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Fields Medalists on School Mathematics

13 Post author: multifoliaterose 11 October 2010 10:06AM

Most people form their impressions of math from their school mathematics courses. The vast majority of school mathematics courses distort the nature of mathematical practice and so have led to widespread misconceptions about the nature of mathematical practice. There's a long history of high caliber mathematicians finding their experiences with school mathematics alienating or irrelevant. I think this should be better known. Here I've collected some relevant quotes.

I'd like to write some Less Wrong articles diffusing common misconceptions about mathematical practice but am not sure how to frame these hypothetical articles. I'd welcome any suggestions.

Acknowledgment - I obtained some of these quotations from a collection of mathematician quotations compiled by my colleague Laurens Gunnarsen.

In Reflections Around the Ramanujan Centenary Fields Medalist Atle Selberg said:

I have talked with many others who became mathematicians, about the mathematics they learned in school. Most of them were not particularly inspired by it but started reading on their own, outside of school by some accident or other, as I myself did.

In his autobiography Ferdinand Eisenstein wrote about how he found his primary school mathematical education tortuous:

During the first years [of elementary school] I acquired my education in the fundamentals: I still remember the torture of completing endless multiplication examples.  From this, you might conclude, erroneously, that I lacked mathematical ability, merely because I showed little inclination for calculating.  In fact the mechanical, always repetitive nature of the procedures annoyed me, and indeed, I am still disgusted with calculations lacking any purpose, while if there was something new to discover, requiring thought and reasoning, I would spare no pains.

There is some overlap between Eisenstein's early school experience and the experience that Fields Medalist William Thurston describes in his essay in Mariana Cook's book Mathematicians: An Outer View of the Inner World:

I've loved mathematics all my life, although I often doubted that mathematics would turn out to be my life's focus even when others thought it obvious.  I hated much of what was taught as mathematics in my early schooling, and I often received poor grades.  I now view many of these early lessons as anti-math: they actively tried to discourage independent thought.  One was supposed to follow an established pattern with mechanical precision, put answers inside boxes, and "show your work," that is, reject mental insights and alternative approaches.  My attention is more inward than that of most people: it can be resistant to being captured and directed externally. Exercises like these mathematics lessons were excruciatingly boring and painful (whether or not I had "mastered the material").

Thurston's quote points to the personal nature of mathematical practice. This is echoed by Fields Medalist Alain Connes in The Unravelers: Mathematical Snapshots

...for me, one starts to become a mathematician more or less through an act of rebellion. In what sense? In the sense that the future mathematician will start to think about a certain problem, and he will notice that, in fact, what he has read in the literature, what he has read in books, doesn't correspond to his personal vision of the problem. Naturally, this is very often the result of ignorance, but that is not important so long as his arguments are based on personal intuition and, of course, on proof. So it doesn't matter, because in this way he'll learn that in mathematics there is no supreme authority! A twelve-year-old pupil can very well oppose his teacher if he finds a proof of what he argues, and that differentiates mathematics from other disciplines, where the teacher can easily hide behind knowledge that the pupil doesn't have. A child of five can say, "Daddy, there isn't any biggest number" and can be certain of it, not because he read it in a book but because he has found a proof in his mind...

In Récoltes et Semailles Fields Medalist Alexander Grothendieck describes an experience of the type that Alain Connes mentions:

I can still recall the first "mathematics essay", and that the teacher gave it a bad mark. It was to be a proof of "three cases in which triangles were congruent." My proof wasn't the official one in the textbook he followed religiously. All the same, I already knew that my proof was neither more nor less convincing than the one in the book, and that it was in accord with the traditional spirit of "gliding this figure over that one." It was self-evident that this man was unable or unwilling to think for himself in judging the worth of a train of reasoning. He needed to lean on some authority, that of a book which he held in his hand. It must have made quite an impression on me that I can now recall it so clearly.

Comments (19)

Comment author: Relsqui 11 October 2010 06:01:52PM 1 point [-]

I like this collection, although I admit I rolled my eyes at this particular snippet:

and "show your work," that is, reject mental insights and alternative approaches. My attention is more inward than that of most people: it can be resistant to being captured and directed externally.

No, the purpose of "showing your work" is not for The Man to keep you down. It's to demonstrate that you didn't just copy your answers from someone else (or from a calculating device). It can be tedious, yes, but so long as students do copy answers I don't grudge the teachers this method of making that more difficult. Additionally, if your difficult-to-capture attention happens to bring you to the wrong answer, having recorded your work will make it much easier for the teacher to nudge you to the right one. So this justification only applies if you're always correct or the teacher is entirely unable or unwilling to teach (granted, sometimes true, but not always). Also that you're not proving anything--because if you can't write down your reasoning, why on earth do you expect anyone to believe your conclusion? That it's difficult is not sufficient reason not to do it.

I realize these aren't your words to defend and hope you don't feel obliged to.

Comment author: multifoliaterose 11 October 2010 07:41:34PM *  1 point [-]

I understand what you're saying here and when I teach calculus I ask my students to show their work. A couple of comments:

No, the purpose of "showing your work" is not for The Man to keep you down. It's to demonstrate that you didn't just copy your answers from someone else (or from a calculating device). It can be tedious, yes, but so long as students do copy answers I don't grudge the teachers this method of making that more difficult.

I agree that this is reasonable given how educational institutions are structured (in particular, given how much emphasis is currently placed on evaluation) but think that there's a general problem of focus on evaluation being detrimental to quality of education. My experience teaching calculus discussion sections has been that my students' focus on getting good grades takes up so much of their attention that they have little attention to bring to the task of actually learning the material. On the flip side circumstances force instructors to devote so much of their time to ensuring that grading is fair that their ability to focus on actually teaching is markedly impaired. I see the common focus on deterring cheating as part of this dynamic. This is not to say that I have a solution, I'm just saying that's Something's Wrong :-).

So this justification only applies if you're always correct or the teacher is entirely unable or unwilling to teach (granted, sometimes true, but not always). Also that you're not proving anything--because if you can't write down your reasoning, why on earth do you expect anyone to believe your conclusion? That it's difficult is not sufficient reason not to do it.

It seems likely to me that Thurston is Generalizing From One Example here. He's legendary for having consistently made obscure and apparently ungrounded statements which have turned out to be fully justified. One could imagine such a background making it easy to forget that not everybody is the same way.

In Jaffe and Quinn's "Theoretical mathematics'': Toward a cultural synthesis of mathematics and theoretical physics the authors say

William Thurston’s “geometrization theorem” concerning structures on Haken three-manifolds is another often-cited example. A grand insight delivered with beautiful but insufficient hints, the proof was never fully published. For many investigators this unredeemed claim became a roadblock rather than an inspiration.

The full proof of the geometrization theorem for Haken manifolds was later published along the lines that Thurston had originally suggested. Thurston responded to Jaffe and Quinn in his essay titled On proofs and progress in mathematics.

Comment author: Relsqui 11 October 2010 07:56:50PM *  1 point [-]

I agree about the overall structure of educational systems vis a vis grades, although there is apparently some evidence that being tested periodically helps one retain information.

It seems likely to me that Thurston is Generalizing From One Example here.

Actually, in fairness, I don't think he's generalizing; I think he's observing one example. My interpretation of the statement as prescriptive may not have been intended. Of course, if it's to be included on this list, it should be expected to be interpreted as prescriptive.

He's legendary for having consistently making obscure and apparently unjustified statements which have turned out to be fully grounded.

I wonder whether my lack of preconceptions about the source of the quote was helpful or harmful here. (My guess is "no." ;))

Comment author: multifoliaterose 12 October 2010 01:07:13AM *  0 points [-]

I agree about the overall structure of educational systems vis a vis grades, although there is apparently some evidence that being tested periodically helps one retain information.

Thanks, I hadn't seen the things that the linked NY Times article discusses before.

Actually, in fairness, I don't think he's generalizing; I think he's observing one example. My interpretation of the statement as prescriptive may not have been intended.

As you remark, equating "show your work" with "reject mental insights and alternative approaches" is too strong. My suggestion was that to the extent that he's drawing such an equivalence, he's likely to be generalizing from one example.

But in line with what you say above, the essay that the quote is from was written as a personal reflection rather than a careful analysis and so the quote is most properly viewed as an offhand remark reflecting on his own experience.

Of course, if it's to be included on this list, it should be expected to be interpreted as prescriptive.

I cited Thurston for a description of his own experience with school math.

I wonder whether my lack of preconceptions about the source of the quote was helpful or harmful here. (My guess is "no." ;))

Sure, makes sense. I was just giving some background in case you're curious. I personally found the essays that I linked above well worth reading.

Comment author: Relsqui 12 October 2010 02:20:17AM 1 point [-]

I cited Thurston for a description of his own experience with school math.

Understood, but if it's included on a list of quotes about math education the overall thrust of which is to show what's wrong with it, the implication is that what he describes is one of the things which is wrong with it.

I hope it's clear that my last line there wasn't any kind of rejection of your explanation. I was just musing which way the bias ran. And I still think this is a worthwhile post overall.

Comment author: multifoliaterose 12 October 2010 02:28:20AM 1 point [-]

Understood, but if it's included on a list of quotes about math education the overall thrust of which is to show what's wrong with it, the implication is that what he describes is one of the things which is wrong with it.

Thanks for pointing this out, you might be surprised to know that it didn't occur to me! I'll bear this point in mind and add some sort of disclaimer if I incorporate the quote into a top level post.

Comment author: James_Miller 12 October 2010 10:39:49PM 1 point [-]

Do you have any book suggestions for mathematically gifted children?

Comment author: cata 14 October 2010 01:36:22PM *  2 points [-]

I could not possibly, conceivably recommend any book in the world for a kindergarten- or elementary-school-aged kid more than Math for Smarty Pants.

Comment author: CronoDAS 15 October 2010 01:46:36AM 0 points [-]

I have that book. It's really good. (The same author also wrote The I Hate Mathematics Book, which is similar.)

Comment author: mindspillage 14 October 2010 06:34:33AM *  2 points [-]

Sideways Arithmetic From Wayside School by Louis Sachar, a puzzle book companion to the silly Wayside School stories; I read it in elementary school to what I think was my great benefit. I posted a bit about it elsewhere--it was my first introduction to some of the concepts of logical thinking I've found generally useful now.

Comment author: mindspillage 14 October 2010 05:32:29PM 1 point [-]

Also, not specific to mathematics (although there is some, as a way of explaining various inventions), but I recommend The Way Things Work to pretty much everyone who asks about good books for children.

Comment author: CronoDAS 15 October 2010 01:49:37AM 0 points [-]

I used to love the television show Square One TV, but I think it's impossible to get videos of.

Comment author: multifoliaterose 13 October 2010 07:38:35AM *  0 points [-]

Yes, this is something that I've thought about a lot. How old and with what background knowledge? I'll eventually be compiling a thorough list, but can mention some things off the cuff with more information.

Comment author: James_Miller 13 October 2010 02:21:03PM 0 points [-]

We were doing 4th grade math together (adding and subtracting large numbers, simple geometric proofs, multiplying one digit numbers, Venn Diagrams) before he lost interest. The 4th grade material seems much less interesting than K-3rd grade stuff and I wonder if lots of kids start to get turned off by math in the 4th grade.

Comment author: multifoliaterose 13 October 2010 02:40:53PM *  2 points [-]

Is he still in 4th grade? I know more about resources for middle/high schoolers than for elementary schoolers. One recommendation that comes to mind is

The Number Devil: A Mathematical Adventure by Hans Mangus Enzenberger --- this has substantive mathematical content and I've found that it's quite popular among gifted children who encounter it.

Three items which may be useful to you in working with him are

  1. Vision in Elementary Mathematics by W. W. Sawyer

  2. Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States by Liping Ma

  3. Arithmetic for Parents: A Book for Grownups about Children's Mathematics by Ron Aharoni.

I haven't had a chance to look at the last one but it's recommended by Alexander Givental, a Berkeley math professor and parent who I have very high regard for.

I think that Raymond Smullyan's books of logic puzzles are good for gifted children.

Comment author: James_Miller 13 October 2010 03:03:39PM 0 points [-]

Thanks,

He hasn't reached the 4th grade but I was teaching him above his grade level math.

Comment author: multifoliaterose 13 October 2010 03:16:34PM 0 points [-]

Right, this makes sense.

Looking at your location, one more thing that I should point out is the Harvard Math Circle. It's probably too far away to be useful considering that the classes for younger children meet on weekdays, but something to keep in mind for the future.

Comment author: nhamann 11 October 2010 09:25:04PM 0 points [-]

I hated much of what was taught as mathematics in my early schooling, and I often received poor grades. I now view many of these early lessons as anti-math: they actively tried to discourage independent thought. One was supposed to follow an established pattern with mechanical precision, put answers inside boxes, and "show your work,"

I guess this puts me out of the running for obtaining a Fields Medal, because I have excelled (and continue to excel) at this kind of work since a young age. :[

Comment author: multifoliaterose 11 October 2010 10:20:53PM *  1 point [-]

Oh no, there's substantial variability among great mathematicians on this point. If I make this article into a top level point I should add qualification to this effect. I don't have the exact quote at hand (will add later), but for counterpoint I'll say that in the same book Don Zagier describes how when he was eleven his teacher told him that if he wanted to, he could read advanced math books in class provided that he precommitted to getting perfect scores on all of his exams. He said

She told me that I could choose whether or not I wanted to accept these conditions, and of course I did. It was very good training because I learned to be quick and careful even in routine calculations, and that was very helpful later.