informs instructional decision making and ultimately, how it propels learning. When a teacher uses problems that elicit multiple levels or types of correct answers, instruction and assessment become a seamless process. Problems with multiple levels of responses provide the opportunity to document a student's initial level of performance while allowing or encouraging the student to adopt more mature perspectives. At the same time, the teacher receives a wide range of responses from the class, which, when ordered by sophistication, provide a picture of the manner in which the students' knowledge develops, and gives the teacher a basis for making instructional decisions.
Finally, the kinds of problems we develop for classroom use and the manner in which we interpret student thinking communicate what we believe about learning mathematics with understanding, about individual differences, about what constitutes good teaching, about the nature of mathematics, and about the individual construction of knowledge. Providing students the flexibility to move around in a mathematical territory and teachers the flexibility to interpret student thinking from a variety of perspectives will provide a more realistic conception of mathematical ability.
Bransford, J. D., Franks, J. J., Vye, N. J., and Sherwood, R. D. ( 1986, June). New approaches to instruction: Because wisdom can't be told. Paper presented at a Conference on Similarity and Analogy, University of Illinois.
Case, R., and Sandieson, R. ( 1988). "A developmental approach to the identification and teaching of central conceptual structures in mathematics and science in the middle grades". In J. Hiebert and M. Behr (Eds.), Number concepts and "operations in the middle grades. Reston, VA: National Council of Teachers of Mathematics and Lawrence Erlbaum Associates; pp. 236-259.
Freudenthal, H. ( 1983). Didactical phenomenology of mathematical structures Dordrecht, Holland: D. Reidel.
Hiebert, J., and Behr, M. ( 1988). "Introduction: Capturing the major themes". In J. Hiebert and M. Behr (Eds.), Number concepts and operatiom in the middle grades. Reston, VA: National Council of Teachers of Mathematics and Lawrence Erlbaum Associates; pp. 1-18.
Hiebert, J. and Wearne, D. ( 1991). "Methodologies for studying learning to inform teaching". In E. Fennema, T. P. Carpenter, and S. J. Lamon (Eds.), Integrating research on teaching and learning mathematics. New York: SUNY Press; pp. 153-176.
Karplus, R., and Peterson, R. W. ( 1970). Intellectual development beyond elementary school IV: Ratio, a survey. Berkeley, CA: University of California, Lawrence Hall of Science.