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The Tools of Mathematical Reasoning

Tamara J. Lakins
American Mathematical Society
Publication Date: 
Number of Pages: 
Pure and Applied Undergraduate Texts
[Reviewed by
Michele Intermont
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Newly arrived to the world of print is Tamara Lakins’ The Tools of Mathematical Reasoning. This text is intended for the sophomore-level student who is first learning to write proofs. Books of this sort come in two basic flavors: those meant to complement another source, such as the book I reviewed here several years ago, and those meant as the primary source for a course. This is of the primary textbook variety.

After an introductory chapter on logic to introduce truth tables, the next two chapters provide exposition and examples of methods of proof. By the fourth chapter, Lakins is organizing the material around mathematical topics rather than methods of proof. The common topics appear: sets, functions, equivalence relations, a bit of number theory. In addition, the book ends with a very brief aside into real analysis.

The text is clearly written and the sections are short enough to be readily digestible. The author is fond of the so-called “Given-Goal diagrams” (introduced by Daniel Velleman in his proof-writing book), often rewriting them once or twice before beginning the polished proof. This translation of assumptions given in words to statements given in symbols is an important idea that almost never appears obvious to those learning to prove mathematical statements. Thus, making this process explicit is valuable. Also valuable is Lakins’ inclusion of “scratchwork” before the presentation of some of the polished proofs. With this, students have the opportunity to see the construction process play out.

Lakins’ book is a nice, solid exposition of the tools students need as they transition to abstract mathematical thinking. In addition to the features mentioned above, the text includes many exercises and a set of guidelines for writing mathematics adapted from Annalisa Crannell’s excellent guidelines (available on Crannell’s website). However, I do struggle to articulate what distinguishes it from other texts of this nature, such as Velleman’s text or the several others on my bookshelf. 


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Michele Intermont is Associate Professor of Mathematics at Kalamazoo College.

See the table of contents in the publisher's webpage.