I have alluded elsewhere to the fact that people learn differently, and that the wise guerrilla scholar will attempt to identify and leverage his or her own unique style of learning to the best advantage. If you’re interested in this subject, take a look at Peak Learning by Ronald Gross, who discusses this in much more detail.

To simplify the idea almost to the point of uselessness, the concept goes something like this: some people like to learn by starting from first principles and working through the subject systematically. They read the textbook starting at page one and proceed page by page, chapter by chapter. Gross calls these people “bottom-up learners”. By contrast, “top-down learners” start with the big picture and start putting together their picture of the subject with a piece here, a piece there, following their interests of the moment and picking up what they need when they need it. This is how most hobbyists learn their favorite subjects. It’s also how human learn their first language.

Now as an aside, it is my observation that the higher up you go in the educational system, the more you see bottom-up learning as opposed to top-down learning. Perhaps this has something to do with the hierarchical nature of a university. Perhaps not. But there it is. I believe that if our methods of learning took greater account of our differences in learning, fewer people would feel intimidated by it.

But one subject in particular—mathematics—by convention is almost always taught in a bottom-up manner. You start with arithmetic, geometry, on to basic algebra, then more advanced “intermediate” and “college” algebra, analytic geometry, trigonometry, and finally basic calculus. The advanced students will get into differential equations, vector calculus, calculus of several variables, and so on.

Rewind back to the early 60’s. When I began elementary school, there was this great pedagogical experiment called “New Math” that was supposed to revolutionize how math was taught by getting kids to think about math the way professional mathematicians did. It was such a total failure that Tom Lehrer (himself a math professor) wrote one of his satirical musical masterpieces to “New Math”. It left me a gibbering idiot where math was concerned, and it took great effort on my part to get past that (which is another story for another time).

But let’s give the inventors of New Math some credit: math is a hard thing. It is one of those subjects that ironically is harder to teach, I suspect, than it is to learn. I would like to know if anyone out there has ever heard of a program that attempts to teach math from a “top-down” perspective. Is this even possible? I am contemplating a personal review of math I’ve learned here and there and perhaps trying to expand what I know to see what can be done on an amateur basis. This isn’t just for kids, this is also for adults who encounter mathematical concepts and wonder if they had grokked math better in school they might get more pleasure out of it.

One other observation is in order, one I pass on from several mathematical-types I’ve known, and that is that what they consider “cool” math, the really interesting stuff, is what you get to after calculus. Maybe in that insight is a key through which one could work backwards to learn the rest. I honestly have no notion of what a curriculum like this would look like. But given the state of math and science in America today, I daresay it’s worth thinking about.