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When asking for answers, an equitable way to include all student thinking and reasoning.  You can also go with good old-fashioned popsicle sticks.

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What To Do Under Quarantine?

You'll have to take our word for it that the wheel landed on Phone an old friend. Your browser cannot display the wheel of choices. Please upgrade to a newer browser that supports HTML5 (most new ones do). Wheel Decide would normally show a wheel right here with choices that you can set (lunch, games, dinner, anything!). You can spin it to get a random decision. Make your own wheel of dinner, wheel of fortune, wheel of lunch, wheel of whatever you want now!

What's Wheel Decide?

Wheel Decide is a free online spinner tool that allows you to create your own digital wheels for decision making, prize giveaways, raffles, games, and more. Browse through our wheels and spin to randomize your life and make the decisions that have no wrong answers.

Decision Making

Let the wheel decide. Save your energy for the big decisions

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Can't decide? Wheel Decide for you! Set your own custom choices and then spin the wheel to make the random decision of lunch, movie, or anything! Make your own wheel of dinner or whatever now!

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Helping students to develop growth mindsets.

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Jo talks about brain pathways during Walk the Graph activity

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Stealing form Language Arts class?

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When math lessons emphasize collaboration and exploration over performance, students become more engaged and confident math thinkers.

Barrie Fanton / agefotostock / Alamy Stock Photo

Writing, discussing, and revising rough drafts is standard practice in language arts classes. But to Amanda Jansen, a junior high math teacher, author, and professor at the University of Delaware's School of Education, "rough draft thinking" is also a powerful tool for making the math classroom an inclusive, engaging place of shared exploration where students feel comfortable thinking aloud as they problem-solve.

Producing rough drafts of math problems, and discussing them in class, is "liberating for students because they can start to see how they're contributing to their colleagues' learning in a lot of different ways," Jansen says in an interview with KQED's MindShift. "After the initial shift, [students] start to get more excited, like, 'Wow, my ideas actually matter.' Everyone feels included in that kind of environment."

When the focus of math class evolves from emphasizing right or wrong answers-and the frequently public judgement that accompanies that high-stakes style of learning-to a collaborative environment where kids feel motivated to explore and take risks, students respond by becoming more engaged and confident about sharing their ideas. "[Students] show up with this sense of just feeling happy to be there," says Jansen. "They are more likely to just put something out there, whether it's in the group work or coming up to the document camera to share their thinking. They tend to feel proud."

Putting rough draft math into practice

When Christine Hubbard, a math teacher in Middletown, Delaware, uses rough draft thinking with her seventh-grade students, she defines the process briefly for her students and then moves directly into a number talk during which she explains the lesson-in this case: visualizing how to chunk sets of dots to make sets easier to count. Then Hubbard briefly shows her students a domino-like figure and asks them how many dots they see.
Courtesy Stenhouse Publishers
"I want you, in your mind, to be thinking about how many dots there are," she says to the class. All her students report seeing eight dots.

Hubbard steers the conversation toward examining students' strategies with questions like "Can anyone be brave enough to share and justify why they think there's eight dots up there?" On a whiteboard-or using Google Slides during synchronous learning-one student draws a set of five dots, like the side of a dice, plus a group of three dots; another student draws two horizontal lines of three dots and an additional two in the middle; and yet another draws a shape that loosely resembles a diamond composed of six dots.
Courtesy of Stenhouse Publishers
Students discuss what to call the diamond-like shape: maybe a "thingy," suggests one student or a "weird rectangle," says another. "Yes, you could describe it that way," Hubbard encourages. Finally, a student suggests that because the shape has two pairs of parallel sides, it could be a parallelogram. Hubbard confirms that yes, it is indeed a parallelogram. Guided by Hubbard, and with multiple students contributing ideas along the way and engaging in the type of verbal exploration that helps learners absorb and connect mathematical concepts, the class arrived at the "correct" answer.

Rough draft thinking can engage more students

Incorporating rough draft thinking into an existing math curriculum can feel like a big lift, but Jansen says the investment pays off: the practice winds up becoming more efficient over time as teachers become more adept at building in revision expectations and experiences.

Getting frustrated students to re-engage with mathematics, after all, is "not as simple as making problems relevant, or connecting them to the real world," writes K-12 math and science coordinator Matthew Beyranevand. "To get students interested in math, we need to provide opportunities for students to authentically inquire. We want kids-not the textbook or the teacher-to ask the questions."

That's where Jansen says framing math as a shared exploration comes into play, giving more students a chance to contribute to collective problem-solving and communal understanding of mathematical concepts. When "everyone's ideas have strengths in them and the teacher points out what's valuable about the drafts, and [students'] peers start to point out what's valuable, then everyone is seeing each other as having some mathematical strengths," Jansen suggests.

It also provides teachers with the opportunity to be more intentional about helping students, who are often riddled with math anxiety and lack confidence in the subject, see themselves as mathematicians. Math is "plagued by stereotypes and many students think math isn't for people who look like them," writes high school math teacher Dylan Kane. "As a math teacher, it's my responsibility to offer counternarratives that help students see themselves as mathematicians and expand what they think it means to 'do mathematics'."

Keeping track of who is being called on in class is critical here so that teachers aren't inadvertently creating new equity dilemmas by positioning the same group of students-girls or Black or LatinX students, for example-as only capable of presenting work that's rough, versus others being consistently recognized for work that's more developed. Refined and brilliant thinking, Jansen says, can come from any student. "We need to make sure not only are [diverse students] having a voice, but their strengths are being really looked at," she says.

For shy, quiet kids who might prefer to keep their math thinking to themselves, Jansen gently challenges them to share in class. She also likes to offer opportunities for writing-usually when the class is working on revision-as a low-stakes way for introverted kids to participate. "Every time we're being asked to articulate our thinking, we make new connections or crystallize our ideas just by trying to put them into words or into writing," Jansen says.

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Rough Draft Thinking Can Make Math Class More Inclusive

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When math lessons emphasize collaboration and exploration over performance, students become more engaged and confident math thinkers. Barrie Fanton / agefotostock / Alamy Stock Photo Writing, discussing, and revising rough drafts is standard practice in language arts classes.

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Great problems for discussing different approaches and making connections.

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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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Fantastic strategy for showing that mistakes are an important part of the learning process.

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Super fun and really tap into creative problem solving and thinking.  #unitchat on Twitter

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Topics: Mental math, numerical fluency; argument & critique
Materials: White board or projector
Common Core: Variable, but especially MP3, MP1, and NBT

Counting with respect to different units.

Unit Chats are one of the great openers developed in the last few years. The emphasis is on different units within the image. Students can answer "how many" with respect to the different options of units presented by the picture, and also relate the units to each other to build the framework for a deeper understanding of multiplication, division, and fractions.

Truly a knockout opener.

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Topics: Mental math, numerical fluency; argument & critique Materials: White board or projector Common Core: Variable, but especially MP3, MP1, and NBT Counting with respect to different units. Unit Chats are one of the great openers developed in the last few years. The emphasis is on different units within the image.

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Love to use this when introducing a new topic or module. Start with questions !

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Developed by the Right Question Institute, the Question Formulation Technique, or QFT, is a structured method for generating and improving questions. It distills sophisticated forms of divergent, convergent, and metacognitive thinking into a deceptively simple, accessible, and reproducible technique.

The QFT builds the skill of asking questions, an essential - yet often overlooked - lifelong learning skill that allows people to think critically, feel greater power and self-efficacy, and become more confident and ready to participate in civic life.

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What is the QFT? - Right Question Institute

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The Question Formulation Technique (QFT) is a simple but rigorous step-by-step process designed to help […]

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Tons of strategies and techniques for implementing cooperative learning in your class

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kagan publishing teaching catalog teacher resources school supplies music games cooperative learning differentiated instruction educational products esl ell foreign language posters multiple intelligences textbooks math books science literature social studies

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Kagan's FREE Articles - Kagan Structures: A Miracle of Active Engagement

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Read Dr. Spencer Kagans provocative and insightful articles on Kagan Structures and how they have the power to transform education. Cooperative learning, multiple intelligences, the brain, character development, Win-Win Discipline.

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Making a mistake while going through a problem to see if a student can catch where there was some sort of mistake and if they can analyze to learn from their mistakes

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I have students write for 1 minute what they think is the answer to the problem. I tell them there is no right answer.

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Instead of asking a class of students if they have any questions - tell the class to ask you 2-3 questions about the lesson

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"Get Rid of the Question" or "Get Rid of the Question and the Numbers": Strategies to help students develop their observations of the problem before being distracted by the formula.

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By now, many educators are familiar with this oft-cited experiment in math education: Ruesser (2000) asked students: 'There are 125 sheep and 5 dogs in a flock. How old is the shepherd?' (p. 23), to which many students presented various answers that represented some relationship between the two numbers in the sentence. Why is this happening? Why would students come up with answers to questions designed to have no answer?

Sensemaking in Math Education

Educators have attributed this phenomenon to a lack of "sensemaking" in mathematics, or "non-sense making." Essentially, students aren't assigning adequate meaning to the elements of math problems as they navigate numbers, words, and symbols to find an answer - they're simply applying familiar operations to numbers in a way that generates another number in the end.

The National Council of Teachers of Mathematics (NCTM) defines sensemaking as:

"...the process of understanding ideas and concepts in order to correctly identify, describe, explain, and apply them. Genuine sense making makes mathematical ideas "feel" clear, logical, valid, or obvious." (Battista, 2017, p.1)

Annie Fetter, a founding staff member of the Math Forum and mathematical consultant, frames sensemaking as the antithesis of "answer-getting." She writes:

Annie Fetter explains sensemaking in a 5-minute Math Forum Ignite talk.

"Many students believe that the goal of math is to get answers really fast. They don't believe that doing math means thinking about situations, pondering different ideas, or even taking time to understand a story before trying to calculate anything." (Fetter, p. 2)

Math teachers can take steps to foster sensemaking habits in students at all grade levels. In addition to improved mastery of critical math concepts, an emphasis on sensemaking in the classroom may also lead to increased student agency, engagement, and even equitable opportunities.

How to Encourage Sensemaking in Math

1. Get Rid of the Question: This strategy calls for educators to present information about a scenario (perhaps a graph, or a few sentences - think, the beginning of a classic math word problem without the actual question). Instructors allow students to make observations and verbalize or document what they know. For more on the process of making observations, see the full white paper.

2. Get Rid of the Question and the Numbers: This approach is similar to the first but goes a step further by first framing the scenario to students without any numbers. The scenario may include words, images, or diagrams - but requires students to begin to make sense of the "story" being told, or the foundation of the math problem.

3. Give Students the Answer: This strategy directly addresses the issue of "answer-getting", because it shifts the unknown element to the process, not the outcome. Examples of this strategy include "In and Out" boxes, where students have a series of numbers to start and a series of corresponding numbers that "came out of the box" and must determine the function that would produce that series of number transformations.

4. Ask About Ideas, Not Answers: To elicit ideas from students about math problems rather than answers, Annie Fetter suggests the phrase "Tell me something about..." in reference to a scenario. There's an element of inclusivity and empowerment to this strategy - rather than implying that someone in the class has the correct answer to the problem, the teacher is implying that everyone in the class has valuable observations about the problem.

In her whitepaper, " Four Ways to Encourage Sensemaking in Math," Annie Fetter outlines a few instructional strategies math teachers can use to encourage students to practice sensemaking:

For detailed examples of each strategy, see the full whitepaper.

Why Sensemaking in Math is Important

Arming students with the skills to make sense of math problems is clearly intended to boost mastery, fluency, and overall mathematical competence - the ultimate goal is simply to help students understand math. But some researchers connect sensemaking in math to other critical elements of students' relationship to math education that are all too often in short supply: including agency, engagement, and even equity.

Sensemaking is tied to student agency. Annie Fetter's strategies for sensemaking illustrate this connection clearly: empowering students to take the time to make observations and tell a story gives them ownership over the problem-solving process. Mathematics scholar Piera Biccard frames agency and sensemaking as the "opportunity that learners will have to see themselves as mathematical thinkers" (Biccard, 2018, p. 4). In order to make sense of math, students must make choices about how to exercise mathematical thinking.

Sensemaking can boost student engagement. Again, the examples from Annie Fetter's paper make this connection intuitive: giving students the room to make decisions about problem-solving also empowers them to stay engaged even when they're struggling. Mathematics educator and researcher Michael Battista reminds educators that math should be an "intellectually satisfying experience" for students, and students who continually come to new understandings about math and believe that they are capable of math are much more likely to stay engaged. Those who continue to fail to understand (let alone produce the "right answer") are more likely to disengage. (Battista, 2017).

Encouraging sensemaking may create a more equitable learning environment. Placing value on the process of understanding math rather than simply producing the right answer is a strong foundation for a more equitable math classroom. In Annie Fetter's "tell me something about..." strategy, all students have the opportunity to share observations and thoughts about mathematics, before anyone has the opportunity to make it to the "answer." Michael Battista points out that sensemaking can align to all three Response to Intervention (RTI) tiers, because teachers can monitor Tier 2 and 3 students' mathematical learning by observing their reasoning through observation and formative assessment (Battista, 2017).

For more on the how and why of sensemaking in math, check out this interview with Annie Fetter:

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What is Sensemaking in Math, and Why is it Important?

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By now, many educators are familiar with this oft-cited experiment in math education: Ruesser (2000) asked students: 'There are 125 sheep...

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The brain dump activity is a great way to get creative juices flowing and help students make sense of a topic/vocab word. Working in groups usually helps kids feel less anxious and more willing to participate.

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I model asking questions so that students can get an idea of what how they can pose questions.

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Constantly tell students to change their statement to a question or they need to ask a question when groups are presenting. This gets them into the habit of asking questions.

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Finding tasks that scaffold off of previously learned material and ask students to build off previous knowledge and ask how they think their previous knowledge would apply to a higher level problem

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Questions are a way that students can verbalize their thinking process to check for understanding or sometimes validate their thinking. Building confidence in students to ask questions is an important trait that teachers should attempt to facilitate for their students.

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Owning up to your own mistakes as a teacher whether it be in a lesson or during note taking time and letting kids know its ok to make mistakes. One of my favorite sayings I repeat over and over to the kids is, "it doesn't have to be perfect, just right" meaning we are going to make mistakes and its ok as long as in the end we work to make things right.

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Have students look at a problem and ask them what do you know that you can apply to a problem. Can it be useful? How would it be applied? Where would you struggle to continue?

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Effective Questions Increase Student Learning

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Effective Questions Increase Student Learning

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Increase your own access to effective questions by structuring the design of the question stem and adding specific go-to question prompts.

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How Asking More Effective Questions Can Increase Student Learning in Math

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Teachers who ask students questions while demonstrating how to solve problems increase engagement and facilitate a deeper understanding.

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I analyze the lesson to determine what skills are going to be needed so I can make a connection to what they have learned.

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Performance can be stressful for students to be evaluated by. The goal that we as math teachers have is to help our students learn. Focus on the learning more so than the outcome.

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Ask students questions that forces them to think in a scaffolding manner

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Math is about creativity and making sense not just right answers!

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Math is about creativity and making sense not just right answers!

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give students projects that allow them to show their growth in learning. Tasks that are scored with rubrics so students can see where they did well and where they can also improve. This allows students to see where they need to learn a little more

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I tell students I always want to see their work because that shows their thinking and I can see where in the process they need help if any.

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Students start the year by setting reasonable / attainable goals starting with the short term. These goals stay in their portfolio as a way to track their progress. Every month we check student progress and uncover what they DID. Positive reinforcement.

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Give students a problem that gives them the right answer, but there are one or two steps in the work that are incorrect. They need to find were that mistake is.

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There's a poster in my classroom that says "It's Okay to Ask for Help" When I see students struggling I point to the poster to remind them that it's okay if they need help with something. I really emphasize that asking questions is how we get better and learn more.

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Shift away assessing from speed and "right-answer-getting" towards open-ended and deeply-provoking questions.

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Mastering the subject has become less about learning and more about performance.

Why do so many students hate math, fear it, or both?

If you ask most students what they think their role is in math classrooms, they will tell you it is to get questions right. Students rarely think that they are in math classrooms to appreciate the beauty of mathematics, to ask deep questions, to explore the rich set of connections that make up the subject, or even to learn about the applicability of the subject; they think they are in math classrooms to perform. This was brought home to me recently when a colleague, Rachel Lambert, told me that her 6-year-old son had come home saying he didn't like math; when she asked him why, he said that math was "too much answer time and not enough learning time."

Students from an early age realize that math is different from other subjects. In many schools across the U.S., math is less about learning than it is about answering questions and taking tests-performing.

More From The Hechinger Report

The testing culture in the U.S., which is more pervasive in math than other subjects, is a large part of the problem. When sixth-graders in my local district came home saying that they had a test on the first day of middle school, it was in one subject only: math. Most students and parents didn't question whether a test was the right way to introduce a new year of mathematics. As one girl said to me, "Well, the teacher was just finding out what we know." But why does this only happen in math? Teachers in history or English don't give tests on the first day to find out what students know. And why do so many math teachers boil the subject down to producing short answers to narrow questions under pressure? It is no wonder that so many students decide mathematics is not for them.

In fact, it is not surprising that teachers test math all the time-in the last decade teachers' jobs have come to depend on student performance on narrow state tests. The Common Core promises an improvement in the types of tests used-with questions that are less narrow and require thinking, instead of choosing a letter (A, B, C or D)-but the bigger problem is the testing culture in classrooms. It is not unusual for high-school math teachers to test children every week, communicating to students that they are constantly being evaluated.
If you ask most students what they think their role is in math classrooms, they will tell you it is to get questions right.
Educators know that the most productive math-learning environments are those in which students receive positive messages about their unlimited potential and work on interesting and complex problems; in which they feel free to try ideas, fail, and revise their thinking. Students with a "growth" mindset are those who believe that their ability is not "fixed" and that failure is a natural part of learning. These are the students who perform at higher levels in math and in life. But students don't get the opportunity to see math as a growth subject if they mainly work on short, closed questions accompanied by frequent tests that communicate to them that math is all about performance and there is no room for failure. When students inevitably struggle, most decide they are not a "math person." The last decade has seen a nation of children emerge from our schools terrified of failing in math and believing that only some students can be good at it-those who can effortlessly achieve on narrow tests.

Teachers see some of the damage caused by our nation's procedural and over-tested math classrooms in the ideas students hold about math. When asked what math is, students typically give descriptions that are very different from those given by experts in the field. Mathematicians define their subject as the study of patterns. They say it is an aesthetic, creative, and beautiful subject (for example, Keith Devlin, " Mathematics: The Science of Patterns"; and Steven Strogatz, " The Joy of x "). Knowledge of mathematical patterns has helped people navigate oceans, chart missions to space, develop technology that powers cellphones and social networks, and create new scientific and medical knowledge. But students will typically say that math is a subject of calculations, procedures, and rules. They believe that the best mathematical thinkers are those who calculate the fastest-that you have to be fast at math to be good at math. Yet mathematicians are often slow with math. I work with many mathematicians and they are simply not fast math thinkers. I don't say this to be disrespectful to mathematicians. They are slow because they think carefully and deeply about mathematics.

Laurent Schwartz won the Fields Medal in mathematics and was one of the greatest mathematicians of his time. But when he was in school he was one of the slowest in his class. In his autobiography, A Mathematician Grappling with His Century, he reflects on his school days and how he felt "stupid" because his school valued fast thinking:

I was always deeply uncertain about my own intellectual capacity; I thought I was unintelligent. And it is true that I was, and still am, rather slow. I need time to seize things because I always need to understand them fully. Towards the end of the eleventh grade, I secretly thought of myself as stupid. I worried about this for a long time.

I'm still just as slow.... At the end of the eleventh grade, I took the measure of the situation, and came to the conclusion that rapidity doesn't have a precise relation to intelligence. What is important 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.

Yet, more than any other subject, mathematics continues to be presented as a speed race: Teachers take answers from the first student to shoot up their hand in class, parents and teachers give timed math tests and drill with flash cards, and math apps race against the clock. It is no wonder that students who think slowly and deeply are put off by mathematics.
When students inevitably struggle, most decide they are not a "math person."
The fact that a narrow and impoverished version of mathematics is taught in many school classrooms cannot be blamed on teachers. Teachers are usually given long lists of content to teach, with hundreds of topics and no time to go into depth on any ideas. When teachers are given these lists, they see a subject that has been stripped down to its bare parts-like a dismantled bike-a collection of nuts and bolts that students are meant to shine and polish all year. Such lists not only take away the connections that weave all through mathematics, but present math as though the connections do not even exist.

I don't want students polishing disconnected bike parts all day. I want them to get onto the bikes and ride freely, experiencing the pleasure of math, the joy of making connections, the euphoria of real mathematical thinking.

When teachers open up mathematics and teach broad, visual, creative math, then they teach math as a learning subject, instead of a performance subject. It is very hard for students to develop a growth mindset if they are only ever answering short questions with right and wrong answers. Such questions themselves transmit fixed messages about math: that you can do it or you cannot. When educators teach open math and ask questions that have many solutions or pathways through them, and give students the opportunity to discuss different mathematical ideas, then students see that learning is possible. To put it simply, math questions should have space inside them for learning, for students to discuss and think about ideas; questions should not simply ask for answers that often require calculations or procedures with no encouragement for broader, engaging thought.

Teachers can see the difference between fixed-math and growth-math questions with elementary- and secondary-school mathematics content. For example, they could ask elementary students to calculate one divided by two-thirds-a fixed question. Or they could encourage them to think creatively and ask them to visually represent one divided by two-thirds and then convince each other why their solution works. Teachers could ask students to find the area of a 12 x 2 rectangle. Or they could pose a more growth-oriented question, asking them to find and draw as many rectangles as you can with an area of 24, which encourages students to think about the relationships between length and width, and represent them visually, instead of simply performing a calculation. At the secondary level, teachers could ask students to prove that the sum of the first n-positive integer cube numbers is the square of the sum of the first n-positive integer numbers. Or they would ask students to make sense of the visual below.

In the first, fixed version of each of these examples, students perform a calculation or move around algebraic symbols. In the second, they are using their own ideas, thinking deeply about math. One version is about performance, the other is about learning.

When educators teach real mathematics-a growth subject of depth and connections-the opportunities for learning increase and classrooms become filled with happy, excited, and engaged math students. Although news sites are filled with opposition to the Common Core, the new curriculum is at least a step in the right direction, as it asks students to engage in in the most mathematical of acts-reasoning. Mathematicians prove ideas by reasoning and justifying their thinking. Those who oppose the Common Core often do so because they want to keep the traditional mathematics approach in classrooms, even though this has turned off millions of students.

Changing classrooms to teach growth-mindset mathematics has a transformative effect on students. Society urgently needs to free our young people from the crippling ideas that they cannot fail, that they cannot mess up, that only some students can be good at math, and that success should be easy and fast, and not involve effort. School teachers and leaders need to introduce students to creative, beautiful mathematics that allows them to ask questions that have not been asked, and to think of ideas that go against traditional and imaginary boundaries.

When instructors encourage open, growth mathematics and the learning messages that support it, they develop our own intellectual freedom, as teachers and parents, and inspire that freedom in others. Now is the time to invite young people onto growth mindset pathways, encouraging them to be the people they should be, free from artificial rules and inspired by the knowledge that they have unlimited mathematics potential. For when school systems open mathematics, and give students the chance to ask their own questions and bring their own natural creativity and curiosity to the foreground as they learn, then they change them as people and the ways they interact with the world. When teachers set students free, beautiful mathematics follows.
This post appears courtesy of The Hechinger Report. It has been excerpted from Jo Boaler's new book, Mathematical Mindsets.

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Why Do Students Struggle So Much in Math Class?

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Mastering the subject has become less about learning and more about performance.

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Give students tasks that are higher level tasks to allow students to reflect on their learning and to allow students to think about how their knowledge can be applied to said task

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A place were students can confidently communicate with each other as there are responses can be give anonymously.

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I sometimes use "foldables" which are little booklets the kids construct with colored paper to take notes on a certain topic. The kids seem to respond differently to taking notes in their self-made booklets rather than in a normal notebook.