How We Learn: Being mixed up

Chapter 8 of How We Learn describes interleaving as a better means of practice. In math education the Saxon textbook is an example. It uses a mix of practice problems, combining everything learned so far, as opposed to typical textbooks where all problems are about one lesson. Not only does this better improve the skills being learned, but students now need to recognize which strategy to use for each problem, and (perhaps as a result) they tend to better apply the skills in other contexts.

It reminds me of my experience with high school math team.  We’d do tests from past years of competitions: 25 questions on a variety of topics. Since these were graded more by participation than percent correct, each person could grow at their own pace. I was able to get through the bulk of tests quickly and spend some time deeply thinking about questions beyond my capacity. One that I always remember is discovering my own approach to trigonometry identities that involved manipulating triangles.

We’d do these tests in the morning and then have people put them up on the board in the afternoon. (Yes, we had two periods of math team plus the regular math class.) I felt this was another benefit – someone around your level could explain a problem as they may have figured it out for the first time. And I would try and fail to explain my homebrew approach to trig.
The chapter also contrasts the conservative and progressive approaches to math education. The progressive supposedly favors conceptual skills like number sense while the conservative builds up from concrete, procedural skills. (I also recommend commentary from Math With Bad Drawings)

The method of learning for math team was strongly in the conservative tradition. We even had a “formula book” that contained formulas that could solve probably 80-90% of the test without much further thought. Sometimes deeper thinking did happen – not just when I will bored and unprepared for trigonometry but when inspired by good question writing that demanded using the material in new ways. (One competition that I highly commend is Mandelbrot.) I don’t think these kind of questions could have been approached without that base of knowledge. Of course the balance is hard to strike: by senior year, some of us were pushing back against being taught with such focus on these more formulaic problems.