Response to the Atlantic article, "Teaching Math to People Who Think They Hate It," a look into Steve Strogatz's adventures teaching math to liberal arts majors at Cornell, has been lovely. I've received a couple of emails asking for more information about Discovering the Art of Mathematics, Julian Fleron and his team at Westfield State, so Steve and I thought it might be fun to post the rest of our interview, as well as a link to the scalene triangle straight-cut origami exercise described in the article (Chapter 3 of Discovering the Art of Mathematics: Art and Sculpture).
Steve is a great interview; he loves what he does, and has a knack for explaining the details of that love to other people. I hated having to cut anything he said, but in order to make my word count, I had to chop, chop, and snip, snip.
If you'd like to read Steve's work, and you have not read his "Elements of Math" series in the New York Times, I highly recommend that series of essays as a starting place. Be sure to start with the first one, "From Fish to Infinity." You can also hear him in his regular appearances on Radiolab.
And with that, here's the rest of our interview, with a few liberties made for the sake of clarity.
Lahey: What prompted you to teach mathematics to liberal arts majors?
Strogatz: For the past few years I've been growing dissatisfied with the results of my usual way of teaching, which is lecturing. Although quite a few of my students seemed to enjoy my lectures, many of them weren't engaging with the material deeply. Just watching a performance, a lecture, and then doing homework, wasn't enough to get them to learn the subject properly, to master it.
So I’d been toying with the idea of trying some more active form of teaching and learning, but I wasn't sure how to start. One day when I was at the big annual math meeting – the “Joint Mathematics Meetings” where the major mathematical societies come together in January – I was wandering around in the exhibition hall and came across an exhibit that caught my eye. There were three or four young professors from Westfield State who were encouraging people to play math games. They were handing out Rubik's cubes, getting people to play a game called hex, or tying knots, or even dancing and making knots with their own bodies in groups. When I took a closer look I noticed that they had workbooks strewn over their table. These were workbooks that they themselves had written for a "math for liberal arts" course that they'd been teaching for the past few years at Westfield State. These workbooks were so attractive, and so filled with interesting activities for students to do, that I started to think this could be a way for me to try teaching in a style where my students would be more active.
When I talked with these faculty from Westfield State, I was struck by their passion for what they called inquiry-based learning. I found myself coming back to their exhibit, over and over again, over the next few days. I kept bringing other colleagues over to their booth to show them what was going on, to show them how cool and exciting it was. Something about it grabbed me.
And I guess what really clinched it was when Julian Fleron, one of the faculty from Westfield State, told me that they had a grant from the National Science Foundation to spread their ideas widely through the math community, and that they would be delighted to come to Cornell to give us a workshop, to show us how to make this style of teaching a reality in our courses. That was an offer I couldn't refuse. So when the time came to choose courses for this year, I asked to teach a course that was already on the books at Cornell called "Mathematical Explorations." It turns out that a course in this active style of learning, this inquiry-based learning, already existed at Cornell and had been taught for a number of years. But I had only recently joined the math department, having spent the first 20 years of my career at Cornell in engineering. So the course was new to me. I asked to teach it.
In mid-August the Westfield State folks came to visit us and give us that workshop. They showed us how to teach in this style, and how to assess our students’ performance, and also how to approach some of the psychological issues that come up with this population of students, issues like math anxiety and math phobia. They also showed us what it would feel like to be a student in such a class. My Cornell colleagues and I were the students, doing a paper-folding-and-cutting game; the Westfield State folks were our teachers. That was important since none of us had ever been students in an inquiry-based learning classroom. We needed to know what it felt like, to have the right kind of empathy for our students.
Lahey: Your usual fodder, as evidenced by your Twitter feed, is higher math. Do you find teaching a more...elementary level of math interesting?
Strogatz: Yes, I find it fascinating and thrilling. This population of students is unlike any I've ever taught before. The course I’m teaching fulfills our "mathematics and quantitative reasoning" requirement at Cornell. That's a requirement to ensure that all students in the College of Arts and Sciences are exposed to some minimal amount of mathematical thinking. As it turned out more than half of the students in my class of about 36 are seniors. In other words, they have been putting off this requirement for as long as they possibly could!
It's what you might imagine – these are students who have had some unpleasant experience with math at some point in their education. For the first assignment I asked them to write their mathematical autobiography, detailing experiences that they had both good and bad in their math education up to this point. I also wanted to hear about any teachers who made an impression on them, positively or negatively, and what other subjects they're interested in and so on. I'm still reading through some of those autobiographies now but what's emerging is that many of the students liked math for several years. These are all very bright students but somewhere along the line they got discouraged. Sometimes it was because of a certain teacher or subject. In other cases everything was fine through high school, but when they took calculus at Cornell, something about that class turned them off.
As for teaching at a more elementary level, well, first of all, math is interesting at every level. Elementary school math is just as interesting as middle school, high school, college, or graduate level math. I love thinking about the fundamentals! So the elementary nature of the subject is not an issue.
Besides, what we’re exploring in this class is not particularly elementary! This week, for example, the week that you're visiting, we’ll be investigating ideas in abstract algebra (in the particular, the subject known as group theory) but we’ll be doing it in an unusual way (at least, unusual for a math class): we'll use dance to explore symmetry. Dance may be a bit of an inflated word for what we’re doing (in fact, I was a little intimidated to start teaching about dance, since I'm such a lousy dancer myself). What we’re doing is more like striking a pose. Or moving very slowly from pose to pose, while a partner tries to follow the leader in mirror-image symmetry, or rotational symmetry, or some other type of symmetry.
I have to say that teaching this class has been a joyful experience in a way that no other class I've ever taught has been. I love teaching, and I certainly love teaching students who already enjoy math – don't get me wrong. But there's something remarkable about working with a group of students who think they hate math or find it boring, and then turning them around, even just a little bit.
For example, the first activity that we worked on was what's known as “straight-cut origami.” Imagine a simple shape, say an equilateral triangle, drawn on a piece of paper. The goal is to cut out the triangle with scissors. Except that you're not allowed to cut out the triangle in the obvious way. Instead you have to fold the paper in such a way that you can cut out the triangle by making a single straight cut. For an equilateral triangle this turns out to be pretty easy and everyone can do it. But if you pick a general triangle – a scalene triangle, meaning one where all three sides are different lengths – then figuring out how to fold the paper in such a way that you can cut out the triangle with a single straight cut turns out to be very difficult. Or, at least, not obvious. I had trouble with it myself for quite a while the first time I tried it.
So while we were working on this in class, with the students seated at tables of four, all discussing the problem, showing each other their ideas, things that had worked or not worked, after they struggled with this for about a half-hour it turned out that only one student out of 36 was able to do it. So at the end of the class, when I noticed that there were only about five minutes left I asked "Would you like a hint?" A few students immediately said yes, but then they were drowned out by the rest of the class, which said no!
I was so proud of them. They were having a true mathematical moment. That is, they were deeply engaged with a puzzle that made sense to them, and they were enjoying the struggle, and no, they did not want a hint! They were feeling what anyone who loves math feels, the pleasure of thinking. The pleasure of wrestling with a problem that fascinates you. No one in the class was asking, “what is this good for?” Or "where will I ever use this?” Those are questions that students ask only when they are not engaged.
I told the students to think about the scalene triangle over the weekend and to try it in their dorm room. Over the weekend I started to get emails from some of them expressing the excitement they felt when they solved it. One student wrote: “I am feeling exceptionally accomplished. I have to admit: this math assignment has made my day. I never thought I would ever be saying this.”
Lahey: There has been a lot of talk lately about approaches to teaching math, particularly as it relates to the Common Core State Standards. Do you have any thoughts about the "critical thinking" approach versus the traditional route toward mathematical fluency via math facts and rote execution of concepts?
Strogatz: On the whole I think we usually go too fast in our teaching of math. There's a big rush to cram all kinds of information into the students’ heads, and get them fluent with certain procedures, at the expense of their understanding what they're doing.
But let me be careful here. It's so easy to cast this discussion in black and white terms, to make one point of view seem ridiculous and the other obvious. I don't want to do that, because of course you need to memorize certain things and of course you need to have an understanding of what you're doing.
It'll probably sound like a wishy-washy answer, but I really want both. I want my students to memorize and know basic facts, and I want them to understand what those facts mean, why they're important, where they come up in the real world, how to calculate efficiently and easily with them, how they developed historically, what their connections are to the arts and humanities and sciences and engineering, where they pop up in daily life and in the universe. I want it all and I think students want it all too.
If we just stick to teaching them rote procedures, math becomes meaningless. That's how it's experienced by many people. So I'm definitely against that.
But likewise if we only teach conceptual approaches to math without developing skill at actually solving math problems, students will feel weak. Their mathematical powers will be flimsy. And if they don't memorize anything, if they don't know the basic facts of addition and multiplication or, later, geometry or still later, calculus, it becomes impossible for them to be creative. They can't take the first step, because they have to rely on their graphing calculator, or look something up in a book. That makes for a student who can never achieve the greatest pleasure or success in math, which is to be inventive, to think of things for yourself. It's like in music. You need to have technique before you can create a composition of your own. But if all we do is teach technique, no one will want to play music at all.
Nothing I'm saying here is very radical or surprising to anyone who actually understands mathematics (or any other creative endeavor). If you want to be a great soccer player, you can't just do drills. You won't even want to play soccer if you're just doing drills all day. You have to get out there and play the game, and learn from your mistakes and then practice. Drills have their place, and so does playing the real game.
We do too much drilling in school, and not enough playing of the real game of math. And as with any game, or playing music or making a piece of art, it's doing the real thing that's inspiring. We need to give students more of a chance to do that. And that’s what I'm trying to do in this class. They are actually making mathematics -- in many cases, for the first time in their lives. And they’re loving it. And why wouldn't they? It’s a joyous, glorious experience. At every level. Little kids can make math. It may be the mathematical equivalent of fingerpainting, but it’s still math. Genuine creativity is required at every level.