Interview with Keith Devlin
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After the recent repeated addition = multiplication kerfuffle, I asked Keith Devlin if I could interview him, and he graciously agreed.
Below is the full interview—just ten questions long. The interview was conducted via E-mail from August 10, 2008 to August 16, 2008. I have not edited either my questions as I originally wrote them or Mr. Devlin's responses in any way. (Update: With permission from Devlin, I have gone through and very lightly copyedited his responses--spelling, punctuation, etc.)
It's important for me to note that I am a newbie at interviewing. I think some of my questions are too complicated and confusing, and I think at times I appear to not understand an answer (though that was something I was trying to do to some degree).
I'll likely have some follow-up comments in a future post, and if you have any reactions, comments, or questions that I could include in that post, please send me an E-mail. For now, enjoy:
(1) I wanted to start by simply asking you about you. I think it's fair to say that you have a greater public profile than many other living, working mathematicians. Many people, including myself, know something about you through your books, your work at NPR, and your speeches and interviews over the years. Yet, in doing a bit of research for this interview, I was surprised to learn that you ride your bike tremendous distances every day, and you also enjoy playing a certain very popular online role-playing game. What else might one not know about Keith Devlin?
Devlin: The cycling is fairly recent. I was a long-distance runner for most of my adult life, including road races, fell running, and cross-country racing in the U.K. and then road running and trail running in the U.S. after I moved here in 1987. I managed to get my marathon time down to 2 hours 41 minutes, but no amount of effort could lower it further. I did best at 10 milers, usually coming in around 52 or 53 minutes. Then about five years ago my knees gave out – I’d run down too many mountain trails I guess – and I switched to cycling. I try to get out two or three days during the week for 15- to 25-mile rides (an hour to an hour-and-a-half), and longer rides at weekends, anything between 60 and 100 miles at a time. My longest, hardest ride was The Death Ride this last summer, a famous 129 miles in the California High Sierras that takes you over five high mountain passes. It took me 11 hours to get round. I’ve ridden in Europe a few times, including two ascents of the infamous Mont Ventoux. If I can’t get out of doors and exercise every few days I start to feel edgy. Mathematics is hard work, but not physically taxing. I took up cycling too late in life to be competitive, even within my age group, but I console myself that my marathon time is still considerably better than Lance Armstrong’s.
The World of Warcraft began as a desire to try to understand better the students I find myself teaching these days, but I soon noticed that such games have HUGE educational potential, especially for mathematics. It’s a new literacy, and teachers need to be familiar with it. Approached in the right way, they are great fun, and way more challenging than is popularly supposed. Since I have limited time to play, it took me 2 years to reach the top level of play in WoW. I’m working on a book about using videogames to teach mathematics.
Early in my career, I wrote a radio play based on mathematics that the BBC performed and broadcast, and worked on a novel that I finished in first draft, but mathematics proved too great a pull. Recently I’ve started to move some of my writing back to a more story-telling mode. My latest book, The Unfinished Game, is a mathematics history, and I am working on another one in the same genre.
Of course, I read all the time, but that’s surely true for everyone in education. We love learning.
Other than that, with a time-consuming job at Stanford, I don’t have time for much else!
(2) In your very first article for your monthly online column, "Devlin's Angle," at The American Mathematical Association's Web site (January, 1996), you wrote to debunk a myth about an alleged computer virus, saying, "There is, you see, no data-destroying "Good Times" virus out there. The warning message is itself the virus." And just before that you wrote, "To date, the only known defense to this virus—information—has failed to stem its spread." Since then, a number of your articles have dealt directly or indirectly with debunking mathematical myths, educating the general public about mathematics, stirring up thinking about innovation in mathematics education, etc.
Jumping twelve years into the future, you wrote the following just nine months ago in an article titled "American Mathematics in a Flat World," regarding the "outsourcing" of mathematics and the need for innovation in mathematics and mathematics education:
"I gave up on the country of my birth (the UK) twenty years ago when it told me it no longer had need for people such as myself (not far from an exact quote from the Vice Chancellor ("President") of the university where I taught, acting under government pressure to reduce its mathematics department by 50%). Having lived through the decline of my home country as a world powerhouse in innovation and economics, I am not about to give up on the country that welcomed me with open arms."
So, my question is, Can one draw a line connecting your commitment to educating the public about mathematics, debunking mathematical myths, provoking thinking about innovation in mathematics and mathematics education, and so on, back to your experience in the UK twenty years ago? Does the United States now have the same "virus" that attacked the UK—one spread by misinformation and misunderstanding about the importance of mathematics?
Devlin: I certainly see signs of the “cult of ignorance” in this country. Right now we are (hopefully) coming off one of the all-time low points in U.S. history, with a President who seems to take pride in his ignorance. Bush has to an extent legitimized ignorance and made it respectable. The U.S. that I immigrated to in the mid-1980s led the world in science, mathematics, technology, and business innovation, all built on a strong commitment to, in particular, higher education and pure research. We need to rekindle that “Can do, will do!” positive attitude that took us to the Moon. The strength of the U.S. is its huge diversity, and I’m hopeful the next few years will see an upsurge in the standards by which Americans conduct their public lives, both at home and abroad. The U.K. was and is far more homogeneous, and I doubt it will ever recover from the damage done to its education system during the 1980s. I had no alternative than to leave the U.K. (except to simply give up, which is what a lot of my colleagues did). So the two situations are very different. Here, we can recover. Still, my experience in the U.K. has perhaps raised my awareness of just how bad things can get, to an extent that I am not going to stop ringing alarm bells here. A country where a reported 50% plus of the population believes in Creationism clearly has problems.
(3) Yet, the countries that consistently outperform everyone else on measures of students' mathematics and science knowledge—Singapore, Japan, South Korea—are barely represented in lists of top universities for technology/mathematics. U.S. universities dominate in this category. Similarly, Singapore, Japan, and South Korea are barely represented, if at all, among Fields Medal winners. Since 1982, the U.S. and U.K. combined can claim about a third of all winners. Is all our worry in the United States about mathematics education somewhat misplaced, since we seem to still dominate at and above the university level?
Devlin: Two things. First, we dominate in part by immigration. I myself am just one of many university mathematicians who were trained abroad and came here because the working conditions are better than in my home country. If you want to find the world-class academics who were educated in those countries you list, just walk onto any major American university campus. There is nothing wrong with that strategy, which the U.S. has used effectively ever since it built its higher education system in the early twentieth century. But as I argued in my MAA “Flat World” column, that approach may not continue in the digitally connected new world. There is an irony here that our very own university system created the technologies that have given us today’s flat world.
Second, the current dominance we have in high-tech industries has its clear origins in massive government funding for “blue skies” university research in the 60s, 70s, and 80s. Without that level of support for basic scientific research, including mathematics, there will be no base for the next wave of innovation. That wave will come, the only question is where. I came to Silicon Valley in 1987 because that is where the action was, and still is, in my own area of interest. The talent will always go to where the opportunities are best. We can keep it here. But only if we have the vision and the will to make it happen.
(4) I understand that you have worked for the U.S. government, using mathematics to help better prevent terrorist attacks. Are you able to describe one or more fundamental problems that mathematicians are confronted with in this area? What might be covered in a Using Math to Prevent Terrorist Attacks 101?
Devlin: Countering terrorism is a two-pronged issue. Primarily, it’s a political problem. The only long-term solution is eliminate the causes of terrorism. In the interim, the only countermeasure is information gathering. It’s not very effective, but it’s the only thing we can do. My own work was in trying to develop protocols for taking massive amounts of information and reasoning about it. I approached it as the development of a “higher order logic.” That is how, and why, I got into that work; the first twenty years of my career I specialized in mathematical logic. In standard logic, we consider that case where a single individual reasons within mathematics. The reasoning considered is context free, ambiguity free, and linear, with limited information, a well defined starting point, and a clear goal. The problem we looked at (I was involved in a big, nationwide project) involved multiple, interacting agents, reasoning over effectively unlimited information, in a fashion that was generally non-linear, often holistic, and where the goal is not always clear. There is no Anti-Terrorism Math 101; it’s a horrendously difficult research area. I like that kind of challenge, but it’s not one you could build an academic career on.
(5) Okay, now here's a tough one for many people, including myself. I'd like to ask you about Cataglyphis fortis. This desert ant wanders aimlessly away from its nest—for hundreds of meters--in search of food, and then, after finding food, it makes what is pretty much a straight line back to its nest. Not only does it know the exact straight-line direction to travel, but it knows the exact distance to travel as well. And, this is the most interesting part, when scientists have tested the ant by moving it after it finds its food, it follows the same "plan" of traveling back to the nest as though it had not been moved.
Now, you have said that Cataglyphis fortis actually uses mathematics to make this work: "This is not implicit mathematics. The mathematics is being done inside that creature. It isn't being done anywhere else . . . It's doing the math. It's been optimized through natural selection to solve that dead-reckoning problem and to solve it with great accuracy."
How can we say that this ant is "doing" math? Isn't it just a simple matter of keeping track of directions and distances?
Devlin: Yes, it’s a matter of keeping track of directions and distances. (“Simple” is a highly subjective term here so I’ll leave it out.) And what word do we have to describe the way humans do that? Mathematics. The point of my book The Math Instinct, where I describe that example and others like it, was to point out that “doing math” is a description we use for certain kinds of “mental” activities. We do this all the time. We talk about a handheld calculator or a supercomputer “computing” or “solving an equation” or whatever, but of course it isn’t. It’s an inanimate device that simply follows the laws of physics. We interpret its behavior as “doing math” because that’s what it looks like to us. Of course, it looks like that to use because we design those devices to appear precisely that way to us. It’s the same with living creatures. Natural selection works by optimization of behavior, and as a result, living creatures usually exhibit one or more behaviors that maximize their survival chances, and in general the natural way for humans to understand such optimized behavior is as “doing (some particular) math”, because that’s how we solve optimization problems. We don’t know exactly how Cataglyphis fortis determines the direction and distance to get home from any point. But then, we don’t know how humans actually solve math problems either. The explanations we give to our students and the arguments we publish in papers are after-the-fact rationalizations. Once you realize that “doing math” is a description of an activity from the outside, largely based on input-output behavior, the Desert Ant and the Fields Medal winner don’t look that different. (Note that I say “don’t look that different”, not “aren’t that different”.)
(6) This idea, that "the explanations we give to our students and the arguments we publish in papers are after-the-fact rationalizations" is extremely interesting to me. It's one I see eye to eye with, but I don't know that everyone in education feels the same way.
We might very well describe most of mathematics education as composed of "explanations we give to our students." Can we gain some insight into teaching by conceiving of explanations as "after-the-fact rationalizations"?
Devlin: First I should clarify the answer I gave to the previous question. In saying that we don’t know how people actually solve mathematics problems, I’m referring to the genuinely creative act of solving a novel problem, a process that inevitably involves the famous “Aha!” moment when we suddenly see the light. That is why I went on to compare the Fields Medal winner (the mathematicians’ equivalent of the Nobel Prize) with the Tunisian Desert Ant. In contrast, a lot of activity that goes under the name of “doing math” is simply the routine application of well established procedures. That kind of activity is entirely transparent both to the doer and to an observer, and it can be learned and practiced to the point of mastery. Creative mathematics usually involves a lot of the routine stuff, but there comes a point where something new is required. Once the key step has been made, we usually have no difficulty finding a rational explanation for what we have done, so that everything appears like a clean, linear, logical progression. But that is not how we got there!
I think many students give up on mathematics because they don’t see how they could possibly come up with the solutions to problems they see in their textbook, or which the teacher gives on the board, or which some of their classmates produce. What they don’t understand is that the clever argument they have just been presented with was not arrived at by deliberate, rational thought. It was constructed after-the-fact. And so the student misses the crucial lesson that the secret to doing mathematics is not an unusual brain but sheer persistence, trying one thing after another and failing each time until eventually the light comes on.
Of course, people will differ as to how successful they can be in solving mathematics problems and in how long it takes them. But that is a matter of degree, not category. Presenting mathematics as a logical progression gives the totally false impression that such is how it is actually DONE. It’s not. The logical progression is an after-the-fact account.
As it happens, I have just completed a book that illustrates this perfectly. It comes out this fall from Basic Books, and is an account of the origins of modern probability theory. Called “The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern”, it examines the detailed correspondence between those two great French mathematicians in which they developed a solution to a hitherto unsolved problem about computing probabilities in a game of dice. The solution they eventually came up with is so “logical and simple” that today it can be taught to middle school pupils. But as the correspondence makes clear, it took two of the greatest mathematical minds of all time a great deal of effort to find that solution. Moreover, it is clear from what they wrote that the solution seemed “logical” only after they had found it and agreed it was correct! I would like to see this book made compulsory reading for all teachers and students of mathematics. Not because I wrote it. I am just the reporter. But because of the story itself -- how it illustrates vividly the way mathematics is really DONE.
(7) Once Pascal and Fermat found their solution, what became part of our mathematics "vocabulary," in large part, was not the effort that got them there, the "doing," but the end-result of that effort, "the meaning." What we try to impart to students when we teach them mathematics is this vocabulary, this collection of end-results, the "meanings." And, of course, we (and in many cases, students) must "do" some mathematics to get to these meanings, by discussion or through clever arguments.
Is there a danger in confusing the "doing" of mathematics with the "meanings" in mathematics?
Devlin: I’m not sure there is a danger of generating confusion among our students all the time the classroom focus remains solely on the finished product. Rather, the loss is that we leave our students ignorant of mathematics as a living, growing discipline. Mathematics, like the natural sciences, history, art, etc. are about DOING, not knowing. Knowing is necessary in order to do, but the goal is the doing. Concentrating on “facts” to learn makes the teacher’s job easier, and makes measuring “progress” straightforward, but it misses what the subject is really about. So yes, there is confusion, but it is in the way the educational curriculum is designed. I think this confusion will persist until more professional mathematicians take an active interest in K-12 education.
(8) You have written before that teachers sometimes present mathematics using "brittle metaphors." Does this relate in some way to the distinction, if any, between the "doing" and "meaning" of mathematics? Doesn't using brittle metaphors help students avoid in many cases the "totally false impression," as you say, that mathematics is done in a logical progression?
Devlin: The problem with metaphors – and this has been noted many times – is that they are helpful only if the user is aware of their limitations from the outset. Otherwise, the metaphor persists and can cause immense problems later. There was a now famous case a few years ago where a TV crew went to Harvard on Commencement Day and interviewed the graduating seniors and some of their professors, asking them, among other things, why it was warmer in the summer than in the winter. Many of those interviewed (I forget the percentages) said this was because the earth was nearer to the sun in the summer months. This is clearly false (consider the southern hemisphere), so the question arises why so many well educated people got it wrong. All of the students had taken obligatory science courses, and all had at some point “learned” the real reason, namely the inclination of the Earth’s axis of rotation to the plane of its orbit. Why did they not recall this simple fact? Because it was over-ridden by a powerful “rule” we all learn when young children and we come close to open fires, hot stoves, and the like; namely, when there is a heat source, it gets stronger the closer we get. That’s a good thing to learn, since it prevents us from burning ourselves. But like all rules, it has its limitations. Of course, this example is about a rule-of-thumb rather than a metaphor, but the message is the same. You need to be aware of the limitations of the rule, metaphor, or whatever that you apply.
Using a brittle metaphor (multiplication is repeated addition, for example) inevitably leads to problems later, when the metaphor no longer holds but gets in the way of a better understanding of the concept. It’s hard enough grasping the abstractions of mathematics without compounding the issue with brittle metaphors. One problem is that metaphors inevitably lead to natural inferences. For instance, thinking of multiplication as repeated addition leads to the belief that multiplication makes things bigger. This false belief often persists throughout people’s lives. It’s particularly hard to eradicate since it is often something the child observes him or herself, and as we all know, knowledge we generate ourselves tends to stick like glue. When it comes to mathematics, I think it is probably always unwise to use metaphors as “interim definitions”, which is what often seems to be done, since they all break sooner or later. Rather we should present the student the same instances, but as motivational and illustrative examples of, not metaphors for. Mathematics is abstract. It does students no good in the long term to present it as something concrete. Moreover, there is no need to do so. There is plenty of evidence that children can handle abstraction, particularly when the learning is scaffolded by a range of concrete examples.
(9) How can educators and the mathematics community come together to improve mathematics education? It seems that, in many cases, we can't understand what mathematicians want, and, on the other hand, mathematicians don't understand the limitations we face in the classroom. Will it ever be possible to bridge these differences?
Devlin: I hope so. For many years, I think the problem was largely on the side of the professional mathematicians, who did not involve themselves in K-12 educational issues. That is changing, but as it does a new problem is arising, namely the huge resistance of some in the teaching profession who, for whatever reason, resent the involvement of mathematicians, who they think have nothing to offer.
I experienced this reaction first hand recently when I wrote a couple of MAA “Devlin’s Angle” columns on the issue of multiplication not being repeated addition. As you know, it generated a huge reaction on various math ed blogs, much of it negative. On a human level, I can understand the reaction. No matter how gently it is put, for many teachers it comes down to being told that they have been doing it wrong all of their careers. But it is only by examining what we do, reflecting on it, and then learning what needs to be changed and how to do it, that we can progress. None of us is perfect. What matters is how we react to learning that what we believed or have been doing was wrong, or could be done better.
Mathematicians are actually well prepared for this. It is notoriously difficult spotting one’s own mistakes in a mathematical argument of any length, so having one’s proofs shot down in flames by others, sometimes in full public view, is part of everyday life for the mathematician. The case of Andrew Wiles’ proof of Fermat’s Last Theorem is a spectacular illustration of this. He did eventually correct his proof, but it took him over a year after his first argument was found by others to be badly flawed. I suspect teachers are less used to this kind of thing, particularly in the U.S., where there is virtually no regular peer review of classroom teaching, no regular testing of teachers’ knowledge of the subject, and no regular in-service education about developments in educational methods — all of which are commonplace in other professions, including colleges and universities.
(10) You have mentioned that you know little about K-12 math education. Any plans in the future to dive more fully into our crazy little world?
Devlin: Well, it’s not true that I don’t know much about mathematics education. I have probably read more research in the math ed field than many practicing teachers, and I try to stay current. The research institute I direct at Stanford includes a world-famous educational research center, the Stanford Center for Innovations in Learning, so I get regular exposure to the latest research in the subject. I also served on MSEB for several years, and I learned a lot then. But having never taught in the K-12 mathematics system, my knowledge is all theoretical. In contrast, I have been active as a professional mathematician for forty years, and as a result have a wide and deep understanding of the subject from the inside. When it comes to mathematics, I am comfortable with being described as an “expert”. But when it comes to K-12 mathematics, I might know a lot, but I am in no way an expert. It’s possible I overplay my lack of experience in K-12 education more than is necessary, but I believe my record in mathematics (and my Stanford affiliation) gives considerable weight to anything I say or write about mathematics, and I think that to many outsiders that weight would extend (unjustifiably) to the K-12 mathematics education field. Math ed has its own (true) experts -- but I am not one of them.
Below is the full interview—just ten questions long. The interview was conducted via E-mail from August 10, 2008 to August 16, 2008. I have not edited either my questions as I originally wrote them or Mr. Devlin's responses in any way. (Update: With permission from Devlin, I have gone through and very lightly copyedited his responses--spelling, punctuation, etc.)
It's important for me to note that I am a newbie at interviewing. I think some of my questions are too complicated and confusing, and I think at times I appear to not understand an answer (though that was something I was trying to do to some degree).
I'll likely have some follow-up comments in a future post, and if you have any reactions, comments, or questions that I could include in that post, please send me an E-mail. For now, enjoy:
(1) I wanted to start by simply asking you about you. I think it's fair to say that you have a greater public profile than many other living, working mathematicians. Many people, including myself, know something about you through your books, your work at NPR, and your speeches and interviews over the years. Yet, in doing a bit of research for this interview, I was surprised to learn that you ride your bike tremendous distances every day, and you also enjoy playing a certain very popular online role-playing game. What else might one not know about Keith Devlin?
Devlin: The cycling is fairly recent. I was a long-distance runner for most of my adult life, including road races, fell running, and cross-country racing in the U.K. and then road running and trail running in the U.S. after I moved here in 1987. I managed to get my marathon time down to 2 hours 41 minutes, but no amount of effort could lower it further. I did best at 10 milers, usually coming in around 52 or 53 minutes. Then about five years ago my knees gave out – I’d run down too many mountain trails I guess – and I switched to cycling. I try to get out two or three days during the week for 15- to 25-mile rides (an hour to an hour-and-a-half), and longer rides at weekends, anything between 60 and 100 miles at a time. My longest, hardest ride was The Death Ride this last summer, a famous 129 miles in the California High Sierras that takes you over five high mountain passes. It took me 11 hours to get round. I’ve ridden in Europe a few times, including two ascents of the infamous Mont Ventoux. If I can’t get out of doors and exercise every few days I start to feel edgy. Mathematics is hard work, but not physically taxing. I took up cycling too late in life to be competitive, even within my age group, but I console myself that my marathon time is still considerably better than Lance Armstrong’s.
The World of Warcraft began as a desire to try to understand better the students I find myself teaching these days, but I soon noticed that such games have HUGE educational potential, especially for mathematics. It’s a new literacy, and teachers need to be familiar with it. Approached in the right way, they are great fun, and way more challenging than is popularly supposed. Since I have limited time to play, it took me 2 years to reach the top level of play in WoW. I’m working on a book about using videogames to teach mathematics.
Early in my career, I wrote a radio play based on mathematics that the BBC performed and broadcast, and worked on a novel that I finished in first draft, but mathematics proved too great a pull. Recently I’ve started to move some of my writing back to a more story-telling mode. My latest book, The Unfinished Game, is a mathematics history, and I am working on another one in the same genre.
Of course, I read all the time, but that’s surely true for everyone in education. We love learning.
Other than that, with a time-consuming job at Stanford, I don’t have time for much else!
(2) In your very first article for your monthly online column, "Devlin's Angle," at The American Mathematical Association's Web site (January, 1996), you wrote to debunk a myth about an alleged computer virus, saying, "There is, you see, no data-destroying "Good Times" virus out there. The warning message is itself the virus." And just before that you wrote, "To date, the only known defense to this virus—information—has failed to stem its spread." Since then, a number of your articles have dealt directly or indirectly with debunking mathematical myths, educating the general public about mathematics, stirring up thinking about innovation in mathematics education, etc.
Jumping twelve years into the future, you wrote the following just nine months ago in an article titled "American Mathematics in a Flat World," regarding the "outsourcing" of mathematics and the need for innovation in mathematics and mathematics education:
"I gave up on the country of my birth (the UK) twenty years ago when it told me it no longer had need for people such as myself (not far from an exact quote from the Vice Chancellor ("President") of the university where I taught, acting under government pressure to reduce its mathematics department by 50%). Having lived through the decline of my home country as a world powerhouse in innovation and economics, I am not about to give up on the country that welcomed me with open arms."
So, my question is, Can one draw a line connecting your commitment to educating the public about mathematics, debunking mathematical myths, provoking thinking about innovation in mathematics and mathematics education, and so on, back to your experience in the UK twenty years ago? Does the United States now have the same "virus" that attacked the UK—one spread by misinformation and misunderstanding about the importance of mathematics?
Devlin: I certainly see signs of the “cult of ignorance” in this country. Right now we are (hopefully) coming off one of the all-time low points in U.S. history, with a President who seems to take pride in his ignorance. Bush has to an extent legitimized ignorance and made it respectable. The U.S. that I immigrated to in the mid-1980s led the world in science, mathematics, technology, and business innovation, all built on a strong commitment to, in particular, higher education and pure research. We need to rekindle that “Can do, will do!” positive attitude that took us to the Moon. The strength of the U.S. is its huge diversity, and I’m hopeful the next few years will see an upsurge in the standards by which Americans conduct their public lives, both at home and abroad. The U.K. was and is far more homogeneous, and I doubt it will ever recover from the damage done to its education system during the 1980s. I had no alternative than to leave the U.K. (except to simply give up, which is what a lot of my colleagues did). So the two situations are very different. Here, we can recover. Still, my experience in the U.K. has perhaps raised my awareness of just how bad things can get, to an extent that I am not going to stop ringing alarm bells here. A country where a reported 50% plus of the population believes in Creationism clearly has problems.
(3) Yet, the countries that consistently outperform everyone else on measures of students' mathematics and science knowledge—Singapore, Japan, South Korea—are barely represented in lists of top universities for technology/mathematics. U.S. universities dominate in this category. Similarly, Singapore, Japan, and South Korea are barely represented, if at all, among Fields Medal winners. Since 1982, the U.S. and U.K. combined can claim about a third of all winners. Is all our worry in the United States about mathematics education somewhat misplaced, since we seem to still dominate at and above the university level?
Devlin: Two things. First, we dominate in part by immigration. I myself am just one of many university mathematicians who were trained abroad and came here because the working conditions are better than in my home country. If you want to find the world-class academics who were educated in those countries you list, just walk onto any major American university campus. There is nothing wrong with that strategy, which the U.S. has used effectively ever since it built its higher education system in the early twentieth century. But as I argued in my MAA “Flat World” column, that approach may not continue in the digitally connected new world. There is an irony here that our very own university system created the technologies that have given us today’s flat world.
Second, the current dominance we have in high-tech industries has its clear origins in massive government funding for “blue skies” university research in the 60s, 70s, and 80s. Without that level of support for basic scientific research, including mathematics, there will be no base for the next wave of innovation. That wave will come, the only question is where. I came to Silicon Valley in 1987 because that is where the action was, and still is, in my own area of interest. The talent will always go to where the opportunities are best. We can keep it here. But only if we have the vision and the will to make it happen.
(4) I understand that you have worked for the U.S. government, using mathematics to help better prevent terrorist attacks. Are you able to describe one or more fundamental problems that mathematicians are confronted with in this area? What might be covered in a Using Math to Prevent Terrorist Attacks 101?
Devlin: Countering terrorism is a two-pronged issue. Primarily, it’s a political problem. The only long-term solution is eliminate the causes of terrorism. In the interim, the only countermeasure is information gathering. It’s not very effective, but it’s the only thing we can do. My own work was in trying to develop protocols for taking massive amounts of information and reasoning about it. I approached it as the development of a “higher order logic.” That is how, and why, I got into that work; the first twenty years of my career I specialized in mathematical logic. In standard logic, we consider that case where a single individual reasons within mathematics. The reasoning considered is context free, ambiguity free, and linear, with limited information, a well defined starting point, and a clear goal. The problem we looked at (I was involved in a big, nationwide project) involved multiple, interacting agents, reasoning over effectively unlimited information, in a fashion that was generally non-linear, often holistic, and where the goal is not always clear. There is no Anti-Terrorism Math 101; it’s a horrendously difficult research area. I like that kind of challenge, but it’s not one you could build an academic career on.
(5) Okay, now here's a tough one for many people, including myself. I'd like to ask you about Cataglyphis fortis. This desert ant wanders aimlessly away from its nest—for hundreds of meters--in search of food, and then, after finding food, it makes what is pretty much a straight line back to its nest. Not only does it know the exact straight-line direction to travel, but it knows the exact distance to travel as well. And, this is the most interesting part, when scientists have tested the ant by moving it after it finds its food, it follows the same "plan" of traveling back to the nest as though it had not been moved.
Now, you have said that Cataglyphis fortis actually uses mathematics to make this work: "This is not implicit mathematics. The mathematics is being done inside that creature. It isn't being done anywhere else . . . It's doing the math. It's been optimized through natural selection to solve that dead-reckoning problem and to solve it with great accuracy."
How can we say that this ant is "doing" math? Isn't it just a simple matter of keeping track of directions and distances?
Devlin: Yes, it’s a matter of keeping track of directions and distances. (“Simple” is a highly subjective term here so I’ll leave it out.) And what word do we have to describe the way humans do that? Mathematics. The point of my book The Math Instinct, where I describe that example and others like it, was to point out that “doing math” is a description we use for certain kinds of “mental” activities. We do this all the time. We talk about a handheld calculator or a supercomputer “computing” or “solving an equation” or whatever, but of course it isn’t. It’s an inanimate device that simply follows the laws of physics. We interpret its behavior as “doing math” because that’s what it looks like to us. Of course, it looks like that to use because we design those devices to appear precisely that way to us. It’s the same with living creatures. Natural selection works by optimization of behavior, and as a result, living creatures usually exhibit one or more behaviors that maximize their survival chances, and in general the natural way for humans to understand such optimized behavior is as “doing (some particular) math”, because that’s how we solve optimization problems. We don’t know exactly how Cataglyphis fortis determines the direction and distance to get home from any point. But then, we don’t know how humans actually solve math problems either. The explanations we give to our students and the arguments we publish in papers are after-the-fact rationalizations. Once you realize that “doing math” is a description of an activity from the outside, largely based on input-output behavior, the Desert Ant and the Fields Medal winner don’t look that different. (Note that I say “don’t look that different”, not “aren’t that different”.)
(6) This idea, that "the explanations we give to our students and the arguments we publish in papers are after-the-fact rationalizations" is extremely interesting to me. It's one I see eye to eye with, but I don't know that everyone in education feels the same way.
We might very well describe most of mathematics education as composed of "explanations we give to our students." Can we gain some insight into teaching by conceiving of explanations as "after-the-fact rationalizations"?
Devlin: First I should clarify the answer I gave to the previous question. In saying that we don’t know how people actually solve mathematics problems, I’m referring to the genuinely creative act of solving a novel problem, a process that inevitably involves the famous “Aha!” moment when we suddenly see the light. That is why I went on to compare the Fields Medal winner (the mathematicians’ equivalent of the Nobel Prize) with the Tunisian Desert Ant. In contrast, a lot of activity that goes under the name of “doing math” is simply the routine application of well established procedures. That kind of activity is entirely transparent both to the doer and to an observer, and it can be learned and practiced to the point of mastery. Creative mathematics usually involves a lot of the routine stuff, but there comes a point where something new is required. Once the key step has been made, we usually have no difficulty finding a rational explanation for what we have done, so that everything appears like a clean, linear, logical progression. But that is not how we got there!
I think many students give up on mathematics because they don’t see how they could possibly come up with the solutions to problems they see in their textbook, or which the teacher gives on the board, or which some of their classmates produce. What they don’t understand is that the clever argument they have just been presented with was not arrived at by deliberate, rational thought. It was constructed after-the-fact. And so the student misses the crucial lesson that the secret to doing mathematics is not an unusual brain but sheer persistence, trying one thing after another and failing each time until eventually the light comes on.
Of course, people will differ as to how successful they can be in solving mathematics problems and in how long it takes them. But that is a matter of degree, not category. Presenting mathematics as a logical progression gives the totally false impression that such is how it is actually DONE. It’s not. The logical progression is an after-the-fact account.
As it happens, I have just completed a book that illustrates this perfectly. It comes out this fall from Basic Books, and is an account of the origins of modern probability theory. Called “The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern”, it examines the detailed correspondence between those two great French mathematicians in which they developed a solution to a hitherto unsolved problem about computing probabilities in a game of dice. The solution they eventually came up with is so “logical and simple” that today it can be taught to middle school pupils. But as the correspondence makes clear, it took two of the greatest mathematical minds of all time a great deal of effort to find that solution. Moreover, it is clear from what they wrote that the solution seemed “logical” only after they had found it and agreed it was correct! I would like to see this book made compulsory reading for all teachers and students of mathematics. Not because I wrote it. I am just the reporter. But because of the story itself -- how it illustrates vividly the way mathematics is really DONE.
(7) Once Pascal and Fermat found their solution, what became part of our mathematics "vocabulary," in large part, was not the effort that got them there, the "doing," but the end-result of that effort, "the meaning." What we try to impart to students when we teach them mathematics is this vocabulary, this collection of end-results, the "meanings." And, of course, we (and in many cases, students) must "do" some mathematics to get to these meanings, by discussion or through clever arguments.
Is there a danger in confusing the "doing" of mathematics with the "meanings" in mathematics?
Devlin: I’m not sure there is a danger of generating confusion among our students all the time the classroom focus remains solely on the finished product. Rather, the loss is that we leave our students ignorant of mathematics as a living, growing discipline. Mathematics, like the natural sciences, history, art, etc. are about DOING, not knowing. Knowing is necessary in order to do, but the goal is the doing. Concentrating on “facts” to learn makes the teacher’s job easier, and makes measuring “progress” straightforward, but it misses what the subject is really about. So yes, there is confusion, but it is in the way the educational curriculum is designed. I think this confusion will persist until more professional mathematicians take an active interest in K-12 education.
(8) You have written before that teachers sometimes present mathematics using "brittle metaphors." Does this relate in some way to the distinction, if any, between the "doing" and "meaning" of mathematics? Doesn't using brittle metaphors help students avoid in many cases the "totally false impression," as you say, that mathematics is done in a logical progression?
Devlin: The problem with metaphors – and this has been noted many times – is that they are helpful only if the user is aware of their limitations from the outset. Otherwise, the metaphor persists and can cause immense problems later. There was a now famous case a few years ago where a TV crew went to Harvard on Commencement Day and interviewed the graduating seniors and some of their professors, asking them, among other things, why it was warmer in the summer than in the winter. Many of those interviewed (I forget the percentages) said this was because the earth was nearer to the sun in the summer months. This is clearly false (consider the southern hemisphere), so the question arises why so many well educated people got it wrong. All of the students had taken obligatory science courses, and all had at some point “learned” the real reason, namely the inclination of the Earth’s axis of rotation to the plane of its orbit. Why did they not recall this simple fact? Because it was over-ridden by a powerful “rule” we all learn when young children and we come close to open fires, hot stoves, and the like; namely, when there is a heat source, it gets stronger the closer we get. That’s a good thing to learn, since it prevents us from burning ourselves. But like all rules, it has its limitations. Of course, this example is about a rule-of-thumb rather than a metaphor, but the message is the same. You need to be aware of the limitations of the rule, metaphor, or whatever that you apply.
Using a brittle metaphor (multiplication is repeated addition, for example) inevitably leads to problems later, when the metaphor no longer holds but gets in the way of a better understanding of the concept. It’s hard enough grasping the abstractions of mathematics without compounding the issue with brittle metaphors. One problem is that metaphors inevitably lead to natural inferences. For instance, thinking of multiplication as repeated addition leads to the belief that multiplication makes things bigger. This false belief often persists throughout people’s lives. It’s particularly hard to eradicate since it is often something the child observes him or herself, and as we all know, knowledge we generate ourselves tends to stick like glue. When it comes to mathematics, I think it is probably always unwise to use metaphors as “interim definitions”, which is what often seems to be done, since they all break sooner or later. Rather we should present the student the same instances, but as motivational and illustrative examples of, not metaphors for. Mathematics is abstract. It does students no good in the long term to present it as something concrete. Moreover, there is no need to do so. There is plenty of evidence that children can handle abstraction, particularly when the learning is scaffolded by a range of concrete examples.
(9) How can educators and the mathematics community come together to improve mathematics education? It seems that, in many cases, we can't understand what mathematicians want, and, on the other hand, mathematicians don't understand the limitations we face in the classroom. Will it ever be possible to bridge these differences?
Devlin: I hope so. For many years, I think the problem was largely on the side of the professional mathematicians, who did not involve themselves in K-12 educational issues. That is changing, but as it does a new problem is arising, namely the huge resistance of some in the teaching profession who, for whatever reason, resent the involvement of mathematicians, who they think have nothing to offer.
I experienced this reaction first hand recently when I wrote a couple of MAA “Devlin’s Angle” columns on the issue of multiplication not being repeated addition. As you know, it generated a huge reaction on various math ed blogs, much of it negative. On a human level, I can understand the reaction. No matter how gently it is put, for many teachers it comes down to being told that they have been doing it wrong all of their careers. But it is only by examining what we do, reflecting on it, and then learning what needs to be changed and how to do it, that we can progress. None of us is perfect. What matters is how we react to learning that what we believed or have been doing was wrong, or could be done better.
Mathematicians are actually well prepared for this. It is notoriously difficult spotting one’s own mistakes in a mathematical argument of any length, so having one’s proofs shot down in flames by others, sometimes in full public view, is part of everyday life for the mathematician. The case of Andrew Wiles’ proof of Fermat’s Last Theorem is a spectacular illustration of this. He did eventually correct his proof, but it took him over a year after his first argument was found by others to be badly flawed. I suspect teachers are less used to this kind of thing, particularly in the U.S., where there is virtually no regular peer review of classroom teaching, no regular testing of teachers’ knowledge of the subject, and no regular in-service education about developments in educational methods — all of which are commonplace in other professions, including colleges and universities.
(10) You have mentioned that you know little about K-12 math education. Any plans in the future to dive more fully into our crazy little world?
Devlin: Well, it’s not true that I don’t know much about mathematics education. I have probably read more research in the math ed field than many practicing teachers, and I try to stay current. The research institute I direct at Stanford includes a world-famous educational research center, the Stanford Center for Innovations in Learning, so I get regular exposure to the latest research in the subject. I also served on MSEB for several years, and I learned a lot then. But having never taught in the K-12 mathematics system, my knowledge is all theoretical. In contrast, I have been active as a professional mathematician for forty years, and as a result have a wide and deep understanding of the subject from the inside. When it comes to mathematics, I am comfortable with being described as an “expert”. But when it comes to K-12 mathematics, I might know a lot, but I am in no way an expert. It’s possible I overplay my lack of experience in K-12 education more than is necessary, but I believe my record in mathematics (and my Stanford affiliation) gives considerable weight to anything I say or write about mathematics, and I think that to many outsiders that weight would extend (unjustifiably) to the K-12 mathematics education field. Math ed has its own (true) experts -- but I am not one of them.
Labels: education, mathematics