*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

*New York Times*

*, a lovely book on math that grew out of his series of postings in the*

*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 1^{st} part is to find good questions. The 2^{nd} 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 1^{st} 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.

PS: Other resources and related thoughts can be found in an earlier blog post of mine here and in a paper I wrote on Quantitative Literacy, for an anthology, available on the MAA site here.

Mr LaPuma

said:Wow, great post! Our problem of practice deals with so many of the topics discussed here….strategic questioning, deep thoughtful dialogue, challenging tasks….

Larry Feldman

said:Great article. The classic book on the teaching of problem posing is The Art of Problem Posing by Stephen I. Brown and Marion I. Walter (3rd ed., 2005). This book is filled with specific ideas for teaching students to become better problem posers, not just problem solvers.

grantwiggins

said:I recall it well! My adviser at Harvard, Israel Scheffler, recommended it to me 30 years ago. (I think one of the authors was a student of his).

HM

said:Awesome post! This is math in my mind!!! But it takes time. “Time we don’t currently have if we are going to reach all of these Common Core standards” I’ve been told. To direct students how to do a problem takes minutes. To pose a problem and let them ask and answer their own questions takes significant time, maybe a couple of class periods, especially if you’re the first teacher to do it. I used to teach this way at least once a week before our “research based practices” push, evaluations, etc. Sad, really. We do need more of this! These are the things that excite me, and used to excite students…”are we doing problems today!?!?!??” “No, we have to move on so we don’t get too far behind.” Because, even then, I took a bit of flack for being behind the other teachers. Who cares if the kids were excited to use thinking and math together. WE WERE BEHIND!!! Research shows if you don’t guarantee a curriculum across all rooms students won’t be bored….wait, I might have that wrong. But now, we spend a week before the standardized tests prepping with test questions. Well, I mean, it’s not like we could be spending that week giving rich tasks like mentioned above. We need to cram their short term memory with boring crap so it looks like we’re awesome.

Salviati

said:Grant,

While I wholeheartedly agree and do my best to implement a problem based approach in my high school physics classes, I do so to my own detriment. I teach in an affluent public high school and what I have found is that the overwhelming majority of students, their parents and school administrators are absolutely obsessed with grades and only grades.

My students are extremely frustrated by my refusal to reduce physics to a series of plugging and chugging, and the parents of the “top students” in my school have been actively trying to change my classroom by putting pressure on school administrators. They believe that since their child “studies”, they are entitled to an “A” or because my class is “more difficult than AP Calculus” that I am somehow to blame. Now personally, I wouldn’t mind giving all my students A’s, grades are fairly meaningless measures anyhow. But this is out of my hands, the administration dictates how much each particular category {Test, Homework, Labs) are worth.

Additionally old habits are hard to break, i.e. students actively resist the problem based approach. Many simply cheat either by plagiarizing what is on the web or copy off of each other. Others continue the approach of plugging and chugging even when it clearly will not work. Yet others, will simply wait for me to explain it in class.

Now I do not believe that my experience is unique and I think it makes a huge contribution to why math/physics instruction is largely plug and chug. Students and their parents want routine, because it guarantees the reward in a grade. Administrators want it too, because it keeps the parents happy. No one really learns and the few students that care to really understand what is going on are quickly shouted down by their peers who have more important things to do, like update their Facebook status. I wish this was not the case, but I can not ignore the evidence before my eyes.

PS: Ironically, my school has implemented UBD.

grantwiggins

said:Sad. No, not unique. But not the only option, either.

I can report to you, for your parents and kids, that Eric Mazur (famed Physics Professor at Harvard) told me personally – and emphatically – that they have no interest at all in AP Physics at Harvard – that their own data shows unequivocally that high AP scores mean nothing as predictors for Harvard success. Indeed, Mazur (who served on the AP Physics Comm. for a while) quit AP work when he could not get the College Board folks to listen to his complaints, which were all buttressed with data.

I know this is easy to say but difficult to do but the situation you describe sounds like a complete failure of school leadership.I do not blame kids and parents for this on the whole because in the absence of a serious school culture, built by strong leadership, what you describe is nervous-nellie default behavior by people who don’t know better.

Put differently, it is imperative that parents be educated to understand the poor returns on such an extrinsic and cynical investment.

I have seen what I am calling for done – in public as well as private schools – all over the world, so I know it can be done. But it takes will, vision, and gumption for when there is pushback.

Salviati

said:Grant,

You are totally correct about the failure of school leadership. I am not so sure it is nervous-nellie behavior on the part of the parents as much as trying to bulldoze over a non-tenured first year teacher in an economic environment where teachers are disposable. My reason for saying this, is that several of these parents did not even bother to contact me, they just went straight to the administrators. They knew what they were doing, but they are also just reacting to larger societal trends.

Along these lines, its a good development that Harvard is putting a dent into the College Board stranglehold over curriculum. Math and physics are majors that are the graveyards of students who scored 5′s on the AP Calc exams.

However, AP Physics C can be structured to be a course that fully prepares students for the rigor of a math and/or physics major if the right textbooks are used. This is one area where I think we may be in disagreement, there are some textbooks that are really phenomenal for learning mathematics and physics. None of them are standard texts in our high schools, but they are all made by world class mathematicians or physicists. If you are interested I can send you some links.

grantwiggins

said:Please! Send the links. My favorite Geometry text is by Harold Jacobs, by the way – alas, few use it…

I was prepared to love Conceptual Physics, but my kids had a weak experience with it…

i thought ChemCom was good back in the day; haven’t seen it in a while.

Salviati

said:The following are three high school level math textbooks written by Israel Gelfand – one of the 20th century’s great mathematicians. Each of these is a gem and should be required reading at the very least for every teacher of mathematics. Here are the amazon links:

Algebra – http://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773/ref=pd_bxgy_b_img_y

Trigonometry – http://www.amazon.com/Trigonometry-I-M-Gelfand/dp/0817639144/ref=pd_sim_b_1

Functions and Graphs – http://www.amazon.com/Functions-Graphs-Dover-Books-Mathematics/dp/0486425649/ref=pd_sim_b_2

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The following book is what should be the standard Calculus book but that will never happen. I will include the Amazon link and the Alibris link because Amazon sells it for an outrageous $222, but you can get an international paperback edition on Alibris for $18

8

Calculus – Tom Apostol – http://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051/ref=pd_bxgy_b_img_y

Calculus – Tom Apostol – International Edition – http://www.alibris.com/booksearch.detail?invid=11561460189&qwork=884894&qsort=&page=1

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The following is the textbook that I use for AP Physics C – Mechanics

Introduction to Mechanics – Kleppner and Kolenkow- http://www.amazon.com/An-Introduction-Mechanics-Daniel-Kleppner/dp/0521198216/ref=pd_sim_b_7

Next year I intend on using the following for AP Physics C – Electricity and Magnetism

Electricity and Magnetism – Purcell and Morin – http://www.amazon.com/Electricity-Magnetism-Edward-M-Purcell/dp/1107014026/ref=pd_sim_b_1

I tell my AP Physics C students that I could care less about the AP exams, my goal is to get them prepared for the rigor of a Physics or Math major. Coincidentally, thank you for the blog posts on Eric Mazur. I spent a couple of hours today watching his Conversion of a Converted Lecturer and checking out the teaching software he helped develop. This will have a major impact on my classes next year.

grantwiggins

said:Cool, thanks. I’ll check them out.