Towards a Theory of When and How Problem Solving Followed by Instruction Supports Learning

Abstract

Recently, there has been a growing interest in learning approaches that combine two phases: an initial problem-solving phase followed by an instruction phase (PS-I). Two often cited examples of instructional approaches following the PS-I scheme include Productive Failure and Invention. Despite the growing interest in PS-I approaches, to the best of our knowledge, there has not yet been a comprehensive attempt to summarize the features that define PS-I and to explain the patterns of results. Therefore, the first goal of this paper is to map the landscape of different PS-I implementations, to identify commonalities and differences in designs, and to associate the identified design features with patterns in the learning outcomes. The review shows that PS-I fosters learning only if specific design features (namely contrasting cases or building instruction on student solutions) are implemented. The second goal is to identify a set of interconnected cognitive mechanisms that may account for these outcomes. Empirical evidence from PS-I literature is associated with these mechanisms and supports an initial theory of PS-I. Finally, positive and negative effects of PS-I are explained using the suggested mechanisms.

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Fig. 1

Notes

  1. 1.

    For a discussion on the different processes triggered by contrasting cases in comparison to rich problems, see Loibl and Rummel (2014b).

  2. 2.

    Naturally, some wording is expected to differ by condition (e.g., “invent a formula/strategy for the following problem” vs. “solve the problem using this formula/strategy”), which is not counted as confound.

  3. 3.

    In the studies in no. 19, the instruction and the worksheets with contrasting cases were held constant across conditions. However, students in the I-PS condition received an additional reminder of the formula and a worked example prior to solving each worksheet, whereas students in the PS-I condition were told what invention means and what they need to invent prior to their first problem-solving attempt.

References

*Papers included in overview (Table 1)

  1. Acuña, S. R., García-Rodicio, H., & Sánchez, E. (2010). Fostering active processing of instructional explanations of learners with high and low prior knowledge. European Journal of Psychology of Education, 26(4), 435–452.

    Article  Google Scholar 

  2. Anderson, J. (1983). The architecture of cognition. Cambridge: Harvard University Press.

    Google Scholar 

  3. *Belenky, D. M., & Nokes-Malach, T. J. (2012). Motivation and transfer: the role of mastery-approach goals in preparation for future learning. Journal of the Learning Sciences, 21(3), 399-432.

  4. Barnett, S. M., & Ceci, S. J. (2002). When and where do we apply what we learn? A taxonomy for far transfer. Psychological Bulletin, 128(4), 612–637.

    Article  Google Scholar 

  5. Booth, J. L., Lange, K. E., Koedinger, K. R., & Newton, K. J. (2013). Using example problems to improve student learning in algebra: differentiating between correct and incorrect examples. Learning and Instruction, 25, 24–34.

    Article  Google Scholar 

  6. Chi, M. T. H. (2000). Self-explaining expository texts: the dual processes of generating inferences and repairing mental models. In R. Glaser (Ed.), Advances in instructional psychology (pp. 161–238). Hillsdale: Lawrence Erlbaum Associates.

    Google Scholar 

  7. Chi, M. T. H., Feltovich, P. J., & Glaser, R. (1981). Categorization and representation of physics problems by experts and novices. Cognitive Science, 5(2), 121–152.

    Article  Google Scholar 

  8. Chi, M. T. H., Slotta, J. D., & de Leeuw, N. (1994). From things to processes: a theory of conceptual change for learning science concepts. Learning and Instruction, 4, 27–43.

    Article  Google Scholar 

  9. DeCaro, D. A., DeCaro, M. S., & Rittle-Johnson, B. (2015). Achievement motivation and knowledge development during exploratory learning. Learning and Individual Differences, 37, 13–26.

  10. *DeCaro, M. S., & Rittle-Johnson, B. (2012). Exploring mathematics problems prepares children to learn from instruction. Journal of Experimental Child Psychology, 113, 552-568

  11. diSessa, A. A., Hammer, D., Sherin, B. L., & Kolpakowski, T. (1991). Inventing graphing: meta-representational expertise in children. The Journal of Mathematical Behavior, 10(2), 117–160.

    Google Scholar 

  12. Duncker, K. (1945). On problem-solving. Psychological Monographs, 58(Whole No. 270).

  13. Durkin, K., & Rittle-Johnson, B. (2012). The effectiveness of using incorrect examples to support learning about decimal magnitude. Learning and Instruction, 22(3), 206–214.

    Article  Google Scholar 

  14. Eva, K. W., & Regehr, G. (2011). Exploring the divergence between self-assessment and self-monitoring. Advances in Health Sciences Education, 16(3), 311–329.

    Article  Google Scholar 

  15. *Fyfe, E. R., DeCaro, M. S. & Rittle-Johnson, B. (2014). An alternative time for telling: when conceptual instruction prior to problem solving improves mathematical knowledge. British Journal of Educational Psychology, 84(3), 502-519.

  16. *Glogger-Frey, I., Fleischer, C., Grüny, L., Kappich, J., & Renkl, A. (2015). Inventing a solution and studying a worked solution prepare differently for learning from direct instruction. Learning and Instruction, 39, 72-87.

  17. Heckler, A. F., Kaminski, J. A., & Sloutsky, V. M. (2008). Learning associations that run counter to biases in learning: overcoming overshadowing and learned inattention. In B. C. Love, K. McRae, & V. M. Sloutsky (Eds.), Proceedings of the 30th Annual Conference of the Cognitive Science Society. (pp. 511-516). Austin, TX: Cognitive Science Society

  18. Holmes, N. G., Day, J., Park, A. H., Bonn, D. A., & Roll, I. (2014). Making the failure more productive: scaffolding the invention process to improve inquiry behaviours and outcomes in productive failure activities. Instructional Science, 42(4), 523–538.

    Article  Google Scholar 

  19. *Hsu, C.-Y., Kalyuga, S. & Sweller, J. (2015). When should guidance be presented in physics instruction? Archives of Scientific Psychology, 3(1), 37-53.

  20. Kalyuga, S., & Singh, A.-M. (2015). Rethinking the boundaries of cognitive load theory in complex learning. Educational Psychology Review. Advance Online Publication. doi:10.1007/s10648-015-9352-0.

    Google Scholar 

  21. *Kapur, M. (2010). Productive failure in mathematical problem solving. Instructional Science, 38(6), 523-550.

  22. *Kapur, M. (2011). A further study of productive failure in mathematical problem solving: unpacking the design components. Instructional Science, 39(4), 561-579.

  23. *Kapur, M. (2012). Productive failure in learning the concept of variance. Instructional Science, 40(4), 651-672.

  24. Kapur, M. (2014). Comparing learning from productive failure and vicarious failure. Journal of the Learning Sciences, 23(4), 651–677.

    Article  Google Scholar 

  25. *Kapur, M. (2014b). Productive failure in learning math. Cognitive Science, 38(5), 1008-1022.

  26. Kapur, M. (2016). Examining productive failure, productive success, unproductive failure, and unproductive success in learning. Educational Psychologist, 51(2), 289–299.

    Article  Google Scholar 

  27. *Kapur, M., & Bielaczyc, K. (2011). Classroom-based experiments in productive failure. In L. Carlson, C. Hoelscher, & T. F. Shipley (Eds.), Proceedings of the 33rd Annual Conference of the Cognitive Science Society (pp. 2812–2817). Austin, TX: Cognitive Science Society.

  28. *Kapur, M., & Bielaczyc, K. (2012). Designing for productive failure. The Journal of the Learning Sciences, 21(1), 45-83.

  29. Koedinger, K. R., Corbett, A. T., & Perfetti, C. (2012). The knowledge-learning-instruction (KLI) framework: bridging the science-practice chasm to enhance robust student learning. Cognitive Science, 36, 757–798.

    Article  Google Scholar 

  30. Leppink, J., Paas, F., Van Gog, T., Van der Vleuten, C., & Van Merrienboer, J. (2014). Effects of pairs of problems and examples on task performance and different types of cognitive load. Learning and Instruction, 30, 32–42.

    Article  Google Scholar 

  31. Linn, M. C. (1995). Designing computer learning environments for engineering and computer science: the scaffolded knowledge integration framework. Journal of Science Education and Technology, 4, 103–126.

    Article  Google Scholar 

  32. *Loehr, A. M., Fyfe, E. R., & Rittle-Johnson, B. (2014). Wait for it … delaying instruction improves mathematics problem solving: a classroom study. Journal of Problem Solving, 7(1), 36-49.

  33. *Loibl, K. & Rummel, N. (2014a). Knowing what you don’t know makes failure productive. Learning and Instruction, 34, 74-85.

  34. *Loibl, K., & Rummel, N. (2014b). The impact of guidance during problem-solving prior to instruction on students’ inventions and learning outcomes. Instructional Science, 42(3), 305-326.

  35. *Matlen, B. J., & Klahr, D. (2013). Sequential effects of high and low instructional guidance on children’s acquisition of experimentation skills: is it all in the timing? Instructional Science, 41(3), 621-634.

  36. Mazziotti, C., Loibl, K., & Rummel, N. (2014). Does collaboration affect learning in a productive failure setting? In: J. L. Polman, E. A. Kyza, D. K. O’Neill, I. Tabak, W. R. Penuel, A. S. Ju-row, K. O’Connor, T. Lee, T., & L. D’Amico, Proceedings of the 11th international conference of the learning sciences (ICLS 2014), Vol. 3 (pp. 1184-1185). International Society of the Learning Sciences, Inc.

  37. Mazziotti, C., Loibl, K., & Rummel, N. (2015). Collaborative or individual learning within productive failure. Does the social form of learning make a difference? In O. Lindwall, P. Häkkinen, T. Koschman, P. Tchounikine, & S. Ludvigsen (Eds.), Exploring the material conditions of learning: the Computer Supported Collaborative Learning (CSCL) Conference 2015 (Vol. 2, pp. 570–575). Gothenburg: ISLS.

    Google Scholar 

  38. Nathan, M. J. (1998). Knowledge and situational feedback in a learning environment for algebra story problem solving. Interactive Learning Environments, 5(1), 135–159.

    Article  Google Scholar 

  39. Nathan, M. J., Alibali, M. W., Masarik, D. K., Stephens, A. C., & Koedinger, K. R. (2010). Enhancing middle school students’ representational fluency: a classroom-based study. WCER Working Paper Series no, 2010-9. Wisconsin Center for Educational Research: Madison, WI. Retrieved June 28, 2013, from http://www.wcer.wisc.edu/Publications/workingPapers/Working_Paper_No_2010_09.pdf.

  40. Needham, D. R., & Begg, I. M. (1991). Problem-oriented training promotes spontaneous analogical transfer: memory-oriented training promotes memory for training. Memory & Cognition, 19(6), 543–57.

    Article  Google Scholar 

  41. Paul, A. M. (2012) Why floundering is good. Time Magazine. Retrieved June 28, 2013, from http://ideas.time.com/2012/04/25/why-floundering-is-good/.

  42. Prediger, S. (2008). The relevance of didactic categories for analysing obstacles in conceptual change: revisiting the case of multiplication of fractions. Learning and Instruction, 18(1), 3–17.

    Article  Google Scholar 

  43. Quilici, J. L., & Mayer, R. E. (1996). Role of examples in how students learn to categorize statistics word problems. Journal of Educational Psychology, 88(1), 144–161.

    Article  Google Scholar 

  44. Quilici, J. L., & Mayer, R. E. (2002). Teaching students to recognize structural similarities between statistics word problems. Applied Cognitive Psychology, 16(3), 325–342.

    Article  Google Scholar 

  45. Reisslein, J., Atkinson, R., Seeling, P., & Reisslein, M. (2006). Encountering the expertise reversal effect with a computer-based environment on electrical circuit analysis. Learning and Instruction, 16, 92–103.

    Article  Google Scholar 

  46. Renkl, A. (2005). The worked-out-example principle in multimedia learning. In R. Mayer (Ed.), Cambridge handbook of multimedia learning (pp. 229–246). Cambridge: Cambridge University Press.

    Google Scholar 

  47. Renkl, A. (2014). Towards an instructionally-oriented theory of example-based learning. Cognitive Science, 38, 1–37.

    Article  Google Scholar 

  48. Rittle-Johnson, B., & Schneider, M. (2014). Developing conceptual and procedural knowledge of mathematics. In R. Cohen Kadosh & A. Dowker (Eds.), Oxford handbook of numerical cognition. Oxford: Oxford University Press.

    Google Scholar 

  49. 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 

  50. *Roll, I., Aleven, V., & Koedinger, K. R. (2009). Helping students know ‘further’—increasing the flexibility of students’ knowledge using symbolic invention tasks. In N.A. Taatgen & H. van Rijn (Eds.), Proceedings of the 31st Annual Conference of the Cognitive Science Society (pp. 1169-1174). Austin: Cognitive Science Society.

  51. *Roll, I., Aleven, V., & Koedinger, K. R. (2011). Outcomes and mechanisms of transfer. In L. Carlson, C. Hölscher, & T. Shipley (Eds.), Proceedings of the 33rd Annual Conference of the Cognitive Science Society (pp. 2824–2829). Austin: Cognitive Science Society.

  52. Roll, I., Holmes, N. G., Day, J., & Bonn, D. (2012). Evaluating metacognitive scaffolding in guided invention activities. Instructional Science, 40(4), 691–710.

    Article  Google Scholar 

  53. Roll, I., Wiese, E., Long, Y., Aleven, V., & Koedinger, K. R. (2014). Tutoring self- and co-regulation with intelligent tutoring systems to help students acquire better learning skills. In R. Sottilare, A. Graesser, X. Hu, & B. Goldberg (Eds.), Design recommendations for adaptive intelligent tutoring systems: volume 2—adaptive instructional strategies (pp. 169–182). Orlando: U.S. Army Research Laboratory.

    Google Scholar 

  54. Sánchez, E., García Rodicio, H., & Acuña, S. R. (2009). Are instructional explanations more effective in the context of an impasse? Instructional Science, 37(6), 537–563.

    Article  Google Scholar 

  55. Schmidt, H. G., De Volder, M. L., De Grave, W. S., Moust, J. H. C., & Patel, V. L. (1989). Exploratory models in the processing of science texts: the role of prior knowledge activation through small-group discussion. Journal of Educational Psychology, 81(4), 610–619.

    Article  Google Scholar 

  56. *Schwartz, D. L., & Bransford, J. D. (1998). A time for telling. Cognition and Instruction, 16(4), 475–522.

  57. *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.

  58. *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.

  59. Schwartz, D. L., Sears, D., & Chang, J. (2007). Reconsidering prior knowledge. In M. C. Lovett & P. Shah (Eds.), Thinking with data (pp. 319–344). New York: Routledge.

    Google Scholar 

  60. Sears, D. A. (2006). Effects of innovation versus efficiency tasks on collaboration and learning. Doctoral dissertation, Stanford University, California. Retrieved November 07, 2012, from http://www.stat.auckland.ac.nz/~iase/publications/dissertations/06.Sears.Dissertation.pdf.

  61. Siegler, R. S. (1983). How knowledge influences learning. American Scientist, 71, 631–638.

    Google Scholar 

  62. Sweller, J. (1988). Cognitive load during problem solving: effects on learning. Cognitive Science, 12(2), 257–285.

    Article  Google Scholar 

  63. Sweller, J., van Merrienboer, J., & Paas, F. (1998). Cognitive architecture and instructional design. Educational Psychology Review, 10(3), 251–296.

    Article  Google Scholar 

  64. Toth, E. E., Klahr, D., & Chen, Z. (2000). Bridging research and practice: a cognitively-based classroom intervention for teaching experimentation skills to elementary school children. Cognition and Instruction, 18(4), 423–459.

    Article  Google Scholar 

  65. Van Gog, T., Kester, L., & Paas, F. (2011). Effects of worked examples, example-problem, and problem example pairs on novices’ learning. Contemporary Educational Psychology, 36, 212–218.

    Article  Google Scholar 

  66. Van Gog, T., & Rummel, N. (2010). Example-based learning: integrating cognitive and social-cognitive research perspectives. Educational Psychology Review, 22(2), 155–174.

    Article  Google Scholar 

  67. VanLehn, K. (1999). Rule learning events in the acquisition of a complex skill: an evaluation of cascade. The Journal of the Learning Sciences, 8(1), 71–125.

    Article  Google Scholar 

  68. VanLehn, K., Siler, S., Murray, C., Yamauchi, T., & Baggett, W. B. (2003). Why do only some events cause learning during human tutoring? Cognition and Instruction, 21(3), 209–249.

    Article  Google Scholar 

  69. Vosniadou, S., & Verschaffel, L. (2004). Extending the conceptual change approach to mathematics learning and teaching. Learning and Instruction, 14(5), 445–451.

    Article  Google Scholar 

  70. Wertheimer, M. (1959). Productive thinking. New York: Harper & Row.

    Google Scholar 

  71. Westermann, K., & Rummel, N. (2012). Delaying instruction: evidence from a study in a university relearning setting. Instructional Science, 40(4), 673–689.

    Article  Google Scholar 

  72. Wiedmann, M., Leach, R. C., Rummel, N., & Wiley, J. (2012). Does group composition affect learning by invention? Instructional Science, 40(4), 711–730.

    Article  Google Scholar 

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Loibl, K., Roll, I. & Rummel, N. Towards a Theory of When and How Problem Solving Followed by Instruction Supports Learning. Educ Psychol Rev 29, 693–715 (2017). https://doi.org/10.1007/s10648-016-9379-x

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Keywords

  • Contrasting cases
  • Invention
  • Learning mechanisms
  • Problem solving
  • Productive Failure
  • Student solutions
  • Compare and contrast