Extending the Frontiers of Mathematics: Inquiries into Proof and Augmentation

by Edward B. Burger

In the real world of research mathematics, mathematicians do not know in advance if their assertions are true or false. Extending the Frontiers of Inquiries into proof and argumentation requires students to develop a mature process that will serve them throughout their professional careers, either inside or outside of mathematics. Its inquiry-based approach to the foundations of mathematics promotes exploring proofs and other advanced mathematical ideas through these

- Puzzles and patterns introduce the pedagogy. These precursors to proofs generate creativity and imagination that the author builds on later

- Prove and extend or disprove and salvage, a consistent format of the text, provides a framework for approaching problems and creating mathematical proofs

- Mathematical challenges are presented which build upon each other, motivate analytical skills, and foster interesting discussion

Paperback

First published January 28, 2006

Reviews

[Charles · Rating 3 out of 5]

Good problems to consider, not much in the way of proof instruction

The second section of the title is a bit of a misnomer in the sense that the inquiry into proof is not all that deep. Each chapter opens with a short section of background material, enough to establish the context of the problem coverage but not enough to bring someone of limited background up to speed. This is followed by a set of theorems/problems that the reader is to prove/solve. If subsequent problems in the chapter are based on other material, it is also briefly introduced.
For the theorems, the instructions to the reader are “Prove and extend or disprove and salvage.” In other words, the reader is not given the implicit and generally significant hint that a theorem is true when it is stated. Furthermore, the words “extend” and “salvage” are nebulous terms with ill-defined mathematical definitions. There are hints to the proofs/solutions in an ending chapter, but they are brief and somewhat indeterminate.
The coverage is also very broad for a book that is this short. There are 19 chapters containing generally distinct material in only 133 pages. Some of the topics are limits, continuity, group theory, paper folding, graph theory, probability, counting principles and modular arithmetic. Clearly, with so many topics in so few pages, if there is depth it must be provided by the reader.
If you are looking for problems that generate exposure to fundamental aspects of mathematics when they are solved, this book will have some value. However, if your search is for assistance in understanding how to do proofs, it will be of little help.

[Teddy · Rating 4 out of 5]

Text for Proofs. Going to kill me before the semester's out.