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Notes

 1 See Harel and Confrey, 1994. Rational numbers, ratios, and proportions, which on the surface are about division, are called multiplicative concepts because any division problem can be rephrased as multiplication. See Chapter 3. 2 Behr, Lesh, Post, and Silver, 1983; Confrey, 1994, 1995; Empson, 1999; Kieren, 1992; Mack, 1990, 1995; Pothier and Sawada, 1983; Streefland, 1991, 1993. 3 Hiebert and Tonnessen, 1978; Pothier and Sawada, 1983. 4 Empson, 1999; Pothier and Sawada, 1983. 5 Confrey, 1994; Pothier and Sawada 1989. 6 Confrey, 1994; Streefland, 1991, 1993. 7 Cramer, Behr, Post, and Lesh, 1997; Empson, 1999; Mack, 1995; Morris, in press; Moss and Case, 1999; Streefland, 1991, 1993. 8 Kouba, Zawojewski, and Strutchens, 1997; Wearne and Kouba, 2000. 9 Behr, Lesh, Post, and Silver, 1983; Behr, Wachsmuth, Post, and Lesh, 1984; Bezuk and Bieck, 1993; Hiebert and Wearne, 1985; Mack, 1990, 1995; Post, Wachsmuth, Lesh, and Behr, 1985; Streefland, 1991, 1993. 10 Kieren, 1976. 11 Kieren, 1976, 1980, 1988. 12 Students not only should “construct relations among them” but should also eventually have some grasp of what is entailed in these relations—for example, that Interpretation D is a contextual instance of E—namely, you multiply the number of green cars by to get the number of red cars, while thinking of as three times (Interpretation A), and thinking of it as 3 divided by 4, is the equation which is basically the associative law for multiplication. 13 Behr, Wachsmuth, Post, and Lesh, 1984; Hiebert and Wearne, 1986. 14 Hiebert and Wearne, 1986; Resnick, Nesher, Leonard, Magone, Omanson, and Peled, 1989. 15 Carpenter, Corbitt, Kepner, Lindquist, and Reys, 1981. 16 Hiebert and Wearne, 1986. 17 Behr, Lesh, Post, and Silver, 1983. 18 Davis, 1988. 19 Behr, Wachsmuth, Post, and Lesh, 1984. 20 Resnick, Nesher, Leonard, Magone, Omanson, and Peled, 1989. 21 Behr, Wachsmuth, Post, and Lesh, 1984. 22 Resnick, Nesher, Leonard, Magone, Omanson, and Peled, 1989. 23 Cramer, Post, Henry, and Jeffers-Ruff, in press; Hiebert and Wearne, 1988; Hunting, 1983; Mack, 1990, 1995; Morris, in press; Moss and Case, 1999; Hiebert, Wearne, and Taber, 1991. 24 Moss and Case, 1999. 25 Behr, Harel, Post, and Lesh, 1992. 26 Davydov and Tsvetkovich, 1991; Morris, in press; Schmittau, 1993. 27 Morris, in press.

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