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Adding + It Up: Helping Children Learn Mathematics
functions, relations, and joint variation; and modeling language. Any one of these characterizations would have led to a somewhat different organization of the research we review.
5.
Lee and Wheeler, 1987.
6.
Boero, 1993.
7.
Pimm, 1995.
8.
Wenger, 1987.
9.
For example, Kirshner and Awtry, in press.
10.
For example, Booth, 1984, and Greeno, 1982.
11.
Kirshner, 1989.
12.
Carry, Lewis, and Bernard, 1980; Wenger, 1987.
13.
Wenger, 1987.
14.
Greeno, 1982.
15.
Lee and Wheeler, 1987.
16.
Nhouyvanisvong, 2001.
17.
Thompson, Philipp, Thompson, and Boyd, 1994.
18.
See Swafford and Langrall, 2000, for research using exponential and inverse variation functions with sixth graders; Rojano, 1996, for research involving systems of linear equations; and Bednarz, Radford, and Janvier, 1995, and Radford, 1994, for research using situations with more than one unknown.
19.
Phillips, Smith, Star, and Herbel-Eisenmann, 1998. For a rationale, see Confrey, 1994, and Confrey and Smith, 1994, 1995.
20.
For example, Heid, 1990. In an historical and theoretical discussion, Kaput, 1994, goes further to argue that, with technology, many of the ideas of calculus are accessible without relying on traditional algebraic skills.
21.
Thompson, Philipp, Thompson, and Boyd, 1994.
22.
Behr, Erlwanger, and Nichols, 1980; Kieran, 1981; Saenz-Ludlow and Walgamuth, 1998.
23.
Vergnaud, Benhadj, and Dussouet, 1979.
24.
Ball and Bass, 1999. See also Ball and Bass, 2001.
25.
Carraher, Brizuela, and Schliemann, 2000; Schliemann, Carraher, and Brizuela, 2000.
26.
Blanton and Kaput, 2000.
27.
Carpenter and Levi, 1999.
28.
For example, Küchemann, 1978, 1981; Kieran, 1983; Wagner, Rachlin, and Jensen, 1984.
29.
Booth, 1984.
30.
Booth, 1984, p. 45.
31.
Booth, 1984, p. 44.
32.
Noss, Hoyles, and Healy, 1997.
33.
Clement, 1982; Clement, Lochhead, and Monk, 1981; Fisher, 1988; Kaput and Sims-Knight, 1983; Lochhead, 1980; MacGregor and Stacey, 1993; Rosnick, 1981; Rosnick and Clement, 1980; Sims-Knight and Kaput, 1983.
34.
By analyzing interview transcripts, Clement, Lockhead, and Monk, 1981, found that some students simply translated the words of the problem directly into mathematical
