# The Calculus of Friendship

In the past few weeks, I’ve read a few books about math.

I don’t have a lot to say about the first book. I learned a few interesting vignettes and a few interesting facts, but overall, I’m not sure I would recommend it to others. The second book was actually incredibly fascinating, and I will maybe write a little somethin’ somethin’ about that later. However, I just finished The Calculus of Friendship and wanted to give it a major shout out.

Let me first tell you how I came upon this book.

Prof. Strogatz emailed me in October of 2008.

I happened across your blog today (isn’t the Internet amazing?) and felt compelled to try contacting you for many reasons.  You seem like a great (former) student who has now turned into a great teacher. That’s wonderful.  Sorry we didn’t overlap at MIT.  And it’s very admirable that you’re now bringing your enthusiasm and training to help inspire high school kids.

The email was longer than that, and incredibly sweet, but what was more amazing than getting this email was the timing of it. I explained in my reply:

You won’t believe how coincidental your email is! In one of my calculus sections this past Friday, we finished the material we needed to go over ten minutes early and we got — somehow, don’t ask me how — on the topic of chaos. So of course I go off on this mini-lesson on the chaotic waterwheel. We watched youtube videos and talked about what it means for something to be chaotic, and how they should understand why weather is so unpredictable from this.

One student came to my office after class, and I showed him your textbook (which I hold up as one paragon of what college math textbooks should strive to be; it was far and away the best math textbook I’ve used, besides the calculus textbook that I used in high school, which will always have a special place in my heart and on my bookshelf).

I loved that.

Then this year, Prof. Strogatz emailed me asking if he can send me a copy of his new book The Calculus of Friendship. The timing of that email was strange too. In my reply email, I said:

Wow! Thank you so much for this super unexpected and thrilling surprise. Talk about things that brighten the day. Something in the zeitgeist must be in sync (ha, groan) because just yesterday I was looking around in our math department for your Chaos DVDs (I asked my department head purchase them last year). They were nowhere to be found. Turns out one of the math teachers took them home over the summer to watch, and her boyfriend then got hooked on them, and that’s why they were missing. Too good! And what a compliment to your digital teaching presence.

I looked at the first few pages of your new book on Amazon.com, and it can’t but help but be an emotional read. Because I suspect that ensconsed in the pages is a portrait of the teacher that I strive to be.

Enough prelude. I finally did get to sit down and read this book over winter break. It’s a short read, only about 150 pages. And it is broken up into sweet little vignettes. Although I could have polished this book off in a few hours, I wanted to savor it, let it linger. So I limited myself to only a dozen or so pages each day.

I was introduced to two characters: a precocious high school student (Strogatz) and a veteran teacher (Joffray). And as I slowly devoured the book, I was taken on an emotional journey about two minds which played off each other, and two lives which slowly and inexorably intertwined with each other. Strogatz has written an honest and critical autobiographical piece, while at the same writing a sublime elegy for his former high school calculus teacher.

The Calculus of Friendship is crafted by the author and narrator, Steve, by analyzing his epistolatory relationship with his teacher, Joff. The letters started after high school and focused on interesting mathematical questions. These letter exchanges continued for decades. What’s interesting is not only the contents of the letters (which I will talk about below), but the changing role that the letters played in the writers’ lives. The meaning of the correspondance between Steve and Joff changed, although the content itself was often intensely and narrowly focused on interesting mathematical problems and solutions. This book is indeed, as the publisher’s blurb says, “an exploration of change. It’s about the transformation that takes place in a student’s heart, as he and his teacher reverse roles, as they age, as they are buffeted by life itself.”

As expected from the author of one of my favorite college textbooks, the actual math is explained clearly. The math problems the two worked on through the years are interesting (chase problems, some fun integrals and series, the gamma function, dimensional analysis, etc.). The problems were different and interesting enough, or the approaches out-of-the-box enough, that I wasn’t bored and didn’t skip any of the math explanations.

Because the epistolatory nature of Steve and Joff’s relationship, and because they were each egging each other on with questions and observations, the puzzle-y aspect of problems solving came to the forefront. Some problems were attacked in a number of different ways, with a few different approaches. (My favorite one was finding $\frac{\sin 1}{1}+\frac{\sin 2}{2}+\frac{\sin 3}{3}+\frac{\sin 4}{4}+...$.)

So yeah, I give the book two thumbs up.

As an aside, this book gave me a thought: a textbook (or unit) written entirely via letters. A fresh back and forth exchange. A little back story to draw in the reader. This approach could showing how math really unfolds, how questions get raised and answered, how some approaches work while other approaches fail, etc. Basically a textbook showing the messy nature that math evolves, because it is written as a dialogue between two people trying to figure something out. Where everything isn’t presented in a sterile, whitewashed way. Where the driving question for a unit is something like “so I was wondering if you can find a curve that goes through the point (2,1) and (4,-2). I’ve figured out how to find a line that goes through these points (which I will explain in this letter below). But what about some other curves? I mean, I can draw an infinite number of curves between these two points. [figure included.] How do I find their equations?”

Okay I should get to bed now. The twilight of my winter break is nigh and my alarm goes off in 7 hours to wake up for my first day back.

1. Matt E says:

In the 2nd year of the PROMYS program, the participants are divided into pairs to do some independent exploratory “research” (not so much looking stuff up as figuring stuff out), which culminates in a paper and a presentation on what had been discovered. They pass around papers from previous years so that everyone can get a sense of what’s expected, and one of them was just such an epistolary style. It was the two participants trading “letters” back and forth about the material, and what they had “discovered”. It was very interesting, and very clever.

I think such a textbook would be awesome. You’re right on about the sterile nature of most textbooks vs the messy nature of actually doing math and solving problems. (Sorry, problem-solving. ;-) ) File the idea somewhere, and when you actually get around to it, I’d love to be part of it!

2. It isn’t exactly what you’re asking for, but Knuth’s book Surreal Numbers has a similar flavor.

3. Sam, what a great post!! On the topic of a syllabus or unit taught entirely through letters, have you heard of the book Sophie’s World? It’s a novel about a high-school aged girl who gets a course in western philosophy via mysterious letters left in her mailbox. So it’s about philosophy, but not math, but defo a very inspiring correspondence of ideas.

Also pedagogically, it seems like teaching through letters/correspondence — little niblets of information — might be less overwhelming than facing a textbook. And it seems like there’s a lot more potential to feel anticipation for a receiving a letter than for cracking open a textbook (though if the textbook is really good, that is also possible.) :)

1. I *loved* Sophie’s World. I definitely have it on my shelf, but haven’t thought of it in forever! That’s exactly the feeling I think would be great.

Sam

1. aww…Sophie’s World! That brings back memories…I read that slowly whenever we had free time in World History my freshman year of high school…might have to dust it off the shelf and give it another read. *smile*

4. John Scammell says:

Thanks for the recommendation. I have just ordered The Calculus of Friendship. I like your idea of a texbook written in a “messy” way. Our texts are far too sterile and only present one way of doing things. Such a book would encourage students to explore, fail, and try again.

5. I just found an interesting new series about math on the NYTimes, written by Steven Strogatz. Here’s the link to the first article.