symbols. Other students who gave the same incorrect answer drew a diagram showing six students and one professor and seemed to think of S and P as units of measure rather than variables. They thought it just as sensible to write 6S=P as to write 12 in.=1 ft.

35.  

Rosnick and Clement, 1980.

36.  

Kaput and Sims-Knight, 1983.

37.  

Soloway, Lochhead, and Clement, 1982.

38.  

Sutherland and Rojano, 1993.

39.  

Sutherland, 1993.

40.  

Kalchman, Moss, and Cass, 2001; Koedinger, Anderson, Hadley, and Mark, 1997.

41.  

Sutherland, 1993.

42.  

For example, Davis and Pobjoy, 1995; Sutherland, Jinich, Mochón, Molyneux, and Rojano, 1996.

43.  

Koedinger, Anderson, Hadley, and Mark, 1997; Moreno, Rojano, Bonilla, and Perrusquia, 1999; Rojano, 1999.

44.  

Koedinger and Anderson, 1998.

45.  

Koedinger, Anderson, Hadley, and Mark, 1997.

46.  

For example, Borba and Confrey, 1996.

47.  

Fey, 1989b; Heid, 1988; Heid, Sheets, Matras, and Menasian, 1988; O’Callaghan, 1998.

48.  

Heid, Sheets, Matras, and Menasian, 1988, p. 2.

49.  

Heid, 1988.

50.  

Huntley, Rasmussen, Villarubi, Sangtong, and Fey, 2000.

51.  

Huntley, Rasmussen, Villarubi, Sangtong, and Fey, 2000.

52.  

Streun, Harskamp, and Suhre, 2000.

53.  

Kieran, 1982, 1984.

54.  

Chaiklin and Lesgold, 1984.

55.  

Collis, 1975.

56.  

Whitman, 1976.

57.  

Kieran, 1988.

58.  

Kieran and Sfard, 1999.

59.  

Linchevski and Vinner, 1990.

60.  

Huntley, Rasmussen, Villarubi, Sangtong, and Fey, 2000, p. 357.

61.  

Kieran and Sfard, 1999; Thompson and Thompson, 1987.

62.  

Usiskin, 1998, pp. 17–18.

63.  

Fey, 1989a, pp. 206–207.

64.  

Huntley, Rasmussen, Villarubi, Sangtong, and Fey, 2000.

65.  

Huntley, Rasmussen, Villarubi, Sangtong, and Fey, 2000, p. 357.

66.  

Lee and Wheeler, 1987, p. 149.

67.  

See “Building Blocks” in Chapter 3 for several other approaches to this problem.

68.  

Arzarello, 1992; Swafford and Langrall, 2000.

69.  

See, for example, Heid, Sheets, Matras, and Menasian, 1988.

70.  

Arcavi, 1994.

71.  

Edwards, 1990.



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