Educational Psychology Review

, Volume 24, Issue 2, pp 251–269 | Cite as

Learning from Comparing Multiple Examples: On the Dilemma of “Similar” or “Different”

  • Jian-Peng GuoEmail author
  • Ming Fai Pang
  • Ling-Yan Yang
  • Yi Ding


Although researchers have demonstrated that studying multiple examples is more effective than studying one example to facilitate learning, the principles found in the literature for designing multiple examples remain ambiguous. This paper reviews variation theory research on example design which sheds light on unclear issues regarding the effects of example variability. First, the distinction of surface/structural should be replaced by critical/uncritical in example study. Aspects and features that are critical to students’ understanding should be identified and compared in example design. Second, variation as well as similarity among examples should be taken into consideration in example design. Certain patterns of variation and invariance should be adopted to systematically determine the variability of examples. Third, students with different levels of prior knowledge perceive different aspects of examples that are critical for their learning. Examples should be designed according to aspects that are critical to specific students.


Multiple examples Variability Comparison Critical aspects Variation theory 



This research was based on the project “An Investigation of Creating Effective Problem Context in Teaching Mathematics” supported by Key Project of Ministry of Education, Plan of National Science of Education of China (GIA117009), and a research grant from the Hong Kong Research Grants Council. We thank the anonymous reviewers for their constructive comments and suggestions.


  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, TX: 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. Bransford, J. D., & Schwartz, D. L. (1999). Rethinking transfer: A simple proposal with multiple implications. In A. Iran-Nejad & P. D. Pearson (Eds.), Review of research in education, 24 (pp. 61–101). Washington, DC: American Educational Research Association.Google Scholar
  5. Catrambone, R. (1994). Improving examples to improve transfer to novel problems. Memory and Cognition, 22(5), 606–615.CrossRefGoogle Scholar
  6. 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
  7. Cheung, W. M. (2005). Describing and enhancing creativity in Chinese writing. Unpublished Ph.D. Dissertation, University of Hong Kong, Hong Kong.Google Scholar
  8. Choy, C. K. (2006). The use of variation theory to improve secondary three students’ learning of the mathematical concept of slope. Unpublished Dissertation, University of Hong Kong, Hong Kong.Google Scholar
  9. 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
  10. 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
  11. Curry, L. A. (2004). The effects of self-explanations of correct and incorrect solutions on algebra problem-solving performance. In K. Forbus, D. Gentner, & T. Regier (Eds.), Proceedings of the 26th Annual Conference of the Cognitive Science Society (p. 1548). Mahwah, NJ: Erlbaum.Google Scholar
  12. Emanuelsson, J., Haggblom, J. et al. (2002). The way we do things in Sweden: On school cultures and math lessons from different perspectives. Paper presented at the Learner’s Perspective Study symposium, Melbourne, Australia.Google Scholar
  13. Fraser, D., Allison, S., Coombes, H., Case, J., & Linder, C. (2006). Using variation to enhance learning in engineering. International Journal of Engineering Education, 22, 102–108.Google Scholar
  14. 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, NH: Erlbaum.Google Scholar
  15. Gentner, D., & Medina, J. (1998). Similarity and the development of rules. Cognition, 65, 263–297.CrossRefGoogle Scholar
  16. Gentner, D., & Namy, L. L. (1999). Comparison in the development of categories. Cognitive Development, 14(4), 487–513.CrossRefGoogle Scholar
  17. Gentner, D., Loewenstein, J., & Thompson, L. (2003). Learning and transfer: A general role for analogical encoding. Journal of Educational Psychology, 95(2), 393–405.CrossRefGoogle Scholar
  18. 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
  19. Gibson, J. J., & Gibson, E. J. (1955). Perceptual learning: Differentiation or enrichment? Psychological Review, 62, 32–41.CrossRefGoogle Scholar
  20. Gick, M. L., & Holyoak, K. J. (1980). Analogical problem solving. Cognitive Psychology, 12, 306–355.CrossRefGoogle Scholar
  21. Gick, M. L., & Holyoak, K. J. (1983). Schema induction and analogical transfer. Cognitive Psychology, 15, 1–38.CrossRefGoogle Scholar
  22. Gick, M. L., & Paterson, K. (1992). Do contrasting examples facilitate schema acquisition and analogical transfer? Canadian Journal of Psychology, 46, 539–550.CrossRefGoogle Scholar
  23. Große, C. S., & Renkl, A. (2006). Effects of multiple solution methods in mathematics learning. Learning and Instruction, 16, 122–138.CrossRefGoogle Scholar
  24. Große, C. S., & Renkl, A. (2007). Finding and fixing errors in worked examples: Can this foster learning outcomes? Learning and Instruction, 17, 612–634.CrossRefGoogle Scholar
  25. Guo, J. P., & Pang, M. F. (2011). Learning a mathematical concept from comparing examples: The importance of variation and prior knowledge. European Journal of Psychology of Education, 26, 495–525.Google Scholar
  26. Hammer, R., Bar-Hillel, A., Hertz, T., Weinshall, D., & Hochstein, S. (2008). Comparison processes in category learning: From theory to behavior. Brain Research, 1225, 102–118.CrossRefGoogle Scholar
  27. Holmqvist, M., Gustavsson, L., & Wernberg, A. (2007). Generative learning: Learning beyond the learning situation. Educational Action Research, 15(2), 181–208.CrossRefGoogle Scholar
  28. Holyoak, K. J. (1985). The pragmatics of analogical transfer. In G. H. Bower (Ed.), The psychology of learning and motivation (pp. 59–87). New York: Academic Press.Google Scholar
  29. Holyoak, K. J., & Koh, K. (1987). Surface and structural similarity in analogical transfer. Memory and Cognition, 15(4), 332–340.CrossRefGoogle Scholar
  30. 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
  31. Kolodner, J. L. (1997). Educational implications of analogy. American Psychologist, 52, 57–66.CrossRefGoogle Scholar
  32. Kotovsky, L., & Gentner, D. (1996). Comparison and categorization in the development of relational similarity. Child Development, 67, 2797–2822.CrossRefGoogle Scholar
  33. Kurtz, K. J., Miao, C., & Gentner, D. (2001). Learning by analogical bootstrapping. The Journal of the Learning Sciences, 10, 417–446.CrossRefGoogle Scholar
  34. Kwong, S. P. E. (2005). The use of variation theory in developing students’ critical thinking skills. Unpublished Dissertation, University of Hong Kong, Hong Kong.Google Scholar
  35. Lam, Y. S. (2005). The use of variation theory to improve the teaching and learning of international trade. Unpublished Dissertation, University of Hong Kong, Hong Kong.Google Scholar
  36. Lave, J. (1996). Teaching, as learning, in practice. Mind, Culture, and Activity, 3(3), 149–164.CrossRefGoogle Scholar
  37. 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
  38. Linder, C., & Marshall, D. (2003). Reflection and phenomenography: Towards theoretical and educational development possibilities. Learning and Instruction, 13(3), 271–284.CrossRefGoogle Scholar
  39. Lo, M. L. (2002). Catering for individual differences: Building on variation: The first findings. Hong Kong: INSTEP.Google Scholar
  40. Lo, M. L., Chik, P., & Pang, M. F. (2006). Patterns of variation in teaching the colour of light to Primary 3 students. Instructional Science, 34(1), 1–19.Google Scholar
  41. Lo, M. L., Marton, F., Pang, M. F., & Pong, W. Y. (2004). Towards a pedagogy of learning. In F. Marton & A.B.M. Tsui (Eds.), Classroom discourse and the space of learning (pp. 189-225). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  42. Lo, M. L., Pong, W. Y., & Chik, P. M. P. (2005). For each and everyone: Catering for individual differences through learning studies. Hong Kong: Hong Kong University Press.Google Scholar
  43. 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
  44. Marton, F. (2006). Sameness and difference in transfer. The Journal of the Learning Sciences, 15(4), 499–535.CrossRefGoogle Scholar
  45. Marton, F., & Booth, S. (1997). Learning and awareness. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
  46. Marton, F., & Morris, P. (2002). What matters? Discovering critical conditions of classroom learning. Kompendiet, Goteborg, Sweden: Acta Universitatis Gothoburgensis.Google Scholar
  47. Marton, F., & Pang, M. F. (2006). On some necessary conditions of learning. Journal of the Learning Sciences, 15(2), 193–220.Google Scholar
  48. 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
  49. Marton, F., & Tsui, A. B. M. (2004). Classroom discourse and the space of learning. Mahwah, N.J.: Lawrence Erlbaum Associates.Google Scholar
  50. 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
  51. Namy, L. L., & Clepper, L. E. (2010). The differing roles of comparison and contrast in children’s categorization. Journal of Experimental Child Psychology, 107, 291–305.CrossRefGoogle Scholar
  52. 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
  53. Ng, F. P., & Lai, M. C. (2002). Learning study 6: P.4 Chinese language lesson on writing of modern poetry. In M. L. Lo (Ed.), Catering for individual differences: Building on variation: The first findings (pp. 74–85). Hong Kong: INSTEP.Google Scholar
  54. Oakes, L. M. (2001). The role of comparison in category formation in infancy. Paper presented at the 68th Anniversary Meeting of the Society for Research in Child Development, Minneapolis, MN.Google Scholar
  55. Oakes, L. M., & Ribar, R. J. (2005). A comparison of infants’ categorization in paired and successive presentation familiarization tasks. Infancy, 7, 85–98.CrossRefGoogle Scholar
  56. 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
  57. Paik, J. H., & Mix, K. S. (2006). Preschoolers’ use of surface similarity in object comparisons: Taking context into account. Journal of Experimental Child Psychology, 95, 194–214.CrossRefGoogle Scholar
  58. 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.Google Scholar
  59. 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
  60. Pang, M. F., & Marton, F. (2003). Beyond lesson study: Comparing two ways of facilitating the grasp of some economic concepts. Instructional Science, 31(3), 175–194.Google Scholar
  61. Pang, M. F., & Marton, F. (2005). Learning theory as teaching resource: Enhancing students’ understanding of economic concepts. Instructional Science, 33(2), 159–191.Google Scholar
  62. Pong, W.Y. (2000). Widening the space of variation—Inter-contextual and intra-contextual shifts in pupils’ understanding of two economic concepts. Unpublished Ph.D. dissertation, University of Hong Kong, Hong Kong.Google Scholar
  63. 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
  64. 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
  65. Reed, S. K. (1989). Constraints on the abstraction of solutions. Journal of Educational Psychology, 81, 532–540.CrossRefGoogle Scholar
  66. 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, NJ: Ablex.Google Scholar
  67. Reed, S. K., & Bolstad, C. A. (1991). Use of examples and procedures in problem solving. Journal of Experimental Psychology: Learning, Memory, and Cognition, 17, 753–766.CrossRefGoogle Scholar
  68. Renkl, A., Stark, R., Gruber, H., & Mandl, H. (1998). Learning from worked-out examples: The effects of example variability and elicited self-explanations. Contemporary Educational Psychology, 23, 90–108.CrossRefGoogle Scholar
  69. Richland, L. E., & McDonough, I. M. (2010). Learning by analogy: Discriminating between potential analogs. Contemporary Educational Psychology, 35, 28–43.CrossRefGoogle Scholar
  70. 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
  71. 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
  72. 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
  73. 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
  74. Ross, B. H. (1984). Remindings and their effects in learning a cognitive skill. Cognitive Psychology, 16, 371–416.CrossRefGoogle Scholar
  75. Ross, B. H. (1987). This is like that: The use of earlier problems and the separation of similarity effects. Journal of Experimental Psychology: Learning, Memory, and Cognition, 13, 629–639.CrossRefGoogle Scholar
  76. Ross, B. H. (1989a). 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
  77. Ross, B. H. (1989b). Remindings in learning and instruction. In S. Vosniadou & A. Ortony (Eds.), Similarity and analogical reasoning (pp. 438–469). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  78. Ross, B. H. (1997). The use of categories affects classification. Journal of Memory and Language, 37(2), 240–267.CrossRefGoogle Scholar
  79. 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
  80. 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
  81. Schwartz, D. L., & Bransford, J. D. (1998). A time for telling. Cognition and Instruction, 16, 475–522.CrossRefGoogle Scholar
  82. Schwartz, D. L., & Martin, T. (2004). Inventing to prepare for future learning: The hidden efficiency of encouraging original student production in statistics instruction. Cognition and Instruction, 22(2), 129–184.CrossRefGoogle Scholar
  83. Siegler, R. S. (2002). Microgenetic studies of self-explanation. In N. Granott & J. Parziale (Eds.), Micro-development: Transition processes in development and learning (pp. 31–58). New York: Cambridge University Press.CrossRefGoogle Scholar
  84. 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. The Journal of Mathematical Behavior, 24, 287–301.CrossRefGoogle Scholar
  85. Sowder, L. (1980). Concept and principle learning. In R. Shumway (Ed.), Research in mathematics education (pp. 244–285). Reston, VA: NCTM.Google Scholar
  86. 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
  87. Sweller, J. (2010). Element interactivity and intrinsic, extraneous, and germane cognitive load. Educational Psychology Review, 22, 123–138.CrossRefGoogle Scholar
  88. 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
  89. Sweller, J., van Merrienboer, J. J. G., & Paas, F. G. (1998). Cognitive architecture and instructional design. Educational Psychology Review, 10(3), 251–296.CrossRefGoogle Scholar
  90. 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
  91. Tennyson, R. D., & Park, O. C. (1980). The teaching of concepts: A review of instructional design research literature. Review of Educational Research, 50(1), 55–70.Google Scholar
  92. Van Dooren, W., de Bock, D., Hessels, A., Janssens, D., & Verschaffel, L. (2004). Remedying secondary school students’ illusion of linearity: A teaching experiment aiming at conceptual change. Learning and Instruction, 14, 485–501.CrossRefGoogle Scholar
  93. VanderStoep, S. W., & Seifert, C. M. (1993). Learning “how” versus learning “when”: Improving transfer of problem-solving principles. The Journal of the Learning Sciences, 3, 93–111.CrossRefGoogle Scholar
  94. Watson, A., & Mason, J. (2002). Student-generated examples in the learning of mathematics. Canadian Journal of Science Mathematics and Technology, 2(2), 237–249.CrossRefGoogle Scholar
  95. Wittwer, J., & Renkl, A. (2010). How effective are instructional explanations in example-based learning? A meta-analytic review. Educational Psychology Review, 22, 393–409.CrossRefGoogle Scholar
  96. Wolff, P., & Gentner, D. (2000). Evidence for role-neutral initial processing of metaphors. Journal of Experimental Psychology: Learning, Memory, and Cognition, 26, 529–541.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Jian-Peng Guo
    • 1
    Email author
  • Ming Fai Pang
    • 2
  • Ling-Yan Yang
    • 3
  • Yi Ding
    • 4
  1. 1.Institute of EducationXiamen UniversityXiamenChina
  2. 2.Faculty of EducationThe University of Hong KongHong Kong S.A.R.China
  3. 3.College of EducationUniversity of IowaIowaUSA
  4. 4.Graduate School of EducationFordham UniversityNew YorkUSA

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