Monday, April 1, 2013

Mathematical Habits of Mind

Karen KingNational Science Foundation 

When I look back on my own mathematical education, I have many people to thank for helping me develop productive mathematical habits of mind. I remember walking to the car with my dad on a bitter cold day on the way home from kindergarten, and I just had to understand how you could do subtraction with regrouping. Instead of brushing off my pesky questioning (and I was pesky), he explained it to me, writing in the frost on the car window to illustrate the ideas. Some years later, Linda Agreen, my Advanced Placement calculus teacher, made sure that I understood why the fundamental theorem of calculus was fundamental, even though that was not going to be on the AP test. These habits of seeking real understanding were solidified in the mathematics department at Spelman College, under Etta Z. Falconer and her colleagues.

Building on the foundation laid by my father and my other mathematics teachers, I learned the mathematical habit of doggedly pursuing a complete understanding of ideas. I also learned how to recognize when my understanding was not complete and the reasoning skills to address the situation.

Unfortunately, too many students of mathematics, whether in college algebra or abstract algebra, do not possess these productive mathematical habits of mind. Instead, they have picked up some bad habits along the way: a tendency to look for the quick answer, a lack of persistence when the answer is not obvious, memorization over understanding.

Why do I keep referring to reasoning skills as “mathematical habits of mind”? Because I believe that if we start thinking about these unproductive practices as habits of mind, it opens up a different set of strategies for addressing the problem. When Al Cuoco, Paul Goldenberg, and June Mark introduced the concept of mathematical habits of mind (The Journal of Mathematical Behavior 15, no. 4 [1996]), it was a powerful concept for rethinking K-12 students’ learning of mathematics.

Habits are behaviors we engage in unconsciously, but they are the result of a long evolution of choices we make at a young age. Habits of mind evolve from the choices that we make about how to think about ideas. Thus, my dad’s early intervention was important. At 5 years old, I was still making choices about how to learn. So were my teachers—in elementary school, high school, and beyond.

But too few students develop the habits of mind needed for more advanced mathematical learning. Presented with a problem with no obvious example to follow, a poorly trained student might start writing things down or try some calculations with no real strategy in mind. Faced with the task of learning to write proofs, a person without sound mathematical habits usually attempts to memorize various arguments instead of re-creating them from their internal logic. These habits may have served them well previously, but no longer.

Habits reflect what a person is likely to do in a given situation, especially a stressful one such as taking a test, and habits are notoriously hard to break. Smokers know that continuing to smoke has a high likelihood of leading to cancer and other diseases, but that knowledge alone is rarely sufficient for those who are trying to quit.

With this in mind, we need to ask whether the way mathematics is currently taught reinforces bad habits of mind. Is it too easy to get by for too long using bad mathematical habits? And where did these bad habits come from in the first place? The likely answer is that there are some entrenched teaching habits in need of attention.

Thinking in terms of habitual behaviors conjures up powerful analogies. How might we change our approach to learning—and teaching—math if we labeled as “unproductive habits of mind” those methods that serve us poorly? Just like the person who finally replaces smoking with a healthier habit—or better yet, who never starts in the first place—we will all be better served with healthier mathematical habits of mind.

Karen King is the former director of research for the National Council of Teachers of Mathematics. She has been a member of the mathematics education faculty at New York University, Michigan State University, and San Diego State University. 

This article was published in the April 2013 issue of Math Horizons.


  1. Dear Karen, in the following paragraph I see a logical misstep:

    "Why do I keep referring to reasoning skills as “mathematical habits of mind”? Because I believe that if we start thinking about these unproductive practices as habits of mind, it opens up a different set of strategies for addressing the problem."

    The first sentence poses a question which the second sentence is supposed to answer, but it does not. It would, had you omitted the word "mathematical".

    The habits of mind you describe are the best possible tools that students could carry out of school, but their acquisition, though, may not necessarily happen in math classes. See, for example,

    A. Bogomolny

    1. Thank you for your comment. I think the focus that I was trying to take is on the word "habits" as opposed to "practices" and the power that the "habits" metaphor brings. So, instead of just calling them reasoning skills, framing these reasoning skills as mathematical habits of mind, "start thinking about these ... as habits of mind" affords us a different set of instructional strategies that can help support students replacing "bad"/unproductive habits of mind for mathematical reasoning with "good"/productive habits of mind for mathematical reasoning (some might call these right/correct).

      I think instead of omitting "mathematical" in the first sentence, it should be added in the second sentence (and was probably removed as I was way over the word limit on my early drafts).

  2. It should be clear from the link I provided in my comment that the habits of mind you describe are just habits of good thinking that can be taught without recourse to any branch of mathematics. The word "mathematical" should be removed from the question, instead of being inserted in the answer. To allow for a slight exaggeration, one can have great thinking skills without even knowing the multiplication table. This should not be confused with the fact that some math knowledge might be handy in various circumstances that require good thinking skills.

    I am sorry your setup does not allow (or I fail to find out how) to leave a signed comment for those who are not members of the selected networks. The least of all I would want is to leave an anonymous comment.

    A. Bogomolny

    1. While I appreciate the notion of good reasoning, I do think mathematical reasoning is different than reasoning in other subject, so I'll have to respectfully disagree. One of the things that I think is interesting is the way in which mathematics standards (such as the NCTM standards or the Common Core State Standards in Mathematics) separate "practices" or "processes" from "content" when for me they are so interwoven, thus habitual. Part of "understanding" congruence or functions is reasoning about them appropriately and using precise language in discussing them (to choose two from those standards of mathematical practice). So, I am not talking about good thinking skills in general, but in mathematics. There are many students who have great thinking skills that they never apply when doing mathematics because they are not part of their mathematical habits of mind but their "not mathematical" habits of mind.