Paul Zorn on Communicating Mathematics V: Rule Books and Tour Guides: Two Live Questions

Posted on October 15, 2013 by Ivars Peterson

Paul Zorn, St. Olaf College

Part I: Introduction

Part II: Mathematics in Literature

Part III: What Is Mathematics?

Part IV: Mathematical Language: Learning from Barbie

Part V: Rule Books and Tour Guides: Two Live Questions

Next, I’d like to acknowledge two general questions that apply broadly in expository writing, but face textbook authors with particular force. First is what I call the rule book question.

How would you help your kid excel at, say, soccer? Hand her a World Cup rule book to study? Or roll a ball out the back door and get out of the way? No sane parent chooses the first plan. But—if Susie shows interest and aptitude—sooner or later the rule book will come into play. The live question is when, and the answer depends on Susie’s talent, interest, and sporting aspirations.

An analogous rule book question applies to mathematical pedagogy: Where and when do theoretical analysis and rigorous argument belong in students’ encounter(s) with, say, single-variable calculus?

The answers depend sensitively, of course, on the students in question, but the rule book question is a live one for almost any student audience. Textbooks I studied in my mathematical youth usually took an “early rule book” approach to their chosen audiences. I think now that students who benefit from this approach will mainly be (in Lake Wobegon parlance) “well above average’’ by the standard of typical beginning college calculus classes. (I was not among these few.)

The needed sophistication for an early rule book approach to calculus involves considerable facility with the algebra of rational functions, basic symbolics of elementary calculus, and—perhaps more important—enough mathematical maturity and linguistic facility to parse and think effectively about such concepts as uniform Lipschitz continuity of a family of functions.

Most students, I now think, will benefit more from an expository approach that blends the rule book with the mathematical analogue of starting by throwing the football out the back door and seeing what happens. With modern mathematical computing, it’s easier than before to make this sort of free-form investigation possible.

A second question that’s especially key to textbook exposition could be called the tour guide question. Like tour guides in, say, Florence, textbook expositors need to make difficult choices of proper subsets of attractive alternatives. You simply can’t see, or say, it all.

The tour guide question arises in milder forms in every genre: Any exposition can smother under its own weight. But tour guide choices may matter most in textbooks, which have to “map” in some sense to terms or semesters of inflexible lengths, and sometime need to anticipate future courses of study.

I’ve found this authorial question especially vexing for textbooks in beginning multivariate and vector calculus. Such courses generally aim for one or more vector variants of the fundamental theorem of calculus: Green’s theorem, the divergence theorem, and Stokes’s theorem in several variables. But getting there in the time available always involves short-shrifting, or ignoring entirely, a lot of attractive alternative destinations: Lagrange multipliers, higher-order linear approximation, differential forms, and important theorems of mathematical physics.

Textbook authors (me, too) often dodge the tour guide question, aiming for the touristic equivalent of “covering” Rome and Florence in a day. But tackling too much leaves tourists, and students, sweaty, dazed, and confused about what any of it means.

Part VI: Lessons from History
Part VII: Valuing Communication