"How much is one coconut?" you ask.
"Thirty-five," he replies with a smile.
You say, "I'd like ten. How much is that?"
The boy pauses for a moment before replying. Thinking out loud, he says: "Three will be 105; with three more, that will be 210. (Pause) I need four more. That is . . . (pause) 315 . . . I think it is 350."
I didn't make up this exchange. It is taken verbatim from a report written some years ago by 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. The three researchers went out into the street markets of Recife with a tape recorder, posing as ordinary market shoppers. At each stall, they presented the young stallholder with a transaction designed to test a particular arithmetical skill.
The purpose of the research was to determine how effective was traditional mathematics instruction, which all the young market traders had received in school since the age of six.
How well did our young coconut seller do?
If you think about it for a moment, it's clear that the boy isn't doing it the quickest way, which is to use the rule that to multiply by 10 you simply add a zero - so 35 becomes 350. The reason he doesn't do it that way is that he doesn't know the rule. He's never learned it. Despite spending six years in school, he has almost no mathematical knowledge at all in the traditional sense. What arithmetical skills he has are self taught at his market stall. Here is how he solves the problem.
Since he often finds himself selling coconuts in groups of two or three, he needs to be able to compute the cost of two or three coconuts; that is, he needs to know the values 2 x 35 = 70 and 3 x 35 = 105. Faced with your highly unusual request for ten coconuts, the young boy proceeds like this. First, he splits the 10 into groups he can handle, namely 3 + 3 + 3 + 1. Arithmetically, he is now faced with the determining the sum 105 + 105 + 105 + 35. He does this is stages. With a little effort, he first calculates 105 + 105 = 210. Then he computes 210 + 105 = 315. Finally, he works out 315 + 35 = 350. Altogether quite an impressive performance for a twelve-year old of supposedly poor education!
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 went back to the subjects and asked each of them to take a pencil-and-paper test that comprised 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 was administered in a one-on-one setting, either at the original location or in the subject's home, and 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 mere 37% when virtually the same problems were presented to them in the form of a straightforward arithmetic test.
The performance of our young coconut seller was typical. One of the questions he had been asked at his market stall, when he was selling coconuts costing 35 cruzeiros each, was: "I'm going to take four coconuts. How much is that?" The boy replied: "There will be one hundred five, plus thirty, that's one thirty- five . . . one coconut is thirty-five . . . that is . . . one forty."
Let's take a look at this solution. Just as he had in the exchange I described earlier, the boy began by breaking the problem up into simpler ones; in this case, three coconuts plus one coconut. This enabled him to start out with the fact he knew, namely that three coconuts cost Cr$105. Then, to add on the cost of the fourth coconut, he first rounded the cost of a coconut to Cr$30 and added that amount to give Cr$135. He then (apparently, though he did not verbalize this step precisely) noted that the "correction factor" for the rounding was Cr$5, and added in that correction factor to give the (correct) answer Cr$140.
On the formal arithmetic test, the boy was asked to calculate 35 x 4. He worked mentally, vocalizing each step as the researcher requested, but the only thing he wrote down was the answer. Here is what he said; "Four times five is twenty, carry the two; two plus three is five, times four is twenty." He then wrote down "200" as his answer.
Despite the fact that, numerically, it was the same problem he had answered correctly at his market stall, he got it wrong. If you follow what he said, it's clear what he was doing and why he went wrong. In trying to carry out the standard right-to-left school method for multiplication, he added the carry from the units-column multiplication (5 x 4) before performing the tens-column multiplication, rather than afterwards, which is the correct way. He did, however, keep track of the positions the various digits should occupy, writing the (correct) 0 from the first multiplication after the (incorrect) 20 from the second, to give his answer 200.
In case after case, Nunes and her colleagues obtained the same results. The children were absolute number wizards when they were at their market stalls, but virtual dunces when presented with the same arithmetic problems presented in a typical school format. The researchers were so impressed - and intrigued - by the children's market-stall performances that they gave it a special name: They called it street mathematics. Since the Recife children demonstrated that they could handle arithmetic in the appropriate context, when the numbers meant something to them, it seems clear that meaning plays a major role in our ability to do arithmetic. When the children carried out computations at their stalls, both the numbers and the operations they performed on them had meaning, and the operations made sense. Indeed, the children were quite literally surrounded by physical meanings of the arithmetical procedures they performed.
In contrast to street mathematics, the essence of school mathematics, which the Recife children were not able to do, is that it is entirely symbolic - i.e., it operates on symbols that are devoid of meaning. In performing a standard school procedure for addition, subtraction, multiplication, or division, you carry out the very same actions, in exactly the same order, regardless of what the actual numbers are or what they measure. This is the whole point. The methods taught in school are supposed to be universal. Learn them once and you can apply them in any particular circumstance, whatever specific numbers are involved.
In the hands of a person who can master the abstract, symbolic procedures taught in school, those procedures are extremely powerful. Indeed, they underpin all our science, technology, and modern medicine, and practically every other aspect of modern life. Their development marks one of the crowning achievements of the human race. But that doesn't make them easy to learn or to apply.
The problem is that humans operate on meanings. In fact, the human brain evolved as a meaning-seeking device. We see, and seek, meaning anywhere and everywhere. We can't avoid it. A computer can be programmed to slavishly follow rules for manipulating symbols, with no understanding of what those symbols mean (in fact no understanding at all), until we tell it to stop. But people can't do that - at least, not to anything like the same extent. With considerable effort, we (or at least some of us) can learn our multiplication tables and train ourselves to follow a small number of arithmetical procedures. But even then, I think meaning is the key. I believe that mastery of school arithmetic involves the acquisition of some kind of meaning for the objects involved and the procedures performed on them. I don't think the human brain can perform genuinely meaningless operations at all.
If I'm right, then that means that a crucial component of mathematics education is making sure that the student is able to construct (or otherwise acquire) appropriate meanings for the various abstract concepts and methods he or she is faced with. A particularly intriguing question is whether that can be done without recourse to plain old rote learning. I don't know the answer to that question. I have some thoughts, and I express them in the latter part of my recent book The Math Instinct, from which this month's column is taken. But now my time is up for this month.