32.  

Siegler, 1995. Alibali and Goldin-Meadow, 1993, showed that in learning to solve problems involving mathematical equivalence, students were most successful when they had passed through a stage of considering multiple solution strategies.

33.  

Carpenter, Ansell, Franke, Fennema, and Weisbeck, 1993; Riley, Greeno, and Heller, 1983; see also Fuson, 1992.

34.  

Riley, Greeno, and Heller, 1983.

35.  

Allardice, 1977; Ginsburg, 1989.

36.  

Alexander, White, and Daugherty, 1997, propose these three conditions for reasoning in young children. There is reason to believe that the conditions apply more generally.

37.  

Shipley and Shepperson, 1990.

38.  

Hughes, 1986; Jordan, Huttenlocher, and Levine, 1992.

39.  

Fuson, Secada, and Hall, 1983.

40.  

See Bowman, Donovan, and Burns, 2001, for a discussion of these ideas.

41.  

National Center for Education Statistics, 2000.

42.  

Wigfield, Eccles, Yoon, Harold, Arbreton, Freedman-Doan, and Blumenfeld, 1997.

43.  

Dweck, 1999; Heyman and Dweck, 1998.

44.  

Heyman, Dweck, and Cain, 1992.

45.  

For example, Jordan, Huttenlocher, and Levine, 1992.

46.  

National Center for Education Statistics, 2000.

47.  

National Center for Education Statistics, 2000.

48.  

National Center for Education Statistics, 2000.

49.  

Griffin, Case, and Siegler, 1994.

50.  

Jordan, Huttenlocher, and Levine, 1992.

51.  

Ginsburg, Klein, and Starkey, 1998.

52.  

Jordan, Levine, and Huttenlocher, 1995.

53.  

Griffin, Case, and Siegler, 1994.

54.  

Fuson, Smith, and Lo Cicero, 1997.

References

Alexander, P.A., White, C.S., & Daugherty, M. (1997). Analogical reasoning and early mathematics learning. In L.D.English (Ed.), Mathematical reasoning: Analogies, metaphors, and images (pp. 117–147). Mahwah, NJ: Erlbaum.

Allardice, B. (1977). The development of written representations for some mathematical concepts. Journal of Children’s Mathematical Behavior, 1(4), 135–148.

Alibali, M.W., & Goldin-Meadow, S. (1993). Gesture-speech mismatch and mechanisms of learning: What the hands reveal about a child’s state of mind. Cognitive Psychology, 25, 468–573.

Antell, S.E., & Keating, D.P. (1983). Perception of numerical invariance in neonates. Child Development, 54, 695–701.


Baroody, A.J. (1987a). Children’s mathematical thinking: A developmental framework for preschool, primary, and special education teachers. New York: Teachers College Press.

Baroody, A.J. (1987b). The development of counting strategies for single-digit addition. Journal for Research in Mathematics Education, 18, 141–157.



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