Forms of Knowing Mathematics: What Preservice Teachers Should Learn



What important ideas about forms of knowing mathematics should be included in mathematics methods courses for preservice teachers? Ideas are proposed that are related to categories in Shulman’s (1986) framework of teacher knowledge. There is a brief discussion of the implications each idea holds for teaching mathematics, and some suggestions are given about experiences that may help preservice teachers appreciate these notions. One portion of Shulman’s pedagogical content knowledge construct is knowing what makes a subject difficult and what preconceptions students are apt to bring. Three of the ideas offered for inclusion in a methods course are related to this aspect of pedagogical content knowledge: (1) Understanding students’ understanding is important, (2) Students knowing in one way do not necessarily know in the other(s), and (3) intuitive understanding is both an asset and a liability. The last two ideas, are related to the other portion of pedagogical content knowledge, knowing how to make the subject comprehensible to learners. These ideas are (4) certain characteristics of instruction appear to promote retention, and (5) providing alternative representations and recognizing and analyzing alternative methods are important. Readers are asked to consider if the suggestions offered are appropriate and how they might best be taught.


Preservice Teacher Conceptual Understanding Prospective Teacher Pedagogical Content Knowledge Procedural Knowledge 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. Bell, A., Brekke, G. and Swan, M.: 1987, ‘Misconceptions, conflict and discussion in the teaching of graphical interpretation’, in J. Novak (ed.), Proceedings of the Second International Seminar on Misconceptions and Educational Strategies in Science and Mathematics, Vol 1, Cornell University, Ithaca, New York, pp. 46–48.Google Scholar
  2. Bell, A., Burkhardt, H. and Swan, M.: 1992, ‘Balanced assessment of mathematical performance’, in R. Lesh and S. Lamon (eds.), Assessment of Authentic Performance in School Mathematics, American Association for the Advancement of Science, Washington, DC, pp. 119–144.Google Scholar
  3. Berger, T. and Keynes, H.: 1995, ‘Everybody counts/everybody else’, CBMS Issues in Mathematics Education 5, 89–110.Google Scholar
  4. Brooks, J. and Brooks, M.: 1993, In Search of Understanding: The Case for Constructivist Classrooms,Association of Supervision and Curriculum Development, Alexandria, Virginia.Google Scholar
  5. Brophy, J. E.: 1991, ‘Conclusion to advances in research on teaching’, in J. Brophy (ed.) Advances in Research on Teaching: Teachers’ Subject-matter Knowledge and Classroom Instruction: VOL. II Teachers’ Knowledge of Subject Matter as it Relates to Teaching Practice,JAI Press, Greenwich, Connecticut, pp. 347–362.Google Scholar
  6. Brown, D. E. and Clement, J.: 1989, ‘Overcoming misconceptions by analogical reasoning: Abstract transfer versus explanatory model construction’, Instructional Science 18,237— 261.Google Scholar
  7. Brown, S., Collins, A. and Duguid, P.: 1989, ‘Situated cognition and the culture of learning,’ Educational Researcher 18 (1), 32–42.Google Scholar
  8. Burns, M.: 1993, Mathematics: Assessing and Understanding. Individual Assessments, Part I. [Videotape]. Available from Cuisenaire/Dale Seymour Publications, P.O. Box 5026, White Plains, New York 10602.Google Scholar
  9. Carpenter, T., Fennema, E., Peterson, P., Chiang, C., and Loef, M.: 1989, ‘Using knowledge of children’s mathematics thinking in classroom teaching: An experimental study’, American Educational Research Journal 26, 499–531.Google Scholar
  10. Carpenter, T. and Moser, J.: 1983, ‘The acquisition of addition and subtraction concepts’, in R. Lesh and M. Landau (eds.), The Acquisition of Mathematics Concepts and Processes, Academic Press, New York, pp. 7–44.Google Scholar
  11. Confrey, J.: 1990, ‘What constructivism implies for teaching’, in R. B. Davis, C. A. Maher and N. Noddings (eds.), Constructivist Views on the Teaching and Learning of Mathematics, National Council of Teachers of Mathematics, Reston, Virginia, pp. 107–122.Google Scholar
  12. Connolly, P. and Vilardi, T.: 1989, Writing to Learn Mathematics and Science, Teachers College Press, New York.Google Scholar
  13. Driver, R.: 1987, ‘Promoting conceptual change in classroom settings: The experiences of the children’s learning science project’, in J. Novak (ed.), Proceedings of the Second International Seminar on Misconceptions and Educational Strategies in Science and Mathematics, Vol 2, Cornell University, Ithaca, New York, pp. 97–107.Google Scholar
  14. Ellerton, N. and Clements, M. A.: 1991, Mathematics in Language: A Review of Language Factors in Mathematics Learning, Deakin University Press, Geelong, Australia.Google Scholar
  15. Fennema, E., Carpenter, T., Franke, M., Levi, L., Jacobs, E. and Empson, S.: 1996, ‘A longitudinal study of learning to use children’s thinking in mathematics instruction’, Journal for Research in Mathematics Education 27 (4), 403–434.CrossRefGoogle Scholar
  16. Fischbein, E.: 1993, ‘The interaction between the formal, the algorithmic and the intuitive components in a mathematical activity’, in R. Biehler, R. Scholz, R. Straser, and B. Winkelmann (eds.), Didactics of Mathematics as a Scientific Discipline, Kluwer, Dordrecht, The Netherlands, pp. 231–245.Google Scholar
  17. Fischbein, E.: 1987, Intuition in Science and Mathematics, D. Reidel, Dordrecht, The Netherlands.Google Scholar
  18. Fuson, K.: 1990, ‘Issues in place-value and multidigit addition and subtraction learning and teaching,’ Journal for Research in Mathematics Education 21 (4), 273–279.CrossRefGoogle Scholar
  19. Fuys, D., Geddes, D. and Tischler, R.: 1988, The van Hiele Model of Thinking in Geometry Among Adolescents (Journal for Research in Mathematics Education Monograph #3), National Council of Teachers of Mathematics, Reston, Virginia.Google Scholar
  20. Gearhart, M., Saxe, G. B. and Stipek, D.: Fall 1995, ‘Helping teachers know more about their students: Findings from the Integrating Mathematics Assessment (IMA) project’, Connections (1), 4–6, 10.Google Scholar
  21. Graeber, A. and Tirosh, D.: 1988, ‘Multiplication and division involving decimals: Pre-service elementary teachers’ performance and beliefs’, The Journal of Mathematical Behavior 7 (3), 263–280.Google Scholar
  22. Hershkowitz, R., Bruckheimer, M. and Vinner, S.: 1987, ‘Activities with teachers based on cognitive research’, in M. Lindquist (ed.), Learning and Teaching Geometry, K-12, 1987 Yearbook, National Council of Teachers of Mathematics, Reston, Virginia, pp. 222–235.Google Scholar
  23. Hiebert, J. and Carpetner, T.: 1992, ‘Learning and teaching with understanding,’ in D. Grouws (ed.), Handbook of Research on Mathematics Teaching and Learning, MacMillan, New York, pp. 65–97.Google Scholar
  24. Hiebert, J. and Lefevre, P.: 1986, ‘Conceptual and procedural knowledge in mathematics: An introductory analysis’, in J. Hiebert (ed.), Conceptual and Procedural Knowledge: The Case of Mathematics, Lawrence Erlbaum, Hillsdale, New Jersey, pp. 1–27.Google Scholar
  25. Hiebert, J. and Wearne, D.: 1986, ’Procedures over concepts: The acquisition of decimal number knowledge’, in J. Hieben (ed.) Conceptual and Procedural Knowledge: The Case of MathematicsLawrence Erlbaum, Hillsdale, New Jersey, pp. 199–224. Google Scholar
  26. Howson, G., Keitel, C. and Kilpatrick, J.:1981, Curriculum Developments in Mathematics,Cambridge University Press, Cambridge, England.Google Scholar
  27. London Mathematical Society, Institute of Mathematics and Its Applications and Royal Statistical Society: October, 1995, Tackling the Mathematics Problem. Google Scholar
  28. Mathematics Science Education Board: 1996, The Preparation of Teachers of Mathematics: Considerations and Challenges, A Letter Report, National Research Council, Washington, DC.Google Scholar
  29. Mestre, J.: 1987, ‘Why should mathematics and science teachers be interested in cognitive research findings?’, Academic Connections 3–5, 8–11.Google Scholar
  30. MPR Associates and The Chief State School Officers: 1997 Item and test specification for the voluntary national test in 8th-grade mathematicsMPR Associates, Washington, DC. Google Scholar
  31. National Council of Teachers of Mathematics: 1991, Professional Standards for Teaching of Mathematics, The Council, Reston, Virginia.Google Scholar
  32. Putnam, R. T. and Leinhardt, G.: 1986. ‘Curriculum scripts and the adjustment of content in mathematics lessons’, paper presented at the annual meeting of the American Educational Research Association, San Francisco, California.Google Scholar
  33. Resnick, L. and Ford, W.: 1981, The Psychology of Mathematics Instruction, Lawrence Erlbaum, Hillsdale, New Jersey.Google Scholar
  34. Rine, S.: 1998, ‘The role of research and teachers’ knowledge base in professional development,’ Educational Researcher 27 (5), 27–31.Google Scholar
  35. Shulman, L. S.: 1986, ‘Those who understand: Knowledge growth in teaching’, Educational Researcher 15 (2), 4–14.Google Scholar
  36. Silver, E. A. and Smith, M. S.: 1996, ’Building discourse communities in mathematics classrooms: A worthwhile but challenging journey’, in P. Elliot (ed.) Communication in Mathematics K-12 and Beyond1996 Yearbook, National Council of Teachers of Mathematics, Reston, Virginia, pp. 20–28. Google Scholar
  37. Skemp, R.: 1978, ‘Relational understanding and instrumental understanding’, Arithmetic Teacher 26 (3), 9–15.Google Scholar
  38. Steife, L.: 1990, ’On the knowledge of mathematics teachers’, in R. B. Davis, C. Maher, and N. Noddings (eds.) Constructivist Views on the Teaching and Learning of MathematicsJRME Monograph #4, National Council of Teachers of Mathematics, Reston, Virginia, pp. 167–184. Google Scholar
  39. Steife, L. and Cobb, P.: 1988, Construction of Arithmetical Meanings and Strategies, Springer-Verlag, New York.CrossRefGoogle Scholar
  40. Swan, M.: 1983, Teaching Decimal Place Value: A Comparative Study of ‘Conflict’ and ’Positive Only’ Approaches, Shell Centre for Mathematics Education, University of Nottingham, Nottingham, England.Google Scholar
  41. Tobias, S.: 1990, ‘They’re not dumb. They’re different: A new “Tier of Talent” for science’, Change 22 (4), 10–30.CrossRefGoogle Scholar
  42. Weame, D. and Hieben, J.: 1988, ’A cognitive approach to meaningful mathematics instruction: Testing a local theory using decimal numbers’ Journal for Research in Mathematics Education 19(5),371–384.Google Scholar
  43. Webb, N. and Romberg, T.: 1992, ‘Evaluation a coat of many colors’, in T. Romberg, (ed.), Mathematics Assessment and Evaluation, State University of New York Press, Albany, New York, pp. 10–36.Google Scholar
  44. Whitehead, A.: 1929, The Aims of Education, MacMillan, New York.Google Scholar
  45. Yackel, E., Cobb, P., Wood, T., Wheatley, G. and Merkel, G.: 1990, ‘The importance of social interaction in children’s construction of mathematical knowledge,’ in T. Cooney and C. Hirsch (eds.), Teaching and Learning Mathematics in the 1990s, 1990 Yearbook, National Council of Teachers of Mathematics, Reston, Virginia, pp. 12–21.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1999

Authors and Affiliations

  1. 1.Department of Curriculum and InstructionUniversity of MarylandCollege ParkUSA

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