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.
