Advertisement

European Journal of Psychology of Education

, Volume 26, Issue 4, pp 495–525 | Cite as

Learning a mathematical concept from comparing examples: the importance of variation and prior knowledge

  • Jian-peng GuoEmail author
  • Ming Fai Pang
Article

Abstract

In experiment 1, novice fourth-grade students (N = 92) who compared multiple examples that separately varied each critical aspect and then simultaneously varied all critical aspects developed better conceptual knowledge about the altitude of a triangle than students who compared multiple examples that did not separately vary each critical aspect but simultaneously varied all critical aspects. In experiment 2, this pattern was the same for fourth-grade students (N = 90) but not for sixth-grade students (N = 94) who had greater prior knowledge about the concept. Aspects that are critical for learning should be varied first separately and then simultaneously, and students with different levels of prior knowledge may perceive different aspects as critical for their learning and thus benefit differently from the identical instruction.

Keywords

Example variability Comparison Conceptual knowledge Mathematics education Positive and negative examples 

Notes

References

  1. Albro, E., Uttal, D., De Loache, J., Kaminski, J. A., Sloutsky, V. M., Heckler, A. F., et al. (2007). Fostering transfer of knowledge in education settings. In D. S. McNamara & G. Trafton (Eds.), Proceedings of the 29th meeting of the Cognitive Science Society (pp. 21–22). Austin: Cognitive Science Society.Google Scholar
  2. Atkinson, R. K., Derry, S. J., Renkl, A., & Wortham, D. (2000). Learning from examples: instructional principles from the worked examples research. Review of Educational Research, 70(2), 181–214.Google Scholar
  3. Bowden, J., & Marton, F. (1998). The University of learning: beyond quality and competence in higher education. London: Kogan Page.Google Scholar
  4. Bruner, J. S., Goodnow, J. J., & Austin, G. A. (1956). A study of thinking. New York: Wiley.Google Scholar
  5. Catrambone, R., & Holyoak, K. J. (1989). Overcoming contextual limitations on problem-solving transfer. Journal of Experimental Psychology. Learning, Memory, and Cognition, 15(6), 1147–1156.CrossRefGoogle Scholar
  6. Chazan, D., & Ball, D. (1999). Beyond being told not to tell. For the Learning of Mathematics, 19, 2–10.Google Scholar
  7. Clarke, T., Ayres, P., & Sweller, J. (2005). The impact of sequencing and prior knowledge on learning mathematics through spreadsheet applications. Educational Technology Research and Development, 53(3), 15–24.CrossRefGoogle Scholar
  8. Cooper, G., & Sweller, J. (1987). Effects of schema acquisition and rule automation on mathematical problem-solving transfer. Journal of Educational Psychology, 79, 347–362.CrossRefGoogle Scholar
  9. Cronbach, L. (1967). How can instruction be adapted to individual differences. In R. Gagne (Ed.), Learning and individual differences (pp. 23–39). Columbus: Merrill.Google Scholar
  10. Cronbach, L., & Snow, R. (1977). Aptitudes and instructional methods: A handbook for research on interactions. New York: Irvington.Google Scholar
  11. Fischbein, E., & Nachlieli, T. (1998). Concepts and figures in geometrical reasoning. International Journal of Science Education, 20(10), 1193–1211.CrossRefGoogle Scholar
  12. Gentner, D. (2005). The development of relational category knowledge. In D. H. Rakison & L. Gershkoff-Stowe (Eds.), Building object categories in developmental time (pp. 245–275). Mahwah: Erlbaum.Google Scholar
  13. Gentner, D., Loewenstein, J., & Hung, B. (2007). Comparison facilitates children’s learning of names for parts. Journal of Cognition and Development, 8(3), 285–307.CrossRefGoogle Scholar
  14. Gentner, D., & Namy, L. L. (1999). Comparison in the development of categories. Cognitive Development, 14(4), 487–513.CrossRefGoogle Scholar
  15. Gibson, J. J., & Gibson, E. J. (1955). Perceptual learning: differentiation or enrichment? Psychological Review, 62, 32–41.CrossRefGoogle Scholar
  16. Gick, M. L., & Holyoak, K. J. (1980). Analogical problem solving. Cognitive Psychology, 12, 306–355.CrossRefGoogle Scholar
  17. Gick, M. L., & Holyoak, K. J. (1983). Schema induction and analogical transfer. Cognitive Psychology, 15, 1–38.CrossRefGoogle Scholar
  18. Gick, M. L., & Paterson, K. (1992). Do contrasting examples facilitate schema acquisition and analogical transfer? Canadian Journal of Psychology, 46, 539–550.CrossRefGoogle Scholar
  19. Gutierrez, A., & Jaime, A. (1999). Pre-service primary teachers’ understanding of the concept of altitude of a triangle. Journal of Mathematics Teacher Education, 2, 253–275.CrossRefGoogle Scholar
  20. Hershkowitz, R. (1989). Visualization in geometry: two sides of the coin. Focus on Learning Problems in Mathematics, 11(1), 61–76.Google Scholar
  21. Hiebert, J., & Wearne, D. (1996). Instruction, understanding, and skill in multidigit addition and subtraction. Cognition and Instruction, 14, 251–283.CrossRefGoogle Scholar
  22. Holmqvist, M., Gustavsson, L., & Wernberg, A. (2007). Generative learning: learning beyond the learning situation. Educational Action Research, 15(2), 181–208.CrossRefGoogle Scholar
  23. Holyoak, K. J., & Koh, K. (1987). Surface and structural similarity in analogical transfer. Memory & Cognition, 15(4), 332–340.CrossRefGoogle Scholar
  24. Kalyuga, S. (2007). Expertise reversal effect and its implications for learner-tailored instruction. Educational Psychology Review, 19(4), 509–539.CrossRefGoogle Scholar
  25. Kalyuga, S., Ayres, P., Chandler, P., & Sweller, J. (2003). The expertise reversal effect. Educational Psychologist, 38(1), 23–31.CrossRefGoogle Scholar
  26. Ki, W. W. (2007). The enigma of Cantonese tones: How intonation language speakers can be assisted to discern them. Unpublished Ph.D. dissertation, University of Hong Kong, Hong Kong.Google Scholar
  27. Kilpatrick, J., Swafford, J. O., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington: National Academy Press.Google Scholar
  28. Klausmeier, H. J. (1992). Concept learning and concept teaching. Educational Psychologist, 27(3), 267–286.CrossRefGoogle Scholar
  29. Kurtz, K. J., Miao, C., & Gentner, D. (2001). Learning by analogical bootstrapping. Journal of the Learning Sciences, 10, 417–446.CrossRefGoogle Scholar
  30. Linder, C., Fraser, D., & Pang, M. F. (2006). Using a variation approach to enhance physics learning in a college classroom. Physics Teacher, 44(9), 589–592.Google Scholar
  31. Lo, M. L., Marton, F., Pang, M. F., & Pong, W. Y. (2004). Toward a pedagogy of learning. In F. Marton & A. B. M. Tsui (Eds.), Classroom discourse and the space of learning (pp. 189–225). Mahwah: Lawrence Erlbaum.Google Scholar
  32. Marton, F. (1999). Variatio est mater Studiorum. In: Opening address delivered to the 8th European Association for Research on Learning and Instruction Biennial Conference, Goteborg, Sweden, August 24–28.Google Scholar
  33. Marton, F., & Booth, S. (1997). Learning and awareness. Mahwah: Lawrence Erlbaum Associates.Google Scholar
  34. Marton, F., & Pang, M. F. (2006). On some necessary conditions of learning. Journal of the Learning Sciences, 15(2), 193–220.CrossRefGoogle Scholar
  35. Marton, F., & Pang, M. F. (2008). The idea of phenomenography and the pedagogy for conceptual change. In S. Vosniadou (Ed.), International handbook of research on conceptual change (pp. 553–559). London: Routledge.Google Scholar
  36. Marton, F., & Tsui, A. B. M. (2004). Classroom discourse and the space of learning. Mahwah: Lawrence Erlbaum Associates.Google Scholar
  37. Merrill, M. D., & Tennyson, R. D. (1978). Concept classification and classification errors as a function of relationships between examples and nonexamples. Improving Human Performance Quarterly, 7, 351–364.Google Scholar
  38. Namy, L. L., & Gentner, D. (2002). Making a silk purse out of two sow’s ears: young children’s use of comparison in category learning. Journal of Experimental Psychology: General, 131(1), 5–15.CrossRefGoogle Scholar
  39. Paas, F., & van Merrienboer, J. (1994). Variability of worked examples and transfer of geometrical problem-solving skills: a cognitive-load approach. Journal of Educational Psychology, 86, 122–133.CrossRefGoogle Scholar
  40. Pang, M. F. (2002). Making learning possible: The use of variation in the teaching of school economics. Unpublished Ph.D. dissertation, University of Hong Kong, Hong Kong.Google Scholar
  41. Pang, M. F. (2003). Two faces of variation: on continuity in the phenomenographic movement. Scandinavian Journal of Educational Research, 47(2), 145–156.CrossRefGoogle Scholar
  42. Pang, M. F., & Marton, F. (2005). Learning theory as teaching resource: enhancing students’ understanding of economic concepts. Instructional Science, 33(2), 159–191.CrossRefGoogle Scholar
  43. Pang, M. F., Linder, C., & Fraser, D. (2006). Beyond lesson studies and design experiments – Using theoretical tools in practice and finding out how they work. International Review of Economics Education, 5(1), 28–45.Google Scholar
  44. Quilici, J. L., & Mayer, R. E. (1996). Role of examples in how students learn to categorize statistics word problems. Journal of Educational Psychology, 88, 144–161.CrossRefGoogle Scholar
  45. Ranzijn, F. J. A. (1991). The number of video examples and the dispersion of examples as instructional design variables in teaching concepts. The Journal of Experimental Education, 59(4), 320–330.Google Scholar
  46. Reed, S. K. (1989). Constraints on the abstraction of solutions. Journal of Educational Psychology, 81, 532–540.CrossRefGoogle Scholar
  47. Reed, S. K. (1993). A schema-based theory of transfer. In D. K. Detterman & R. J. Sternberg (Eds.), Transfer on trial: Intelligence, cognition, and instruction (pp. 39–67). Norwood: Ablex.Google Scholar
  48. Renkl, A., Stark, R., Gruber, H., & Mandl, H. (1998). Learning from worked-out examples: the effects of example variability and elicited self-explanations. Cotemporary Educational Psychology, 23, 90–108.CrossRefGoogle Scholar
  49. Richland, L. E., Holyoak, K. J., & Stigler, J. W. (2004). Analogy use in eighth-grade mathematics classrooms. Cognition and Instruction, 22, 37–60.CrossRefGoogle Scholar
  50. Richland, L. E., Zur, O., & Holyoak, K. J. (2007). Cognitive supports for analogies in the mathematics classroom. Science, 316, 1128–1129.CrossRefGoogle Scholar
  51. Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99(3), 561–574.CrossRefGoogle Scholar
  52. Rittle-Johnson, B., & Star, J. R. (2009). Compared with what? The effects of different comparisons on conceptual knowledge and procedural flexibility for equation solving. Journal of Educational Psychology, 101(3), 529–544.CrossRefGoogle Scholar
  53. Rittle-Johnson, B., Star, J. R., & Durkin, K. (2009). The importance of prior knowledge when comparing examples: influences on conceptual and procedural knowledge of equation solving. Journal of Educational Psychology, 101(4), 836–852.CrossRefGoogle Scholar
  54. Ross, B. H. (1989a). Remindings in learning and instruction. In S. Vosniadou & A. Ortony (Eds.), Similarity and analogical reasoning (pp. 438–469). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  55. Ross, B. H. (1989b). Distinguishing types of superficial similarities: different effects on the access and use of earlier problems. Journal of Experimental Psychology. Learning, Memory, and Cognition, 15, 456–468.CrossRefGoogle Scholar
  56. Ross, B. H. (1997). The use of categories affects classification. Journal of Memory and Language, 37(2), 240–267.CrossRefGoogle Scholar
  57. Ross, B., & Kennedy, P. (1990). Generalizing from the use of earlier examples in problem solving. Journal of Experimental Psychology. Learning, Memory, and Cognition, 16(1), 42–55.CrossRefGoogle Scholar
  58. Ross, B. H., & Kilbane, M. C. (1997). Effects of principle explanation and superficial similarity on analogical mapping in problem solving. Journal of Experimental Psychology. Learning, Memory, and Cognition, 23(2), 427–440.CrossRefGoogle Scholar
  59. Schwartz, D. L., & Bransford, J. D. (1998). A time for telling. Cognition and Instruction, 16, 475–522.CrossRefGoogle Scholar
  60. Silver, E. A., Ghousseini, H., Gosen, D., Charalambous, C., & Strawhun, B. (2005). Moving from rhetoric to praxis: issues faced by teachers in having students consider multiple solutions for problems in the mathematics classroom. Journal of Mathematical Behavior, 24, 287–301.CrossRefGoogle Scholar
  61. Snow, R., & Lohman, D. (1984). Toward a theory of cognitive aptitude for learning from instruction. Journal of Educational Psychology, 76, 347–376.CrossRefGoogle Scholar
  62. Star, J. R., & Rittle-Johnson, B. (2009). It pays to compare: an experimental study on computational estimation. Journal of Experimental Child Psychology, 102(4), 408–426.CrossRefGoogle Scholar
  63. Svensson, L. (1984). Människobilden i INOM-gruppens forskning: Den lärande människan. [The view of man in the research of the INOM-group: The learning man]. Goteborg: Institutionen för pedagogik, Göteborgs universitet.Google Scholar
  64. Sweller, J., & Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition and Instruction, 2, 59–89.CrossRefGoogle Scholar
  65. Tennyson, R. D. (1973). Effect of negative instances in concept acquisition using a verbal learning task. Journal of Educational Psychology, 64, 247–260.CrossRefGoogle Scholar
  66. Tennyson, R. D., & Cocchiarella, M. J. (1986). An empirically based instructional design theory for teaching concepts. Review of Educational Research, 56, 40–71.Google Scholar
  67. Tennyson, R. D., Wooley, F. R., & Merrill, M. D. (1972). Exemplar and nonexemplar variables which produce correct concept classification behavior and specified classification errors. Journal of Educational Psychology, 63, 144–152.CrossRefGoogle Scholar
  68. Tennyson, R. D., Youngers, J., & Suebsonthi, P. (1983). Acquisition of mathematical concepts by children using prototype and skill development presentation forms. Journal of Educational Psychology, 75, 280–291.CrossRefGoogle Scholar
  69. VanderStoep, S. W., & Seifert, C. M. (1993). Learning “how” versus learning “when”: improving transfer of problem-solving principles. Journal of the Learning Sciences, 3, 93–111.CrossRefGoogle Scholar
  70. Vinner, S., & Hershkowitz, R. (1983). On concept formation in geometry. Zentralbl. Didakt. Math., 1, 20–25.Google Scholar

Copyright information

© Instituto Superior de Psicologia Aplicada, Lisboa, Portugal and Springer Science+Business Media BV 2011

Authors and Affiliations

  1. 1.Institute of Education, Xiamen UniversityXiamenChina
  2. 2.Faculty of EducationThe University of Hong KongHong Kong SARChina

Personalised recommendations