Educational Studies in Mathematics

, Volume 40, Issue 1, pp 71–90 | Cite as

Reducing Abstraction Level When Learning Abstract Algebra Concepts

  • Orit Hazzan


How do undergraduate students cope with abstract algebra concepts? How should we go about researching this question? Based on interviews with undergraduate students and on written questionnaires, a theoretical framework evolved which could coherently account for most of the data. According to this theoretical framework, students' responses can be interpreted as a result of reducing the level of abstraction. In this paper, the theme of reducing abstraction is examined, based on three interpretations for levels of abstraction discussed in mathematics education research literature. From these three perspectives on abstraction, ways in which students reduce abstraction level are analyzed and exemplified.


Undergraduate Student Mathematics Education Education Research Research Literature Abstraction Level 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Asiala, M., Brown, A., DeVries, D., Dubinsky, E., Mathews, D. and Thomas, K.: 1996, ‘A framework for research and curriculum development in undergraduate mathematics education’, Research in Collegiate Mathematics Education II, The American Mathematics Society, pp. 1–32.Google Scholar
  2. Asiala, M., Brown, A., Kleiman, J. and Mathews, D.: 1998, ‘The development of students' understanding of permutations and symmetries’, International Journal of Computers for Mathematical Learning 3(1), 13–43.Google Scholar
  3. Asiala, M., Dubinsky, E., Mathews, D. M., Morics, S. and Oktaç, A.: 1997, ‘Development of students' understanding of cosets, normality, and quotient groups’, Journal of Mathematical Behavior 16(3), 241–309.Google Scholar
  4. Beth, E. W. and Piaget, J.: 1966, Mathematical Epistemology and Psychology, D. Reidel Publishing Company, Dordrecht, The Netherlands.Google Scholar
  5. Brown, A., DeVries, D. J., Dubinsky, E. and Thomas, K.: 1997, ‘Learning binary operations, groups and subgroups’, Journal of Mathematical Behavior 16(3), 187–239.Google Scholar
  6. Buchthal, D. C.: 1977, ‘Operative and nonoperative group definitions’, Mathematics Teacher 70(3), 262–263.Google Scholar
  7. Confrey, J.: 1990, ‘Chapter 8: What constructivism implies for teaching’, in R. B. Davis, C. A. Maher and N. Noddings (eds.), Journal for Research in Mathematics Education, Monograph Number 4, Constructivist Views on the Teaching and Learning of Mathematics, The National Council of Teachers of Mathematics, Inc., pp. 107–122.Google Scholar
  8. Davis, R. B., Maher, C. A. and Noddings, N.: 1990, ‘Chapter 12: Suggestions for the improvement of mathematics education’, in R. B. Davis, C. A. Maher and N. Noddings (eds.), Journal for Research in Mathematics Education, Monograph Number 4, Constructivist Views on the Teaching and Learning of Mathematics, The National Council of Teachers of Mathematics, Inc., pp. 187–191.Google Scholar
  9. Dreyfus, T.: 1990, ‘Advanced mathematical thinking’, in P. Nesher and J. Kilpatrick (eds.), Mathematics and Cognition: A research Synthesis by the International Group for the Psychology of Mathematical Education, Cambridge University Press, pp. 113–134.Google Scholar
  10. Dreyfus, T.: 1995, ‘Guest editorial, special issue — Advanced mathematical thinking’, Educational Studies in Mathematics 29, 93–95.Google Scholar
  11. Dubinsky, E., Elterman, F. and Gong, C.: 1988, ‘The student's construction of quantification’, For the Learning of Mathematics 8(2), 44–51.Google Scholar
  12. Dubinsky, E.: 1991, ‘Reflective abstraction in advanced mathematical thinking’, in D. Tall (ed.), Advanced Mathematical Thinking, Kluwer Academic Press, pp. 95–123.Google Scholar
  13. Dubinsky, E.: 1997, ‘On learning quantification’, Journal of Computers in Mathematics and Science Teaching 16(2/3), 335–362.Google Scholar
  14. Dubinsky, E., Dautermann, J., Leron, U. and Zazkis, R.: 1994, ‘On learning fundamental concepts of group theory’, Educational Studies in Mathematics 27, 267–305.Google Scholar
  15. Dubinsky, E. and Leron, U.: 1994, Learning Abstract Algebra with Isetl, Springer-Verlag.Google Scholar
  16. Freedman, H.: 1983, ‘A way of teaching abstract algebra’, American Mathematical Monthly 90(9), 641–644.Google Scholar
  17. Frorer, P., Hazzan, O. and Manes, M.: 1997, ‘Revealing the faces of abstraction’, The International Journal of Computers for Mathematical Learning 2(3), 217–228.Google Scholar
  18. Gallian, J. A.: 1990, Contemporary Abstract Algebra, second edition, D. C. Heath and Company.Google Scholar
  19. Glaser, B. and Strauss, A. L.: 1967, The Discovery of Grounded Theory: Strategies for Qualitative Research, Chicago, Aldine.Google Scholar
  20. Harnik, V.: 1986, ‘Infinitesimals from Leibniz to Robinson, Time to bring them to school’, The Mathematical Intelligencer 8(2), 41–47, 63.Google Scholar
  21. Hart, E.: 1994, ‘A conceptual analysis of the proof-writing performance of expert and novice students in elementary group theory’, in J. Kaput and E. Dubinsky (eds.), Research Issues in Mathematics Learning: Preliminary Analyses and Results, MAA notes, No. 33.Google Scholar
  22. Hazzan, O.: 1994, ‘A students' belief about the solutions of the equation x=x−1 in a group’, Proceedings of the 18th international conference for the Psychology of Mathematics Education, Lisbon, Portugal, Vol. III, pp. 49–56.Google Scholar
  23. Hazzan, O.: 1995, Undergraduate Students' Understanding of Abstract Algebra Concepts. Ph.D. thesis, Technion — Israel Institute of Technology. Unpublished (Hebrew).Google Scholar
  24. Hazzan, O. and Leron, U.: 1996, ‘Students' use and misuse of Mathematical Theorems: The Case of Lagrange's Theorem’, For the Learning of Mathematics 16, 23–26.Google Scholar
  25. Herstein, I. N.: 1986, Abstract Algebra, Macmillan Publishing Company.Google Scholar
  26. Kilpatrick, J.: 1987, ‘What constructivism might be in mathematics education’, Proceedings of the Eleventh International Conference for the Psychology of Mathematical Education, Volume I, 3–27.Google Scholar
  27. Kleiner, I.: 1991, ‘Rigor and proof in mathematics: A historical perspective’, Mathematics Magazine 64(5), 291–314.Google Scholar
  28. Leron, U.: 1987, ‘Abstraction barriers in mathematics and computer science’, Proceedings of the Third International Conference for the Psychology of Mathematical Education, Montreal.Google Scholar
  29. Leron, U. and Dubinsky, E.: 1995, ‘An abstract algebra story’, The American Mathematical Monthly 102(3), 227–242.Google Scholar
  30. Leron, U., Hazzan, O. and Zazkis, R.: 1994, ‘Students constructions of group isomorphism’, Proceedings of the Eleventh International Conference for the Psychology of Mathematical Education 3, 152–159.Google Scholar
  31. Leron, U., Hazzan, O. and Zazkis, R.: 1995, ‘Learning group isomorphism: A crossroads of many concepts’, Educational Studies in Mathematics 29, 153–174.Google Scholar
  32. Lesh, R. A.: 1976, ‘The influence of two types of advanced organizers on an instructional unit about finite groups’, Journal for Research in Mathematics Education 7(2), 87–91.Google Scholar
  33. Lichtenberg, D. R.: 1981, ‘A group whose elements are functions’, Mathematics Teacher 74(7), 521–523.Google Scholar
  34. Macdonald, I. D.: 1976, ‘Modern algebra in the nineteenth century’, Australian Mathematics Teacher 32(1), 33–38.Google Scholar
  35. Nesher, P. and Teubal, E.: 1975, ‘Verbal cues as an interfering factor in verbal problem solving’, Educational Studies in Mathematics 6, 41–51.Google Scholar
  36. Nesher, P.: 1980, ‘The stereotyped nature of school word problems’, For the Learning of Mathematics 1(1), 41–48.Google Scholar
  37. Noss, R. and Hoyles, C.: 1996, Windows on Mathematical Meanings — Learning Cultures and Computers, Kluwer Academic Publishers, Dordrecht, The Netherlands.Google Scholar
  38. Papert, S.: 1980, Mindstorms — Children, Computers and Powerful Ideas, Basic Books, Inc., Publishers / New York.Google Scholar
  39. Pedersen, J. J.: 1972, ‘Sneaking up on a group’, Two Year College Mathematics Journal 3(1), 9–12.Google Scholar
  40. Petricig, M.: 1988, ‘Combining individualized instruction with the traditional lecture method in a college algebra course’, Mathematics Teacher 81, 385–387.Google Scholar
  41. Piaget, J.: 1977, ‘Problems of Equilibration’, in M. H. Appel and L. S. Goldberg (eds.), Topics in Cognitive Development, Volume 1: Equilibration: Theory, Research and Application, Plenum Press, NY, pp. 3–13.Google Scholar
  42. Polya, G.: 1973, How to Solve it?, (Second edition), Princeton University Press, Princeton, New Jersey.Google Scholar
  43. Quadling, D.: 1978, ‘A contorted isomorphism’, Mathematics Teaching 85, 48–49.Google Scholar
  44. Rumelhart, D. E.: 1989, ‘Toward a microstructural account of human reasoning’, in S. Vosniadou and A. Ortony (eds.), Similarity and Analogical Reasoning, Cambridge University Press, pp. 298–312.Google Scholar
  45. Schoenfeld, A. H.: 1985, Mathematical Problem Solving, Academic press, Inc.Google Scholar
  46. Selden, A. and Selden, J.: 1987, ‘Errors and misconceptions in college level theorem proving’, Proceedings of the Second International Seminar on Misconceptions and Educational Strategies in Science and Mathematics, Cornell University, Vol. III, pp. 456–471.Google Scholar
  47. Selden, A. and Selden, J.: 1993, ‘Collegiate mathematics education research: What would that be like?’ The College Mathematics Journal 24(5), 431–445.Google Scholar
  48. Sfard, A.: 1991, ‘On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin’, Educational Studies in mathematics 22, 1–36.Google Scholar
  49. Sfard, A.: 1992, ‘Operational origins of mathematical objects and the quandary of reification — The case of function’, in E. Dubinsky and G. Harel (eds.), The Concept of Function — Aspects of Epistemology and Pedagogy, MAA Notes.Google Scholar
  50. Simmonds, G.: 1982, ‘Computer discovery of a theorem of modern algebra’, Mathematics and Computer Education 16(1), 58–61.Google Scholar
  51. Sinclair, H.: 1987, ‘Constructivism and the psychology of mathematics’, Proceedings of the Eleventh International Conference for the Psychology of Mathematical Education, Volume I, pp. 28–41.Google Scholar
  52. Smith, J. P., diSessa, A. A., Roschelle, J.: 1993, ‘Misconceptions reconceived: A constructivist analysis of knowledge in transition’, The Journal of the Learning Sciences 3(2), 115–163.Google Scholar
  53. Staub, F. C. and Stern, E.: 1997, ‘Abstract reasoning with mathematical constructs’, International Journal of Educational Research 27(1), 63–75.Google Scholar
  54. Tall, D.: 1991 (ed.), Advanced Mathematical Thinking, Kluwer Academic Press, Dordrecht, The Netherlands.Google Scholar
  55. Thompson, P. W.: 1985, ‘Experience, problem solving, and learning mathematics: Considerations in developing mathematics curricula’, in E. A. Silver (ed.), Teaching and Learning Mathematical Problem Solving: Multiple Research Perspective, Hillsdale, NJ, pp. 189–236.Google Scholar
  56. Thompson, P. W.: 1993, ‘Yes, Virginia, some children do grow up to be mathematicians’, Journal for Research in Mathematics Education 24(3), 279–284.Google Scholar
  57. Thrash, K. R. and Walls, G. L.: 1991, ‘A classroom note on understanding the concept of group isomorphism’, Mathematics and Computer Education 25(1), 53–55.Google Scholar
  58. Wilensky, U.: 1991, ‘Abstract meditations on the concrete and concrete implications for mathematical education’, in I. Harel and S. Papert (eds.), Constructionism, Ablex Publishing Corporation, Norwood, NJ, pp. 193–203.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Orit Hazzan
    • 1
  1. 1.Department of Education in Technology and Science TechnionIsrael Institute of TechnologyHaifaIsrael

Personalised recommendations