Abstract
In this introductory article, after some initial discussion of an appropriate approach to mathematics as a curriculum subject, we sketch a theory for designing teaching, based on mathematical activity, situations, tasks, and interventions, exposing and resolving cognitive conflicts, changes of structure and context, feedback, reflection and review. We next review the main psychological principles underlying this theory, then consider some examples of teaching designs in the light of the theory. Thus we open the discussion of the theme of this issue, which continues with the fuller discussion of other examples in the remaining articles.
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References
Balacheff, N.: 1982, “Preuves et demonstrations en mathématiques au collège”,Recherches en Didactiques des Mathematiques 3(3), 204–261.
Balacheff, N.: 1990, “Towards a problematique for research on mathematics teaching”,Journal for Research in Mathematics Education 21(4), 258–272.
Bartlett, F. C.: 1932,Remembering, Cambridge University Press, Cambridge.
Bassford, D.: 1988,A Comparison of Teaching Methods in the Teaching of Fractions, Shell Centre for Mathematical Education, University of Nottingham.
Bell, A.: 1986–87, “Diagnostic teaching 1–3”,Mathematics Teaching Nos. 115, 116, 118.
Bell, A.: 1992, “Problem solving, mathematical activity and learning”, in J. P. Ponte (ed),Mathematical Problem Solving and Information Processing: Research in Contexts of Practice, Springer, Berlin.
Bell, A.: 1993, “Some experiments in diagnostic teaching”,Educational Studies in Mathematics 24, 115–137.
Bell, A. W. and Bassford, D.: 1989, “Conflict and guided discovery in the teaching of fractions”,Proceedings of PME 13 1, 125–132.
Bell, A. and Birks, D.: 1990, “Diagnosis and response in transformation Geometry,Proceedings of PME 14,2, 219–228.
Bell, A. W., Costello, J., and Küchemann, D.: 1983,A Review of Research in Mathematical Education, Part A: Research on Learning and Teaching, NFER-Nelson, London.
Bell, A., Fischbein, E., and Greer, B.: 1984, “Choice of operation in verbal arithmetic problems: The effects of number size, problem structure and context”,Educational Studies in Mathematics 15, 129–147.
Bell, A., Greer, B., Grimison, L., and Mangan, C.: 1989, “Children's performance on multiplicative word problems: Elements of a descriptive theory,Journal for Research in Mathematics Education 20(5), 434–449.
Bell, A., Swan, M., Onslow, B., Pratt, K., and Purdy, D.: 1985,Diagnostic Teaching: Teaching for Long Term Learning (Report of an ESRC Project), Shell Centre for Mathematical Education, University of Nottingham.
Bell, A. and Onslow, B.: 1987, “Multiplicative structures: Development of the understanding of rates/Intensive quantities”,Proceedings of PME 11,2, 275–281.
Birks, D.: 1987,Reflections: A Diagnostic Teaching Experiment, Shell Centre for Mathematical Education, University of Nottingham.
Brink, van den, F. J.: 1974, “Numbers in contextual frameworks”,Educational Studies in Mathematics 5, 239–259.
Brousseau, G.: 1981, ‘Problèmes de didactique des d écimaux’,Recherches en Didactiques des Mathématiques 2(1), 37–127.
Brownell, W. A. and Chazal, C. B.: 1935, “The effects of premature drill in third-grade arithmetic”,Journal of Educational Research 29, 17–28.
Carraher, T. N., Carraher, D. W., and Schliemann, A.: 1985, “Mathematics in the streets and in schools”,British Journal of Developmental Psychology 3(1), 21–30.
Comiti, C. and Bessot, A.: 1987, “A study of children's strategies for the comparison of numerals”,For the Learning of Mathematics 7(1), 39–47.
Craik, F.I.M. and Lockhart, R. S.: 1972, “Levels of processing: A framework for memory research”,Journal of Verbal Learning and Verbal Behaviour 11, 671–684.
Freudenthal, H.: 1968, “Why to teach mathematics so as to be useful”,Educational Studies in Mathematics 1, 3–8.
Gattegno, C.: 1963,For the Teaching of Mathematics, Educational Explorers, Reading, UK.
Gelman, R.: 1969, “Conservation acquisition: A problem of learning to attend to relevant attributes”,Journal of Experimental Child Psychology 7(2), 167–187.
Hartley, J.: 1980,Using the computer to study and assist the learning of Mathematics. Paper presented to the British Society for the Learning of Mathematics, Shell Centre for Mathematical Education, University of Nottingham.
Jones, D. A.: 1975, “Don't just mark the answer — Have a look at the method”,Mathematics in School 4(3), 29–31.
Johnson, D. C.: 1989,Children's Mathematical Frameworks, NFER/Nelson, London.
Johnson-Laird, P. N., Legrenzi, P., and Legrenzi, M. S.: 1972, “Reasoning and a sense of reality”,British Journal of Psychology 63, 395–400.
Johnson-Laird, P. N and Wason, P.C.: 1970, “A theoretical analysis of insight into a reasoning task”,Cognitive Psychology 1, 134–148.
Lave, J.: 1988,Cognition in Practice: Mind, Mathematics and Culture, Cambridge University Press, Cambridge.
McIntosh, A.: 1978, “Some subtractions: What do you think you are doing?”Mathematics Teaching 83, 17–19.
Onslow, B.: 1986,Overcoming Conceptual Obstacles Concerning Rates. Shell Centre, University of Nottingham.
Peel, E. A.: 1971, Psychological and educational research bearing on school mathematics, in W. Servais and T. Varga (eds.),Teaching School Mathematics, Penguin Educational for UNESCO.
Plunkett, S.: 1979, “Decomposition and all that rot”,Mathematics in School 8(3), 2–5.
Rouchier, A.: 1980, “Situations et processus didactiques dans l'étude des nombres rationnels positifs”,Recherches en Didactique des Mathématiques 1(2), 225–227
Schoenfeld, A. H.: 1985,Mathematical Problem Solving, Academic Press, New York.
Shulman, H. G.: 1971, “Similarity effects in short-term memory”,Psychological Bulletin 75, 399–415.
Sutherland, R.: 1993, “Connecting theory and practice”,Educational Studies in Mathematics 24, 95–113
Swan, M. B.: 1983a,Teaching Decimal Place Value: A Comparative Study, Shell Centre for Mathematical Education, University of Nottingham.
Swan, M. B.: 1983b, “Teaching decimal place value: A comparative study of conflict and positive only approaches”,Proceedings of PME 7, 211–216.
Trembath, R. J. and White, R. T.: 1979, “Mastery achievement of intellectual skills”,Journal of Experimental Education 47, 247–252.
Tresselt, M. E. and Mayzner, M. S.: 1960, “A study of incidental learning”,Journal of Psychology 50, 339–347.
Wason, P. C.: 1966, “Reasoning”, in B. Foss (ed.),New Horizons in Psychology, Penguin, Harmondsworth.
Wason, P. C. and Shapiro, D.: 1971, “Natural and contrived experience in a reasoning task”,Quarterly Journal of Experimental Psychology 23, 63–71.
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Bell, A. Principles for the design of teaching. Educ Stud Math 24, 5–34 (1993). https://doi.org/10.1007/BF01273293
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DOI: https://doi.org/10.1007/BF01273293