Tuesday, December 10, 2013

Reading mathematics

I know students do not read mathematics textbooks, or if they try, they are usually confused. Part of the reason is that most textbooks are too wordy. Students struggle with the sentence length and the density of words on the page.

Part of the reason is that most textbooks tend to get too technical too fast. Unusual words, coupled with odd sentence structures ("Given that...," "it follows that..."), give students trouble.

Students bear some of the blame, because they think math can be read empty-handed and passively. "No," I exclaim, "You must have a pencil and paper out and work the examples as you read! No one can read passively and just absorb it."

These are known flaws.

But the main problem is that most textbooks do not have a good narrative structure to the lesson. By narrative structure, I do not mean personable characters or real-world inspired problems. I mean that the problems unfold in a good, compelling psychological order. A lesson should be almost like a puzzle in a murder mystery; each step provides another clue, the solution seems nearly in reach, and most important, what we want to solve is always clear (there is a known murder victim and a limited cast of suspects). A good lesson presents step-by-step clues to solve a straightforward but unknown problem.

A good lesson must be psychologically compelling, not logically ordered. For example, it makes sense to introduce definition at the end, once students have wrestled with examples and have an intuition about what counts and what does not and why. This is the best psychological order but backwards from logical order.

In a way, step-by-step clues is like reading a multi-layer G.I.S. map. If all the layers are visible at once, the map is overwhelming and incomprehensible. Instead, a good lesson shows students one layer of information at a time. "Look at this. Notice __ . Now look at this. How are these related?" The connections between different layers is how new knowledge is built -- this is all-important work. And the students must do the thinking on their own to make the deep connections. Yet a good lesson does not simply dump all the layers of information on them at once, and the sequence with which layers of information are revealed is critically important to making the lesson work.

Another key to a good textbook lesson is recognizing that students will quickly develop heuristics -- some good, some bad. The bad ones are over-simplifications that under-appreciate the complexity of a concept. The good ones reflect an idea accurately. (On a related note, I think a concept is about replacing specific details with key terminology or new ideas. A concept is like an svg, as opposed to a bitmap. A bitmap is dumb; an svg requires more "software" to understand -- but a lot more information can be compressed.) Anyway, the point is that a good lesson anticipates and immediately confronts and confounds bad heuristics.

My ideal textbook would look like what is below: straightforward questions, unfolding complexity, a goal in mind from the beginning, and -- best of all -- students are required to actively participate.

This seems a lot better than most textbooks at introducing a concept!

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