This is a rather broad (and perhaps too philosophical) question about undergraduate and graduate mathematics education.

Gowers, in his article "the two cultures of mathematics", observes differences between the problem solving and theory building aesthetics in mathematics. To slightly perturb an analogy of Alexander Grothendieck, one might view the problem solving culture as focusing more on a "Yang" or "attacking" mathematical aesthetic, and the theory building culture as focusing on a more "Yin" or "yielding" aesthetic. Of course, I have taken liberties in my perturbation of this analogy, and the distinction is probably best understood by reading Gowers's article linked to above.

Of course, the whole of mathematics involves an interplay of the "Yin" and "Yang" of the two cultures, and every mathematician is at any moment looking at things more as a problem solver than as a theory builder or the other way around. Subjects emphasizing the different styles mostly take care of sorting mathematicians into appropriate communities for their tastes, but I can't help but wonder if there are better ways to recognize the diversity of mathematical tastes in my classroom.

Some students simply love to read and learn about mathematics, and gain a very strong base of ideas to think about, and there is a danger of these students becoming depressed by seeing a problem solver work incisively on a problem.

Some students are excellent at problem solving, but become depressed when looking at classmates who have a broader knowledge base. Many such clever students are turned off by routine mathematics classwork aimed at building up their knowledge base.

The above perceptions extend rather far into one's career. Once, a problem-solver friend of mine was shocked when he overheard a discussion between a few mathematicians (one of whom had a Fields Medal) and noticed that the participants didn't have a solid grasp of a certain definition they were talking about. (I think it might be common among some theory-building types to try to tease new perceptions from an ill-defined understanding of a definition.)

On the other hand, a former graduate student of an IMO Gold Medalist who is now primarily a theory-builder told me stories about the disdain his advisor had for results that were "just combinatorics".

What concretely can we do to better embrace the diversity of mathematical tastes in our departments? What can we concretely do as mathematics educators to encourage communication between the two cultures?

(EDIT: Here is an example of the sort of thing I'm hoping to see, ideas for concrete pedagogical practices that encourage either "culture" or perhaps how to concretely achieve a balance.)

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    $\begingroup$ You might also be interested in Freeman Dyson's "Birds and Frogs" talk: ams.org/notices/200902/rtx090200212p.pdf $\endgroup$ May 8, 2014 at 8:16
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    $\begingroup$ @Benjamin: The question could have been "A healthy classroom for birds and frogs" $\endgroup$
    – Jon Bannon
    May 8, 2014 at 11:14
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    $\begingroup$ for one, actually test on the proofs of theorems on occasion. $\endgroup$ May 8, 2014 at 16:03
  • $\begingroup$ I think the question is a it too broad to respond to thoughtfully. Picking a main area and responding to it, I'd say that foxyness is overrated and overemphasized in pop ed research. The more we learn from careful psychrometric studies (not ed school stuff) as well as even in some cases brain imaging, the more we we see that problem solving tends to be domain specific. Also, for practical reasons of efficiency and audience, I think you're better off emphasizing more imitation and practice and only very small hurdles with requiring innovations. $\endgroup$ Apr 1, 2023 at 0:26

2 Answers 2


Perhaps a bit of a rant...:

As long as we condone "mathematics" being treated as a competitive sport, in whatever mode, it becomes a vehicle for ego and machismo.

"Math contests" stir passions, but by appealing to some unfortunate human weaknesses. Similarly, conducting mathematics courses in a fashion allowing or encouraging "competition", as opposed to cooperation to move toward better collective understanding, is merely a milder form of "contest". If one wants to claim that this is a necessary evil, then it is inevitable that the "losers" (in a given sub-contest) will "feel bad".

Perpetuating the supposed "two cultures" or "birds and frogs" talk makes things worse exactly by suggesting that there is conflict between the two alleged "types", and that supposedly people tend to one or the other extreme. This is akin to the fake distinction between "pure" and "applied" math, whose most inimical feature is creating labels to which people can declare allegiance, and then be polarized from the "opposing" camp.

That is, the problem is the perpetuation of the description, and perpetuation of talking about mathematical activity in quasi-violent or quasi-sexualized/genderized terms, and in allowing mathematics to lose its own meaning, but become a vehicle for various eternal human-weakness/conflict issues.

I would take issue with the premise that routine classwork is the way to "get understanding", so, then, kids who are impatient with it are entirely reasonable. It is entirely reasonable to try to make the subject "more meaningful" than the standard curriculum model may give a student. Both "broader understanding" and "problem-solving" allow more exercise of judgement, more genuine intellectual activity, than the standard model which emphasizes obedience and suspension of critical judgement.

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    $\begingroup$ I enjoyed the rant! $\endgroup$
    – Jon Bannon
    May 8, 2014 at 23:34
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    $\begingroup$ :) .............. $\endgroup$ May 8, 2014 at 23:57

This is somewhat tangential to your thrust, but I do believe that there is a distinction between—and a continuum between—the hedgehog and fox mentalities described in Isaiah Berlin's famous essay. I know mathematicians who essentially have no focus other than solving one problem after another, where "solving one problem" may require an investment of as much as five years. Others dig in the same hole for their entire careers, repeatedly finding new tangents and avenues to explore, but constantly returning to some overarching question.

Berlin, Isaiah. The hedgehog and the fox: An essay on Tolstoy's view of history. Simon and Schuster, 1966.

If one accepts a spread along this dimension as fundamental, then the avenue to reach students is via problems wherever one sits on the continuum. But "problems" here is decidedly not "contest problems," but real conundrums, open, unsolved, perhaps amorphous problems. Fortunately, with effort, accessible problems can be found on the frontiers of mathematics reachable by college students, and more rarely, even accessible to high school students.

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    $\begingroup$ Yes, certainly both that there's a continuum between extremes... and that artificial problems are unconvincing, at best. $\endgroup$ May 9, 2014 at 0:11
  • $\begingroup$ Learning known skills still has huge motivation. I don't need to be the first person to compete a quadruple back flip. Just learning a flip (if I don't know one yet) is incredible motivating. Humans have evolved to have very strong learning/teaching skills. It's what enables us to have culture and technology (yes, even rock chipping and fish cleaning)! I would urge against emphasizing new to the world Andrew Wiles challenges for undergraduates. Just learning a topic that they didn't know is a challenge and an achievement And success is motivational! $\endgroup$ Apr 1, 2023 at 0:30

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