Math as Art

A Mathematician’s Lament was Slashdotted weeks ago, but I finally sat down and read my way through the whole thing. Lockhart, a math professor who returned to elementary and high school math education, writes about the fundamental flaws he sees in how we approach teaching math, particularly at the youngest levels. He opens with two stories that describe in his view what music and art education would be like if they were taught in the same way math is taught:

I was surprised to find myself in a regular school classroom— no easels, no tubes of paint.
“Oh we don’t actually apply paint until high school,” I was told by the students. “In seventh
grade we mostly study colors and applicators.” They showed me a worksheet. On one side were
swatches of color with blank spaces next to them. They were told to write in the names. “I like
painting,” one of them remarked, “they tell me what to do and I do it. It’s easy!”
After class I spoke with the teacher. “So your students don’t actually do any painting?” I
asked. “Well, next year they take Pre-Paint-by-Numbers. That prepares them for the main
Paint-by-Numbers sequence in high school. So they’ll get to use what they’ve learned here and
apply it to real-life painting situations— dipping the brush into paint, wiping it off, stuff like that.
Of course we track our students by ability. The really excellent painters— the ones who know
their colors and brushes backwards and forwards— they get to the actual painting a little sooner,
and some of them even take the Advanced Placement classes for college credit. But mostly
we’re just trying to give these kids a good foundation in what painting is all about, so when they
get out there in the real world and paint their kitchen they don’t make a total mess of it.”

The heart of the article is an argument for mathematics as an art, where creativity is central. Lockhart acknowledges that most people do not understand what mathematics really is – because they never get to really do mathematics in school. Math is taught as facts to be known rather than celebrating the ideas and insights that led to those facts being established. The drive to make math practical and relevant to our lives has taken out the part of math that is fun and interesting. I love the example that he gives of a good math question – one that asks you to think and have ideas:

Suppose I am given the sum and difference of two numbers. How
can I figure out what the numbers are themselves?

As he notes, if you know algebra you can apply it to this problem, but even if you do not know algebra you can think about it and come up with an answer and test it out. It is a puzzle that encourages creative thinking instead of applications of formulas to problems that follow a predictable template. Ultimately Lockhart doesn’t argue that we shouldn’t teach students to learn arithmetic and algebra, but that those things should come out of real, interesting problems that the students are trying to solve.
For me, I think about how different my introductory programming class would be if math was taught this way. I watch students struggle with how to solve a problem based on the things they know how to do, and a number of them want to be told what process to follow to come up with an answer – what similar problem that they have solved before can they slightly adjust to come up with an answer to this problem. Exploring ideas, some of which will fail, seems to be an unfamiliar process for a lot of them. Which is sad, because that is part of what is so fun about programming – you can have an idea and then test it out right there and see what happens. The whole process hinges on having learned at some point to think “what if I do this” and see where it takes you. If math was taught in school the way Lockhart describes, the transition to programming would simply be one of adding a bit of syntax so the computer can understand your ideas.
If you are all at all curious about the article but daunted by its length (it is very long), read the first section with the “dreams” and then just skim through the dialogs – they are a nice Socratic summation of Lockhard’s argument.

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