24.  

Carpenter and Moser, 1984.

25.  

Fuson, 1986b; Fuson and Fuson, 1992.

26.  

Armstrong, 1990/1991; Huinker, 1990/1991; Rathmell and Huinker, 1989.

27.  

Resnick, 1983.

28.  

Baroody, 1999a.

29.  

Mulligan and Mitchelmore, 1997; Steffe, 1994. Lemaire and Siegler, 1995, found similar results with French second graders. Brownell, 1944, showed that, from grades 3 to 5, students became faster at multiplication combinations because they progressively used more efficient strategies.

30.  

Thornton, 1978; Baroody, 1987a, 1999b.

31.  

Carpenter, Fennema, Peterson, Chiang, and Loef, 1989, found that when instruction focused on problem solving, children not only became better problem solvers but also mastered more combinations than did children whose instruction focused on drill and practice of basic facts.

32.  

Brownell and Chazal, 1935, found that drill on arithmetic facts does not necessarily lead to recall. In spite of drill, children tend to maintain whatever procedures have satisfied their number needs. Drill does not supply children with more mature ways of dealing with number combinations. Brownell and Chazal argue that drill must be preceded by sound instruction.

33.  

Carnine and Stein, 1981; Cook and Dossey, 1982; Rathmell, 1978; Thornton, 1978.

34.  

See Rathmell, 1978.

35.  

Bergeron and Herscovics, 1990.

36.  

Brownell and Chazal, 1935.

37.  

Davydov and Andronov, 1981; Fuson and Kwon, 1992b; Saxe, 1982.

38.  

Bergeron and Herscovics, 1990; Fuson, 1988; Steffe, Cobb, and von Glasersfeld, 1988.

39.  

Geary and Brown 1991; Siegler, 1996, pp. 61–71.

40.  

Siegler, 1996, p. 97.

41.  

Siegler and Jenkins, 1989.

42.  

Geary, 1994; Ginsburg and Allardice, 1984.

43.  

Carnine and Stein, 1981; Cook and Dossey, 1982; Thornton, 1978.

44.  

Fuson, 1986b; Fuson and Fuson, 1992; Fuson and Willis, 1988.

45.  

Siegler and Jenkins, 1989; Siegler, 1996, pp. 61–71.

46.  

For example, see Ron, 1998, for a discussion of a European-Latino subtraction algorithm; Fuson and Kwon, 1992a, for a Korean subtraction algorithm; and Chapter 3 of this volume for various multiplication algorithms learned by teachers in this country.

47.  

Siegler, in press.

48.  

Brown and Van Lehn, 1980.

49.  

For a synthesis on the relationship between conceptual and procedural knowledge for multidigit addition and subtraction, see Rittle-Johnson and Siegler, 1998. For a specific study, see Hiebert and Wearne, 1996.

50.  

Fuson and Briars, 1990; Fuson, Wearne, Hiebert, Murray, Human, Olivier, Carpenter, Fennema, 1997; Hiebert and Wearne, 1996.



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