Math education and the illusion of certainty

Two articles were the inspiration for this musing. Firstly, there was this oped in the LA Times titled 'Definitional Drift: Math goes Postmodern'. The premise:

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.
[...]
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.

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.

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.