Mathematical Habits of Mind

look for patterns:    to look for patterns amongst a set of numbers or figures

tinker:    to play around with numbers, figures, or other mathematical expressions in order to learn something more about them or the situation; experiment

describe:    to describe clearly a problem, a process, a series of steps to a solution; modulate the language (its complexity or formalness) depending on the audience

visualize:    to draw, or represent in some fashion, a diagram in order to help understand a problem; to interpret or vary a given diagram

represent symbolically:    to use algebra to solve problems efficiently and to have more confidence in one’s answer, and also so as to communicate solutions more persuasively, to acquire deeper understanding of problems, and to investigate the possibility of multiple solutions

prove:    to desire that a statement be proved to you or by you; to engage in dialogue aimed at clarifying an argument; to establish a deductive proof; to use indirect reasoning or a counter-example as a way of constructing an argument

check for plausibility:    To routinely check the reasonableness of any statement in a problem or its proposed solution, regardless of whether it seems true or false on initial impression; to be particularly skeptical of results that seem contradictory or implausible, whether the source be peer, teacher, evening news, book, newspaper, internet or some other; and to look at special and limiting cases to see if a formula or an argument makes sense in some easily examined specific situations.

take things apart:    to break a large or complex problem into smaller chunks or cases, achieve some understanding of these parts or cases, and rebuild the original problem; to focus on one part of a problem (or definition or concept) in order to understand the larger problem

conjecture:    to generalize from specific examples; to extend or combine ideas in order to form new ones

change or simplify the problem:    change some variables or unknowns to numbers; change the value of a constant to make the problem easier; change one of the conditions of the problem; reduce or increase the number of conditions; specialize the problem; make the problem more general

work backwards:    to reverse a process as a way of trying to understand it or as a way of learning something new; to work a problem backwards as a way of solving

re-examine the problem:    to look at a problem slowly and carefully, closely examining it and thinking about the meaning and implications of each term, phrase, number and piece of information given before trying to answer the question posed

change representations:    to look at a problem from a different perspective by representing it using mathematical concepts that are not directly suggested by the problem; to invent an equivalent problem, about a seemingly different situation, to which the present problem can be reduced; to use a different field (mathematics or other) from the present problem’s field in order to learn more about its structure

create:    to invent mathematics both for utilitarian purposes (such as in constructing an algorithm) and for fun (such as in a mathematical game); to posit a series of premises (axioms) and see what can be logically derived from them

One Comment

  1. @hodteacher
    Posted August 12, 2013 at 5:31 am | Permalink | Reply

    I read your blog with interest, having recently written about ‘mathematical habits’. Your practical approach in embedding the theory in every day practice is something I am striving towards in my own practice. I would appreciate your comments on my future blog posts as I attempt to do something similar in the UK.

6 Trackbacks

  1. […] the problem sets as they develop, but I think they’re going to be heavily related to the habits of mind work that the Park School of Baltimore engages in. I honestly think that when their curriculum is […]

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  4. By Mathematical Habits of Mind | lttmaths on May 18, 2013 at 6:22 am

    […] Independent school in Baltimore who developed their own curriculum with a focus on a set of Mathematical Habits of Mind. Inspiring stuff – the faculty publish their teaching reflections on the school website and […]

  5. […] of Mind posters were made by stealing some of Bryan Meyer’s words and some phrases from the Park School of Mathematics. The posters themselves can be found […]

  6. By Finding the answers you need | O'Shea Physics 180 on October 12, 2014 at 9:13 am

    […] This problem (first, find the size of the third square given the first two so that the corners are collinear (as shown)—then find the size of the second square in terms of the first two) was solved in several different ways by different groups in the class. Moving from numbers to variables was a tricky step for us. I’ve realized that I have a class filled with great and dedicated tinkerers—which is actually really cool and fun and impressive. The only problem right now is that they usually only want to reach for that tool—so now we’re starting to develop some other habits and tools so that we can solve more problems more efficiently. (Thanks, Park School Math!) […]

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