Wednesday, October 16, 2013

Math professors on making math fun

Certain math professionals are convinced that there must be better ways out there to help the rest of us appreciate their favorite subject. Generally, their ideas involve comparing the beauty of mathematics to that of the arts. Two of the best-known math promoters, NPR “math guy” Keith Devlin and "Lockhart’s Lament" author Paul Lockhart, have analogized math appreciation to music appreciation.

Recently, two other would-be math popularizers have entered the public eye. There’s Berkeley math professor Edward Frenkel, the writer, director, and star of the movie “Rites of Love and Math." And there’s University of Maryland math professor Manil Suri, the author of the novels “The Death of Vishnu,” “The Age of Shiva,” and “The City of Devi.” In August, Frenkel did an interview on loving math with the Wall Street Journal. In September, Suri wrote an OpEd piece on math appreciation in the New York Times. But rather than music, each has chosen the visual arts as their chosen vehicle.

Like Devlin and Lockhart, however, both Suri and Frenkel begin their arguments with assertions about traditional math and drill-oriented arthmetic being boring. Frenkel alludes to “the boring way that math is traditionally taught in schools,” while Suri notes how “in schools, as I’ve heard several teachers lament, the opportunity to immerse students in interesting mathematical ideas is usually jettisoned to make more time for testing and arithmetic drills.”

Both these impressions appear to be based on hearsay: neither Frenkel nor Suri grew up in this country, and neither appears (from their entries in Wikipedia) to have school-aged children. That doesn’t stop Frenkel from diagnosing the problem:

It's like teaching an art class where they only tell you how to paint a fence but they never show you Picasso. People say, “I'm bad at math,” but what they're really saying is “I was bad at painting the fence.”
Suri thinks that painting the fence is completely unnecessary:
Think of it this way: you can appreciate art without acquiring the ability to paint, or enjoy a symphony without being able to read music.
Analogously:
One can develop a fairly good understanding of the power and elegance of calculus, say, without actually being able to use it to solve scientific or engineering problems.    
 It’s noteworthy that Suri shies away from what would be a more precise analogy:
One can develop a fairly good understanding of the power and elegance of calculus without actually being able use it to solve calculus problems.  
Few people would dispute the idea that one can appreciate a great deal about art without being able to do real art. But how much can one appreciate about math without being able to do real math? On this question, someone who is unusually skilled in doing real math, i.e., a math professor, may not be the best informant.

As I’ve argued earlier, there’s only so far the comparison between math and art goes. Consider Suri’s examples of the beauty of math. First, there’s this Youtube video of the construction of the integers from operations on the empty set.

Then:
For a more contemplative example, gaze at a sequence of regular polygons: a hexagon, an octagon, a decagon and so on. I can almost imagine a yoga instructor asking a class to meditate on what would happen if the number of sides kept increasing indefinitely. Eventually, the sides shrink so much that the kinks start flattening out and the perimeter begins to appear curved. And then you see it: what will emerge is a circle, while at the same time the polygon can never actually become one. The realization is exhilarating — it lights up pleasure centers in your brain. This underlying concept of a limit is one upon which all of calculus is built.
Finally:
…fractal images — those black, amoebalike splotches surrounded by bands of psychedelic colors — hardly qualifies as making a math connection. But suppose you knew that such an image (for example, the Julia Set) depicts a mathematical rule that plucks every point from its spot in the plane and moves it to another location. Imagine this rule applied over and over again, so that every point hops from location to location. Then the “amoeba” comprises those well-behaved points that remain hopping around within this black region, while the colored points are more adventurous and all lope off toward infinity. Not only does the picture acquire more richness and meaning with this knowledge, it suddenly churns with drama, with activity.
 Suri asks:
Would you be intrigued enough to find out more — for instance, what the different shades of color signified? Would the Big Bang example make you wonder where negative numbers came from, or fractions or irrationals? Could the thrill of recognizing the circle as a limit of polygons lure you into visualizing the sphere as a stack of its circular cross sections, as Archimedes did over 2,000 years ago to calculate its volume?
Yes, yes, and yes—assuming I’m a mathematically-inclined person who’s had a solid foundation in math at least all the way through arithmetic. Suri, however, isn’t preaching to the choir, but to students in general:
If the answer is yes, then math appreciation may provide more than just casual enjoyment: it could also help change negative attitudes toward the subject that are passed on from generation to generation. Students have a better chance of succeeding in a subject perceived as playful and stimulating, rather than one with a disastrous P.R. image.
The problem is that Suri has it backwards. The boring foundations must come first before you can appreciate—let alone do—the fun stuff.

Furthermore, the more we move away from visualizable geometry (the math promoter’s favorite topic) to the more abstract, symbolic math that dominates the field, the less well the math-appreciation-as-art-appreciation comparison hold up. As I argue earlier:
There's an important difference between math and music... Music has a privileged place in subjective experience. Along with sensations like color, taste, and smell, it produces in us a characteristic, irreducible, qualitative impression—an instance of what philosophers call "qualia." Just as there's no way to capture the subjective impression of "redness" with a graph of its electromagnetic frequency, or of "chocolate" with a 3-D model of its molecular structure, so, too, with the subjective feeling of a tonic-dominant-submediant-mediant-subdominant-tonic-subdominant-dominant chord progression. Embedded in what makes music what it is to us is the qualia of its chords and melodies.
Like most other, more abstract concepts ("heliocentric," "temporary"), mathematic concepts don't generally evoke this qualia sensation. What makes math beautiful are things like eloquence, patterns, and power. Unlike a Bach fugue translated homomorphically into, say, a collage of shapes, mathematical concepts can be translated into different representational systems without losing their essence and beauty.
Math is much more appropriately compared with thoughts than with music. This means that there is a much closer connection between passive appreciation and active skill. You can appreciate a fugue without being able to compose it; you can’t appreciate a thought if you don’t understand it well enough paraphrase it.

For those who aren’t up to understanding set theory or geometric limits or the question of where numbers come from, there are still plenty ways to have fun with math. As I discussed in an earlier post, however, they involve math as puzzles rather than math as art, and they’re all in the realm of active doing; not of passive appreciating. And to get there, for most ordinary human beings, a certain amount of “boring drill” is absolutely necessary.

Frenkel’s strategy for math appreciation is less about appreciating the “art” in math, and more about “putting love into math.” “Everyone loves love,” he points out. As the Wall Street Journal puts it, “Mr. Frenkel, a youthful, puckish 45-­year-­old with a slight Russian accent and a flair for fitted shirts and tailored jeans, hopes to be math's next leading man.”

Whence Frenkel’s above-mentioned cinematic endeavor, a 28 minute erotic movie that features a naked lover, played by Frenkel himself, and his naked and love interest, played by an actress young enough to be his student. The film culminates with the math professor tattooing the mathematical formula for love on the latter’s naked back. You can watch a trailer on Youtube.

The Wall Street Journal notes that YouTube videos of Frenkel’s lectures at UC Berkeley are “viewed by hundreds of thousands of people," with Mr. Frenkel adding “and that's even the most boring stuff.” And actually, his math videos aren’t that interesting. I can think of several math professors I know who are much more dynamic lecturers; they just don't post themselves on Youtube. Perhaps seeing the naked Frenkel do the more interesting, i.e. erotic, stuff has sparked interest in seeing the clad Frenkel do the math stuff. Perhaps more math professors should try their hands at erotica.

To get students interested in math, though, it really shouldn’t be necessary to take your clothes off. A more promising approach, once again, involves fun problems of the sort that reward perseverance and clever thinking—just like the kinds of puzzles that all sorts of people readily do for fun.

4 comments:

C T said...

Math Professor: "Everyone should LUUUURVE math as much as I do. Real analysis and differential equations are actually quite easy if you just get them the way I do!"

Everyone else: "Dude, you're a freak. Seriously, a statistical freak of nature."

Everyone else's future employers: "Please, can we just find some workers who can do basic arithmetic in their heads?"

Katharine Beals said...

Brilliant!

Kim said...

Heres' the thing. I appreciate art because I understand what goes into it. I can barely draw a stick figure that looks like a stick figure, so when I look at a work of art I see the beauty in it and recognize the skill and imagination that it took to create such a piece. It's the same thing when I see a skilled dancer, musician, or football player. Part of appreciating what they do so well is recognizing how much work it took to make them the works of art they are.

The thing is - none of them - the artist, the dancer, the musician, the football player - not a single one of them became what they are just by observing and appreciating art, dancing, music, or football. They had to learn about it first. They had to learn how to hold a paintbrush, how to stand on their toes, how to play a note, and how to catch a ball. Lots of other people tried, just like them, but didn't have the gift they do. That's what makes their gift so valuable - the fact that not many of us can do what they do.

The same thing applies to math. You have to learn how to add, subtract, multiply, and divide to understand and be in awe of things like pi.

And...just like artists, musicians, dancers, and football players are freaks of nature, so too are the gifted mathematicians who imagine that the swirling hexagon / decagon exercise would do anything other than give most of us a headache.

Anonymous said...

The difference is that incredible artists, musicians, dancers, and football players don't imagine everybody else could easily do what they do if they just followed their simple educational program.

Too smart by half, those math geniuses.