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Distilling Ideas: An Introduction to Mathematical Thinking

Brian P. Katz and Michael Starbird
Mathematical Association of America
Publication Date: 
Number of Pages: 
Mathematics Through Inquiry
[Reviewed by
Annie Selden
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The authors of this book, as they explain in their first chapter, are taking readers/students on a journey — one that will “add new, powerful inquiry skills” to a student’s repertoire: abstraction, exploration, conjecture, justification, application, and extension. The book is clearly intended for inquiry-based mathematics courses. It is replete with explorations to undertake, definitions to explore, and theorems to prove, but there are no sample proofs or answers.

The book has five chapters, with the middle three chapters on graphs, groups, and \(\varepsilon\)-\(\delta\) calculus constituting the bulk of the book. To me, it is clear that any one of these middle chapters by itself has the potential to be used for an entire one-semester course. There is really enough material in the \(\varepsilon\)-\(\delta\) calculus chapter for an introductory junior level real analysis course. Similarly, the group theory chapter gets to Sylow’s Theorem and has more in it than most students could cover in one-semester when doing all the work themselves. Perhaps the same is true of the graph theory chapter, but I cannot personally vouch for that. While a teacher can “choose to select only some of the exercises and theorems in a given unit [chapter]”, the book would be slow going for students who are expected to come up with all their own examples, solutions, and proofs. However, as has been said of such courses, in the end, “less is more.” The students gain self-confidence, persistence, and skills they rarely learn in lecture-based courses.

The back of the book contains an annotated index (a.k.a., glossary) that “rather than containing precise definitions … gives reminders of the terms and links to [page numbers for] their precise definitions”. For example, one such glossary entry states, “An automorphism is an isomorphism of an object with itself, possibly in a non-trivial fashion.” There is also a four-page list of symbols, and their meanings, that includes: for sets, symbols like those for union, intersection, and ordered pairs; for groups, symbols like those for coset, symmetric group, and stabilizer; and for calculus, symbols like those for sup, inf, derivative, and integral.

The chapters on graphs and \(\varepsilon\)-\(\delta\) calculus are introduced using fictionalized characters and scenes. For example, the graph theory chapter begins, “One day, Königsberg resident Friedrich ran into his friend Otto at the local Sternbuck’s coffee shop. Otto bet Friedrich a Venti Raspberry Mocha Cappucino that Friedrich could not leave the café, walk over all seven bridges without crossing over the same bridge twice (without swimming or flying) and return to the café. Friedrich set out, but he never returned.” The characters in the \(\varepsilon\)-\(\delta\) calculus chapter are Zeno, Isaac (Newton) and Gottfried (Leibniz) with Zeno being an archer at the Summer Olympics and Isaac and Gottfried being two referees charged with deciding whether Zeno’s arrow actually struck the bullseye. In contrast, the chapter on groups has a more no-nonsense tone. As a result, one gets the impression that the chapters on graphs and the \(\varepsilon\)-\(\delta\) calculus were written by one of the two authors, whereas the chapter on groups was written by the other.

Who might this book be for? Although I often teach inquiry-based courses, I prefer to write my own notes for them, and I think many others who already teach such courses would similarly prefer to write their own notes. However, some teachers may not have the time to, or may not feel equipped to, write their own notes, and perhaps that is where this book could play a role. And although the authors claim the book could be used for individual study, I find it hard to believe that any student would come to write proofs in the style of mathematicians, without a single sample proof included. I think a lot of the effectiveness of such a textbook/course depends on its implementation by a perceptive, knowledgeable teacher who can steer classroom discussions in helpful, and correct, mathematical directions. In my opinion, this is definitely not a self-help book for individual students.

This small paperback comes with a high price tag — $45 for just 171 pages, at the MAA member price, and $54 for everyone else. It is in the MAA Textbook Series, and according to the Preface, is part of a “collection of resources” called, Through Inquiry, which is “intended to provide instructors with the flexibility to create textbooks supporting a whole range of different courses to fit a variety of instructional needs. To date, the series has four e-units [chapters] treating graphs, groups, \(\varepsilon\)-\(\delta\) calculus, and number theory”. While these four units [chapters] are briefly foreshadowed in the Preface, the unit [chapter] on number theory is nowhere to be found, nor is it listed in the Table of Contents. The Preface also states that “Sample, specific threads are described in the Instructor’s Resource”; however, that instructor’s resource is also nowhere in the book. If it exists, there should be an indication of where. Such anomalies could be fixed in a second printing.

Annie Selden is Adjunct Professor of Mathematics at New Mexico State University and Professor Emerita of Mathematics from Tennessee Technological University. She regularly teaches graduate courses in mathematics and mathematics education. In 2002, she was recipient of the Association for Women in Mathematics 12th Annual Louise Hay Award for Contributions to Mathematics Education. In 2003, she was elected a Fellow of the American Association for the Advancement of Science. She remains active in mathematics education research and curriculum development. 

1. Introduction

2. Graphs

3. Groups

4. Calculus

5. Conclusion

Annotated Index

List of Symbols

About the Authors