# Mathematical Habits of Mind

We all have them, some good and some bad. We pick them up from friends, family, and even strangers. But we may not recall who we picked them up from or when they began.

Because we've practiced them over and over, these seemingly thoughtless repeated habits or behaviors, the pathways in our brain have become so broad, fast, and efficient in carrying them out that we do them automatically without even thinking. Yet these unconscious habits and behaviors add structure and order to our lives and help us to make sense of the world we live in.

Our classrooms are full of them. We teachers are pros when it comes to employing and modeling good habits and routines that enable us to manage and carry out the many tasks and demands of teaching. And when it comes to teaching mathematics, we model and teach our students how to carry out procedures and algorithms flawlessly. But why is it that these same students often struggle when confronted with a problem to which the immediate answer is unknown?

### Experienced vs. Inexperienced Problem Solvers

According to Levasseur and Cuoco (PDF, 80KB), it's the mathematical habits of mind, or modes of thought, that enable us to reason about the world from a quantitative and spatial perspective, and to reason about math content that empowers us to use our mathematical knowledge and skills to make sense of and solve problems. It's these habits that separate the "experienced" from the "inexperienced" problem solver.

Far too many students can be classified as inexperienced problem solvers who don't know what to do when they don't know what to do. These are the students who lack experiences in making sense of and solving problems, and in communicating and using precise, appropriate mathematics and mathematics language. They have never developed the overarching habits of a productive mathematical thinker.

When they're presented with a problem to which they don't know the immediate answer, their bad habits often rise to the occasion. They may begin with no real strategy in mind, they look for key words and terms that can sometimes be misleading, they want a quick answer, they resort to memorization over understanding, or they simply make no effort at all in finding the solution to the problem.

On the other hand, experienced problem solvers know what to do when they don't know what to do! They practice perseverance and automatically employ the mathematical habits of mind of a productive problem solver in their quest for a problem solution. They automatically engage in Polya’s four-step problem-solving process to find a solution to the problem.

No one has to remind experienced problem solvers to first make an attempt at understanding the problem and ask clarifying questions that help them interpret and understand the conditions of the problem. They instinctively try to make sense of the mathematical situation at hand and choose a strategy, tools, and/or models, often based upon previous learning experiences or problems they've solved -- experiences that they see as relevant and applicable in solving the problem.

If they try a strategy and it doesn't work, they try a different one – all part of perseverance in problem solving. And like any good problem solver, once they have a solution, they take one last look to determine if their solution is a reasonable solution to the problem.

### Practice and Experience

But these habits of mind can’t be learned and developed by simply talking about them or memorizing the definition. Students immersed in classroom experiences that let them engage in learning mathematics concepts through problem solving, making and using abstractions, and developing and applying mathematical theories have greater opportunities for developing mathematical habits of mind. Classrooms steeped in the Common Core Standards for Mathematical Practice (MP) provide maximum opportunities for students to learn and develop these habits as they:

Teachers who saturate learning experiences with the Standards for Mathematical Practice offer students a variety of palatable problems. Presenting relevant, interesting, and challenging problems that can be approached and solved with different strategies and tools gives students choices that best fit their learning preferences. This approach is more inviting to them and leads them to reason about, look for, and make use of structure. In turn, it increases their chances of understanding important math concepts and skills, which in turn increases their ability to explain their thinking and reasoning, and to critique the reasoning of others.

You play a vital role in the development of mathematical habits by conveying to your students a mindset that fosters productive struggle and emphasizes the possible value of wrong answers. You set the stage by embracing the importance and value of grappling with problems. The more you engage your students in learning and doing mathematics, the greater the likelihood of their developing the mathematical habits of mind of a productive mathematical thinker -- and becoming experienced problem solvers who *know* what to do when they don't know what to do.

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Kia,

I'm glad you like the problem solving web and I'd love to see your 4th grade recreation. In regards to your question, you may find these additional Edutopia resources helpful https://www.edutopia.org/blogs/tag/resilience-grit and https://www.edutopia.org/blog/growth-mindset-common-core-math-cindy-bryant. I've also found that varying the activities can be a key factor. Find something of interest to students and they are more likely to put forth more effort. Best, Cindy!

Just curious, what if you, yourself are not familiar or don't know how to convey that mindset or even know how to go about helping your students gain it?

I happen to enjoy math to some degree, so I do continue, or persevere until I figure an answer out. I am not at the point where I try to find multiple ways, but I try to not give up or give in.

How do i help my students with that?

Dear FernM16,

Thank you for your feedback and questions. There are several things teachers can do to support the development of a productive struggle and perseverance in problem solving mindset. It begins by offering a variety of tasks/problems - different problems grab different students. Offering different types of problems provides opportunities for them to use different problem solving strategies. Creative Problem Solving in Mathematics by Dr. George Lenchner is an excellent resource that provides examples of problems and using the various problem solving strategies to solve problems.

It's also important to consider the responses and encouragement we provide to students. Responding in a manner that keeps the focus on thinking and reasoning, rather than only the right answer is key. And praising students for the strategies they use, the specific work they do, or their effort is also critical in creating that supportive climate. Letting students know that it's acceptable to make mistakes and providing them with appropriate feedback helps them make new brain connections and learn from their mistakes. My Favorite No: Learning from Mistakes https://www.teachingchannel.org/videos/class-warm-up-routine is a beautiful example of what one teacher does to provide students opportunities to learn from their mistakes in a non-threatening manner.

You may also find these two additional Edutopia resources helpful.

http://www.edutopia.org/math-underachieving-mathnext-rutgers-newark

http://www.edutopia.org/blog/productive-group-work-andrew-miller

Please feel free to contact me at cindy (at) learnbop (dot) com if I can be of any further assistance.

Good reminder of a well-established protocol for problem solving. Also nice graphic on the various strategies for problem solving. The SMPs clearly play a huge role in developing proficient problem solvers. Your article linked to Polya's Four Principles http://www.math.wichita.edu/history/men/polya.html. The fourth is not bolded. Could www.witchita.edu reformat their HTML?

Victoria, thank you for your positive feedback. I'm fine with the reformatting to bold the fourth principle if it's acceptable with the Edutopia team.

Victoria, since that page is on Wichita State University's website, they're the ones who can change it.

The page looks to be part of a larger project, which you can find here: http://www.math.wichita.edu/history/index.html. You might try contacting the professor in charge.

I really like the problem-solving web. This year our school is focusing much more on the process of solving problems. When I click on Polya's 4-step problem-solving process, it does not open. Could you send that to me in a different format? Thanks!

Hi Susan,

Thank you for your positive feedback. Please email me at cindy@learnbop.com and I'll be happy to send you the problem-solving web in a different format.

Best,

Cindy

This is very interesting. I might add a more specific mathematical problem solving process that I have tried with my own kids (grown now). As part of the first step (understanding the problem), you must name variables. For each variable indicate whether its value is known or not. Also, indicate whether that variable is the answer you are looking for. Then, as the problem statement is understood more deeply, record relationships within the variables using math. This method works with algebraic problems (if this were geometry you would need to make drawings and can use the method after that). In the end, you have a set of equations (and possibly inequations) you need to solve. Depending on the topic, you may need to add something not explicit in the problem but that you must know. For example, if we are solving something about a triangle and the angles in question were named alfa, beta and gamma, then we might need to add the equation alfa+beta+gamma=180o.

Pablo, Thank you for taking the time to share your insightful remarks regarding the naming of the variables and determining whether the value is known or not process that you used with your own children. Always good to know more about what has been successful in helping students solve problems!

Best,

Cindy

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