why it actually matters

06Apr10

It’s been a while since I’ve posted anything here – I’ve been pretty busy with work.

In earlier posts I’d mentioned interesting articles on mathematics by Bill Thurston, Terry Tao, and Paul Lockhart. Today I’m going to try and relate these articles to the topic of educating children in mathematics generally.

Let me start with Lockhart’s article. In that article, Paul takes the aggressive position that math is more like art than anything else; because we treat it in a rote fashion, we rob children of the beauty of the subject matter. I find this an interesting point because I think it perfectly demonstrates our collective bipolar attitude towards math. There are two simultaneous memes about mathematics out there:

1) Math is rigorous, precise, mechanical, devoid of spirituality, dry, cold, logical, spock-ish, etc.

2) Math is beautiful, creative, artistic, elegant, wondrous, God keeps its most amazing results in a little book, etc.

I assert that most people have had exposure to *both* memes; unfortunately, most people merely have evidence for the first meme, and not the second, and so the first one is perhaps the more commonly-held belief.

But my main point here is that both memes are one-dimensional: they fail to capture the entirety of mathematics, and (more importantly) they fail to capture the spectrum of reasons why children should learn math.

There are at least two dimensions to this problem. One dimension might represent the range of attractions of mathematics itself: the stereotypical engineer loves the pragmatic value of the Fast Fourier Transform or of error correcting codes; the stereotypical head-in-the-clouds number theorist loves ancient riddles about whole numbers (ok, yes, this is foreshadowing – turns out the lover of error-correcting codes now has a lot in common with the lover of arithmetic – but that’ll be a *much* later post!).

But the other dimension to this problem is that there’s a hierarchy of specialization: not every citizen needs to get a PhD in mathematics to be deemed mathematically literate. So, in effect, a math curriculum will by necessity partition the body of math knowledge into ~3 strata:

1) The math that is valuable for the general public (basic math literacy)

2) The math that is needed for certain vocations (say, for undergrad majors in engineering, science, etc)

3) The math that is needed for specialists in mathematics itself.

Paul makes a very compelling argument that, in partitioning math as above, we’ve inadvertently also removed all the interesting and beautiful stuff about math from the first and second tiers of this strata. I would generally agree with this point. But there is a flip side to this: focusing on the beautiful patterns, and on selling the wondrous-ocity of these patterns to kids. In my experience, middle-school kids can be quite pragmatic, too, and you want to appeal to that side as well.

Richard Feynman once wrote that he’d aimed his physics lectures at both the theorist and the experimentalist: he wanted to provide something for each personality type. Similarly, if you teach math purely as an art form, and fail to note the “unreasonable effectiveness of mathematics,” then some of your class will miss the point. BTW, I’m very confident that Paul personally provides something for everyone in his class, I am pointing this out mainly to reference what I think is a weakness in his thesis.

To expound on this point some more, let me suggest that math is in many ways like English. Consider the problem of teaching English to the masses. There are two parallel goals: basic proficiency with the language (both literacy and writing skills) and exposure to literature as an art. Great English teachers understand that reading and discussing great works of literature is a powerful and effective way of developing a child’s ability to understand the existing corpus and reason effectively about it. At the same time, diagramming sentences and “mechanically” writing 5 paragraph essays – these are great etudes, a focused bit of deliberate practice that (when properly motivated) hone a student’s craft. Great English teachers juxtapose both kinds of work throughout their curriculum.

The same is of course true for music: a great music education trains you in musicianship, in instrument-specific technique, in theory and in developing a rich understanding of the history of music and the body of work that has been done before. You study the great works, and understand why they are wonderful, and you strive to copy them, and then later to find your own musical voice.

So, big surprise: math education is like many other human intellectual endeavors, and getting great at it requires developing facility with the tools and techniques of the trade, while also deepening one’s conceptual understanding of the subject. This conceptual understanding is in turn achieved thru a combination of carefully studying specific examples in great depth, along with then abstracting out core principles, patterns, and connections to other topics. At various times, the subject of mathematics has been developed and expanded by folks who were motivated in dramatically different ways: some were motivated by very pragmatic, even mundane or quotidian problems; others were driven purely by aesthetic considerations, whether it was beauty, fun, farce, or perhaps even a combination of these and other driving forces.

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