We conclude that during the past 25 years mathematics instruction in U.S. schools has not sufficiently developed mathematical proficiency in the sense we have defined it. It has developed some procedural fluency, but it clearly has not helped students develop the other strands very far, nor has it helped them connect the strands. Consequently, all strands have suffered. In the next four chapters, we look again at students’ learning. We consider not just performance levels but also the nature of the learning process itself. We describe what students are capable of, what the big obstacles are for them, and what knowledge and intuition they have that might be helpful in designing effective learning experiences. This information, we believe, reveals how to improve current efforts to help students become mathematically proficient.

Notes

1.  

Brownell, 1935.

2.  

National Assessment Governing Board, 2000.

3.  

See Hiebert and Carpenter, 1992, for a discussion of the ways that cognitive science informs mathematics education on the nature of conceptual understanding. For views about learning in general, see Bransford, Brown, and Cocking, 1999; Donovan, Bransford, and Pellegrino, 1999. For discussion of learning in early childhood, see Bowman, Donovan, and Burns, 2001.

4.  

Bransford, Brown, and Cocking, 1999; Carpenter and Lehrer, 1999; Greeno, Pearson, and Schoenfeld, 1997; Hiebert, 1986; Hiebert and Carpenter, 1992. For a broader perspective on classrooms that promote understanding, see Fennema and Romberg, 1999.

5.  

See, for example, Hiebert and Carpenter, 1992, pp. 74–75; Hiebert and Wearne, 1996. For work in psychology, see Baddeley, 1976; Bruner, 1960, pp. 24–25; Druckman and Bjork, 1991, pp. 30–33; Hilgard, 1957; Katona, 1940; Mayer, 1999; Wertheimer, 1959.

6.  

Geary, 1995.

7.  

Hiebert and Wearne, 1986; Kilpatrick, 1985.

8.  

Bransford, Brown, and Cocking, 1999.

9.  

Hiebert and Wearne, 1996.

10.  

Steinberg, 1985; Thornton and Toohey, 1985.

11.  

Fuson, 1990, 1992b; Fuson and Briars, 1990; Fuson and Burghardt, 1993; Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Olivier, and Human, 1997; Hiebert and Wearne, 1996; Resnick and Omanson, 1987.

12.  

Brownell, 1956/1987; Wu, 1999.

13.  

Brownell, 1935; Carpenter, Franke, Jacobs, Fennema, and Empson, 1998; Hatano, 1988; Wearne and Hiebert, 1988; Mack, 1995; Rittle-Johnson and Alibali, 1999.

14.  

Pesek and Kirshner, 2000.

15.  

Fuson and Briars, 1990; Fuson, Carroll, and Landis, 1996.

16.  

Resnick and Omanson, 1987.



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