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How to Think Like a Mathematician: A Companion to Undergraduate Mathematics

Kevin Houston
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Michele Intermont
, on

I wish I had seen this book a few months ago! I would have enthusiastically used it as assigned reading in my introduction to proof course. As it is, it will become required reading for those students doing undergraduate theses under my direction.

Houston has written a book with a very conversational tone aimed at helping students navigate the transition from writing down calculus homework to writing up formal proofs. And that’s not all. In addition to focusing plenty of attention on the mechanics and style of writing, he also probes the questions one begins to ask, the reasoning one begins to employ, in this evolution.

The subtitle of the book is “A Companion to Undergraduate Mathematics” and that is how I best envision using this text. Houston employs mathematical content to provide illustrations of his points and exercises, but he does not concentrate on explaining subject matter. While the fifth chapter of the book contains “mathematics that all good mathematicians need”, the other five chapters build on material which will be familiar to most students (with all necessary definitions and theorems included), and focus instead on what a mathematical statement is and how to parse one, how to read definitions and theorems, outlining standard methods of proof, etc. Of course one needs mathematics to work with in order to accomplish this, but by choosing not to write a text with a particular mathematical content goal, I think Houston succeeds in making the book user-friendly. It will be a nice supplement for a class which already has specific mathematical content. Because the mathematics will generally not be new to them, students can concentrate on the connective tissue of the mathematical thought process which is exactly the author’s hope.

One of the reasons I am so taken with this book is that the author really does call attention to the writing of mathematics and not just the logic of constructing proofs. For example, he talks about the role of punctuation, the fact that mathematical style tends to be terse. He mentions good practice and, when necessary, where that differs from standard practice. I think one of his underlying beliefs is that by making explicit what the conventions of mathematical writing are and why they are that way, students can more easily become better readers of mathematics and grasp content slightly more easily. At least, that’s what resonates with me.

In sum, How to Think Like a Mathematician is a wonderful resource. Houston claims that “the aim of this book is to divulge the secrets of how a mathematician actually thinks” and I think he does just this.

Michele Intermont is an associate professor of mathematics at Kalamazoo College in Kalamazoo, MI.


0. Preface; Part I. Study Skills For Mathematicians: 1. Sets and functions; 2. Reading mathematics; 3. Writing mathematics I; 4. Writing mathematics II; 5. How to solve problems; Part II. How To Think Logically: 6. Making a statement; 7. Implications; 8. Finer points concerning implications; 9. Converse and equivalence; 10. Quantifiers – For all and There exists; 11. Complexity and negation of quantifiers; 12. Examples and counterexamples; 13. Summary of logic; Part III. Definitions, Theorems and Proofs: 14. Definitions, theorems and proofs; 15. How to read a definition; 16. How to read a theorem; 17. Proof; 18. How to read a proof; 19. A study of Pythagoras’ Theorem; Part IV. Techniques of Proof: 20. Techniques of proof I: direct method; 21. Some common mistakes; 22. Techniques of proof II: proof by cases; 23. Techniques of proof III: Contradiction; 24. Techniques of proof IV: Induction; 25. More sophisticated induction techniques; 26. Techniques of proof V: contrapositive method; Part V. Mathematics That All Good Mathematicians Need: 27. Divisors; 28. The Euclidean Algorithm; 29. Modular arithmetic; 30. Injective, surjective, bijective – and a bit about infinity; 31. Equivalence relations; Part VI. Closing Remarks: 32. Putting it all together; 33. Generalization and specialization; 34. True understanding; 35. The biggest secret; Appendices: A. Greek alphabet; B. Commonly used symbols and notation; C. How to prove that …; Index.