One such scholar is Jean Lave, an anthropologist in the Department of Education at the University of California at Berkeley. Between 1973 and 1978, when she was on the faculty at the University of California at Irvine, Lave spent time among a group of apprentice tailors in Liberia, studying the way the young tailors learned and used the arithmetic they required for their work. Her goal was to compare the way people learned in school -- where the learning takes place out of any context of use and without any immediate use for what is learned -- with the way they learned particular skills on the job, when they really needed what they learned. (In her own terminology, she wanted to compare "formal" learning with "informal" learning.) She chose arithmetic not because she had any particular interest in mathematics, but because it was easy to test people's ability and compare the results.
One particular hypothesis Lave wanted to test was that arithmetical skills learned on the job will be learned much better than skills learned at school. But when she presented the young apprentices with (practical) problems designed to test their arithmetical skills, she found there was little difference between the two. The tailors had as much difficulty solving the problems designed to test the arithmetic they learned and used at work as they did with the problems designed to test what they had learned at school. No matter whether the students learned a particular arithmetic skill at school or at work, they seemed unable to apply that skill in a different, real-life context. Lave published her findings in her 1988 book Cognition in Practice: Mind, Mathematics and Culture in Everyday Life. I re-read this book recently, and this column is based on what she said there.
Of course, there are plenty of circumstances where people do apply in their work things they learned in a classroom. For example, scientists and doctors make regular use of knowledge they acquire as students in the classroom. But in these cases, either the job has a recognizable "scholastic" nature, or the training is specifically geared to preparation for work, or both. Lave's question was getting at something more general: Is it possible to acquire general skills out of context in a classroom, and then use them in any real-life situation where they are in principal applicable. Arithmetic, as traditionally taught in schools, is a prime example of a collection of general skills that are taught in classrooms all over the world, and which should, in principal, be applicable in a wide variety of real-life situations.
Lave's initial study led her to the tentative conclusion that when people acquired skills out of context in the classroom, they could not, in general, apply those skills in real-life situations for which they were theoretically appropriate. But that did not prevent them for developing their own, often highly effective, ways of doing things. One example that Lave quotes (Lave 1988, p.65) comes from a series of verbal purchase transactions taped by researchers in a Brazil market. The subject is a twelve-year old boy manning a stall selling coconuts.
Customer: How much is one coconut?
Customer: I'd like ten. How much is that?
Boy: (Pause) 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.
As a mathematician, my initial reaction on reading the above passage was, "How silly! The quickest way to arrive at the answer is to use the rule that to multiply by 10 you simply add a zero -- so 35 becomes 350." Indeed, that is by far the quickest method, provided you know the rule. If I did not know any better, I might have gone on to conclude: "That young boy obviously doesn't know his multiplication table, not even the easiest case!" But having read Lave (and other authors who make essentially the same point), instead I asked myself, how does the situation appear if I put myself in the position of the young market trader? Presumably, he often finds himself selling coconuts in groups of two or three. So 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 the highly unusual request for ten coconuts -- the shopper was in fact a researcher who set out to see how the young traders could cope with transactions involving such arithmetical problems -- the young boy actually adopts an approach that, in its own way, is mathematically very sophisticated. 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 performs 210 + 105 = 315. Finally, he works out 315 + 35 = 350. Altogether quite an impressive performance for a twelve-year old of supposedly poor education.
To give you a taste of the kinds of things Lave observed in the AMP, a male dieter preparing a meal was faced with having to measure out 3/4 of the 2/3 of a cup of cottage cheese stipulated in the recipe he was using. Before reading on, how would you go about this?
Here is what the subject did. (Lave 1988, p.165.) He measured out 2/3 of a cup of cheese using his measuring device, and spread it out on a chopping board in the shape of a circle. He then divided the circle into four quarters, removed one quarter and returned it to the container, leaving on the board the desired 3/4 of 2/3 a cup. Perfectly correct.
What is my reaction as a mathematician? That there is a much "easier" way: When you multiply the fraction 3/4 by 2/3 the 3 cancels and you end up with 2/4, which simplifies to 1/2; so all the subject needed was 1/2 a cup of cheese, which he could have measured out directly. Simple. But our man did not see this solution. Nevertheless he clearly knew what the concept "three-quarters" means, and was able to use that knowledge to solve the problem in his own way.
The AMP studied thirty-five subjects in Orange County, in Southern California. The subjects varied considerably in terms of education and family income, and included some people of poor educational background and low income, for whom buying groceries economically was very important. Twenty-five subjects were in the supermarket study, ten in the dieting cooks study.
Since the idea was to examine the way ordinary people actually used mathematics in their everyday lives, the researchers could not simply test them with questions such as "If you were faced with three kinds of frozen french-fries with the following weights and prices, how would you decide which is the most economical?" As Lave and her colleagues showed, the answer that people give to such a question has very little to do with what they would actually do in a real shopping situation. To put it bluntly, "What if?" questions don't work.
Instead, the researchers chose to follow the subjects around and observe them, taking copious notes, occasionally asking the subjects to explain their reasoning out loud as they went about their shopping or food preparation, and sometimes asking for explanations just after the transaction had been completed. Of course, this procedure is highly contrived. The very presence of the observers changes the experience of shopping or of preparing a meal, as does the request that the subjects describe what they are doing. Thus, to some extent the study is not really one of people "in their normal, everyday activities." But it's probably as close as you can get. Moreover, anthropologists have developed ways of going about such work so as to minimize any influences their presence has on their subjects' behavior.
Each of the researchers spent a total of about forty hours with each of her or his subjects, including time spent interviewing subjects to determine their backgrounds (education, occupation, etc.). Though most of the subjects were women, there were some men in the group. However, the researchers noticed no difference in the mathematical performances of the men and the women in either the supermarket study or the dieting cooks study, so gender did not seem to be a significant factor.
Out of a total of around 800 individual purchases that the subjects made in the course of the study, just over 200 involved some arithmetic -- which the researchers defined to be "an occasion on which a shopper associated two or more numbers with one or more arithmetical operations: addition, subtraction, multiplication, or division." (Lave 1988, p.53.) The shoppers varied enormously in the frequency with which they used mathematics. One shopper used none whatsoever, while three of the subjects performed calculations in making over half their purchases. On average, 16% of purchases involved arithmetic.
One interesting observation that came out of the study was that the attitudes the subjects reported having had towards mathematics when they had been at school had no effect on their arithmetical performance in the supermarket or the kitchen.
Another fascinating result is that, in comparing competing products to decide which was the best buy, shoppers made relatively little use of the unit price printed on the label -- an item of information included on the label by law specifically to enable shoppers to compare prices. The most likely explanation is that the unit price is essentially abstract, arithmetical information. Unless the product is something that the shopper either buys or uses in, say, single ounce units, then the price per ounce has no concrete significance for that shopper. Thus, even though direct comparison of unit prices is the most direct way to determine value-for-money, shoppers often ignore it. And, needless to say, they are even less likely to calculate it.
A common approach was to calculate ratios between prices and quantities in a way that made direct comparison possible. This could be done if the quantities were in a simple ratio to one another, say 2:1 or 3:1. For example, if product A cost $5 for 5 oz and product B cost $9 for 10 oz, the comparison was easy. A typical shopper would reason like this: "Product A would cost $10 for 10 oz, and product B is $9 for 10 oz, hence product B is the cheaper buy."
Notice that the kind of argument just described does not involve the explicit calculation of unit prices, although mathematically it is entirely equivalent. One difference is that the unit price provides a single figure associated with each item, thereby allowing the comparison of any two similar items in the store. But shoppers generally only resorted to arithmetic when making a comparison between two (or sometimes three) items, and then they made whatever transformations would bring the price information for those two items into a form that made comparison possible.
Another advantage of working with the actual amounts that might be bought is that the price comparison is often just one part of a more complex decision making process, in which the shopper's storage capacity, size of family, likely rate of usage, and the estimated storage period before a particular item might spoil all play a part. As the AMP researchers observed time and again, what shoppers did was to juggle all of these variables in order to reach a decision, thinking about the purchase options first one way, then another, then another. The price-comparison arithmetic was certainly a part of this process, but it was by no means the only part. Despite the complexity of this process, shoppers did not have to expend any great effort. Indeed, they were not aware that they were "thinking" much at all; they were "just shopping."
Another method that many shoppers used to decide between two options was to calculate the price differential, a procedure that requires just two subtractions. For example, faced with a choice between a 5 oz packet costing $3.29 and a 6 oz packet priced at $3.59, the shopper would argue, "If I take the larger packet, it will cost me 30 cents for an extra ounce. Is it worth it?"
Among the arithmetical techniques that the researchers observed shoppers performing were estimation, rounding (say to the nearest dollar or the nearest dollar and a half), and left-to-right calculation (as opposed to the right-to-left calculation taught in school). What seemed to be absent, however, were most of the techniques the shoppers had been taught in school. Lave and her colleagues set out to investigate where the school math had gone.
In fairness to the subjects, it has to be said that, despite the surrounding circumstances, the "math test" did have all the elements of a typical school arithmetic test: questions involving whole numbers, both positive and negative, fractions, decimals, addition, subtraction, multiplication, and division. All the problems were, however, designed to test the kind of arithmetical skill that the researchers had observed the subjects using (in context) in the supermarket. For instance, having observed that shoppers frequently compared prices of competing products by comparing price-to-quantity ratios, the researchers included some problems to see how the subjects fared with abstract versions of such problems. For example, faced with an item costing $4 for a 3 oz packet and a larger packet costing $7 for 6 oz, many shoppers would -- in effect -- compare the ratios 4/3 and 7/6 to see which was the larger. So the researchers would include on the test the question: "Circle the larger of 4/3 and 7/6." One obvious difference between such a question in the formal test and its equivalent in real life was that the subjects took the test question as requiring a precise calculation, whereas they were much more likely to use estimation in real life, though often with considerable accuracy.
Perhaps the greatest surprise in the study was the huge disparity between how the subjects performed in the real life situations and their results on the test. Although the arithmetic test was designed to test the very mathematics that the subjects used in the supermarket, assuming they performed the calculations using the method they learned in school, the shoppers performance was rated at an average 98% in the supermarket as against a mere 59% average on the test. In other words, the shoppers in the supermarket were probably not using the arithmetical skills they learned in school. Rather, they were solving the problem another way.
This last conclusion is supported by the fact that performance on the test was higher the longer subjects had studied math at school and the more recently they had finished school, whereas neither length of schooling nor the time since schooling had any measurable effect on how well they did in the supermarket. Thus, school math classes seem to teach people how to perform on school math tests, but not how to solve real-life problems that involve math.
Since the subjects were highly successful when it came to performing arithmetical tasks in real-life situations -- regardless of their schooling history -- one obvious question is, how were they doing it?
Of course, part of the difference in performance might have been due to the difference between actually being in the store as against "taking a test." As we observed a moment ago, the subjects could not help viewing the arithmetic test as a "school quiz," with all the psychological stress that might entail. But that did not seem to be the major factor. Rather, what seemed to make the biggest difference was the kind of test the subjects were asked to take and the manner in which the questions were presented. This was shown by a further test the researchers put the subjects through: a shopping simulation, where, in their homes, the subjects were presented with simulated best-buy shopping problems, based on the very best-buy problems the researchers had observed the subjects resolve in the supermarket. In some of these simulations, the subjects were presented with actual cans, bottles, jars, and packets of various items taken from the supermarket and asked to decide which to buy among competing brands; in others they were presented with the price and quantity information printed on cards. In this simulation, which was evidently a kind of "test" situation, but with the questions clearly of a shopping nature as opposed to being school-like "math questions," the subjects scored an average of 93%. (The fact that the simulation was done in the subject's home, carried out by the researcher who had accompanied the subject on the shopping trip, also seems to have been a significant factor.)
To put this in terms of a specific example, a subject would perform extremely well (in the 93% success rate category) in the home shopping simulation, when presented with a card that said 3 oz of product A cost $4 and another card that said 6 oz of product B cost $7 and asked which was the best buy; but in the context of being presented with a list of arithmetic problems, the same subject would do far worse (in the 59% success rate category) when asked to circle the larger of 4/3 and 7/6. And yet, the arithmetic problem underlying the two questions is exactly the same!
The conclusion seems to be not that people can't do math; rather they can't do school math. When faced with a real-life task that requires elementary arithmetic, most people do just fine -- indeed, 98% success is virtually error free. And yet, school math -- at least the more elementary parts -- is supposed to provide us with just the arithmetic skills we need in our everyday lives. Indeed, the methods taught in school are supposed to be the simplest and the best -- that's why they are taught!
Incidentally, though a number of the AMP shoppers did carry a calculator with them, only on one occasion during the entire project did one shopper take it out and use it in order to carry out a price comparison. And no one ever used a pencil and paper to carry out a calculation. When the calculation became too hard to do in their heads, they simply resorted to other criteria on which to base their decision.
Does Lave's research have implications for the way we should teach basic arithmetic in our schools? You bet it does. But that's another story, one that deserves -- and requires -- far more space than I have left in this issue of Devlin's Angle. I'll come back to it at a later date.
Nunes T, Schliemann A, and Carraher D (1993) Street Mathematics and School Mathematics, Cambridge, UK: Cambridge University Press.