How a popular math book taught me some things about teaching.
I’m reading David Foster Wallace’s Everything and More: A Compact History of Infinity.
It’s a well-written and enjoyable history of the infinity from the Greeks to Cantor’s set theory. He doesn’t hide the math, so you need a bit of a background of math, at least Calculus and probably a first course in Analysis. Also, he has a distinctive1 writing style, which I very much enjoy, but it takes some devotion to keep up with it. Once I finish, I hope to post a more complete review. But from a teaching standpoint, he’s reminded me of three things:
(1) Have One Theme and Point it Out: Early in the book, Wallace establishes the basic idea from Zeno’s and Aristotle’s writings that he carries through the rest of the book. In each part of the book, representing a different time of history, he relates the issues of the time back to the original ideas. It’s not just that he relates it, but that he relates it by recognizing that we the readers will be seeing how the arguments he constructs and the problems he faces are very similar to ones he presented earlier. He both respects the intelligence of the reader, and includes the reader into the various issues of the times.
(2) Be an Enthusiast: Wallace is not a mathematician, but he is a fan. He honestly enjoys mathematics, and he can’t help to share his enjoyment through his writing. His enthusiasm is contagious. Also, he obviously has more to say, but is able to keep and progress the main parts of his arguments in the primary part of the text, relegating the extra bits to footnotes (labeled IYI = if you’re interested) and other asides. This provides for a rich (or as rich as you want) exploration of the topic.
(3) Have a Story with Interesting Characters: The book is a story of an idea, but it is also much about the people struggling with that idea. Those people, however, are not the only (nor the most interesting) characters. Early on he introduces Mr. G, his high school math teacher. Wallace uses Mr. G, and the other characters, to make the writing more personal and to provide an opening for some non-standard mathematics and off-topic references.
Altogether these make the book (and teaching) more engaging as it keeps the intent clear, and respects and provides opportunities for the students.
1If you are aware of his style, you will recognize this as a major understatement2.
2And you’ll recognize these footnotes as a (pale) tribute to his style.