Skip to main content
SpringerLink
Log in

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

  • Published:
European Journal of Psychology of Education Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • 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 

  • 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 

  • Bowden, J., & Marton, F. (1998). The University of learning: beyond quality and competence in higher education. London: Kogan Page.

    Google Scholar 

  • Bruner, J. S., Goodnow, J. J., & Austin, G. A. (1956). A study of thinking. New York: Wiley.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Chazan, D., & Ball, D. (1999). Beyond being told not to tell. For the Learning of Mathematics, 19, 2–10.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Cooper, G., & Sweller, J. (1987). Effects of schema acquisition and rule automation on mathematical problem-solving transfer. Journal of Educational Psychology, 79, 347–362.

    Article  Google Scholar 

  • 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 

  • Cronbach, L., & Snow, R. (1977). Aptitudes and instructional methods: A handbook for research on interactions. New York: Irvington.

    Google Scholar 

  • Fischbein, E., & Nachlieli, T. (1998). Concepts and figures in geometrical reasoning. International Journal of Science Education, 20(10), 1193–1211.

    Article  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • Gentner, D., & Namy, L. L. (1999). Comparison in the development of categories. Cognitive Development, 14(4), 487–513.

    Article  Google Scholar 

  • Gibson, J. J., & Gibson, E. J. (1955). Perceptual learning: differentiation or enrichment? Psychological Review, 62, 32–41.

    Article  Google Scholar 

  • Gick, M. L., & Holyoak, K. J. (1980). Analogical problem solving. Cognitive Psychology, 12, 306–355.

    Article  Google Scholar 

  • Gick, M. L., & Holyoak, K. J. (1983). Schema induction and analogical transfer. Cognitive Psychology, 15, 1–38.

    Article  Google Scholar 

  • Gick, M. L., & Paterson, K. (1992). Do contrasting examples facilitate schema acquisition and analogical transfer? Canadian Journal of Psychology, 46, 539–550.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Hershkowitz, R. (1989). Visualization in geometry: two sides of the coin. Focus on Learning Problems in Mathematics, 11(1), 61–76.

    Google Scholar 

  • Hiebert, J., & Wearne, D. (1996). Instruction, understanding, and skill in multidigit addition and subtraction. Cognition and Instruction, 14, 251–283.

    Article  Google Scholar 

  • Holmqvist, M., Gustavsson, L., & Wernberg, A. (2007). Generative learning: learning beyond the learning situation. Educational Action Research, 15(2), 181–208.

    Article  Google Scholar 

  • Holyoak, K. J., & Koh, K. (1987). Surface and structural similarity in analogical transfer. Memory & Cognition, 15(4), 332–340.

    Article  Google Scholar 

  • Kalyuga, S. (2007). Expertise reversal effect and its implications for learner-tailored instruction. Educational Psychology Review, 19(4), 509–539.

    Article  Google Scholar 

  • Kalyuga, S., Ayres, P., Chandler, P., & Sweller, J. (2003). The expertise reversal effect. Educational Psychologist, 38(1), 23–31.

    Article  Google Scholar 

  • 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.

  • Kilpatrick, J., Swafford, J. O., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington: National Academy Press.

    Google Scholar 

  • Klausmeier, H. J. (1992). Concept learning and concept teaching. Educational Psychologist, 27(3), 267–286.

    Article  Google Scholar 

  • Kurtz, K. J., Miao, C., & Gentner, D. (2001). Learning by analogical bootstrapping. Journal of the Learning Sciences, 10, 417–446.

    Article  Google Scholar 

  • 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 

  • 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.

  • 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.

  • Marton, F., & Booth, S. (1997). Learning and awareness. Mahwah: Lawrence Erlbaum Associates.

    Google Scholar 

  • Marton, F., & Pang, M. F. (2006). On some necessary conditions of learning. Journal of the Learning Sciences, 15(2), 193–220.

    Article  Google Scholar 

  • 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 

  • Marton, F., & Tsui, A. B. M. (2004). Classroom discourse and the space of learning. Mahwah: Lawrence Erlbaum Associates.

    Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

  • Pang, M. F. (2003). Two faces of variation: on continuity in the phenomenographic movement. Scandinavian Journal of Educational Research, 47(2), 145–156.

    Article  Google Scholar 

  • Pang, M. F., & Marton, F. (2005). Learning theory as teaching resource: enhancing students’ understanding of economic concepts. Instructional Science, 33(2), 159–191.

    Article  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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 

  • Reed, S. K. (1989). Constraints on the abstraction of solutions. Journal of Educational Psychology, 81, 532–540.

    Article  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • Richland, L. E., Holyoak, K. J., & Stigler, J. W. (2004). Analogy use in eighth-grade mathematics classrooms. Cognition and Instruction, 22, 37–60.

    Article  Google Scholar 

  • Richland, L. E., Zur, O., & Holyoak, K. J. (2007). Cognitive supports for analogies in the mathematics classroom. Science, 316, 1128–1129.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Chapter  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Ross, B. H. (1997). The use of categories affects classification. Journal of Memory and Language, 37(2), 240–267.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Schwartz, D. L., & Bransford, J. D. (1998). A time for telling. Cognition and Instruction, 16, 475–522.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Snow, R., & Lohman, D. (1984). Toward a theory of cognitive aptitude for learning from instruction. Journal of Educational Psychology, 76, 347–376.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

  • 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.

    Article  Google Scholar 

  • Tennyson, R. D. (1973). Effect of negative instances in concept acquisition using a verbal learning task. Journal of Educational Psychology, 64, 247–260.

    Article  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Vinner, S., & Hershkowitz, R. (1983). On concept formation in geometry. Zentralbl. Didakt. Math., 1, 20–25.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Education, Xiamen University, Xiamen, China

    Jian-peng Guo

  2. Faculty of Education, The University of Hong Kong, Hong Kong SAR, China

    Ming Fai Pang

Authors
  1. Jian-peng Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ming Fai Pang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jian-peng Guo.

Additional information

Jian-peng Guo. Institute of Education, Xiamen University, China. E-mail: guojp@xmu.edu.cn

Current themes of research:

Curriculum and instruction. Educational psychology. Math education.

Ming Fai Pang. Faculty of Education, The University of Hong Kong, Hong Kong S.A.R. China. E-mail: pangmf@hkucc.hku.hk

Current themes of research:

Curriculum and instruction. Teaching and learning. Teacher education.

Pang, M. F. (2003). Two faces of variation: On continuity in the phenomenographic movement. Scandinavian Journal of Educational Research, 47(2), 145–156.

Pang, M. F. & F. Marton (2003). “Beyond Lesson Study: Comparing Two Ways of Facilitating the Grasp of Some Economic Concepts.” Instructional Science 31(3), p. 175–94.

Pang, M. F., & Marton, F. (2005). Learning theory as teaching resource: Enhancing students’ understanding of economic concepts. Instructional Science, 33(2), 159–191.

Marton, F., & Pang, M. F. (2006). On some necessary conditions of learning. Journal of the Learning Sciences, 15(2), 193–220.

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.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Guo, Jp., Pang, M.F. Learning a mathematical concept from comparing examples: the importance of variation and prior knowledge. Eur J Psychol Educ 26, 495–525 (2011). https://doi.org/10.1007/s10212-011-0060-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10212-011-0060-y

Keywords

Search

Navigation