**An Interview with Keith Devlin: An Ambassador of Mathematics for Many Eras**

Dr. Keith Devlin is a co-founder and Executive Director of the university’s H-STAR institute, a co-founder of the Stanford mediaX research network, and a Senior Researcher at CSLI. He is a World Economic Forum Fellow, a Fellow of the American Association for the Advancement of Science, and a Fellow of the American Mathematical Society. His current research is focused on the use of different media to teach and communicate mathematics to diverse audiences. In this connection, he is a co-founder and President of an educational technology company, BrainQuake, that creates mathematics learning video games. He also works on the design of information/reasoning systems for intelligence analysis. Other research interests include the theory of information, models of reasoning, applications of mathematical techniques in the study of communication, and mathematical cognition. He has written 33 books and over 80 published research articles. Recipient of the Pythagoras Prize, the Peano Prize, the Carl Sagan Award, and the Joint Policy Board for Mathematics Communications Award. In 2003, he was recognized by the California State Assembly for his “innovative work and longtime service in the field of mathematics and its relation to logic and linguistics.” He is “the Math Guy” on National Public Radio.

**Michael Paul Goldenberg:** Welcome to The Math Blog, Keith. Thank you for joining us this month.

**Keith Devlin: **Happy to be here. Thank you for the invitation

**MPG:** Let me start with something from your latest book, *Finding Fibonacci*. You said:

*“**By good fortune, I had very good mathematics and science teachers in high school, and that, coupled with my early desire to ‘go into space science,’ meant that my school mathematics and science education continually stimulated my interests. I never found the subjects easy; to this day I find doing mathematics hard. That, to my mind, is what makes it so interesting. Anyone who thinks that doing math is easy does not know what math really is— just as the middle- school me did not really know there is more to history and literature than simply memorizing facts. But though I always found mathematics hard, I never lost interest in what was self- evidently (to me, and I suspect to anyone **who overcomes any negative effects of bad school mathematics classes) a towering edifice of collective human thought over several thousand years.”* (p.221)

I’d like you to comment further on what you mean when you say that you find that mathematics is hard: isn’t that odd coming from a professional mathematician?

**KD: **Coming from you, I’ll take the question as rhetorical. But for many people, it would not be (including many parents and some teachers I have interacted with), and that indicates the major disconnect we have between the professional mathematical community and the rest of society.

Mathematics is frequently taught as a collection of rules and procedures to be mastered and practiced. If that is someone’s only connection with the subject, it is natural to be left with the impression that once mastery has been achieved, there is nothing more to it apart from perhaps getting faster. But that would be like thinking that being a pianist amounts to being able to play scales quickly and fluently. The rules and procedures we teach in K-8, and to some extent beyond the early grades, are important, but they are just the raw ingredients you need to do mathematics.

Mathematics is a way of thinking that we humans have developed and learned to use over thousands of years to make sense of our world and to do things in our world. It is a way of thinking that the human brain does not find natural. Our brains are good at ** an**alogical thinking, but mathematics is predominantly logical. This is why mathematical thinking is such a crucial skill—it allows us to extend the capacity of our minds beyond what they evolved to do.

Professional mathematicians spend their time applying the mathematical approach to new problems. It would be a waste of time, and deeply boring, to tackle a problem that we find easy. So, we spend our time—and the money of those who employ us—working on new problems. And they never get any easier.

Speed has nothing to do with math by the way; like many successful mathematicians, I have always been slow. But understanding the fundamental concepts is key to doing math.

** ****MPG: **Do you think it’s possible to convey your ideas above to decision-makers at the federal, state, and district levels who are not mathematicians and may have the impression that mathematics is easy for people who are “good at math,” and those who find math to be hard must by definition be bad at math?

**KD:** What you refer to seems to be a widespread misconception, in my experience. In fact, the opposite is the case. If your experience doing mathematics is that it is easy, you are working way below your ability level, and that means you are wasting your time and the time/money of anyone you are working for. Mathematics is the result of efforts by thousands of people over several millennia to develop the powerful, versatile, and incredibly useful way of thinking we call mathematics. We should not squander it by using it on problems we find “easy”. Its power is that it can take us beyond the easy—sometimes well beyond easy!

** ****MPG:** One of your books, *Mathematics Education for a New Era: Video Games as a Medium for Learning*, explores the potential for learning math through video games. Could you share some of your conclusions about the potential for students to benefit through that medium and whether the growth of smart phones or any software you’ve seen in the last decade has changed your thinking in that regard?

**KD: **Video-game learning, when done well, exploits the human brain’s innate capacity to learn from experience. For things we ** do**, (as opposed to simply know), be it playing music, athletic pursuits, riding a bike, driving a car, flying a plane, acting, debating with others, mastering a foreign language, etc., by far the most efficient way to learn is to do it, not to read about it, attend a lecture, or watch a video. Those all have a role to play, but nothing beats live interaction within the activity.

I wrote that book you mentioned when I was working with a large video game studio to try to develop an immersive world—the hugely successful role-playing game *World of Warcraft* was our model—to try to build a math learning simulation in which kids (and adults) could learn to think mathematically by engaging with math in the same way they learn how to drive a car by driving around with an instructor, or pilots learn to fly by going in a flight simulator. The game we set out to build was intended to be a “math simulator”. I remain convinced to this day that, if built, such a learning environment would be hugely successful. But the cost grew to such an extent that it was clearly not possible as a commercial enterprise.

With the rise of mobile gaming, however, it’s possible to take a different approach. Instead of taking flight simulators as a model, you can take the musical instrument approach. In a math simulator, the learner enters a mathematical world. In the instrument approach, the learner plays on a mathematical instrument. So, I took everything I learned from trying to build a math simulator and co-founded a company (BrainQuake) to build math instruments that people can learn from by playing them the way they learn to play a musical instrument.

[It’s interesting. You get high-level engagement and interaction both when you are immersed in an environment and when you hold an environment on your hands. Lectures and videos lie in between, which is one reason why they are not as efficient for learning.]

Given everything we know about the power of learning by experience, I was not surprised when independent university studies showed that our math instruments were effective. What did surprise me was that you can get significant positive learning effects after as little as two hours play.

For sure, learning math through playing math instruments is only part of what is required to learn how to think mathematically. It is fine for providing the basic mathematical knowledge required for everyday life in today’s society, but to go beyond that, a learner eventually needs to master the symbolic notations and formal, abstract reasoning. I believe that step will be easier once the learner has mastered the basic concepts using instruments like ours, but the research on that has yet to be done.

Having said all of that, I should note that there are north of 20,000 interactive math apps available for download onto your mobile device, but at most, maybe 20 or so are designed to provide “math instruments” of the kind I am talking about. The vast majority focus on mastery (and execution speed) of basic math facts and rules. As I already indicated, that has little to do with being a competent mathematical thinker. That does not make them a total waste, since doing mathematics does require mastery of a few basics, and video games provide an efficient way to achieve that. But remember, there are apps on our smartphones that perform those activities for us more quickly and more accurately than we can. Consequently, in today’s world, we don’t need to master the basics in order to perform basic calculations in our head or on paper, but rather to make efficient use of the technologies that are available.

**MPG:** You’ve taught many iterations of a MOOC [Massive Online Open Course] in mathematical thinking via Stanford University and Coursera. (I took the course early on, had to drop because of work conflicts, then took it again about 18 months ago and completed the full course). What has changed in your thinking about the potential for such courses as you’ve gained more experience with them?

**KD:** I remain of the same mind as when I first offered the course in 2012, when MOOCS were a new thing. I designed my course as an interactive experience, where students around the world could interact online, in almost-real-time, with one another, with me, and with a small army of tutors I recruited.

The video lectures and online quizzes that form the most visible aspect of a MOOC are just mechanisms to set the stage for students to work on problems alone, in pairs, or in groups, co-located or over social media, consulting with experts when required.

In other words, I set out to take online, to a larger audience, as best I could, the successful interactive classes, based on exploration and discussion, that you find at the best colleges and universities. It worked pretty well for a while, but then the need to develop a sustaining funding model forced Coursera to modify the platform to one that essentially provides “learning on demand”. All the apparatus of a fixed course schedule, with submission deadlines, that was essential to keep as many as 80,000 students all working on the same problems at the same time to force intense, active discussions, was discarded. So, after the eighth session, I and my tutors pretty well dropped out. (It’s not clear how long we could have continued anyway. It was a lot of work!)

The course still runs, with a new session starting immediately the previous one ends, and I know from emails I receive occasionally from students who take it that some, at least, find it beneficial. But to me, it is now really just an interactive textbook. That’s not to denigrate it. I have spent my life writing textbooks. And I think an interactive textbook with video instruction is probably more effective than a print textbook, particularly for math. So I am all for textbooks, and for MOOCs as they have now become. But the new MOOCs cannot come close to the full social experience that is a living math class. In particular, I think that to be successful, the vast majority of students in such a MOOC will need a tutor, both for pedagogic and psychological reasons. I fear the very best MOOCs came and went in a brief, three-year span. But what is left is still better than we had before!

** ****MPG: **Let’s talk about multiplication and repeated addition. Nine years ago, you wrote a column that shook up a lot of people. In “It Ain’t No Repeated Addition,” you said:

*Multiplication simply is not repeated addition, and telling young pupils it inevitably leads to problems when they subsequently learn that it is not. Multiplication of natural numbers certainly gives the same result as repeated addition, but that does not make it the same. Riding my bicycle gets me to my office in about the same time as taking my car, but the two processes are very different. Telling students falsehoods on the assumption that they can be corrected later is rarely a good idea. And telling them that multiplication is repeated addition definitely requires undoing later.*

That alone engendered a lot of flack. One of the more vociferous reactions came from some know-it-all blogger named “Michael Paul Goldenberg.” My first piece, “Devlin On Multiplication (or What is the meaning of “is”?)” was a less-than-diplomatic attempt to suggest that you were out of your element trying to tell K-6 teachers their business, though I should say that reading it now, I was less awful than I feared. As it happened, reading posts by other bloggers and from a more reasonable subset of their commentators very quickly led me to do something my greatest detractors believed impossible: I wrote a retraction of my critique of your post and became quite an advocate for the notion that “multiplication IS repeated addition” (MIRA) is a problem.

You subsequently wrote more pieces on this issue for your column[“It’s Still Not Repeated Addition,” “SpellMultiplication and Those Pesky British Spellings,” “Repeated Addition – One More Spin,” “What Exactly is Multiplication?” and “How multiplication is really defined in Peano arithmetic“. What are your thoughts about it now?

**KD: **Interestingly, you were one of the few detractors who I responded to. [Which I guess led, many years later, after various exchanges I hope have been as valuable to you as to me, to me writing these replies!] Most respondents simply spammed me, demonstrating their mathematical ignorance in every sentence. You were wrong but in a thoughtful (and civilized) way.

I actually tried to go easy on “telling teachers their job.” Rather, I was writing from the perspective of my job—a professional mathematician. From which perspective, multiplication simply is not repeated addition. Period. In fact, unlike multiplication, “repeated addition” is not even a mathematical operation, it’s a procedural schema that lives in the meta-mathematical world.

I was genuinely surprised at the violent reaction, particularly from teachers. I had gone along happily in my mathematical world thinking everyone in the mathematics business knew that addition, multiplication, and exponentiation, are three fundamental, binary operations on the real numbers, none of which can be defined in terms of the others. (We’re talking operations here; when you move into the world of mathematical models, such as functions, the situation gets more complicated, but my initial, light-hearted, off-the-cuff observation was aimed at the world of K-8 teaching, so functions are way off in the collegiate future.)

My naïve assumption about what the majority of teachers knew was no doubt buttressed by my having served on the Mathematical Science Education Board at the time the NRC volume *Adding It Up* came out. My exposure—through my MSEB membership—to the work of some of the leading scholars in mathematics education actually led me to a more sophisticated understanding of multiplication and the problems it presents to K-8 teaching. Since *Adding It Up* was intended to be the “bible” for K-12 math teaching in the US, I assumed it consolidated in one place what all math teachers knew. I made my throwaway, online remark at the end of a “Devlin’s Angle” post for the MAA thinking I was directing it to a tiny few who had, as it were, not got the NRC memo. Then the sky fell in!

In the end, as you note, I wrote a whole series of articles about the true nature of multiplication, with the later posts providing citations to the extensive literature on multiplication (citations were not common in “Devlin’s Angle”), to counter the frequent response that I was just “giving my opinion”. To be sure, a lot of what I write in my column is opinion, but my pieces about multiplication were not at all opinion. I was describing, as simply as I could, the rigorous definitions of the fundamental operations of arithmetic, ** as accepted and used by the mathematical community the world over**.

I would hope that many people who read those columns—and the many online discussions they generated—did eventually correct their erroneous conception of multiplication. We are talking accepted definitions here, not some “self-evident truths” that might cause us embarrassment if we get them wrong. If any of us has an incorrect concept, something went wrong with the way we were taught. Okay, that’s no big deal; we correct it and move on.

For my part, I learned a lot about how poor is the mathematics education of many math teachers. Again, it’s not their fault! The system failed them!

Mathematics is a particularly difficult subject to teach well, and multiplication is, in fact, one of the most difficult concepts, having many subtleties. Referring to basic arithmetic as “elementary mathematics” is a mistake.

To return to Fibonacci, where we started, it’s worth noting that in his mammoth arithmetic textbook *Liber abbaci*, the book that brought modern arithmetic to the West, Leonardo (to use his real name) covered multiplication before addition, and devoted a lot more space to it. He knew that mastery of an efficient method for performing multiplication was the key to modern trade, commerce, and finance. Addition just doesn’t cut it. Even if you repeat it! 🙂

**MPG: **I think readers here will be interested in your analogies in *The Man of Numbers *and *Finding Fibonacci *between Leonardo and Steven Jobs. Could you share some of that here?

** ****KD: **Yes, that similarity struck me as I was flying back and forth between Tuscany and Silicon Valley working on my Fibonacci history. Both Leonardo and Steve (“Leonardo and Steve” is the title of my e-book companion to *The Man of Numbers*, where I laid out my observations) saw something when they were young that others had developed, but had not fully exploited—indeed, in each case did not appear to see the development’s world-changing potential.

The traders in the Arabic-speaking world of North Africa inherited (from the Indians) and developed modern arithmetic and numerical algebra, but did not fully exploit its enormous potential, and the researchers at Xerox PARC in Silicon Valley inherited (in part from SRI) and developed the modern WIMPS computer interface, but Xerox did not recognize its enormous potential. Leonardo saw the former, Steve saw the latter. Both immediately moved heaven and earth to turn what they had seen into a consumer product.

In both cases, it was an interface issue, to make computing more readily accessible to ordinary people. Leonardo wrote *Liber abbaci* and then a more popular version *Book for Merchants*; Steve developed the Lisa personal computer and then a much cheaper, easier-to-use version, the Macintosh. It was all about recognizing the potential of the invention and then using good audience design and product marketing to bring the new invention to the consumer market. The result in both cases was a personal computing revolution. I describe the full story in all three of my Fibonacci history books. The parallels are truly uncanny.

** ****MPG: **What projects and books are you working on next?

**KD: **My ed tech startup BrainQuake takes a lot of my time. We won a $1M award from the Department of Education a year ago that is funding us for two years, which enables us to (1) build two new math learning games, (2) develop an adaptive engine to deliver puzzles at the right level of difficulty for the player, (3) produce some game-related classroom apps for teachers to use to assist kids transfer what they learn using our games (which, with a nod to my Leonardo and Steve work, provide novel interfaces to mathematics that are optimized for learning) to the familiar paper-and-pencil interface to mathematics required for more advanced work, and (4) create an online course to help teachers use our products more effectively. We are working with an educational assessment organization, WestEd, to run efficacy studies of our products.

I keep getting attracted to go back to the kind of work I did for the Defense Department early post-9/11 on improving intelligence analysis. It’s a fascinating topic, with many potential ramifications. (My video game company is in some ways a derivative of that project. Exercise for the reader: explain the connection.) The problem for me is that I can do that kind of work only when I trust my government. Post-Snowden, I don’t; indeed, my trust in my government has sunk even lower over the past few months. I hope things will change. In the interim, I am at the early stages of resuming similar work for the commercial world with a colleague in the UK.

I’m also working on a new mathematical thinking course for adults to run at Stanford as a Continuing Studies course this fall, and a book about what it takes to become world class in mathematics. Both of those projects are with my friend Gary Antonick, the former editor of the New York Times “Numberplay” column.

**MPG: **Thank you so much for speaking with me. I am sure that our readers will be intrigued by what you’ve had to say.

** **

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Hi Michael,

I enjoyed your interview with Keith. I’m glad he’s still pursuing his Apollo program though on a much smaller scale. Jason Dyer blogged about Keith’s vision back in 2012:

Keith Devlin has proposed a “simulator” [Apollo project] to teach mathematics via an integrated videogame world. Essentially he wants to tap the same effect seen in a study of Brazilian street mathematics where mathematics done in a natural environment gives a high success rate even when the exact same test subjects do badly on a formal test of the same concepts.

Its too bad the price tag for Keith’s group to build it is astronomical.

Best wishes,

Ihor

Thank you Sir

Keith Devlin (and Michael Goldenberg) make some interesting comments about multiplication as not ’repeated addition.’ Let me forego comments about age appropriate ways of learning multiplication (learning, that is, over time) and try to do, so to speak, a mathematical resuscitation of what might be called ‘repeated addition.’ I need, however, to note in advance that I have neither read Keith Devlin’s writings on this subject or heard there was a furor (that is possibly not quite true as once a mathematician acquaintance burst into my office concerned about children experiencing/being taught multiplication in an ‘improper’ fashion and that may have had its genesis in something Keith Devlin wrote).

Much of what I write here has its roots in, for example, Leon Cohen and Gertrude Erlich’s ‘The Structure of the Real Number System’ or Jean Rubin’s ‘Set Theory for Mathematicians,’ but, my impression (not having looked into these things for awhile) what I will sketch is not especially unusual. I’m not going to give page numbers or a complete development; if you find that necessary you are on your own. Okay, I define m x n as

m x n = p(m,n)

where for m and n in the naturals (yes, I know people disagree about whether 0 is a natural number)

p(m,0) = 0

p(m,S(n)) = p(m.n) + m [Here S is the successor function (i.e. S(n) = n+1)]

So what is

4 x 5

Well 5 = S(4) so by definition

4 x 5 = p(4,4) + 4

= (p(4,3) + 4) + 4

= ((p(4,2) + 4) + 4) + 4)

= (((p(4,1) + 4 ) + 4) + 4) + 4

= ((((p(4,0) + 4) + 4) + 4) + 4) + 4

Sure looks like ‘repeated addition’ to me (I’m cumulatively adding the 4 ‘five’ times (I know, I know, people quibble at the language of “4 ‘five’ times,” but you know what I mean). In fact I often use such addition to compute products; for instance, I used to forget 6 x 7 so I did 6 x 6 + 6. Oh, you say “I’m not using the above definition; I’m just using the distributive law.” Actually the distributive law is a consequence of the definition and, no, I don’t, in this case, use the distributive law.

I can hear somebody else saying, “But what about the integers. This ‘repeated addition’ stuff isn’t of much use there. First, I remark that the integers are an extension of the naturals so there is going to be some interesting behavior when we look at the embedding of the naturals. There are a number of ways to go about this, but, for fun, let me try to extend p (i.e. the foregoing is possibly a flight of fancy and may have serious flaws so read carefully).

Okay, we can certainly define the successor function on the integers (e.g.. S(-4) = -3). So, for example, if I was to retain my definition of p then

p(m,0) = 0

p(m,0) = m + p(m,-1)

and, hence, p(m,-1) = -m. A little computation gives, for example,

p(m,-2) = -m + -m

and so forth. Finally, there seems no large problem when m≤0; the usual sign rules are, as they should be, just a consequence of the definition extended to the integers.

I can hear somebody saying, “Now , I expect you are going to say that division is repeated subtraction. That is not only not right, it is not wrong.” Well, now that you mention it (smile).

Quote from the article: “I had gone along happily in my mathematical world thinking everyone in the mathematics business knew that addition, multiplication, and exponentiation, are three fundamental, binary operations on the real numbers, none of which can be defined in terms of the others. (We’re talking operations here; when you move into the world of mathematical models, such as functions, the situation gets more complicated, but my initial, light-hearted, off-the-cuff observation was aimed at the world of K-8 teaching, so functions are way off in the collegiate future.)”

Commentators on the original posts kept coming back with discussions of recursive definitions of functions on the natural numbers. All good fun for math majors, but kinda missing the point about teaching young kids what the basic operations of arithmetic are! (Also, doing those recursions correctly is tricky, and when you do it right you don’t get “repeated addition” because there is no such function.)

“I can hear somebody saying, “Now, I expect you are going to say that division is repeated subtraction.”

Ed, I find repeated subtraction a useful model when I take on why division by 0 is undefined. It’s not the only way I go at it: the first is just to look at what happens if we assume that n /0 = x for non-zero n, and then solve for n. Obviously, we run into the uncomfortable conclusion that 0*x = n, when n was assumed to be non-zero (what to think about the situation when n does equal 0 is another interesting conversation).

But with “division can be thought of as ‘repeated subtraction’,” I’ve been able to help students and one math-phobic friend of 50+ years feel comfortable with stating that division of a non-zero integer is undefined (I’m not getting into non-integers with them or you here).

Take as a working model “Division of n by non-zero integer d asks ‘How many times can we subtract d from n until we have 0 or less than d remaining?'” Using say 6 for n, I start with d = 6, then 3, 2, and 1. Then I suggest we try making d = 1/2, then 1/4, 1/100, 1/1,000,000, with most people seeing that the result gets larger and larger the closer we make d to 0 without reaching it. For a lot of people, that makes sense enough. Sometimes I ask, “If we made d equal to 0, would we ever get any closer to the goal of having 0 left?” Some people find that helpful; others just get confused. But almost always, one of these approaches is helpful for getting an intuitive sense of why dividing by 0 doesn’t make sense, without my having to resort to algebra, which makes some people’s eyes start swimming immediately.

So I wouldn’t want to abandon the repeated subtraction metaphor. But I would always eschew stating “division IS repeated subtraction.” Ditto saying “multiplication IS repeated addition.” Your mileage may vary, of course.

It’s fine to teach repeated addition. It leads to tiling, arrays and equal group models. Just don’t say repeated addition IS multiplication because it’s not!

Division CANNOT be thought of as ‘repeated subtraction’. Repeated subtraction is an ALGORITHM or recipe for doing division. It’s like saying travel can be done by walking so travel can be thought of as walking. OK, let’s drive. Straight away our definition that travel is walking comes unstuck. Division can’t be thought of as ‘repeated subtraction’ because as any ancient Egyptian will tell you, division is done via addition!

Multiplication and division are concerned with similarity. For multiplication, as the unit is to the multiplier, the multiplicand is to the product. For division, as the divisor is to the unit, the dividend is to the quotient. Such concepts hold from the Naturals to the Reals. Why? Because math is not just about patterns, it is about RELATIONSHIPS.

Exponentiation is NOT repeated multiplication. Again, RM is just an algorithm. Exponentiation (related to logs) is about growth and decay.

Invoking Peano Arithmetic to justify the belief Multiplication IS Repeated Addition (MIRA) myth, is a waste of time and energy. Too many involved with math education prefer to baffle with BS rather than think deeply through the eyes and intuitions of the customer – the student. With PA, (and likely set theory) the only logical conclusion you can reach is that multiplication is successive counting. Why? Oh, you must define the numbers in 4 multiplied by 5. What is 4? Ah, the successor of 3! But what is 3? Ah, the successor of 2, and so on.

Number theory, when limited to the positive Naturals, will always reduce to counting. Not very useful, and problematic as soon as more interesting numbers are enCOUNTered.

I enjoyed your interview with Keith. I’m glad he’s still pursuing his Apollo program though on a much smaller scale. Jason Dyer blogged about Keith’s vision back in 2012: