Making Math Fun: The Power of Mathematical Visualization—The Torch Podcast

An Interview with Professor James Tanton, Ph.D.

Making math fun? Is it possible? On this Episode of the Torch we examine how seeing math problems in a new way—by constructing pictures in your mind—will make finding the answers more obvious than by conventional means you learned in school.

Here to discuss the tools for making math fun is James Tanton Ph.D., Mathematician at Large at The Mathematical Association of America.

The following transcript has been edited slightly for readability.

The Great Courses: A lot of people think that they’re just bad at math, right? It’s a common perception.

James Tanton: Indeed.

The Great Courses: Indeed, I know I feel it, but what if you could learn to see the problems in mathematics in a new helpful way? Mathematicians have worked out some special ways to construct pictures in your mind so that the answers to many mathematical ideas become a little bit more obvious than if you try to do it otherwise. Here to explain, James Tanton, professor of our geometry course, and now you have another one.

James Tanton: Indeed, exactly.

The Great Courses: Good to see you again. The new course is called The Power of Mathematical Visualization.

James Tanton: Indeed it is.

The Great Courses: We’re going to talk about that in a second, but first of all, I want to say that I’ve always loved your title. You are Mathematician in Residence at the Mathematical Association of America.

James Tanton: Actually, I’ve moved.

The Great Courses: Have you moved?

James Tanton: I’m now the Mathematician At Large.

The Great Courses: At Large?

James Tanton: Talk about a better title than that. I get to be the Mathematician At Large.

The Great Courses: In residence, I always perceive that you’d be strolling along the walkways.

James Tanton: I know. I’m now roaming free, wreaking havoc wherever I go, mathematical, all places.

The Great Courses: At large, you’re everywhere. You’ve blown up. We’re going to talk about the new course, but I want to start with a quote that we have on, your professor bio page. I just love this quote, and I think it says so much about you and how you do your work.

James Tanton: I wonder what it is…

The Great Courses: “Our complex society demands not only mastery of quantitative skills, the math, but also the confidence to ask new questions to explore, wonder, flail, to rely on one’s wits, and to innovate,” and then you say, “Let’s teach joyous and successful thinking.” I love that!

Let's teach joyous and successful thinking. Click To Tweet

James Tanton: I’m all for that absolutely, and here we are with the The Power of Visualization in Mathematics.

Learn More: The Power of a Mathematical Picture

The Great Courses: Right, so let’s talk about some joyous and successful thinking in mathematical visualization. What are we talking about when we say that, use that term?

Moving Past Memorization

James Tanton: Let’s think about a typical experience. Most of us have gone through schooling in mathematics, and many people, from decades past in particular, say that it might not have been the most enlightened experience of math education, in their day.

A lot of rote memorization, drill, skill, practice, do it under speed. All about “what” questions, what’s 17 times 18, what’s 34 times 92, over and over again. No “why” questions, no “what else” questions.

Better yet, no “what if” questions, because the “what if” questions, which is about science, business technology, needing to innovate, answers those “what if” questions.

The Great Courses: Give me an example of a “what if” question in mathematics.

James Tanton: All right, I’ll do it all in one fell swoop. We’ve all gone through, in our lives, probably grade 4, 5, 6, somewhere around there, long multiplication. We’ve all learned to do 17 by 18, and we learned a particular algorithm, which we’re taught, we do it, 17, 18, lined up by columns, a multiplication sign.

Then, 17, 18, we read one seven, but we do the algorithm backwards, from right to left, which is weird, because in math class, you’re suddenly reading from right to left, though you’re taught to read left to right everywhere else.

You don’t question that, because we’re so used to it. You do things like 7 times 8 is 56, but you don’t allow yourself to write down 56. If you remember, you write down a 6 and do this weird thing with carrying a 5. You just do it, you’re often just taught to do it, and it’s deeply mysterious.

The thing is, we’re so familiar with it, that we forget that we don’t understand it. We often equate familiarity with understanding. To really do more, let’s understand it. I’m going to take 17 time 18, and push it to the max for you.

I’m going to do it in a way that doesn’t require memorization, in fact, I’ll go back a step first and come to the 17 and 18s, and then I’ll play with it in a “what if” way.

Many kids, early on, have to memorize the multiplication tables.

Image of girl writing Multiplication tables for article on Making Math Fun

Making Math Fun: Multiplication

The Great Courses: Right, a lot of memorization.

James Tanton: A lot of memorization, and actually, it worries me, because that induced a lot of fear, so why equate fear with mathematics quite early on? That’s the emotion people have with mathematics.

Yes, you need to know your multiplication tables at some point, but you don’t need to memorize all of them, because here’s a cute trick to do with your hands to get up to ten times tables.

Why equate fear with mathematics? Click To Tweet

The Great Courses: With your hands?

James Tanton: With your hands. I mentioned 7 times 8, I’m going to do 7 times 8 with my hands. A closed fist is going to represent 5. To make it 7, add two more digits, so 5 plus 2 makes 7. Closed fist represents five.

Closed fist is 5, add 3 more digits, make it 8, so here’s 7, here’s 8. This little trick I’m about to do, each digit raised is worth ten, so right now I’ve got 5 digits raised, 1, 2, 3, 4, 5, so each worth 10, 50.

All I need to know is, up to my 5 times tables for what we’re doing now, which is multiply 3, three digits down on my right, 2 digits down on my left, 3 times 2 is 6, I claim 7 times 8 is 56.

he Great Courses: Oh, wow.

James Tanton: I’ll do another one. 9, a closed fist is 5, add 4 more fingers, makes it 9. Closed fist is 5, one more finger makes it 6. Each digit up is worth 10, we are working with 10 fingers after all. 50, multiply the digits down, 1 down, 4 down, 1 times 4 is 4, 6 times 9 is 54. Now, getting to your area.

The Great Courses: Bravo.

James Tanton: All right. Now, let’s get weird.  That was just fingers. We’ve got fingers and toes. So, let me do 17 times 18 this way, without a lick of memorization.

On my right side, 10 digits down on this side, I’m going to make it 17. Up 5 toes, you can’t see it, and up 2 more fingers. Right now I’ve got 7 digits raised on my right side. 18, everything down is now worth ten, we have 5 toes, 3 fingers, 5 and 3 makes this 18. 7 digits up on this side, 8 digits up on this side, great.

Now, each digit up is now worth 20, because I’m using all 20 digits. 7 up on this side, 8 up on this side, I’ve got 15 digits up, each worth 20, so that’s 300, and multiply the digits down, 3 and 2, 3 times 2 is six. 17 times 18 is 306.

Learn More: Visualizing Extraordinary Ways to Multiply

The Great Courses: Wow, we’re going to have to have kids to math in an aerobics studio.

James Tanton: That would be kind of cool.

The Great Courses: That would be kind of cool, right, innovation.

James Tanton: Here’s the thing. What I’ve just taught is yet another algorithm for you. The real question is, why does that work?

The Great Courses: Yes, why does it?

James Tanton: Because then, don’t memorize it, because if you understand what’s going on, then it gets really cool, and that’s where the visualization math comes in. What is multiplication, really? It’s about the area of a rectangle.

17 times 18 is for a rectangle 17 wide, 18 high, the area of that rectangle is 17 times 18, and I say, don’t memorize a thing, because the natural thing to do with 17 times 18 is chop it into pieces, probably 10 and 7, 10 and 8, and they could be 4 pieces. I could draw the picture on camera, but it’s actually easy to see the answers to, just add them up, and you’ll get 306.

The Great Courses: Is this new thinking, or did we somehow, was there a war at some point between visual styles of math and non-visual styles?

James Tanton: Something’s going on there. There’s nothing actually new here, because of course multiplication is just an area problem. What I think happens is, that somewhere in that school experience, we associate visualization, pictures, with being very low-level work. I don’t know why.

The Great Courses: Yeah, elementary.

James Tanton: I don’t know why, because then when you get to high school, you’re meant to be very analytical and algebraic. There’s nothing but formulas and equations, and we lose sight of the visual side of things. I don’t know why, because now we can get to the “what if” parts.

Portrait of pensive student visualizing math problem The Great Courses: You said you were going to do it all holistically, in a way that only an at large mathematician can do, so keep going.

James Tanton: Thank you. What’s the most annoying question out there about numbers? Why is negative times a negative positive? I’m going to take my picture of 17 times 18. Instead of splitting the 17 as 10 and 7, and the 18 as 10 and 8, how about I get quirky, and split this 20 and negative 3, and 20 and negative 2, 17 and 18?

There’ll be a little piece there that wants to be a negative times a negative, and the math will now show you, wow, it has to be positive 6, you can just see what negative times negative has to be positive, from that picture of multiplication. Well in this course, that’s one set of lectures, and we go from there.

Learn More: Visualizing Negative Numbers

The Great Courses: That is awesome. By the way “What are the most challenging mathematical concepts to visualize?”  You just did multiplication. So what about something a little more complex? What are some tough ones to visualize?

James Tanton: That’s the wonderful thing I love. I was a university professor for a number of years, and I got very interested in the state of school education. I became high school teacher because of that very challenge.

You think, algebra 2, how do you possibly visualize quadratic equations? How can you possibly visualize that? Yes, we draw graphs of things, great. That’s really not that deep, but that’s been my beautiful challenge. How can I go through that entire school curriculum and say, actually, there is a visual way to think about it. It is hard, there are some challenges.

That’s what I love about mathematics. Muddle stuff, and things come to you after awhile, and that’s actually what I’ve been doing for the last 20 good years of my life, that lead to this. Some are definitely more challenging than others, no doubt about it, but actually, I think visuals are at the basis of most everything.

I think so much of mathematics is actually visual in fact, it’s hard for me to pick things out where it’s not.

I think so much of math is actually visual. Click To Tweet

The Great Courses: You used your body to visualize this. What are some other tools that you use for visualization?

James Tanton: Actually, of course, we do actual objects. In fact, a very good way to think of negative numbers, I mentioned already, is to actually think of it being a sandbox. What do you do, you make piles of sand, so I’ve discovered, the counting up of piles of sand.

With the sandbox, you can do something magical, you can do the opposite of a pile, you can dig a hole. Now you’ve got piles and holes, put them together, and they annihilate.

There’s a visual image that actually explains all of the annoying arithmetic of negative numbers. All those minus signs you lost track of in algebra class actually makes sense. Just keep piles and holes in your brain, and it all makes sense, all of a sudden. Distributing a negative sign becomes a non-issue. Don’t even call it that, just do it. If you see the picture, you just do it without thinking.

Teaching Teachers

The Great Courses: You teach teachers.

James Tanton: Indeed I do.

The Great Courses: Is it a challenge, because teachers come in with the preconceived, older ideas?

James Tanton: Actually, you know what, it is. I mean, yes and no, everyone’s their own person, but the thing is, that what I said before, we often equate familiarity with understanding. We also know that long multiplication algorithm that it’s hard for us to realize that we don’t actually understand it, because we can do it instinctively.

Doing it instinctively doesn’t mean you understand it. That means, as a teacher, when you’re trying to teach, we forget that students are seeing it for the first time, so actually, it is bizarre to them. It looks bizarre, and we adults forget that. That’s the hard part for me with the teachers, to forget what it’s like not to know it so well, and then realize, you actually don’t know it, like really know it.

The Great Courses: One of the questions that people will want to understand about the new courses, how difficult is the math that you’re doing? What levels are we going to go into?

James Tanton: Here’s the beautiful thing about mathematics. All mathematics, no matter what level it is, what seems extraordinarily elementary, or more advanced, serves as a portal to great, wondrous, delightful thinking. I actually start with the really beginning, literally. Lecture one is about counting, the counting numbers, 1, 2, 3. You think there might be just, what could he possibly say about the counting numbers of interest?

The Great Courses: Do people think that’s too remedial?

James Tanton: Maybe, but however, you look at it. You’ll see that the counting numbers serve as a portal to incredible, incredible work. We’ll get to something called the Galilean Ratios in that particular lecture, just from the simplicity of counting. Then we’ll go to negative numbers, and finally understand them.

This course actually works from the very beginning, and it works its way all the way through, and we do get to some quite complex stuff, so there are challenges in there. Don’t you worry.

Image of Leonardo_da_Pisa Fibonnaci
Leonardo da Pisa Fibonnaci

In fact, I’ll even tell you, one of my lectures, the lectures on Fibonacci numbers, I actually present, I think for the first time, to the world, a new result on Fibonacci numbers, in there, because I’ve not seen it, I’ve not seen it in the literature. I’ve talked to my colleagues, no one’s known this. Maybe I’m missing it, but this is my public debut of this one particular mathematical result about Fibonacci numbers.

The Great Courses: Amazing, all right, you heard it here first, folks. I think everyone at The Great Courses is going to try to catch you on that, so we will all verify that.

James Tanton: I know. I think, I have to do a proper literature search, but I haven’t come across it.

Learn More: Visualizing the Fibonnaci Numbers

Is the Current Curriculum Working?

The Great Courses: You bring mathematics so to life. How are we doing in math education in the United States?

James Tanton: Oh, that’s so interesting, yes, I know. Of course, the big fuss right now is about the common core, and that’s really interesting. They really are trying to bring thinking into the curriculum. I don’t know what you feel about it. I have to say, I can only applaud some group of people trying to bring thinking into the state of mathematics education, so I say, yes.

Maybe the troubles we’re experiencing are about the implementation of it, not so much the deal in itself. I think things are actually much more light, and things are actually moving forward in a good way, but it’s very hard, because again, we have this fear that if we do something different, it’s wrong, because it’s not familiar.

I do worry about the political arguments that go along, we get the back to basics crowd, please memorize everything, because that’s how we did it as adults, it worked for us. Don’t fix it if it ain’t broken.

The Great Courses: That’s before they see you whip out the toes.

James Tanton: Well, there we go. I think it’s a blend of both. It’s not a binary item, and that’s what frustrates me. Of course we need to get some stuff in our heads, of course we can just play and understand and work with both. It’s a blend.

The Great Courses: The field has always been at least perceived as more of a male dominated field, especially in the higher levels. Are you seeing more gender equality?

James Tanton: Me personally? Absolutely, in my circles, yes. I actually have an argument for that one. I’ve had a feeling that this test taking strategy is all about doing things under speed, in a competition-like way.

I’ve a feeling that’s a little bit gender biased, therefore, if we’ve got an education system based on doing things under speed, maybe that appeals to a particular gender. I have to then question, why do you always have to do things under speed in mathematics?

The Great Courses: That’s a great point, by the way.

James Tanton: The mind, the mulling is the beautiful part.

The Great Courses: And in the mulling come the breakthroughs, right?

The mulling is the beautiful part, that's where the epiphanies lie. Click To Tweet

James Tanton: The epiphanies, that’s where the epiphanies lie. Let our children have epiphanies. Let our audience have epiphanies. Epiphanies, one and all.

The Great Courses: I love that word.

James Tanton: Yes.

The Great Courses: Have there been any great epiphanies lately in mathematics that you can make us neophytes aware of?

James Tanton: Yeah, there’s a lot of great work going on. People still don’t understand the prime numbers, for example. The numbers 2, 3, 5, 7, 11, and so forth, that factor only the most simplistic of ways.

People wish they knew a pathway to them, and so forth, but people have started finding structure in those prime numbers, left, right, and center. Every now and then, people are finding…

I mean, there’s something called the twin prime conjecture. Two primes that are very close to each other, like 3 and 5, just one little number in between them. 11 and 13. No one knows if there are infinitely many examples of those, but the bound on how long those examples could be, could there be like a triple of them, could be … We’ve got twin, there could be triples, and so forth. That’s typing up as we speak, and many people work on this stuff.

It’s kind of exciting, and you wonder, why does it even matter for the world? First of all, poetry, it’s beautiful. Why not just enjoy it for the sake of being poetry? But the funny thing is, we like prime numbers, for all our encryption systems that we use. Credit card security and all the rest, based on large primes. Anything we understand about primes is good for the security of the world, so there we are.

The Great Courses: By the way, is this what happens, if I’m going to get a beer with mathematicians? You’re having this discussion?

James Tanton: Correct. On napkins, usually. Lots of paper stuff. Lots of diagrams, because we’re official

The Great Courses: Napkins are filled with figures. The course, not remedial. Is it good for people that have very astute knowledge of mathematics, as well as people who struggle with it?

James Tanton: I would absolutely say yes, because I think most people find … My brain seems to work in an interesting way, to take a subject that seems very familiar, and basically turn it in a way that something just brings new light onto it.

Of course, like I said, even the most, what you think is an elementary topic can serve as a portal to so much more. Yes, let’s go back and look at counting numbers, see if I can push them in a direction you haven’t thought of before.

The Great Courses: The man is definitely teaching joyous and successful thinking.

James Tanton: Thank you.

More episodes:
Searching For The Theory Of Everything
Popular Brain Myths We All Thought Were True
Debate Tactics That Work

From the Lecture Series: The Power of Mathematical Visualization
Taught by Professor James Tanton, Ph.D.
By Hans-Peter Postel (Own work) [CC BY 2.5 (], via Wikimedia Commons