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

European Journal of Psychology of Education volume 26, pages 495–525 (2011)Cite this 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.

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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
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  2. Ming Fai Pang
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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.

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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

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

Keywords

  • Example variability
  • Comparison
  • Conceptual knowledge
  • Mathematics education
  • Positive and negative examples