Learning and re-learning

Textbooks are to learn from, of course.  But who is doing the learning, and when?

One of our consultants, who is tutoring High School students evenings and on weekends, brings us reports of secondary school as it is done these days.  (Our own experience is long past and is getting foggy in our memories.)  The subject matter, at least in mathematics and science, seems to be very much the same as we remember.  Textbooks have changed somewhat.  They’re all large and hardcover, which seems unnecessary: paperbacks would last the term or the year through, and students hardly need them longer than that.  There are many more illustrations.  Most have links to on-line material.  All (that we’ve seen) have computer-generated cartoon people in them, mostly children of the age of the target students.  We’ve not seen these serve any purpose in the exposition; maybe their only role is to convince the students that people actually do mathematics or science.

Writing this kind of textbook must be a tremendous undertaking, gathering lots of material and presenting it systematically–to say nothing of generating thousands of problems for the students to solve.  And that doesn’t even touch on the main difficulty: how to present the subject so that students actually learn.  A logically clear and mathematically airtight exposition, so long the ideal of textbook writers (from Euclid’s Elements up through at least the middle of the last century), may be useless for actually getting in students’ heads and staying there.  No doubt the people who write these things have some idea of how to go about it.

This brought to mind our own experience with teaching and with textbooks, at the undergraduate and graduate level.  Pulling a few of them out, we sorted them into different kinds.  These are our names for the categories:

Students’ textbooks are the best for students to learn from.  That is, the explanations and presentation are easiest for someone who had never seen the material before.  These are probably the hardest to write, and require a deep insight into students as well as the subject.

Aspirational textbooks have an approach and a level of treatment that all teachers (and most others who deal with the subject) admire.  Unfortunately, they are too difficult for students who haven’t seen the material before, and too elementary for those who have.  Most teachers have them on their shelves, though, and leaf through them now and then to find a useful approach for a particular lecture, or another way to think about things.  Lectures on Physics (Feynman, Leighton & Sands) is in this category.

Reference textbooks are all but impossible to learn from, even for very bright students.  However, they cover all aspects of the subject in great detail, so that once you’ve mastered it you can go to these books to fill in any gaps.  They’re particularly useful because, where other books only mention “it can be shown that. . .”, they show it.

See-also textbooks are generally undistinguished, but survive because they have one or a few sections that are uniquely clear or complete in a given area.  You wouldn’t actually buy one of them, because all textbooks are expensive, but you would like to have access to one now and then

We won’t actually mention rite-of-passage textbooks, whose only function is to separate out the weak and unfit, as a strictly graduate-school phenomenon.