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 with positive rational numbers over 2000 years before negative numbers became accepted. See also Behr, Harel, Post, and Lesh, 1992. 8. The rules are in a sense guided by the fractional notation, a/b. In other notational systems, such as decimal representation, the rules will look somewhat different, although they will be equivalent. 9. These numbers (and many others) are not rational because they cannot be expressed as fractions with integers in the numerator and denominator. 10. In the number-line illustrations throughout this chapter, the portion displayed and the scale vary to suit the intent of the illustration. That is reasonable not just because one can imagine moving a “lens” left and right and zooming in and out, but also because the ideas are independent of the choice of origin and unit. 11. Bruner, 1966 (pp. 10–11), suggests three ways of transforming experience into models of the world: enactive, iconic, and symbolic representations. Enactively, addition might be the action of combining a plate of three cookies with a plate of five cookies; iconically, it might be represented by a picture of two plates of cookies; symbolically, it might be represented as 5 cookies plus 3 cookies, or merely 5+3. 12. Greeno and Hall, 1997. 13. Pimm, 1995, suggests that people seek representational systems in which they can operate on the symbols as though the symbols were the mathematical objects. 14. Duvall, 1999. 15. Kaput, 1987, argues that much of elementary school mathematics is not about numbers but about a particular representational system for numbers. See Cuoco, 2001, for detailed discussions of various ways representations come into play in school mathematics. 16. See Lakoff and Núñez, 1997, and Sfard, 1997, for detailed discussion of the metaphoric nature of mathematics. 17. Sfard, 1997, p. 36, emphasis in original. 18. “I remember as a child, in fifth grade, coming to the amazing (to me) realization that the answer to 134 divided by 29 is 134/29 (and so forth). What a tremendous labor-saving device! To me, ‘134 divided by 29’ meant a certain tedious chore, while 134/ 29 was an object with no implicit work. I went excitedly to my father to explain my discovery. He told me that of course this is so, a/b and a divided by b are just synonyms. To him it was just a small variation in notation” (Thurston, 1990, p. 847). 19. Grouping is a common approach in measurement activities. For example, in measuring time, there are 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day, approximately 30 days in a month, 12 months in a year, and so on. For distance, the customary U.S. system uses inches, feet, yards, and miles, and the metric system uses centimeters, meters, and kilometers. 20. For example, IX means nine (that is, one less than ten), whereas XI means eleven (one more than ten). 21. This generality was a significant accomplishment. In the third century B.C. in Greece, with its primitive numeration system, a subject of debate was whether there even existed a number large enough to describe the number of grains of sand in the universe. The issue was serious enough that Archimedes, the greatest mathematician

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