Devlin's Angle

December 2008

How Do We Learn Math?

"God made the integers; all else is the work of man." Probably one of the most famous mathematical quotations of all time. Its author was the German mathematician Leopold Kronecker (1823-1891). Though sometimes interpreted (erroneously) as a theological claim, Kronecker was articulating an intellectual thrust that dominated a lot of mathematics through the second half of the nineteenth century, to reduce the real number system first to whole numbers and ultimately to formal logic. Motivated in large part by a desire to place the infinitesimal calculus on a "sound, logical footing," it took many years to achieve this goal. The final key step, from the perspective of number systems, was the formulation by the Italian mathematician Giuseppe Peano (1858-1932) of a set of axioms (more precisely - and imprecise formulation turned out to be a dangerous rock on which many a promising advance floundered - an infinite axiom schema) that determines the additive structure of the positive whole numbers. (The rest of the reduction process showed how numbers can be defined within abstract set theory, which in turn can be reduced to formal logic.)

Looked at as a whole, it's an impressive piece of work, one of humankind's greatest intellectual achievements many would say. I am one such; indeed, it was that work as much as anything that led me to do my doctoral work - and much of my professional research thereafter - in mathematical logic, with a particular emphasis on set theory.

Mathematical logic and set theory are two of a small group of subjects that generally go under the name "Foundations of Mathematics." When I started out on my postgraduate work, the mathematical world had just undergone another of a whole series of "crises in the foundations," in that case Paul Cohen's 1963 discovery that there were specific questions about numbers that provably could not be answered (on the basis of the currently accepted axioms).

Now there was something odd about all of those crises. (An earlier one was Bertrand Russell's 1901 disicovery of the paradox named after him, that destroyed Frege's attempt to ground mathematics in elementary set theory.) While the mathematical community had no hesitation recognizing the importance of those discoveries as precisely that - new mathematical discoveries (Cohen was awarded the Fields Medal for his theorem) - mathematicians did not modify their everyday mathematical practice one iota. They continued exactly as they had before.

It is, therefore, an odd notion of "foundations" that, no matter how much they are shaken or even proved untenable and eventually replaced, life in the building supposedly erected on top of them goes on as if nothing had happened.

There's something else odd about these particular foundations as well. They were constructed after the mathematics supposedly built on top of them.

In what sense, then, are formal logic, abstract set theory, the Peano axioms, and all the rest, "foundational"? The answer - clear to all of us who have lived in the modern mathematical world long enough - is that they are the start of a logical chain of development, where each new link in the chain - or each new floor of the building if you prefer the construction metaphor implied by the word "foundations" - that, if you follow it far (or high) enough, eventually gives you all of mathematics.

Looking back, most of the math courses I received as a student, and the many more I gave over several decades of teaching university mathematics majors and graduate students, followed the same logical structure implicit in the foundational view of mathematics. I would start with the basics - the definitions and the axioms - and then build everything up from there. It was very much a synthetic view of mathematics. Among those courses was one called "Real Analysis", which, starting from some clearly specified first principles, builds up the concept of continuity and the basic elements of the differential and integral calculi. Occasionally I would note to myself how totally inappropriate was the name "analysis" for a course that was out-and-out synthesis. But I knew the historical reason for the name. The subject arose as a result of a long struggle to analyze the real number system.

But if that is the case, and it is, then why don't we typically teach it in a fashion that follows the historical development? In order words, why don't we teach it as a process of analysis (of an intuitive notion of a continuous real line with an arithmetic structure)? Well, some people do, or at least have. But most of us don't, and the reason I think (for sure my reason) is that it is simply way more efficient to follow the inherent logical-mathematical structure rather than the historical thread.

"People's earlier, intuitive notions of continuity (for example) were just wrong," many would say, "So why waste time raking over the coals of history? Just give the student the correct definition and move on." That worked for me, both as a student and a professor, and it worked for most of my professional colleagues. Along the way to becoming a professional, however, a lot of my fellow student travelers dropped by the wayside. The approach that worked for me did not appear to suit everyone.

In my more recent years in the profession, I have become more interested in issues of mathematical cognition. (Some sixteen years separate my weighty tome Contructibility, published in 1983 and about as synthetic, foundational a treatment of mathematics as you can get, from my far more accessible (I hope) 2000 book The Math Gene, where I present an evolutionary account of the development of mathematical ability in the human brain.) That change in focus has led me to reflect on the relationship between the synthetic approach to mathematics that dominates the way mathematics majors and postgraduate mathematics students are taught, and the historical/cognitive development, both of Homo sapiens the species, and of young children learning mathematics.

In both cases, evolutionary cognitive development and mathematics learning, my reflections have been, of necessity, those of an outsider, albeit one who has spent his professional life working in the domain of interest, to whit, mathematics. I am not a cultural anthropologist or an evolutionary biologist, I am not trained in the methods of cognitive psychology, and my only experience of elementary mathematics teaching was as an enforced recipient of the process more years ago than I care to remember. Still, over the past twenty years I've read a ton of research in all those domains - enough to realize that we know far, far less about how the brain does mathematics, how it acquired that ability, and how young children learn it, than we do about the subject itself.

A consequence of that lack of current scientific knowledge has an obvious consequence: we don't know the best way to teach math!

"Well, ain't that a surprise!" you say.

No really. I'm not just talking about how to introduce particular topics or whether it is important that students master the long division algorithm. It's more fundamental than that. We don't know what view of mathematics on which to base our instruction! In fact, as far as I can tell from the emails I receive - and I get a fair number - many US educators are unaware that there could be an alternative to the one we automatically assume and (implicitly) use.

That approach, the one that is prevalent in the US, and the one that was implicit in the way I was taught math, is that the beginning math student abstracts mathematical concepts from his or her everyday experience. As far as we know, this was how the concept of (positive, whole) numbers arose in Sumeria between 8,000 and 5,000 B.C. (I describe this fascinating story in my books Mathematics: The Science of Patterns and in The Math Gene.) The assumption behind today's standard US K-12 math curriculum is that the student then builds on his or her intuition-grounded, world-abstracted, reality-based understanding of the counting numbers to develop concepts and procedures for handling fractions and negative numbers - the exact order of introduction here is not clear - and then eventually the real numbers. (The complex number system, the "end point" of the development from a mathematical perspective, is left to the university level. I'll come back to complex numbers later.)

[I said "today's standard US math curriculum" in the above paragraph. Some years ago, geometry was also a standard part of the curriculum, but that was eventually abandoned in order to concentrate on the number systems and algebra believed to be more important for life in today's society. I'll come back to that later as well.]

This view of the acquisition of mathematical knowledge and ability is implicit in the account I give in The Math Gene and was made abundantly explicit in Lakoff and Nunez's book Where Mathematics Comes From, which, although published just after mine, by the same publisher, and seemingly an immediate sequel to mine, was written completely independently, though at the same time.

I confess that, as something of a Lakoff-metaphor fan, and a one-time colleague of Nunez, the first time I read their book, I agreed enthusiastically with everything they said. But on reflection, followed by a second and then a third reading, together with discussions with colleagues - particularly the Israeli mathematics education specialist Uri Leron - the doubts began to set in. The picture Lakoff and Nunez paint of the acquisition of new mathematical concepts and knowledge, is one of iterated metaphor building, where each new concept is created from the body of knowledge already acquired through the construction of a new metaphor.

Now, Lakoff and Nunez do not claim that these metaphors - mappings from one domain to another - are deliberate or conscious, though some may be. Rather, they seek to describe a mechanism whereby the brain, as a physical organ, extends its domain of activity. My problem, and that of others I talked to, was that the process they described, while plausible (and perhaps correct) for the way we learn elementary arithmetic and possibly other more basic parts of mathematics, does not at all resemble the way (some? many? most? all?) professional mathematicians learn a new advanced field of abstract mathematics.

Rather, a mathematician (at least me and others I've asked) learns new math the way people learn to play chess. We first learn the rules of chess. Those rules don't relate to anything in our everyday experience. They don't make sense. They are just the rules of chess. To play chess, you don't have to understand the rules or know where they came from or what they "mean". You simply have to follow them. In our first few attempts at playing chess, we follow the rules blindly, without any insight or understanding what we are doing. And, unless we are playing another beginner, we get beat. But then, after we've played a few games, the rules begin to make sense to us - we start to understand them. Not in terms of anything in the real world or in our prior experience, but in terms of the game itself. Eventually, after we have played many games, the rules are forgotten. We just play chess. And it really does make sense to us. The moves do have meaning (in terms of the game). But this is not a process of constructing a metaphor. Rather it is one of cognitive bootstrapping (my term), where we make use of the fact that, through conscious effort, the brain can learn to follow arbitrary and meaningless rules, and then, after our brain has sufficient experience working with those rules, it starts to make sense of them and they acquire meaning for us. (At least it does if those rules are formulated and put together in a way that has a structure that enables this.)

This, as I say, is the way I, and (at least some, if not most or all) other professional mathematicians, learn new mathematics. (Not in every case, to be sure. Sometimes we see from the start what the new game is all about.) Often, after we have learned the new stuff in a rule-determined manner, we can link it to things we knew previously. We can, in other words, construct a metaphor map linking the new to the old. But that is possible only after we have completed the bootstrap. It's not how we learned it. Similarly, expert chess players often describe their play in terms of military metaphors, using terms like "threat", "advance", "retreat", and "reinforce". But none of those make sense when a beginner is first learning how to play. The real-world metaphor here depends upon a fairly advanced understanding of chess, it does not lead to it.

Well, so far, this all sounds like an interesting discussion for the coffee room in the university math department. But here's the rub. If learning advanced mathematics is more akin to learning chess than it is to, say, learning to walk, learning to play tennis, or learning to ride a bike - where we start with our native abilities and refine and practice them - at what point in the K-university curriculum does this "different" kind of mathematics begin?

Leron, who I mentioned earlier, and others, have produced some convincing evidence that it certainly begins - or has begun - when the student meets the concept of a mathematical function. As Leron and others have shown, a significant proportion of university mathematics students do not have the correct concept of a function.

Do you? Here is a simple test. (This one is far simpler than the more penetrating ones Leron used.) Consider the "doubling function" y = 2x (or, if you prefer more sophisticated notation, f(x) = 2x.) Question: When you start with a number, what does this function do to it?

If you answered, "It doubles it," you are wrong. No, no going back now and saying "Well what I really meant was ..." That original answer was wrong, and shows that, even if you "know" the correct definition, your underlying concept of a function is wrong. Functions, as defined and used all the time in mathematics, don't do anything to anything. They are not processes. They relate things. The "doubling function" relates the number 14 to the number 7, but it doesn't do anything to 7. Functions are not processes but objects in the mathematical realm. A student who has not fully grasped and internalized that, whose underlying concept of a function is a process, will have difficulty in calculus, where functions are very definitely treated as objects that you do things do - at least sometimes you do things to them; more often, you apply other functions to them, so there is no doing, just more relating. Note that I am not claiming, and nor is Leron, that those students do not understand the difference between the two alternative possible notions of a function, or that they do not understand the correct (by agreed definition) concept. The issue is, what is their concept of a function?

This is not a trivial issue. As mathematicians learned over many centuries, definitions matter. Fine distinctions matter. Concepts matter. Having the right concept matters. If you make a small change in one of the rules of chess you will end up with a different game, and the same in the (rule-based) game we call mathematics. In both cases, the alternate game is likely to be uninteresting and useless.

Okay, we've picked a topic in the mathematics curriculum, functions, and found that many people - I suspect most people - have an "incorrect" concept of a function. But "incorrect" here means it is not the one mathematicians use (in calculus and all that builds upon it, which covers most of science and engineering, so we are not talking about something that is largely irrelevant). Is it really a problem if the majority of citizens think of functions as processes? Well, it is a problem they have to overcome if they want to go on and become scientists, engineers, or whatever, and as the Leron and similar studies have shown, changing a basic concept once it has been acquired, internalized, and assimilated is no easy matter. But how about the rest? The ones who do not go to university and study a scientific subject.

Well, having an incorrect function concept might not be a problem for most people, but the function concept was simply an example. We still have not answered the original question: Where does the "abstracted from everyday experience and developed by iterated metaphors" mathematics end and the "rule-based mathematics that has to be bootstrapped" begin?

What if the mathematics that has to be bootstrapped in order to be properly mastered includes the real numbers? What if it includes the negative integers? What if it includes the concept of multiplication (a topic of three of my more recent columns)? What if teaching multiplication as repeated addition (see those previous columns) or introducing negative numbers using an everyday (explicit) metaphor (such as owing money) results in an incorrect concept that leads to increased difficulty later when the child needs to move on in math?

Even if there is a problem somewhere down the educational line, is there anything we can do about it? Is there any alternative to using the "abstract it from everyday experience" approach that we in the US accept as the only way to ground K-8 mathematics? Is that really the only way for young children to learn it? And if not the only way, is it the best way, given the goal of getting as many children as possible as far along the mathematical path as possible?

Perhaps the ultimate, and maybe the most startling question: Do Kronecker's words apply when it comes to mathematics education? Is starting with the counting numbers the only, or the best, way to teach mathematics to young children in today's world?

Answering those questions will be the focus of next month's column (where I'll also be true to my promise to come back to geometry and complex numbers in mathematics education). The only clue I'll give now is that in the above discussion I kept referring to "US" education.

And no, I am not setting up to advocate a particular philosophy of mathematics education. As I have stated on several occasions before, I am neither trained in nor do I have first-hand experience in elementary mathematics education. But I can and do read the words of those who do have such expertise. At least one other approach has been developed elsewhere in the world, by people with the aforementioned necessary expertise and experience, and there is some evidence to suggest that the alternative may be better than the one we use here. I say "may be better," note. The evidence is good, but as yet there is not enough of it, and as always it is tricky interpreting experimental results in education. But I do take what evidence there is as indication that we should at the very least discuss and evaluate that alternative approach, even if we start out skeptical of where it might lead. Yet, as far as I can tell, the mathematics education community in the US has so far acted as though this other approach simply did not exist. That may, of course, simply be due to, in the words of the prison guard in the classic 1967 Paul Newman movie Cool Hand Luke, a failure to communicate. (Either between one part of the world and another, or between our math ed community and the rest of us.) If so, then my goal is to try to fix it.


Devlin's Angle is updated at the beginning of each month. Devlin's most recent book is The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern, published by Basic Books.

Mathematician Keith Devlin (email: devlin@stanford.edu) is the Executive Director of the Human-Sciences and Technologies Advanced Research Institute (H-STAR) at Stanford University and The Math Guy on NPR's Weekend Edition.