26.  

Hart, 1988; Karplus and Peterson, 1970; Karplus et al., 1981, 1983; Cramer et al., 1993; Noelting, 1980a, 1980b.

27.  

National Council of Teachers of Mathematics, 1989, 2000; National Research Council, 2001.

28.  

Ball, 1993.

29.  

Sowder, 1988.

30.  

Baroody, 1999; Heibert, 1992; Hiebert and Wearne, 1986; Moss and Case, 1999; Post et al., 1993.

31.  

Armstrong and Bezuk, 1995; Ball, 1993; Hiebert and Wearne, 1986; Mack, 1990, 1993; Markovits and Sowder, 1991, 1994; Sowder, 1995.

32.  

Confrey, 1994, 1995; Kieren, 1994, 1995; Post et al., 1993; Streefland, 1991, 1993.

33.  

Kieren, 1994, 1995; Mack, 1993, 1995; Sowder, 1995; Streefland, 1993.

34.  

Kieren, 1994, p. 389.

35.  

Kieren, 1992, 1995.

36.  

Confrey, 1995.

37.  

Lachance and Confrey, 1995.

38.  

Streefland, 1991, 1993.

39.  

Mack, 1990, 1993.

40.  

Lamon, 1993, 1994, 1999.

41.  

As of this writing, this curriculum is being implemented with students of low socioeconomic status in a grade 7 and 8 class. Preliminary analyses have shown that it is highly effective in helping struggling students relearn this number system and gain a stronger conceptual understanding.

42.  

Kalchman et al., 2000; Moss, 1997, 2000, 2001, 2003; Moss and Case, 1999.

43.  

National Research Council, 2001.

44.  

Parker and Leinhardt, 1995.

45.  

Case, 1985; Noelting, 1980a; Nunes and Bryant, 1996; Spinillo and Bryant, 1991.

46.  

Confrey, 1994; Kieren, 1994.

47.  

Resnick and Singer, 1993.

48.  

Case, 1985.

49.  

Confrey, 1994; Kieren, 1993.

50.  

Case and Okomoto, 1996.

51.  

Case, 1998; Kalchman et al., 2000.

52.  

Parker and Leinhardt, 1995.

53.  

Lembke and Reys, 1994.

54.  

While the activities and lessons we designed are organized in three phases, the actual order of the lessons and the pacing of the teaching, as well as the particular content of the activities described below, varied in different classrooms depending on the needs, capabilities, and interests of the participating students.

55.  

These materials are available at any building supply store.

56.  

Parker and Leinhardt, 1995.

57.  

Hiebert et al., 1991.

58.  

Resnick et al., 1989.



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