Devlin's Angle

October 2007

Kinds of Math

Small children learn to understand and produce the language they hear spoken around them rapidly and with no conscious effort, but it takes a lengthy period of struggle to learn to read and write that same language. Yet the written language is merely a symbolic representation on paper (or some other medium) of the structure they hear and speak.

Do you see anything in that last sentence that strikes you as perhaps wrong? I do. It's that word "merely". If the written form of language were merely a symbolic representation of the language, it should not be so hard to learn it - so hard that many people never achieve mastery. Spoken language is a system that evolved with our species, over a three million year period culminating, we think, about 100,000 years ago, and to a great extent is characteristic of Homo sapiens. Written language, on the other hand, is a system that our ancestors invented some time around 10,000 B.C. From a cognitive standpoint, written language and spoken language are clearly very different processes.

I believe the same holds for mathematics, at least the parts of mathematics that relate directly to, and indeed are abstracted directly from, the real world in which we live (specifically number, elementary arithmetic, and basic ideas of geometry and trigonometry).

Studies have demonstrated repeatedly that when people find themselves in a situation where they need basic math skills in their everyday lives, they pick them up fairly quickly, and rapidly become fluent.

In one such study, carried out in the early 1990s, three researchers, Terezinha Nunes of the University of London, England, and Analucia Dias Schliemann and David William Carraher of the Federal University of Pernambuco in Recife, Brazil went out into the street markets of Recife with a tape recorder, posing as ordinary market shoppers. Their target subjects were young children aged between 8 and 14 years of age who were looking after their parents' stalls while the latter were away. At each stall, the researchers presented the young stallholders with transactions designed to test various arithmetical skills.

Working entirely in their heads, with no paper and pencil, let alone a hand calculator, the young children got the correct answer 98% of the time. But posing as customers was just the first stage of the study Nunes and her colleagues carried out. About a week after they had "tested" the children at their stalls, they visited the subjects in their homes and asked each of them to take a pencil-and-paper test that included exactly the same arithmetic problems that had been presented to them in the context of purchases the week before. The investigators took pains to give this second test in as non-threatening a way as possible. It included both straightforward arithmetic questions presented in written form and verbally presented word problems in the form of sales transactions of the same kind the children carried out at their stalls. The subjects were provided with paper and pencil, and were asked to write their answer and whatever working they wished to put down. They were also asked to speak their reasoning aloud as they went along.

Although the children's arithmetic was practically faultless when they were at their market stalls (just over 98% correct), they averaged only 74% when presented with market-stall word problems requiring the same arithmetic, and a staggeringly low 37% when the same problems were presented to them in the form of a straightforward symbolic arithmetic test.

The researchers noted that the methods the children used - to great effect - in the street-market were not the ones they had been taught (and were still being taught) in school. Rather, in the market they applied methods they had picked up working alongside their parents and friends. Clearly, "street mathematics," as Nunes and her colleagues called the mental activity they had observed in the marketplace, was quite different from the symbolic symbol system the children encountered in school.

If you want to see details of the methods the children used in the market, and examine the mistakes they made when trying to carry out paper-and-pencil calculations, see the book Street Mathematics and School Mathematics (Learning in Doing: Social, Cognitive and Computational Perspectives) that Nunes and her colleagues wrote about their study.

A description of a similar kind of study carried out in the U.S., this time of price-conscious, adult supermarket shoppers, with similar results, was given by Jean Lave in her book Cognition in Practice: Mind, Mathematics and Culture in Everyday Life (Learning in Doing).

I summarized both studies in my more recent book The Math Instinct.

While we do not yet understand how the human brain does mathematics, either mentally in a real-world environment such as a street market or a supermarket, or in a symbolic fashion using paper and pencil, it seems pretty clear that the two activities are at least as different from each other as are written and spoken language. Written, symbolic mathematics is not "merely" a physically represented version of the mental activity Nunes et al dubbed "street mathematics".

While modern technology means that a mastery of accurate mental arithmetic skills is nothing like as important today as it used to be, it is generally accepted that a good understanding of number and quantity - what is often called numeracy or quantitative literacy - is absolutely crucial for an individual to be a properly functioning member of present-day society. Recognition of this fact has led to the production of a number of textbooks designed to teach people this important skill.

But wait a minute. Doesn't something strike you as odd about that development? We don't use books to teach young children to understand and speak their native language - a Catch 22 challenge if ever there were one; rather, we allow them to pick it up in the environment in which they are exposed to it. Nor do we teach people to read and write musical notation in order for them to enjoy music, to sing, to play a musical instrument, or even to create music; instead, we expose them to music and let them develop singing and playing skill by doing. Why then should be believe for one minute that providing a child with a math textbook will lead to numeracy - to their becoming competent at what I prefer to call "everyday math"?

In fact, I don't think we do believe that such an approach will work. (Of course, given the variation among people, pretty well any approach is likely to work for some, but the studies by Nunes et al, by Lave, and by others show that all but a tiny minority of individuals become proficient at street mathematics when put in a real-world environment where it is important to them and in which they are exposed to it.) Rather, we write books because that is the dominant technology for recording mathematical knowledge and disseminating it widely. Indeed, until very recently it was our only technology for those purposes.

The textbook approach worked (for many people) in the days when there was sufficient motivation for (many) people to put in the enormous effort required to make it work - as evidenced by the huge effect Leonardo's book Liber abaci had on western civilization after it appeared in 1202. But it patently does not work in today's western societies.

And I am leaving aside the question of how we measure whether an individual has achieved the desired level of skill in everyday math. If the evaluation method is to give them a written test, then the child stallholders in Recife would be classified as innumerate, which patently they were not!

As I see it, today, technology and an increased understanding of how people learn provide us with a number of alternatives for teaching and evaluating mastery of everyday math, some computer-based, others focused on physical activities, some making use of both. But examining those alternatives is not the purpose of this essay. I'll address that exciting future in a later column. Rather, my present point is that we need to recognize the fundamental fact that written, symbolic mathematics is not "merely" a written version of the mental activity I am calling "everyday math." (Not that I am the first to use that term, nor by any means the only one to use it.) Indeed, from a cognitive standpoint, I don't think symbolic math is a variant of everyday math at all; I believe it is a very different mental activity. (I have lived long enough to regret my telling generations of college-level students that mathematics is just "formalized common sense". Everyday math is, but symbolic math is most definitely not.)

Such recognition does not imply that symbolic mathematics is not important. Heavens, our present society depends on it in spades! Our very survival requires a steady supply of individuals with differing degrees of mastery of symbolic math, all the way up to fully-fledged expert. But by conflating two very different kinds of mental activity under the common term "(basic) math", and tacitly thinking of one as just a written version of the other, I believe we shoot ourselves in both feet when it comes to teaching either kind.

Devlin's Angle is updated at the beginning of each month.
Mathematician Keith Devlin (email: [email protected]) is the Executive Director of the Center for the Study of Language and Information at Stanford University and The Math Guy on NPR's Weekend Edition. Devlin's most recent book, Solving Crimes with Mathematics: THE NUMBERS BEHIND NUMB3RS, is the companion book to the hit television crime series NUMB3RS, and is co-written with Professor Gary Lorden of Caltech, the lead mathematics adviser on the series. It was published last month by Plume.