit, you should break up the larger number as illustrated below:

If one number is much larger than the other, it often pays to break up the larger number, even if it has a larger last digit. You will see what I mean when you try the following problem two different ways:

Did you find the first method to be faster than the second? I did.

Here's another exception to the rule of breaking up the number with the smaller last digit. When you multiply a number in the fifties by an even number, you'll want to break up the number in the fifties:

The last digit of the number 84 is smaller than the last digit of 59, but if you break up 59, your product will be a multiple of 100, just as 4200 is, in the example above. This makes the subsequent addition problem much easier.

Now try an easy problem of a different sort:

Though the calculation above is pretty simple, there is an even easier and faster way to multiply any two-digit number by 11. This is mathemagics at its best: you won't believe your eyes when you see it (unless you remember it from Chapter 0)!

Here's how it works. Suppose you have a two-digit number whose digits add up to 9 or less. To multiply this number by 11, merely add the two digits together and insert the total between the original two digits. For example, to do 42 × 11, first do 4 + 2 = 6. If you place the 6 between the 4 and the 2, you get 462, the answer to the problem!

Try 54 × 11 by this method.

What could be simpler? All you had to do was place the 9 between the 5 and the 4 to give you the final answer of 594.

You may wonder what happens when the two numbers add up to a number larger than 9. In such cases, increase the tens digit by 1, then insert the last digit of the sum between the two numbers, as before. For example, when multiplying 76 × 11, 7 + 6 = 13, you increase the 7 in 76 to 8, and then insert the 3 between the 8 and the 6, giving you 836, the final answer. See the diagrammed answer below:

Your turn. Try 68 × 11.

Once you get the hang of this trick, you will never multiply by 11 any other way. Try a few more problems and then check your answers at the back of the book.

Returning to the addition method, the next problem is a real challenge the first time you try it. Try solving 89 × 72 in your head, looking back at the problem if necessary. If you have to start over a couple of times, that's okay.

If you got the right answer the first or second time, pat yourself on the back. The 2-by-2 multiplication problems really do not get any tougher than this. If you did not get the answer right away, don't worry. In the next two sections, I'll teach you some much easier strategies for dealing with problems like this. But before you read on, practice the addition method on the following multiplication problems.
The Subtraction Method
The subtraction method really comes in handy when one of the numbers you want to multiply ends in 8 or 9. The following problem illustrates what I mean:

Although most people find addition easier than subtraction, it is usually easier to subtract a small number than to add a big number. (If we had done this problem by the addition method, we would have added 850 + 153 = 1003.)

Now let's do the challenging problem from the end of the last section:

Wasn't that a whole lot easier? Now, here's a problem where one number ends in 8:

In this case you should treat 88 as 90 − 2, then multiply 90 × 23 = 2070. But you multiplied by too much. How much? By 2 × 23, or 46 too much. So subtract 46 from 2070 to arrive at 2024, the final answer.

I want to emphasize here that it is important to work out these problems in your head and not simply look to see how we did it in the diagram. Go through them and say the steps to yourself or even out loud to reinforce your thoughts.

Not only do I use the subtraction method with numbers that end in 8 or 9, but also for numbers in the high 90s because 100 is such a convenient number to multiply. For example, if someone asked me to multiply 96 × 73, I would immediately round up 96 to 100:

When the subtraction component of a multiplication problem requires you to borrow a number, using complements (as we learned in Chapter 1) can help you arrive at the answer more quickly. You'll see what I mean as you work your way through the problems below. For example, subtract 340 − 78. We know the answer will be in the 200s. The difference between 40 and 78 is 38. Now take the complement of 38 to get 62. And that's the answer, 262!

Now let's try another problem:

There are two ways to perform the subtraction component of this problem. The "long" way subtracts 200 and adds back 48:

The short way is to realize that the answer will be 66 hundred and something. To determine something, we subtract 52 − 40 = 12 and then find the complement of 12, which is 88. Hence the answer is 6688.

Try this one.

Again, you can see that the answer will be 3900 and something. Because 67 − 20 = 47, the complement 53 means the answer is 3953.

As you may have realized, you can use this method with any subtraction problem that requires you to borrow a number, not just those that are part of a multiplication problem. All of this is further proof, if you need it, that complements are a very powerful tool in mathemagics. Master this technique and, pretty soon, people will be complimenting you!
The Factoring Method
The factoring method is my favorite method of multiplying two-digit numbers since it involves no addition or subtraction at all. You use it when one of the numbers in a two-digit multiplication problem can be factored into one-digit numbers.

To factor a number means to break it down into one-digit numbers that, when multiplied together, give the original number. For example, the number 24 can be factored into 8 × 3 or 6 × 4. (It can also be factored into 12 × 2, but we prefer to use only single-digit factors.)

Here are some other examples of factored numbers:

42 = 7 × 6
63 = 9 × 7
84 = 7×6×2 or 7×4×3

To see how factoring makes multiplication easier, consider the following problem:

Previously we solved this problem by multiplying 46 × 40 and 46 × 2 and adding the products together. To use the factoring method, treat 42 as 7 × 6 and begin by multiplying 46 × 7, which is 322. Then multiply 322 × 6 for the final answer of 1932. You already know how to do 2-by-1 and 3-by-1 multiplication problems, so this should not be too hard:

46 × 42 = 46 × (7 × 6) = (46 × 7) × 6 = 322 × 6 = 1932

Of course, this problem could also have been solved by reversing the factors of 42:

46 × 42 = 46 × (6 × 7) = (46 × 6) × 7 = 276 × 7 = 1932

In this case, it is easier to multiply 322 × 6 than it is to multiply 276 × 7. In most cases, I like to use the larger factor in solving the initial 2-by-1 problem and to reserve the smaller factor for the 3-by-1 component of the problem.

Factoring results in a 2-by-2 multiplication problem being simplified to an easier 3-by-1 (or sometimes 2-by-1) multiplication problem. The advantage of the factoring method in mental calculation is you do not have to hold much in memory. Let's look at another example, 75 × 63:

75 × 63 = 75 × (9 × 7) = (75 × 9) × 7 = 675 × 7 = 4725

As before, you simplify this 2-by-2 problem by factoring 63 into 9 × 7 and then multiplying 75 by these factors. (By the way, the reason we can shift parentheses in the second step is the associative law of multiplication.)

63 × 75 = 63 × (5 × 5 × 3) = (63 × 5) × 5 × 3 = 315 × 5 × 3 = 1575 × 3 = 4725

Try the following problem for practice:

57 × 24 = 57 × 8 × 3 = 456 × 3 = 1368

You could have factored 24 as 6 × 4 for another easy computation:

57 × 24 = 57 × 6 × 4 = 342 × 4 = 1368

Compare this approach with the addition method:

With the addition method, you have to perform two 2-by-1 problems and then add. With the factoring method, you have just two multiplication problems: a 2-by-1 and a 3-by-1, and then you are done. The factoring method is usually easier on your memory.

Remember that challenging multiplication problem earlier in this chapter? Here it is again:

We tackled that problem easily enough with the subtraction method, but factoring works even faster:

89 × 72 = 89 × 9 × 8 = 801 × 8 = 6408

The problem is especially easy because of the 0 in the middle of 801. Our next example illustrates that it sometimes pays to factor the numbers in an order that exploits this situation. Let's look at two ways of computing 67 × 42:

67 × 42 = 67 × 7 × 6 = 469 × 6 = 2814
67 × 42 = 67 × 6 × 7 = 402 × 7 = 2814

Ordinarily you should factor 42 into 7 × 6, as in the first example, following the rule of using the larger factor first. But the problem is easier to solve if you factor 42 into 6 × 7 because it creates a number with a 0 in the center, which is easier to multiply. I call such numbers friendly products.

Look for the friendly product in the problem done two ways below:

43 × 56 = 43 × 8 × 7 = 344 × 7 = 2408
43 × 56 = 43 × 7 × 8 = 301 × 8 = 2408

Did you think the second way was easier?

When using the factoring method, it pays to find friendly products whenever you can. The following list should help. I don't expect you to memorize it so much as to familiarize yourself with it. With practice you will be able to nose out friendly products more often, and the list will become more meaningful.
Numbers with Friendly Products
12: 12 × 9= 108

13: 13 × 8= 104

15: 15 × 7= 105

17: 17 × 6= 102

18: 18 × 6 = 108

21: 21 × 5 = 105

23: 23 × 9 = 207

25: 25 × 4 = 100, 25 × 8 = 200

26: 26 × 4 = 104, 26 × 8 = 208

27: 27 × 4 = 108

29: 29 × 7 = 203

34: 34 × 3 = 102, 34 × 6 = 204, 34 × 9 = 306

35: 35 × 3 = 105

36: 36 × 3 = 108

38: 38 × 8 = 304

41: 41 × 5 = 205

43: 43 × 7 = 301

44: 44 × 7 = 308

45: 45 × 9 = 405

51:51 × 2 = 102, 51 × 4 = 204, 51 × 6 = 306, 51 × 8 = 408

52: 52 × 2 = 104, 52 × 4 = 208

53: 53 × 2 = 106

54: 54 × 2 = 108

56: 56 × 9 = 504

61: 61 × 5 = 305

63: 63 × 8 = 504

67: 67 × 3 = 201, 67 × 6 = 402, 67 × 9 = 603

68: 68 × 3 = 204, 68 × 6 = 408

69: 69 × 3 = 207

72: 72 × 7 = 504

76: 76 × 4 = 304, 76 × 8 = 608

77: 77 × 4 = 308

78: 78 × 9 = 702

81: 81 × 5 = 405

84: 84 × 6 = 504

86: 86 × 7 = 802

88: 88 × 8 = 704

89: 89 × 9 = 801

Previously in this chapter you learned how easy it is to multiply numbers by 11. It usually pays to use the factoring method when one of the numbers is a multiple of 11, as in the examples below:

52 × 33 = 52 × 11 × 3 = 572 × 3 = 1716
83 × 66 = 83 × 11 × 6 = 913 × 6 = 5478
APPROACHING MULTIPLICATION CREATIVELY
I mentioned at the beginning of the chapter that multiplication problems are fun because they can be solved any number of ways. Now that you know what I mean, let's apply all three methods explained in this chapter to a single problem, 73 × 49. We'll begin by using the addition method:

Now try the subtraction method:

Note that the last two digits of the subtraction could be obtained by adding 50 + (complement of 73) = 50 + 27 = 77 or by simply taking the complement of (73 − 50) = complement of 23 = 77.

Finally, try the factoring method:

73 × 49 = 73 × 7 × 7 = 511 × 7 = 3577

Congratulations! You have mastered 2-by-2 multiplication and now have all the basic skills you need to be a fast mental calculator. All you need to become a lightning calculator is more practice!

EXERCISE: 2-BY-2 GENERAL MULTIPLICATION-ANYTHING GOES!

Many of the following exercises can be solved by more than one method. Try computing them in as many ways as you can think of, then check your answers and computations at the back of the book. Our answers suggest various ways the problem can be mathemagically solved, starting with what I think is the easiest method.

The following 2-by-2s occur as subproblems to problems appearing later when we do 3-by-2s, 3-by-3s, and 5-by-5s. You can do these problems now for practice, and refer back to them when they are used in the larger problems.
THREE-DIGIT SQUARES
Squaring three-digit numbers is an impressive feat of mental prestidigitation. Just as you square two-digit numbers by rounding up or down to the nearest multiple of 10, to square three-digit numbers, you round up or down to the nearest multiple of 100. Take 193:

By rounding up to 200 and down to 186, you've transformed a 3-by-3 multiplication problem into a far simpler 3-by-1 problem. After all, 200 × 186 is just 2 × 186 = 372 with two zeros attached. Almost done! Now all you have to add is 72 = 49 to arrive at 37,249.

Now try squaring 706:

Rounding down by 6 to 700 requires you to round up by 6 to 712. Since 712 × 7 = 4984 (a simple 3-by-1 problem), 712 × 700 = 498,400. After adding 62 = 36, you arrive at 498,436.

These last problems are not terribly hard because there is no real addition involved. Moreover, you know the answers to 62 and 72 by heart. Squaring a number that's farther away from a multiple of 100 is a tougher proposition. Try your hand at 3142:

For this three-digit square, go down 14 to 300 and up 14 to 328, then multiply 328 × 3 = 984. Tack on two 0s to arrive at 98,400. Then add the square of 14. If 142 = 196 comes to you in a flash (through memory or calculation), you're in good shape. Just add 98,400 + 196 to arrive at 98,596. If you need time to compute 142, repeat the number 98,400 to yourself a few times before you go on. (Otherwise you might compute 142 = 196 and forget what number to add it to.)

The farther away you get from a multiple of 100, the more difficult squaring a three-digit number becomes. Try 5292:

If you have an audience you want to impress, you can say 279,000 out loud before you compute 292. But this will not work for every problem. For instance, try squaring 636:

Now your brain is really working, right? The key here is to repeat 403,200 to yourself several times. Then square 36 to get 1,296 in the usual way. The hard part comes in adding 1,296 to 403,200. Do it one digit at a time, left to right, to arrive at your answer of 404,496. Take my word that as you become more familiar with two-digit squares, these three-digit problems get easier.

Here's an even tougher problem, 8632:

The first problem is deciding what numbers to multiply together. Clearly one of the numbers will be 900, and the other number will be in the 800s. But what number? You can compute it two ways:

1. The hard way: the difference between 863 and 900 is 37 (the complement of 63). Subtract 37 from 863 to arrive at 826.

2. The easy way: double the number 63 to get 126, and take the last two digits to give you 826.

Here's why the easy way works. Because both numbers are the same distance from 863, their sum must be twice 863, or 1726. One of your numbers is 900, so the other must be 826.

You then compute the problem like this:

If you find it impossible to remember 743,400 after squaring 37, fear not. In a later chapter, you will learn a memory system that will make remembering such numbers much easier.

Try your hand at squaring 359, the hardest problem yet:

To obtain 318, either subtract 41 (the complement of 59) from 359, or multiply 2 × 59 = 118 and use the last two digits. Next multiply 400 × 318 = 127,200. Adding 412, or 1,681, gives you 128,881. Whew! They don't get much harder than that! If you got it right the first time, take a bow!

Let's finish this section with a big problem that is easy to do, 9872:

What's Behind Door Number 1?

The mathematical chestnut of 1991 that got everyone hopping mad was an article in Parade magazine by Marilyn vos Savant, the woman listed by the Guinness Book of World Records as having the world's highest IQ. The paradox has come to be known as the Monty Hall problem, and it goes like this.

You are a contestant on Let's Make a Deal. Monty Hall allows you to pick one of three doors; behind one of these doors is the big prize, behind the other two are goats. You pick Door Number 2. But before Monty reveals the prize of your choice, he shows you what you didn't pick behind Door Number 3. It's a goat. Now, in his tantalizing way, Monty gives you another choice. Do you want to stick with Door Number 2, or do you want to risk a chance to see what's behind Door Number 1? What should you do? Assuming that Monty is only going to reveal where the big prize is not, he will always open one of the consolation doors. This leaves two doors, one with the big prize and the other with another consolation. The odds are now 50-50 for your choice, right?

Wrong! The odds that you chose correctly the first time remain 1 in 3. The probability that the big prize is behind the other door increases to 2 in 3 because the probability must add to 1.

Thus, by switching doors, you double the odds of winning! (The problem assumes that Monty will always give a player the option to switch, that he will always reveal a nonwinning door, and that when your first pick is correct he will choose a nonwinning door at random.) Think of playing the game with ten doors and after your pick he reveals eight other nonwinning doors. Here, your instincts would probably tell you to switch. People confuse this problem for a variant: if Monty Hall does not know where the grand prize is, and reveals Door Number 3, which happens to contain a goat (though it might have contained the prize), then Door Number 1 has a 50 percent chance of being correct. This result is so counterintuitive that Marilyn vos Savant received piles of letters, many from scientists and even mathematicians, telling her she shouldn't write about math. They were all wrong.
CUBING
We end this chapter with a new method for cubing two-digit numbers. (Recall that the cube of a number is that number multiplied by itself twice. For example, 5 cubed-denoted 53-is equal to 5 × 5 × 5 = 125.) As you will see, this is not much harder than multiplying two-digit numbers. The method is based on the algebraic observation that

A3 = (A - d)A(A + d) + d2A

where d is any number. Just like with squaring two-digit numbers, I choose d to be the distance to the nearest multiple of ten. For example, when squaring 13, we let d = 3, resulting in:

133 = (10 × 13 × 16) + (32 × 13)

Since 13 × 16 = 13 × 4 × 4 = 52 × 4 = 208, and 9 × 13 = 117, we have

133 = 2080 + 117 = 2197

How about the cube of 35? Letting d = 5, we get

353 = (30 × 35 × 40) + (52 × 35)

Since 30 × 35 × 40 = 30 × 1,400 = 42,000 and 35 × 5 × 5 = 175 × 5 = 875, we get

353 = 42,000 + 875 = 42,875

When cubing 49, we let d = 1 in order to round up to 50. Here

493 = (48 × 49 × 50) + (12 × 49)

We can solve 48 × 49 by the factoring method, but for this sort of problem, I prefer to use the close-together method, which will be described in Chapter 8. (Go ahead and look at it now, if you'd like!) Using that method, we get 48 × 49 = (50 × 47) + (1 × 2) = 2352. Multiplying this by 50, we get 117,600 and therefore

493 = 117,600 + 49 = 117,649

Here's a larger one. Try the cube of 92.

923 = (90 × 92 × 94) + (22 × 92)

If you are fast at squaring two-digit numbers, then you could solve 92 × 94 = 932 − 1 = 8648, or you could use the close-together method, resulting in 92 × 94 = (90 × 96) + (2 × 4) = 8648. Multiplying this by 9 (as described in the beginning of Chapter 8), we obtain 9 × (8,600 + 48) = 77,400 + 432 = 77,832, and therefore 90 × 92 × 94 = 778,320. Since 4 × 92 = 368, we get

923 = 778,320 + 368 = 778,688

We note that when the close-together method is used for the multiplication problems that arise when cubing two-digit numbers, the small product being added will either be (depending on whether d = 1, 2, 3, 4, or 5) 1 × 2 = 2, 2 × 4 = 8, 3 × 6 = 18, 4 × 8 = 32, or 5 × 10 = 50. Let's finish with the cube of 96.

963 = (92 × 96 × 100) + (42 × 96)

The product 92 × 96 = 8,832 can be done many different ways. To celebrate the end of this chapter, let's do some of them. I'll start with what I consider to be the hardest way, and end with what I consider the easiest way. By the addition method, (90 + 2) × 96 = 8,640 + 192 = 8,832; by the subtraction method, 92 × (100 − 4) = 9,200 − 368 = 8,832; by the factoring method, 92 × 6 × 4 × 4 = 552 × 4 × 4 = 2,208 × 4 = 8,832; by squaring, 942 − 22 = 8,836 − 4 = 8,832; by the close-together method with a base of 90, (90 × 98) + (2 × 6) = 8,820 + 12 = 8,832; and by the close-together method with a base of 100, (100 × 88) + (−8 × −4) = 8,800 + 32 = 8,832.

The product 42 × 96 = 1,536 can also be done a few different ways, such as 96 × 4 × 4 = 384 × 4 = 1,536, or as 16 × (100 − 4) = 1,600 − 64 = 1,536. Finally, since 8,832 × 100 = 883,200, we have

963 = 883,200 + 1,536 = 884,736
Chapter 4 Divide and Conquer:
Mental Division
Mental division is a particularly handy skill to have, both in business and in daily life. How many times a week are you confronted with situations that call on you to evenly divide things, such as a check at a restaurant? This same skill comes in handy when you want to figure out the cost per unit of a case of dog food on sale, or to split the pot in poker, or to figure out how many gallons of gas you can buy with a $20 bill. The ability to divide in your head can save you the inconvenience of having to pull out a calculator every time you need to compute something.

With mental division, the left-to-right method of calculation comes into its own. This is the same method we all learned in school, so you will be doing what comes naturally. I remember as a kid thinking that this left-to-right method of division is the way all arithmetic should be done. I have often speculated that if the schools could have figured out a way to teach division right-to-left, they probably would have done so!
ONE-DIGIT DIVISION
The first step when dividing mentally is to figure out how many digits will be in your answer. To see what I mean, try on the following problem for size:

179 ÷ 7

To solve 179 ÷ 7, we're looking for a number, Q, such that 7 times Q is 179. Now, since 179 lies between 7 × 10 = 70 and 7 × 100 = 700, Q must lie between 10 and 100, which means our answer must be a two-digit number. Knowing that, we first determine the largest multiple of 10 that can be multiplied by 7 whose answer is below 179. We know that 7 × 20 = 140 and 7 × 30 = 210, so our answer must be in the twenties. At this point we can actually say the number 20 since that part of our answer will certainly not change. Next we subtract 179 − 140 = 39. Our problem has now been reduced to the division problem 39 ÷ 7. Since 7 × 5 = 35, which is 4 away from 39, we have the rest of our answer, namely 5 with a remainder of 4, or 5 and . Altogether, we have our answer, 25 with a remainder of 4, or if you prefer, 25 . Here's what the process looks like:

Let's try a similar division problem using the same method of mental computation:

675 ÷ 8

As before, since 675 falls between 8 × 10 = 80 and 8 × 100 = 800, your answer must be below 100 and therefore is a two-digit number. To divide 8 into 675, notice that 8 × 80 = 640 and 8 × 90 = 720. Therefore, your answer is 80 something. But what is that "something"? To find out, subtract 640 from 675 for a remainder of 35. After saying the 80, our problem has been reduced to 35 ÷ 8. Since 8 × 4 = 32, the final answer is 84 with a remainder of 3, or 84⅜. We illustrate this problem as follows:

Like most mental calculations, division can be thought of as a process of simplification. The more you calculate, the simpler the problem becomes. What began as 675 ÷ 8 was simplified to a smaller problem, 35 ÷ 8.

Now let's try a division problem that results in a three-digit answer:

947 ÷ 4

This time, your answer will have three digits because 947 falls between 4 × 100 = 400 and 4 × 1000 = 4000. Thus we must first find the largest multiple of 100 that can be squeezed into 947. Since 4 × 200 = 800, our answer is definitely in the 200s, so go ahead and say it! Subtracting 800 from 947 gives us our new division problem, 147 ÷ 4. Since 4 × 30 = 120, we can now say the 30. After subtracting 120 from 147, we compute 27 ÷ 4 to obtain the rest of the answer: 6 with a remainder of 3. Altogether, we have 236 with a remainder of 3, or 236¾.

The process is just as easy when dividing a one-digit number into a four-digit number, as in our next example.

2196 ÷ 5

Here the answer will be in the hundreds because 2196 is between 5 × 100 = 500 and 5 × 1000 = 5000. After subtracting 5 × 400 = 2000, we can say the 400, and our problem has reduced to 196 ÷ 5, which can be solved as in the previous examples.

Actually, there is a much easier way to solve this last problem. We can simplify our problem by doubling both numbers. Since 2196 × 2 = 4392, we have 2196 ÷ 5 = 4392 ÷ 10 = 439.2 or . We'll see more division shortcuts in the next section.
THE RULE OF "THUMB"
When dividing in your head instead of on paper, you may find it difficult to remember parts of the answer as you continue to calculate. One option, as you've seen, is to say the answer out loud as you go. But for greater dramatic effect, you may prefer, as I do, to hold the answer on your fingers and say it all together at the end. In that case, you may run into problems remembering digits greater than five if, like most of us, you have only five fingers on each hand. The solution is to use a special technique, based on sign language, which I call the Rule of "Thumb." It is most effective for remembering three-digit and greater numbers. This technique not only is useful in this chapter but also will come in handy (pardon the pun) in subsequent chapters dealing with larger problems and longer numbers to remember.

You already know that to represent numbers 0 through 5, all you have to do is raise the equivalent number of fingers on your hand. When you get your thumb involved, it's just as easy to represent numbers 6 through 9. Here are the Rules of "Thumb":

* To hold on to 6, place your thumb on top of your pinky.

* To hold on to 7, place your thumb on top of your ring finger.

* To hold on to 8, place your thumb on top of your middle finger.

* To hold on to 9, place your thumb on top of your index finger.

With three-digit answers, hold the hundreds digit on your left hand and the tens digit on your right. When you get to the ones digit, you've reached the end of the problem (except for a possible remainder). Now say the number on your left hand, the number on your right hand, the one-digit you've just computed, and the remainder (in your head). Presto-you've said your answer!

For practice, try to compute the following four-digit division problem:

In using the Rule of "Thumb" to remember the answer, you'll hold the 7 on your left hand by placing your thumb and ring finger together and the 6 on your right hand by placing your thumb and little finger together. Once you've calculated the ones digit (which is 3) and the remainder (which is 1), you can "read" the final answer off your hands from left to right: "seven ... six ... three with a remainder of one, or one-sixth."

Some four-digit division problems yield four-digit answers. In that case, since you have only two hands, you will have to say the thousands digit of the answer out loud and use the rule of thumb to remember the rest of the answer. For example:

8352 ÷ 3

For this problem, you divide 3 into 8 to get your thousands digit of 2, say "two thousand" out loud, then divide 3 into 2352 in the usual way.
TWO-DIGIT DIVISION
This section assumes you already have mastered the art of dividing by a one-digit number. Naturally, division problems become harder as the number you divide by gets larger. Fortunately, I have some magic up my sleeve to make your life easier.

Let's start with a relatively easy problem first:

597 ÷ 14

Since 597 lies between 14 × 10 and 14 × 100, the answer (also called the quotient) lies between 10 and 100. To determine the answer, your first step is to ask how many times 14 goes into 590. Because 14 × 40 = 560, you know that the answer is 40 something, and so you can say "forty" out loud.

Next, subtract 560 from 597, which is 37 and reduces your problem to dividing 14 into 37. Since 14 × 2 = 28, your answer is 42. Subtracting 28 from 37 leaves you a remainder of 9. The process of deriving the solution to this problem may be illustrated as follows:

The following problem is slightly harder because the two-digit divisor (here 23) is larger.

682 ÷ 23

In this problem, the answer is a two-digit number because 682 falls between 23 × 10 = 230 and 23 × 100 = 2300. To figure out the tens digit of the two-digit answer, you need to ask how many times 23 goes into 680. If you try 30, you'll see that it's slightly too much, as 30 × 23 = 690. Now you know that the answer is 20 something, and you can say so. Then subtract 23 × 20 = 460 from 682 to obtain 222. Since 23 × 9 = 207, the answer is 29 with a remainder of 222 − 207 = 15.

Now consider:

491 ÷ 62

Since 491 is less than 62 × 10 = 620, your answer will simply be a one-digit number with a remainder. You might guess 8, but 62 × 8 = 496, which is a little high. Since 62 × 7 = 434, the answer is 7 with a remainder of 491 − 434 = 57, or

Actually, there's a nifty trick to make problems like this easier. Remember how you first tried multiplying 62 × 8, but found it came out a little high at 496? Well, that wasn't a wasted effort. Aside from knowing that the answer is 7, you can also compute the remainder right away. Since 496 is 5 more than 491, the remainder will be 5 less than 62, the divisor. Since 62 − 5 = 57, your answer is 7 . The reason this trick works is because 491 = (62 × 8) − 5 = 62 × (7 + 1) − 5 = (62 × 7 + 62) − 5 = (62 × 7) + (62 − 5) = 62 × 7 + 57.

Now try 380 ÷ 39 using the shortcut we just learned. So 39 × 10 = 390, which is too high by 10. Hence the answer is 9 with a remainder of 39 − 10 = 29.

Your next challenge is to divide a two-digit number into a four-digit number:

3657 ÷ 54

Since 54 × 100 = 5400, you know your answer will be a two-digit number. To arrive at the first digit of the answer, you need to figure how many times 54 goes into 3657. Since 54 × 70 = 3780 is a little high, you know the answer must be 60 something.

Next, multiply 54 × 60 = 3240 and subtract 3657 − 3240 = 417. Once you say the 60, your problem has been simplified to 417 ÷ 54. Since 54 × 8 = 432 is a little too high, your last digit is 7 with the remainder 54 − 15 = 39.

Now try your hand at a problem with a three-digit answer:
Simplifying Division Problems
If by this point you're suffering from brain strain, relax. As promised, I want to share with you a couple of tricks for simplifying certain mental division problems. These tricks are based on the principle of dividing both parts of the problem by a common factor. If both numbers in the problem are even numbers, you can make the problem twice as easy by dividing each number by 2 before you begin. For example, 858 ÷ 16 has two even numbers, and dividing each by 2 yields the much simpler problem of 429 ÷ 8:

As you can see, the remainders 10 and 5 are not the same; but if you write the remainder in the form of a fraction, you get that is the same as ⅝. Therefore, when using this method, you must always express the answer in fractional form.

We've done both sets of calculations for you to see how much easier it is. Now you try one for practice:

The problem on the right is much easier to calculate mentally. If you're really alert, you could divide both sides of the original problem by 18 to arrive at an even simpler problem: 201 ÷ 3 = 67.

Watch for problems that can be divided by 2 twice, such as 1652 ÷ 36:

I usually find it easier to divide the problem by 2 twice than to divide both numbers by 4. Next, when both numbers end in 0, you can divide each by 10:

But if both numbers end in 5, double them and then divide both by 10 to simplify the problem. For example:

Finally, if the divisor ends in 5 and the number you're dividing into ends in 0, multiply both by 2 and then divide by 10, just as you did above:

EXERCISE:TWO-DIGIT DIVISION

Here you'll find a variety of two-digit division problems that will test your mental prowess, using the simplification techniques explained earlier in this chapter. Check the end of the book for answers and explanations.
MATCHING WITS WITH A CALCULATOR:
LEARNING DECIMALIZATION
As you might guess, I like to work some magic when I convert fractions to decimals. In the case of one-digit fractions, the best way is to commit the following fractions-from halves through elevenths-to memory. This isn't as hard as it sounds. As you'll see below, most one-digit fractions have special properties that make them hard to forget. Anytime you can reduce a fraction to one you already know, you'll speed up the process.

Chances are you already know the decimal equivalent of the following fractions:

Likewise:

The fifths are easy to remember:

The sixths require memorizing only two new answers:

I'll return to the sevenths in a moment. The eighths are a breeze:

The ninths have a magic all their own:

where the bar indicates that the decimal repeats. For instance, The tenths you already know:

For the elevenths, if you remember that = .0909, the rest is easy:

The sevenths are truly remarkable. Once you memorize , you can get all the other sevenths without having to compute them:

Note that the same pattern of numbers repeats itself in each fraction. Only the starting point varies. You can figure out the starting point in a flash by multiplying .14 by the numerator. In the case of , 2 × .14 = .28, so use the sequence that begins with the 2, namely .285714. Likewise with , since 3 × .14 = .42, use the sequence that begins with 4, namely .428571. The rest follow in a similar way.

You will have to calculate fractions higher than as you would any other division problem. However, keep your eyes peeled for ways of simplifying such problems. For example, you can simplify the fraction by dividing both numbers by 2, to reduce it to , which is easier to compute.

If the denominator of the fraction is an even number, you can simplify the fraction by reducing it in half, even if the numerator is odd. For example:

Dividing the numerator and denominator in half reduces it to a sevenths fraction. Although the sevenths sequence previously shown doesn't provide the decimal for , once you begin the calculation, the number you memorized will pop up:

As you can see, you needn't work out the entire problem. Once you've reduced it to dividing 3 by 7, you can make a great impression on an audience by rattling off this long string of numbers almost instantly!

When the divisor ends in 5, it almost always pays to double the problem, then divide by 10. For example,

Numbers that end in 25 or 75 should be multiplied by 4 before dividing by 100.

You can even put this trick to use in the middle of a problem. If your fraction is look what happens:

Once the problem is reduced to , you can further reduce it to , which you know to be .875. Thus = .1875.

To solve the following problems, don't forget to employ the various one-digit fractions you already know as decimals. Wherever appropriate, simplify the fraction before converting it to a decimal.
TESTING FOR DIVISIBILITY
In the last section, we saw how division problems could be simplified when both numbers were divisible by a common factor. We end this chapter with a brief discussion of how to determine whether one number is a factor of another number. Being able to find the factors of a number helps us simplify division problems and can speed up many multiplication problems. This will also be a very useful tool when we get to advanced multiplication, as you will often be looking for ways to factor a two-, three-, or even a five-digit number in the middle of a multiplication problem. Being able to factor these numbers quickly is very handy. And besides, I think some of the rules are just beautiful.

It's easy to test whether a number is divisible by 2. All you need to do is to check if the last digit is even. If the last digit is 2, 4, 6, 8, or 0, the entire number is divisible by 2.

To test whether a number is divisible by 4, check if the two-digit number at the end is divisible by 4. The number 57,852 is a multiple of 4 because 52 = 13 × 4. The number 69,346 is not a multiple of 4 because 46 is not a multiple of 4. The reason this works is because 4 divides evenly into 100 and thus into any multiple of 100. Thus, since 4 divides evenly into 57,800, and 4 divides into 52, we know that 4 divides evenly into their sum, 57,852.

Likewise, since 8 divides into 1000, to test for divisibility by 8, check the last three digits of the number. For the number 14,918, divide 8 into 918. Since this leaves you with a remainder (918 ÷ 8 = 114 ), the number is not divisible by 8. You could also have observed this by noticing that 18 (the last two digits of 14,918) is not divisible by 4, and since 14,918 is not divisible by 4, it can't be divisible by 8 either.

When it comes to divisibility by 3, here's a cool rule that's easy to remember: A number is divisible by 3 if and only if the sum of its digits are divisible by 3-no matter how many digits are in the number. To test whether 57,852 is divisible by 3, simply add 5 + 7 + 8 + 5 + 2 = 27. Since 27 is a multiple of 3, we know 57,852 is a multiple of 3. The same amazing rule holds true for divisibility by 9. A number is divisible by 9 if and only if its digits sum to a multiple of 9. Hence, 57,852 is a multiple of 9, whereas 31,416, which sums to 15, is not. The reason this works is based on the fact that the numbers 1, 10, 100, 1,000, 10,000, and so on, are all 1 greater than a multiple of 9.

A number is divisible by 6 if and only if it is even and divisible by 3, so it is easy to test divisibility by 6.

To establish whether a number is divisible by 5 is even easier. Any number, no matter how large, is a multiple of 5 if and only if it ends in 5 or 0.

Establishing divisibility by 11 is almost as easy as determining divisibility by 3 or 9. A number is divisible by 11 if and only if you arrive at either 0 or a multiple of 11 when you alternately subtract and add the digits of a number. For instance, 73,958 is not divisible by 11 since 7−3 + 9 − 5 + 8 = 16. However, the numbers 8,492 and 73,194 are multiples of 11, since 8 − 4 + 9 − 2 = 11 and 7 − 3 + 1 − 9 + 4 = 0. The reason this works is based, like the rule for 3s and 9s, on the fact that the numbers 1, 100, 10,000, and 1,000,000 are 1 more than a multiple of 11, whereas the numbers 10, 1,000, 100,000, and so on are 1 less than a multiple of 11.

Testing for divisibility by 7 is a bit trickier. If you add or subtract a number that is a multiple of 7 to the number you are testing, and the resulting number is a multiple of 7, then the test is positive. I always choose to add or subtract a multiple of 7 so that the resulting sum or difference ends in 0. For example, to test the number 5292, I subtract 42 (a multiple of 7) to obtain 5250. Next, I get rid of the 0 at the end (since dividing by ten does not affect divisibility by seven), leaving me with 525. Then I repeat the process by adding 35 (a multiple of 7), which gives me 560. When I delete the 0, I'm left with 56, which I know to be a multiple of 7. Therefore, the original number 5292 is divisible by 7.

This method works not only for 7s, but also for any odd number that doesn't end in 5. For example, to test whether 8792 is divisible by 13, subtract 4 × 13 = 52 from 8792, to arrive at 8740. Dropping the 0 results in 874. Then add 2 × 13 = 26 to arrive at 900. Dropping the two 0s leaves you with 9, which is clearly not a multiple of 13. Therefore, 8792 is not a multiple of 13.

In this final set of exercises, be especially careful when you test for divisibility by 7 and 17. The rest should be easy for you.
FRACTIONS
If you can manipulate whole numbers, then doing arithmetic with fractions is almost as easy. In this section, we review the basic methods for adding, subtracting, multiplying, dividing, and simplifying fractions. Those already familiar with fractions can skip this section with no loss of continuity.
Multiplying Fractions
To multiply two fractions, simply multiply the top numbers (called the numerators), then multiply the bottom numbers (called the denominators). For example,

What could be simpler! Try these exercises before going further.
Dividing Fractions
Dividing fractions is just as easy as multiplying fractions. There's just one extra step. First, turn the second fraction upside down (this is called the reciprocal) then multiply. For instance, the reciprocal of ⅘ is . Therefore,

Now it's your turn. Divide these fractions.
Simplifying Fractions
Fractions can be thought of as little division problems. For instance, is the same as 6 ÷ 3 = 2. The fraction is the same as 1 ÷ 4 (which is .25 in decimal form). Now we know that when we multiply any number by 1, the number stays the same. For example, = × 1. But if we replace 1 with , we get = × 1 = × = . Hence, = Likewise, if we replace 1 with , we get . In other words, if we multiply the numerator and denominator by the same number, we get a fraction that is equal to the first fraction.

For another example,

It is also true that if we divide the numerator and denominator by the same number, then we get a fraction that is equal to the first one.

For instance,

This is called simplifying the fraction.

For the fractions below, can you find an equal fraction whose denominator is 12?

Simplify these fractions.
Adding Fractions
The easy case is that of equal denominators. If the denominators are equal, then we add the numerators and keep the same denominators.

For instance,

Sometimes we can simplify our answer. For instance,

The trickier case: unequal denominators. When the denominators are not equal, then we replace our fractions with fractions where the denominators are equal.

For instance, to add

we notice that

Therefore,

To add

we notice that

So,

To add

we see that

So,
Subtracting Fractions
Subtracting fractions works very much like adding them. We have illustrated with examples and provided exercises for you to do.
Chapter 5 Good Enough: The Art
of "Guesstimation"
So far you've been perfecting the mental techniques necessary to figure out the exact answers to math problems. Often, however, all you'll want is a ballpark estimate. Say you're getting quotes from different lenders on refinancing your home. All you really need at this information-gathering stage is a ballpark estimate of what your monthly payments will be. Or say you're settling a restaurant bill with a group of friends and you don't want to figure each person's bill to the penny. The guesstimation methods described in this chapter will make both these tasks-and many more just like them-much easier. Addition, subtraction, division, and multiplication all lend themselves to guesstimation. As usual, you'll do your computations from left to right.
ADDITION GUESSTIMATION
Guesstimation is a good way to make your life easier when the numbers of a problem are too long to remember. The trick is to round the original numbers up or down:

George Parker Bidder: The Calculating Engineer

The British have had their share of lightning calculators, and the mental performances of George Parker Bidder (1806-1878), born in Devonshire, were as impressive as any. Like most lightning calculators, Bidder began to try his hand (and mind) at mental arithmetic as a young lad. Learning to count, add, subtract, multiply, and divide by playing with marbles, Bidder went on tour with his father at age nine.

Almost no question was too difficult for him to handle. "If the moon is 123,256 miles from the earth and sound travels four miles a minute, how long would it take for sound to travel from the earth to the moon?" The young Bidder, his face wrinkled in thought for nearly a minute, replied, "Twenty-one days, nine hours, thirty-four minutes." (We know now that the distance is closer to 240,000 miles and sound cannot travel through the vacuum of space.) At age ten, Bidder mentally computed the square root of 119,550,669,121 as 345,761 in a mere thirty seconds. In 1818, Bidder and the American lightning calculator Zerah Colburn were paired in a mental calculating duel in which Bidder, apparently, "outnumbered" Colburn.

Riding on his fame, George Bidder entered the University of Edinburgh and went on to become one of the more respected engineers in England. In parliamentary debates over railroad conflicts, Bidder was frequently called as a witness, which made the opposition shudder; as one said, "Nature had endowed him with particular qualities that did not place his opponents on a fair footing." Unlike Colburn, who retired as a lightning calculator at age twenty, Bidder kept it up for his entire life. As late as 1878, in fact, just before his death, Bidder calculated the number of vibrations of light striking the eye in one second, based on the fact that there are 36,918 waves of red light per inch, and light travels at approximately 190,000 miles per second.

Notice that we rounded the first number down to the nearest thousand and the second number up. Since the exact answer is 14,186, our relative error is small.

If you want to be more exact, instead of rounding off to the nearest thousand, round off to the nearest hundred:

The answer is only 14 off from the exact answer, an error of less than .1%. This is what I call a good guesstimation!

Try a five-digit addition problem, rounding to the nearest hundred:

By rounding to the nearest hundred, our answer will always be off by less than 100. If the answer is larger than 10,000, your guesstimate will be within 1% of the exact answer.

Now let's try something wild:

If you round to the nearest million, you get an answer of 31 million, off by roughly 285,000. Not bad, but you can do better by rounding to the nearest hundred thousand, as we've shown in the right-hand column. Again your guesstimate will be within 1% of the precise answer. If you can compute these smaller problems exactly, you can guesstimate the answer to any addition problem.
Guesstimating at the Supermarket
Let's try a real-world example. Have you ever gone to the store and wondered what the total is going to be before the cashier rings it up? For estimating the total, my technique is to round the prices to the nearest 50¢. For example, while the cashier is adding the numbers shown below on the left, I mentally add the numbers shown on the right:

My final figure is usually within a dollar of the exact total.
SUBTRACTION GUESSTIMATION
The way to guesstimate the answers to subtraction problems is the same-you round to the nearest thousand or hundreds digit, preferably the latter:

You can see that rounding to the nearest thousand leaves you with an answer quite a bit off the mark. By rounding to the second digit (hundreds, in the example), your answer will usually be within 3% of the exact answer. For this problem, your answer is off by only 52, a relative error of 2%. If you round to the third digit, the relative error will usually be below 1%. For instance:

By rounding the numbers to the third digit rather than to the second digit, you improve the accuracy of the estimate by a significant amount.
DIVISION GUESSTIMATION
The first, and most important, step in guesstimating the answer to a division problem is to determine the magnitude of the answer:

The next step is to round off the larger numbers to the nearest thousand and change the 57,867 to 58,000. Dividing 6 into 58 gives you 9 with a remainder. But the most important component in this problem is where to place the 9.

For example, multiplying 6 × 90 yields 540, while multiplying 6 × 900 yields 5,400, both of which are too small. But 6 × 9,000 = 54,000, which is pretty close to the answer. This tells you the answer is 9,000 and something. You can estimate just what that something is by first subtracting 58 − 54 = 4. At this point you could bring down the 0 and divide 6 into 40, and so forth. But if you're on your toes, you'll realize that dividing 6 into 4 gives you .667. Since you know the answer is 9,000 something, you're now in a position to guess 9,667. In fact, the actual answer is 9,645-darn close!

Division on this level is simple. But what about large division problems? Let's say we want to compute, just for fun, the amount of money a professional athlete earns a day if he makes $5,000,000 a year:

First you must determine the magnitude of the answer. Does this player earn thousands every day? Well, 365 × 1000 = 365,000, which is too low.

Does he earn tens of thousands every day? Well, 365 × 10,000 = 3,650,000, and that's more like it. To guesstimate your answer, divide the first two digits (or 36 into 50) and figure that's or Since 18 goes into 70 about 4 times, your guess is that the athlete earns about $14,000. The exact answer is $13,698.63 per day. Not a bad estimate (and not a bad salary!).

Here's an astronomical calculation for you. How many seconds does it take light to get from the sun to the earth? Well, light travels at 186,282 miles per second, and the sun is (on average) 92,960,130 miles away. I doubt you're particularly eager to attempt this problem by hand. Fortunately, it's relatively simple to guesstimate an answer. First, simplify the problem:

Now divide 186 into 930, which yields 5 with no remainder. Then append the two 0s you removed from 93,000 and you get 500 seconds. The exact answer is 499.02 seconds, so this is a very respectable guesstimate.
MULTIPLICATION GUESSTIMATION
You can use much the same techniques to guesstimate your answers to multiplication problems. For example,

Rounding up to the nearest multiple of 10 simplifies the problem considerably, but you're still off by 252, or about 5%. You can do better if you round both numbers by the same amount, but in opposite directions. That is, if you round 88 by increasing 2, you should also decrease 54 by 2:

Instead of a 1-by-1 multiplication problem, you now have a 2-by-1 problem, which should be easy enough for you to do. Your guesstimation is off by only 1.5%.

When you guesstimate the answer to multiplication problems by rounding the larger number up and the smaller number down, your guesstimate will be a little low. If you round the larger number down and the smaller number up so that the numbers are closer together, your guesstimate will be a little high. The larger the amount by which you round up or down, the greater your guesstimate will be off from the exact answer. For example:

Since the numbers are closer together after you round them off, your guesstimate is a little high.

Since the numbers are farther apart, the estimated answer is too low, though again, not by much. You can see that this multiplication guesstimation method works quite well. Also notice that this problem is just 672 and that our approximation is just the first step of the squaring techniques. Let's look at one more example:

We observe that the approximation is most accurate when the original numbers are close together. Try estimating a 3-by-2 multiplication problem:

By rounding 63 down to 60 and 728 up to 731, you create a 3-by-1 multiplication problem, which puts your guesstimate within 2004 of the exact answer, an error of 4.3%.

Now try guesstimating the following 3-by-3 problem:

You will notice that although you rounded both numbers up and down by 8, your guesstimate is off by over 1000. That's because the multiplication problem is larger and the size of the rounding number is larger, so the resulting estimate will be off by a greater amount. But the relative error is still under 1%.

How high can you go with this system of guesstimating multiplication problems? As high as you want. You just need to know the names of large numbers. A thousand thousand is a million, and a thousand million is a billion. Knowing these names and numbers, try this one on for size:

As before, the objective is to round the numbers to simpler numbers such as 29,000,000 and 14,000. Dropping the 0s for now, this is just a 2-by-2 multiplication problem: 29 × 14 = 406 (29 × 14 = 29 × 7 × 2 = 203 × 2 = 406). Hence the answer is roughly 406 billion, since a thousand million is a billion.
SQUARE ROOT ESTIMATION: DIVIDE AND AVERAGE
The quantity , the square root of a number n, is the number which, when multiplied by itself, will give you n For example, the square root of 9 is 3 because 3 × 3 = 9. The square root is used in many science and engineering problems and is almost always solved with a calculator. The following method provides an accurate estimate of the answer.

In square root estimation your goal is to come up with a number that when multiplied by itself approximates the original number. Since the square root of most numbers is not a whole number, your estimate is likely to contain a fraction or decimal point.

Let's start by guesstimating the square root of 19. Your first step is to think of the number that when multiplied by itself comes closest to 19. Well, 4 × 4=16 and 5 × 5 = 25. Since 25 is too high, the answer must be 4 point something. Your next step is to divide 4 into 19, giving you 4.75. Now, since 4 × 4 is less than 4 × 4.75 = 19, which in turn is less than 4.75 × 4.75, we know that 19 (or 4 × 4.75) lies between 42 and 4.752. Hence, the square root of 19 lies between 4 and 4.75.

I'd guess the square root of 19 to be about halfway between, at 4.375. In fact, the square root of 19 (rounded to three decimal places) is 4.359, so our guesstimate is pretty close. We illustrate this procedure as follows:

Actually, we can obtain this answer another way, which you might find easier. We know 4 squared is 16, which is shy of 19 by 3. To improve our guess, we "add the error divided by twice our guess." Here, we add 3 divided by 8 to get 4 = 4.375. We note that this method will always produce an answer that is a little higher than the exact answer.

Now you try a slightly harder one. What's the square root of 87?

First come up with your ballpark figure, which you can get fairly quickly by noting that 9 × 9 = 81 and 10 × 10 = 100, which means the answer is 9 point something. Carrying out the division of 9 into 87 to two decimal places, you get 9.66. To improve your guesstimate, take the average of 9 and 9.66, which is 9.33-exactly the square root of 87 rounded to the second decimal place! Alternatively, our guesstimate is 9 + (error)/18 = .

Using this technique, it's pretty easy to guesstimate the square root of two-digit numbers. But what about three-digit numbers? Actually, they are not much harder. I can tell you right off the bat that all three-digit and four-digit numbers have two-digit square roots before the decimal point. And the procedure for computing square roots is the same, no matter how large the number. For instance, to compute the square root of 679, first find your ballpark figure. Because 20 squared is 400 and 30 squared is 900, the square root of 679 must lie between 20 and 30.

When you divide 20 into 679, you get approximately 34. Averaging 20 and 34 gives you a guesstimate of 27, but here's a better estimate. If you know that 25 squared is 625, then your error is 679 − 625 = 54. Dividing that by 50, we have = 1.08. Hence our improved guesstimate is 25 + 1.08 = 26.08. (For an even better estimate, if you know that 26 squared is 676, your error is 3, so add .06 to get 26.06.) The exact answer is 26.06, rounded to two decimal places.

To guesstimate the square root of four-digit numbers, look at the first two digits of the number to determine the first digit of the square root. For example, to find the square root of 7369, consider the square root of 73. Since 8 × 8 = 64 and 9 × 9 = 81, 8 must be the first digit of the square root. So the answer is 80 something. Now proceed the usual way. Dividing 80 into 7369 gives 92 plus a fraction, so a good guesstimate is 86. If you squared 86 to get 7396, you would be high by 27, so you should subtract .16 for a better guesstimate of 85.84, which is right on the money.

To guesstimate the square root of a six-digit number like 593,472 would seem like an impossible task for the uninitiated, but for you it's no sweat. Since 7002 = 490,000, and 8002 = 640,000, the square of 593,472 must lie between 700 and 800. In fact, all five-digit and six-digit numbers have three-digit square roots. In practice, you only need to look at the square root of the first two digits of six-digit numbers (or the first digit of five numbers). Once you figure out that the square root of 59 lies between 7 and 8, you know your answer is in the 700s.

Now proceed in the usual manner:

The exact square root of 593,472 is 770.37 (to five places), so you're pretty close. But you could have been closer, as the following trick demonstrates. Note that the first two digits, 59, are closer to 64 (8 × 8) than they are to 49 (7 × 7). Because of this you can start your guesstimation with the number 8 and proceed from there:

Just for fun, let's do a real whopper-the square root of 28,674,529. This isn't as hard as it might seem. Your first step is to round to the nearest large number-in this case, just find the square root of 29.

All seven-digit and eight-digit numbers have four-digit square roots, so 5.4 becomes 5400, your estimate. The exact answer is slightly greater than 5354.8. Not bad!

This wraps up the chapter on guesstimation math. After doing the exercises below, turn to the next chapter on pencil-and-paper math, where you will learn to write down answers to problems, but in a much quicker way than you've done on paper before.

The Mathematical Duel of Évariste Galois

The tragic story of the French mathematician Évariste Galois (1811-1832) killed at the age of twenty in a duel over "an infamous coquette" is legendary in the annals of the history of mathematics. A precociously brilliant student, Galois laid the foundation for a branch of mathematics known as group theory. Legend has it that he penned his theory the night before the duel, anticipating his demise and wanting to leave his legacy to the mathematics community. Hours before his death, on May 30, 1832, Galois wrote to Auguste Chevalier: "I have made some new discoveries in analysis. The first concerns the theory of equations, the others integral functions." After describing these, he asked his friend: "Make a public request of Jacobi or Gauss to give their opinions not as to the truth but as to the importance of these theorems. After that, I hope some men will find it profitable to sort out this mess."

Romantic legend and historical truth, however, do not always match. What Galois penned the night before his death were corrections and editorial changes to papers that had been accepted by the Academy of Sciences long before. Further, Galois's initial papers had been submitted three years prior to the duel, when he was all of seventeen! It was after this that Galois became embroiled in political controversy, was arrested, spent time in a prison dungeon, and, ultimately, got himself mixed up in a dispute over a woman and killed.

Aware of his own precocity, Galois noted, "I have carried out researches which will halt many savants in theirs." For over a century that proved to be the case.
MORE TIPS ON TIPS
As we indicated in Chapter 0, it is easy to figure out tips for most situations. For example, to figure out a 10% tip, we merely multiply the bill by 0.1 (or divide the bill by 10). For example, if a bill came to $42, then a 10% tip would be $4.20. To determine a 20% tip, you simply multiply the bill by 0.2, or double the amount of a 10% tip. Thus a 20% tip on a $42 bill would be $8.40.

To determine a 15% tip, we have a few options. If you have mastered the techniques of Chapter 2, and are comfortable with multiplying by 15 = 5 × 3, you can simply multiply the bill by 15, then divide by 100. For example, with a $42 bill, we have 42 × 15 = 42 × 5 × 3 = 210 × 3 = 630, which when divided by 100, gives us a tip of $6.30. Another method is to take the average of a 10% tip and a 20% tip. From our earlier calculation, this would be

Perhaps the most popular approach to taking a 15% tip is to take 10% of the bill, cut that amount in half (which would be a 5% tip), then add those two numbers together. So, for instance, with a $42 bill, you would add $4.20 plus half that amount, $2.10, to get

Let's use all three methods to compute 15% on a bill that is $67. By the direct method, 67 × 3 × 5 = 201 × 5 = 1005, which when divided by 100 gives us $10.05. By the averaging method, we average the 10% tip of $6.70 with the 20% tip of $13.40, to get

Using the last method, we add $6.70 to half that amount, $3.35 to get

$6.70 + $3.35 = $10.05

Finally, to calculate a 25% tip, we offer two methods. Either multiply the amount by 25, then divide by 100, or divide the amount by 4 (perhaps by cutting the amount in half twice). For example, with a $42 bill, you can compute 42 × 25 = 42 × 5 × 5 = 210 × 5 = 1050, which when divided by 100 produces a tip of $10.50. Or you can divide the original amount directly by 4, or cut in half twice: half of $42 is $21, and half of that is $10.50. With a $67 bill, I would probably divide by 4 directly: since 67 = 4 = 16¾, we get a 25% tip of $16.75.
NOT-TOO-TAXING CALCULATIONS
In this section, I will show you my method for mentally estimating sales tax. For some tax rates, like 5% or 6% or 10%, the calculation is straightforward. For example, to calculate 6% tax, just multiply by 6, then divide by 100. For instance, if the sale came to $58, then 58 × 6 = 348, which when divided by 100 gives the exact sales tax of $3.48. (So your total amount would be $61.48.)

But how would you calculate a sales tax of 6½% on $58? I will show you several ways to do this calculation, and you choose the one that seems easiest for you. Perhaps the easiest way to add half of a percent to any dollar amount is to simply cut the dollar amount in half, then turn it into cents. For example, with $58, since half of 58 is 29, simply add 29 cents to the 6% tax (already computed as $3.48) to get a sales tax of $3.77.

Another way to calculate the answer (or a good mental estimate) is to take the 6% tax, divide it by 12, then add those two numbers. For example, since 6% of $58 is $3.48, and 12 goes into 348 almost 30 times, then add 30 cents for an estimate of $3.78, which is only off by a penny. If you would rather divide by 10 instead of 12, go ahead and do it. You will be calculating 6.6% instead of 6.5% (since = 0.6) but that should still be a very good estimate. Here, you would take $3.48 and add 34¢ to get $3.82.

Let's try some other sales tax percentages. How can we calculate 7¼% sales tax on $124? Begin by computing 7% of 124. From the methods shown in Chapter 2, you know that 124 × 7 = 868, so 7% of 124 is $8.68. To add a quarter percent, you can divide the original dollar amount by 4 (or cut it in half, twice) and turn the dollars into cents. Here, 124 = 4 = 31, so add 310 to $8.68 to get an exact sales tax of $8.99.

Another way to arrive at 31¢ is to take your 7% sales tax, $8.68, and divide it by 28. The reason this works is because = ¼ For a quick mental estimate, I would probably divide $8.68 by 30, to get about 29¢, for an approximate total sales tax of $8.97.

When you divide by 30, then you are actually computing a tax of %, which is approximately 7.23% instead of 7.25%.

How would you calculate a sales tax of 7.75%? Probably for most approximations, it is sufficient to just say that it is a little less than 8% sales tax. But to get a better approximation, here are some suggestions. As you saw in the last example, if you can easily calculate a ¼% adjustment, then simply triple that amount for the ¾% adjustment. For example, to calculate 7.75% of $124, you first calculate 7% to obtain $8.68. If you calculated that ¼% is 31¢, then ¾% would be 93¢, for a grand total of $8.68 + 0.93 = $9.61. For a quick approximation, you can exploit the fact that = .777 is approximately .75, so you can divide the 7% sales tax by 9 to get slightly more than the .75% adjustment. In this example, since $8.68 divided by 9 is about 96¢, simply add $8.68 + 0.96 = $9.64 for a slight overestimate.

We can use this approximation procedure for any sales tax. For a general formula, to estimate the sales tax of $A.B%, first multiply the amount of the sale by A%. Then divide this amount by the number D, where A/D is equal to 0.B. (Thus D is equal to A times the reciprocal of B.) Adding these numbers together gives you the total sales tax (or an approximate one, if you rounded D to an easier nearby number). For instance, with 7.75%, the magic divisor D would be , which we rounded down to 9. For sales tax of %, first compute the sales tax on 6%, then divide that number by 16, since . (To divide a number by 16, divide the number by 4 twice, or divide the number by 8, then by 2.) Try to come up with methods for finding the sales tax in the state that you live in. You will find that the problem is not as taxing as it seems!
SOME "INTEREST-ING" CALCULATIONS
Finally, we'll briefly mention some practical problems pertaining to interest, from the standpoint of watching your investments grow, and paying off money that you owe.

We begin with the famous Rule of 70, which tells you approximately how long it takes your money to double: To find the number of years that it will take for your money to double, divide the number 70 by the rate of interest.

Suppose that you find an investment that promises to pay you 5% interest per year. Since 70 ÷ 5 = 14, then it will take about 14 years for your money to double. For example, if you invested $1000 in a savings account that paid that interest, then after 14 years, it will have $1000(1.05)14 = $1979.93. With an interest rate of 7%, the Rule of 70 indicates that it will take about 10 years for your money to double. Indeed, if you invest $1000 at that annual interest rate, you will have after ten years $1000(1.07)10 = $1967.15. At a rate of 2%, the Rule of 70 says that it should take about 35 years to double, as shown below:

$1000(1.02)35 = $1999.88

A similar method is called the Rule of 110, which indicates how long it takes for your money to triple. For example, at 5% interest, since 110 ÷ 5 = 22, it should take about 22 years to turn $1000 into $3000. This is verified by the calculation $1000(1.05)22 = $2925.26. The Rule of 70 and the Rule of 110 are based on properties of the number e = 2.71828 ... and "natural logarithms" (studied in precalculus), but fortunately we do not need to utilize this higher mathematics to apply the rules.

Now suppose that you borrow money, and you have to pay the money back. For example, suppose that you borrow $360,000 at an annual interest rate of 6% per year (which we shall interpret to mean that interest will accumulate at a rate of 0.5% per month) and suppose that you have thirty years to pay off the loan. About how much will you need to pay each month? First of all, you will need to pay $360,000 times 0.5% = $1,800 each month just to cover what you owe in interest. (Although, actually, what you owe in interest will go down gradually over time.) Since you will make 30 × 12 = 360 monthly payments, then paying an extra $1,000 each month would cover the rest of the loan, so an upper bound on your monthly payment is $1,800 + $1,000 = $2,800. But fortunately you do not need to pay that much extra. Here is my rule of thumb for estimating your monthly payment.

Let i be your monthly interest rate. (This is your annual interest rate divided by 12.) Then to pay back a loan of $P in ? months, your monthly payment M is about

In our last example, ? = $360,000, and i = 0.005, so our formula indicates that our monthly payment should be about

Notice that the first two numbers in the numerator multiply to $1,800. Using a calculator (for a change) to compute (1.005)360 = 6.02, we have that your monthly payment should be about $1,800(6.02)/5.02, which is about $2,160 per month.

Here's one more example. Suppose you wish to pay for a car, and after your down payment, you owe $18,000 to be paid off in five years, with an annual interest rate of 4%. If there was no interest, you would have to pay $18,000/60 = $300 per month. Since the first year's interest is $18,000(.04) = $720, you know that you will need to pay no more than $300 + $60 = $360. Using our formula, since the monthly interest rate is i = .04/12 = 0.00333, we have and since (1.00333)60 = 1.22, we have a monthly payment of about $60(1.22)/(0.22) = $333.

We conclude with some exercises that we hope will hold your interest.
GUESSTIMATION EXERCISES
Go through the following exercises for guesstimation math; then check your answers and computations at the back of the book.

Round these numbers up or down and see how close you come to the exact answer:

Mentally estimate the total for the following column of prices by rounding to the nearest 50¢:

Estimate the following subtraction problems by rounding to the second or third digit:

Adjust the numbers in a way that allows you to guesstimate the following division problems:

Adjust the numbers in a way that allows you to guesstimate the following multiplication problems:

Estimate the square roots of the following numbers using the divide and average method:

1. Compute 15% of $88.

2. Compute 15% of $53.

3. Compute 25% of $74.

4. How long does it take an annual interest rate of 10% to double your money?

5. How long does it take an annual interest rate of 6% to double your money?

6. How long does it take an annual interest rate of 7% to triple your money?

7. How long does it take an annual interest rate of 7% to quadruple your money?

8. Estimate the monthly payment to repay a loan of $100,000 at an interest rate of 9% over a ten-year period.

9. Estimate the monthly payment to repay a loan of $30,000 at an interest rate of 5% over a four-year period.
Chapter 6 Math for the Board:
Pencil-and-Paper Math
In the Introduction to this book I discussed the many benefits you will get from being able to do mental calculations. In this chapter I present some methods for speeding up pencil-and-paper calculations as well. Since calculators have replaced much of the need for pencil-and-paper arithmetic in most practical situations, I've chosen to concentrate on the lost art of calculating square roots and the flashy criss-cross method for multiplying large numbers. Since these are, admittedly, mostly for mental gymnastics and not for some practical application, I will first touch on addition and subtraction and show you just a couple of little tricks for speeding up the process and for checking your answers. These techniques can be used in daily life, as you'll see.

If you are eager to get to the more challenging multiplication problems, you can skip this chapter and go directly to Chapter 7, which is critical for mastering the big problems in Chapter 8. If you need a break and just want to have some fun, then I recommend going through this chapter-you'll enjoy playing with pencil and paper once again.
COLUMNS OF NUMBERS
Adding long columns of numbers is just the sort of problem you might run into in business or while figuring out your personal finances. Add the following column of numbers as you normally would and then check to see how I do it.

When I have pencil and paper at my disposal, I add the numbers from top to bottom and from right to left, just as we learned to do in school. With practice, you can do these problems in your head as fast or faster than you can with a calculator. As I sum the digits, the only numbers that I "hear" are the partial sums. That is, when I sum the first (rightmost) column 8 + 4 + 0 + 7 + 7 + 5, I hear 8 ... 12 ... 19 ... 26 ... 31. Then I put down the 1, carry the 3, and proceed as usual. The next column would then sound like 3 ... 5 ... 13 ... 15 ... 22 ... 23 ... 25. Once I have my final answer, I write it down, then check my computation by adding the numbers from bottom to top and, I hope, arrive at the same answer.

For instance, the first column would be summed, from bottom to top, as 5 + 7 + 7 + 0 + 4 + 8 (which in my mind sounds like 5 ... 12 ... 19 ... 23 ... 31). Then I mentally carry the 3, and add 3 + 2 + 1 + 7 + 2 + 8 + 2, and so on. By adding the numbers in a different order, you are less likely to make the same mistake twice. Of course, if the answers differ, then at least one of the calculations must be wrong.
MOD SUMS
If I'm not sure about my answer, I sometimes check my solution by a method I call mod sums (because it is based on the elegant mathematics of modular arithmetic). This method also goes by the names of digital roots and casting out nines. I admit this method is not as practical, but it's easy to use.

With the mod sums method, you sum the digits of each number until you are left with a single digit. For example, to compute the mod sum of 4328, add 4 + 3 + 2 + 8 = 17. Then add the digits of 17 to get 1 + 7 = 8. Hence the mod sum of 4328 is 8. For the previous problem the mod sums of each number are computed as follows:

As illustrated above, the next step is to add all the mod sums together (8 + 2 + 8 + 1 + 5 + 5). This yields 29, which sums to 11, which in turn sums to 2. Note that the mod sum of 8651, your original total of the original digits, is also 2. This is not a coincidence! If you have computed the answer and the mod sums correctly, your final mod sums must be the same. If they are different, you have definitely made a mistake somewhere: there is about a 1 in 9 chance that the mod sums will match accidentally. If there is a mistake, then this method will detect it 8 times out of 9.

The mod sum method is more commonly known to mathematicians and accountants as casting out nines because the mod sum of a number happens to be equal to the remainder obtained when the number is divided by 9. In the case of the answer above-8651-the mod sum was 2. If you divide 8651 by 9, the answer is 961 with a remainder of 2. In other words, if you cast out 9 from 8651 a total of 961 times, you'll have a remainder of 2. There's one small exception to this. Recall that the sum of the digits of any multiple of 9 is also a multiple of 9. Thus, if a number is a multiple of 9, it will have a mod sum of 9, even though it has a remainder of 0.
SUBTRACTING ON PAPER
You can't, of course, subtract columns of numbers the same way you add them. Rather, you subtract them number by number, which means that all subtraction problems involve just two numbers. Once again, with pencil and paper at our disposal, it is easier to subtract from right to left. To check your answer, just add the answer to the second number. If you are correct, then you should get the top number.

If you want, you can also use mod sums to check your answer. The key is to subtract the mod sums you arrive at and then compare that number to the mod sum of your answer:

There's one extra twist. If the difference in the mod sums is a negative number or 0, add 9 to it. For instance:
PENCIL-AND-PAPER SQUARE ROOTS
With the advent of pocket calculators, the pencil-and-paper method of calculating square roots has practically become a lost art. You've already seen how to estimate square roots mentally. Now I'll show you how to do it exactly, using pencil and paper.

Remember how in guesstimating square roots you calculated the square root of nineteen? Let's look at that problem again, this time using a method that will give you the exact square root.

I will describe the general method that fits all cases, and illustrate it with the above example.

Step 1. If the number of digits to the left of the decimal point is one, three, five, seven, or any odd number of digits, the first digit of the answer (or quotient) will be the largest number whose square is less than the first digit of the original number. If the number of digits to the left of the decimal point is two, four, six, or any even number of digits, the first digit of the quotient will be the largest number whose square is less than the first two digits of the dividend. In this case, 19 is a two-digit number, so the first digit of the quotient is the largest number whose square is less than 19. That number is 4. Write the answer above either the first digit of the dividend (if odd) or the second digit of the dividend (if even).

Step 2. Subtract the square of the number in step 1, then bring down two more digits. Since 42 = 16, we subtract 19 − 16 = 3. We bring down two 0s, leaving 300 as the current remainder.

Step 3. Double the current quotient (ignoring any decimal point), and put a blank space in following it. Here 4 × 2 = 8. Put down 8_ x_ to the left of the current remainder, in this case 300.

Step 4. The next digit of the quotient will be the largest number that can be put in both blanks so that the resulting multiplication problem is less than or equal to the current remainder. In this case the number is 3, because 83 × 3 = 249, whereas 84 × 4 = 336, which is too high. Write this number above the second digit of the next two numbers; in this case the 3 would go above the second 0. We now have a quotient of 4.3.

Step 5. If you want more digits, subtract the product from the remainder (i.e., 300 − 249 = 51), and bring down the next two digits; in this case 51 turns into 5100, which becomes the current remainder. Now repeat steps 3 and 4.

To get the third digit of the square root, double the quotient, again ignoring the decimal point (i.e., 43 × 2 = 86). Place 86_ x_ to the left of 5100. The number 5 gives us 865 × 5 = 4325, the largest product below 5100. The 5 goes above the next two numbers, in this case two more 0s. We now have a quotient of 4.35. For even more digits, repeat the process as we've done in the example.

Here's an example of an odd number of digits before the decimal point:

Next, we'll calculate the square root of a four-digit number. In this case-as with two-digit numbers-we consider the first two digits of the problem to determine the first digit of the square root:

Finally, if the number for which you are calculating the square root is a perfect square, you will know it as soon as you end up with a remainder of 0 and nothing to bring down. For example:
PENCIL-AND-PAPER MULTIPLICATION
For pencil-and-paper multiplication I use the criss-cross method, which enables me to write down the entire answer on one line without ever writing any partial results! This is one of the most impressive displays of mathemagics when you have pencil and paper at your disposal. Many lightning calculators in the past earned their reputations with this method. They would be given two large numbers and write down the answer almost instantaneously. The criss-cross method is best learned by example.

Step 1. First, multiply 4 × 7 to yield 28, write down the 8, and mentally carry the 2 to the next computation, below.

Step 2. Following the diagram, add 2 + (4 × 4) + (3 × 7) = 39, write down the 9, and carry the 3 to the final computation, below.

Step 3. End by adding 3 + (3 × 4) = 15 and writing down 15 to arrive at your final answer.

You have now written the answer: 1598.

Let's solve another 2-by-2 multiplication problem using the criss-cross method:

The steps and diagrams appear as follows:

Step 1. 5 × 3 = 15

Step 2. 1 + (5 × 8) + (6 × 3) = 59

Step 3. 5 + (6 × 8) = 53

Answer: 5395

The criss-cross method gets slightly more complicated with 3-by-3 problems.

We proceed as suggested by our pattern below:

Step 1. 2 × 3 = 6

Step 2. (2 × 5) + (6 × 3) = 28

Step 3. 2 + (2 × 8) + (7 × 3) + (6 × 5) = 69

Step 4. 6 + (6 × 8) + (7 × 5) = 89

Step 5. 8 + (8 × 7) = 64

Answer: 649,986

Notice that the number of multiplications at each step is 1, 2, 3, 2, and 1 respectively. The mathematics underlying the criss-cross method is nothing more that the distributive law. For instance, 853 × 762 = (800 + 50 + 3) × (700 + 60 + 2) = (3 × 2) + [(5 × 2) + (3 × 6)] × 10 + [(8 × 2) + (5 × 6) + (3 × 7)] × 100 + [(8 × 6) + (5 × 7)] × 1000 + (8 × 7) × 10,000, which are precisely the calculations of the criss-cross method.

You can check your answer with the mod sum method by multiplying the mod sums of the two numbers and computing the resulting number's mod sum. Compare this number to the mod sum of the answer. If your answer is correct, they must match. For example:

If the mod sums don't match, you made a mistake. This method will detect mistakes, on average, about 8 times out of 9.

In the case of 3-by-2 multiplication problems, the procedure is the same except you treat the hundreds digit of the second number as a 0:

Step 1. 7 × 6 = 42

Step 2. 4 + (7 × 4) + (3 × 6) = 50

Step 3. 5 + (7 × 8) + (0 × 6) + (3 × 4) = 73

Step 4. 7 + (3 × 8) + (0 × 4) = 31

Step 5. 3 + (0 × 8) = 3

Answer: 31,302

Of course, you would normally just ignore the multiplication by zero, in practice.

You can use the criss-cross to do any size multiplication problem. To answer the 5-by-5 problem below will require nine steps. The number of multiplications in each step is 1, 2, 3, 4, 5, 4, 3, 2, 1, for twenty-five multiplications altogether.

Step 1. 9 × 7= 63

Step 2. 6 + (9 × 6) + (4 × 7) = 88

Step 3. 8 + (9 × 8) + (0 × 7) + (4 × 6) = 104

Step 4. 10 + (9 × 2) + (2 × 7) + (4 × 8) + (0 × 6) = 74

Step 5. 7 + (9 × 4) + (5 × 7) + (4 × 2) + (2 × 6) + (0 × 8) = 98

Step 6. 9 + (4 × 4) + (5 × 6) + (0 × 2) + (2 × 8) = 71

Step 7. 7 + (0 × 4) + (5 × 8) + (2 × 2) = 51

Step 8. 5 + (2 × 4) + (5 × 2) = 23

Step 9. (5 × 4) + 2 = 22

Answer: 2,231,184,483

You can check your answer by using the mod sums method.
CASTING OUT ELEVENS
To double-check your answer another way, you can use the method known as casting out elevens. It's similar to casting out nines, except you reduce the numbers by alternately subtracting and adding the digits from right to left, ignoring any decimal point. If the result is negative, then add eleven to it. (It may be tempting to do the addition and subtraction from left to right, as you do with mod sums, but in this case you must do it from right to left for it to work.) For example:

The same method works for subtraction problems:

Shakuntala Devi: That's Incalculable!

In 1976 the New York Times reported that an Indian woman named Shakuntala Devi (b. 1939) added 25,842 + 111,201,721 + 370,247,830 + 55,511,315, and then multiplied that sum by 9,878, for a correct total of 5,559,369,456,432, all in less than twenty seconds. Hard to believe, though this uneducated daughter of impoverished parents has made a name for herself in the United States and Europe as a lightning calculator.

Unfortunately, most of Devi's truly amazing feats not done by obvious "tricks of the trade" are poorly documented. Her greatest claimed accomplishment-the timed multiplication of two thirteen-digit numbers on paper-has appeared in the Guinness Book of World Records as an example of a "Human Computer." The time of the calculation, however, is questionable at best. Devi, a master of the criss-cross method, reportedly multiplied 7,686,369,774,870 × 2,465,099,745,799, numbers randomly generated at the computer department of Imperial College, London, on June 18, 1980. Her correct answer of 18,947,668,177,995,426,773,730 was allegedly generated in an incredible twenty seconds. Guinness offers this disclaimer: "Some eminent mathematical writers have questioned the conditions under which this was apparently achieved and predict that it would be impossible for her to replicate such a feat under highly rigorous surveillance." Since she had to calculate 169 multiplication problems and 167 addition problems, for a total of 336 operations, she would have had to do each calculation in under a tenth of a second, with no mistakes, taking the time to write down all 26 digits of the answer. Reaction time alone makes this record truly in the category of "that's incalculable!"

Despite this, Devi has proven her abilities doing rapid calculation and has even written her own book on the subject.

It even works with multiplication:

If the numbers disagree, you made a mistake somewhere. But if they match, it's still possible for a mistake to exist. On average, this method will detect an error 10 times out of 11. Thus a mistake has a 1 in 11 chance of sneaking past the elevens check, a 1 in 9 chance of sneaking past the nines check, and only a 1 in 99 chance of being undetected if both checks are used. For more information about this and other fascinating mathemagical topics, I would encourage you to read any of Martin Gardner's recreational math books.

You are now ready for the ultimate pencil-and-paper multiplication problem in the book, a 10-by-10! This has no practical value whatsoever except for showing off! (And personally I think multiplying five-digit numbers is impressive enough since the answer will be at the capacity of most calculators.) We present it here just to prove that it can be done. The criss-crosses follow the same basic pattern as that in the 5-by-5 problem. There will be nineteen computation steps and at the tenth step there will be ten criss-crosses! Here you go:

Here's how we do it:

Step 1. 6 × 1=6

Step 2. (6 × 5) + (1 × 1) = 31

Step 3. 3 + (6 × 4) + (2 × 1) + (1 × 5) = 34

Step 4. 3 + (6 × 9) + (5 × 1) + (1 × 4) + (2 × 5) = 76

Step 5. 7 + (6 × 2) + (7 × 1) + (1 × 9) + (5 × 5) + (2 × 4) = 68

Step 6. 6 + (6 × 8) + (5 × 1) + (1 × 2) + (7 × 5) + (2 × 9) + (5 × 4) = 134

Step 7. 13 + (6 × 6) + (5 × 1) + (1 × 8) + (5 × 5) + (2 × 2) + (7 × 4) + (5 × 9) = 164

Step 8. 16 + (6 × 6) + (2 × 1) + (1 × 6) + (5 × 5) + (2 × 8) + (5 × 4) + (5 × 2) + (7 × 9) = 194

Step 9. 19 + (6 × 7) + (4 × 1) + (1 × 6) + (2 × 5) + (2 × 6) + (5 × 4) + (5 × 8) + (5 × 9) + (7 × 2) = 212

Step 10. 21 + (6 × 2) + (4 × 1) + (1 × 7) + (4 × 5) + (2 × 6) + (2 × 4) + (5 × 6) + (5 × 9) + (7 × 8) + (5 × 2) = 225

Step 11. 22 + (1 × 2) + (4 × 5) + (2 × 7) + (4 × 4) + (5 × 6) + (2 × 9) + (7 × 6) + (5 × 2) + (5 × 8) = 214

Step 12. 21 + (2 × 2) + (4 × 4) + (5 × 7) + (4 × 9) + (7 × 6) + (2 × 2) + (5 × 6) + (5 × 8) = 228

Step 13. 22 + (5 × 2) + (4 × 9) + (7 × 7) + (4 × 2) + (5 × 6) + (2 × 8) + (5 × 6) = 201

Step 14. 20 + (7 × 2) + (4 × 2) + (5 × 7) + (4 × 8) + (5 × 6) + (2 × 6) = 151

Step 15. 15 + (5 × 2) + (4 × 8) + (5 × 7) + (4 × 6) + (2 × 6) = 128

Step 16. 12 + (5 × 2) + (4 × 6) + (2 × 7) + (4 × 6) = 84

Step 17. 8 + (2 × 2) + (4 × 6) + (4 × 7) = 64

Step 18. 6 + (4 × 2) + (4 × 7) = 42

Step 19. 4 + (4 × 2) = 12

If you were able to negotiate this extremely difficult problem successfully the first time through, you are on the verge of graduating from apprentice to master mathemagician!

PENCIL-AND-PAPER MATHEMATICS EXERCISES

Sum the following column of numbers, checking your answer by reading the column from bottom to top, and then by the mod sums method. If the two mod sums do not match, recheck your addition:

Sum this column of dollars and cents:

Subtract the following numbers, using mod sums to check your answer and then by adding the bottom two numbers to get the top number:

For the following numbers, compute the exact square root using the doubling and dividing technique.

To wrap up this exercise session, use the criss-cross method to compute exact multiplication problems of any size. Place the answer below the problems on one line, from right to left. Check your answers using mod sums.
Chapter 7 A Memorable Chapter:
Memorizing Numbers
One of the most common questions I am asked pertains to my memory. No, I don't have an extraordinary memory. Rather, I apply a memory system that can be learned by anyone and is described in the following pages. In fact, experiments have shown that almost anyone of average intelligence can be taught to vastly improve their ability to memorize numbers.

In this chapter we'll teach you a way to improve your memory for numbers. Not only does this have obvious practical benefits, such as remembering dates or recalling phone numbers, but it also allows the mathemagician to solve much larger problems mentally. In the next chapter we'll show you how to apply the techniques of this chapter to multiply five-digit numbers in your head!
USING MNEMONICS
The method presented here is an example of a mnemonic-that is, a tool to improve memory encoding and retrieval. Mnemonics work by converting incomprehensible data (such as digit sequences) to something more meaningful. For example, take just a moment to memorize the sentence below:

"My turtle Pancho will, my love, pick up my new mover, Ginger."

Read it over several times. Look away from the page and say it to yourself over and over until you don't have to look back at the page, visualizing the turtle, Pancho, picking up your new mover, Ginger, as you do so. Got it?

Congratulations! You have just memorized the first twenty-four digits of the mathematical expression p (pi). Recall that p is the ratio of the circumference of a circle to its diameter, usually approximated in school as 3.14, or . In fact, p is an irrational number (one whose digits continue indefinitely with no repetition or pattern), and computers have been used to calculate p to billions of places.
THE PHONETIC CODE
I'm sure you're wondering how "My turtle Pancho will, my love, pick up my new mover, Ginger" translates into 24 places of p.

To do this, you first need to memorize the phonetic code below, where each number between 0 and 9 is assigned a consonant sound.

1 is the t or d sound.

2 is the n sound.

3 is the m sound.

4 is the r sound.

5 is the l sound.

6 is the j, ch, or sh sound.

7 is the k or hard g sound.

8 is the f or v sound.

9 is the p or b sound.

0 is the z or s sound.

Memorizing this code isn't as hard as it looks. For one thing, notice that in cases where more than one letter is associated with a number, they have similar pronunciations. For example, the k sound (as it appears in words like kite or cat) is similar to the hard g sound (as it appears in such words as goat). You can also rely on the following tricks to help you memorize the code:

1 A typewritten t or d has just 1 downstroke.

2 A typewritten n has 2 downstrokes.

3 A typewritten m has 3 downstrokes.

4 The number 4 ends in the letter r.

5 Shape your hand with 4 fingers up and the thumb at a 90-degree angle-that's 5 fingers in an L shape.

6 A J looks sort of like a backward 6.

7 A K can be drawn by laying two 7s back to back.

8 A lowercase f written in cursive looks like an 8.

9 The number 9 looks like a backward p or an upside-down b.

0 The word zero begins with the letter z.

Or you can just remember the list in order, by thinking of the name Tony Marloshkovips!

Practice remembering this list. In about ten minutes you should have all the one-digit numbers associated with consonant sounds. Next, you can convert numbers into words by placing vowel sounds around or between the consonant sounds. For instance, the number 32 can become any of the following words: man, men, mine, mane, moon, many, money, menu, amen, omen, amino, mini, minnie, and so on. Notice that the word minnie is acceptable since the n sound is only used once.

The following words could not represent the number 32 because they use other consonant sounds: mourn, melon, mint. These words would be represented by the numbers 342, 352, and 321, respectively. The consonant sounds of h, w, and y can be added freely since they don't appear on the list. Hence, 32 can also become human, woman, yeoman, or my honey.

The following list gives you a good idea of the types of words you can create using this phonetic code. Don't feel obligated to memorize it-use it as inspiration to explore the possibilities on your own.
The Number-Word List
For practice, translate the following numbers into words, then check the correct translation below. When translating numbers into words, there are a variety of words that can be formed:

Here are some words that I came up with:

As an exercise, translate each of the following words into its unique number:

Answers:

Although a number can usually be converted into many words, a word can be translated only into a single number. This is an important property for our applications as it enables us to recall specific numbers.

Using this system you can translate any number or series of numbers (e.g., phone numbers, Social Security numbers, driver's license numbers, the digits of p) into a word or a sentence. Here's how the code works to translate the first twenty-four digits of p:

Remember that, with this phonetic code, g is a hard sound, as in grass, so a soft g (as in Ginger) sounds like j and is represented by a 6. Also, the word will is, phonetically, just L, and is represented by 5, since the consonant sound w can be used freely. Since this sentence can only be translated back to the twenty-four digits above, you have effectively memorized p to twenty-four digits!

There's no limit to the number of numbers this code will allow you to memorize. For example, the following two sentences, when added to "My turtle Pancho will, my love, pick up my new mover, Ginger," will allow you to memorize the first sixty digits of p:

And:

What the heck, let's go for a hundred digits:

You can really feel proud of yourself once these sentences roll trippingly off your tongue, and you're able to translate them quickly back into numbers. But you've got a ways to go for the world record. Hiroyuki Goto of Japan recited p to 42,195 places, from memory, in seventeen hours and twenty-one minutes in 1995.
HOW MNEMONICS MAKES MENTAL CALCULATION EASIER
Aside from improving your ability to memorize long sequences of numbers, mnemonics can be used to store partial results in the middle of a difficult mental calculation. For example, here's how you can use mnemonics to help you square a three-digit number:

A Piece of Pi for Alexander Craig Aitken

Perhaps one of the most impressive feats of mental calculation was performed by a professor of mathematics at the University of Edinburgh, Alexander Craig Aitken (1895-1967), who not only learned the value of p to 1,000 places but, when asked to demonstrate his amazing memory during a lecture, promptly rattled off the first 250 digits of p. He was then challenged to skip ahead and begin at the 551st digit and continue for another 150 places. This he did successfully without a single error.

How did he do it? Aitken explained to his audience that "the secret, to my mind, is relaxation, the complete antithesis of concentration as usually understood." Aitken's technique was more auditory than usual. He arranged the numbers into chunks of fifty digits and memorized them in a sort of rhythm. With undaunted confidence he explained, "It would have been a reprehensibly useless feat had it not been so easy."

That Aitken could memorize p to a thousand places does not qualify him as a lightning calculator. That he could easily multiply five-digit numbers against each other does. A mathematician named Thomas O'Beirne recalled Aitken at a desk calculator demonstration. "The salesman," O'Beirne wrote, "said something like 'We'll now multiply 23,586 by 71,283.' Aitken said right off, 'And get ...' (whatever it was). The salesman was too intent on selling even to notice, but his manager, who was watching, did. When he saw Aitken was right, he nearly threw a fit (and so did I)."

Ironically, Aitken noted that when he bought a desk calculator for himself, his own mental skills deteriorated quickly. Anticipating what the future might hold, he lamented, "Mental calculators may, like the Tasmanian or the Maori, be doomed to extinction. Therefore you may be able to feel an almost anthropological interest in surveying a curious specimen, and some of my auditors here may be able to say in the year A.D. 2000, 'Yes, I knew one such.' " Fortunately, history has proved him wrong!

As you recall from Chapter 3, to square 342 you first multiply 300 × 384, for 115,200, then add 422. But by the time you've computed 422 = 1,764, you may have forgotten the earlier number, 115,200. Here's where our memory system comes to the rescue. To store the number 115,200, put 200 on your hand by raising two fingers, and convert 115 into a word like title. (By the way, I do not consider storing the 200 on fingers to be cheating. After all, what are hands for if not for holding on to digits!) Repeat the word title to yourself once or twice. That's easier to remember than 115,200, especially after you start calculating 422. Once you've arrived at 1,764, you can add that to title 2, or 115,200, for a total of 116,964.

Here's another:

After multiplying 300 × 246 = 73,800, convert 73 into gum and hold 800 on your hand by raising eight fingers. Once you've computed 272 = 729, just add that to gum 8, or 73,800, for a total of 74,529. This may seem a bit cumbersome at first, but with practice the conversion from numbers to words and back to numbers becomes almost second nature.

You have seen how easily two-digit numbers can be translated into simple words. Not all three-digit numbers can be translated so easily, but if you're at a loss for a simple word to act as a mnemonic, an unusual word or a nonsense word will do. For example, if no simple word for 286 or 638 comes quickly to mind, use a combination word like no fudge or a nonsense word like jam-off. Even these unusual words should be easier to recall than 286 or 638 during a long calculation. For some of the huge problems in the next chapter, these memory tricks will be indispensable.
MEMORY MAGIC
Without using mnemonics, the average human memory (including mine) can hold only about seven or eight digits at a time. But once you've mastered the ability to change numbers into words, you can expand your memory capacity considerably. Have someone slowly call out sixteen digits while someone else writes them down on a blackboard or a piece of paper. Once they are written down, you repeat them back in the exact order they were given without looking at the board or the piece of paper! At a recent lecture demonstration, I was given the following series of numbers:

1, 2, 9, 7, 3, 6, 2, 7, 9, 3, 3, 2, 8, 2, 6, 1

As the numbers were called out, I used the phonetic code to turn them into words and then tied them together with a nonsensical story. In this case, 12 becomes tiny, 97 becomes book, 362 becomes machine, 793 becomes kaboom, 32 becomes moon, and 8261 becomes finished.

As the words were being created, I linked them together to form a silly story to help me remember them. I pictured finding a tiny book, which I placed inside a machine. This caused the machine to go kaboom, and tossed me to the moon, where I was finished. This story may sound bizarre, but the more ridiculous the story, the easier it is to remember-and besides, it's more fun.
Chapter 8 The Tough Stuff Made Easy:
Advanced Multiplication
At this point in the book-if you've gone through it chapter by chapter-you have learned to do mental addition, subtraction, multiplication, and division, as well as the art of guesstimation, pencil-and-paper mathemagics, and the phonetic code for number memory. This chapter is for serious, die-hard mathemagicians who want to stretch their minds to the limits of mental calculation. The problems in this chapter range from four-digit squares to the largest problem I perform publicly-the multiplication of two different five-digit numbers.

In order to do these problems, it is particularly important that you are comfortable and reasonably fast using the phonetic code. And even though, if you glance ahead in this chapter, the problems seem overwhelming, let me restate the two fundamental premises of this book: (1) all mental calculation skills can be learned by almost anyone; and (2) the key is the simplification of all problems into easier problems that can be done quickly. There is no problem in this chapter (or anywhere else) that you will encounter (comparable to these) that cannot be mastered and learned through the simplification techniques you've learned in previous chapters. Because we are assuming that you've mastered the techniques needed for this chapter, we will be teaching primarily by illustration, rather than walking you through the problems word for word. As an aid, however, we remind you that many of the simpler problems embedded within these larger problems are ones you have already encountered in previous chapters. We begin with four-digit squares. Good luck!
FOUR-DIGIT SQUARES
As a preliminary skill for mastering four-digit squares, you need to be able to do 4-by-1 problems. We break these down into two 2-by-1 problems, as in the examples below:

Once you've mastered 4-by-1s you're ready to tackle four-digit squares. Let's square 4,267. Using the same method we used with two-digit and three-digit squares, we'll square the number 4,267 by rounding down by 267 to 4,000, and up by 267 to 4,534, multiplying 4,534 × 4,000 (a 4-by-1), and then adding the square of the number you went up and down by, or 2672, as illustrated below:

Now, obviously there is a lot going on in this problem. I realize it's one thing to say, "Just add the square of 267" and quite another to actually do it and remember what it was you were supposed to add to it. First of all, once you multiply 4,534 × 4 to get 18,136, you can actually say the first part of the answer out loud: "Eighteen million ..." You can do so because the original number is always rounded to the nearest thousand. Thus the largest three-digit number you will ever square in the next step is 500. The square of 500 is 250,000, so as soon as you see that the rest of your answer (in this case, 136,000) is less than 750,000, you know that the millions digit will not change.

Once you've said "eighteen million ...," you need to hold on to 136,000 before you square 267. Here's where our mnemonics from the last chapter comes to the rescue! Using the phonetic code, you can convert 136 to damage (1 = d, 3 = m, 6 = j). Now you can work on the next part of the problem and just remember damage (and that there are three zeros following-this will always be the case). If at any time in the computations you forget what the original problem is, you can either glance at the original number or, if it isn't written down, ask the audience to repeat the problem (which gives the illusion you are starting the problem over from scratch when, in fact, you have already done some of the calculations)!

You now do the three-digit square just as you learned to do before, to get 71,289. I used to have trouble remembering the hundreds digits of my answer (2, in this case). I cured this by raising my fingers (two fingers here). If you forget the last two digits (89), you can go back to the original number (4,267), square its last two digits (672 = 4,489), and take the last two digits of that.

To compute the final answer, you now add 71,289 to damage (which translates back to 136,000) resulting in 207,289, which you may now say out loud.

Let's do one more four-digit square-8,4312:

Without repeating all the detailed steps as we did in the last problem, I will point out some of the highlights of this problem. After doing the 8 × 8,862 = 70,896, note that 896 is above 750, so a carry is possible. In fact, since 4312 is greater than 4002 = 160,000, there definitely will be a carry when you do the addition to the number 896,000. Hence we can safely say aloud, "Seventy-one million ..." at this point.

When you square 431, you get 185,761. Add the 185 to the 896 to get 1,081, and say the rest of the answer. But remember that you already anticipated the carry, so you just say, "... 81 thousand ... 761." You're done!

We illustrate one more fine point with 2,7532:

Since you are rounding up to 3,000, you will be multiplying 3,000 by another number in the 2,000s. You could subtract 2,753 − 247 = 2,506, but that's a little messy. To obtain the last three digits, double 753 to get 1,506. The last three digits of this number, 506, are the last three digits of the 2,000 number: 2,506! This works because the two numbers being multiplied must add to twice the original number.

Then proceed in the usual manner of multiplying 3,000 × 2,506 = 7,518,000, convert 518 to the words light off, and say the first part of the answer out loud as "Seven million ..." You can say this with confidence since 518 is below 750, so there will not be a carry.

Thomas Fuller: Learned Men and Great Fools

It would be hard to top the physical handicap on learning experienced by Helen Keller, though the social handicap imposed on Thomas Fuller, born in Africa in 1710, comes close. He was not only illiterate; he was forced to work in the fields of Virginia as a slave and never received a single day of education. The "property" of Mrs. Elizabeth Cox, Thomas Fuller taught himself to count to 100, after which he increased his numerical powers by counting such items at hand as the grains in a bushel of wheat, the seeds in a bushel of flax, and the number of hairs in a cow's tail (2,872).

Extrapolating from mere counting, Fuller learned to compute the number of shingles a house would need to cover its roof, the number of posts needed to enclose it, and other numbers relevant to materials needed in construction. His prodigious skills grew, and with them his reputation. In his old age, two Pennsylvanians challenged him to compute, in his head, numbers that would challenge the best lightning calculators. For example, they asked: "Suppose a farmer has six sows, and each sow has six female pigs the first year, and they all increase in the same proportion, to the end of eight years, how many sows will the farmer then have?" The problem can be written as 78 × 6, that is, (7 × 7 × 7 × 7 × 7 × 7 × 7 × 7)X 6. Within ten minutes Fuller gave his response of 34,588,806, the correct answer.

Upon Fuller's death in 1790, the Columbian Centinel reported that "he could give the number of poles, yards, feet, inches, and barleycorns in any given distance, say the diameter of the earth's orbit; and in every calculation he would produce the true answer, in less time than ninety-nine men in a hundred would take with their pens." When Fuller was asked if he regretted having never gained a traditional education he responded: "No, Massa-it is best I got no learning: for many learned men be great fools."

Next, you add the square of 247. Don't forget that you can derive 247 quickly as the complement of 753. Then proceed to the final answer as you did in the previous four-digit problems.
3-BY-2 MULTIPLICATION
We saw in the 2-by-2 multiplication problems that there are several different ways to tackle the same problem. The variety of methods increases when you increase the number of digits in the problem. In 3-by-2 multiplication I find that it pays to take a few moments to look at the problem to determine the method of calculation that will put the least amount of strain on the brain.
Factoring Methods
The easiest 3-by-2 problems to compute are those in which the two-digit number is factorable.

For example:

These are great because you don't have to add anything. You just factor 56 into 8 × 7, then do a 3-by-1 (637 × 8 = 5,096), and finally a 4-by-1 (5,096 × 7 = 35,672). There are no addition steps and you don't have to store any partial results.

More than half of all two-digit numbers are factorable into numbers 11 and below, so you'll be able to use this method for many problems. Here's an example:

To multiply 853 × 11, you treat 853 as 850 + 3 and proceed as follows:

Now multiply 9,383 × 4 by treating 9,383 as 9,300 + 83, as follows:

If the two-digit number is not factorable into small numbers, examine the three-digit number to see if it can be factored:

Notice that the sequence of the multiplication problems is a 2-by-1, a 3-by-1, and finally a 4-by-1. Since these are all problems you can now do with considerable ease, this type of 3-by-2 problem should be no problem at all.

Here's another example where the two-digit number is not factorable but the three-digit number is:

Here the sequence is a 2-by-2, a 3-by-1, and a 4-by-1, though when the three-digit number is factorable by 11, you can use the elevens method for a very simple 2-by-2 (53 × 11 = 583). In this case it pays to be able to recognize when a number is divisible by 11, as described in Chapter 4.

If the two-digit number is not factorable, and the three-digit number is factorable into only a 2-by-1, the problem can still be handled easily by a 2-by-2, followed by a 4-by-1:

Here you need to see that 423 is divisible by 9, setting up the problem as 83 × 47 × 9. This 2-by-2 is not so easy, but by treating 83 as 80 + 3, you get:

Then do the 4-by-1 problem of 3,901 × 9 for your final answer of 35,109.
The Addition Method
If the two-digit number and the three-digit number cannot be conveniently factored in the 3-by-2 problem you're doing, you can always resort to the addition method:

The method requires you to add a 2-by-2 (times 10) to a 2-by-1. These problems tend to be more difficult than problems that can be factored since you have to perform a 2-by-1 while holding on to a five-digit number, and then add the results together. In fact, it is probably easier to solve this problem by factoring 721 into 103 × 7, then computing 37 × 103 × 7 = 3,811 × 7 = 26,677.

Here's another example:

Though you will usually break up the three-digit number when using the addition method, it occasionally pays to break up the two-digit number instead, particularly when the last digit of the two-digit number is a 1 or a 2, as in the following example:

This reduces the 3-by-2 to a 3-by-1, made especially easy since the second multiplication problem involves a 1. Note, too, that we were aided in multiplying a 5 by an even number, which produces an extra 0 in the answer, so there is only one digit of overlap in the addition problem.

Another example of multiplying a 5 by an even number is illustrated in the following problem:

When you multiply the 6 in 60 by the 5 in 835, it generates an extra 0 in the answer, making the addition problem especially easy.
The Subtraction Method
As with 2-by-2s, it is sometimes more convenient to solve a 3-by-2 with subtraction instead of addition, as in the following problems:

By contrast, you can compare the last subtraction method with the addition method, below, for this same problem:

My preference for tackling this problem would be to use the subtraction method because I always try to leave myself with the easiest possible addition or subtraction problem at the end. In this case, I'd rather subtract 86 than add 344, even though the 2-by-2 multiplication problem in the subtraction method above is slightly harder than the one in the addition method.

The subtraction method can also be used for three-digit numbers below a multiple of 100 or close to 1,000, as in the next two examples:

The last three digits of the answers were obtained by taking the complements of 609 − 100 = 509 and 816, respectively.

Finally, in the following illustration we break up the two-digit number using the subtraction method. Notice how we subtract 736 by subtracting 1,000 and adding back the complement:

Solve the 3-by-2 problems below, using the factoring, addition, or subtraction method. Factoring, when possible, is usually easier. The solutions appear in the back of the book.

The following 3-by-2s will appear in the five-digit squares and the 5-by-5 multiplication problems that follow.
FIVE-DIGIT SQUARES
Mastering 3-by-2 multiplication takes a fair amount of practice, but once you've managed that, you can slide right into doing five-digit squares because they simplify into a 3-by-2 problem plus a two-digit square and a three-digit square. Watch:

To square the following number:

46,7922

Treat it as:

Using the distributive law we can break this down to:

This can be stated more simply as:

4622 1 million + (46)(792)(2000) + 79222

But I do not do them in this order. In fact, I start in the middle, because the 3-by-2 problems are harder than the two-digit and three-digit squares. So in keeping with the principle of getting the hard stuff out of the way first, I do 792 × 46 × 2 and attach three zeroes on the end, as follows:

Using the subtraction method, as shown above, compute 792 × 46 = 36,432, then double that number to get 72,864. Using the phonetic code from the last chapter on the number 864 allows you to store this number as 72 Fisher.

The next step is to do 462 × 1 million, which is 2,116,000,000.

At this point, you can say, "Two billion ..."

Recalling the 72 of 72 Fisher, you add 116 million to get 188 million. Before saying this number aloud, you need to check ahead to see if there is a carry when adding Fisher, or 864, to 7922. Here you don't actually calculate 7922; rather, you determine that its product will be large enough to make the 864,000 carry. (You can guesstimate this by noting that 8002 is 640,000, which will easily make the 864,000 carry, thus you bump the 188 up a notch and say, "... 189 million.... ")

Still holding on to Fisher, compute the square of 792, using the three-digit square method (rounding up and down by 8, and so on) to get 627,264. Finally, add 627 to Fisher, or 864, to get 1,491. But since you already made the carry, drop the 1 and say, "... 491 thousand 264."

Sometimes I forget the last three digits of the answer because my mind has been so preoccupied with the larger computations. So before doing the final addition I will store the 2 of 264 on my fingers and try to remember the 64, which I can usually do because we tend to recall the most recent things heard. If this fails, I can come up with the final two digits by squaring the final two digits of the original number, 922, or 8,464, the last two digits of which are the last two digits I'm looking for: 64. (Alternatively, you can convert 264 into a word like nature.)

I know this is quite a mouthful. To reiterate the entire problem in a single illustration, here is how I computed 46,7922:

Let's look at another five-digit square example:

83,5222

As before, we compute the answer in this order:

83 × 522 × 2,000, 832 × 1 million, then 5222

For the first problem, notice that 522 is a multiple of 9. In fact, 522 = 58 × 9. Treating 83 as 80 + 3, we get:

Doubling 43,326 results in 86,652, which can be stored as 86 Julian. Since 832 = 6,889, we can say, "Six billion ..."

Adding 889 + 86 gives us 975. Before saying 975 million we check to see if Julian (652,000) will cause a carry after squaring 522. Guesstimating 5222 as about 270,000 (500 × 540), you see it will not carry. Thus you can safely say, "... 975 million ..."

Finally, square 522 in the usual way to get 272,484 and add that to Julian (652,000) for the rest of the answer: "... 924,484."

Illustrated, this problem looks like:
3-BY-3 MULTIPLICATION
In building to the grand finale of 5-by-5 multiplication, 3-by-3s are the final hurdle. As with 3-by-2s, there are a variety of methods you can use to exploit the numbers in the simplification process.
Factoring Method
We'll begin with the factoring method. Unfortunately, most three-digit numbers are not factorable into one-digit numbers, but when they are, the calculation is not too bad.

Notice the sequence involved. You simplify the 3-by-3 (829 × 288) to a 3-by-1-by-1-by-1 through the factoring of 288 into 9 × 8 × 4. This then turns into a 4-by-1-by-1 (7,461 × 8 × 4), and finally into a 5-by-1 to yield the final answer of 238,752. The beauty of this process is that there is nothing to add and nothing to store in memory. When you get to the 5-by-1, you are one step away from completion.

The 5-by-1 problem can be solved in two steps by treating 59,688 as 59,000 + 688, then adding the results of the 2-by-1 (59,000 × 4) and the 3-by-1 (688 × 4), as below:

If both three-digit numbers are factorable into 2-by-1s, then the 3-by-3 can be simplified to a 2-by-2-by-1-by-1, as in the following problem:

As usual, it is best to get the hard part of the problem over first (the 2-by-2). Once you've got this, the problem is then reduced to a 4-by-1, then to a 5-by-1.

More often than not, only one of the three-digit numbers will be factorable, in which case it is reduced to a 3-by-2-by-1, as in the following problem:

The next 3-by-3 is really just a 3-by-2 in disguise:

By doubling the 435 and cutting 624 in half, we obtain the equivalent problem:
Close-Together Method
Are you ready for something easier? The next multiplication shortcut, which we introduced in Chapter 0, is based on the following algebraic formula:

(x + a)(z + b) = z2 + za + zb + ab

Which we rewrite as:

(x + a)(z + b) = z(z + a + b) + ab

This formula is valid for any values of z, a, and b. We shall take advantage of this whenever the three-digit numbers to be multiplied (z + a) and (z + b) are both near an easy number z (typically one with lots of zeroes in it). For example, to multiply:

We treat this problem as (100 + 7)(100 + 11).

Using z = 100, a = 7, b = 11, our formula gives us:

100(100 + 7 + 11) + 7 × 11 = 100 × 118 + 77 = 11,877

I diagram the problem this way:

The numbers in parentheses denote the difference between the number and our convenient "base number" (here, z = 100). The number 118 can be obtained either by adding 107 + 11 or 111 + 7. Algebraically these sums are always equal since (z + a) + b = (z + b) + a.

With fewer words this time, here's another quickie:

Neat!

Let's up the ante a little with a higher base number:

Although this method is usually used for three-digit multiplication, we can use it for 2-by-2s as well:

Here, the base number is 70, which we multiply by 81 (78 + 3). Even the addition component is usually very simple.

We can also apply this method when the two numbers are both lower than the base number, as in the following problem where both numbers are just under 400:

The number 383 can be obtained from 396 − 13, or from 387 − 4. I would use this method for 2-by-2 problems like the ones below:

In our next example, the base number falls between the two numbers:

The number 409 is obtained from 396 + 13, or 413 − 4. Notice that since −4 and 13 are of opposite signs we must subtract 52 here.

Let's raise the ante higher still, to where the second step requires a 2-by-2 multiplication:

We note here that step 1 in the multiplication problem (600 × 658) is itself a reasonable estimate. Our method enables you to go from an estimate to the exact answer.

Also notice that in all these examples, the numbers we multiply in the first step have the same sum as the original numbers. For example, in the problem above, 900 + 829 = 1729 just as 876 + 853 = 1729. This is because:

x + [(x + a) + b] = (x + a) + (x + b)

Thus, to obtain the number to be multiplied by 900 (which you know will be 800 plus something), you need only look at the last two digits of 76 + 53 = 129 to determine 829.

In the next problem, adding 827 + 761 = 1588 tells us that we should simply multiply 800 × 788, then subtract 27 × 39 as follows:

This method is so effective that if the 3-by-3 problem you are presented with has numbers that are not close together, you can sometimes modify the problem by dividing one and multiplying the other, both by the same number, to bring them closer together. For instance, 672 × 157 can be solved by:

When the numbers being multiplied are the same (you can't get any closer than that!), notice that the close-together method produces the exact same calculations you did in our traditional squaring procedure:
Addition Method
When none of the previous methods work, I look for an addition-method possibility, particularly when the first two digits of one of the three-digit numbers is easy to work with. For example, in the problem below, the 64 of 641 is factorable into 8 × 8, so I would solve the problem as illustrated:

In a similar way, in the next problem the 42 of 427 is factorable into 7 × 6, so you can use the addition method and treat 427 as 420 + 7:

Often I break the last addition problem into two steps, as follows:

Since addition-method problems can be very strenuous, I usually go out of my way to find a method that will produce a simple computation at the end. For example, the above problem could have been done using the factoring method. In fact, that is how I would choose to do it:

The simplest addition-method problems are those in which one number has a 0 in the middle, as below:

These problems tend to be so much easier than other addition-method problems that it pays to see whether the 3-by-3 can be converted to a problem like this. For instance, 732 × 308 could have been obtained by either of the "non-zero" problems below:

We mention that another way to do this problem is by 308 × 366 × 2, and take advantage of the closeness of 308 and 366. Let's do one more toughie:
Subtraction Method
The subtraction method is one that I sometimes use when one of the three-digit numbers can be rounded up to a convenient two-digit number with a 0 at the end, as in the next problem:

Likewise in the following example:
When-All-Else-Fails Method
When all else fails, I use the following method, which is foolproof when you can find no other method to exploit the numbers. In the when-all-else-fails method, the 3-by-3 problem is broken down into three parts: a 3-by-1, a 2-by-1, and a 2-by-2. As you do each computation, you sum the totals as you go. These problems are difficult, especially if you cannot see the original number. In my presentation of 3-by-3s and 5-by-5s, I have the problems written down, but I do all the calculations mentally.

Here's an example:

In practice, the calculation actually proceeds as shown below. Sometimes I use the phonetic code to store the thousands digits (e.g., 447 = our rug) and put the hundreds digit (1) on my fingers:

Let's do another example, but this time I'll break up the first number. (I usually break up the larger one so that the addition problem is easier.)

The following problems are embedded in the 5-by-5 multiplication problems in the next section:
5-BY-5 MULTIPLICATION
The largest problem we will attempt is the mental multiplication of two five-digit numbers. To do a 5-by-5, you need to have mastered 2-by-2s, 2-by-3s, and 3-by-3s, as well as the phonetic code. Now it is just a matter of putting it all together. As you did in the five-digit square problems, you will use the distributive law to break down the numbers. For example:

Based on this you can break down the problem into four easier multiplication problems, which I illustrate below in criss-cross fashion as a 2-by-2, two 3-by-2s, and finally a 3-by-3, summing the results for a grand total. That is,

(27 × 52) million
+ [(27 × 196) + (52 × 639)] thousand
+ (639 × 196)

As with the five-digit squares, I begin in the middle with the 3-by-2s, starting with the harder 3-by-2:

Committing 33,228 to memory with the mnemonic Mom, no knife, you then turn to the second 3-by-2:

2. 27 × 196 = 27 × (200 − 4) = 5,400 − 108 = 5,292

and add it to the number that you are storing:

for a new total, which we store as:

"Movie lines" (38 million, 520 thousand)

Holding on to movie lines, compute the 2-by-2:

4. 52 × 27 = 52 × 9 × 3 = 1,404

At this point, you can give part of the answer. Since this 2-by-2 represents 52 × 27 million, 1,404 represents 1 billion, 404 million. Since 404 million will not cause a carry, you can safely say, "One billion ..."

5. 404 + "Movie" (38) = 442

In this step you add 404 to movie (38) to get 442, at which point you can say, "... 442 million...." You can say this because you know 442 will not carry-you've peeked ahead at the 3-by-3 to see whether it will cause 442 to carry to a higher number. If you found that it would carry, you would say, "443 million." But since lines is 520,000 and the 3-by-3 (639 × 196) will not exceed 500,000 (a rough guesstimate of 600 × 200 = 120,000 shows this), you are safe.

6. 639 × 196 = 639 × 7 × 7 × 4 = 4,473 × 7 × 4 = 31,311 × 4 = 125,244

While still holding on to lines, you now compute the 3-by-3 using the factoring method, to get 125,244. You might convert 244 into a word like nearer. The final step is a simple addition of:

7. 125,244 + "Lines" (520,000)

This allows you to say the rest of the answer: "... 645,244." Since a picture is worth a thousand calculations, here's our picture of how this would look:

I should make a parenthetical note here that I am assuming in doing 5-by-5s that you can write down the problem on a blackboard or piece of paper. If you can't, you will have to create a mnemonic for each of the four numbers. For example, in the last problem, you could store the problem as:

Then you would multiply lion × jump, dopish × neck, lion × neck, and finally dopish × jump. Obviously this would slow you down a bit, but if you want the extra challenge of not being able to see the numbers, you can still solve the problem.

We conclude with one more 5-by-5 multiplication:

The steps are the same as those in the last problem. You start with the harder 3-by-2 and store the answer with a mnemonic:

Then you compute the other 3-by-2.

2. 838 × 45 = 838 × 5 × 9 = 4,190 × 9 = 37,710

Summing the 3-by-2s you commit the new total to memory.

4. 79 × 45 = 79 × 9 × 5 = 711 × 5 = 3,555

The 2-by-2 gives you the first digit of the final answer, which you can say out loud with confidence: "Three billion ..."

5. 555 + "Face" (80) = 635

The millions digits of the answer involve a carry from 635 to 636, because Panama (923) needs only 77,000 to cause it to carry, and the 3-by-3 (838 × 547) will easily exceed that figure. So you say: "... 636 million ..."

The 3-by-3 is computed using the addition method:

And in the next step you add this total to Panama (923,000):

Since you've already used the 1 in the carry to 636 million, you just say the thousands: "... 381 thousand ... 386," and take a bow!

This problem may be illustrated in the following way:
Chapter 9 Presto-digit-ation: The Art of
Mathematical Magic
Playing with numbers has brought me great joy in life. I find that arithmetic can be just as entertaining as magic. But understanding the magical secrets of arithmetic requires algebra. Of course, there are other reasons to learn algebra (SATs, modeling real-world problems, computer programming, and understanding higher mathematics, to name just a few), but what first got me interested in algebra was the desire to understand some mathematical magic tricks, which I now present to you!
PSYCHIC MATH
Say to a volunteer in the audience, "Think of a number, any number." And you should also say, "But to make it easy on yourself, think of a one-digit or two-digit number." After you've reminded your volunteer that there's no way you could know her number, ask her to:

1. double the number,

2. add 12,

3. divide the total by 2,

4. subtract the original number.

Then say, "Are you now thinking of the number six?" Try this one on yourself first and you will see that the sequence always produces the number 6 no matter what number is originally selected.
Why This Trick Works
This trick is based on simple algebra. In fact, I sometimes use this as a way to introduce algebra to students. The secret number that your volunteer chose can be represented by the letter x. Here are the functions you performed in the order you performed them:

1. 2x (double the number)

2. 2x + 12 (then add 12)

3. (2x + 12) ÷ 2 = × + 6 (then divide by 2)

4. x + 6 - x = 6 (then subtract original number)

So no matter what number your volunteer chooses, the final answer will always be 6. If you repeat this trick, have the volunteer add a different number at step 2 (say 18). The final answer will be half that number (namely 9).
THE MAGIC 1089!
Here is a trick that has been around for centuries. Have your audience member take out paper and pencil and:

1. secretly write down a three-digit number where the digits are decreasing (like 851 or 973),

2. reverse that number and subtract it from the first number,

3. take that answer and add it to the reverse of itself.

At the end of this sequence, the answer 1089 will magically appear, no matter what number your volunteer originally chose. For example:
Why This Trick Works
No matter what three-digit number you or anyone else chooses in this game, the final result will always be 1089. Why? Let abc denote the unknown three-digit number. Algebraically, this is equal to:

100a + 10b + c

When you reverse the number and subtract it from the original number, you get the number cba, algebraically equal to:

100c + 10b + a

Upon subtracting cba from abc, you get:

Hence, after subtracting in step 2, we must have one of the following multiples of 99: 198, 297, 396, 495, 594, 693, 792, or 891, each one of which will produce 1089 after adding it to the reverse of itself, as we did in step 3.
MISSING-DIGIT TRICKS
Using the number 1,089 from the last effect, hand a volunteer a calculator and ask her to multiply 1089 by any three-digit number she likes, but not to tell you the three-digit number. (Say she secretly multiplies 1,089 × 256 = 278,784.) Ask her how many digits are in her answer. She'll reply, "Six."

Next you say: "Call out five of your six digits to me in any order you like. I shall try to determine the missing digit."

Suppose she calls out, "Two ... four ... seven ... eight ... eight." You correctly tell her that she left out the number 7.

The secret is based on the fact that a number is a multiple of 9 if and only if its digits sum to a multiple of 9. Since 1 + 0 + 8 + 9 = 18 is a multiple of 9, then so is 1089. Thus 1089 times any whole number will also be a multiple of 9. Since the digits called out add up to 29, and the next multiple of 9 greater than 29 is 36, our volunteer must have left out the number 7 (since 29 + 7= 36).

There are more subtle ways to force the volunteer to end up with a multiple of 9. Here are some of my favorites:

1. Have the volunteer randomly choose a six-digit number, scramble its digits, then subtract the smaller six-digit number from the larger one. Since we're subtracting two numbers with the same mod sum (indeed, the identical sum of digits), the resulting difference will have a mod sum of 0, and hence be a multiple of 9. Then continue as before to find the missing digit.

2. Have the volunteer secretly choose a four-digit number, reverse the digits, then subtract the smaller number from the larger. (This will be a multiple of 9.) Then multiply this by any three-digit number, and continue as before.

3. Ask the volunteer to multiply one-digit numbers randomly until the product is seven digits long. This is not "guaranteed" to produce a multiple of 9, but in practice it will do so at least 90% of the time (the chances are high that the one-digit numbers being multiplied include a 9 or two 3s or two 6s, or a 3 and a 6). I often use this method in front of mathematically advanced audiences who might see through other methods.

There is one problem to watch out for. Suppose the numbers called out add up to a multiple of 9 (say 18). Then you have no way of determining whether the missing digit is 0 or 9. How do you remedy that? Simple-you cheat! You merely say, "You didn't leave out a zero, did you?" If she did leave out a 0, you have completed the trick successfully. If she did not leave out the 0, you say: "Oh, it seemed as though you were thinking of nothing! You didn't leave out a one, two, three, or four, did you?" She'll either shake her head, or say no. Then you follow with, "Nor did you leave out a five, six, seven, or eight, either. You left out the number nine, didn't you?" She'll respond in the affirmative, and you will receive your well-deserved applause!
LEAPFROG ADDITION
This trick combines a quick mental calculation with an astonishing prediction. Handing the spectator a card with ten lines, numbered 1 through 10, have the spectator think of two positive numbers between 1 and 20, and enter them on lines 1 and 2 of the card. Next have the spectator write the sum of lines 1 and 2 on line 3, then the sum of lines 2 and 3 on line 4, and so on as illustrated below.

Finally, have the spectator show you the card. At a glance, you can tell him the sum of all the numbers on the card. For instance, in our example, you could instantly announce that the numbers sum up to 671 faster than the spectator could do using a calculator! As a kicker, hand the spectator a calculator, and ask him to divide the number on line 10 by the number on line 9. In our example, the quotient = 1.616.... Have the spectator announce the first three digits of the quotient, then turn the card over (where you have already written a prediction). He'll be surprised to see that you've already written the number 1.61!
Why This Trick Works
To perform the quick calculation, you simply multiply the number on line 7 by 11. Here 61 × 11 = 671. The reason this works is illustrated in the table below. If we denote the numbers on lines 1 and 2 by × and y, respectively, then the sum of lines 1 through 10 must be 55x + 88y, which equals 11 times (5x + 8y), that is, eleven times the number on line 7.

As for the prediction, we exploit the fact that for any positive numbers, a, b, c, d, if a/b * c/d, then it can be shown that the fraction you get by "adding fractions badly" (i.e., adding the numerators and adding the denominators) must lie in between the original two fractions. That is,

Thus the quotient of line 10 divided by line 9 (21x + 34y)/(13% + 21y), must lie between

Hence, the ratio must begin with the digits 1.61, as predicted.

In fact, if you continue the leapfrog process indefinitely, the ratio of consecutive terms gets closer and closer to

a number with so many amazingly beautiful and mysterious properties that it is often called the golden ratio.
MAGIC SQUARES
Are you ready for a challenge of a different sort? Below you will find what is called a magic square. There has been much written on magic squares and how to construct them, going back as far as ancient China. Here we describe a way to present magic squares in an entertaining fashion. This is a routine I've been doing for years.

I bring out a business card with the following written on the back:

I say, "This is called a magic square. In fact, it's the smallest magic square you can create, using the numbers one through sixteen. You'll notice that every row and every column adds to the same number-thirty-four. Now I've done such an extensive study on magic squares that I propose to create one for you right before your very eyes."

I then ask someone from the audience to give me any number larger than 34. Let's suppose she says 67.

I then bring out another business card and draw a blank 4-by-4 grid, and place the number 67 to the right of it. Next I ask her to point to the squares, one at a time, in any order. As she points to an empty box, I immediately write a number inside it. The end result looks like this:

I continue: "Now with the first magic square, every row and column added to thirty-four. [I usually put the thirty-four card away at this point.] Let's see how we did with your square." After checking that each row and column adds up to 67, I say: "But I did not stop there. For you, I decided to go one step further. Notice that both diagonals add up to sixty-seven!" Then I point out that the four squares in the upper left corner sum to 67 (16 + 19 + 22 + 10 = 67), as do the other three four-square corners, the four squares in the middle, and the four corner squares! "They all sum to sixty-seven. But don't take my word for any of this. Please keep this magic square as a souvenir from me-and check it out for yourself!"
HOW TO CONSTRUCT A MAGIC SQUARE
You can create a magic square that sums to any number by taking advantage of the original magic square that sums to 34. Keep that square within eyeshot while you construct the volunteer's magic square. As you draw the 4-by-4 grid, mentally perform the calculations of steps 1 and 2:

1. Subtract 34 from the given number (e.g., 67 − 34 = 33).

2. Divide this number by 4 (e.g., 33 ÷ 4 = 8 with a remainder of 1). The quotient is the first "magic" number. The quotient plus the remainder is the second "magic" number. (Here our magic numbers are 8 and 9.)

3. When the volunteer points out a square, inconspicuously look at the 34-square and see what is in the corresponding square. If it is a 13, 14, 15, or 16, add the second number to it (e.g., 9). If not, add the first magic number (e.g., 8).

4. Insert the appropriate number until the magic square is completed.

Note that when the given number is even, but not a multiple of 4, then your first and second magic numbers will be the same, so you'll have just one magic number to add to the numbers in your 34-square.
Why This Trick Works
The reason this method works is based on the fact that every row, column, diagonal (and more) from the originally displayed magic square sums to 34. Suppose the given number had been 82. Since 82 − 34 = 48 (and 48 ÷ 4 = 12), we would add 12 to each square. Then every group of four that had previously summed to 34 would add to 34 + 48 = 82. See the magic square below.

On the other hand, if the given number were 85, our magic numbers would be 12 and 15, so we would be adding 3 more to the squares showing 13, 14, 15, and 16. Since each row, column, and group of four contains exactly one of these numbers, each group of four would now add to 34 + 48 + 3 = 85 in the following magic square.

As an interesting piece of mathemagical trivia, let me point out another astonishing property of the famous 3-by-3 magic square below.

Not only do the rows, columns, and diagonals add up to 15, but if you treat the rows of the magic square as three-digit numbers, you can verify on your calculator that 4922 + 3572 + 8162 = 2942 + 7532 + 6182. Also, 4382 + 9512 + 2762 = 8342 + 1592 + 6722. If you are curious about why this property happens, you might want to explore my paper Magic "Squares" Indeed! (included in the bibliography).
QUICK CUBE ROOTS
Ask someone to select a two-digit number and keep it secret. Then have him cube the number; that is, multiply it by itself twice (using a calculator). For instance, if the secret number is 68, have the volunteer compute 68 × 68 × 68 = 314,432. Then ask the volunteer to tell you his answer. Once he tells you the cube, 314,432, you can instantly reveal the original secret number, the cube root, 68. How?

To calculate cube roots, you need to learn the cubes from 1 to 10:

Once you have learned these, calculating the cube roots is as easy as p. For instance, with this example problem:

What is the cube root of 314,432?

Seems like a pretty tough one to begin with, but don't panic, it's actually quite simple. As usual, we'll take it one step at a time:

1. Look at the magnitude of the thousands number (the numbers to the left of the comma), 314 in this example.

2. Since 314 lies between 63 = 216 and 73 = 343, the cube root lies in the 60s (since 603 = 216,000 and 703 = 343,000). Hence the first digit of the cube root is 6.

3. To determine the last digit of the cube root, note that only the number 8 has a cube that ends in 2 (83 = 512), so the last digit ends in 8.

Therefore, the cube root of 314,432 is 68. Three simple steps and you're there. Notice that every digit, 0 through 9, appears once among the last digits of the cubes. (In fact, the last digit of the cube root is equal to the last digit of the cube of the last digit of the cube. Go figure out that one!)

Now you try one for practice:

What is the cube root of 19,683?

1. 19 lies between 8 and 27 (23 and 33).

2. Therefore the cube root is 20-something.

3. The last digit of the problem is 3, which corresponds to 343 = 73, so 7 is the last digit.

The answer is 27.

Notice that our derivation of the last digit will only work if the original number is the cube root of a whole number. For instance, the cube root of 19,684 is 27.0004572 ... definitely not 24. That's why we included this in our mathemagical magic section and not in an earlier chapter. (Besides, the calculation goes so fast, it seems like magic!)
SIMPLIFIED SQUARE ROOTS
Square roots can also be calculated easily if you are given a perfect square. For instance, if someone told you that the square of a two-digit number was 7569, you could immediately tell her that the original number (the square root) is 87. Here's how.

1. Look at the magnitude of the "hundreds number" (the numbers preceding the last two digits), 75 in this example.

2. Since 75 lies between 82 (8 × 8 = 64) and 92 (9 × 9 = 81), then we know that the square root lies in the 80s. Hence the first digit of the square root is 8. Now there are two numbers whose square ends in 9: 32 = 9 and 72 = 49. So the last digit must be 3 or 7. Hence the square root is either 83 or 87. Which one?

3. Compare the original number with the square of 85 (which we can easily compute as 80 × 90 + 25 = 7225. Since 7569 is larger than 7225, the square root is the larger number, 87.

Let's do one more example.

What is the square root of 4761?

Since 47 lies between 62 = 36 and 72 = 49, the answer must be in the 60s. Since the last digit of the square is 1, the last digit of the square root must be 1 or 9. Since 4761 is greater than 652 = 42 25, the square root must be 69. As with the previous cube root trick, this method can be applied only when the original number given is a perfect square.
AN "AMAZING" SUM
The following trick was first shown to me by James "the Amazing" Randi, who has used it effectively in his magic. Here, the magician is able to predict the total of four randomly chosen three-digit numbers.

To prepare this trick you will need three sets of nine cards each, and a piece of paper with the number 2247 written down on it and then sealed in an envelope. Next, on each of the three sets of cards do the following:

On Set A write the following numbers, one number on each card:

4286 5771 9083 6518 2396 6860 2909 5546 8174

On Set A write the following numbers:

5792 6881 7547 3299 7187 6557 7097 5288 6548

On Set C write the following numbers:

2708 5435 6812 7343 1286 5237 6470 8234 5129

Select three people in the audience and give each one a set of cards. Have each of your volunteers randomly pick one of the nine cards they hold. Let's say they choose the numbers 4286, 5792, and 5435. Now, in the sequence, have each one call out one digit from the four-digit number, first person A, then person B, and finally person C. Say they call out the numbers 8, 9, and 5. Write down the numbers 8, 9, and 5 (895) and say, "You must admit that this number was picked entirely at random and could not possibly have been predicted in advance."

Next, have the three people call out a different number from their cards. Say they call out 4, 5, and 3. Write 453 below 895. Then repeat this two more times with their remaining two numbers, resulting in four three-digit numbers, such as:

Next have someone add the four numbers and announce the total. Then have someone open the envelope to reveal your prediction. Take a bow!
Why This Trick Works
Look at the numbers in each set of cards and see if you can find anything consistent about them. Each set of numbers sums to the same total. Set A numbers total to 20. Set B numbers total to 23. Set C numbers total to 17. With person C's numbers in the right column totaling to 17, you will always put down the 7 and carry the 1. With person B's numbers totaling to 23, plus the 1, you will always put down the 4 and carry the 2. Finally with person A's numbers totaling to 20, adding the 2 gives you a total of 2247!
A DAY FOR ANY DATE
We conclude our book with one of the classic feats of mental calculation: how to figure out the day of the week of anyone's birthday. This is actually a very practical skill. It's not every day that someone will ask you to square a three-digit number, but hardly a day goes by without somebody mentioning a date to you in the past or future. With just a little bit of practice, you can quickly and easily determine the day of the week of practically any date in history.

First we assign a code number to every day of the week. They are easy to remember:

I have found the above list to be easy to remember, especially since 2's day is Tuesday. The other days can be given similar mnemonics: "1 day is Monday, 4's day is Thursday, and 5 day is Friday. For Wednesday, notice that if you hold up three fingers, you get the letter w. For the weekends (at the end of our list), you might think of Saturday as 6-urday and Sunday as 7 day (especially if you pronounce the ? softly). Or to remember the zero, you could think of Sunday as none-day or nun-day!

Next we need a code for every month of the year. These month codes are used for every year, with two exceptions. In a leap year (like 2000 or 2004 or 2008 or ...) the month code for January is 5, and the month code for February is 1. To make the month codes easier to remember, we provide a table of mnemonics below.

Now let's calculate the day of the week for any date in 2006. After that, we will describe 2007, then 2008, and so on, for the rest of your life. Once all future dates are taken care of, we can look back into the past and determine the days of the week for any date in the 1900s or any other century.

Every year is assigned a code number, and for 2006 that year code happens to be 0 (see this page).

Now, to calculate the day of the week, you simply add the month code plus the date code plus the year code. Thus for December 3, 2006, we compute

Month Code + Date + Year Code = 4 + 3 + 0 = 7

Hence, this date will be on Day 7, which is Sunday.

How about November 18, 2006? Since November has a month code of 2, we have

Month Code + Date + Year Code = 2 + 18 + 0 = 20.

Now since the week repeats every seven days, we can subtract any multiple of 7 from our answer (7, 14, 21, 28, 35, ...) and this will not change the day of the week. So our final step is to subtract the biggest multiple of 7 to get 20 − 14 = 6. Hence November 18, 2006, occurs on Saturday.

How about 2007? Well, what happens to your birthday as you go from one year to the next? For most years, there are 365 days, and since 364 is a multiple of 7 (7 × 52 = 364), then the day of the week of your birthday will shift forward by one day in most years. If there are 366 days between your birthdays, then it will shift forward by two days. Hence, for 2007 we calculate the day of the week just as before, but now we use a year code of 1. Next, 2008 is a leap year. (Leap years occur every four years, so 2000, 2004, 2008, 2012, ..., 2096 are the leap years of the twenty-first century.) Hence, for 2008, the year code increases by two, so it will have a year code of 3. The next year, 2009, is not a leap year, so the year code increases by one, to 4.

Thus, for example, May 2, 2007, has

Month Code + Date + Year Code = 0 + 2 + 1=3

so this date is a Wednesday.

For September 9, 2008, we have

Month Code + Date + Year Code = 4 + 9 + 3= 16

Subtracting the biggest multiple of 7, we have 16 − 14 = 2, so this date is a Tuesday.

On the other hand, January 16, 2008, is a leap year January, so its month code will be 5 instead of 6. Thus, we have

Month Code + Date + Year Code = 5 + 16 + 3 = 24

and therefore occurs on day 24 − 21 = 3, which is Wednesday. For your reference, we have listed all of the year codes for the twenty-first century in the figure on the following page.

The good news is that we do not have to memorize this table. We can mentally calculate the year code for any date between 2000 and 2099. For the year code of 2000 + x, we simply take the number x/4 (ignoring any remainder) and add that to x. The year code can be reduced by subtracting any multiple of 7.

For example, with 2061, we see that = 15 (with a remainder of 1 that we ignore). Thus 2061 has a year code of 61 + 15 = 76. And since we can subtract any multiple of 7, we use the simpler year code of 76 − 70 = 6.

Hence March 19, 2061, has

Month Code + Date + Year Code = 2 + 19 + 6 = 27

Subtracting 27 − 21 = 6 tells us that this date will take place on Saturday.

What about birth dates between 1900 and 1999? Do the problem exactly as in the previous calculation, but shift the final answer forward by one day (or simply add 1 to the year code). Thus March 19, 1961, occurred on a Sunday.

For the date December 3, 1998, we have = 24 (with a remainder of 2 that we ignore). Thus 1998 has year code 98 + 24 + 1 = 123, where the plus one is added for dates in the 1900s. Next, subtract the biggest multiple of 7. For handy reference, here are the multiples of 7 that you might need:

7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, ...

Since 123 − 119 = 4, 1998 has a year code of 4. Therefore, December 3, 1998, has

Month Code + Date + Year Code = 4 + 3 + 4 = 11

and 11 − 7 = 4, so this date occurred on a Thursday.

For dates in the 1800s, we add 3 to the year code. For example, Charles Darwin and Abraham Lincoln were both born on February 12, 1809. Since 2009 has a year code of 4, then 1809 has a year code of 4 + 3 = 7, which can be reduced to 0. Thus, February 12, 1809, has

Month Code + Date + Year Code = 2 + 12 + 0= 14

and 14 − 14 = 0, so they were both born on a Sunday.

For dates in the 2100s, we add 5 to the year code (or, equivalently, subtract 2 from the year code). For example, since 2009 has a year code of 4, then 2109 has a year code of 4 + 5 = 9, which, after subtracting 7 is the same as a year code of 2. Dates in the 1700s are treated just like in the 2100s (by adding 5 or subtracting 2) but we need to be careful. The days that we are calculating are based on the Gregorian calendar, established in 1582. But this calendar was not adopted by England (and the American colonies) until 1752, when Wednesday, September 2, was followed by Thursday, September 14. Let's verify that September 14, 1752, was indeed a Thursday. Since 2052 has a year code of 2 (from this page, or by 52 + 13 − 63 = 2), then 1752 has a year code of 0. Thus September 14, 1752, has

Month Code + Date + Year Code = 4 + 14 + 0= 18

and 18 − 14 = 4, so it was indeed a Thursday. However, our formula will not work for earlier dates (which were governed by the Julian calendar).

Finally, we remark that under the Gregorian calendar, a leap year occurs every four years, with the exception of years that are divisible by 100, although there is an exception to the exception: years divisible by 400 are leap years. Thus 1600, 2000, 2400, and 2800 are leap years, but 1700, 1800, 1900, 2100, 2200, 2300, and 2500 are not leap years. In fact, the Gregorian calendar repeats every 400 years, so you can convert any future date into a date near 2000. For example, March 19, 2361, and March 19, 2761, will have the same day of the week as March 19, 1961, which, as we already determined, is a Sunday.

Determine the days of the week for the following dates:

1. January 19, 2007

2. February 14, 2012

3. June 20, 1993

4. September 1, 1983

5. September 8, 1954

6. November 19, 1863

7. July 4, 1776

8. February 22, 2222

9. June 31, 2468

10. January 1, 2358

*In a leap year, the code for January is 5 and the code for February is 1.
Chapter ∞ Epilogue: How Math Helps Us Think About Weird Things by Michael Shermer
As the publisher of Skeptic magazine, the executive director of the Skeptics Society, and a Scientific American editor with a monthly column entitled "Skeptic," I receive volumes of mail from people who challenge me with stories about their unusual experiences, such as haunted houses, ghosts, near-death and out-of-body experiences, UFO sightings, alien abductions, death-premonition dreams, and much more.

The most interesting stories to me are those about highly improbable events. The implication of the letter writer's tale is that if I cannot offer a satisfactory natural explanation for that particular event, the general principle of supernaturalism is preserved. A common story is the one about having a dream about the death of a friend or relative, then a phone call comes the next day about the unexpected death of the person in the dream. What are the odds of that? I am asked.

Here is where math comes in to play in our thinking and reasoning. I don't want to pontificate about how mathematics in school teaches students to think critically, because that has probably been said by nearly every math teacher in nearly every math class in nearly every school in America, at least once a year. I want to give some specific examples of how I use very simple math to help me on the job in explaining why weird things happen to people.

Although I cannot always explain such specific occurrences, a principle of probability called the Law of Large Numbers shows that an event with a low probability of occurrence in a small number of trials has a high probability of occurrence in a large number of trials. Or, as I like to say, million-to-one odds happen 295 times a day in America.

Let's begin with death premonitions. Here is a little "back-of-the-envelope" calculation I did. Psychologists tell us that the average person has about five dreams per day, which equals 1,825 dreams per year. Even if we remember only one out of ten dreams, that still results in 182.5 remembered dreams a year. There are 295 million Americans, so that means there will be 53.8 billion remembered dreams per year. Now, anthropologists and sociologists tell us that each of us knows about 150 people fairly well (that is, the average person has about 150 names in his or her address book about which can be said something significant). That means there is a network grid of 44.3 billion personal relationships among those 295 million Americans. The annual U.S. death rate from all causes across all ages is .008, or 2.6 million per year. It is inevitable that some of those 53.8 billion remembered dreams will be about some of these 2.6 million deaths among the 295 million Americans and their 44.3 billion relationships. It would be a miracle, in fact, if some "death premonition" dreams did not come true.

Even if my numbers are off, even way off, the point still stands. What are the odds of a death premonition dream coming true? Pretty darn good.

There is an additional psychological factor at work here called the confirmation bias, where we notice the hits and ignore the misses in support of our favorite beliefs. The confirmation bias explains how conspiracy theories work, for example. People who adhere to a particular conspiracy theory (9/11 was orchestrated by the Bush administration in order to launch a war in the Middle East), will look for and find little factoids here and there that seem to indicate that it might be true (Bush sat in that classroom reading to the children about goats as if he knew he was safe), while ignoring the vast body of evidence that points to another more likely explanation (Osama bin Laden and his band of international terrorists orchestrated 9/11). The confirmation bias also helps explain how astrologers, tarot-card readers, and psychics seem so successful at "reading" people. People who get a reading are likely to remember the handful of hits and forget the countless misses. When such hits and misses are actually counted-which I once did for an ABC television special on psychics-it turns out that there is nothing more than guessing and random chance at work.

In the case of the death-premonition dream, if just a couple of these people who have such dreams recount their miraculous tales in a public forum (next on Oprah!), the paranormal seems vindicated. In fact, it is nothing more than the laws of probability writ large.

This mathematical process of thinking about weird things led me to another back-of-the-envelope calculation about miracles. People typically invoke the term miracle to describe really unusual events, events whose odds of occurring are a "million to one." Okay, let's take that as our benchmark definition. A miracle is an event whose odds of occurrence are a million to one. Now, as we go about our day, we see and hear things happen about once per second. That is, data from the world and events around us are pouring in through our senses at a rate of about one per second. If we are awake and alert and out in the world for, say, eight hours a day, that means there are thirty thousand bits of data per day, or one million events per month that we take in. The vast majority of these data and events are completely meaningless, of course, and our brains are wired to filter out and forget the vast majority of them because we would be overwhelmed otherwise. But, in the course of a month, we would expect million-to-one odds to happen at least once. Add to that the confirmation bias where we will remember the most unusual events and forget all the rest, and it is inevitable that someone somewhere will report a miracle every month. And the tabloids will be there to record it!

This is a short primer on how science works. In our quest to understand how the world works, we need to determine what is real and what is not, what happens by chance and what happens because of some particular predictable cause. The problem we face is that the human brain was designed by evolution to pay attention to the really unusual events and ignore the vast body of data flowing by; as such, thinking statistically and with probabilities does not come naturally. Science, to that extent, does not come naturally. It takes some training and practice.

In addition, there are those pesky cognitive biases I mentioned, such as the confirmation bias. And there are others. The data do not just speak for themselves. Data are filtered through very subjective and biased brains. The self-serving bias, for example, dictates that we tend to see ourselves in a more positive light than others see us: national surveys show that most business people believe they are more moral than other business people, while psychologists who study moral intuition think they are more moral than other such psychologists. In one College Entrance Examination Board survey of 829,000 high school seniors, 0 percent rated themselves below average in "ability to get along with others," while 60 percent put themselves in the top 10 percent (presumably not all were from Lake Woebegone). And according to a 1997 U.S. News & World Report study on who Americans believe are most likely to go to heaven, 52 percent said Bill Clinton, 60 percent thought Princess Diana, 65 percent chose Michael Jordan, 79 percent selected Mother Teresa, and, at 87 percent, the person most likely to go to heaven was the survey taker!

Princeton University psychology professor Emily Pronin and her colleagues tested a bias called blind spot, in which subjects recognized the existence and influence in others of eight different cognitive biases, but they failed to see those same biases in themselves. In one study on Stanford University students, when asked to compare themselves to their peers on such personal qualities as friendliness and selfishness, they predictably rated themselves higher. Even when the subjects were warned about the better-than-average bias and were asked to reevaluate their original assessments, 63 percent claimed that their initial evaluations were objective, and 13 percent even claimed that they were originally too modest! In a second study, Pronin randomly assigned subjects high or low scores on a "social intelligence" test. Unsurprisingly, those given the high marks rated the test fairer and more useful than those receiving low marks. When asked if it was possible that they had been influenced by the score on the test, subjects responded that other participants had been far more biased than they were. In a third study in which Pronin queried subjects about what method they used to assess their own and others' biases, she found that people tend to use general theories of behavior when evaluating others, but use introspection when appraising themselves; however, in what is called the introspection illusion, people do not believe that others can be trusted to do the same. Okay for me but not for thee.

The University of California at Berkeley psychologist Frank J. Sulloway and I made a similar discovery of an attribution bias in a study we conducted on why people say they believe in God, and why they think other people believe in God. In general, most people attribute their own belief in God to such intellectual reasons as the good design and complexity of the world, whereas they attribute others' belief in God to such emotional reasons as it is comforting, gives meaning, and that they were raised to believe. Political scientists have made a similar discovery about political attitudes, where Republicans justify their conservative attitudes with rational arguments but claim that Democrats are "bleeding-heart liberals," and where Democrats claim that their liberal attitudes are the most rational but claim that Republicans are "heartless."

How does science deal with such subjective biases? How do we know when a claim is bogus or real? We want to be open-minded enough to accept radical new ideas when they occasionally come along, but we don't want to be so open-minded that our brains fall out. This problem led us at the Skeptics Society to create an educational tool called the Baloney Detection Kit, inspired by Carl Sagan's discussion of how to detect "baloney" in his marvelous book The Demon-Haunted World. In this Baloney Detection Kit, we suggest ten questions to ask when encountering any claim that can help us decide if we are being too open-minded in accepting it or too closed-minded in rejecting it.

1. How reliable is the source of the claim? As Daniel Kevles showed so effectively in his 1999 book The Baltimore Affair, in investigating possible scientific fraud there is a boundary problem in detecting a fraudulent signal within the background noise of mistakes and sloppiness that is a normal part of the scientific process. The investigation of research notes in a laboratory affiliated with Nobel laureate David Baltimore by an independent committee established by Congress to investigate potential fraud revealed a surprising number of mistakes. But science is messier than most people realize. Baltimore was exonerated when it became clear that there was no purposeful data manipulation.

2. Does this source often make similar claims? Pseudoscientists have a habit of going well beyond the facts, so when individuals make numerous extraordinary claims, they may be more than just iconoclasts. This is a matter of quantitative scaling, since some great thinkers often go beyond the data in their creative speculations. Cornell's Thomas Gold is notorious for his radical ideas, but he has been right often enough that other scientists listen to what he has to say. Gold proposes, for example, that oil is not a fossil fuel at all, but the by-product of a deep hot biosphere. Hardly any earth scientists I have spoken with take this thesis seriously, yet they do not consider Gold a crank. What we are looking for here is a pattern of fringe thinking that consistently ignores or distorts data.

3. Have the claims been verified by another source? Typically pseudoscientists will make statements that are unverified, or verified by a source within their own belief circle. We must ask who is checking the claims, and even who is checking the checkers. The biggest problem with the cold fusion debacle, for example, was not that scientists Stanley Pons and Martin Fleischman were wrong; it was that they announced their spectacular discovery before it was verified by other laboratories (at a press conference no less), and, worse, when cold fusion was not replicated, they continued to cling to their claim.

4. How does the claim fit with what we know about how the world works? An extraordinary claim must be placed into a larger context to see how it fits. When people claim that the pyramids and the Sphinx were built more than ten thousand years ago by an advanced race of humans, they are not presenting any context for that earlier civilization. Where are the rest of the artifacts of those people? Where are their works of art, their weapons, their clothing, their tools, their trash? This is simply not how archaeology works.

5. Has anyone gone out of the way to disprove the claim, or has only confirmatory evidence been sought? This is the confirmation bias, or the tendency to seek confirmatory evidence and reject or ignore disconfirmatory evidence. The confirmation bias is powerful and pervasive and is almost impossible for any of us to avoid. It is why the methods of science that emphasize checking and rechecking, verification and replication, and especially attempts to falsify a claim are so critical.

6. Does the preponderance of evidence converge to the claimant's conclusion, or a different one? The theory of evolution, for example, is proven through a convergence of evidence from a number of independent lines of inquiry. No one fossil, no one piece of biological or paleontological evidence has "evolution" written on it; instead there is a convergence of evidence from tens of thousands of evidentiary bits that adds up to a story of the evolution of life. Creationists conveniently ignore this convergence, focusing instead on trivial anomalies or currently unexplained phenomena in the history of life.

7. Is the claimant employing the accepted rules of reason and tools of research, or have these been abandoned in favor of others that lead to the desired conclusion? UFOlogists suffer this fallacy in their continued focus on a handful of unexplained atmospheric anomalies and visual misperceptions by eyewitnesses, while conveniently ignoring the fact that the vast majority (90 to 95 percent) of UFO sightings are fully explicable with prosaic answers.

8. Has the claimant provided a different explanation for the observed phenomena, or is it strictly a process of denying the existing explanation? This is a classic debate strategy-criticize your opponent and never affirm what you believe in order to avoid criticism. But this stratagem is unacceptable in science. Big Bang skeptics, for example, ignore the convergence of evidence of this cosmological model, focus on the few flaws in the accepted model, and have yet to offer a viable cosmological alternative that carries a preponderance of evidence in favor of it.

9. If the claimant has proffered a new explanation, does it account for as many phenomena as the old explanation? The HIV-AIDS skeptics argue that lifestyle, not HIV, causes AIDS. Yet, to make this argument they must ignore the convergence of evidence in support of HIV as the causal vector in AIDS, and simultaneously ignore such blatant evidence as the significant correlation between the rise in AIDS among hemophiliacs shortly after HIV was inadvertently introduced into the blood supply. On top of this, their alternative theory does not explain nearly as much of the data as the HIV theory.

10. Do the claimants' personal beliefs and biases drive the conclusions, or vice versa? All scientists hold social, political, and ideological beliefs that could potentially slant their interpretations of the data, but how do those biases and beliefs affect their research? At some point, usually during the peer-review system, such biases and beliefs are rooted out, or the paper or book is rejected for publication. This is why one should not work in an intellectual vacuum. If you don't catch the biases in your research, someone else will.

There is no definitive set of criteria we can apply in determining how open-minded we should be when encountering new claims and ideas, but with mathematical calculations on the odds of weird things happening and with an analysis of the sorts of questions we should ask when we encounter weird things, we have made a start toward coming to grips with our weird and wonderful world.
Answers CHAPTER 1: A LITTLE GIVE AND TAKE Two-Digit Addition (this page)
1. 23 + 16 = 23 + 10 + 6 = 33 + 6 = 39

2. 64 + 43 = 64 + 40 + 3 = 104 + 3 = 107

3. 95 + 32 = 95 + 30 + 2 = 125 + 2 = 127

4. 34 + 26 = 34 + 20 + 6 = 54 + 6 = 60

5. 89 + 78 = 89 + 70 + 8 = 159 + 8 = 167

6. 73 + 58 = 73 + 50 + 8 = 123 + 8 = 131

7. 47 + 36 = 47 + 30 + 6 = 77 + 6 = 83

8. 19 + 17 = 19 + 10 + 7= 29 + 7= 36

9. 55 + 49 = 55 + 40 + 9 = 95 + 9 = 104 10. 39 + 38 = 39 + 30 + 8 = 69 + 8 = 77
Three-Digit Addition (this page)
1. 242 + 137 = 242 + 100 + 30 + 7 = 342 + 30 + 7 = 372 + 7 = 379

2. 312 + 256 = 312 + 200 + 50 + 6 = 512 + 50 + 6 = 562 + 6 = 568

3. 635 + 814 = 635 + 800 + 10 + 4 = 1435 + 10 + 4 = 1445 + 4 = 1449

4. 457 + 241 = 457 + 200 + 40 + 1 = 657 + 40 + 1 = 697 + 1 = 698

5. 912 + 475 = 912 + 400 + 70 + 5 = 1312 + 70 + 5 = 1382 + 5 = 1387

6. 852 + 378 = 852 + 300 + 70 + 8 = 1 152 + 70 + 8 = 1222 + 8 = 1230

7. 457 + 269 = 457 + 200 + 60 + 9 = 657 + 60 + 9 = 717 + 9 = 726

8. 878 + 797 = 878 + 700 + 90 + 7 = 1578 + 90 + 7 = 1668 + 7 = 1675 or 878 + 797 = 878 + 800 − 3 = 1678 − 3 = 1675

9. 276 + 689 = 276 + 600 + 80 + 9 = 876 + 80 + 9 = 956 + 9 = 965

10. 877 + 539 = 877 + 500 + 30 + 9 = 1 377 + 30 + 9 = 1407 + 9 = 1416

11. 5400 + 252 = 5400 + 200 + 52 = 5600 + 52 = 5652

12. 1800 + 855 = 1800 + 800 + 55 = 2600 + 55 = 2655

13. 6120 + 136 = 6120 + 100 + 30 + 6 = 6220 + 30 + 6 = 6250 + 6 = 6256

14. 7830 + 348 = 7830 + 300 + 40 + 8 = 81 30 + 40 + 8 = 8170 + 8 = 8178

15. 4240 + 371 = 4240 + 300 + 70 + 1 = 4540 + 70 + 1 = 4610 + 1 = 4611
Two-Digit Subtraction (this page)
1. 38 − 23 = 38 − 20 − 3 = 18 − 3 = 15

2. 84 − 59 = 84 − 60 + 1 = 24 + 1 =25

3. 92 − 34 = 92 − 40 + 6 = 52 + 6 = 58

4. 67 − 48 = 67 − 50 + 2 = 17 + 2 = 19

5. 79 − 29 = 79 − 20 − 9 = 59 − 9 = 50 or 79 − 29 = 79 − 30 + 1 = 49 + 1 = 50

6. 63 - 46 = 63 − 50 + 4 = 13 + 4 = 17

7. 51 - 27 = 51 − 30 + 3 = 21 + 3 = 24

8. 89 − 48 = 89 − 40 − 8 = 49 − 8 = 41

9. 125 − 79 = 125 − 80 + 1 = 45 + 1 = 46

10. 148 − 86 = 148 − 90 + 4 = 58 + 4 = 62
Three-Digit Subtraction (this page)
1. 583 − 271 = 583 − 200 − 70 − 1 = 383 − 70 − 1 = 313 − 1 = 312

2. 936 − 725 = 936 − 700 − 20 − 5 = 236 − 20 − 5 = 216 − 5 = 211

3. 587 − 298 = 587 − 300 + 2 = 287 + 2 = 289

4. 763 − 486 = 763 − 500 + 14 = 263 + 14 = 277

5. 204 − 185 = 204 − 200 + 15 = 04 + 15 = 19

6. 793 − 402 = 793 − 400 − 2 = 393 − 2 = 391

7. 219 − 176 = 219 − 200 + 24 = 19 + 24 = 43

8. 978 − 784 = 978 − 800 + 16 = 178 + 16 = 194

9. 455 − 319 = 455 − 400 + 81 = 55 + 81 = 136

10. 772 − 596 = 772 − 600 + 4 = 172 + 4 = 176

11. 873 − 357 = 873 − 400 + 43 = 473 + 43 = 516

12. 564 − 228 = 564 − 300 + 72 = 264 + 72 = 336

13. 1428 − 571 = 1428 − 600 + 29 = 828 + 29 = 857

14. 2345 − 678 = 2345 − 700 + 22 = 1645 + 22 = 1667

15. 1776 − 987 = 1776 − 1000 + 13 = 776 + 13 = 789
CHAPTER 2: PRODUCTS OF A MISSPENT YOUTH 2-by-1 Multiplication (this page) 3-by-1 Multiplication (this page) Two-Digit Squares (this page) CHAPTER 3: NEW AND IMPROVED PRODUCTS Multiplying by 11 (this page) 2-by-2 Addition-Method Multiplication Problems (this page) 2-by-2 Subtraction-Method Multiplication Problems (this page) 2-by-2 Factoring-Method Multiplication Problems (this page)
1. 27 × 14 = 27 × 7 × 2 = 189 × 2 = 378 or 14 × 27 = 14 × 9 × 3 = 126 × 3 = 378

2. 86 × 28 = 86 × 7 × 4 = 602 × 4 = 2408

3. 57 × 14 = 57 × 7 × 2 = 399 × 2 = 798

4. 81 × 48 = 81 × 8 × 6 = 648 × 6 = 3888 or 48 × 81 = 48 × 9 × 9 = 432 × 9 = 3888

5. 56 × 29 = 29 × 7 × 8 = 203 × 8 = 1624

6. 83 × 18 = 83 × 6 × 3 = 498 × 3 = 1494

7. 72 × 17 = 17 × 9 × 8 = 153 × 8 = 1224

8. 85 × 42 = 85 × 6 × 7 = 510 × 7 = 3570

9. 33 × 16 = 33 × 8 × 2 = 264 × 2 = 528 or 16 × 33 = 16 × 11 × 3 = 176 × 3 = 528

10. 62 × 77 = 62 × 11 × 7 = 682 × 7 = 4774

11. 45 × 36 = 45 × 6 × 6 = 270 × 6 = 1620 or 45 × 36 = 45 × 9 × 4 = 405 × 4 = 1620 or 36 × 45 = 36 × 9 × 5 = 324 × 5 = 1620 or 36 × 45 = 36 × 5 × 9 = 180 × 9 = 1620

12. 37 × 48 = 37 × 8 × 6 = 296 × 6 = 1776
2-by-2 General Multiplication-Anything Goes! (this page) Three-Digit Squares (this page) Two-Digit Cubes (this page)
1. 123 = (10 × 12 × 14) + (22 × 12) = 1680 + 48 = 1728

2. 173 = (14 × 17 × 20) + (32 × 17) = 4760 + 153 = 4913

3. 213 = (20 × 21 × 22) + (12 × 21) = 9240 + 21 = 9261

4. 283 = (26 × 28 × 30) + (22 × 28) = 21,840 + 112 = 21,952 5 . 333 = (30 × 33 × 36) + (32 × 33) = 35,640 + 297 = 35,93 7

6. 393 = (38 × 39 × 40) + (12 × 39) = 59,280 + 39 = 59,319

7. 403 = 40 × 40 × 40 = 64,000

8. 443 = (40 × 44 × 48) + (42 × 44) = 84,480 + 704 = 85,184

9. 523 = (50 × 52 × 54) + (22 × 52) = 140,400 + 208 = 140,608

10. 563 = (52 × 56 × 60) + (42 × 56) = 174,720 + 896 = 175,616

11. 653 = (60 × 65 × 70) + (52 × 65) = 273,000 + 1,625 = 274,625

12. 713 = (70 × 71 × 72) + (12 × 71) = 357,840 + 71 = 357,911

13. 783 = (76 × 78 × 80) + (22 × 78) = 474,240 + 312 = 474,552

14. 853 = (80 × 85 × 90) + (52 × 85) = 612,000 + 2,125 = 614,125

15. 873 = (84 × 87 × 90) + (32 × 87) = 657,720 + 783 = 658,503

16. 993 = (98 × 99 × 100) + (12 × 99) = 970,200 + 99 = 970,299
CHAPTER 4: DIVIDE AND CONQUER One-Digit Division (this page) Two-Digit Division (this page) Decimalization (this page) Testing for Divisibility (this page) Multiplying Fractions (this page) Dividing Fractions (this page) Simplifying Fractions (this page) Adding Fractions (Equal Denominators) (this page) Adding Fractions (Unequal Denominators) (this page) Subtracting Fractions (this page) CHAPTER 5: GOOD ENOUGH
Addition Guesstimation (this page)
Subtraction Guesstimation (this page) Division Guesstimation (this page) Multiplication Guesstimation (this page) Square Root Guesstimation (this page) Everyday Math (this page)
1. $8.80 + $4.40 = $13.20

2. $5.30 + $2.65 = $7.95

3. $74 ÷ 2 ÷ 2 = $37 ÷ 2 = $18.50

4. Since 70 ÷ 10 = 7, seven years

5. Since 70 ÷ 6 = 11 .67, it will take twelve years to double

6. Since 110 ÷ 7= 15.714, it will take sixteen years to triple

7. Since 70 ÷ 7 = 10, it will take ten years to double, then another ten years to double again. Thus it will take twenty years to quadruple.
CHAPTER 6: MATH FOR THE BOARD Columns of Numbers (this page) Subtracting on Paper (this page) Square Root Guesstimation (this page) Pencil-and-Paper Multiplication (this page) CHAPTER 8: THE TOUGH STUFF MADE EASY Four-Digit Squares (this page) 3-by-2 Exercises Using Factoring, Addition, and Subtraction Methods (this page) Five-Digit Squares (this page) 3-by-3 Multiplication (this page) 5-by-5 Multiplication (this page) A Day for Any Date (this page)
1. January 19, 2007, is Friday: 6 + 19 + 1 = 26;26 − 21 = 5

2. February 14, 2012, is Tuesday: 1 + 14 + 1 = 16; 16 − 14 = 2

3. June 20, 1993, is Sunday: 3 + S + 20 = 28; 28 − 28 = 0

4. September 1, 1983, is Thursday: 4 + 1 + 6=11;11−7 = 4

5. September 8, 1954, is Wednesday: 4 + 8 + 5= 17; 17 − 14=3

6. November 19, 1863, is Thursday: 2 + 19 + 4 = 25; 25 − 21 =4

7. July 4, 1776, is Thursday: 5 + 4 + 2= 11I;11 − 7 = 4

8. February 22,2222, is Friday: 2 + 22 + 2 = 26; 26 − 21 =5

9. June 3 1,2468, doesn't exist (only 30 days in June)! But June 30, 2468, is Saturday, so the next day would be Sunday.

10. January 1, 2358, is Wednesday: 6 + 1 + 3= 10; 10−7=3
Bibliography RAPID CALCULATION
Cutler, Ann, and Rudolph McShane. The Trachtenberg Speed System of Basic Mathematics. New York: Doubleday, 1960.

Devi, Shakuntala. Figuring: The Joys of Numbers. New York: Basic Books, 1964.

Doerfler, Ronald W. Dead Reckoning: Calculating Without Instruments. Houston: Gulf Publishing Company, 1993.

Flansburg, Scott, and Victoria Hay. Math Magic. New York: William Morrow and Co., 1993.

Handley, Bill. Speed Mathematics: Secrets of Lightning Mental Calculation. Queensland, Australia: Wrightbooks, 2003.

Julius, Edward H. Rapid Math Tricks and Tips: 30 Days to Number Power. New York: John Wiley & Sons, 1992.

Lucas, Jerry. Becoming a Mental Math Wizard. Crozet, Virginia: Shoe Tree Press, 1991.

Menninger, K. Calculator's Cunning. New York: Basic Books, 1964.

Smith, Steven B. The Great Mental Calculators: The Psychology, Methods, and Lives of Calculating Prodigies, Past and Present. New York: Columbia University Press, 1983.

Sticker, Henry. How to Calculate Quickly. New York: Dover, 1955.

Stoddard, Edward. Speed Mathematics Simplified. New York: Dover, 1994.

Tirtha, Jagadguru Swami Bharati Krishna, Shankaracharya of Govardhana Pitha. Vedic Mathematics or "Sixteen Simple Mathematical Formulae from the Vedas." Banaras, India: Hindu University Press, 1965.
MEMORY
Lorayne, Harry, and Jerry Lucas. The Memory Book. New York: Ballantine Books, 1974.

Sanstrom, Robert. The Ultimate Memory Book. Los Angeles: Stepping Stone Books, 1990.
RECREATIONAL MATHEMATICS
Gardner, Martin. Magic and Mystery. New York: Random House, 1956.

---. Mathematical Carnival. Washington, D.C.: Mathematical Association of America, 1965.

---. Mathematical Magic Show. New York: Random House, 1977.

---. The Unexpected Hanging and Other Mathematical Diversions. New York: Simon & Schuster, 1969.

Huff, Darrell. How to Lie with Statistics. New York: Norton, 1954.

Paulos, John Allen. Innumeracy: Mathematical Illiteracy and Its Consequences. New York: Hill and Wang, 1988.

Stewart, Ian. Game, Set, and Math: Enigmas and Conundrums. New York: Penguin Books, 1989.
ADVANCED MATHEMATICS (BY ARTHUR BENJAMIN)
Benjamin, Arthur T., and Jennifer J. Quinn. Proofs That Really Count: The Art of Combinatorial Proof. Washington: Mathematical Association of America, 2003.

Benjamin, Arthur T., and Kan Yasuda. "Magic 'Squares' Indeed!," The American Mathematical Monthly 106, no. 2 (February 1999): 152−56.
About the Authors
DR. ARTHUR BENJAMIN is a professor of mathematics at Harvey Mudd College in Claremont, California, having received his PhD in mathematical sciences from Johns Hopkins University in 1989. In 2000, the Mathematical Association of America awarded him the Haimo Prize for Distinguished College Teaching. He is also a professional magician and frequently performs at the Magic Castle in Hollywood. He has demonstrated and explained his calculating talents to audiences all over the world. In 2005, Reader's Digest called him "America's Best Math Whiz."

DR. MICHAEL SHERMER is a contributing editor to and monthly columnist for Scientific American, the publisher of Skeptic magazine (www.skeptic.com), the executive director of the Skeptics Society, and the host of the Caltech public science lecture series. He is the author of numerous science books, including Why People Believe Weird Things, How We Believe, The Science of Good and Evil, The Borderlands of Science, and Science Friction.

Publication Date: 09-20-2016

All Rights Reserved

Dedication:
These simple math secrets and tricks will forever change how you look at the world of numbers. Secrets of Mental Math will have you thinking like a math genius in no time. Get ready to amaze your friends-and yourself-with incredible calculations you never thought you could master, as renowned "mathemagician" Arthur Benjamin shares his techniques for lightning-quick calculations and amazing number tricks. This book will teach you to do math in your head faster than you ever thought possible, dramatically improve your memory for numbers, and-maybe for the first time-make mathematics fun. Yes, even you can learn to do seemingly complex equations in your head; all you need to learn are a few tricks. Youâ€™ll be able to quickly multiply and divide triple digits, compute with fractions, and determine squares, cubes, and roots without blinking an eye. No matter what your age or current math ability, Secrets of Mental Math will allow you to perform fantastic feats of the mind effortlessl

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Secrets of Mental Math (Arthur Benjamin)

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Copyright © 2006 by Arthur Benjamin and Michael Shermer All rights reserved. Published in the United States by Three Rivers Press, an imprint of the Crown Publishing Group, a division of Random House, Inc., New York. www.crownpublishing.com Originally published in different form as Mathemagics by Lowell House, Los Angeles, in 1993.

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Can a student's math achievement in prekindergarten predict whether they will graduate from college with a bachelor's degree or higher?

Math is so much more than just numbers and symbols. It's logic and critical thinking. It's problem solving. It helps us understand our world in deeper, more meaningful ways. And it may affect how successful we are in that world.

For example, one researcher analyzed the educational outcomes and math scores of several thousand students. After crunching the numbers, he declared that, even without considering any other factors, math achievement alone explained "about a 30% to 60% range of variance in the chance of being on track to college readiness." ¹ In his view, math wasn't the only factor influencing college readiness, but it was one of the more important ones.

This researcher even claimed he could use a student's math achievement in prekindergarten to determine whether they would stay on track for enrollment at a two-year college, completion of two-year college with an associate's degree or above, enrollment at a four-year college, or completion of four-year college with a bachelor's degree or above. As students moved through the grades, he seemed better able to predict how they would track for college outcomes. He concluded that "school math achievement is a good predictor of whether students in P-12 education stay on track toward two-year or four-year college education."

While no other studies have reached the same conclusion-and until much more research is done on the topic, we remain rather doubtful that prekindergarten or even elementary math achievement can accurately predict college graduation-it is clear that there is often a link between math achievement and overall student success.

Considering the critical role math may play in student outcomes, what are the key milestones that students need to stay on track for success?

Elementary math predicts high school achievement

Let's start at the beginning. Which types of mathematics content knowledge are most essential for elementary-aged children?

The answer is fractions and whole-number division. Mastering these two concepts in elementary school can help students get on the right track for success in high school.

This was the conclusion reached by one study that analyzed the math scores of more than 4,000 children in the United States (ages 10-12) and the United Kingdom (age 10 only) to identify the types of mathematics content knowledge that best predicted students' long-term learning. In a multiple-regression analysis, fractions and whole-number division were the only two early math concepts that had a significant correlation with students' overall mathematics achievement in high school in both countries. ²

In addition to finding that knowledge of fractions and whole-number division were better predictors of high school math achievement than whole-number addition, subtraction, or multiplication, the study also discovered that knowledge of fraction and whole-number division had a stronger relation to overall math achievement than family income level did.

That's not to say that educators can skip addition and subtraction and go straight to division and fractions. Addition and subtraction are prerequisites that must be mastered first, before students can advance to more difficult concepts like division and fractions. Instead, these results indicate that students can't stop at addition and subtraction in elementary; they must strive toward more advanced concepts.

Furthermore, while early knowledge of fractions and division were found to be good predictors of students' overall mathematics achievement in high school, they were poor predictors of skill in other areas, such as spelling or passage comprehension. The authors also concluded that the "greater predictive power of knowledge of fractions and knowledge of division was not due to their generally predicting intellectual outcomes more accurately." In other words, measuring these two concepts did not simply identify high-performing students who succeeded in all areas, but predicted math achievement for students of all different ability levels.

Middle school math predicts high school graduation

If elementary school math can predict high school math, what can middle school math predict?

One longitudinal study examined the graduation patterns of nearly 13,000 students in an urban district to discover how middle school factors related to high school graduation. The results showed that only 13% of students who failed a math course in sixth grade graduated on time. An additional 6% graduated one year late. The remaining 81% had not graduated by the time the study ended. ³

Moreover, failing a sixth-grade math course was a better predictor of failure to graduate than either a student's race or status as an English Language Learner. In comparison, end-of-sixth-grade test scores were poor predictors of whether a student would drop out.

High school math predicts career goals

When it comes to math achievement, the high school years may be some of the most critical in a student's academic career. We're not trying to sound overdramatic, but research indicates a student's math performance in these grades may affect the rest of their life.

To start with, there appears to be a relationship between high school students' math achievement and their hopes and dreams for the future. A longitudinal study examined the career aspirations of "high-achieving" ninth graders (who attained 75% or better in their math courses) and "low-achieving" ninth graders (who attained less than 75% in their math courses).

After controlling for students' perceptions of their own math competence, their overall GPA, and the education levels of their parents, the authors found ninth-grade math achievement had a significant effect on career aspirations-not just in ninth grade but even after high school graduation. ⁴ Math achievement, specifically, seemed to affect career goals above and beyond overall academic achievement.

Low-math-achieving students not only started out with lower occupational aspirations, but also experienced a dramatic decline in their aspirations from mid-high school onward. High-math-achieving students also saw a decline in career aspirations, but started with higher overall aspirations and saw a more gradual decline throughout high school and postsecondary years, ending with career goals notably higher than their low-achieving peers.

The authors suggested that ninth-grade math achievement is a "critical filter" for students' career goals and "low math achievement early in high school may steer adolescents away from the advanced math courses required by postsecondary education programs that lead to many higher-prestige career paths, not only in the physical science and technology fields, but also in health, commerce, and many social sciences."

As we're about to see, those advanced math courses in high school are particularly important when it comes to college and career outcomes.

High school math predicts college graduation, postsecondary degrees, and career earnings

A report that examined the long-term outcomes of thousands of students across the United States found clear connections between the highest level of math completed by the end of high school and the highest degree of education attained 10 years later. ⁵ Only 3% of students who had completed vocational math but no higher-level math courses had earned a bachelor's degree or higher, whereas 73% of students who had completed at least one calculus course held a bachelor's degree or higher.

Furthermore, students who did not complete more-advanced math courses such as algebra or calculus were much more likely to drop out of high school or fail to earn a high school diploma.

For all ethnicities and all income levels, the higher the level of math courses a student completed, the more likely they were to graduate college with a bachelor's degree. Completion of more-advanced math courses increased the predicted probability of college graduation even when the authors controlled for demographic traits, socioeconomic status, family and school characteristics, and overall measures of math ability (i.e., math GPA and grade 10 math test score).

Even 10 years after high school, math curriculum had an effect on annual earnings. Students who had completed a calculus course earned 65% more than students whose math education did not progress past a vocational math course. After accounting for ethnicity, gender, and other demographic traits; parental education and income; and high school characteristics, more-advanced mathematics courses still had an effect on annual earnings.

Moreover, math curriculum continued to have "striking" effects on earnings even after controlling for highest degree earned and college major as well as demographics, family, and school, suggesting that "there is a direct effect of math curriculum on labor market productivity, which works independently of the final degree attained." Following this logic, if there were two individuals of the same ethnicity and gender, who come from similar families and attended the same high school, who both have the same degree in the same field (say a bachelor's degree in biology), the one who completed a calculus course by the end of high school would likely earn more money than the one who did not!

On top of that, math courses still seemed to have an effect on earnings when courses taken in other subjects were considered. Advanced-level English courses increased earnings, but not as much as advanced algebra or calculus did. In addition, algebra/geometry, intermediate algebra, advanced algebra, and calculus all appeared to increase earnings more than average-level English courses did.

Note that for all these analyses, the total number of math courses completed wasn't the key factor-it was the level of those courses. As the authors put it, "it is not simply the number of math courses that matters; what matters more is the extent to which students take the more demanding courses, such as algebra/geometry."

Math's path to college and career readiness

We've seen how math milestones can help students get on the path to college and career success. From mastering fractions and whole-number division by the end of elementary school to completing more-advanced math courses by the end of high school, each milestone a student achieves is one step closer to success in school and in life.

What does the overall path look like for students? Returning to the first study examined in this post, that author concluded that state standards (as they existed in 2007) were a good starting point. ¹ When he looked at the math achievement trajectories that led to completion of college, he found that attaining the typical state standard of math proficiency would put students on track for earning an associate's degree or above from a two-year college. Students who met or exceeded National Assessment of Educational Progress (NAEP) proficient level for math or Trends in International Mathematics and Science Study (TIMSS) high benchmark for math were likely to earn a bachelor's degree or above from a four-year college.

Considering that only 40% of fourth-grade students, 33% of eighth-grade students, and 25% of twelfth-grade students scored proficient or above on the 2015 NAEP math assessment, ⁶ this may seem like a high bar to reach-but it's not impossible.

Students can move toward college and career readiness incrementally, one math concept at a time. An analysis of students in grades 3-8 using a research-based math practice program found that 42% of the students who mastered an average of one to three subskills weekly reached college- and career-readiness benchmarks. In comparison, only 33% of the typical students-who did not use the program-were college and career ready. The highest level of college and career readiness was seen among students using the program who averaged mastery of four or more subskills per week; 58% of these students met proficiency benchmarks-almost double the rate of the typical students.

Imagine what your students could achieve by mastering just one or two more subskills each week. For many, that small change could have big effects. However, there are hundreds of different math skills and even more math subskills. How do you know which ones are the most important for students to learn? Get the answer in an on-demand webinar on achieving math milestones.
To access this free webinar, click the banner below.

References

¹Lee, J. (2012). College for all: Gaps between desirable and actual P-12 math achievement trajectories for college readiness. Educational Researcher, 41(2), 43-55.
²Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., Susperreguy, M. I., & Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological Science 23(7), 691-697.
³Balfanz, R., Herzog, L., & Mac Iver, D. (2007). Preventing student disengagement and keeping students on the graduation path in urban middle-grades schools: Early identification and effective interventions. Educational Psychologist, 42(4), 223-235.
⁴Shapka, J. D., Domene, J. F., & Keating, D. P. (2006). Trajectories of career aspirations through adolescence and young adulthood: Early math achievement as a critical filter. Educational Research and Evaluation, 12(4), 347-358.
⁵Rose, H., & Betts, J. R. (2001). Math matters: The links between high school curriculum, college graduation, and earnings. San Francisco, CA: Public Policy Institute of California.
⁶National Center for Education Statistics. (2015). National Assessment of Educational Progress (NAEP) 2015 mathematics assessments. Digest of Education Statistics. Washington, DC: U.S. Department of Education Institute of Education Sciences. Retrieved from: https://www.nationsreportcard.gov/
⁷Renaissance. (2017). Trends in student outcome measures: The role of individualized math practice. Wisconsin Rapids, WI: Author.

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Math milestones: The critical role of math achievement in student success | Renaissance

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What are the math milestones that students need to achieve in elementary, middle school, and high school in order to stay on track for college and career readiness? Discover the how math achievement relates to high school graduation, college degrees, and good salaries in this research-driven post.

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School Readiness Study - Greg Duncan, et al.

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Interview skills to help you get the job

Your interview is a chance to showcase your hard and soft skills and prove you're the right person for the job. Because the stakes are high, preparing an interview action plan will boost your confidence and smooth your communication skills to ensure you're presenting the best version of yourself.

Practice the following six essential interview skills to nail your next job interview.

1. Clarify interview questions during the job interview.

Most people are hesitant to ask an interviewer to clarify a question, according to Sara Sutton, CEO and founder of the job listing website FlexJobs. They may be concerned the interviewer will think they're being inattentive. However, ensuring you thoroughly understand the question can help you give a thoughtful, relevant response. "Try to paraphrase the question and say, 'Is this what you're asking?'" Sutton advised.

You also can use question clarification as an opportunity to turn the tables on interviewers and ask them questions. By treating the interview like a two-way conversation and asking intelligent and thoughtful questions, candidates can gauge whether the company is a good fit for them, according to Dana Leavy-Detrick, a personal brand strategist and career consultant at Brooklyn Resume Studio.

"Candidates should pose questions that will ultimately provide them with deeper insight into the company's values, culture and even challenges," Leavy-Detrick said. For example, candidates can ask an interviewer how they mapped their career path with the company. They can also ask interviewers to describe a typical day or highlight qualities that make someone successful in the organization.

2. Think out loud during the job interview.

A common job interview mistake many interviewees make is to stall when they don't have a ready answer to a question, or responding with "I don't know," according to Shon Burton, CEO of social recruiting tool HiringSolved. Thinking aloud is an excellent tactic for combating this problem.

"The best approach is to have humble confidence," Burton said. "Repeat the interviewer's question, and work through your thought process out loud. The interviewer may give you a hint if you're actively thinking instead of stalling."

3. Communicate nonverbally during the job interview.

You're communicating with the interviewer even when you're not speaking. It's crucial to be aware of your interview body language and work on ways to keep it under control to appear calmer and more confident.

"Good nonverbal communication speaks volumes about a candidate," said Jonna Myers, an instructor at the Everett Dobson School of Business and Technology at Southwestern Oklahoma State University. "It's something most people don't practice, but it makes it very evident when you're nervous."

Myers recommended conducting mock interviews with a friend or in front of a mirror to practice your eye contact, posture and other body language confidence indicators.

"Since there's no quicker way to kill an interview than to come across disinterested, bored or unengaged, having an opportunity to practice and get feedback will help you overcome the little nuances of your interview persona that you may not be aware of yet," said Sanjay Sathe, founder and former CEO of the outplacement firm RiseSmart.

One of the best ways to keep your body language under control is to take a deep breath before you step into the interview room and ensure you keep breathing deeply and steadily. This type of breathing helps your body feel more in control and can help keep anxiety at bay.

4. Know your resume in depth before your job interview.

You've likely put immense time and effort into creating a great resume, but do you know it inside and out? While your resume speaks for itself, you'll add to your credibility when you can easily provide measurable specifics about the accomplishments documented in it.

"Whenever possible, include a statistic to put your accomplishments in perspective," Sathe recommended. It's much more compelling to say you delivered customer service to more than 120 customers per week and achieved a 75% resolution rating than simply saying you provided customer service and resolved issues, he said.

"Whatever your contributions were, quantifying them will legitimize your accomplishments," Sathe added.

5. Tell a compelling story during your job interview.

Hiring managers can collect bits and pieces of information from social media screenings and your resume, but these details won't always represent you accurately. The interview is an opportunity to create a more comprehensive narrative of who you are and what you can offer an employer.

"The interview is your opportunity to tie everything together and tell a cohesive and compelling story about yourself and your brand," Sathe said. "Creating vignettes that tell a story of your accomplishments and your career trajectory that is tied into the needs of the employer is considerably more compelling [than] a disjointed list of past job responsibilities and a description of your skills."

6. Leverage your knowledge of the company and the interviewer during the interview.

Every job seeker knows they should thoroughly research the company and position they're interviewing for. However, using that information to your advantage is just as crucial. Myers recommended researching the company's community, location, job description and organization.

"It's very impressive when a candidate can talk about why [they are] a good fit for the position, as well as things that are going on in the company's community," Myers said.

Burton added that using LinkedIn to research the hiring manager and anyone else you might be speaking to can give you an understanding of each person's background and potentially some common ground to spark a discussion.

Additionally, following the company on Facebook and Twitter and setting a Google alert to receive notices whenever the company appears in the news will help you thoroughly research the company and its industry. By immersing yourself in this information, you can shape interview responses and talking points that speak specifically and intelligently to the business you want to join.

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Interview Skills You Need to Get the Job

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Nailing a job interview may come down to interview skills and how you present yourself. Learn six skills to help you better prepare for a job interview.