delanceyplace.com 8/8/12 - thinking about numbers

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In today's excerpt - even though most of us have learned to think about numbers as evenly spaced, as if on a ruler or number line, is has long been more natural and common for people to think about numbers logarithmically:

"In one of his most fascinating experiments, [scientists and linguist Pierre] Pica examined the [South American Munduruku] Indians' spatial understanding of numbers. How did they visualize numbers spread out on a line? In the modern world, we do this all the time—on tape measures, rulers, graphs and house numbers along a street. Since the Munduruku don't have numbers, Pica tested them using sets of dots on a screen. Each volunteer was shown an unmarked line on the screen. To the left side of the line was one dot, to the right ten dots. Each volunteer was then shown random sets of between one and ten dots. For each set the subject had to point at where on the line he or she thought the number of dots should be located. Pica moved the cursor to this point and clicked. Through repeated clicks, he could see exactly how the Mundu­ruku spaced numbers between one and ten.

"When American adults were given this test, they placed the num­bers at equal intervals along the line. They re-created the number line we learn at school, in which adjacent digits are the same distance apart as if measured by a ruler. The Munduruku, however, responded quite differ­ently. They thought that intervals between the numbers started large and became progressively smaller as the numbers increased. For example, the distances between the marks for one dot and two dots, and two dots and three dots, were much larger than the distance between seven and eight dots, or eight and nine dots.

"The results were striking. It is generally considered a self-evident truth that numbers are evenly spaced. We are taught this at school and we accept it easily. It is the basis of all measurement and science. Yet the Munduruku do not see the world like this. They visualize magnitudes in a completely different way.

"When numbers are spread out evenly on a ruler, the scale is called linear. When numbers get closer as they get larger, the scale is called logarithmic* It turns out that the logarithmic approach is not exclusive to Amazonian Indians. We are all born conceiving of numbers this way. In 2004, Robert Siegler and Julie Booth at Carnegie Mellon University in Pennsylvania presented a similar version of the number line experiment to a group of kindergarten pupils (with an average age of 5.8 years), first graders (6.9) and second graders (7.8). The results showed in slow-motion how familiarity with counting molds our intuitions. The kindergarten pupil, with no formal math education, maps numbers out logarithmically. By the first year at school, when the pupils are being introduced to num­ber words and symbols, the graph is straightening. And by the second year at school, the numbers are at last evenly laid out along the line.

"Why do Indians and children think that higher numbers are closer together than lower numbers? There is a simple explanation. In the experi­ments, the volunteers were presented with a set of dots and asked where this set should be located in relation to a line with one dot on the left and ten dots on the right. (Or, in the children's case, 100 dots.) Imagine a Munduruku is presented with five dots. He will study them closely and see that five dots are five times bigger than one dot, but ten dots are only twice as big as five dots. The Munduruku and the children seem to be making their decisions about where numbers lie by estimating the ratios between amounts. In considering ratios, it is logical that the distance between five and one is much greater than the distance between ten and five. And if you judge amounts using ratios, you will always produce a logarithmic scale.

"It is Picas belief that understanding quantities approximately in terms
of estimating ratios is a universal human intuition. In fact, humans who do not have numbers -- like Indians and young children -- have no alterna­tive but to see the world in this way. By contrast, understanding quanti­ties in terms of exact numbers is not a universal intuition; it is a product of culture. The precedence of approximations and ratios over exact num­bers, Pica suggests, is due to the fact that ratios are much more important for survival in the wild than the ability to count. Faced with a group of spear-wielding adversaries, we needed to know instantly whether there were more of them than us. When we saw two trees we needed to know instantly which had more fruit hanging from it. In neither case was it nec­essary to enumerate every enemy or every fruit individually. The crucial thing was to be able to make quick estimates of the relative amounts. ...

"Exact numbers provide us with a linear framework that contradicts our logarithmic intuitions. Indeed, our proficiency with exact numbers means that the logarithmic intuition is overruled in most situations. But it is not eliminated altogether. We live with both a linear and a logarithmic understanding of quantity. ... Our deep-seated logarithmic instinct surfaces most clearly when it comes to thinking about very large numbers. For example, we can all understand the difference between one and ten. It is unlikely we would confuse one pint of beer and ten pints of beer. Yet what about the difference between a billion gallons of water and ten billion gallons of water? Even though the difference is enormous, we tend to see both quantities as quite similar-as very large amounts of water. Likewise, the terms "millionaire" and "billionaire" are thrown around almost as synonyms -- as if there is not so much differ­ence between being very rich and being very, very rich. Yet a billionaire is a thousand times richer than a millionaire. The higher numbers are, the closer together they feel."

*In fact, numbers need to get closer in a certain way for the scale to be logarithmic. For a fuller discussion of the logarithmic scale, see page 130.