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Cryptarithms - Alphametics - Cryptarithms.com

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Cryptarithms - Alphametics - Cryptarithms.com

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Cryptarithms - Alphametics - Cryptarithms.com

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Cryptarithms are mathematical puzzles in which the digits in an arithmetic expression are replaced by letters of the alphabet. The object is to work out which numbers correspond to each letter. Generally the letters are chosen so that the sum also spells out some related words or a short message - an Alphametic puzzle.

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You weren’t bad at maths — you just weren’t looking at it the right way

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You weren’t bad at maths — you just weren’t looking at it the right way

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Mental representations in mathematics

What does mathematics look like to you? Do you see a wondrous landscape filled with connected ideas, or a sprawling mess of symbols? The distinction matters a great deal, because your mathematical worldview is inextricably tied to your success in the subject.

We are all familiar with the multiplication grid, a centrepiece of classrooms and home studies the world over:

You cannot fault this image for accuracy. There is a bluntness to the grid; a seemingly disconnected array of factual truths. They are to be learned, memorised, recalled. To the trained eye, however, patterns and structure abound. Just as Cypher doesn't even see the code anymore in The Matrix, a mathematician sees much more than 100 solitary numbers.

Perhaps they see this - a scaled drawing of the multiplication grid:

With this simple tweak, the grid is beginning to speak to us. It conveys size and proportion rather than just numerical outputs. It connects otherwise disconnected topics, binding together number with geometry; multiplication with area.

With deliberate use of colour, we can extract a different type of structure - here we see the multiplication grid as a nested collection of smaller grids:

The size and shape of each layer reveals new truths. Often, it reveals old truths in new ways - can you see why summing the first n odd integers results in n²?

Having the right mental representations is the key to developing your mathematical potential.

The road to expertise is paved with 10,000 hours of practice, so claims Malcolm Gladwell in . Gladwell's claim is a gross simplification. Anders Ericsson, the man behind the original research, has set the record straight in . He duly rebukes Gladwell for imposing an arbitrary threshold of 10,000 hours (it actually varies within and between disciplines). More importantly, Gladwell does not distinguish between different types of practice. Ericsson's research, and the weight of his book, is premised on the principles of deliberate practice.

The chief virtue of deliberate practice is that it helps us develop rich mental representations: pre-existing patterns of information that sit in our long-term memory.

The stronger and more numerous our representations, the more we can draw on them to connect ideas, develop intuitions and solve problems.

Mental representations anchor us to our worldview of mathematics. The more we have, the better. Sticking with one-word titles, Adam Grant's lays out the key characteristics of creative geniuses. One such characteristic is that they can form more ideas when approaching a task. In mathematics, it stands to reason: you are more likely to solve a problem if you have different lines of attack. These ideas spring from our mental representations.

But just as not all practice is created equal, some mental representations are stronger than others. This explains why many people leave school fluent in the facts of multiplication but without any real intuition or sense of number. They possess weak representations that privilege recall over understanding. When students are drip-fed multiplication facts as a collection of disconnected truths, it is hardly surprising that they struggle to connect those facts to broader mathematical themes. The best mathematicians understand multiplication as part of a rich tapestry of concepts. For them, fluency and understanding are mutually reinforcing.

Mathematics can seem lifeless when our representations are weak.

Prime numbers do not get the respect they deserve in the curriculum. For many, primes are simply a collection of numbers that happen to have two divisors. Another item on the syllabus, useful for calculating HCFs, LCMs and other arbitrary values that frequent exam papers. To the mathematician, however, primes are the DNA of the subject. This biological metaphor is no accident, nor a triviality.

The Fundamental Theorem of Arithmetic tells us that every whole number is a unique product of primes. The more you dwell on this revelation, the more you will be drawn to the supremacy of primes. How appropriate then, that these objects are shrouded in mystery, giving rise to some of the deepest unresolved problems in mathematics.

The fascination with primes is not the preserve of professional mathematicians. With the right mental representations, they can delight, intrigue and perplex us all at once. So here is the 100-square, as you have never seen it before, courtesy of Daniel Finkel:

As Finkel himself advises, let this grid speak to you. Explore the patterns. Play with its structure. Sink into the depths of multiplication, and discover the starring role primes play in our times tables.

If maths was nothing more than a sprawling mess of symbols for you at school, take solace in the fact that an enthralling, quite separate universe awaits you. Maths is replete with rich and wonderful mental representations - the kinds that foster understanding and forge connections between ideas. This is the maths that mathematicians fall in love with.

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You weren't bad at maths - you just weren't looking at it the right way

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Mental representations in mathematics What does mathematics look like to you? Do you see a wondrous landscape filled with connected ideas, or a sprawling mess of symbols? The distinction matters a great deal, because your mathematical worldview is inextricably tied to your success in the subject.

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Research-Based Strategies for Problem-Solving in Mathematics

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Research-Based Strategies for Problem-Solving in Mathematics

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Educational Leadership:Making Math Count:Nine Ways to Catch Kids Up

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Educational Leadership:Making Math Count:Nine Ways to Catch Kids Up

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November 2007 | Volume 65 | Number 3
Making Math Count Pages 16-21

Marilyn Burns

Paul, a 4th grader, was struggling to learn multiplication. Paul's teacher was concerned that he typically worked very slowly in math and "didn't get much done." I agreed to see whether I could figure out the nature of Paul's difficulty. Here's how our conversation began:

Marilyn: Can you tell me something you know about multiplication?

Paul: [Thinks, then responds] 6 × 8 is 48.

Marilyn: Do you know how much 6 × 9 is?

Paul: I don't know that one. I didn't learn it yet.

Marilyn: Can you figure it out some way?

Paul: [Sits silently for a moment and then shakes his head.]

Marilyn: How did you learn 6 × 8?

Paul: [Brightens and grins] It's easy-goin' fishing, got no bait, 6 × 8 is 48.

As I talked with Paul, I found out that multiplication was a mystery to him. Because of his weak foundation of understanding, he was falling behind his classmates, who were multiplying problems like 683 × 4. Before he could begin to tackle such problems, Paul needed to understand the concept of multiplication and how it connects to addition.

Paul wasn't the only student in this class who was floundering. Through talking with teachers and drawing on my own teaching experience, I've realized that in every class a handful of students are at serious risk of failure in mathematics and aren't being adequately served by the instruction offered. What should we do for such students?

Grappling with Interventions

My exchange with Paul reminded me of three issues that are essential to teaching mathematics:

It's important to help students make connections among mathematical ideas so they do not see these ideas as disconnected facts. (Paul saw each multiplication fact as a separate piece of information to memorize.)

It's important to build students' new understandings on the foundation of their prior learning. (Paul did not make use of what he knew about addition to figure products.)

It's important to remember that students' correct answers, without accompanying explanations of how they reason, are not sufficient for judging mathematical understanding. (Paul's initial correct answer about the product of 6 × 8 masked his lack of deeper understanding.)

For many years, my professional focus has been on finding ways to more effectively teach arithmetic, the cornerstone of elementary mathematics. Along with teaching students basic numerical concepts and skills, instruction in number and operations prepares them for algebra. I've developed lessons that help students make sense of number and operations with attention to three important elements-computation, number sense, and problem solving. My intent has been to avoid the "yours is not to question why, just invert and multiply" approach and to create lessons that are accessible to all students and that teach skills in the context of deeper understanding. Of course, even well-planned lessons will require differentiated instruction, and much of the differentiation needed can happen within regular classroom instruction.

But students like Paul present a greater challenge. Many are already at least a year behind and lack the foundation of mathematical understanding on which to build new learning. They may have multiple misconceptions that hamper progress. They have experienced failure and lack confidence.

Such students not only demand more time and attention, but they also need supplemental instruction that differs from the regular program and is designed specifically for their success. I've recently shifted my professional focus to thinking about the kind of instruction we need to serve students like Paul. My colleagues and I have developed lessons that provide effective interventions for teaching number and operations to those far behind. We've grappled with how to provide instruction that is engaging, offers scaffolded instruction in bite-sized learning experiences, is paced for students' success, provides the practice students need to cement fragile understanding and skills, and bolsters students' mathematical foundations along with their confidence.

In developing intervention instruction, I have reaffirmed my longtime commitment to helping students learn facts and skills-the basics of arithmetic. But I've also reaffirmed that "the basics" of number and operations for all students, including those who struggle, must address all three aspects of numerical proficiency-computation, number sense, and problem solving. Only when the basics include understanding as well as skill proficiency will all students learn what they need for their continued success.

Essential Strategies

I have found the following nine strategies to be essential to successful intervention instruction for struggling math learners. Most of these strategies will need to be applied in a supplementary setting, but teachers can use some of them in large-group instruction.

1. Determine and Scaffold the Essential Mathematics Content

Determining the essential mathematics content is like peeling an onion-we must identify those concepts and skills we want students to learn and discard what is extraneous. Only then can teachers scaffold this content, organizing it into manageable chunks and sequencing these chunks for learning.

For Paul to multiply 683 × 4, for example, he needs a collection of certain skills. He must know the basic multiplication facts. He needs an understanding of place value that allows him to think about 683 as 600 + 80 + 3. He needs to be able to apply the distributive property to figure and then combine partial products. For this particular problem, he needs to be able to multiply 4 by 3 (one of the basic facts); 4 by 80 (or 8 × 10, a multiple of 10); 4 and by 600 (or 6 × 100, a multiple of a power of 10). To master multidigit multiplication, Paul must be able to combine these skills with ease. Thus, lesson planning must ensure that each skill is explicitly taught and practiced.

2. Pace Lessons Carefully

We've all seen the look in students' eyes when they get lost in math class. When it appears, ideally teachers should stop, deal with the confusion, and move on only when all students are ready. Yet curriculum demands keep teachers pressing forward, even when some students lag behind. Students who struggle typically need more time to grapple with new ideas and practice new skills in order to internalize them. Many of these students need to unlearn before they relearn.

3. Build in a Routine of Support

Students are quick to reveal when a lesson hasn't been scaffolded sufficiently or paced slowly enough: As soon as you give an assignment, hands shoot up for help. Avoid this scenario by building in a routine of support to reinforce concepts and skills before students are expected to complete independent work. I have found a four-stage process helpful for supporting students.

In the first stage, the teacher models what students are expected to learn and records the appropriate mathematical representation on the board. For example, to simultaneously give students practice multiplying and experience applying the associative and commutative properties, we present them with problems that involve multiplying three one-digit factors. An appropriate first problem is 2 × 3 × 4. The teacher thinks aloud to demonstrate three ways of working this problem. He or she might say,

I could start by multiplying 2 × 3 to get 6, and then multiply 6 × 4 to get 24. Or I could first multiply 2 × 4, and then multiply 8 × 3, which gives 24 again. Or I could do 3 × 4, and then 12 × 2. All three ways produce the same product of 24.

As the teacher describes these operations, he or she could write on the board:

It's important to point out that solving a problem in more than one way is a good strategy for checking your answer.

In the second stage, the teacher models again with a similar problem-such as 2 × 4 × 5-but this time elicits responses from students. For example, the teacher might ask, "Which two factors might you multiply first? What is the product of those two factors? What should we multiply next? What is another way to start?" Asking such questions allows the teacher to reinforce correct mathematical vocabulary. As students respond, the teacher again records different ways to solve the problem on the board.

During the third stage, the teacher presents a similar problem-for example, 2 × 3 × 5. After taking a moment to think on their own, students work in pairs to solve the problem in three different ways, recording their work. As students report back to the class, the teacher writes on the board and discusses their problem-solving choices with the group.

In the fourth stage, students work independently, referring to the work recorded on the board if needed. This routine both sets an expectation for student involvement and gives learners the direction and support they need to be successful.

4. Foster Student Interaction

We know something best once we've taught it. Teaching entails communicating ideas coherently, which requires the one teaching to formulate, reflect on, and clarify those ideas-all processes that support learning. Giving students opportunities to voice their ideas and explain them to others helps extend and cement their learning.

Thus, to strengthen the math understandings of students who lag behind, make student interaction an integral part of instruction. You might implement the think-pair-share strategy, also called turn and talk. Students are first asked to collect their thoughts on their own, and then talk with a partner; finally, students share their ideas with the whole group. Maximizing students' opportunities to express their math knowledge verbally is particularly valuable for students who are developing English language skills.

5. Make Connections Explicit

Students who need intervention instruction typically fail to look for relationships or make connections among mathematical ideas on their own. They need help building new learning on what they already know. For example, Paul needed explicit instruction to understand how thinking about 6 × 8 could give him access to the solution for 6 × 9. He needed to connect the meaning of multiplication to what he already knew about addition (that 6 × 8 can be thought of as combining 6 groups of 8). He needed time and practice to cement this understanding for all multiplication problems. He would benefit from investigating six groups of other numbers-6 × 2, 6 × 3, and so on-and looking at the numerical pattern of these products. Teachers need to provide many experiences like these, carefully sequenced and paced, to prepare students like Paul to grasp ideas like how 6 × 9 connects to 6 × 8.

6. Encourage Mental Calculations

Calculating mentally builds students' ability to reason and fosters their number sense. Once students have a foundational understanding of multiplication, it's key for them to learn the basic multiplication facts-but their experience with multiplying mentally should expand beyond these basics. For example, students should investigate patterns that help them mentally multiply any number by a power of 10. I am concerned when I see a student multiply 18 × 10, for example, by reaching for a pencil and writing:

Revisiting students' prior work with multiplying three factors can help develop their skills with multiplying mentally. Helping students judge which way is most efficient to multiply three factors, depending on the numbers at hand, deepens their understanding. For example, to multiply 2 × 9 × 5, students have the following options:

Guiding students to check for factors that produce a product of 10 helps build the tools they need to reason mathematically.

When students calculate mentally, they can estimate before they solve problems so that they can judge whether the answer they arrive at makes sense. For example, to estimate the product of 683 × 4, students could figure out the answer to 700 × 4. You can help students multiply 700 × 4 mentally by building on their prior experience changing three-factor problems to two-factor problems: Now they can change a two-factor problem-700 × 4-into a three-factor problem that includes a power of 10-7 × 100 × 4. Encourage students to multiply by the power of 10 last for easiest computing.

7. Help Students Use Written Calculations to Track Thinking

Students should be able to multiply 700 × 4 in their heads, but they'll need pencil and paper to multiply 683 × 4. As students learn and practice procedures for calculating, their calculating with paper and pencil should be clearly rooted in an understanding of math concepts. Help students see paper and pencil as a tool for keeping track of how they think. For example, to multiply 14 × 6 in their heads, students can first multiply 10 × 6 to get 60, then 4 × 6 to get 24, and then combine the two partial products, 60 and 24. To keep track of the partial products, they might write:

14 × 6

10 × 6 = 60

4 × 6 = 24

60 + 24 = 84

They can also reason and calculate this way for problems that involve multiplying by three-digit numbers, like 683 × 4.

8. Provide Practice

Struggling math students typically need a great deal of practice. It's essential that practice be directly connected to students' immediate learning experiences. Choose practice problems that support the elements of your scaffolded instruction, always promoting understanding as well as skills. I recommend giving assignments through the four-stage support routine, allowing for a gradual release to independent work.

Games can be another effective way to stimulate student practice. For example, a game like Pathways (see Figure 1 for a sample game board and instructions) gives students practice with multiplication. Students hone multiplication skills by marking boxes on the board that share a common side and that each contain a product of two designated factors.

Figure 1. Pathways Multiplication Game

9. Build In Vocabulary Instruction

The meanings of words in math-for example, even, odd, product, and factor -often differ from their use in common language. Many students needing math intervention have weak mathematical vocabularies. It's key that students develop a firm understanding of mathematical concepts before learning new vocabulary, so that they can anchor terminology in their understanding. We should explicitly teach vocabulary in the context of a learning activity and then use it consistently. A math vocabulary chart can help keep both teacher and students focused on the importance of accurately using math terms.

When Should We Offer Intervention?

There is no one answer to when teachers should provide intervention instruction on a topic a particular student is struggling with. Three different timing scenarios suggest themselves, each with pluses and caveats.

While the Class Is Studying the Topic

Extra help for struggling learners must be more than additional practice on the topic the class is working on. We must also provide comprehensive instruction geared to repairing the student's shaky foundation of understanding.

The plus: Intervening at this time may give students the support they need to keep up with the class.

The caveat: Students may have a serious lack of background that requires reaching back to mathematical concepts taught in previous years. The focus should be on the underlying math, not on class assignments. For example, while others are learning multidigit multiplication, floundering students may need experiences to help them learn basic underlying concepts, such as that 5 × 9 can be interpreted as five groups of nine.

Before the Class Studies the Topic

Suppose the class is studying multiplication but will begin a unit on fractions within a month, first by cutting out individual fraction kits. It would be extremely effective for at-risk students to have the fraction kit experience before the others, and then to experience it again with the class.

The plus: We prepare students so they can learn with their classmates.

The caveat: With this approach, struggling students are studying two different and unrelated mathematics topics at the same time.

After the Class Has Studied the Topic.

This approach offers learners a repeat experience, such as during summer school, with a math area that initially challenged them.

The plus: Students get a fresh start in a new situation.

The caveat: Waiting until after the rest of the class has studied a topic to intervene can compound a student's confusion and failure during regular class instruction.

How My Teaching Has Changed

Developing intervention lessons for at-risk students has not only been an all-consuming professional focus for me in recent years, but has also reinforced my belief that instruction-for all students and especially for at-risk students-must emphasize understanding, sense making, and skills.

Thinking about how to serve students like Paul has contributed to changing my instructional practice. I am now much more intentional about creating and teaching lessons that help intervention students catch up and keep up, particularly scaffolding the mathematical content to introduce concepts and skills through a routine of support. Such careful scaffolding may not be necessary for students who learn mathematics easily, who know to look for connections, and who have mathematical intuition. But it is crucial for students at risk of failure who can't repair their math foundations on their own.

Marilyn Burns is Founder of Math Solutions Professional Development, Sausalito, California; 800-868-9092; mburns@mathsolutions.com.

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November 2007 | Volume 65 | Number 3 Making Math Count Pages 16-21 Marilyn Burns Paul, a 4th grader, was struggling to learn multiplication. Paul's teacher was concerned that he typically worked very slowly in math and "didn't get much done." I agreed to see whether I could figure out the nature of Paul's difficulty.

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The Contrasting Cases Curriculum in Five Sections: Materials, Assessments, Teacher Professional Development, Interview Protocol, and Legacy Curriculum

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The Fraction Splat! Series | Steve Wyborney's Blog: I'm on a Learning Mission.

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The Fraction Splat! Series | Steve Wyborney's Blog: I'm on a Learning Mission.

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Steve Wyborney's Blog: I'm on a Learning Mission.

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Fraction Splats! have arrived! The original Splat! post rocketed up to over 35,000 hits in less than a month, which included over 19,000 lesson downloads! All 50 lessons in the Splat! post involved whole numbers. In this post, I've turned my attention to fractions, and have created 20 new lessons for you to download! This is The Fraction Splat! Series. Watch the video, download the lessons, and try them in your classroom!
Note: I've created the lessons in PowerPoint, so I recommend using PowerPoint to view them. Note: These lessons use a single splat along with a mixture of unit fractions and whole numbers. In Level 11, you control the Splat!
Instant Fraction Splat!
Note: These lessons use a single Splat! which that appears at the same moment as the whole numbers and fractions. Each question instantly presents missing information.
Multiple Splats! with Fractions
Note: These lessons include multiple Splats! and you control the timing of those Splats! The entire screen can be seen, the total can be determined, and then Splat! Remember: Each Splat! that is the same color must be covering the same number.
Multiple Splats! with Fractions
Note: These lessons use multiple Splats! which appear at the same moment as the whole numbers and fractions. Each question instantly presents missing information. Remember: Each Splat! that is the same color must be covering the same number.
Other Posts and Resources Which May Be of Interest

Splat! The original 50 downloadable Splat! lessons include 10 levels and variations.

Tiled Area Questions is a colorful, powerful, highly visual strategy for decomposing numbers - and offers many downloadable pages.

The Animated Multiplication Table remains one of the most popular downloaded resources from this blog, and this post includes several ideas about how to use it.

The Maze Hundreds Chart is another downloadable resource that can lead to some highly intriguing questions and rich discourse.

You may also be interested in Provide Massive Space to Notice.

Please take a moment to subscribe to the blog so you'll be notified each time new resources are posted!

I would treasure hearing about your experiences using The Fraction Splat! Series to promote number sense in your classroom!

All my best,
Steve

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Fraction Splats! have arrived! The original Splat! post rocketed up to over 35,000 hits in less than a month, which included over 19,000 lesson downloads! All 50 lessons in the Splat! post involved whole numbers. In this post, I've turned my attention to fractions, and have created 20 new lessons for you to download!

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Sign up for the Knowre Math Comic Digest! Comics for Pre-Algebra, Algebra 1, Geometry, and Algebra 2 delivered right to your inbox.

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Explore these patterns with your students and watch their natural tendencies to see patterns morph into powerful algebraic thinking and reasoning. It’s an ideal routine to foster mathematical practice #7 - look for and make use of structure.

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This Puzzling Math Brain Teaser Has a Simple Solution | Mental Floss

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This Puzzling Math Brain Teaser Has a Simple Solution | Mental Floss

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In the days before widescreen televisions, most of the movies you watched on VHS or on cable looked a little different than their big-screen versions. The sides of the image had to be cropped out so that you could watch a movie made for a rectangular screen on the small screen. Today, those little black bars on the top and bottom of the screen that allow you to watch the same movie scaled to any shape of screen are everywhere. But it turns out, cropping for aspect ratios is alive and well-on HBO, as YouTube film vlogger Patrick Willems explains.

In his latest video, which we spotted on Digg, Willems explains why aspect ratios matter, and how the commonly used aspect ratios can fundamentally change a movie.

Most old-school televisions have 4:3 aspect ratios, meaning movies had to be significantly cropped to fit wide-screen films on the small screen. Now, most computers and televisions use 16:9 aspect ratios, which is approximately the same as the one used for movies, typically 1.85:1, so many movies expand to fit TV screens perfectly. The catch: Some Hollywood movies are shot with even wider angles to show even more of an image at once. And even though viewers are familiar with the sight of those black bars, it seems the streaming sites are determined to limit their use, even for movies that don't fit on a normal screen. As a result, you may only be seeing the central part of the image, not the whole thing. You could be missing characters, action, and landscape that's happening on the far sides of the screen.

Since 1993, the Motion Picture Association of America has mandated that any film that's been altered in a way that changes the original vision of its creators-say, to edit out swear words, adjust the run time, or to make it fit a certain screen-run with a disclaimer that says as much. That's why before movies run on TV, they usually show a note that says something like "This film has been modified from its original version. It has been formatted to fit this screen." But this doesn't seem to apply to streaming.

In 2013, Netflix was accused of cropping films, too, showing wide-angle movies to fit the standard 16:9 screen instead of running the original version with black bars. The streaming giant claimed it was a mistake due to distributors sending them the cropped version, and those films would be replaced with the originals. However, as of 2015, users were still complaining of the problem. According to Willems, it's a problem that still plagues not just HBO, but Starz and Hulu, too, and there isn't any clear rationale for it other than that perhaps people don't like looking at black bars. But frankly, that seems better than seeing a version of a film that the director never intended.

You can get all the details in the video below:

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This Puzzling Math Brain Teaser Has a Simple Solution

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In the days before widescreen televisions, most of the movies you watched on VHS or on cable looked a little different than their big-screen versions. The sides of the image had to be cropped out so that you could watch a movie made for a rectangular screen on the small screen.

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Photo by Stephen Lovekin/Getty Images for Robert F. Kennedy Center For Justice And Human Rights

People who teach math, like me, hate it when students ask us, "When am I going to use this?" We don't hate it because it's a bad question. We hate it because it's a really good question, and one that our curriculum isn't really set up to answer. And that's a problem.

In my day job I'm a pure mathematician, specializing in the most abstract parts of number theory and geometry. But I've also been writing here at Slate for more than a decade about the connections between the mathematical world and the things we think about every day. The foreignness of mathematical language, from arithmetic to calculus, can create among outsiders the misapprehension that these spheres of thought are totally alien and contrary to how most people navigate the world around them.

That's exactly wrong. Math is built on our natural ability to reason. Despite the power of mathematics, and despite its sometimes forbidding notation and abstraction, the actual mental work involved doesn't differ much at all from the way we think about more down-to-earth problems. Rather, mathematics is like an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength.

In my new book, How Not to Be Wrong, I write about the many ways math is woven into our thinking, covering everything from lottery schemes to the obesity apocalypse to the Supreme Court's view of crime and punishment to the existence of God. It's not the kind of book where math is a big floating blob to be admired (or feared) from afar. We get right up next to it and get our hands dirty. Because that's what math is about.

This week, I'll be math-blogging at Slate-sometimes about stuff from the book, other times about math that's in the news right now. Let's get started right now, because I have something to get off my chest about mathematics education.

We are angry about the way math is taught. We have always been angry about the way math is taught. One day Pythagoras started talking up his theorem, and the next day, people were saying, When I was a kid we just measured the hypotenuse. Easy peasy. Why do you have to make things so complicated, Pythagoras?

Nobody's expressed this anger more precisely or more hilariously than Tom Lehrer did in his song "New Math" (from the early 1960s, in case you're under the mistaken impression the math wars are a contemporary development). "But in the new approach, as you know," Lehrer says, "the important thing is to understand what you're doing, rather than to get the right answer."

At the moment, the anger is centered on the Common Core, a fairly mild-mannered suite of goals and standards that's become a stand-in in the popular mind for everything from overreliance on standardized tests (if you're a Democrat) to jackbooted federal intrusion into local culture (if you're a Republican). A Florida legislator said it would turn our kids gay. Louis C.K. said it made his kid cry.

And then there's the "number sentence." OK, number sentences aren't actually mentioned anywhere in the Common Core, and they've been part of the math curriculum for decades. They sound weird and unfamiliar, though, which makes them fair game for schools-these-days tsk-tsk-ery. A New York principal argued in the Washington Post that the number sentence concept was too sophisticated for young children. And Stephen Colbert mocked the phrase, suggesting that students should also have the option of using "a word equation or formula paragraph."

The phrase "number sentence" was new to me, too, when my 8-year-old son brought it home from school this year. But I didn't make fun of it. I cheered it! When we call an equation like

a "number sentence" we're not dumbing down. We're telling it like it is. That equation is a sentence. It's a sequence of symbols that makes an assertion about the world. "Dave" is not a sentence; it's a proper noun. "Vsdsgs" is not a sentence; it's a meaningless glurg. But "Dave is my brother" is a sentence. It has a subject ("Dave"), a verb, ("is"), and an object ("my brother"). Just like 2 + 3 = 5: the subject is "2+3," the verb is "=," and the object is "5."

What's wrong with calling 2 + 3 = 5 a good old "equation"? First of all, not all number sentences are equations. A number sentence is a sentence about numbers, which could be an equation, like 2 + 3 = 5, but also an inequality, like 5 > 3.

Worse, "equation" has been around so long that it's lost contact with its literal meaning, "something that equates." Students in my courses routinely refer to "5 > 3" as an equation. The same with a mathematical expression like " x^2 + y^2," which is no more an equation than "Dave" is a sentence.

When we call 2 + 3 = 5 a "sentence" we engage in the radical act of insisting that mathematics has meaning. That shouldn't be a radical act. But, too often, we teach our students that "doing mathematics" means "manipulating clusters of digits according to rules presented to us by the teacher."

That's not math. And when we teach our students to do that, and only that, we are training them to be slow, buggy versions of Excel. What's the point?

If a student doesn't truly grasp that "2 + 3 = 5" is a sentence, a statement about the world that might be true or false, it's hard to see how, when algebra comes around, they can grasp that

is a sentence, too, one which is true for precisely two values of x (namely, the positive and negative square roots of 2) and false for all the rest.

One solution, of course, is to double down, addressing algebra, too, in a purely algorithmic way. You have an equation involving x, certain modifications of the equation are allowed (because the teacher says they're allowed) and when you get to something that has just " x =" on the other side, you've won the game. If you can do this, you can get an A on a typical algebra test. But can you do algebra? I'm not so sure.

My kids really liked a tablet game called DragonBox, which, according to its promotional material, "secretly teaches algebra to your children." Kind of like the way you can secretly feed kale to your children by grinding it up and hiding it in a meatball. But what DragonBox actually teaches are the rules of algebra: that you're allowed to add the same symbol to both sides, that multiplying a sum by a symbol requires you to multiply each summand by the same symbol, and so on. Getting these rules in muscle memory is what you need if you want to get fluent in algebraic computations.

There's just one thing missing, but it's a big thing: the fact that algebra is made of sentences, that it means something, that it refers to something outside itself. An algebraic statement isn't just a string of symbols with an x stuck in there somewhere. It's an assertion about a relationship between quantities (or, when you get to more advanced algebra, between functions, or operations, or even other assertions.) Without that animating idea, algebra is a dead and empty exercise.

To be fair, Jean-Baptiste Huynh, who created DragonBox, gets this. He says in an interview that "DragonBox does 50 percent of the job. We need to teach the rest." Fifty percent is not such a small number, and it sounds about right to me. Computation is important! We lose just as badly if we generate students who have some wispy sense of mathematical meaning but who can't work examples swiftly and correctly. A math teacher's least favorite thing to hear from a student is "I get the concept, but I couldn't do the problems." Though the student doesn't know it, this is shorthand for "I don't get the concept." The ideas of mathematics can sound abstract, but they make sense only in reference to concrete computations. William Carlos Williams put it crisply: no ideas but in things.

Because math is about things. Not, despite how it sometimes looks, about itself, and not about getting a good score on the test. It's about which things-which sentences -are right, and which are wrong, and how to tell the difference.

Update, June 12, 2014: I should have said that my thoughts about DragonBox, formalism, and algebra were touched off by this 2013 John Holbo post at Crooked Timber and the associated comment thread, whose existence I'd forgotten about when I was writing the piece.

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Stephen Colbert Thinks "Number Sentences" Are Silly. They're Not.

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People who teach math, like me, hate it when students ask us, "When am I going to use this?" We don't hate it because it's a bad question. We hate it because it's a really good question, and one that our curriculum isn't really set up to answer. And that's a problem.

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by Janet Pinto, Chief Academic Officer at Curriki

I wish geometry was taught in this way when I was in school.

Curriki's new Project-based Learning (PBL) high school Geometry course is now available. For so many students, it's difficult to make a real-life connection between math and their everyday lives. By adopting a Project-Based Learning (PBL) approach, students learn that geometry is not only theoretical, but practical and necessary. Students will move beyond a basic understanding of concepts to an enjoyment of discovery.

This free Geometry course not only leverages the popular PBL "active" approach but is also aligned to Common Core State Standards. Thom Markham, Ph.D., President of PBL International, is the designer of the course. He notes that Project-Based Learning points us toward the future of education. "It's a proven method for integrating the 21st century skills of communication and teamwork into the delivery of core subjects."

Available online at the Curriki site, Curriki Geometry is designed to meet the needs of students raised in a global, interactive, digitally-connected world through the use of real-world examples, engaging projects, interactive technologies, videos and targeted feedback. Developed with funding from AT&T, the Curriki Geometry course is modular, so can be used as the foundation for students' Geometry 1 curriculum, as a supplementary resource, in an after-school program, or in a homeschool environment.

The course units have been designed with carefully selected Curriki instructional materials, interactive content, videos and other materials that students can explore and use as they solve the series of challenges in each unit. The course structure is based on the six Common Core High School Geometry topics, organized in six projects:
Project-Based Learning in Math: 6 Examples
1. Selling Geometry

This project introduces students to a brief history of geometry, geometric terms, geometric shapes, and transformation and manipulation of shapes through reflections, tessellations, and dilations. Students will form marketing teams to "sell" geometry by explaining key terms, demonstrating key shapes, and describing the significance of geometry to an audience

2. Designing a Winner

This project allows students to apply the geometric principles of triangles, volume, and coordinates to the mapping and design of a multi-purpose arena in a limited area. The arena must serve as a venue that can be easily converted for use by two sports, plus serve as a concert venue if needed. Students will form design teams to create a map showing how the arena can be converted for the two sports. The map will be accompanied by a hand-built model or a digital design using Google SketchUp or other tool. In the final presentations, the map and design ideas will be presented to the local City Council.

3. What's Your Angle, Pythagoras?

One real-world task students must learn is the ability to explain what you know to others. The challenge in this project is for older students to find common examples of right-angle geometry and use their geometric knowledge to create a lesson that explains Pythagorean principles to younger students in a way that is engaging, understandable, meaningful, and relevant.

4. TED Talk: House of the Future

The challenge in this project is for students to examine trends in housing, extrapolate that information to predict the future, and use their geometric modeling skills to design a house that supports their predictions. Students will create a floor plan and basic model of a house of the future that reflects four trends and then deliver their design and give evidence of their thinking in the form of a 10 minute presentation about why their house will be necessary and useful in the future. The format for the talk can vary, but it is suggested that the presentation follow the guidelines for a TED-like talk in which presenters give short presentations that focus on the future and innovation

5. The Art of Triangles

This project teaches students the basics of triangles, such as types, congruence, proofs, and similarity, by asking each student to create a poster, drawing, or personal adornment such as a fingernail design, piece of jewelry, or tattoo that uses at least two different triangular shapes. The product must meet the criteria for 'beautiful'-that is, it must display symmetry or other design elements that make it attractive.

6. How Random is My Life?

This project is designed to have students work through and master basic problems in probability, and then apply that knowledge to authentic issues in which statistics and probability play large roles. The topics should apply to teen interests and concerns. Students will be given the opportunity to research and choose their topic. However, topics may also be assigned by the teacher in one of four areas:

1. Driving and cell phone use
2. Diet and health
3. Professional athletics
4. Costs associated with a college education

After researching the issue, students are expected to explain how probability affects their lives in each of these areas, and to make recommendations to other teens on actions that can improve their chances of making good decisions on each issue

The Common Core Math Practice Standards are woven through each unit, with projects and explorations designed to develop students' mathematical "habits of mind", problem-solving and reasoning skills, and abilities to use mathematical models and technology tools strategically.
Project-Based Learning in Math: 6 Examples; image attribution flickr user vancouverfilmschool

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Project-Based Learning in Math: 6 Examples

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by Janet Pinto, Chief Academic Officer at Curriki I wish geometry was taught in this way when I was in school. Curriki's new Project-based Learning (PBL) high school Geometry course is now available. For so many students, it's difficult to make a real-life connection between math and their everyday lives.

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Kid Snippets: "Math Class" (Imagined by Kids)

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How digital can reshape the way we see mathematics

Humility. It's all I can feel after watching 3Blue1Brown's latest maths video, in which Grant Sanderson and his team provide the most novel of approaches to solving 2D equations, using colour. It is the second time 3Blue1Brown has blown my mind to pieces in just a few weeks, following their earlier video that solved the Basel problem with light.

While I was familiar with the results/methods in both cases, the representations came as a complete surprise. The 3Blue1Brown videos are far from perfect (they proceed at a cadence that sometimes proves frustrating, lack interaction and, in the most recent example, may be of little use to the colour-blind), yet they remind us that our understanding of mathematics will never be complete. However well you think you understand a concept, there will always be new representations to deepen your thinking and force you to confront familiar truths in unfamiliar ways. Mathematics is never a done deal.

Humility is what keeps us open minded to new and novel representations of old concepts. It stands opposed to the arrogance of attaching oneself to singular representations, and to assuming that there is only one way to engage with a given maths concept.

Take bar models: a particular representation that is in vogue here in the UK as the poster child of maths mastery. A key tenet of mastery-based approaches is that mastery is never fully acquired; our understanding of mathematical concepts runs infinitely deep. There are always more problems to solve, more reasoning to be articulated. How sad then, that the implementation of mastery is often restricted to the specific representation of bar models. Don't get me wrong - I have much love for the bars: Cuisenaire rods and Base 10 blocks are among my most commonly used manipulatives, not least because they can showcase a range of concepts. But bar modelling is often embraced so rigidly that the attachment borders on ideological; I have reluctantly indulged in debates with educators who refuse to entertain any alternatives to the bar model representation.

An exclusive focus on any representation - bar model or otherwise - only defeats the pursuit of mastery because it induces procedural fixedness: students may acquire fluency with a particular representation, only to come unstuck when asked to apply their knowledge to less familiar contexts. Flexible thinking comes from having multiple representations. The more ways we have of seeing a particular concept, the exponentially more likely we are to apply that knowledge in new situations.

The digital medium is poised to serve up a multitude of previously unconceived representations. 3Blue1Brown is an all too rare of EdTech shunning the temptation to imitate static, textbook representations. Instead, Sanderson has illuminated my understanding of mathematical concepts in a manner so dynamic that one can scarcely imagine his content in print form. This is not to discard print representations altogether -  Sanderson himself lists a slew of textbooks as his inspiration. But to achieve our deepest understanding of mathematics, we have to embrace digital, print and whatever other media are available to us to build new pathways to mathematical enlightenment.

If the notion of changing up our representations of maths unsettles you, just remember that our mathematical worldview has always been in flux. While mathematical truths are cast in stone, the way in which we interact with those truths depends on the technology available to us. You may, for example, assume that Algebra is necessarily symbolic: it is hard to imagine an algebraic problem that does not involve an x. Yet the original architects of Algebra did not employ such formal expressions. The Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī did not solve quadratic equations in notation that you or I would recognise. He resorted instead to tirelessly writing out his problems of interest (motivated mainly by trade and inheritance laws) in narrative form.

The symbolic form that we so closely associate with Algebra today owes much to the invention of the printing press, which gave mathematicians unprecedented opportunities to share their ideas across cultures and civilisations at rapid speed. The need for a common language, and to avoid ambiguity in translation, became ever more paramount and so entered the x'sand y's we take for granted today.

Keith Devlin is among those who have argued that the digital technologies of today will occasion a shift towards representations that are both efficient and intuitive, arguing that video games are the natural medium for 21st century mathematics. It's an idea worth taking seriously given the confusion and anxiety that symbol-rich representations induce.

My greatest hope for EdTech is that it inspires new representations of mathematics. I would wager that for a good chunk of mathematiccal concepts, the most illuminating representations still await our discovery. The ascendance of dynamic modelling apps such as Desmos and GeoGebra, not to mention the breathtaking videos of 3Blue1Brown, are already reshaping the way we see and interact with mathematics.

The mathematics that awaits us may only be limited by our imagination. Thank goodness we have people like Grant Sanderson dreaming up new ways of presenting old ideas.

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The best representations of mathematics may not exist yet

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Humility. It's all I can feel after watching 3Blue1Brown's latest maths video, in which Grant Sanderson and his team provide the most novel of approaches to solving 2D equations, using colour. It is the second time 3Blue1Brown has blown my mind to pieces in just a few weeks, following their earlier video that solved the Basel problem with light.

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Every teacher knows the frustration of spending a sufficient amount of time teaching and reviewing a concept, only to have students still not "get" how to do the work.

I am constantly asked by my fellow math teachers the following question, "I've taught it over and over again, and spent more than enough time on this skill, yet several students still don't 'get it.' What do I do?"

We've all felt that pain.

Below are 7 strategies I use that have eliminated this pain from my life. At the end of the article, I will share the easy to implement strategy that will help you do all 7 strategies at once, and effectively reach all your students.

1) It's all About Number Sense

Math is not easy for everyone, and there are a lot of rules and formulas to memorize. Students often mix them up or struggle remembering all of them.

To help them with this, you need to teach number sense. Explain to students WHY a formula is what it is. And spend the extra time helping students make sense of it, and how to deduce it if they can't memorize the formula.

For example, I can never remember if a number to the power of zero equals 1 or 0, so I'm sure my students struggle with it as well. So I teach this by modeling a simple thought experiment. I ask the class what 10 ² is. Most of them know that 10 2 is 10 x 10, and most know that 10 x 10 = 100. I then ask what 10 3 is, and guide them to the realization that it is 10 x 10 x 10, which equals 1,000. I continue asking what 10 to the next power is, and drawing the pattern on the board:

Once the pattern has been realized, I ask the students to work backwards, and simply add 10 1 to the top. They all realize it equals 10, and then we talk about why. We can then add 10 0 above that, and deduce that it is 1, simply based on the logic of the pattern.

By teaching my students it this way, I am not only teaching the skill, but I am teaching them HOW to think mathematically. Furthermore, they are being given the tools they will need to find the answer if they can't memorize the formula.

2) Give a ReTeach, Not Another Worksheet

I am always frustrated when I observe a teacher tell their class, "you guys did really poor on this skill on the last test, so we're going to practice it some more," and then that teacher hands the class a worksheet.

For the record, I am guilty of doing it too.

But, honestly, this makes no sense. If the student(s) didn't understand it on the test, why would they all of a sudden know how to do it now? Obviously there is something that they have not grasped, or are doing incorrectly, that needs fixing.

An effective way to help your students is to give a ReTeach of the skill. Again model the work, point out common mistakes, and try to teach WHY certain processes work.

3) The Secret to Committing a New Skill to Memory

When learning a new skill, you have to practice it repetitively to commit it to memory.

Yet, often, we as teachers model a problem several times, and then give the students 5 to 10 practice problems expecting the students to fully grasp the concept.

Even if they do all the practice problems correctly, if the student has not practiced the skill enough, that knowledge will be forgotten in a short period of time.

Give your students LOTS of practice. And if you have a student who is weaker mathematically, he will need more practice then most, so give it to him.

4) Practice Does NOT Make Perfect

Many times I see teachers do stations with their class, and one of the stations is a worksheet. Often, I will go over to the students in that station, and look at their work. I'm always discouraged to see students, who have done all the work, but done every problem wrong.

Practice does not make perfect. Perfect practice makes perfect

There's no value in working many math problems, but doing everyone incorrectly. This does not help student learn, and it wastes time.

A better strategy is to give feedback after each question, so that students can recognize when they make mistakes. This also will allow them the opportunity to learn from their mistakes.

To accomplish this, I try to assign work on websites like IXL and KhanAcademy as much as possible. But if I cannot get sufficient technology in the classroom, I put answer sheets with the assigned work, so that students may go over each problem after they have attempted it.

5) Don't Penalize Their Failures

We know that failure is part of the learning process, yet teachers often penalize students for making them, or don't show them the error in time to help them learn from them.

Giving immediate feedback will help them recognize their mistakes so that they can learn from them. Not penalizing the student for making the mistake will give them the courage to attempt the work, so that they can learn the skill.

6) Take as Long as You Need

Many math skills build upon each other. For example, you can't do two step equations, if you haven't already mastered one step equations!

If a student has not mastered a skill, and then the math teacher progresses him onto the next skill, the student has no chance of mastering it.

Moving students on to more advanced skills before they are ready has not only guaranteed their failure, but it will also lead to feelings of resentment towards math, school, and you.

I give students as much practice as they need for each skill, and allow them to spend as much time as they need on that skill. Yes, sometimes that has resulted in students spending exorbitant amounts of time on one standard, and yes, they did not get exposed to all the skills that year. But I would rather my students learn three skills in one year, and master them, then me teach them 15 skills, and they learn none of them.

7) Celebrate Their Success

One of the greatest joys you will ever have is when a student has struggled, and finally "gets it."

These should be exciting moments in their life! So make it that way.

In our class, I assess students every 4 days with a quiz on the standard that they have been working on. Which means we celebrate success every 4 days! When a student demonstrates mastery on the skill they are working on, we announce it to the class, we play music, we clap and cheer, and we put that students name under a banner that announces their success! (I also send home a certificate so they can get the same treatment from their parents).
When students can see their success, it encourages them to want to do better. It also helps them measure their accomplishments and build positive sentiments towards school and math.
Take some time to contemplate how you will measure if your student has succeeded at one of the math standards in your class, and how will you celebrate that success?

Here's what my class looks like:

I start with a short lesson on a standard that we are focusing on. Including exposure to test style questions and a review of their data.

We then move into our station work. Each station consists of 4 days of work on one skill that the student needs to develop (so every student may be working on a different skill). The station is built with a reteach, as well as sufficient practice to allow the student to demonstrate mastery.

The station is about 20-30 minutes each day, and it lasts for 3 days. On the fourth day, I assess them for mastery. If they have learned the skill, they move on to the next station. If they haven't mastered it, they repeat the station until they learn the skill and demonstrate proficiency.

At the end of the period we always focus on test design, data, and word problems to help my students prepare for the assessment (the stations are to help them with the math skill).

RethinkMathTeacher.com has a free teaching series to help you develop your own learning stations.

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What to Do When Your Students Still Don't Get It - RETHINK Math Teacher

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Every teacher knows the frustration of spending a sufficient amount of time teaching and reviewing a concept, only to have students still not "get" how to do the work. I am constantly asked by my fellow math teachers the following question, "I've taught it over and over again, and spent more than enough time on this skill, yet several students still don't 'get it.'

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This Measuring Angles Project provides the instructions for students to design a putt-putt golf course that incorporates requirements for acute, obtuse, right, straight, and reflex angles.

There is a list of specific measurements to use as well as parallel and perpendicular lines and shapes to include such as a right, isosceles, equilateral, and scalene triangle; and other geometric shapes.

Examples are given that show some finished products and how to label them. A 'blueprint' with labels and a finished product are recommended for this project. The lesson is available in printable PDF format.

Your students will be designing a Full 9 Hole Putt-Putt Golf Course! Follow the instructions below to get started.

This project can be done as a solo project if you have a class full of class clowns or even honors students. I prefer to have the student work on their courses in groups of 2 or 3. Anything more than that and they will have too many ideas and it will slow down the course creation.

This project can be done 3 different ways.

1. Split your class into 9 groups. Have each group create a real Miniature Golf Course Hole. They can use things like wood, cardboard, or foam board. There are tons of options for this. Let them be creative and use what they want or what is available. Things they try will work and won't work. When ideas fail they are good teaching points. When they complete the course let them actually play it. You'll need putters and golf balls. (My advice would be to use plastic golf clubs and balls. The ones toddlers use growing up.)

2. The second option would be to have them create the course using some form of technology. There are tons of apps and software that can be used for this. One free option would be have them create the course using Canva or even something like Geogebra.

3. The last option would be to have them draw it up on paper or poster board. This is less fun for them but still gets the lesson across with the same result. This also takes less class time to finish. Each group will also be able to create all 9 holes instead of just one.

The Course Must Contain Each of These Ball Path Angles...

1. 30 Degree Angle

2. 45 Degree Angle

3. 60 Degree Angle

4. 90 Degree Angle

5. 270 Degree Angle

6. 300 Degree Angle

7. 330 Degree Angle

Course Must Include At Least Each of These Shapes...

1. 9+ Circles

2. 8 Squares

3. 7 Rhombus

4. 6 Rectangles

5. 5 Kites

6. 4 Trapezoids

7. 3 Equilateral Triangles

8. 9 Right Triangles

9. 4 Acute Triangles

10. 4 Obtuse Triangles

11. 4 Isosceles Triangle

12. 4 Scalene Triangle

Each Individual Holes Boundaries Must Contain...

1. At least one set of Parallel Lines

2. At least one set of Perpendicular Lines

3. At Least one Acute Angle

4. At least one Obtuse Angle

5. At least one Right Angle

6. At least one Reflex Angle

Putt-Putt Course Design
Here are some samples of individual hole designs...

Here is the link to download the:

Measuring Angels - Putt-Putt Course Project Rubric
Measuring Angels - Putt-Putt Course Project Rubric
You will be designing a Full 9 Hole Putt-Putt Golf Course! Follow the instructions below to get started.

Project Requirements
The Course Must Contain Each of These Ball Path Angles... ( 1 Points Each = 7 TOTAL )

30 Degree Angle

45 Degree Angle

60 Degree Angle

90 Degree Angle

270 Degree Angle

300 Degree Angle

330 Degree Angle

Course Must Include At Least Each of These Shapes... ( 1 Point Each = 70 TOTAL )

9+ Circles

8 Squares

7 Rhombus

6 Rectangles

5 Kites

4 Trapezoids

3 Equilateral Triangles

9 Right Triangles

5 Acute Triangles

5 Obtuse Triangles

5 Isosceles Triangle

4 Scalene Triangle

Each Individual Holes Boundaries Must Contain... ( 1 Point Each = 54 TOTAL )

At least one set of Parallel Lines

At least one set of Perpendicular Lines

At Least one Acute Angle

At least one Obtuse Angle

At least one Right Angle

At least one Reflex Angle

Neatness
(9 Points Each Hole = 9 TOTAL)
Creativity and Accuracy
(60 Points TOTAL)
** TOTAL POINTS FOR PROJECT = 200! **

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Measuring Angles Putt-Putt Course Design Project ⋆ GeometryCoach.com

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This Measuring Angles Project provides the instructions for students to design a putt-putt golf course that incorporates requirements for acute, obtuse, right, straight, and reflex angles. There is a list of specific measurements to use as well as parallel and perpendicular lines and shapes to include such as a right, isosceles, equilateral, and scalene triangle; and other geometric shapes.

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Mathematics education in the United States is broken. Open any newspaper and stories of math failure shout from the pages: low international rankings, widespread innumeracy in the general population, declines in math majors. Here's the most shocking statistic I have read in recent years: 60 percent of the 13 million two-year college students in the U.S. are currently placed into remedial math courses; 75 percent of them fail or drop the courses and leave college with no degree.

We need to change the way we teach math in the U.S., and it is for this reason that I support the move to Common Core mathematics. The new curriculum standards that are currently being rolled out in 45 states do not incorporate all the changes that this country needs, by any means, but they are a necessary step in the right direction.

I have spent years conducting research on students who study mathematics through different teaching approaches-in England and in the U.S. All of my research studies have shown that when mathematics is opened up and broader math is taught-math that includes problem solving, reasoning, representing ideas in multiple forms, and question asking-students perform at higher levels, more students take advanced mathematics, and achievement is more equitable.

One of the reasons for these results is that mathematical problems that need thought, connection making, and even creativity are more engaging for students of all levels and for students of different genders, races, and socio-economic groups. This is not only shown by my research but by decades of research in our field. When all aspects of mathematics are encouraged, rather than procedure execution alone, many more students contribute and feel valued. For example, some students are good at procedure execution, but may be less good at connecting methods, explaining their thinking, or representing ideas visually. All of these ways of working are critical in mathematical work and when they are taught and valued, many more students contribute, leading to higher achievement. I refer to this broadening and opening of the mathematics taught in classrooms as mathematical democratization. When we open mathematics we also open the doors of math achievement and many more students succeed.

In mathematics education we suffer from the widespread, distinctly American idea that only some people can be " math people." This idea has been disproved by scientific research showing the incredible potential of the brain to grow and adapt. But the idea that math is hard, uninteresting, and accessible only to "nerds" persists. This idea is made even more damaging by harsh stereotypical thinking-mathematics is for select racial groups and men. This thinking, as well as the teaching practices that go with it, have provided the perfect conditions for the creation of a math underclass. Narrow mathematics teaching combined with low and stereotypical expectations for students are the two main reasons that the U.S. is in dire mathematical straights.

This summer I taught a course through Stanford's open online platform explaining research evidence on ability and the brain and on good mathematics teaching, for teachers and parents. The course had a transformative effect. It was taken by 40,000 people, and 95 percent said they would change their teaching or parenting as a result. Hundreds wrote telling me that the ideas in the course had been life-changing for them. Teachers and parents are open to research, and new technologies are finally providing a way that important research evidence, on mathematics, learning, and the brain, can reach the audiences that need them.

Conrad Wolfram, cofounder of Wolfram-Alpha, one of the world's most important mathematical companies, has spoken widely about the mismatch between the math that people need in the 21st century and the math they spend most of their time on in classrooms: computing by hand. The Common Core helps to correct this problem by embracing broader mathematics and requiring the use of advanced technology, such as dynamic geometry software. Students in the Common Core will spend less time practicing isolated methods and more time solving applied problems that involve connecting different methods, using technology, understanding multiple representations of ideas, and justifying their thinking.

For example, consider the following two published test questions. The first comes from California's old standards, the second from the Common Core.

1. Which of the following best describes the triangles shown below?

A both similar and congruent

B similar but not congruent

C congruent but not similar

D neither similar nor congruent
California Standards Test, released test questions, geometry, 2009
2. Triangle ABC undergoes a series of some of the following transformations to become triangle DEF:

Rotation

Reflection

Translation

Dilation

Is DEF always, sometimes, or never congruent to ABC? Provide justification to support your conclusion.

Common Core Smarter Balanced Grade 8 Sample Item, 2013
The second question, from one of the Common Core assessment teams, does not simply test a mathematical definition, as the first does. It requires that students visualize a triangle, use transformational geometry, consider whether different cases satisfy the mathematical definition, and then justify their thinking. It combines different areas of geometry and asks students to problem solve and justify. It does not offer four multiple-choice options. Common Core mathematics is more challenging than the mathematics it will replace. It is also more interesting for students and many times closer to the mathematics that is needed in 21st-century life and work.

An important requirement in the Common Core is the need for students to discuss ideas and justify their thinking. There is a good reason for this: Justification and reasoning are two of the acts that lie at the heart of mathematics. They are, in many ways, the essence of what mathematics is. Scientists work to prove or disprove new theories by finding many cases that work or counter-examples that do not. Mathematicians, by contrast prove the validity of their propositions through justification and reasoning.

Mathematicians are not the only people who need to engage in justification and reasoning. The young people who are successful in today's workforce are those who can discuss and reason about productive mathematical pathways, and who can be wrong, but can trace back to errors and work to correct them. In our new technological world, employers do not need people who can calculate correctly or fast, they need people who can reason about approaches, estimate and verify results, produce and interpret different powerful representations, and connect with other people's mathematical ideas.
The new Common Core curriculum gives more time for depth and exploration than the curricula it has replaced
Another problem addressed by the Common Core is the American idea that those who are good at math are those who are fast. Speed is revered in math classes across the U.S., and students as young as five years old are given timed tests-even though these have been shown to create math anxiety in young children. Parents use flash cards and other devices to promote speed, not knowing that they are probably damaging their children's mathematical development. At the same time mathematicians point out that speed in math is irrelevant. One of the world's top mathematicians, Laurent Schwartz, reflected in his memoir that he was made to feel unintelligent in school because he was the slowest math thinker in his class. But he points out that what is important in mathematics "is to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn't really relevant." It is fortunate for Schwartz, and all of us, that he did not grow up in the speed- and test-driven classrooms of the last decade that have successfully dissuaded any child that thinks deeply or slowly from pursuing mathematics or even thinking of themselves as capable.

The new Common Core curriculum gives more time for depth and exploration than the curricula it has replaced by removing some of the redundant methods students will never need or use. Sadly it does not go far enough in this regard, and the high-school grades in particular are still packed with obsolete content. But educational progress is rarely fast and the changes implemented in the Common Core are a step in the right direction.

The U.S. does not need fast procedure executors anymore. We need people who are confident with mathematics, who can develop mathematical models and predictions, and who can justify, reason, communicate, and problem solve. We need a broad and diverse range of people who are powerful mathematical thinkers and who have not been held back by stereotypical thinking and teaching. Common Core mathematics, imperfect though it may be, can help us reach those goals.

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The Stereotypes That Distort How Americans Teach and Learn Math

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Mathematics education in the United States is broken. Open any newspaper and stories of math failure shout from the pages: low international rankings, widespread innumeracy in the general population, declines in math majors. Here's the most shocking statistic I have read in recent years: 60 percent of the 13 million two-year college students in the U.S.

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I have worked with a lot of teachers who tell stories of being pushed toward one single correct way of representing their math work by their teachers. If we teach that way, we are probably teaching our students to be too inflexible, too rigid with their mathematical representations. Typically, we focus on the product (written) of the representation, rather than the entire mental process that leads to representation (on paper).

That said, some representations, and tools for representation, are better than others, and we can help kids choose between them. Representing fractions with a circle area model is generally less effective than representing fractions using a rectangular area model, for example. Number lines are great for showing addition of integers, but a trickier for subtraction of integers. Two colour tiles are useful for illustrating the zero principle, that every integer has its opposite, that is an equal distance from zero.

One conflation we struggle with is that representation is the same as "drawing a picture". Clearly, if I show you subtraction with tiles, that's my representation, and perhaps there is no need to "draw it out" on paper. That might be a waste of time. Sometimes kids do just "'draw a picture", because they are encouraged to do so. By picture, I think, we mean "a picture that has mathematical content", or, a "representation". A picture with no mathematical content is just a picture.

Infamously, standardized testing here used to ask kids for answers in "numbers, pictures, AND words". Not only is this a recipe for getting kids to hate math, it may lead them to draw some bad, bad pictures, where pictures are not what is needed to represent the mathematical content of the problem.

Purposeful representation of the mathematics in the problem is what is needed. Here is a helpful definition of representations.

Representations are tools for thinking, explaining, justifying. NB: Pape/Tchoshanov, 2001. pic.twitter.com/Nux9jP7Z4b

- Matthew Oldridge (@MatthewOldridge)

The Pape/Tchoshanov paper is instructive for teachers in getting kids to develop their own mental representations, and to get them out into the physical world. (This may or may not mean drawing them on paper. The physical act of representation with thinking tools or manipulatives may be what''s needed).

In the authors' words:

"We use the term representation(s) to refer to both the internal and external manifestations of mathematical concepts."

This is a huge argument for paying attention to your students' thinking, in order to make it visible. We need to help kids visualize representations of interesting mathematical concepts, in order to imagine them into being.

So we have two complementary aspects of mathematical representations:

-the act of representing (verb)

-the representation itself (noun)

This quite neatly corresponds to both NCTM and Ontario curriculum definitions of representation: as both process and product.

The authors propose a "'zone" of interaction between mental and physical interactions.

The nice example used is of "six"-a kid develops an internal mental image of "'six', then connects it to sets of six objects, the numeral 6, the written word "six", the spoken word "six", and so on.

As kids advance in math, we should encourage representational thinking. That is, building and having a repertoire of ways to externalize math concepts. Tables, graphs, tree diagrams, area models, arrays- these are all examples. Physical manipulation of objects like snap cubes is another.

The authors put it nicely:

"the development of students' thinking skills requires a multiple representational approach."

Put simply: representations are tools for reasoning.

Two things to think about.

Kids often mathematize math tasks by drawing pictures with little or no mathematical content. Do these count as representations?

"Students often produce representations that lack meaning."

Manipulatives are generally a good thing, as long as they are accurate mathematical representations of the concept, the mathematics you are working on. Daniel Willingham presents this caveat here.

Manipulatives, if not chosen wisely, can get in the way of mathematical representation. Let the tool fit the task (and the math).

The end goal of representation is good communication. Can the student communicate their thinking, in words, or on paper, using mathematical representations?

Even better, can they start to see how different representations are connected? Do they see a table of values, a pattern made with cubes, a graph, and an algebraic expression as fundamentally the same thing?

How do they show the same math in different ways?

Fluency with multiple representations should be a goal of thinking math classrooms; we can help kids to visualize, conceptualize, and make real their understanding of powerful mathematics.

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Mathematical Representations as Products of Thinking

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I have worked with a lot of teachers who tell stories of being pushed toward one single correct way of representing their math work by their teachers. If we teach that way, we are probably teaching our students to be too inflexible, too rigid with their mathematical representations.