Math Teaching and Learning at UBC

I'm going to venture out on a limb here with this rather sweeping generalization: most first year university/college math textbooks are frustratingly difficult to study from. Let's call this a hunch at this point, based on decades of trying to help students learn this material, using dozens of different textbooks, and a few days of a close, critical reading of a number of texts. I further suspect that the situation may be the same for other first year technical subjects.

My explanation/current working hypothesis is this: these books are not a natural genre. They attempt to introduce mathematical ideas using an "adapted" version of idiomatic mathematical language. This language has evolved as part of mathematical culture, and thus has many complex cultural aspects associated with it, including aesthetic values. We all have papers and books which we can point to and say "this is beautiful mathematical exposition". With a very small number of exceptions, there are no first year Calculus texts on these lists of favourite books. Possible exceptions: Spivak, Courant, Apostol, Hardy, Zorich. I'd love to hear from you if you have other examples to add to this list.

I've not had a close look at the classics recently, and I should say I've only flipped through a few preview pages of Zorich online. I optimistically suspect that this latter may be interesting as an example of a somewhat larger class of good books that were first published in languages other than English.

I'm not suggesting that these be widely adopted. In my estimation, the current first year students would not have an easier time with these books, though their difficulties would be different. So, these are non-starters for institutional reasons, and because of the level of readiness of the students we get. I think we need to take at least part of the blame for both of these factors. We have failed to prepare any but the best of our incoming students to read the best introductions to Calculus.

In eduspeak, I'd refer to these texts as great models for product. If a student came to me and asked, "what should great Calculus look like?", these are the books I'd suggest as examples.

So, we're in a situation where we have to abandon the authentic product because it's too difficult for our learners. I'm willing to accept this, if I think about what we teach instead as being an intermediate stage, a bridge to the final product. Unfortunately, most of the other texts we might choose would not help to prepare the student who succeeds in studying their content to now engage with any of the truly worthwhile texts. This is an unfortunate state of affairs.

Let's assume for a moment that the content of all first year Calculus texts is comparable: the definitions, theorems, types of examples, problems, and so on don't vary all that much. Then the remaining pedagogical axis to investigate is process. How are the students to learn, and how will this shape them in their practice of mathematics?

Unfortunately, the few texts that I've studied closely don't seem to explicitly develop process, either in terms of the structure of the examples/worked problems, or in terms of larger processes such as problem solving, logical argumentation and analysis, abstraction, modelling - i.e., reading and writing mathematics, engaging in mathematical dialogue, and all of the other aspects of doing mathematics.

No wonder students frequently comment "I don't know how to do this!". I've answered this in the past by saying "Did you look at the example in the book?" I realize now that this isn't helpful. They're not asking "What should my solutions look like?" (though some of them should ;) - they're asking "By what process can I arrive at an understanding of the problem and the steps of the solution?".

Here's what I think may be going on. The cultural values pertaining to written mathematics don't commonly encourage narrative personal explanation, or a description of the (often roundabout) process of exploration or discovery. So, in abandoning the "model" product in our textbooks, we haven't gone far enough, to take the extra step and embrace on the written page (for certain textbooks) some of the same modalities that explicate process in our spoken culture.

In summary, textbooks fall short of their potential usefulness because they are models of neither product nor process. They are shackled by an aesthetic which, I would argue, is not appropriate to the constrained, somewhat artificial genre that they are situated in.