Cognitive Science and Mathematics Education

By Alan H. Schoenfeld | Go to book overview

discipline composed of relations on the relations on the relations in ascending order. When Jim talks about learning representations, I get a little bit anxious. There are trade offs, trade offs in the ability to generalize and extend to higher levels of thinking. Earlier Jim Kaput gave the example of how spreadsheet programs can be used to solve maximum/ minimum calculus problems. It's true that that's an acceptable representation with which to solve those problems. The question for me is what one gains by using that representation, other than having established an ad hoc alternative process for solving a particular type of problem. The key issue is the relationship among representations. Do we simply build different representations for different topics? How are the representations related? Are some better than others for certain purposes? Is there some coherency underlying the diversity?


REFERENCES

Anderson J. R. ( 1983). The architecture of cognition. Cambridge, MA: Harvard University Press.

Booth L. R. ( 1984). Algebra: Children's strategies and errors: A report of the strategies and errors in secondary mathematics project. Windsor, Eng.: Neer Nelson.

Brown A. L., & DeLoache J. S. ( 1978). "Skills, plans and self-regulation". In R. S. Siegler (Ed.), Children's thinking: What develops (pp. 3-35). Hillsdale, NJ: Lawrence Erlbaum Associates.

Brown J. S., & Burton R. B. ( 1978). "Diagnostic models for procedural bugs in basic mathematical skills". Cognitive Science, 2, 155-192.

Bundy A. ( 1979). A treatise on elementary equation solving. (DAI Working Paper No. 51). University of Edinburgh, Department of Artificial Intelligence.

Bundy A. ( 1983). The computer modelling of mathematical reasoning. London: Academic Press.

Bundy A., & Sterling L. S. ( 1981). Meta-level inference in algebra. (Research Paper 164). University of Edinburgh, Department of Artificial Intelligence.

Carry L. R., Lewis C., & Bernard J. E. ( 1981). Psychology of equation solving: An information processing study. University of Texas at Austin, Department of Curriculum & Instruction.

Clement J., Lochhead J., & Soloway E. ( 1980). Positive effects of computer programming on students' understanding of variables and equations. Proceedings of the National ACM Conference, Nashville, TN.

Davis R. B. ( 1984). Learning mathematics: The cognitive science approach to mathematics education. London: Croom Helm.

Gentner D., & Stevens A. L. ( 1983). Mental models. Hillsdale, NJ: Lawrence Erlbaum Associates.

Greeno J. G. ( 1978). "Understanding and procedural knowledge in mathematics education". Educational Psychologist, 12, 262-283.

Heid M. K. ( 1983). "Calculus with muMath: Implications for curriculum reform". The Computing Teacher, Nov. 6, p. 46-49.

Kieran C. ( 1979a). "Children's operational thinking within the context of bracketing and the order of operations", Proceedings of the 3rd International Conference for the Psychology of Mathematics Education, Warwick, Eng. (pp. 128-132).

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