Math as Art

The director of my school sent along another article about how math teachers should be showing kids the beauty of mathematics. It’s an argument that is being made widely: I’ve seen it in the NY Times, in Frenkel‘s wonderful Love & Math, and in the education literature.

The structure of the argument is to draw a metaphor to the fine arts: kids love music/painting/film because they get to experience Beethoven/van Gogh/Spielberg. Math also has some beautiful pieces, so why don’t we show those to students, and they’ll love math, too?

There are two important flaws in this argument:

1. Of the beautiful aspects of mathematics, what is accessible is often trivial, and what is profound is usually inaccessible. For example, Noether’s Theorem is a beautiful connection between symmetries in nature and conservation laws in physics — but explaining it is beyond the scope even of a top-notch Calculus 101 course (here’s my attempt). Beautiful and meaningful, but inaccessible. On the other hand, the proof that there is no highest prime is accessible to a high school student: but in my experience, students say, “meh”. Accessible, but not profound in their cosmologies.
2. There are plenty of sources of beauty in the world. Depending on your school system, these might all fall under the purview of the art department. Or perhaps (like fire-dancing, electronic circuit design, and sunset-watching) your school doesn’t feel a responsibility to teach students about every conceivable source of beauty in the world. Math has a unique position in the educational ziggurat because of its value in preparing students for STEM-related careers. If we wish to push math as primarily an aesthetic enterprise, we must reconsider its usefulness: does a knowledge of group theory prepare students for the knowledge economy better than the ability to draw freehand?

I think the analogy with the fine arts is a poor one. A much better comparison can be drawn with the subject that is sometimes known as “English Language Arts” or, less precisely, Literature.

A couple decades ago, we started moving away from teaching English through rote grammar recitations. Instead, there is a focus on reading and responding to authentic texts, with a great deal of freedom given to the teacher to choose the material. A cynical view is to say that the only meaningful outcome of these courses, nowadays, is the ability to write essays.

Compare with math, which is stuck in the “grammar recitation” era. Both mathematics and communication are deemed essential, yet the latter was allowed to evolve according to progressive ideals of education, which the former stagnated.

Just like the fine arts above, and math, literature has elements of deep and meaningful beauty. For high school students, connecting with these is rare (required readings rarely bring about the emotional response intended by the teacher!). Like math, high school literature eschews the cheap beauty of the fine arts.

So what would need to happen for a high school math curriculum to work in the same was as an English Language Arts scheme? First, administrators would need to trust math teachers to construct a meaningful scheme of study. Second, math teachers would need to up their game dramatically: no more reliance on worksheet, textbooks, and answer keys. Third, the support infrastructure would need to build resources for active math learning, including a system to track and monitor prior learning (Salman Khan‘s vision of math education comes to mind).

But perhaps most importantly, we would need to accept that high school math does not serve the needs of the majority of the students who study it. The benefits of math education dry up some time during middle school, for students who learn the curriculum the first time through. Advanced levels of numeracy, of the sort that we will require for future citizens, are as likely to emerge from the current pedagogical morass as 21st-century communication skills would be derived from grammar worksheets.

So here’s my response to these calls to teach the beauty of mathematics: don’t teach it, but let it serve as a guide. Use emotional touchstones to drive the curriculum. Let’s demand the same high level curriculum planning from math teachers that we ask of English teachers, and let’s give them the freedom to do so.

The alternative is not teaching budding artists brush strokes without ever seeing a masterpiece: instead, the alternative is teaching budding poets by doing nothing but close-reading sonnets.