A TeXas-Style Introduction to Proof

This slim paperback book is intended for a one-semester IBL-style transition-to-proof course, and no doubt it could be used that way. I, however, have some reservations about it. The authors are, respectively, a 2018 Haimo Award winner and a 2015 Alder Award recipient, so are thus considered to be very good teachers. Still, perhaps there is something amiss in the translation of their teaching talents to the printed page. As a teacher of transition-to-proof courses out of way too many textbooks and my own notes, I find some things distracting. For example, I think there are way too many footnotes (91 in all, with as many as 6 on one page) and way too many interesting asides. Thinking back to my long-ago student days, I think I would have wanted to get to the “heart of the matter” and not have to keep looking down at the bottom of the page to see whether a particular footnote was relevant or not. I would also not have wanted so many quotations inserted in the text, which I would have to read over and decide whether they were relevant. There are quotes from such luminaries as Paul Halmos, Mark Twain, and Georg Cantor, along with at least one Chinese proverb.

As my husband, also a mathematician who also looked over the text, said, “R. L. Moore would not have recognized it as a Texas-style course,” meaning a traditional Moore Method course, which we have each taken and taught. In addition, the play on words in the title will pass right over the students’ heads, and only mathematics professors looking for a potential textbook are likely to get the double-entendre. This sort of humor, together with asides, while no doubt well intended, is found throughout the book. I think many students will find this distracting, as I did. A textbook should be for students to learn from, and distractions in textbooks, just like distractions in proofs themselves, are to be avoided. This textbook also contains lots of well-intentioned advice. For example, in Section 2.1 on Variable Names, after about a half page “word of caution” on variables and the introduction of the idea of “fixed but arbitrary,” the text continues, “You will see this in action as you go along.” This is followed by a short paragraph alluding to Shakespeare’s Romeo and Juliet and the immateriality of who plays the roles of Romeo and Juliet.

The topics covered in the first five chapters — logic, proof methods, mathematical induction, sets, functions and relations — are standard at the beginning of many transition-to-proof course textbooks. The remaining chapters are where most such textbooks begin to differ. In this case, the final two chapters are on counting and axiomatics. There follow three appendices on mathematical writing, style, and LaTeX. The final 14-page appendix on LaTeX is just about all the help students will get with formatting proofs for submission in LaTeX. I agree that reading hand-written proofs can be a painful experience for the teacher, and it seems like a good idea to have students submit their proofs using LaTeX, so that they will at least have to read them over and check them before submission. Yet I wonder whether learning LaTeX will put an extra burden on students, especially preservice secondary teachers. The authors clearly think not and give students a bit of advice in Section 0.4. From what they write here, they are clearly not expecting students to have had any previous experience with LaTeX. Indeed, they are at pains to point out that LaTeX is not a word processor, and as a result, it is not self-contained. That is, LaTeX does not produce “WYSIWYG”, with a footnote that explains “What You See Is What You Get”.

Despite my reservations concerning this textbook, I think that “the proof of the pudding is in the eating,” so the number of satisfied, adopting teachers will be the ultimate judge of its success as an IBL-text for an introduction-to-proof course.

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.