Monday, July 26, 2010

Teaching yourself mathematics -- a footnote for future posts

I'm pretty sure I'm going to be making this claim repeatedly so I might as well take a few minutes to put it down in a linkable form for future use.

Of all the subjects a student is likely to encounter after elementary school, mathematics is by far the easiest to teach yourself. With the appropriate attitude and assumptions, adequate motivation and a simple and easily mastered set of skills the majority of students can take themselves from pre-algebra through calculus.

What is it that makes math teachers so expendable? Part of the answer lies in mathematics position on the fact/process spectrum. Viewed in sufficiently general terms, all subjects start with giving the student a set of facts and ideally end with the student performing some process using those facts. In subjects like history and to a slightly lesser extent, science, most of the early stages of mastering the subject center around absorbing the facts. On the other end of the spectrum, subjects like music, writing and mathematics involve a relatively small set of facts*. Students studying these subjects tend to focus primarily on process almost from the beginning.

Put another way, at some point all disciplines require the transition from passive to active and that transition can be challenging. In courses like high school history and science, the emphasis on passively acquiring knowledge (yes, I realize that students write essays in history classes and apply formulas in science classes but that represents a relatively small portion of their time and, more importantly, the work those students do is fundamentally different from the day-to-day work done by historians and scientists). By comparison, junior high students playing in an orchestra, writing short stories or solving math problems are almost entirely focused on processes and those processes are essentially the same as those engaged in by professional musicians, writers and mathematicians.

Unlike music and writing, however, mathematics starts out as a convergent process. It doesn't stay that way. By the time a student gets to upper level math courses like real analysis or applied subjects like statistics ** there are any number of valid proofs for theorems and approaches to problems, but for anything before, say, differential equations, most math problems have only one solution and students are able to quickly and accurately check their work. Compare this to writing. There is no quick or accurate way to gauge the quality of a piece of prose or, worse yet, verse. Writers spend years refining their editing skills and even then they still generally seek out other critics to help them assess their own work.

This unique position of mathematics allows for any number of easy and effective self-study techniques. One of the simplest is the approach that got me through a linear algebra section taught by the worst college level instructor I have ever encountered (and that, my friends, covers some territory).

All you need is a textbook and a few sheets of scratch paper. You cover everything below the paragraph you're reading with the sheet of paper. When you get to an example, leave the solution covered and try the problem. After you've finished check your work. If you got it right you continue working your way through the section. If you got it wrong, you have a few choices. If you feel you basically understood the solution and see where you made your mistake, you might simply want to go on; if you're not quite sure about some of the steps in the solution, you should probably go back to the beginning of the section; if you're really lost, you should go back to the preceding section and/or the previous sections that introduced the concepts you're having trouble with.

Once you've worked through all the examples, start on the odd numbered problems and check your answers as you go. If you're feeling confident, you can skip to the difficult problems but if you make a mistake or get stuck you should probably go back to number 1.

Don't get me wrong. I'm not saying this is the only technique, let alone the best, for teaching yourself mathematics. Nor am I suggesting that we make a practice of dumping student in sink-or-swim situations. I think we should provide students with the best teachers and support system possible, but even under those conditions, the internal resources needed to teach yourself mathematics are tremendously valuable to all students and are absolutely essential to anyone who has to use sophisticated analytical reasoning.

Tragic postscript: In what I can only assume is an idiotic attempt raise standards, most books have stopped giving answers to odd-numbered problems. Under the old system you would assign odd problems when you wanted the students to be able to check their work and even problems when you wanted to make sure they weren't just looking answers up in the back of the book. It was a simple, flexible, and effective system that encouraged students to be independent and resourceful. No wonder it was such a prime target for reform.

* I heard a story (possibly apocryphal) of a professor who walked into an upper level math class, wrote properties of real numbers on the board, told the class that was all they needed to know to prove all the theorems in the book, then walked out.

** The relationship between mathematics and statistics is particularly complex, far too complex to discuss in a blog post.