MY MATH HABITS OF MIND

A student guide

A four-step problem-solving approach

George Polya was a mathematician who suggested that there are four steps to follow when solving a mathematical problem:

1. Understand the problem

2. Make a plan

3. Carry out the plan

4. Look back at your work

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But what should you do if those four steps fail you?

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“If you can’t solve a problem,” Polya wrote, “then there is an easier problem you can solve: find it.”

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You can solve anything.

As you work to understand a problem, devise a plan, carry out your plan, and look back at your work, you can use the Math Habits of Mind to think like a mathematician. They’ll also help you to clearly explain your thinking and problem-solving process to others.

Developing the skills to think like a mathematician will give you new ways to think about work challenges, social issues, and personal and professional decisions. The Math Habits of Mind are really about the skills of exploration and investigation, of problem-solving using symbols (numbers in this case). If you really develop these skills, they will shift how you think about yourself as a learner and how you tackle challenging tasks. Give it a try.

There are six Math Habits.

In this Student Guide to Math Habits of Mind, you’ll have the chance to learn about and practice each math habit. There are six:

CONJECTURE

TINKER

VISUALIZE

FIND PATTERNS

GATHER AND ORGANIZE INFORMATION

EXPERIMENT

GATHER AND ORGANIZE INFORMATION

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GETTING STARTED:

WHAT IS GATHERING & ORGANIZING INFORMATION?

WHAT DOES IT MEAN?

When you gather information you look at the problem and draw out the information you need, noting both what you know and what you need to find out. You also consider what information you have elsewhere—such as in your head or in your text or other resources—and how you can access it.

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Organizing information is one of the first steps in making sense of what you’ve gathered. You think about how you can keep track of what you know. You separate the relevant and irrelevant, group related things, and begin to infer other relationships in the information you have.

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GETTING STARTED:

GATHER:

• Record what you notice and what you wonder about the problem.
• Rephrase the problem verbally or in writing, swapping out non-math terms for math terms, or vice versa.
• Make sense of the problem by removing the numbers and thinking about what the problem is really asking you to do. Predict what form the solution will take.
• Look at your notes, textbook, and other resources to identify formulas and tools that you can use to solve the problem.

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ORGANIZE:

• Use a T-chart or organized list to sort the important information and potential distractors in the problem. One option: “I need this information because …” and “I don’t need this information because …”
• Use a quantity-value table or data table to organize the information you have, the information you need to find, and your solutions.
• In both non-math and math terms, describe the steps for the problem.

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HOW DO I GATHER & ORGANIZE INFORMATION?

WHAT TO DO:

As you look at a math problem, begin by identifying all of the information you have and look for a way to keep track of it.

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Mathematicians use some common tools for organizing information. AND, they also create their own tools and strategies. You may already have tools and strategies that work for you. Try them, AND try these.

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MY TURN:

DIRECTIONS: Choose a challenging problem to experiment with. Identify a “gathering” strategy and an “organizing” strategy that you think might help you understand the problem more clearly. You may need to try different strategies until you have gathered and organized the most useful information. Remember: these strategies may only be a starting point. From here you’ll probably have to move into using other Habits of Mind to actually solve the problem… Be Persistent!

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HOW DO I GATHER & ORGANIZE INFORMATION?

VISUALIZE

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GETTING STARTED:

WHAT IS VISUALIZING?

WHAT DOES IT MEAN?

Visualizing creates a sensory connection, allowing you to step into a problem, model it, and plan your next steps. Creating a picture in your mind, or on a piece of paper, or a screen, is just one way to visualize. You can also use manipulatives, tokens, coins, or other tangibles (things you can touch) to make a model to visualize a problem. By visualizing, you identify the key components of the problem and the relationships among them. You can “see” and describe the structure of the problem, which can help you to identify similarities and differences, make connections, and communicate your understanding of the problem and its solution.

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GETTING STARTED:

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• Draw or copy the objects, shapes, tables, or graphs presented in the problem, and refer to them as you solve the problem.
• Use graphic organizers—such as Venn diagrams, factor trees, T-charts, two-way tables, flowcharts—to help you visualize relationships.
• Make connections between problems and similarly structured situations to create visual analogies—such as chains, input/output machines, road maps, trees—to visualize processes.
• Imagine the problem unfolding as a movie or create a storyboard to visualize change.
• Use manipulatives, tokens, or other tangible objects to create a model of the problem.
• “See” a proof or mathematical argument by visualizing a chain made up of the components that will make it logically sound.

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WHAT TO DO:

As you look at a math problem, try to “see” what it describes in your mind, on paper or screen, or using something tangible.

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You may already have visualizing tools and strategies that work for you. Try them, try these, and don’t be afraid to create your own!

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HOW DO I VISUALIZE?

MY TURN:

DIRECTIONS: Choose a challenging problem to experiment with. Experiment with a visualizing strategy. Keep trying different ways to visualize the problem until you discover one that helps you to develop a new or better understanding. Remember that visualizing can help throughout the problem-solving process, from understanding the problem, to planning, to checking and communicating your solution. You may need to move to using other Habits of Mind, too… Remind yourself that challenging problems usually require us to use multiple strategies! Don’t give up if visualizing only gets you part of the way there!

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HOW DO I VISUALIZE?

FIND PATTERNS

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GETTING STARTED:

WHAT IS FINDING PATTERNS?

WHAT DOES IT MEAN?

Finding patterns means looking for regularity, symmetry, and repetition in problems and their solutions. Sometimes it is stepping back to see how the smaller pieces make up a larger whole. Sometimes it is zooming in to see how the larger whole is made up of smaller pieces. At other times, you find patterns by comparing two things that seem like they shouldn’t have anything in common, but then you make a connection that helps you see that they do. Finding patterns helps you to make sense of complexity and ambiguity in problems and allows you to identify shortcuts that make problem solving simpler.

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GETTING STARTED:

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• Write data, numbers, and expressions on sticky notes and sort them different ways.
• As you look at a problem, make note of what changes and what stays the same.
• Use proportional reasoning by thinking about how something smaller can represent something larger, or vice versa.
• Look for and make use of structure by seeing wholes made up of parts and parts as making up a whole.
• Generalize by asking “Does this apply to other examples, situations, or problems?”
• Describe the problem, data, expressions, or numbers without using numbers or other mathematical language. Then, if you can, describe the same thing using numbers and mathematical language.

WHAT TO DO:

As you look at a math problem, a set of numbers, or other information, look for repetition, symmetry, and regularity.

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The tools on the right are just some ideas to get you started.

Create your own tools and strategies and use the tools and strategies you’ve used in the past that work for you. Try them, AND try these.

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HOW DO I FIND PATTERNS?

MY TURN:

DIRECTIONS: Choose a challenging problem to experiment with. Experiment with a pattern-finding strategy. Use different ways of finding patterns until you discover one that helps you to see the repetition, symmetry, or regularity. Patterns can exist in problems and their solutions, so be sure to look for patterns throughout your problem-solving process. You may need to move to using other Habits of Mind, too… Remind yourself that challenging problems usually require us to use multiple strategies! Sometimes a pattern isn’t obvious, so it’s okay to step back, take a break, and try again with a fresh mind!

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HOW DO I FIND PATTERNS?

TINKER

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GETTING STARTED:

WHAT IS TINKERING?

WHAT DOES IT MEAN?

Tinkering begins with a question: “How does this work?” To tinker is to take something apart, to look at the pieces, and to put it back together. But in doing so, maybe you move something around, leave something out, put something else in…and when you do that, sometimes you end up with something new that works better than what you had before.

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And when it doesn’t work, that’s okay too, because tinkering is also looking for productive mistakes—those “wrong” answers that help you understand the problem more deeply and move you closer to discovering the “right” answer.

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GETTING STARTED:

• Take apart the problem or solution by writing each on a separate sticky note. Rearrange, recombine, sort…and think about the role each plays in the problem or solution.
• Write down the first answer that comes to your mind, then check it, looking for where you were correct and where you missed the mark.
• When you find a wrong answer ask, “How do I know this is wrong?”
• As you read a problem or evaluate a solution, ask “What are the parts of this problem or solution? How can I take them apart, move them around, and recombine them to better understand what I see?”
• Use a strategy or tool that you think won’t work for the problem you’re trying to solve. If it works or if it doesn’t work, ask yourself “Why?”
• Ask, “Is there a similar, simpler problem that I can solve to figure out how to solve this problem?”
• Play with the problem by changing the numbers, figures, or other mathematical expressions in order to gain more understanding or to generate novel solutions.

WHAT TO DO:

As you look at a math problem or solution, think about the building blocks that make it up. Move those blocks around and see what happens.

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Try some of the tools listed on the right, which are some common tools for tinkering mathematicians use. But they also create their own tools and strategies—and so can you!

WHAT IS TINKERING?

MY TURN:

DIRECTIONS: Choose a challenging problem to experiment with a tinkering strategy. Try different ways of breaking down the problem and recombining the pieces. Tinkering can lead to many false starts, so remember that mistakes can be productive too! You may need to use other Habits of Mind, too as you tinker…Remind yourself that wrong answers can provide insight into problems if you take the time to look at why something you tried didn’t work! Keep at it and have fun!

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HOW DO I TINKER?

CONJECTURE

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GETTING STARTED:

WHAT IS CONJECTURING?

WHAT DOES IT MEAN?

Conjecturing is taking the information you have—whether complete or incomplete—and coming to a conclusion about what it means. Similar to making an inference when you read, conjecturing requires you to combine the information in the problem or solution and your understanding of it with your previous experience. Mixing them together allows you to claim, through your conjecture, “I believe this to be true, but I have no proof yet.”

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GETTING STARTED:

• As you read a problem, extend your understanding by making a T-chart with two columns: “What the problem says” and “What this means”.
• Generate claims by using sentence stems such as “Because it is true that … I believe it is also true that …”
• Organize evidence for your conjectures into two columns: information in the problem or solution and things you know from previous experience.
• See if your conjecture passes the sniff test by verbally sharing your reasoning with a critical partner.
• Think about how you would convince yourself, a friend, and a skeptic to believe your conjecture is true. Add more detail as you move through the list.
• Listen to the arguments of others, noting when they make sense and noting when they don’t. Look for the same in your own conjectures.
• Ask yourself, “If this is true, does it mean it’s always true?” Try to find counterexamples, then use those to revise your conjecture.

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WHAT TO DO:

As you look at a math problem, think about what the solution might look like, what it might be. Consider the evidence you have for your conjecture, and be honest about why you might not be correct.

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Think about other times, outside of math, when you’ve had to use evidence to make and support a claim. Those tools can be useful in math too! Try them out and try the ones listed here.

WHAT IS CONJECTURING?

MY TURN:

DIRECTIONS: Choose a challenging problem to experiment with. Experiment with a conjecturing strategy to help you solve the problem. Consider what you know and how you can extend that to new situations or to develop a deeper understanding of the problem. You may need to use other Habits of Mind as you conjecture, so keep your mind open for those possibilities. Remember that when you conjecture, being wrong can be helpful too, since it helps you rethink what you believed you knew. Don’t give up!

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HOW DO I CONJECTURE?

EXPERIMENT

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GETTING STARTED:

WHAT IS EXPERIMENTING?

WHAT DOES IT MEAN?

Experimenting is about purposefully trying different things and recording the results you get. “What if we do this? What will happen then?” the experimenter asks. “Let’s try this and see what we get…and then compare it to what we got last time,” the experimenter says. Rather than throwing a bunch of different tools at a problem, the experimenter develops a plan, executes that plan, and then considers why or why it didn’t work.

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When you encounter new problems, experimenting is also using the strategies, tools, and techniques that were successful in the past to see if they’ll work this time too.

GETTING STARTED:

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• Ask “What if we do this? What will happen then?” Then, test your idea.
• Develop a plan to solve the problem. After you’ve executed the plan, look back and ask “What can I learn from what went right and what didn’t go as I planned?”
• Modify the task to explore other patterns, other outcomes, and other strategies; then, compare the outcomes.
• Play around with numbers, figures, or other mathematical expressions in order to learn something more about them or the situation.
• Think about what worked in the past and adjust your problem-solving strategy based on prior efforts.
• Be skeptical by asking “Just because this worked this time, does that mean it will work every time?” Look for ways to “break” your solution.
• Run “thought experiments” by thinking through a problem, process, or system before you write anything down. Use other tools to then verify or refute your thinking.

WHAT TO DO:

Bring all of your tools together to try something new—a new approach, a new solution, a new way of looking at a problem—and record your results.

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You can try some of the tools listed here, as well as tools you’ve acquired outside of math class. Don’t be afraid to experiment with your experimentation!

WHAT IS EXPERIMENTING?

MY TURN:

DIRECTIONS: It may have occurred to you that you’ve already been experimenting as you worked with the other strategies. But don’t forget, experimenting with a problem isn’t just trying different strategies—experimenting has its own unique set of tools, too. Remember that solving problems requires the use of multiple strategies. Be an experimenter! Don’t give up when things don’t work the first time…keep experimenting and you’ll find what works!

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HOW DO I EXPERIMENT?

GATHER AND ORGANIZE INFORMATION

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VISUALIZE

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FIND PATTERNS

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TINKER

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CONJECTURE

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EXPERIMENT

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