I have a treat for readers today, an interview I did recently with Steven Strogatz, mathematician and writer on math extraordinaire. Strogatz is the Schurman Professor of applied mathematics at Cornell University. He is the author, most recently, of The Joy of x, a lovely book on math that grew out of his series of postings in the New York Times called the Elements of Math. He recently concluded his second series in the Times.
How do I know Steve? I was his teacher in high school! We have remained in touch over the years, and he graciously consented to spend an hour on the phone with me recently to discuss math and math education.
GRANT: So, Steve, talk to me about the interesting part of math, the creative side. So many kids think math is just drudgery plug-and-chug work. What does it mean to be creative as a mathematician?
STEVE: Well, there’s a question part and an answer part to what we do. The 1st part is to find good questions. The 2nd part is to turn well-formed questions into answers. Both demand some creativity, but it’s the questioning part that needs more emphasis in schools.
How do I know what to investigate or think about? Most people would be puzzled – “Isn’t math already done? Don’t we know all the numbers? Are you trying to think of bigger and bigger numbers or new kinds of shapes?” Well, no. There are all sorts of interesting theoretical and applied problems out there.
Math is not just what we heard about in high school, the known and straightforward part of the subject. For example, calculus has all kinds of logical difficulties in it about handling infinity. Infinity, which is central to the calculus, is very problematic! And, thus a new entire branch of math grew up in 1800s, analysis, to handle these kinds of problems.
For me, I try to think about mathematizing parts of sciences that haven’t been understood mathematically, e.g. of social networks. A really interesting question that I have been working on, for example, involves people who sit on boards of directors, and the math of connections of those people. There is a practical issue of how to get the greatest connectivity between members of Boards who serve on many different Boards. But it generalizes beyond corporate governance issues to disease propagation, and Google algorithms. It’s the application of linear algebra. (I wrote a Chapter in the Joy of X on this).
GRANT: What then separates good from so-so mathematicians?
STEVE: The quality of their creativity and the quality of their technique. Most mathematicians are good at one or the other. Great ones are good at both. So, it becomes a self-knowledge issue, too. Just like any artist, you have to think – what problems will you work on? Are you comfortable on incremental or revolutionary issues? In terms of technical expertise: how strong are you at solving problems that are now more sharply posed? Etc.
I am more of a creative type than a technical type. And here again we find laypeople puzzled – what could possibly be creative about finding problems? Well, there is a huge amount of creativity in posing mathematically tractable Qs. Mathematical modeling – a key phrase in the new Common Core math standards in k-12 education – is, at its heart, the ability to spot interesting potential issues and pose them as problems that mathematicians can address.
GRANT: I think people would find that funny – that you are better at framing than actually solving as a mathematician, and can get paid for that.
STEVE: Here’s what makes me say that. The research I’m probably best known for is my work with my former student Duncan Watts on “small-world networks.” We were curious about the math behind “six degrees of separation”. How could it be that in a world of billions of people, we’re all just a few handshakes apart? We weren’t experts in network theory, and neither of us was a technical powerhouse… but we did manage to convince our colleagues that there was a whole new field here, just waiting to be investigated. We also gave evidence that the small world property might be universal for networks, by demonstrating that it occurred in three disparate systems: the power grid of the western United States; the nervous system of a simple worm; and the network of Hollywood actors. In the years since our paper came out in Nature magazine in 1998, it’s been cited by other researchers more than 17,000 times. Our contribution was mainly to phrase the question in a way that others could address it mathematically. It was an act of synthesis. Of course, there are other kinds of researchers who contribute by drilling deep, by focusing on solving small specialty problems. That approach – analysis as opposed to synthesis — is another way to make a mark. Fortunately, there are a lot of ways to express yourself.
GRANT: In terms of working with Cornell students: how do you get them to think more creatively (especially since their training is not ideal for it)?
STEVE: I spend a lot of my time with students about how to ask good Qs, and to get more in touch with their own curiosity and questioning. What are your sources of inspiration? What paradoxes might you consider? Paradoxes are very fruitful! Something puzzling – how can everyone on the planet be 6 handshakes apart, (as we just mentioned)? – has rich potential as a problem. Recognizing it as a paradox is a key 1st step, then thinking about it endlessly is the next part.
GRANT: Say more about the adequacy of preparation for real math in college.
STEVE: Well, almost all students have no conception of their strengths and weaknesses in math in terms of creativity and technique. Since almost every school emphasizes only the procedural side, how could they? The idea that you would find and formulate your own problems is unknown to most students. So, this is vital in school math: students have to practice and improve at finding and framing problems. It’s a habit, a skill; you can’t just teach ‘math modeling’ and expect them to be able to do this.
My old HS teacher [and Grant’s former colleague, Don Joffray, about whom Steve wrote a touching book on their correspondence] would take us out to the football field and set up a problem. Should you kick the field goal when you are close but way off to the side? Or take a 5-yard penalty which, while making it longer for the kicker, seems to give a much better angle. As soon as you start talking, you are modeling, you are practicing problem framing. I just don’t see students coming in with this ability. If this were more regularly done it would be very helpful to me and my colleagues.
GRANT: Say more, specifically, about the deficits of incoming students.
STEVE: There is an almost universal thoughtlessness, the feeling that this is all mechanical, very robotic thinking, that you can only handle already-well-formed problems. If you ask Qs that depart from that, well, the students are brittle, they have no suppleness to think about it. (Getting good at this is like getting good at word problems in school, and those are the ones that students often dislike the most). Happens a lot students confront a novel problem and protest: “we didn’t cover that.” [Grant: this is of course central to Understanding by Design and our emphasis on transfer.]
Another big stumbling block is all the misconceptions students bring to the work, misconceptions that have to be rooted out in discussion. This is why it is important to get at what they think they know and what they think they don’t know. They often think they know something that is not true, in fact. I have found that’s very important, it’s not ignorance and just learning a right way to do a problem. Until you root out the misconceptions and misunderstandings (which they are often reluctant to share because they start to feel dumb), they can’t move forward. So there has to be empathy and a questioning spirit in the class. They have to trust you enough to be able to admit an idea – a shaky feeling – that they think might be wrong. Until it gets laid out on the table they cannot advance. Good math teaching is a bit like surgery, it’s a little like removing a tumor. That may not be the right metaphor, but it captures how I think about my need to have their misconceptions brought to the light to be removed thru back and forth with me and with peers.
Students need to constantly confront problems that have 4-5 plausible ways of looking at and framing them; and they need to see that sometimes a technique works and sometimes it doesn’t. For me the rush to more AP and more content is just not helpful. We don’t need more sophisticated content in school courses that students don’t really get, we need better problem solvers.
Of course, this generation of teachers hasn’t been taught how to think about and find such problems readily. Nor do most of them have first-hand experience in thinking about real problems day after day, not much personal experience really doing math.
GRANT: Then, aside from such classics as Polya’s How to Solve It, what are some great resources for math teachers in how to get kids to become better problem solvers?
STEVE: Two great books are Guesstimation 2.0 and Streetfighting Mathematics. And of course, as we discussed [in another part of the conversation not provided here], all the Car Talk puzzlers and Martin Gardner books!
Going back to the idea of paradoxes, there are some in high school math that can be addressed by teachers:
- Why can’t you divide by zero? (Many teachers think this is an arbitrary edict!)
- Why is a negative times a negative a positive?
- Is .99999999… the same as 1? Is infinity a number?
- What is zero to the zero power?
GRANT: These are great, Steve. And so are your other reflections. Thanks so much for sharing your thoughts with readers on mathematics and math education.