A colleague at a US university once quipped to me, “Before we let anybody study math at the university, we first make them study engineering for two years.” Indeed, the calculus, differential equations, and basic linear algebra courses are deeply different in culture from the upper-level analysis, group and ring theory, geometry, and various other courses within the math major program proper.

An Introduction to Proof course is intended to bridge this gap. At most institutions, there is a standard core of material: naive set theory, Boolean connectives, truth tables, quantifiers, induction, injection/surjection/bijection, and equivalence structures. In addition, there are significant areas of flexibility: shall the examples come from elementary number theory, combinatorics, graph theory, real analysis, or something else? Most fundamentally, though, the purpose of the course is to get students accustomed to reading, thinking about, and writing proofs (and, to a lesser extent, definitions and theorems).

There is no shortage of textbooks for such a course, and they are deeply varied. Indeed, my own department has used no fewer than four different texts in recent years, depending on the taste of the instructor. Many are similar, but there has been a remarkable degree of innovation in this space in recent years.

The classic format for such a book starts with a chapter or two on sets and logic, and then takes on the rest of the subjects in some order, giving a somewhat terse explanation of each point, followed by an appropriate selection of exercises. I happen to have at hand Chartrand, Polimeni, and Zhang, $\boldsymbol {Mathematical \ Proofs}, \ 2^{nd}$ edition, Pearson, 2008, but there are many others. This method approaches the core cultural adjustment problem of the course by making the textbook look a good deal like what students have seen before.

A second family of texts for the course, exemplified by Burger’s $\boldsymbol {Extending \ the \ Frontiers \ of} \boldsymbol {Mathematics}$ , Key College Publishing, 2007 and Taylor and Rault’s $\boldsymbol {A \ TeXas}\textbf{-}\boldsymbol {Style} \boldsymbol {Introduction\ to \ Proof}$ , MAA Press, 2017, leans heavily into the cognitive dissonance of the course by posing the most minimal definitions and structuring the entire book as a carefully sequenced collection of problems that take students on a fight-for-every-step tour of the material. They tend to use terms like “Texas Style” or “Inquiry.” I have typically used one of these texts for the course.

The book under review provides a fresh new option for approaching the problem of cultural transition in the introduction to proofs class. It is a highly readable book that starts with the familiar and explains why the additional tools are needed. This struck me immediately when the first section of the first chapter covered neither sets nor logic, but the quadratic formula—complete with a proof.

It is no small advantage to a book in this category that a student who enjoyed earlier math courses could pick up Fuchs’s book and immediately understand not only what they’re reading but why they’re reading it. The exposition is lucid and transitions are smooth.

The text has other interesting pedagogical choices, as well, which seem to align well with the overarching conceit that students’ prior knowledge matters (see Dochy, Segers, and Buehl, The Relation between Assessment Practices and Outcomes of Studies: The Case of Research on Prior Knowledge, $\boldsymbol {Review \ of \ Educational \ Research}$ , vol. 69 (1999), pp. 145–186). Functions are treated before, and separately from, relations. As a logical matter, of course, a function is a special kind of relation, but psychologically most students come to the course with an idea of a function that takes some work to reconcile with the formal set-theoretic definition.

The early treatment of functions allows them to come before induction, and allows a rather full world of math to be in play by the time induction is introduced. In my experience, this is the section that students find the most technically demanding, so I was glad to see that this section retained the high motivation and readability of the rest of the text. The theory and examples are routine, but very well explained, as opposed to merely exposed.

The transition-out chapters on combinatorics, real analysis, complex numbers, and linear algebra are more diverse and perhaps more complete than is usual in these books. Collectively, they account for over a third of the page count. The student of this book both comes from somewhere (e.g., calculus) and is going somewhere (e.g., analysis).

The book under review raises the important question of what we want a mid-level undergraduate textbook to do in the modern age.

The traditional role of the text has been as a reference. As students proceed through the course, they can look up the detailed formal statement of each point. And, of course, access a bank of exercises keyed to the topics.

Certainly by the early 1980s there were already serious questions about a student’s ability to learn meaningfully by simply reading a systematic exposition of all the material to be learned (see, for instance, Reder and Anderson, Effects of spacing and embellishment on memory for the main points of a text, $\boldsymbol {Memory \ and \ Cognition}$ , vol. 10 (1982), pp. 97–102). With a near-universal acceptance now of the importance of active learning (see Freeman et al., Active learning increases student performance in science, engineering, and mathematics, $\boldsymbol {Proceedings \ of \ the \ National \ Academy \ of \ Sciences}$ , vol. 111 (2014), pp. 8410–8415), the question is sharpened: What sort of value is the book supposed to add to the learning experience?

Certainly one answer, that of Burger as well as Taylor and Rault, is that the textbook forms the skeleton of the experience, with the students producing the flesh of the course as they go—almost what we used to call a “workbook.” This answer is an attractive one, but it should not be the only innovative answer on the table.

Fuchs provides the first text I have seen for this course with a different novel answer: The text is an independent guide, a source of motivation and explanation beyond mere technique. And this is a valuable thing for a textbook to do.