My guest this month is James S. Tanton, a mathematician, math educator, and author. He was awarded the Kidder Faculty Prize for his teaching at the The St. Mark’s Math Institute and is currently visiting scholar at the Mathematical Association of America. He is the author of over many books on mathematics, curriculum, and education, and creator of videos about mathematics available free on YouTube and his own websites.
Michael Paul Goldenberg: Thanks for joining me, James, and a pleasure to ask you about mathematics and mathematics education. You have a doctorate in pure mathematics from Princeton, but you spent many years teaching high school mathematics near Boston. What made you decide to do that instead of working as a university mathematician or in the private sector?
James S. Tanton: A pleasure to be here.
Let’s see: my mathematical life story.
It really begins with my own school experience of mathematics back in Australia during the 1970s and very early 1980s. I have to say it really was quite the unenlightened curriculum, full of much rote procedure: just memorize and do. I found it joyless, by and large, and just uninteresting. I was good at it, nonetheless, and I seemed to develop the reputation of being able to help others make sense of it (so I guess I did gather or figure out some storyline to it all). This was my first glimpse that teaching is in my bones.
Next stop, university. There I ran away from mathematics – but not too far, clearly: I worked in theoretical physics. But it was an Abstract Algebra class that turned me onto what mathematics really is, and what it could have been and should have been during my school experience: exploration and the play of ideas, the pursuit of what else and why questions, and even fabulous what if questions. I then knew I was a mathematician and so I pursued the subject all the way through.
Next stop, graduate school. That brought me to the U.S. in 1988. And yes, it was the high-powered research world of Princeton. But that teaching thing really was in me. I managed to finagle some ways to teach undergraduate courses, despite it being against department policy at the time. (I even won a professorial teaching award, which I thought was ironic!)
After graduating, I decided to work in the Liberal Arts College environment, for that balance of research and teaching.
Life then took me to Boston where I discovered Math Circles with Bob and Ellen Kaplan. This had me working with children of all ages–which I just loved–and it also led me to some conducting professional development work for teachers. That made me look at the U.S. school curriculum and, when I did, I was shocked! I saw the same unenlightened experience being offered to U.S. children that I had received some 30 years earlier in Australia. I felt morally compelled to try to do something about it.
But I felt I had to be clear and honest and to try to truly understand the state of affairs on mathematics educational matters. I wanted to know the reality of the situation for teachers, and the limitations and barriers to actually teaching mathematics with joy, freedom, confidence, play, and human learning.
I saw problems and worries within all three levels of the school curriculum –elementary school, middle school, and high school–but I could see some good work being done to try to rethink and revamp the first two levels of education. It seemed to me that high school mathematics was being left out of these discussions, and that troubled me. I feared that any good work and joy being brought to students in the K-8 world was going to be ignored, perhaps even squelched, for the sake of “rigor” and the seriousness of the high-school curriculum: “No more fun-and-games; we’ve got to get these kids into college.” It seemed to me that the high-school experience was the toughest of challenges, and so I went for it. I became a high-school teacher for almost 10 years before life pulled me away from Boston.
I should point out that since the start of all this, the nation has done a considerable amount of reflection on its mathematical education program at all three levels, and matters are now very different from the early 2000s when I came on the scene. I even see the word “joy” now bandied about in discussions of mathematics education matters. I clearly wasn’t the only one seeing and feeling concern.
MPG: What are some things from your work at St. Mark’s School of which you are most proud?
JST: So, with a Ph.D. in Mathematics from Princeton University, I was not actually qualified to teach in the public school arena. Private schools, on the other hand, can hire whoever they desire. I accepted a position at St. Mark’s School in Southborough, MA.
St. Mark’s is a boarding school and its faculty are required to live on campus and follow the traditional “triple threat” model: teach full-time, be a dorm parent, and coach sports. The first two requirements, no worries. The third: Me? No way! I am the most non-athletic person on this planet. So I cooked up a deal with the Heads of School to start a mathematics outreach Institute in lieu of coaching. That was exciting.
I hooked up with Northeastern University and designed and regularly taught through the Institute five graduate mathematics content courses for in-service teachers. (So even though I was not qualified to teach in the public school system, I could teach teachers how to teach in the public school system. What fabulous irony!) I also offered extracurricular research classes for kids all around the Boston area, gave public lectures, wrote newsletters and books, and basically did whatever I could to promote joyous math thinking and doing.
But I am most proud of those extracurricular classes with kids. We’d meet one time a week in the evening for ten weeks and explore a single unsolved mathematics question, at least one that I devised that was accessible and, as far as I was aware, was new to the world. Sometimes we’d stick with that specific question or sometimes a lass or a lad would ask about a variant question that turned out to be more exciting and fun. Anyway, we’d discuss a question and see where we’d get with it after ten weeks. Sometimes nowhere. (Welcome to the world of original research.) Sometimes somewhere! And sometimes those somewheres led to published papers.
I am most proud of redefining what success in mathematics means in the environment I worked. Being a co-author on a published paper is way more exciting and so much more to be proud of than yet another test score. What an item for one’s college resume!
But the truth is I am personally proud of the “failures” of those classes too. What a lesson to realize that not getting anywhere after ten weeks of very hard thinking is okay too!
MPG: When I first met Bob & Ellen Kaplan at Northwestern University’s Math Club in 2002, I asked them what sort of people taught for them in their Math Circles classes. Bob told me, “In Boston, it’s easy to find people who know enough mathematics. What’s difficult is finding people who know the mathematics and can keep their mouths shut!” And that was when I first heard your name. While I am pretty sure I know what he meant, would you care to comment on that?
JST: Let me say this first. Parents were most welcome to come to my extracurricular research classes with their kids, but the rule was that they had to sit at the back and remain quiet. Not easy!
In the K-12 math teaching world, the notion of “extended mulling” is a foreign concept. Mathematicians mull. When they are stuck on a problem, they go for a walk. They sleep on it. They just let it go for a while. There is no time for such a thing in the teaching world. There is a curriculum to get through and there is no time to not be getting through it. This is very sad and is very much the antithesis of mathematical practice. It is often seen as appropriate to bring students to the “right” mode of thinking as quickly as you can and have them then practice that work for swift fluency. The practice here as the math leader is to always speak up, and push ideas forward, quickly.
But kids are kids. They are smart thinkers. They do naturally wonder, and toy with ideas, and mull, and work to “bend the rules” just as is the wont of mathematicians. The Math Circle practice is to let space and silence come to the fore.
It is true that I might have 40 years more experience mulling on mathematics than do the kids, and so I might indeed readily see “pits” and “traps” in our thinking we might want to steer from. But, actually, sometimes we need to fall into pits and traps to know what one is and why we want to avoid it.
But the true answer for wanting to say little in the Math Circle is a selfish one: One can learn so much from kids thinking. Did I say they are brilliant thinkers? They will take a problem that seems so familiar and standard to you and see it in a light that puts a weird, wild, wonderful twist on, one so wild that it just knocks your socks off. That invariably happens.
So my advice to Math Leaders of all types is to reveal little: for both the kids’ sake and for your own. The human interaction of conversation of mathematics is an uplifting experience for the intellect and for the soul. If you make it one-sided, you miss out on it. Don’t!
MPG: Dan Meyer has written about the need for teachers to “Be less helpful” and to stop doing all the heavy lifting for students. I have taught, supervised student teachers, and done content coaching for novice and veteran teachers in grades 4 through 12. In each of these roles, I’ve found myself coming back to the same issues that Dan raises. You are doing a great deal of professional development work these days in your presentations about Exploding Dots and in your role as a visiting scholar for the Mathematics Association of America. Would you comment on how you balance the hands off approach you used when working with the Math Circle and the work you do with teachers?
JST: In my work, this question chiefly refers to the high-school curriculum, the high-pressured thing that one simply must “get through” with the kids. How do you bring the joy, the discovery, the back-and-forth of deep learning, and the intellectual play into that cultural mindset?
My personal approach was to be honest about the content. Really, very little of it, if any of it, is actually important. I cannot recall a single time in my personal life that I needed knowledge of the quadratic formula (and I am not even sure if I’ve actually needed in my professional mathematical work either)? I’ve never had to divide polynomials, and if someone wants a graph of a rational function, I recommend Wolfram|Alpha.
So, for the most of it, the mathematical content of the high school curriculum is not actually important. However, I still believe in teaching it. Teach that content as a vehicle for beautiful, human thinking and problem-solving. Use the content to teach the confidence to rely on your wits, use common sense, and have the perseverance to just “nut your way” through challenges.
The story of an algebra unit on quadratics is not “how to get answers to certain types of equations you weren’t asking about in the first place,” but rather a story of the power of symmetry in mathematical thinking. Once we learn that all quadratics graphs have a symmetric U-shapes, then let’s learn how to use that to our advantage.
A quadratic graph passes through the points (2,10), (5, 87), (12, 10). Where is its line of symmetry?
Memorizing x = -b/2a does me no good here. Common sense, on the other hand, does! We have two symmetrical points (2,10) and (12, 10) on a symmetrical graph. The line of symmetry must be right between them at x = 7.
Sketch a graph of y = (x-3)(x-5) + 7,
is a piece of cake. We can’t help but notice that x=3 and x=5 both seem interesting. In fact, they both give y = 7. Ah! Two symmetrical points on a symmetrical graph. The line of symmetry is at x = 4. And put in x=4 to get y = 6. That’s enough information to sketch the beast.
And now, the real joy of being a teacher:
Can you sketch y = x^2 – 2x + 5 with the same ease?
This is the Tanton version of that “heavy lifting” at the fore now. I am inviting everyone now to think like a mathematician. We have a problem. I want to avoid hard work. Let’s take the time to mull on this and see if we can avoid hard work.
The key is that we liked noticing obvious (and symmetrical) x-values. So can we do anything to this equation to highlight some obvious x-values? The only thing I can personally think to do right now is to rewrite it as:
y = x(x-2) + 5.
Ah. Now I see that x=0 and x=2 are interesting, and I am good to go!
Let’s teach all of high school mathematics this way, as a conversation of thinking principles, of being human, of puzzling, of mulling, and of taking the time to indeed just nut things out. My approach is Math Circle-y for sure. I certainly set the stage up for “problems” to mull on. But most of all, I’ve set the stage for letting the power of common sense reign.
This is what I loved about being a high school teacher: taking those curriculum topics and really work them back to teaching life thinking skills. Along with that I wanted to bring in the historical and mathematical stories to them all, and let each topic be an illustration of the human it deserves to be. I’ve said that English departments teach both grammar and poetry. Mathematics departments should, too.
MPG: You have a vast array of videos available on YouTube and via your own and other websites, all of them free. On the G’Day Math site, you have a series of short courses that include Exploding Dots, one on Quadratics, and one on Combinations and Permutations. Each of these has what I think of as a certain “Tantonesque” spin to it, where you come at a subject differently from what people are likely to have picked up in a typical American K-12 mathematics education or beyond. Is that something you’re consciously aware of trying to accomplish? Each of the courses I’ve mentioned seems to represent a sort of deconstruction of some assumptions students tend to bring to these topics based on typical curriculum and instruction. In using the section on solving quadratic equations from your course with my adult students for the past three and a half years, I’ve had a very large percentage of them ask why this topic wasn’t taught to them that way in school. I’ve also deepened my own understanding of the usual method of completing the square by looking at it through the lens of your course. Do you find yourself frequently at odds with the typical math content and teaching methods American teachers are expected to get their students to learn?
JST: I didn’t read ahead to upcoming questions! I think my previous answer attended to this question too, except for the “at odds” part.
Even though I had the freedom in my classroom to teach topics to all my students in any way I wished (and I know that is a supreme luxury in the K-12 world), I had little or no control over the common exams these students were to take. So I still had to make sure students could pass those exams with comfort and ease.
I lucked out in that the department did not ask “solve this equation by the XXX method” questions, otherwise I would have been forced to teach the XXX method and have students memorize the name of the XXX method (rather than just do it!). Assessment really does dictate the culture of the classroom practice.
I was fully aware that my students knew that students in other sections were of the same class were being taught various methods that had names and procedures so, to contend with FOMO (fear-of-missing-out) we’d discuss those methods too, often realise that is was often just easier to solve the problem rather than follow a prescribed set of steps, and not worry about having to have names in our heads. My students, once we learned how to solve quadratics, derived the quadratic formula themselves. They could choose to memorize and use it if it suited their style, or they could just continue to solve quadratics without invoking it directly.
In the K-8 world, following “alternative paths” to mathematics can be unsettling to many parents. What is not familiar is sometimes seen as dangerous and wrong. (“I learned to do long multiplication this way. What my child is doing is different. This is not math.”) Sometimes it is easy to mistakenly equate familiarity with understanding. Try explaining the long multiplication algorithm: why work from right to left? Why, sometimes in it, are you allowed to write down 3 x 5 is 15 and other times you can only write 5 and carry the one? Why does the algorithm for multiplication end with the task of addition? And so on. Often the response is “But it works. It gives the right answer. That’s what counts.” But if the goal is to get the answer, with no thinking involved along the way, then do the smart thing: pull out the calculator on your smart phone. That’s what I would personally do.
In the high-school world, parental oversight of the mathematics being taught is generally non-existent. (Is this a statement of the effectiveness of the previous generations’ educational experiences?) As a teacher, it was indeed a nominal issue for me. (But I have since run into the “back-to-the-basics” mantra in my general consulting work.)
And finally, there is the general issue of humans and change. It’s hard! Although I work with teachers with open minds and eager efforts, it’s hard to turn around a familiar approach to a classroom topic. That’s okay. All I say is: be kind to yourself. Try a small change or two in a direction that feels right and comfortable to your human relationship with mathematics and with your students. Be true to your human self. And if that small change resonates well, just expand on it a bit. That’s all.
MPG: Could you tell our readers about your latest project? How do you see it fitting into the overall arc of your work thus far?
JST: That would be the Global Math Project: www.theglobalmathproject.org).
Maybe I’ll cheat a bit here and share with you a pre-written blurb about it:
Here is a bold and audacious plan: Let’s create a fundamental paradigm shift as to how the world perceives and enjoys mathematics. Let’s have each and every person on this planet come to see mathematics as creative, meaningful, intriguing, delightful, relevant, and uplifting. Let’s prove to the world that mathematics—true, joyful mathematics—is a fundamental shared, human, experience.
Welcome to the Global Math Project!
Modifying the model of an Hour of Code, the Global Math Project invites teachers and students together (including math clubs, home school groups, math circle groups, etc.) to engage in a common piece of mathematics for just one class period during Global Math Week starting 10.10.2017, and to share their experiences — thoughts, photos, videos, extensions, inspirations — with the world on a special social media platform. That is, we invite the world to take part in a global conversation about a piece of joyous mathematics. The Global Math Project is based on the belief that meaningful mathematics can transcend borders and unite communities.
The roll-out topic for our inaugural 2017 Global Math Week is EXPLODING DOTS, a topic consistently described as a “mind-blowing” story of mathematics. We will offer the entire Exploding Dots curriculum—along with videos, texts, teaching guides, interactive web apps—completely freely, in perpetuity. We ask teachers, students, and math enthusiasts to just have a first Exploding Dots experience during Global Math Week.
To learn more about Global Math Week and the topic of Exploding Dots, see this video: https://www.youtube.com/watch?v=OyaeeVLdBi0).
To help spread the word about the project, consider becoming a Global Math Project Ambassador https://www.theglobalmathproject.org/ambassadors).
Basically, a team of seven of us, want to prove to the world that all mathematics, even curriculum mathematics, serves as a portal to human wonder and delight. And it is important we attend to curriculum mathematics.
So much of math enrichment works to show that mathematics is fun (true), creative (true), to be found in interesting places (true), comes in many forms (true), and so on. We play with soap bubbles for minimal surfaces, build Sierpinski tetrahedra with business cards, and roll weird solids of constant height. But, in the end, the reality of math, the school day-to-school day experience of mathematics student can hold on to is curriculum mathematics. It, too, can be fun, creative, surprising, multi-faceted, and so on. Let’s not give the implied message: “Just wade your way through 12 years of this other stuff, and if you make it, you get to do all this fun stuff later.”
The Global Math Project wants to turn that around. All mathematics should be a meaningful and uplifting story, a piece of poetry for the soul.
And as I mentioned earlier, Exploding Dots, our rollout topic for 2017, is the exemplar of all curriculum-relevant, mathematical poetry. (You can see them at https://gdaymath.com/courses/exploding-dots/.)
And how does this fit into the arc of my career? Well, I think it is clear that the mission of this work has been the driving force behind my career from the get-go. Now I and the team are being bold about it!
MPG: Finally, what do you see yourself doing ten years from now regarding mathematics and mathematics education?
Well, I truly hope that the 2017 Global Math Project is deemed a success as we would like to do it again with a brand new topic in 2018, and then again in 2019, and so on. All the materials we produce will be available, for free, in perpetuity. So in another ten years, it sounds like we might have a significant swag of a joyful K-12 curriculum familiar to the world.
Next step ….?
I’ll be around for it!
MPG: Let’s hope we all are. It’s an exciting time to be learning and teaching mathematics despite what the nay-sayers would have us believe. I feel extremely fortunate to be around at a time when so many good people are working to invite everyone to sit at the table of mathematics while making that invitation more than just lip-service through offering a host of inspiring and insightful content, free via the Internet. You are certainly one of the gifted mathematicians and teachers who are making the idea of “Math For All” more than an empty slogan. It’s been a privilege to share your thoughts with our readers. Thank you for taking the time to do so.
What’s great about math is that it’s sort of “reasoning” without meaning. And math problems and puzzles for us to use our reasoning systems in all sorts of clever ways, independent of the meaning.
This gives math-skilled people the ability to reason about life using logic, even when meaning opposes it. Sometimes this is entertaining or even silly. And sometimes it leads us to realize that life is filled with cultural myths that violate logic. Suddenly, much more is possible than an ordinary life, confined by the bounds of culture.
Mr Tanton, Michael introduced me to exploding dots several years ago and subsequently I have opened up my math course in 5th grade by playing “exploding dots”, complete with the little “pow”. I always love moving to the “1-10” machine, which they predict will be really challenging, only to find out that the “code” for say, 247 is… 247.
Later when we teach exponents, we start back with exploding dots. And when we solve the chances of correctly picking every team in the NCAA basketball tournament or solving the Tower of Hanoi, we have exploding dots to fall back upon for understanding.
Your paragraph about mathematicians mull but the reality of curriculum leads to “getting through” it without mulling is one I have shared with those who will listen. Also for the need to understand the logic in computation algorithms. Thank you for sharing.
I am so glad to actually know that I am so close to the way someone somewhere so far away is teaching math.