# Why so many books on introduction or bridges to proofs for undergraduates?

The two quotes that I embolded below substantiate there are too many books that allegedly assist undergraduates to transition to proofs. If these authors and publishers are desperate for income, wouldn't they profit more from writing solutions (like John Weatherwax Ph.D. (MIT)) to books that don't come with official solutions manual like Sheldon Ross's *An Elementary Introduction to Mathematical Finance*? Or Schaum's Outlines or problem books with full solutions?

OUP published *Visual Complex Analysis* in 1999. MAA published *Visual Group Theory* in 2009. Wouldn't these authors profit more from visual counterparts to other undergraduate subjects like Combinatorics, Geometry, Matrix Theory, Number Theory, Real Analysis, Rings Fields Modules, Topology?

Joseph Rotman. *Journey Into Mathematics: An Introduction to Proofs* (2007).

With the current surge of interest in undergraduate proof production, bridge courses, and how best to teach undergraduates to perform proof, the re-issue of Rotman's

Journeyis certainly timely, since Rotman intended this book to be a text for just such courses and uses.

Ethan Bloch. *Proofs and Fundamentals: A First Course in Abstract Mathematics* (2011 2 edn).

My feeling is that this sort of material is not best for a wide variety of students who have just finished calculus. The elevation of rigor above all other mathematical virtues gives a skewed view of what mathematics is. My experience with other texts from this genre tells me that the danger of alienating students is very real.

Andrew Wohlgemuth. *Introduction to Proof in Abstract Mathematics* (2011).

Introduction to Proof in Abstract Mathematicsis unlike any other book on writing proofs that I have encountered.

Jeffrey Meyer. *Passage to Abstract Mathematics* (2011).

The exercises are of varying degrees of difficulty, but the authors have deliberately provided no solutions or hints.

Daniel Cunningham, *A Logical Introduction to Proof* (2012).

This is yet another textbook for a “transition-to-proofs” course. The last time I counted (in 2009) there were at least 25 such textbooks, some of which may now be out of print. Why another such book? Clearly, the author, who has taught the course using both Velleman’s

How to Prove It: A Structured Approachand Epp’sDiscrete Mathematics with Applications, thinks he has a new approach.

Larry Gerstein. *Introduction to Mathematical Structures and Proofs* (2012).

It’s clearly a huge cash-cow for publishers, since when one integrates over the nation’s mathematics and computer science majors, one gets a very decent population to hit. [Emphasis mine]And there are accordingly scores and scores of text books aimed at guiding our post-calculus students in the direction of genuine mathematical reasoning, i.e. the business of doing proofs — so many sow’s ears, so few silk purses, but try we must.

Steven Krantz. *Elements of Advanced Mathematics* (2012).

In a book that is meant to form a bridging course, Steven Krantz’s coverage of the real number system is, in part, too formal, and it sometimes introduces ideas with little motivation.

Aristides Mouzakitis. *Bridge to Abstract Mathematics* (2012).

One weak area in the present book is motivation: There’s not much explanation of why proofs are important or why they are the central feature of advanced mathematics.

Béla Bajnok. *An Invitation to Abstract Mathematics* (2013).

There are dozens of textbooks designed to help mathematics majors “transition” to higher-level mathematics and mathematical thinking, but Bajnok’s new book truly

invitesstudents to enjoy the beauty, power, and challenge of abstract mathematics.

Michael Cullinane. *A Transition to Mathematics with Proofs* (2013).

In short, this is a good, if not particularly groundbreaking, text.

Michael Starbird. *Distilling Ideas: An Introduction to Mathematical Thinking* (2013).

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.

Rosenthal's. *A Readable Introduction to Real Mathematics* (2014).

I think the big weakness of using this book for a bridge course is that it never talks about proof per se; there is no discussion of what a proof is, no consideration of proof strategies, and only a couple of examples of fallacious proofs. It would have to be supplemented a good bit for a bridge course; as it stands, the students would have to learn proofs by osmosis, which defeats the purpose of a bridge course. Two better books for this purpose are Beck & Geoghegan’s

The Art of Proofand Rotman’sJourney into Mathematics. Both cover much real mathematics but are much stronger on proof techniques, and both have a great deal of coverage of continuous math.

Suely Oliveira. *Building Proofs: A Practical Guide* (2015).

The only complaint I have is that proof by contraposition and proof by contradiction are considered the same technique.

Tamara Lakins. *The Tools of Mathematical Reasoning* (2016).

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. [Emphasis mine]

Vladimir Lepetic. *Principles of Mathematics: A Primer* (2016).

Books that introduce undergraduate students to higher mathematics are numerous. One common complaint against a significant portion of them is that they talk a lot about how to prove theorems without actually proving anything interesting. It is as if a dinner host discussed the details of eating a nice meal, but only served chips and water.

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