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Journey Into Mathematics: An Introduction to Proofs
Joseph J. Rotman
Publisher: Dover Publications (2007)
Details: 237 pages, Paperback
Topics: Proofs and Logic, Transition to Advanced Mathematics
This book is in the MAA's basic library list.
MAA Review[Reviewed by Brian Rogers, on 04/10/2007]
With the current surge of interest in undergraduate proof production, bridge courses, and how best to teach undergraduates to perform proof, the re-issue of Rotman's Journey is certainly timely, since Rotman intended this book to be a text for just such courses and uses. Besides that, as I read Journey, I kept finding things that seemed to reflect current research findings on undergraduate proof schemes, methods of teaching that are effective, and even embodied cogntive research that has only been published in the last year. Impressive for a book originally copyrighted in 1998.
Unlike the other books I have seen which are intended for bridge courses, Journey begins with proofs: proofs for students to read and proofs for them to do. Don't get me wrong; the material on propositional calculus and first-order logic is there; it just is not the first thing introduced. I believe this to be salutary: too often, I think we give students tools before they need them. It's like pneumatic nail guns. Certainly they are essential tools for any professional rough carpenter or even finish carpenter, but the average householder can get by fine with a 16-oz hammer — and be a lot safer, too.
Rotman starts with mathematics the average college sophomore math major already knows, so the only new thing to learn is how to do the proofs that establish and explain such knowledge. Tools are introduced sparingly, and only as students encounter proofs that need them. By the time the book reaches chapter 4, however, the basic concepts of analysis have been explored, as well as those of algebra. Both, by the way, are explored along the lines of the historical development of the concept, so the student is allowed to first learn the less-sophisticated answers upon which today's answers were built.
I've been looking for a textbook or supplement to use in classes and workshops I'm currently designing in a project related to my dissertation, and I believe I've found a top-tier candidate. I recommend this as a textbook or supplemental textbook for such classes, especially if you're less than satisfied with current approaches to teaching undergraduate proof-production. I also suggest it for all of those who are curious about other possible approaches to teaching such classes. Rotman writes in a clear, informal style that will be well within the reach of students, and they will be able to learn some on their own from Journey. Combined with effective teaching, this could be a winning combination.
Brian Rogers is a doctoral student in Educational Mathematics at the University of Northern Colorado, Greeley, Colorado. After nearly a half century of teaching English, programming computers, and teaching others to program computers, he feels he is finally where he belongs. Kind comments or questions may be directed to Brian.Rogers@unco.edu; it has not been decided what to do with other types of communication, so if this applies to you, please forget the e-mail address above.
BLL** — The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.