NOTES

1.  

The study of functions, as we define it here, overlaps substantially with the topic of “algebra” traditionally taught in the United States in ninth grade, though national and many state standards now recommend that aspects of algebra be addressed in earlier grades (as is done in most other countries). Although functions are a critical piece of algebra, other aspects of algebra, such as equation solving, are not addressed in this chapter.

2.  

Thomas, 1972, p. 17.

3.  

Goldenberg, 1995; Leinhardt et al., 1990; Romberg et al., 1993.

4.  

Nathan and Koedinger, 2000.

5.  

Koedinger and Nathan, 2004.

6.  

Koedinger and Nathan, 2004.

7.  

Koedinger et al., 1997.

8.  

Kalchman, 2001.

9.  

Schoenfeld et al., 1993.

10.  

Schoenfeld et al., 1987.

11.  

Schoenfeld et al., 1998, p. 81.

12.  

Chi et al., 1981.

13.  

Chi et al., 1981; Schoenfeld et al., 1993.

14.  

Kalchman, 2001.

REFERENCES

Chi, M.T.H., Feltovich, P.J., and Glaser, R. (1981). Categorization and representation of physics problems by experts and novices. Cognitive Science, 5, 121-152.


Goldenberg, E.P. (1995). Multiple representations: A vehicle for understanding. In D. Perkins, J. Schwartz, M. West, and M. Wiske (Eds.), Software goes to school: Teaching for understanding with new technologies (pp. 155-171). New York: Oxford University Press.


Kalchman, M. (2001). Using a neo-Piagetian framework for learning and teaching mathematical functions. Doctoral Dissertation, Toronto, Ontario, University of Toronto.

Koedinger, K.R., and Nathan, M.J. (2004). The real story behind story problems: Effects of representations on quantitative reasoning. Journal of the Learning Sciences, 13(2).

Koedinger, K.R., Anderson, J.R., Hadley, W.H., and Mark, M.A. (1997). Intelligent tutoring goes to school in the big city. International Journal of Artificial Intelligence in Education, 8, 30-43.


Leinhardt, G., Zaslavsky, O., and Stein, M. (1990). Functions, graphs, and graphing: Tasks, learning, and teaching. Review of Educational Research, 60(1), 1-64.


Nathan, M.J., and Koedinger, K.R. (2000). Teachers’ and researchers’ beliefs of early algebra development. Journal for Research in Mathematics Education, 31(2), 168-190.


Romberg, T., Fennema, E., and Carpenter, T.P. (1993). Integrating research on the graphical representation of functions. Mahwah, NJ: Lawrence Erlbaum Associates.



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