multidigit quantities that can enable their calculation methods to become personally meaningful. Mathematical proficiency with whole numbers depends on all five strands developing together.

Notes

1.  

Griffin, Case, and Siegler, 1994; Fuson, Smith, and Lo Cicero, 1997.

2.  

Baroody, 1984b, 1985; LeFevre, Bisanz, Daley, Buffone, Greenham, and Sadesky, 1996; LeFevre, Sadesky, and Bisanz, 1996.

3.  

Baroody, 1985, 1994. Basic number combinations may be represented not as a table of facts but as a network of facts and interconnecting relations (e.g., Baroody, 1985, 1987b, 1992). This idea is consistent with research in cognitive science, which says that expert knowledge is organized and connected (Bransford, Brown, and Cocking, 1999).

4.  

This observation was suggested by Jerman, 1970, and later verified by the work of Baroody, 1999a, 1999b, and many others.

5.  

Baroody, 1999a, 1999b.

6.  

Carpenter, Ansell, Franke, Fennema, Weisbeck, 1993; Carpenter and Moser, 1984; Carpenter, Moser, and Romberg, 1982; Fuson, 1992a, 1992b; Riley and Greeno, 1988; Siegler, in press; Verschaffel and De Corte, 1993.

7.  

Carpenter, Ansell, Franke, Fennema, and Weisbeck, 1993.

8.  

Carpenter, 1985; Fuson, 1992a, 1992b.

9.  

For a detailed analysis of multiplication and division problems, see Greer, 1992; Nesher, 1992; Vergnaud, 1983; Harel and Confrey, 1994.

10.  

Carpenter, Ansell, Franke, Fennema, and Weisbeck, 1993; Kouba, 1989.

11.  

Kouba, 1989.

12.  

For example, Brownell, 1956/1987.

13.  

Fuson, 1992b.

14.  

Fuson and Secada, 1986; Leutzinger, 1979; Steinberg, 1985; Thornton, 1978.

15.  

Baroody, 1996; Resnick and Ford, 1981.

16.  

Fuson and Kwon, 1992b; Fuson, Stigler, and Bartsch, 1988; Geary, 1994; Matsushita, 1994. Hatano, 1988, discusses a “complementary number-to-10” strategy used by Japanese students on an abacus. When there are not enough beads available to add 8, for example, the student adds 10 and subtracts 2.

17.  

Fuson and Kwon, 1991.

18.  

See, for example, the 1999 edition of Scott Foresman-Addison Wesley Math, Grade 1.

19.  

Carpenter and Moser, 1984; Fuson, 1992a, 1992b.

20.  

See reviews in Fuson, 1992a, 1992b.

21.  

Fuson and Secada, 1986; Fuson and Fuson, 1992. Both studies included students with learning disabilities and English-language learners (K.C.Fuson, personal communication, Northwestern University, 2000).

22.  

Carpenter, Ansell, Franke, Fennema, and Weisbeck, 1993; Carpenter, Moser, and Romberg, 1982; Fuson, 1992a; Riley and Greeno, 1988; Siegler, in press.

23.  

See, for example, Baroody, 1984a.



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