Instructional Science

, Volume 46, Issue 1, pp 61–76 | Cite as

Examining the preparatory effects of problem generation and solution generation on learning from instruction

Article

Abstract

The goal of this paper is to isolate the preparatory effects of problem-generation from solution generation in problem-posing contexts, and their underlying mechanisms on learning from instruction. Using a randomized-controlled design, students were assigned to one of two conditions: (a) problem-posing with solution generation, where they generated problems and solutions to a novel situation, or (b) problem-posing without solution generation, where they generated only problems. All students then received instruction on a novel math concept. Findings revealed that problem-posing with solution generation prior to instruction resulted in significantly better conceptual knowledge, without any significant difference in procedural knowledge and transfer. Although solution generation prior to instruction plays a critical role in the development of conceptual understanding, which is necessary for transfer, generating problems plays an equally critical role in transfer. Implications for learning and instruction are discussed.

Keywords

Problem posing Preparatory activities Math learning Transfer 

Notes

Acknowledgements

A shorter version of this manuscript has been submitted to the 2017 Annual Meeting of the Cognitive Science Society. The author would like to thank the principal, teachers and students for their support and participation, as well as the research assistants who helped with data collection and coding.

References

  1. Alfieri, L., Nokes-Malach, T. J., & Schunn, C. D. (2013). Learning through case comparisons: A meta-analytic review. Educational Psychologist, 48(2), 87–113.CrossRefGoogle Scholar
  2. Bransford, J. D., & Schwartz, D. L. (1999). Rethinking transfer: A simple proposal with multiple implications. Review of Research in Education, 24, 61–100.Google Scholar
  3. Brown, S. I., & Walter, M. I. (2005). The art of problem posing (3rd ed.). Hillsdale, NJ: L. Erlbaum Associates.Google Scholar
  4. Burns, B. D., & Vollmeyer, R. (2002). Goal specificity effects on hypothesis testing in problem solving. The Quarterly Journal of Experimental Psychology, 55A, 241–261.CrossRefGoogle Scholar
  5. Cifarelli, V., & Sheets, C. (2009). Problem posing and problem solving: A dynamic connection. School Science and Mathematics, 109, 245–246.CrossRefGoogle Scholar
  6. DeCaro, M. S., & Rittle-Johnson, B. (2012). Exploring mathematics problems prepares children to learn from instruction. Journal of Experimental Child Psychology, 113(4), 552–568.CrossRefGoogle Scholar
  7. Duncker, K. (1945). On problem solving. Psychological Monographs, 58(5), i.CrossRefGoogle Scholar
  8. Einstein, A., & Infeld, L. (1938). The Evolution of Physics. New York: Simon and Schuster.Google Scholar
  9. Ellerton, N. F. (1986). Children’s made-up mathematics problems: A new perspective on talented mathematicians. Educational Studies in Mathematics, 17, 261–271.CrossRefGoogle Scholar
  10. English, L. D. (1998). Children’s problem posing within formal and informal contexts. Journal for Research in mathematics Education, 29, 83–106.CrossRefGoogle Scholar
  11. Frank, M. C., & Ramscar, M. (2003). How do presentation and context influence representation for functional fixedness tasks? In Proceedings of the 25th Annual Meeting of the Cognitive Science Society.Google Scholar
  12. Kapur, M. (2012). Productive failure in learning the concept of variance. Instructional Science, 40(4), 651–672.CrossRefGoogle Scholar
  13. Kapur, M. (2013). Comparing learning from productive failure and vicarious failure. The Journal of the Learning Sciences, 23(4), 651–677.CrossRefGoogle Scholar
  14. Kapur, M. (2014). Productive failure in learning math. Cognitive Science, 38(5), 1008–1022.CrossRefGoogle Scholar
  15. Kapur, M. (2015). The preparatory effects of problem solving versus problem posing on learning from instruction. Learning and Instruction, 39, 23–31.CrossRefGoogle Scholar
  16. Kapur, M. (2016). Examining productive failure, productive success, unproductive failure, and unproductive success in learning. Educational Psychologist, 51(2), 289–299.CrossRefGoogle Scholar
  17. Kapur, M., & Bielaczyc, K. (2012). Designing for productive failure. The Journal of the Learning Sciences, 21(1), 45–83.CrossRefGoogle Scholar
  18. Kilpatrick, J. (1987). Problem formulating: Where do good problems come from. Cognitive Science and Mathematics Education (pp. 123–147).Google Scholar
  19. Loibl, K., & Rummel, N. (2013). Delaying instruction alone doesn’t work: Comparing and contrasting student solutions is necessary for learning from problem-solving prior to instruction. In N. Rummel, M. Kapur, M. Nathan, & S. Puntambekar (Eds.), Proceedings of the 10th international conference on computer-supported collaborative learning (CSCL 2013) (Vol. 1, pp. 296–303). International Society of the Learning Sciences, Inc.Google Scholar
  20. Loibl, K., Roll, I., & Rummel, N. (2017). Towards a theory of when and how problem solving followed by instruction support learning. Educational Psychology Review, 29, 693–715.CrossRefGoogle Scholar
  21. Lurchins, A. S., & Lurchins, E. H. (1959). Rigidity of behaviour: A variational approach to the effects of einstellung. Eugene: University of Oregon Books.Google Scholar
  22. Mawer, R. F., & Sweller, J. (1982). Effects of subgoal density and location on learning during problem solving. Journal of Experimental Psychology. Learning, Memory, and Cognition, 8, 252–259.CrossRefGoogle Scholar
  23. Miller, C. S., Lehman, J. F., & Koedinger, K. R. (1999). Goals and learning in microworlds. Cognitive Science, 23(3), 305–336.CrossRefGoogle Scholar
  24. Moses, B., Bjork, E., & Goldenberg, E. P. (1990). Beyond problem solving: Problem posing. In T. J. Cooney & C. R. Hirsch (Eds.), Teaching and learning mathematics in the 1990s (pp. 82–91). Reston, VA: NCTM.Google Scholar
  25. Paas, F. (1992). Training strategies for achieving transfer of problem-solving skill in statistics: A cognitive load approach. Journal of Educational Psychology, 84(4), 429–434.CrossRefGoogle Scholar
  26. Perrin, J. R. (2007). Problem posing at all levels in the calculus classroom. School Science and Mathematics, 107(5), 182–192.CrossRefGoogle Scholar
  27. Rittle-Johnson, B., & Star, J. R. (2009). Compared to 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
  28. Roll, I., Aleven, V., & Koedinger, K. (2011). Outcomes and mechanisms of transfer in invention activities. In L. Carlson, C. Hölscher, & T. Shipley (Eds.), Proceedings of the 33rd annual conference of the cognitive science society (pp. 2824–2829). Austin, TX: Cognitive Science Society.Google Scholar
  29. Schwartz, D. L., & Bransford, J. D. (1998). A time for telling. Cognition and Instruction, 16(4), 475–522.CrossRefGoogle Scholar
  30. Schwartz, D. L., Chase, C. C., Oppezzo, M. A., & Chin, D. B. (2011). Practicing versus inventing with contrasting cases: The effects of telling first on learning and transfer. Journal of Educational Psychology, 103(4), 759–775.CrossRefGoogle Scholar
  31. 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
  32. Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28.Google Scholar
  33. Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM, 29(3), 75–80.Google Scholar
  34. Silver, E. A., & Cai, J. (1996). An analysis of arithmetic problem posing by middle school students. Journal for Research in Mathematics Education, 27, 521–539.CrossRefGoogle Scholar
  35. Singer, F. M., Ellerton, N., & Cai, J. (2013). Problem-posing research in mathematics education: New questions and directions. Educational Studies in Mathematics, 83, 1–7.CrossRefGoogle Scholar
  36. Sweller, J., Mawer, R. F., & Howe, W. (1982). Consequences of history-cued and means-end strategies in problem solving. American Journal of Psychology, 95, 455–483.CrossRefGoogle Scholar
  37. Sweller, J., Mawer, R. F., & Ward, M. R. (1983). Development of expertise in mathematical problem solving. Journal of Experimental Psychology: General, 112, 463–474.Google Scholar
  38. Vollmeyer, R., Burns, B. D., & Holyoak, K. J. (1996). The impact of goal specificity and systematicity of strategies on the acquisition of problem structure. Cognitive Science, 20, 75–100.CrossRefGoogle Scholar
  39. Wiedmann, M., Leach, R. C., Rummel, N., & Wiley, J. (2012). Does group composition affect learning by invention? Instructional Science., 40, 711–730.CrossRefGoogle Scholar
  40. Wirth, J., Kunsting, J., & Leutner, D. (2009). The impact of goal specificity and goal type on learning outcome and cognitive load. Computers in Human Behavior.  https://doi.org/10.1016/j.chb.2008.12.004.Google Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Institute of Learning Sciences and Higher EducationETH ZurichZurichSwitzerland

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