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• #### Index 441-454

branches of mathematics. It is these connections that give mathematics much of its power. If students are to become proficient in mathematics by eighth grade, they need to be proficient with the numbers and operations discussed in this chapter, as well as with beginning algebra, measure, space, data, and chance—all of which are intricately related to number.

### Notes

 1 Some authors (see, e.g., Russell, 1919, p. 3; Freudenthal, 1983, pp. 77ff) call these the natural numbers. We are adopting the common usage of the U.S. mathematics education literature, in which the natural numbers begin 1, 2, 3, and so on, and the whole numbers include zero. 2 The recognition that zero should be considered a legitimate number—rather than the absence of number—was an important intellectual achievement in the history of mathematics. Zero (as an idea) is present in the earliest schooling, but zero (as a number) is a significant obstacle for some students and teachers. “Zero is nothing,” some people say. “How can we ask whether it is even or odd?” 3 “To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his [or her] most important tools: analogy, generality, and simplicity” (Stewart, 1989, p. 291). 4 Although negative numbers are quite familiar today, and part of the standard elementary curriculum, they are quite a recent development in historical terms, having become common only since the Renaissance. Descartes, who invented analytic geometry and after whom the standard Cartesian coordinate system on the plane is named, rejected negative numbers as impossible. (His coordinate axes had only a positive direction.) His reason was that he thought of numbers as quantities and held that there could be no quantity less than nothing. Now, however, people are not limited to thinking of numbers solely in terms of quantity. In dealing with negative numbers, they have learned that if they think of numbers as representing movement along a line, then positive numbers can correspond to movement to the right, and negative numbers can represent movement to the left. This interpretation of numbers as oriented length is subtly different from the old interpretation in terms of quantity, which would here be unoriented length, and gives a sensible and quite concrete way to think about these numbers that Descartes thought impossible. 5 Freudenthal, 1983, suggests that “negative numbers did not really become important until they appeared to be indispensable for the permanence of expressions, equations, formulae in the ‘analytic geometry’” (p. 436). “Later on arguments of content character were contrived…although some of them are not quite convincing (positive-negative as capital-debt, gain-loss, and so on)” (p. 435). 6 See Freudenthal, 1983, p. 435. 7 Although rational numbers seem to present more difficulties for students than negative integers, historically they came well before. The Greeks were comfortable

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