This is the first in a series of author interviews. Enjoy!
Suggestions for further mathematics author interviews can be made via the comments or email to exh@ams.org.
The 
Tools of Mathematical Reasoning by Tamara Lakins was published in the AMS Undergraduate Textbooks Series earlier this year. In the few months since it has appeared, the book has already received 22 desk copy requests, and has been acclaimed for its “crystal clear exposition and abundance of exercises” and its careful attention to the language of mathematics. Here are the author’s responses to a list of questions we posed to her by email.
Tamara Lakins
• What made you decide to write the book? Was there a gap in the literature you were trying to fill? Did you use existing notes from teaching?
I imagine that my motivation behind writing my book was similar to that of many authors. I had been teaching Allegheny’s “Foundations of Mathematics” (introduction to proofs) course for many years, without being able to find a textbook that I was completely happy with. My experience teaching from various textbooks helped clarify in my mind what I wanted in a textbook, such as a quick path to proofs and an emphasis on the process of finding a proof. So, I began by converting my existing notes from teaching into a very early draft of the textbook.
• What was the writing process like? Did you write every day on a set schedule, or did you have periods of setting it aside? Did you find ways to get feedback while writing your book or was it a solitary effort?
I did my writing in “spurts” and “sprints”. I began converting my teaching notes into textbook form during a sabbatical about 8 years ago, with the goal of using that early draft as a textbook when I taught the course in the following fall. I updated the draft often during that fall semester, as I discovered what parts of the book were working well for students, and what parts weren’t. After that, I sent the manuscript to several colleagues at other institutions for their feedback, which was encouraging. I was also fortunate that, over the next several years, several members of my department used my manuscript as their textbook for the introduction to proofs class. I benefitted greatly from my colleagues’ feedback. The feedback from my colleagues, both at Allegheny and at other institutions, is what gave me the courage to proceed with my plan to try to publish the textbook. Except for reacting to feedback and making corrections as I and my colleagues continued to use my manuscript as a textbook, I essentially set the manuscript aside to await my next sabbatical. I returned to the manuscript about a year before my next sabbatical, to prepare it for submission to a publisher. When the manuscript was accepted by the AMS, I worked on it (thankfully while on sabbatical) for many hours a day, almost every day, until it was due, which was about three months later.
• What did you focus on the most when writing? What was the most challenging aspect? What came easily?
The early chapters of the textbook, on the introductory logic and discussion of proof techniques, came most easily to me because I had been thinking about how to best teach these concepts for about 15 years. I found the material on sizes of sets and the foundations of analysis (which I don’t have much time for in class) very difficult to write. I am a logician by training, but I didn’t want the material on sizes of sets to start sounding like a course in set theory; my goal was to focus on what the typical math major needed to know about sizes of sets. A similar tension existed in the chapter on the foundations of analysis, where I wanted to spend some time discussing the question “what is a real number?”.
• What were the positives and negatives of the experience? Did anything about the experience surprise you? Did time pressure or other responsibilities help or hurt your writing?
When my manuscript was accepted by the AMS, I believed that it was essentially in its final form, with the exception of the chapters on sizes of sets and the foundations of analysis. Those two chapters were in very rough form (as I didn’t have much time in class to devote to this material), and I was expecting to have to spend a lot of time not only writing, but also thinking about the organization of, those chapters. But I was surprised at how much time I spent also carefully reviewing and revising the other chapters, partially in response to the reviewers’ comments, but also because this was my “last chance” to “get it right”.
I had been planning to work on the manuscript during an entire spring semester, and I was surprised at how much earlier the AMS suggested I set my deadline. In many respects, the time pressure helped keep me focused, although I did find that I made more typos and other errors when working late hours after my son was finally asleep!
One of the best aspects of completing the textbook was that it inspired many stimulating conversations with my husband (who is also a mathematician and who was also on sabbatical) about the intro to proofs course, teaching, and my vision for the textbook.
I was surprised at how much still needed to be done after my “deadline”, when I thought that my part of the process was complete. I was very fortunate that the anonymous reviewers carefully read that “final” draft, providing me with many valuable comments and suggestions for improving the exposition of my manuscript and the exercises. Expanding the exercises in the textbook beyond what I normally assign in class was surprisingly time consuming. So, I continued to work on my manuscript for several months after it was “due”.
I think of myself as a very careful writer and proofreader, but I learned that there are always typos that one misses each time one proofreads!
• How did you choose a publisher? What was important to you when you made the choice?
It seemed to me that the AMS Pure and Applied Undergraduate Texts, with its focus on post-calculus courses, was a good fit. It was also important to me that my textbook be affordable to students.
• What advice would you give to new authors?
Write what you are passionate about. Teaching the introduction to proofs class is one of my passions, and I believe that passion was essential to all phases (beginning, development, and completion) of writing my manuscript.