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Later this year, high school teachers from the region will visit our university for a day and be given special lectures on different topics.

I've been asked to give a talk with the title "How to think mathematically?". In addition to mathematical examples, they also want to hear about the newest Math Ed research regarding students who struggle to think mathematically. The talk is supposed to be light and inspiring rather than heavy and theoretical. Other than the above, I haven't been given much to go on.

Could you give me any ideas I could use? Do you have inspiring examples of mathematical thinking and/or references to Math Ed research about the topic?

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I believe the classic reference from the mathematics education literature is:

Cuoco, A., Goldenberg, E. P., & Mark, J. (1996). Habits of mind: An organizing principle for mathematics curricula. The Journal of Mathematical Behavior, 15(4), 375-402. Link (no pay-wall).

The authors are all out of EDC (Cuoco and Goldenberg on the linked page; Mark on the next one).

Before listing the habits of mind, they write:

enter image description here

See the paper for more information (including examples) but the habits of mind, in order, are:

  • Students Should Be Pattern Sniffers

  • Students Should Be Experimenters

  • Students Should Be Describers

  • Students Should Be Tinkerers

  • Students Should Be Inventors

  • Students Should Be Visualizers

  • Students Should Be Conjecturers

  • Students Should Be Guessers

The next section is entitled, Mathematical Approaches to Things.

Again, I suggest reading the paper for examples, but here are the headers:

  • Mathematicians Talk Big and Think Small

  • Mathematicians Talk Small and Think Big

  • Mathematicians Use Functions

  • Mathematicians Use Multiple Points of View

  • Mathematicians Mix Deduction and Experiment

  • Mathematicians Push the Language

  • Mathematicians Use Intellectual Chants

Next, the article provides approaches that are specific to geometric thinking, and then they do the same for algebraic thinking.

Finally, you can find plenty more by checking the papers that have cited this piece (google scholar).

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    $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. // I moved the comments above to chat, as indicated by the auto-generated part providing a link, where the discussion that while interesting seems to veer off-topic could be continued. // Meanwhile I removed the comments here; (They are preserved in chat.) $\endgroup$ – quid Oct 9 '15 at 22:37
  • $\begingroup$ A lot has changed since 1996. $\endgroup$ – Andrew Oct 13 '15 at 13:02
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I've found "The 5 Elements of Effective Thinking" to be a good start. It was originally written to address exactly what you ask about: to explain how mathematicians think. However (according to the author whom I heard speak once) the editors quickly recognized that the ideas and methods were good ways of thinking for more than just mathematicians and so the text was re-written with a broader slant. Knowing that, however, you can still read it to learn about how mathematicians think.

In a similar vain, I've heard good things about "How Not to Be Wrong: The Power of Mathematical Thinking" but haven't taken the chance to read it myself yet.

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  • $\begingroup$ This remind me of a related thread that is not yet linked: MESE 294 What are some great books for exploring mathematics? (not kids' books and not textbooks) (I put down the "How Not to Be Wrong" reference there; +1 for suggesting it here, too!) $\endgroup$ – Benjamin Dickman Oct 8 '15 at 19:56
  • $\begingroup$ The first is a very nice book, in that it really captures some general "mental states" one finds oneself in while doing mathematics. The book does a great job of conveying these states without talking about much mathematics at all. $\endgroup$ – Jon Bannon Oct 21 '15 at 19:38
  • $\begingroup$ (One year later:) I really enjoyed Ellenberg's book, but The 5 Elements of Effective Thinking was too much of a self-help book for my personal taste. $\endgroup$ – Dag Oskar Madsen Oct 3 '16 at 12:38
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This seems appropriate, but I haven't read it yet:

Byers, William. How mathematicians think: Using ambiguity, contradiction, and paradox to create mathematics. Princeton University Press, 2010.


          BookCover
          (Book cover image from Princeton Press.)


Donal O'Shea: "Ambitious, accessible and provocative...[In] How Mathematicians Think, William Byers argues that the core ingredients of mathematics are not numbers, structure, patterns or proofs, but ideas..."

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  • $\begingroup$ That book is very interesting, but I'm not sure it gets to everyday thinking as readily as it should. $\endgroup$ – Jon Bannon Oct 6 '15 at 17:00
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    $\begingroup$ @JonBannon: I guess I should read it to find out for myself! It may be a good book but not an answer to the OP's query. $\endgroup$ – Joseph O'Rourke Oct 6 '15 at 20:28
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I recently bought Thinking Mathematically by John Mason with Leone Burto and Kaye Stacey. I have the revised edition, but I believe there is a newer edition. I am an educator who loves math, but I am not a math educator. I've found this book invaluable. There are many simple concepts that I was never taught as a high school student despite being someone who did well in math and took calculus in high school. I wish I had a formal course that taught these courses when I was younger. I am finding it much easier to learn higher level math now as an adult because I started with bigger picture issues like how to think mathematically first.

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To some, "mathematics" is foreign "language." One way to teach mathematics is through an "immersion" program, whereby people express themselves "mathematically," 24/7, even when they are having dinner, or going about their "normal" activities.

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  • $\begingroup$ Can you provide concrete examples in this direction? The closest that comes to [my] mind is work on Math Trails (the link is to a pdf of a COMAP book on this topic; I highly recommend pdf 103/136 = p. 99 for the section on American Flags...) $\endgroup$ – Benjamin Dickman Oct 8 '15 at 19:59
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I like to think that a subject in its own right is "how to find an easy way", where the goal can be not just to solve but to explain. This leads to several possibilities for both, which is why mathematicians cover the same problems over and over again, looking for an easier way.

If you have loads of time, you could guide students through a couple of ways in the problem I present below. Take ten trees and plant them in five rows with four trees in each row. A really careful and interactive discussion of this would take twenty minutes. If you have only five minutes or less to devote (and are willing to have the problem spoiled), I would recommend you cover the following points in this order.

  • What happens when you try planting trees? Can you find a solution easily?
  • Try planting rows instead. What should a row look like?
  • Can you have two rows or more that do not share a tree?
  • Try drawing three rows, each of which share a tree.
  • See if you can extend this to four and then five rows.

The upshot is that taking the perspective of placing lines rather than points (rows rather than individual trees) leads to an easy and general solution (five lines of which no two are parallel and no three concurrent) to the problem, and that one part of mathematical thinking (in both solving and education) is to shift perspectives, always looking for "the easy way".

Gerhard "Sometimes Finds The Easy Way" Paseman, 2015.10.06

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