Abstract
From 1984 through 1988, the authors worked with teachers using an inquiry approach to teach high school geometry courses with the aid of the GEOMETRIC SUPPOSERS. Problems are a critical component of the approach, as they are of any instructional process, because they focus attention and energy and guide students in the application, integration, and extension of knowledge. Inquiry problems differ from traditional, single-answer textbook exercises in that they must leave room for student initiative and creativity. The observations presented in this paper about the delicate balance between specifying too much instruction and too little, which is part of creating and posing inquiry problems, are based on careful examination of students' papers and classroom observations. The paper closes with speculations on whether these observations suggest general lessons for those seeking practical and successful strategies to introduce student inquiry into classrooms, with the hope of stimulating interest in and discussion of such strategies.
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Ball, D. (1990). Teaching mathematics for understanding: what do teachers need to know about the subject matter? In Competing visions of teacher knowledge: proceedings from a National Center for Research on Teacher Education seminar for education policymakers. National Center for Research on Teacher Education, Michigan State University.
Brown, S. and Walter, M. (1983). The art of problem posing. Philadelphia: The Franklin Institute Press.
Carpenter, T. P. and Moser, J. M. (1984). The acquisition of addition and subtraction concepts in grades 1–3. Journal for Research in Mathematics Education, 15, 179–202.
Collins, A. (1986a). Different goals of inquiry teaching. Technical Report No. 6458, Bolt, Beranek and Newman Laboratories, Inc., Cambridge, MA.
Collins, A. (1986b). A sample dialogue based on a theory of inquiry teaching. Technical Report No. 367, Bolt, Beranek and Newman Laboratories, Inc., Cambridge, MA.
DiSessa, A. A. (1985). Learning About knowing. In E. L.Klein (Ed.), Children and computers. San Francisco: Jossey-Bass.
Gillman, L. (1990 January-February). Teaching programs that work. FOCUS: The Newsletter of the Mathematical Association of America, 10 (1). 7–10.
Goldin, G. and McClintock, C. E. (1984). Task variables in mathematical problem solving. Philadelphia: The Franklin Institute Press.
Greeno, J. G. (1976). Indefinite goals in well-structured problems. Psychological Review, 83 (6), 479–491.
Harel, G., Behr, M., Post, T. and Lesh, R. (1989). Fishbein's Theory: a further consideration. In G.Vergnaud, M.Artigue and J.Rogalski, (Eds.), Proceedings of the 13th Annual meeting of the International Group for the Psychology of Mathematics Education (Vol 2, 52–60). Paris, France: International Group for the Psychology of Mathematics Education.
Hoffer, A. (1981). Geometry is more than a proof. Mathematics Teacher, 74, 11–18.
Jensen, R. J. (1986). Microcomputer-based conjecturing environments. In G.Lappan and R.Even (Eds.), Proceedings of the Eighth Annual Meeting PME-NA. East Lansing, MI: International Group for the Psychology of Mathematics Education-North American Branch.
Kaput, J. J. (1986). Information technology and mathematics: opening new representational windows. Educational Technology Center Topical Paper 86–3. Cambridge, MA: Harvard Graduate School of Education.
Kilpatrick, J. (1987). Problem formulating: Where do good problems come from? In A. H.Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 123–148). Hillsdale, NJ: Lawrence Erlbaum Associates.
Krutetskii, V. A. (1969a). An investigation of mathematics abilities in school children. In J.Kilpatrick and I.Wirzup (Eds.), Soviet Studies in the Psychology of Learning and Teaching Mathematics (Vol. 2). Chicago, IL: University of Chicago Press.
Krutetskii, V. A. (1969b). An analysis of the individual structure of mathematics abilities in school children. In J.Kilpatrick and I.Wirzup (Eds.), Soviet Studies in the Psychology of Learning and Teaching Mathematics (Vol. 2). Chicago, IL: University of Chicago Press.
Lampert, M. (1988). Teachers' thinking about students' thinking about geometry: the effects of new teaching tools. Educational Technology Center Technical Report TR88-1. Harvard Graduate School of Education.
Lampert, M. (1990). When the problem is not the question and the solution is not the answer: mathematical knowing and teaching. Unpublished manuscript. Institute for Research on Teaching. Michigan State University.
Polya, G. (1962). Mathematical discovery: on understanding, learning and teaching problem solving. New York: John Wiley.
Rakow, S. (1986). Teaching Science as inquiry. Bloomington, IN: Phi Delta Kappa Educational Foundation. (Fastback 246).
Rissland (Michener), E. (1978). The structure of mathematical knowledge. Artificial Intelligence. Cambridge, MA.: Massachusetts Institute of Technology, Technical Report 472.
Schoenfeld, A. H. (1983). Problem solving in the mathematics curriculum: a report, recommendationsand and annotated bibliography. The Committee on the Undergraduate Program in Mathematics of the Mathematical Association of America.
Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, FL: Academic Press.
Schoenfeld, A. H. (1988). When good teaching leads to bad results: the disaster of “well taught” mathematics courses. In P. Peterson and T. Carpenter (Eds.), Learning through instruction: the study of students' thinking during instruction in mathematics. Special issue of Educational Psychologist, 23 (2), 145–166.
Schramm, R. (1988). How to develop the students' capacity for dealing with problems of proof? Zentralblatt für Didaktik der Mathematik, 88(3), 104–107.
Schwab, J. (1962). The teaching of science as enquiry. In J. J.Schwab and P. F.Brandwein (Eds.), The Teaching of Science. Cambridge, MA: Harvard University Press.
Schwartz, J. L. and Yerushalmy, M. (1985). THE GEOMETRIC SUPPOSERS (Computer Software). Pleasantville, NY: Sunburst Communications, Inc.
Schwartz, J. L. and Yerushalmy, M. (1987a). The Geometric Supposer: the computer as intellectual prosthetic for the making of conjectures. The College Mathematics Journal, 18, (1), 5–12.
Schwartz, J. L. and Yerushalmy, M. (1987b). The Geometric Supposer: Using microcomputers to Restore Invention of the Learning of Mathematics. In D.Perkins, J.Lockhead and JBishop (Eds.), The Second International Conference on Thinking. Hillsdale, NJ: Erlbaum.
Shulman, L. and Keislar, E. (Eds.) (1966). Learning by discovery: a critical appraisal. Chicago: Rand McNally.
Shulman, L. S. (1985). On teaching problem solving and solving the problems of teaching. In E. A.Silver (Ed.), Teaching and learning mathematical problem solving. Hillsdale, NJ: Lawrence Erlbaum Associates.
Suchman, J. R. (1961). Inquiry training: building skills for autonomous discovery. Merill Palmer Quarterly, Behavior Development, 73, 147–169.
Usiskin, Z. (1980). What should not be in the Algebra and Geometry curricula of average collegebound students? Mathematics Teacher, 73, 413–424.
Wittrock, M. C. (1966). The Learning-by-Discovery Hypothesis. In L.Shulman and E.Keslar, (Eds.), Learning by discovery: a critical appraisal. Chicago: Rand McNally.
Yerushalmy, M. (1986). Induction and generalization: an experiment in teaching and learning high school geometry. Unpublished doctoral thesis, Harvard Graduate School of Education.
Yerushalmy, M., Chazan, D. and Gordon, M. (1987). Guided inquiry and technology: a year-long study of children and teachers using the Geometric Supposer. Educational Technology Center Technical Report TR88-6, Harvard Graduate School of Education.
Yerushalmy, M. and Houde, R. (1987). Geometry problems and projects: triangles. Pleasantville. NY: Sunburst Communications, Inc.
Yerushalmy, M. and Houde, R. (1988a). Geometry problems and project: quadrilaterals. Pleasantville, NY: Sunburst Communications, Inc.
Yerushalmy, M. and Houde, R. (1988b). Geometry problems and projects: circles. Pleasantville, NY: Sunburst Communications, Inc.
Yerushalmy, M. and Chazan, D. (1990). Overcoming visual obstacles with the aid of the Geometric Supposer. Educational Studies in Mathematics, 21(3), 199–219.
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Yerushalmy, M., Chazan, D. & Gordon, M. Mathematical problem posing: Implications for facilitating student inquiry in classrooms. Instr Sci 19, 219–245 (1990). https://doi.org/10.1007/BF00120197
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DOI: https://doi.org/10.1007/BF00120197