Mathematics for High School Teachers: An Advanced Perspective

Rating:
Summary: Sui Generis
Review: This book is one of a kind. It affords an integrated perspective of traditional high school mathematics, making explicit the intimate relationships between arithmetic, algebra, and geometry. Additionally, it indicates and suggests lines of development that are pursued in undergraduate courses. Both purposes - showing the unity of the subject, and indicating further development - are accomplished by placing traditional high school topics in a broader conceptual and historical perspective.

The book is divided into two parts; the first, titled "Algebra and Analysis with Connections to Geometry", deals with numbers, functions, equations, polynomials, and number systems. The second, titled "Geometry with Connections to Algebra and Analysis", deals with congurence, symmetry, similarity, area annd volume, axiomatics, and trigonometry.

To give some idea of coverage, the second chapter (on real and complex numbers) discusses irrational numbers, a proof of the irrationality of e, the nested intervals property of the reals, countable and uncountable sets, and the diagonal proof of the uncountability of the reals. The chapter on equations briefly discusses cubic and quartic equations and states the unsolvability of the general quintic; the names of Gauss, Ruffini and Galois are mentioned. The chapter on integers and polynomials discusses induction, recursive definitions, simple diophantine equations and the fundamental theorem of arithmetic. It also indicates the analogies between the integers and the set of polynomials (both are integral domains). The chapter on number system structures discusses modular arithmetic, the Chinese remainder theorem, and gives examples of number fields other than the real and complex number systems (e.g. quadratic fields, and finite fields).

The projects at the end of each chapter extend the material covered in a natural way, and are challenging. To give some stray examples, the coordinatisation of the Riemann sphere, the Cardano-Tartaglia method for solving cubic equations, Fermat's last theorem for n = 4, constructible numbers, and the impossibility of squaring the circle and doubling the cube.

The chapter bibliographies are annotated, up-to-date, and list excellent books for further study.

I have a few criticisms. The first is that surjective functions are not discussed, and in this connection the Schroder-Bernstein theorem does not get mentioned or proved. A second and more serious criticism is the slender coverage of analytic geometry. Only five or six pages are devoted to this. As a consequence, the authors cannot discuss the rich field of algebraic curves in particular, and algebraic geometry in general. There is also no mention of projective transformations (i.e. projective geometry) or continuous transformations (i.e. topology). Finally, there is no mention of Klein's Erlanger program.

These quibbles aside, the book is well-conceived and well-written. It can join Courant and Robbins' "What is Mathematics", and Stillwell's "Mathematics and its History" as a book that gives a bird's eye perspective of (part of) the discipline.

Professors teaching undergrad courses would want this book on their shelves; it shows some of the connections between high school material and the relatively abstract courses taught at college (e.g. Galois theory, group theory, algebraic number theory, and real and complex analysis). Undergrad students might want this book for the same reasons. High school teachers who want a bird's eye perspective of high school mathematics from a sophisticated point of view might also want a copy; suggested lines of development can be used as enrichment topics.