utes to their learning becoming generative. Teachers are not given readymade solutions to teaching problems or prescriptions for practice. Instead, they adapt what they are learning and engage in problem solving to deal with the situations that arise when they attempt to use what they learn.

Professional development beyond initial preparation is critical for developing proficiency in teaching mathematics. However, such professional development requires the marshalling of substantial resources. One of the critical resources is time. If teachers are going engage in inquiry, they need repeated opportunities to try out ideas and approaches with their students and continuing opportunities to discuss their experiences with specialists in mathematics, staff developers, and other teachers. These opportunities should not be limited to a period of a few weeks or months; instead, they should be part of the ongoing culture of professional practice. Through inquiry into teaching, teacher learning can become generative, and teachers can continue to learn and grow as professionals.

Notes

1.  

Shulman, 1987.

2.  

Cohen and Ball, 1999, 2000.

3.  

Ball, 1991; Ma, 1999; Post, Harel, Behr, and Lesh, 1991; Tirosh, Fischbein, Graeber, and Wilson, 1999.

4.  

Ball, 1991; Ma, 1999.

5.  

Post, Harel, Behr, and Lesh, 1991.

6.  

Ball, 1988; Martin and Harel, 1989; Simon and Blume, 1996.

7.  

Ball, 1990, 1991.

8.  

Mullens, Murnane, and Willet, 1996; but see Begle, 1972.

9.  

Monk, 1994.

10.  

Begle, 1979.

11.  

Begle, 1979, p. 51.

12.  

The Longitudinal Study of American Youth (LSAY) was conducted in the late 1980s and early 1990s with high school sophomores and juniors. Student achievement data were based on items developed for NAEP.

13.  

Monk, 1994, p. 130.

14.  

Hawkins, Stancavage, and Dossey, 1998.

15.  

In fact, it appears that sometimes content knowledge by itself may be detrimental to good teaching. In one study, more knowledgeable teachers sometimes overestimated the accessibility of symbol-based representations and procedures (Nathan and Koedinger, 2000).

16.  

Ball and Bass, 2000; Ma, 1999.

17.  

Carpenter, Fennema, and Franke, 1996; Carpenter, Fennema, Franke, Empson, and Levi, 1999; Greer, 1992.



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