How Do We Learn Math

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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 mathematicalquotations of all time. Its author was the German mathematician Leopold Kronecker (1823-1891).Though sometimes interpreted (erroneously) as a theological claim, Kronecker was articulating anintellectual thrust that dominated a lot of mathematics through the second half of the nineteenthcentury, 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," ittook many years to achieve this goal. The final key step, from the perspective of number systems, wasthe formulation by the Italian mathematician Giuseppe Peano (1858-1932) of a set of axioms (moreprecisely - and imprecise formulation turned out to be a dangerous rock on which many a promisingadvance floundered - an infinite axiom schema) that determines the additive structure of the positivewhole numbers. (The rest of the reduction process showed how numbers can be defined withinabstract 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 intellectualachievements many would say. I am one such; indeed, it was that work as much as anything that ledme 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 thename "Foundations of Mathematics." When I started out on my postgraduate work, the mathematicalworld had just undergone another of a whole series of "crises in the foundations," in that case PaulCohen's 1963 discovery that there were specific questions about numbers that provably could not beanswered (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 1901disicovery of the paradox named after him, that destroyed Frege's attempt to ground mathematics inelementary set theory.) While the mathematical community had no hesitation recognizing theimportance of those discoveries as precisely that - new mathematical discoveries (Cohen was awardedthe Fields Medal for his theorem) - mathematicians did not modify their everyday mathematicalpractice 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 evenproved untenable and eventually replaced, life in the building supposedly erected on top of them goeson as if nothing had happened.

There's something else odd about these particular foundations as well. They were constructed after themathematics 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 longenough - 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.


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Looking back, most of the math courses I received as a student, and the many more I gave overseveral decades of teaching university mathematics majors and graduate students, followed the samelogical structure implicit in the foundational view of mathematics. I would start with the basics - thedefinitions and the axioms - and then build everything up from there. It was very much a syntheticview of mathematics. Among those courses was one called "Real Analysis", which, starting fromsome 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 wasthe name "analysis" for a course that was out-and-out synthesis. But I knew the historical reason forthe 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 thehistorical development? In order words, why don't we teach it as a process of analysis (of an intuitivenotion 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 efficientto 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, "Sowhy 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 professionalcolleagues. Along the way to becoming a professional, however, a lot of my fellow student travelersdropped 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 mathematicalcognition. (Some sixteen years separate my weighty tome Contructibility , published in 1983 andabout as synthetic, foundational a treatment of mathematics as you can get, from my far moreaccessible (I hope) 2000 book The Math Gene , where I present an evolutionary account of thedevelopment of mathematical ability in the human brain.) That change in focus has led me to reflecton the relationship between the synthetic approach to mathematics that dominates the waymathematics majors and postgraduate mathematics students are taught, and the historical/cognitivedevelopment, both of Homo sapiens the species, and of young children learning mathematics.

In both cases, evolutionary cognitive development and mathematics learning, my reflections havebeen, of necessity, those of an outsider, albeit one who has spent his professional life working in thedomain of interest, to whit, mathematics. I am not a cultural anthropologist or an evolutionarybiologist, 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 Icare 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 acquiredthat 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'tknow 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 thatstudents master the long division algorithm. It's more fundamental than that. We don't know whatview of mathematics on which to base our instruction! In fact, as far as I can tell from the emails Ireceive - and I get a fair number - many US educators are unaware that there could be an alternative tothe 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 wastaught math, is that the beginning math student abstracts mathematical concepts from his or hereveryday experience. As far as we know, this was how the concept of (positive, whole) numbers arosein Sumeria between 8,000 and 5,000 B.C. (I describe this fascinating story in my books Mathematics:


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The Science of Patterns and in The Math Gene .) The assumption behind today's standard US K-12math 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 handlingfractions and negative numbers - the exact order of introduction here is not clear - and then eventuallythe real numbers. (The complex number system, the "end point" of the development from amathematical 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 onthe number systems and algebra believed to be more important for life in today's society. I'll comeback to that later as well.]

This view of the acquisition of mathematical knowledge and ability is implicit in the account I give inThe Math Gene and was made abundantly explicit in Lakoff and Nunez's book Where MathematicsComes From , which, although published just after mine, by the same publisher, and seemingly animmediate 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 firsttime I read their book, I agreed enthusiastically with everything they said. But on reflection, followedby a second and then a third reading, together with discussions with colleagues - particularly theIsraeli mathematics education specialist Uri Leron - the doubts began to set in. The picture Lakoff andNunez paint of the acquisition of new mathematical concepts and knowledge, is one of iteratedmetaphor building, where each new concept is created from the body of knowledge already acquiredthrough 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 wherebythe brain, as a physical organ, extends its domain of activity. My problem, and that of others I talkedto, was that the process they described, while plausible (and perhaps correct) for the way we learnelementary arithmetic and possibly other more basic parts of mathematics, does not at all resemble theway (some? many? most? all?) professional mathematicians learn a new advanced field of abstractmathematics.

Rather, a mathematician (at least me and others I've asked) learns new math the way people learn toplay chess. We first learn the rules of chess. Those rules don't relate to anything in our everydayexperience. They don't make sense. They are just the rules of chess. To play chess, you don't have tounderstand the rules or know where they came from or what they "mean". You simply have to followthem. In our first few attempts at playing chess, we follow the rules blindly, without any insight orunderstanding 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. Notin 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 itreally does make sense to us. The moves do have meaning (in terms of the game). But this is not aprocess of constructing a metaphor. Rather it is one of cognitive bootstrapping (my term), where wemake use of the fact that, through conscious effort, the brain can learn to follow arbitrary andmeaningless rules, and then, after our brain has sufficient experience working with those rules, it startsto make sense of them and they acquire meaning for us. (At least it does if those rules are formulatedand 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 newgame is all about.) Often, after we have learned the new stuff in a rule-determined manner, we canlink it to things we knew previously. We can, in other words, construct a metaphor map linking thenew to the old. But that is possible only after we have completed the bootstrap. It's not how welearned it. Similarly, expert chess players often describe their play in terms of military metaphors,


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using terms like "threat", "advance", "retreat", and "reinforce". But none of those make sense when abeginner is first learning how to play. The real-world metaphor here depends upon a fairly advancedunderstanding 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 mathdepartment. But here's the rub. If learning advanced mathematics is more akin to learning chess than itis to, say, learning to walk, learning to play tennis, or learning to ride a bike - where we start with ournative 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 certainlybegins - or has begun - when the student meets the concept of a mathematical function. As Leron andothers have shown, a significant proportion of university mathematics students do not have the correctconcept 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 Ireally meant was ..." That original answer was wrong, and shows that, even if you "know" the correctdefinition, your underlying concept of a function is wrong. Functions, as defined and used all the timein 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. Functionsare not processes but objects in the mathematical realm. A student who has not fully grasped andinternalized 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 dothings 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 differencebetween 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. Finedistinctions matter. Concepts matter. Having the right concept matters. If you make a small change inone of the rules of chess you will end up with a different game, and the same in the (rule-based) gamewe 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 - Isuspect 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 andengineering, 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 studieshave shown, changing a basic concept once it has been acquired, internalized, and assimilated is noeasy matter. But how about the rest? The ones who do not go to university and study a scientificsubject.

Well, having an incorrect function concept might not be a problem for most people, but the functionconcept 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 realnumbers? What if it includes the negative integers? What if it includes the concept of multiplication (atopic of three of my more recent columns)? What if teaching multiplication as repeated addition (see


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those previous columns) or introducing negative numbers using an everyday (explicit) metaphor (suchas owing money) results in an incorrect concept that leads to increased difficulty later when the childneeds 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 USaccept as the only way to ground K-8 mathematics? Is that really the only way for young children tolearn 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 itcomes to mathematics education? Is starting with the counting numbers the only, or the best, way toteach 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 mypromise to come back to geometry and complex numbers in mathematics education). The only clue I'llgive 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 havestated on several occasions before, I am neither trained in nor do I have first-hand experience inelementary mathematics education. But I can and do read the words of those who do have suchexpertise. At least one other approach has been developed elsewhere in the world, by people with theaforementioned necessary expertise and experience, and there is some evidence to suggest that thealternative 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 ineducation. But I do take what evidence there is as indication that we should at the very least discussand evaluate that alternative approach, even if we start out skeptical of where it might lead. Yet, as faras I can tell, the mathematics education community in the US has so far acted as though this otherapproach simply did not exist. That may, of course, simply be due to, in the words of the prison guardin the classic 1967 Paul Newman movie Cool Hand Luke , a failure to communicate. (Either betweenone part of the world and another, or between our math ed community and the rest of us.) If so, thenmy goal is to try to fix it.

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

Mathematician Keith Devlin (email: [email protected] ) is the Executive Director of the Human-

Sciences and Technologies Advanced Research Institute ( H-STAR ) at Stanford University and TheMath Guy on NPR's Weekend Edition .
mailto:[email protected]://www.stanford.edu/~kdevlin/MathGuy.htmlhttp://www.amazon.com/Unfinished-Game-Pascal-Fermat-Seventeenth-Century/dp/0465009107/ref=sr_1_1?ie=UTF8&s=books&qid=1218025192&sr=8-1http://hstar.stanford.edu/http://www.npr.org/programs/wesat

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