Sanjoy Mahajan in Freakonomics:
…good estimators possess a clear mental number line — one in which numbers are evenly spaced, or linear, rather than a logarithmic one in which numbers crowd closer together as they get bigger. Most schoolchildren start out with the latter understanding, shedding it as they grow more experienced with numbers.
I do agree that children start out with a logarithmic understanding. I first learned this idea from a wonderful episode of WNYC’s Radio Lab on “Innate numbers” (Nov. 30, 2009). The producers had asked Stanislas Dehaene to discuss his research on innate number perception.
One of his studies involved an Indian tribe in the Amazon. This tribe does not have words for numbers beyond five, and does not have formal teaching of arithmetic. But the members have a sophisticated understanding of numbers. They were given the following problem: on a line with one object at one end, and nine objects at the other end, they were asked, “How many objects would you place directly in the middle of the line?”
What number would you choose?
Twenty years ago I would have chosen five, for five is the average of one and nine: It is larger than one by four, and smaller than nine also by four. Thus, it is halfway on a linear (or additive) number line. Almost all Western students also choose five. However, the members of the Indian tribe chose three. That choice also makes sense, but in a different way: Three is larger than one by a factor of 3, and smaller than nine also by a factor of 3. Thus, three is halfway between one and nine on a multiplicative or logarithmic number line.
Dehaene concludes, and I agree, that our innate perception of numbers is logarithmic (or multiplicative); and that we learn our linear (or additive) scale through our culture.
More here. [Thanks to Annie Murphy Paul.]