Al Sethuraman has solved an important problem encountered by each of us whenever we’re faced with the prospect of teaching serious mathematics to future high school teachers: what do you teach them, given the initial and boundary conditions that almost always apply?
Sethuraman’s solution is algebraic, so to speak, and consists in driving at a leisurely but non-trivial pace toward the destination of doing justice to a beautiful and meaningful problem, very famous at that, getting out along the way at various rest-stops to explore some of the by-ways and lovely vistas. The final focus of Rings, Fields, and Vector Spaces is disclosed by its subtitle: “an introduction to abstract algebra via geometric constructability,” and, to be sure, Sethuraman achieves his objective with gusto: he starts with a thorough treatment of divisibility in Z, goes on to develop a good deal of nice algebra, and finishes with a discussion of the fact that an angle of 60º cannot be trisected with a compass and a straightedge.
At the end of the (sixteen-week?) trip the class of future high school teachers will have seen some very pretty (and pretty deep) mathematics, learned how to view some of the hidden background of the fruitier stuff of high school algebra vom höheren Standpunkte aus (gratia Felix Klein), and, presumably, become able to answer most of the corresponding questions by the smart-aleck kid in the front row. So, Rings, Fields, and Vector Spaces is indeed tailor-made for a “topics in algebra” course aimed at future teachers, replete with a component on how to learn mathematics (à la Sethuraman, of course, but I don’t think there are any pedagogical heresies here — see e.g. p. 6: “… you observe patterns…, you ask yourself questions, and you try to answer these questions on your own. In the process you discover most of the mathematics yourselves…” Fair enough.)
Of course, Sethuraman’s perspective on this question of how to go about teaching teachers, while surely the most popular one, is not the only one. Another viable approach consists in going for more breadth, if you will, which is to say to approach the subject more kaleidoscopically, introducing, say, a half-dozen themes (e.g. groups, rings, fields, vector spaces, categories, special topics), present some interesting material under each heading, and quickly explicate something deep but relatively accessible. Recently I did just that, albeit at the master’s degree level, and even managed to weave the common thread of Emmy Noether’s first isomorphism theorem through the first five themes mentioned above. This (qualified) success vindicated an approach which was in part motivated by the unfortunate heterogeneity of my audience as far as their earlier training was concerned, which appears to be a universal contemporary problem these days. Oh, well…
Still, most of us who are called to introduce future teachers to mathematics properly so-called would invariably opt for Al Sethuraman’s approach, and Rings, Fields, and Vector Spaces serves the cause very well indeed.
Michael Berg is Professor of Mathematics at Loyola Marymount University in California.