Induction or deduction
We compare two approaches to teaching mathematics.
Our tutor has students at a variety of levels of mathematics, from several schools, taking a number of different classes. (As he has pointed out before, he prefers the higher levels, not because more elementary math is easier to teach or less important; neither is true; but because he is less familiar with the difficulties younger students have, and the best strategies to deal with them.) This term he has students using textbooks that take almost opposite approaches in presenting their material.
The first is deductive. In each section of the chapter, the question to be dealt with is initially stated in all generality and rigor, using formulas and equations that the student will never actually apply. Only later are restrictions and further assumptions brought in, so that techniques can be developed to deal with particular types of problem and useful results obtained. The student using this textbook, bright and capable but not used to thinking like a mathematician, was often confused by the beginning of each section. Only later she learned just to skim it and maybe go back later when she saw the point the authors were leading toward. Her questions to our tutor were generally along the lines of, “This is all very well, but why? I can figure the divergence of this vector field, but what does it do for me?” In the context of the course, the answer was that these are interesting mathematical objects about which much could be learned. More satisfying was the fact that our tutor was able to come up with examples from mathematical physics in which the divergence gives very useful information indeed. So far he’s been able to produce similar justification for many strange objects that have popped up in mathematics.
The second approach is inductive. This textbook starts with a number of examples, suggesting that there might be similarities or systematic behavior, and tries to lead the student into stating a general law. This is much closer to the way research is actually done, that is, the way people come up with mathematics that wasn’t known before (it also applies to science, of course). In principle this is a very powerful method. Not only does the student learn some math, but she also learns how to learn something new in almost any kind of situation. “Discovery learning” can be highly motivating, and is as far away from the tedium of rote learning as it’s possible to be. In practice, there are serious drawbacks. Not all students are good at mathematical or scientific investigation. Even those who are, sometimes cannot see the pattern or come up with the basic law. This approach avoids tedium, but often replaces it with simple incomprehension.
We’ve commented on the different types of textbook before. They are not easy to write. The connection between the authors, who know their material thoroughly, and the students, who by definition do not, can be very difficult to make. And of course everyone learns differently. But we think that a combination of deduction and induction would work better than a concentration on either alone.
