Algebra: Abstract and Concrete

Frederick M. Goodman

Pedagogical Perspective:

This text is an introduction to modern algebra for undergraduate students, published by Prentice Hall in August, 1997. The book addresses the expected topics -- groups, rings, and fields -- with symmetry as a unifying theme. Presenting these topics is no doubt important, as the subject matter is central and ubiquitous in modern mathematics.

However, the more important goal of this book is to introduce students to the active practice of mathematics and to draw them away from the view of mathematics as a system of rules and procedures. Students are asked to participate and investigate, starting on the first page.

The first part of this text is suitable for beginners. Any course on abstract algebra for beginners faces an enormous pedagogical challenge: to bring students who have been raised on procedural mathematics to start thinking about mathematics like mathematicians. It has to be expected that students will have great difficulty in making this transition, just as a person untrained in art has great difficulty to begin to see and draw like an artist. The author is convinced that the apprentice artist had better learn to draw by drawing, and the apprentice mathematician had better learn to do mathematics by starting, immediately, to do mathematics. This thought guides especially the crucial introductory sections of the text, which consist of a concrete exploration of the symmetries of simple geometric figures. The concreteness of this investigation is a great challenge to students, who have learned to prefer an abstract and formal procedure to a concrete problem!