The Mathematical Encounter That Changed My Life

By Art Duval, Contributing Editor, University of Texas at El Paso

I just returned from an all-years reunion of the Hampshire College Summer Studies in Mathematics (HCSSiM) program, a six-week program I attended during the summer between my sophomore and junior years of high school.  It has been run by David C. Kelly, whom everyone refers to just as Kelly, since he started it in 1971.  There are several other summer high school math programs around the country (a good start is this list from the AMS), which likely share some characteristics with Hampshire, but since Hampshire is the one I have personal experience with, this is the one I am compelled to talk about.  And while several people and experiences were instrumental in my path to becoming a mathematician, Hampshire is the one that stands out most prominently in my mind, the one mathematical encounter that changed my life.  And from talking to other people at the reunion last weekend, I know that many other program alumni feel the same way.

There are other accounts written about Hampshire.  The AMS has a nice commentary on Hampshire and other similar programs, and Jim Propp has a very nice blog post about it, from the perspective of someone who has been a student, and has taught at the program as junior staff and senior staff.  I was only a student one summer, and never taught there, but the program had such a profound effect on me that I want to share my personal reflection on the experience. Now over 34 years later I am a professor at UTEP, and I hope that the benefit of my looking back with the hindsight of years of learning and teaching math will outweigh the loss of some details through those same years.  But, as with any transformative experience, some details remain crystal clear.

Apparently you can go home again, at least for the weekend. (Photo: Owen Michals)

I’d loved math from an early age, but when I showed up at Hampshire as a rising high school junior, the only proofs I’d seen were the two-column proofs in high school geometry that previous year.  From the outset, it was clear that this program was going to be … different from high school.  The first day of class (we met in 4 classes with about 17 students for 4 hours in the morning, 6 days a week; each class with different instructors, but discussing similar topics), Kelly started immediately with a problem, which I still remember: If we have a (three-dimensional) hunk of cheese, and we slice it with some number of cuts with a knife, how many regions will we have?  This was not at all like my math classes in school, and a little like some of the puzzles I read about in mathematical puzzle books on my own, but it was somehow a little deeper than those puzzles, and it was an entire class spent exploring the problem together.

We spent almost all morning (four hours!) on just this one problem.  It served as a vehicle to develop for ourselves (with guiding questions from the staff, to be sure) mathematical problem-solving strategies we were to return to all summer: Work examples and gather data; formulate your assumptions carefully (early idea, soon discarded: What if all the cuts are parallel? That’s not an interesting problem anymore); don’t assume that patterns always continue (in this case, 1,2,4,8 was followed not by 16, but by 15); when problems are too hard, try a simpler problem first (we moved from 3-dimensions to 2-dimensions); draw pictures; make conjectures; try to prove your conjectures, or make new ones if necessary.

The rest of the summer proceeded similarly.  Topics were introduced by way of problems, some of them imaginative, which we discussed and refined, with lots of student input.  Methods of proof were woven into discussions about how to verify our conjectures.  Classes were engaging, and even fun, because of the interesting problems and because of the interaction among students and staff. Looking back at it now I would describe it as active learning, with an additional ingredient:  It never felt like anyone (student or staff) was doing something because they had to, or for a grade, or really for any reason other than that it was inherently interesting.  But at the time, I only knew that I really liked it.  If this was math, I could spend all day doing math!

Evenings were devoted to 3-hour problem sessions.  Problem sets ranged in difficulty from working examples to writing proofs.  Problems reviewed that day’s material, expanded on ideas, or let us play with ideas that had come up during the day.  Sometimes they previewed upcoming material.  During the problem sessions, we could work on whichever problems struck our fancy, and, when we solved something we were proud of, we could turn it in for constructive feedback.  In a change from my previous learning/student experience, nothing was ever awarded a grade.  (Instead, at the end of summer, Kelly wrote a detailed descriptive letter of recommendation for each student.)  When I looked back many years later at some of the proofs I’d written, they looked very rudimentary to my more experienced eyes, but I know that by the end of the summer I really had learned at least the basics of writing proofs, and that I loved it.

Another remarkable difference from my previous experiences was that we were strongly encouraged to work in groups on problems.  In fact, throughout the program there was a strong sense of cooperation instead of competition.  I quickly grew to embrace this cooperative view of mathematics and of education, and never turned back.

Afternoons were free time, but I spent most of my afternoons working with other students on the weekly “program journal”, hanging out in the room where we put it together.  Like many aspects of the program, it was almost entirely student-run.  We wrote summaries of the week’s activities in the classes and of the daily “Prime Time Theorem” lectures (self-contained hour-long talks given by visiting mathematicians or by the staff), and ran a problems section (pose problems, solicit solutions, print solutions the next week).  Of course, we also had some less serious features, such as reviews of the weekly math movies, cartoons, and silly math songs.  I wrote some of the Prime Time Theorem write-ups, and I distinctly remember noting that by the final week of the program I was paying more attention to the precision I had to use to get the details correct.

Eventually we saw many different topics, none requiring much prerequisites beyond some high school algebra and, more importantly, an intense curiosity and a willingness to experiment and learn.  I don’t recall precisely all the topics, but we certainly covered a lot of number theory, the basics of group theory, and some combinatorics and probability, some topology, some notions of infinity.  It very much felt like that any subject might come up on any day.  Halfway through the program, we finished the overview class, and each student could pick a class focused on one of four specific topics; I picked the class on large prime numbers and factoring large numbers, but later wished I’d picked the class on group theory centered around Rubik’s Cube (a few months before the Cube became wildly popular in the United States). I got to see lots of other things I would later take for granted (the geometry of how complex numbers multiply; how to think of GCD in terms of buying postage stamps; etc.).  It took me some time to realize that not all math students learned these things in high school!

As with any good educational experience, I also learned a lot from my fellow students.  Many were attending selective high schools in New York City and elsewhere (I was attending the public high school in my suburban town), and they generally had much higher expectations than I’d even thought about.  They went to national math competitions, and did well.  They planned to go to Ivy League colleges.  Being around them raised my own expectations of college and my future.  I entered Hampshire thinking I would be a meteorologist (because I liked looking at clouds), and left thinking I would be, well, if not a mathematician, at least an engineer.  But I was also certain I would take any math class I could.  (Which I did, and then eventually switched to math.)  And I also believed I could go to the best programs in the country to pursue further education.

Even though we were studying serious and advanced mathematics (even without having taken calculus!), everything was infused with a sense of playfulness.  From Kelly on down, staff conveyed the idea that what we were doing was inherently interesting, and that it was fun to just play around with ideas, and problems, and explore ideas.  Though there were jokes and kidding around, it was the wonder of the mathematics that always took center stage.  One of the ways in which this playfulness was transmitted was through the program’s adopted mascot and number.

Just some of Kelly’s yellow pigs (photo: Owen Michals)

You probably were expecting me to get to this part if you have heard of Hampshire before.  We quickly found out that Kelly, and everyone else at the program, had a thing about yellow pigs and the number 17.  Yellow pigs were everywhere, including on our t-shirts once Yellow Pigs Day happened on July 17, when Kelly gave his talk on the mathematical and social history of 17.   (For instance, a regular 17-sided polygon can be constructed with ruler and compass because 17 is a Fermat prime, \(17=2^{2^n}+1\) with \(n=2\).) Soon all the students found and used YP’s and 17’s everywhere (for instance, look again carefully at the first sentence of this paragraph, especially the first two words and the number of words).

Of course, one purpose of yellow pigs and 17 was for a program identity and cohesion (and for alumni to be able to recognize one another), but 17 had another useful purpose.  If you wanted to give a proof, but start with an example, you could pick 17 as the value of a variable such as \(n\).  Everyone (at Hampshire) would recognize you were using it just as a placeholder, and the next step, replacing all the 17’s by \(n\)’s might not be too hard.  (Kelly also showed us how this transition could be achieved typographically; see below illustration.)

Generalizing from 17 to $n$, typographically (photo: Art Duval)

Some years after I’d been teaching at the university level, I realized that almost every innovation I tried to implement was based on some aspect of Hampshire.  Well, I didn’t try to use yellow pigs, but I do use 17 in examples in class when I can.  (And I do reflexively look for 17’s everywhere.)  More than that, the Hampshire way of thinking about mathematics remains at the core of how I approach mathematics and education.  It remains remarkable to me that just six weeks could change my life so profoundly, but I remain eternally grateful to Kelly and HCSSiM that it did.

Let us know in the comments what your most significant mathematical experience was, and what effect it had on your life.