18.  

Carpenter, Fennema, Franke, Empson, and Levi, 1999.

19.  

Rowan, Chiang, and Miller, 1997.

20.  

Ball, 1991; Leinhardt and Smith, 1985.

21.  

Borko, Eisenhart, Brown, Underhill, Jones, Agard, 1992.

22.  

Leinhardt and Smith, 1985; Putnam, Heaton, Prawat, and Remillard, 1992.

23.  

Ball, 1991; Fernandez, 1997.

24.  

Lubinski, Otto, Rich, and Jaberg, 1998; Thompson and Thompson, 1994, 1996.

25.  

Kieran, 1981; Matz, 1982.

26.  

Behr, Erlwanger, and Nichols,1976, 1980; Erlwanger and Berlanger, 1983; Kieran, 1981; Saenz-Ludlow and Walgamuth, 1998.

27.  

Falkner, Levi, and Carpenter, 1999.

28.  

Ball and Bass, 2000; Putnam and Borko, 2000.

29.  

Leinhardt and Smith, 1985; Schoenfeld, 1998.

30.  

Carpenter, 1988.

31.  

Clark and Peterson, 1986.

32.  

Schon, 1987.

33.  

Brown, Collins, and Duguid, 1989; Lewis and Ball, 2000; Schon, 1987.

34.  

Franke, Carpenter, Fennema, Ansell, and Behrent, 1998; Franke, Carpenter, Levi, and Fennema, in press.

35.  

For an example of how such study might be conducted, see Ma, 1999.

36.  

National Research Council, 2000.

37.  

Franke, Carpenter, Fennema, Ansell, and Behrend, 1998; Franke, Carpenter, Levi, and Fennema, in press; Little, 1993; Sarason, 1990, 1996.

38.  

Franke, Carpenter, Levi, and Fennema, in press.

39.  

These programs share the idea that professional development should be based upon the mathematical work of teaching. For more examples, see National Research Council, 2001. A comprehensive guide for designing professional development programs can be found in Loucks-Horsley, Hewson, Love, Stiles, 1998.

40.  

Franke, Carpenter, Levi, and Fennema, in press.

41.  

Cognitively Guided Instruction (CGI) is a professional development program for teachers that focuses on helping them construct explicit models of the development of children’s mathematical thinking in well-defined content domains. No instructional materials or specifications for practice are provided in CGI; teachers develop their own instructional materials and practices from watching and listening to their students solve problems. Although the program focuses on children’s mathematical thinking, teachers acquire a knowledge of mathematics as they are learning about children’s thinking by analyzing structural features of the problems children solve and the mathematical principles underlying their solutions. A major thesis of CGI is that children bring to school informal or intuitive knowledge of mathematics that can serve as the basis for developing much of the formal mathematics of the primary school mathematics curriculum. The development of children’s mathematical thinking is portrayed as the progressive abstraction and formalization of children’s informal attempts to solve problems by constructing models of problem situations.

42.  

Carpenter and Levi, 1999.



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