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math, representations, understanding math

Three kinds of mathematical thinkers

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This is a continuation of my post Syntactic and semantic thinkers, in which I mentioned Leone Burton’s book [1] but hadn’t read it yet.  Well, now it is due back at the library so I’d better post about it!

I recommend this book for anyone interested in knowing more about how mathematicians think about and learn math.  The book is based on in-depth interviews with seventy mathematicians.  (One in-depth interview is worth a thousand statistical studies.)   On page 53, she writes

At the outset of this study, I had two conjectures with respect to thinking style.  The first was that I would find the two different thinking styles,the visual and the analytic, well recorded in the literature… The second was that research mathematicians would move flexibly between the two.  Neither of these conjectures were confirmed.

What she discovered was three styles of mathematical thinking:

Style A: Visual (or thinking in pictures, often dynamic)

Style B: Analytic (or thinking symbolically, formalistically)

Style C: Conceptual (thinking in ideas, classifying)

Style B corresponds more or less with what was called “syntactic” in [3] (based on [2]).  Styles A and C are rather like the distinctions I made in [3] that I called “conceptual” and “visual”, although I really want Style A to communicate not only “visual” but “geometric”.

I recommend jumping through the book reading the quotes from the interviews.  You get a good picture of the three styles that way.

Visual vs. conceptual

I had thought about this distinction before and have had a hard time explaining what “conceptual” means, particularly since for me it has a visual component.  I mentioned this in [3].  I think about various structures and their relationship by imagining them as each in a different part of a visual field, with the connections as near as I can tell felt rather than seen.  I do not usually think in terms of the structures’ names (see [4]).  It is the position that helps me know what I am thinking about.

When it comes time to write up the work I am doing, I have to come up with names for things and find words to describe the relationships that I was feeling. (See remark (5) below).  Sometimes I have also written things down and come up with names, and if this happened very much I invariable get a clash of notation that didn’t bother me when I was thinking about the concepts because the notations referred to things in different places.

I would be curious if others do math this way.  Especially people better than I am.  (Clue to a reasonable research career:  Hang around people smarter than you.)

Remarks

1) I have written a lot about images and metaphors [5], [6].  They show up in the way I think about things sometimes.  For example, when I am chasing a diagram I am thinking of each successive arrow as doing something.  But I don’t have any sense that I depend a lot on metaphors.  What I depend on is my experience with thinking about the concept!

2) Some of the questions on Math Overflow are of the “how do I think about…” type (or “what is the motivation for…”).  Some of the answers have been Absolutely Entrancing.

3) Some of the respondents in [1] mentioned intuition, most of them saying that they thought of it as an important part of doing math.  I don’t think the book mentioned any correlation between these feelings and the Styles A, B, C, but then I didn’t read the book carefully.  I never read any book carefully. (My experience with Style B of the subtype Logic Rules diss intuition. But not analysts of the sort who estimate errors and so on.)

4) Concerning A, B, C:  I use Style C (conceptual) thinking mostly, but a good bit of Style (B) (analytic) as well.  I think geometrically when I do geometry problems, but my research has never tended in that direction.  Often the analytic part comes after most of the work has been done, when I have to turn the work into a genuine dry-bones proof.

5) As an example of how I have sometimes worked, I remember doing a paper about lifting group automorphisms (see [7]), in which I had a conceptual picture with a conceptual understanding of the calculations of doing one transformation after another which produced an exact sequence in cohomology.  When I wrote it up I thought it would be short.  But all the verifications made the paper much longer.  The paper was conceptually BigChunk BigChunk BigChunk BigChunk … but each BigChunk required a lot of Analytic work.  Even so, I missed a conceptual point (one of the groups involved was a stabilizer but I didn’t notice that.)

References

[1] Leone Burton, Mathematicians as Enquirers: Learning about Learning Mathematics.  Kluwer, 2004.

[2] Keith Weber, Keith Weber, How syntactic reasoners can develop understanding, evaluate conjectures, and generate counterexamples in advanced mathematics. Proof copy available from Science Direct.

[3] Post on this blog: Syntactic and semantic thinkers.

[4] Post: Thinking without words.

[5] Post: Proofs without dry bones.

[6] Abstractmath.org article on Images and Metaphors.

[7] Post: Automorphisms of group extensions updated.

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abstractmath.org, math, representations, understanding math

Syntactic and semantic thinkers

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A paper by Keith Weber

Reidar Mosvold’s math-ed blog recently provided a link to an article by Keith Weber (Reference [2]) about a very good university math student he referred to as a “syntactic reasoner”.  He interviewed the student in depth as the student worked on some proofs suitable to his level.  The student would “write the proofs out in quantifiers” and reason based on previous steps of the proof in a syntactic way rather than than depending on an intuitive understanding of the problem, as many of us do (the author calls us semantic reasoners).  The student didn’t think about specific examples —  he always tried to make them as abstract as possible while letting them remain examples (or counterexamples).

I recommend this paper if you are at all interested in math education at the university math major level — it is fascinating.  It made all sorts of connections for me with other ideas about how we think about math that I have thought about for years and which appear in the Understanding Math part of abstractmath.org.  It also raises lots of new (to me) questions.

Weber’s paper talks mostly about how the student comes up with a proof.  I suspect that the distinction between syntactic reasoners and semantic reasoners can be seen in other aspects of mathematical behavior, too, in trying to understand and explain math concepts.  Some thoughts:

Other behaviors of syntactic reasoners (maybe)

1) Many mathematicians (and good math students) explain math using conceptual and geometric images and metaphors, as described in Images and metaphors in abstractmath.org.   Some people I think of as syntactic reasoners seem to avoid such things. Some of them even deny thinking in images and metaphors, as I discussed in the post Thinking without words.   It used to be that even semantic reasoners were embarassed to used images and metaphors when lecturing (see the post How “math is logic” ruined math for a generation).

2) In my experience, syntactic reasoners like to use first order symbolic notation, for example eq0001MP

and will often translate a complicated sentence in ordinary mathematical English into this notation so they can understand it better.  (Weber describes the student he interviewed as doing this.)  Furthermore they seem to think that putting a formula such as the one above on the board says it all, so they don’t need to draw pictures, wave their hands [Note 1], and so on.  When you come up with a picture of a concept or theorem that you claim explains it their first impulse is to say it out in words that generally can be translated very easily into first order symbolism, and say that is what is going on.  It is a matter of what is primary.

The semantic reasoners of students and (I think) many mathematicians find the symbolic notation difficult to parse and would rather have it written out in English.  I am pretty good at reading such symbolic notation [Note 2] but I still prefer ordinary English.

3) I suspect the syntactic reasoners also prefer to read proofs step by step, as I described in my post Grasshoppers and linear proofs, rather than skipping around like a grasshopper.

And maybe not

Now it may very well be that syntactic thinkers do not all do all those things I mentioned in (1)-(3).  Perhaps the group is not cohesive in all those ways.  Probably really good mathematicians use both techniques, although Weyl didn’t think so (quoted in Weber’s paper).   I think of myself as an image and metaphor person but I do use syntax, and sometimes even find that a certain syntactic explanation feels like a genuinely useful insight, as in the example I discussed under conceptual in the Handbook.

Distinctions among semantic thinkers

Semantic thinkers differ among themselves.  One demarcation line is between those who use a lot of visual thinking and those who use conceptual thinking which is not necessarily visual.  I have known grad students who couldn’t understand how I could do group theory (that was in a Former Life, before category theory) because how could you “see” what was happening?  But the way I think about groups is certainly conceptual, not syntactic.  When I think of a group acting on a space I think of it as stirring the space around.  But the stirring is something I feel more than I see.  On the other hand, when I am thinking about the relationships between certain abstract objects, I “see” the different objects in different parts of an interior visual space.  For example, group is on the right, stirring the space-acted-upon on the left, or the group is in one place, a subgroup is in another place while simultaneously being inside the group, and the cosets are grouped (sorry) together in a third place, being (guess what) stirred around by the group acting by conjugation (Note [3]).

This distinction between conceptual and visual, perhaps I should say visual-conceptual and non-visual-conceptual, both opposed to linguistic or syntactic reasoning, may or may not be as fundamental as syntactic vs semantic.   But it feels fundamental to me.

Weber’s paper mentions an intriguing sounding book (Reference [1]) by Burton which describes a three-way distinction called conceptual, visual and symbolic, that sounds like it might be the distinction I am discussing here.  I have asked for it on ILL.

Notes

  1. Handwaving is now called kinesthetic communication.  Just to keep you au courant.
  2. I took Joe Shoenfield’s course in logic when his book  Mathematical Logic [3] was still purple.
  3. Clockwise for left action, counterclockwise for right action.  Not.

References

  1. Leone L. Burton, Mathematicians as Enquirers: Learning about Learning Mathematics.  Springer, 2004.
  2. Keith Weber, How syntactic reasoners can develop understanding, evaluate conjectures, and generate counterexamples in advanced mathematics. Proof copy available from Science Direct.
  3. Joseph Shoenfield, Mathematical logic, Addison-Wesley 1967, reprinted 2001 by the Association for Symbolic Logic.
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