A baker knows when a loaf of bread is done and a builder knows when a house is finished. Yogi Berra told us "it ain't over till it's over," which implies that at some point it is over. But in mathematics things aren't so simple. Increasingly, mathematicians are confront ing problems wherein it is not clear whether it will ever be over.The second article is a note published by Thurston in the Notices of the AMS in 1990, which was then reprinted in the arXiv. It's on math education, and I found the following excerpt quite salient:

People appreciate and catch on to a mathematical theory much better after they have first grappled for themselves with the questions the theory is designed to answer.Insofar as these comments relate to math, they do apply (in a lesser degree) to theoryCS as well. Probably the biggest misconception about mathematics is that it is about calculations and formulae. Thurston's article refers to the common student desire to "know the formula to get the right answer", and Becky Hirta and TDM have written countless posts on this topic.

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As mathematicians, we know that there will never be an end to unanswered questions. In contrast, students generally perceive mathematics as something which is already cut and dried—they have just not gotten very far in digesting it.

The sense of determinism that lies at the heart of these assumptions (and which is exemplified, by a rhetorical contrast, in the LA Times piece) is the idea that mathematics is a dead subject full of cranks that you turn to spit out answers to questions. Want a derivative ? Here's a chain rule for you. How do I solve this quadratic equation ? Apply this formula.

What is the consequence ? You come to math expecting to memorize expressions, and do things A CERTAIN WAY. For various pedagogical reasons, none of which I am competent to discuss, teachers often encourage this approach, and students, while gaining some appreciation of the "scaffolding" of mathematics, miss out on the deeper meanings that make the study of abstract structures a far more beautiful and satisfying endeavour than it appears on the outside.

Math textbooks also don't help the matter much because, as Thurston points out:

The best psychological order for a subject in mathematics is often quite different from the most efficient logical order.I have found this to be the case myself, when trying to pick up some new math. The intuition behind the definition of an affine connection is far more likely to stick in my head than the actual definition of a covariant derivative, and is crucial to understanding how the definitions come to be in the first place (the whole 'teach a man to fish' argument).

Growing up learning mathematics without acquiring an appreciation for the true underlying dynamics causes the kind of misconceptions perpetuated even on a show like Numb3rs, where the whizkid mathematicians just does calculations all the time ! And this is one of the better shows in terms of depiction of mathematicians. A mindset that views mathematical work as deterministic number crunching is unable to understand or 'grok' the incredible amount of creativity, artistry, and aesthetics that go into mathematical work and in fact are important reasons for why we do mathematics at all.

Of course, it is easy to say, and harder to do. Thurston's article makes for excellent reading, but is light on actual implementable suggestions. But the basic point of the article is a good one: that at a basic level, students are not getting a fundamental appreciation of what mathematics is all about. Consider this juxtaposition:

From the LA Times piece:

Increasingly, mathematicians are confronting problems wherein it is not clear whether it will ever be over.From Thurston's article:

As mathematicians, we know that there will never be an end to unanswered questions.Source: Mathforge.

Update: more on this topic from Tea Total.

This is also true in how physics is taught. When I took high school physics i had the option of 'conceptual physics' or 'mathematical physics.' Being more math inclined I opted for the mathematical version, as did many of my like-minded peers. However, it turned out that the mathematical physics meant that we learned the basic formulas and then solved problems by plugging in the correct numbers, whereas in the conceptual physics, the students learned the intuition for the formulas and were then shown them. As someone mathematically inclined, I could have easily plugged numbers into the formulas and would have more easily understood how to with the proper intuition.

Jeff

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I have also always lamented this approach to mathematics. Whenever I've read theorem A or B it just seems so desolate not knowing WHY someone was motivated to come up with this. They only give you the end result and almost never the justification for it.

Anonymous

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"You come to math expecting to memorize expressions, and do things A CERTAIN WAY. For various pedagogical reasons, none of which I am competent to discuss, teachers often encourage this approach..."

JeffE

ReplyDeleteI believe the word you're looking for is INCOMPETENCE.

I did a bit of high-school math tutoring when I was in college. I was always amazed how quickly the kids would pick up a new topic once we found/derived the right way for THEM to think about it. ("Why didn't they just explain it that way the first time?") Unfortunately, I had to warn them to translate their answers into the form the teacher expected, or they woudn't get credit.

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