- Note to searchers: “Just before Christmas a piece appeared in the Guardian discussing cases where people had dreamed that they had uncovered the secret of the universe, only to waken next morning and find they could not remember it. One classic instance, reported by the psychologist William James, was that of a man who repeatedly had this dream and finally managed to write the formula down before he went back to sleep. Next morning he found he had written:
**“A smell of petroleum prevails throughout.”**Another involved an opium addict who jotted the secret down, only to read when he came to full consciousness: “The banana is great, but the skin is greater.” — Downhill to Mandalay

### Illuminating but unsound ideas on pedagogy

In my algebra class, I harp a lot on the associative, distributive and commutative laws. I started doing this because it seemed to me that most of the students’ systemic errors came from misunderstanding these basics. I suppose that’s almost a tautology. When I took linear algebra in college, we did a systematic development of algebra from first principles. I found this tremendously helpful. Much that had been confusing to me became clear, and I hoped that at least some of my students would experience a similar epiphany.

Anyway I wonder if what I’m doing, with my students and the commutative law, is trying to get them adopt a better ontology. The discussion about taxonomy at Faith in Fiction; something I read on Slashdot about folk taxonomies on the web; something my uncle told me; These all came together to make me wonder: How would my students classify algebraic expressions? I read about an exercise in website usability where you give to a few of your users index cards with all your web pages on them, one web page to a card. Ask your users to sort them into categories. Look at what they’ve done, and organize your site accordingly. How would my students classify equations? Expressions? Word problems? (“Johnny has a pie… So divide? Or square it, because it’s a pie…”)

At Faith in Fiction, a weblog by a fiction acquisitions editor, Dave is trying to construct a taxonomy of writing that doesn’t lead to the same old genres. In an off-hand comment, I suggested that maybe you could divide novels up as character-centered (*Great Expectations*) vs. plot-centered (*Jurassic Park* or *The Hunt for Red October*). Thinking about it, I’m not sure this is a good idea. You might end up putting the readers into categories, instead of the books. But that discussion put the idea in my mind.

In my senior year of high school, my uncle told me that “the purpose of college is to make you conform.” He meant this in the sense of adopting the outlook of the ruling classes, i.e. the college graduates who ran things from his point of view. He advised me to cooperate in the process. If he’d been less plain-spoken, he might have said, “Go to college to learn and adopt our society’s dominant ontology.” (I ignored his advice, and would have no matter how he phrased it. Like the scorpion said, It’s my nature.)

### Well, *Somebody* has a skull full of mush anyway…

So all this stuff was in my mind, and I thought of my math students. Of course, they’re never far from my mind. That’s just the kind of guy I am. Conscientious to a fault, and self-effacing to boot. I wondered how my students see algebra. (Through a glass darkly, in some cases.) How do students categorize expressions? What taxonomy do they use? For some, a poor one that leads them into error. I want to replace theirs with a better one. I want them to adopt the outlook of the people who can do algebra; To leave thinking like a lawyer.

Could I do this more effectively if I set out purposely to do so? And how did the students acquire this bad ontology, the one that I’m trying to replace with a better? Have there been any studies of early math education and how students develop their ontology? Something backed up with solid diagnostic testing? If we could use that research and get to them in the early grades…

Then I realized: This is how we end up with stuff like new math, or whole-language learning, or outcomes-based education.

Maybe in the course of a good education students do adopt the outlook of other well-educated people. Certainly math education could be done better, especially for those who aren’t going to go on to be mathematicians or engineers. But there’s no royal road to geometry, to algebra, or to anywhere worth going.

It’s not rocket science, nor yet philosophy. It takes practice. To learn algebra, students need to show up to class, ask questions, think about what they’re doing; They need to read the text book, work through the examples, and do their homework.

And I need to stop blogging and write up the next test. There **will** be a question on the distributive law.