Of Minds and Language by Piattelli-Palmarini Massimo; Uriagereka Juan; Salaburu Pello

Author:Piattelli-Palmarini, Massimo; Uriagereka, Juan; Salaburu, Pello
Language: eng
Format: epub
Publisher: Oxford University Press
Published: 2009-08-20T16:00:00+00:00

15.5 Rational numbers are hard

I will conclude now with two contrasting numerical concepts: the successor principle and rational numbers. The successor principle captures the idea that there is always another cardinal number after the one just counted or thought about. This is because addition is closed under the natural numbers. As expected, when Hartnett and Gelman (1998) asked children ranging in age from about 6 years to 8 years of age if they could keep adding 1 to the biggest number that they could or were thinking about, a surprising number indicated that they could. Even when we suggested that a googol or some other very large cardinal number was the biggest number there could be, we were challenged by the child, who noted it was possible to add another 1 to even our number.

The successor principle is seldom taught in elementary school, whereas notions about fractions are. However, when it comes to moving on to considering rational numbers, and the idea that one integer divided by another is a rational number, we run into another example of a HoW domain. This perhaps is not surprising since there is no unique number between a pair of rational numbers. Formally, there is an infinite number of rational numbers between any two pairs of this kind of number. There is more to say about this, but I think that starts to give you the flavor that we really have moved into a different domain and that we may have a case of a conceptual change.

To end this presentation, I illustrate the kind of errorful but systematic patterns of responses we have obtained from school-aged children asked to place in order, from left to right, a series of number symbols, each one of which is on a separate card. Keep in mind that these children were given practice at placing sticks of different lengths on an ordering cloth; they were even told that it was acceptable to put sticks there of the same length but different colors and to move sticks, and then the test cards, until they were happy with their placement order. Careful inspection of the placements reveals that the children invented natural number solutions. For example, an 8-year-old started by placing each of three cards left to right as follows: 1/2, 2/2, 2 1/2, etc. The following interpretation captures these and all further placements. The child took the cards as an opportunity to apply his knowledge of natural number addition:

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