Mathematics Author of the Month - August: Neil Nicholson
CRC Press is pleased to share with you our author Q&A session with Neil Nicholson!
A Transition to Proof : An Introduction to Advanced Mathematics, 1st Edition
Author(s): Neil R. Nicholson
Price: $115.00
Cat. #: K418481
ISBN: 9780367201579
Publication Date: April 02, 2019
Binding: Hardback
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Q&A with Neil Nicholson
To what extent did your own students and their experiences inspire you to write this book?
My students and their experiences played a major inspirational role in writing this book. From the very first time I taught Nature of Proof in Mathematics, I was amazed at how different students approached the material in such varying ways. Going into the class, I was of the expectation that “this is how you write a proof.” I’d teach them, they’d mirror what I did, and they’d become decent proof-writers. But that was very much not so. Much like how everyone has “their way” to write expository pieces, I noticed nuances, some small and some big, in every students’ writing and I often found myself thinking, “That’s an interesting approach, and there is nothing wrong with it. Clever!”
So I began to incorporate the development of one’s own style into that course (and also in other higher-level proof-based courses). Over the years and hundreds of students later I’d collected a multitude of “approaches” to such personalization and it made me think, “I bet others would find this useful.” In particular, to the budding mathematician who notices that their proofs don’t always look the same as those in texts, I wanted to reassure them that they very well could be writing exemplary mathematics!
As the text started to get drafted out, I realized that there were two parts to the proof-writing and introduction-to-abstract-mathematics process that needed addressed. The aforementioned personalization of a proof’s final draft was one of them, but before students could get to that “last step” in the writing process, they needed to learn how to construct that actual machinery that the proof was based on. For many students in the course for which this text would be used, this is their first time actually having to “figure out how the proof works.” That is not an easy process.
But again, I relied on what my students have exhibited. Working those approaches, tips, suggestions, and thought-processed into the book should, I hope, help develop the “mathematical thinking” of up-and-coming mathematicians.
Whilst you were researching and writing the book did you come across any information which surprised you or challenged your previous approach to writing proofs?
Some of the things that I take for granted as a mathematician surprised me. For example, mathematical induction is a proof technique. “Of course it is a valid approach.” But I never stopped to ask myself why that is or is it really valid to use it without proper justification. That realization that it must be stated as an axiom, clearly and up front, so that the reader really understands why it is allowed jumps out as such a topic.
There are places in the text where I try to exhibit “bad” proofs. I was surprised at the difficulty in coming up with good examples of these. One would think that I’d have many after teaching so many students. It was challenging to construct proofs that, except for one error, were decent.
Lastly, a challenge to my own proof writing would come into play as I tried to constantly vary the style of proofs in the book. A theme of the book, as mentioned above, is that there is not just one way to write proofs. In the book, then, there are probably over one hundred proofs. I couldn’t do the reader justice if I simply stuck to “my style” for all of those proofs. I had to stretch my mind, think about unique approaches past students had taken, look at minute changes that have substantial impact. I constantly had to keep my guard up and not fall into a stylistic rut, writing proof after proof all in the same manner.
What did you enjoy writing about the book?
I am a very detail-oriented person who loves making lists, staying organized, outlining my daily and weekly to-do’s. Writing this book forced me to do this, especially considering when the bulk of my writing took place. My son Zeke (first and only child) was born in January, and during that calendar year, I would say 75% of the book actually took shape. On top of it, for some reason I thought it would be a good idea to run my first 100-mile ultramarathon during the fall of that year. Needless to say, between learning how to be a dad, running 80 miles a week and writing my first book, I sharpened my organizational and time management skills. This is probably a great place to say that NONE of this would have been possible without the support of my amazingly talented and even-more-organized wife.
What do you think are your most significant research accomplishments to date?
I approach this question as to, “What of my research am I most proud of?” Two things come to mind, and they are very much related. First, the fact that I’ve sustained somewhat of a research agenda is something that I would not have predicted twelve years ago when I took my first academic position. I did research because I had to do it to get a Ph.D., but it was not something that I loved in the way I loved working with students in and out of the classroom. Even during my first year or two in academia, I did research because I thought I could get it “in the bank” for tenure. But then I discovered the fun of collaborating with students on original research projects, and that is the second part of my answer to this question.
I just happened to stumble upon an original research question and a student to attack it with early on in my career. It was true collaboration rather than simply overseeing him doing the research. And that was exciting. It created a different type of dynamic with students, helping them see that they had the tools necessary for graduate school. Since that time I’ve constantly looked for projects and students to collaborate with. I can count upwards of twenty students I’ve collaborated with. We don’t always get tangible “public facing” results, but about half the time we have. It’s never cutting-edge, high-ranked journal quality research, but it is original and it is serving a much bigger purpose, in my opinion, than simply expanding the breadth of mathematical knowledge. It’s influencing, mentoring, and impacting students in ways I don’t think are possible inside the classroom. This is my most significant research accomplishment.
