# A further study of productive failure in mathematical problem solving: unpacking the design components

- 1.2k Downloads
- 59 Citations

## Abstract

This paper replicates and extends my earlier work on productive failure in mathematical problem solving (Kapur, doi: 10.1007/s11251-009-9093-x, 2009). One hundred and nine, seventh-grade mathematics students taught by the same teacher from a Singapore school experienced one of three learning designs: (a) traditional lecture and practice (LP), (b) productive failure (PF), where they solved complex problems in small groups without any instructional facilitation up until a teacher-led consolidation, or (c) facilitated complex problem solving (FCPS), which was the same as the PF condition except that students received instructional facilitation throughout their lessons. Despite seemingly failing in their collective and individual problem-solving efforts, PF students significantly outperformed their counterparts in the other two conditions on both the well-structured and higher-order application problems on the post-test, and demonstrated greater representation flexibility in working with graphical representations. The differences between the FCPS and LP conditions did not reach significance. Findings and implications of productive failure for theory, design of learning, and future research are discussed.

## Keywords

Failure Complex problems Mathematical problem solving Persistence Multiple representations## References

- Bielaczyc, K., & Kapur, M. (in press). Playing epistemic games in science and mathematics classrooms.
*Educational Technology*.Google Scholar - Brown, A. L. (1992). Design experiments.
*The Journal of the Learning Sciences,**2*(2), 141–178.CrossRefGoogle Scholar - Bruner, J. (1986).
*Actual minds, possible worlds*. Cambridge, MA: Harvard University Press.Google Scholar - Carey, S. (1999). Sources of conceptual change. In E. K. Scholnick, K. Nelson, & P. Miller (Eds.),
*Conceptual development: Piaget’s legacy*(pp. 293–326). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar - Chi, M. T. H., Feltovich, P. J., & Glaser, R. (1981). Categorization and representation of physics problems by experts and novices.
*Cognitive Science,**5*, 121–152.CrossRefGoogle Scholar - Chi, M. T. H., Glaser, R., & Farr, M. J. (1988).
*The nature of expertise*. Hillsdale, NJ: Erlbaum.Google Scholar - Clifford, M. M. (1984). Thoughts on a theory of constructive failure.
*Educational Psychologist,**19*(2), 108–120.CrossRefGoogle Scholar - Cobb, P., Wood, T., & Yackel, E. (1993). Discourse, mathematical thinking and classroom practice. In E. Forman, N. Minick, & C. Stone (Eds.),
*Contexts for learning: Sociocultural dynamics in children’s development*(pp. 91–119). New York: Oxford University Press.Google Scholar - diSessa, A. A. (2006). A history of conceptual change research: Threads and fault lines. In K. Sawyer (Ed.),
*Cambridge handbook of the learning sciences*. Cambridge, UK: Cambridge University Press.Google Scholar - diSessa, A. A., Hammer, D., Sherin, B., & Kolpakowski, T. (1991). Inventing graphing: meta-representational expertise in children.
*Journal of Mathematical Behavior,**10*(2), 117–160.Google Scholar - Goldin, G. A. (2008). Perspectives on representation in mathematical learning and problem solving. In L. D. English (Ed.),
*Handbook of international research in mathematics education*(pp. 176–201). New York, NY: Taylor & Francis.Google Scholar - Janvier, C. (1987).
*Problems of representation in the teaching and learning of mathematics*. Hillsdale, NJ: Erlbaum.Google Scholar - Kapur, M. (2008). Productive failure.
*Cognition and Instruction,**26*(3), 379–424.CrossRefGoogle Scholar - Kapur, M. (2009). Productive failure in mathematical problem solving.
*Instructional Science*. doi: 10.1007/s11251-009-9093-x. - Kapur, M., Dickson, L., & Toh, P. Y. (2008). Productive failure in mathematical problem solving. In B. C. Love, K. McRae, & V. M. Sloutsky (Eds.),
*Proceedings of the 30th annual conference of the cognitive science society*(pp. 1717–1722). Austin, TX: Cognitive Science Society.Google Scholar - Kapur, M., Hung, D., Jacobson, M., Voiklis, J., Kinzer, C., & Chen, D.-T. (2007). Emergence of learning in computer-supported, large-scale collective dynamics: A research agenda. In C. A. Clark, G. Erkens, & S. Puntambekar (Eds.),
*Proceedings of the international conference of computer-supported collaborative learning*(pp. 323–332). Mahwah, NJ: Erlbaum.Google Scholar - Kapur, M., & Kinzer, C. (2007). The effect of problem type on interactional activity, inequity, and group performance in a synchronous computer-supported collaborative environment.
*Educational Technology, Research and Development, 55*(5), 439–459.CrossRefGoogle Scholar - Kapur, M., & Kinzer, C. (2009). Productive failure in CSCL groups.
*International Journal of Computer-Supported Collaborative Learning (ijCSCL),**4*(1), 21–46.CrossRefGoogle Scholar - Kapur, M., & Lee, J. (2009). Designing for productive failure in mathematical problem solving. In N. Taatgen & V. R. Hedderick (Eds.),
*Proceedings of the 31st annual conference of the cognitive science society*(pp. 2632–2637). Austin, TX: Cognitive Science Society.Google Scholar - Kapur, M., & Lee, J. (2010). Productive failure in learning the concept of variance. In R. Catrambone & S. Ohlsson (Eds.),
*Proceedings of the 32nd annual conference of the cognitive science society*. Austin, TX: Cognitive Science Society (in press).Google Scholar - Kapur, M., & Rummel, N. (2009). The assistance dilemma in CSCL. In
*Proceedings of the computer-supported collaborative learning conference*. Rhodes, GreeceGoogle Scholar - Kapur, M., Voiklis, J., Kinzer, C., & Black, J. (2006). Insights into the emergence of convergence in group discussions. In S. Barab, K. Hay, & D. Hickey (Eds.),
*Proceedings of the international conference on the learning sciences*(pp. 300–306). Mahwah, NJ: Erlbaum.Google Scholar - Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work.
*Educational Psychologist,**41*(2), 75–86.CrossRefGoogle Scholar - Koedinger, K. R., & Aleven, V. (2007). Exploring the assistance dilemma in experiments with cognitive tutors.
*Educational Psychological Review,**19*(3), 239–264.CrossRefGoogle Scholar - Lesh, R. R., & Doerr, H. M. (2003).
*Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching*. Mahwah, NJ: Erlbaum.Google Scholar - Pea, R. D. (2004). The social and technological dimensions of scaffolding and related theoretical concepts of learning, education, and human activity.
*Journal of the Learning Sciences,**13*(3), 423–451.CrossRefGoogle Scholar - Piaget, J. (1963).
*The psychology of intelligence*. New York: Routledge.Google Scholar - Puntambekar, S., & Hübscher, R. (2005). Tools for scaffolding students in a complex learning environment: What have we gained and what have we missed?
*Educational Psychologist,**40*(1), 1–12.CrossRefGoogle Scholar - Schmidt, R. A., & Bjork, R. A. (1992). New conceptualizations of practice: Common principles in three paradigms suggest new concepts for training.
*Psychological Science,**3*(4), 207–217.CrossRefGoogle Scholar - Schoenfeld, A. H. (2008). Research methods in mathematics education. In L. D. English (Ed.),
*Handbook of international research in mathematics education*(pp. 467–519). New York, NY: Taylor & Francis.Google Scholar - Schwartz, D. L., & Bransford, J. D. (1998). A time for telling.
*Cognition and Instruction,**16*(4), 475–522.CrossRefGoogle Scholar - 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 - Sweller, J. (2010). What human cognitive architecture tells us about constructivism. In S. Tobias & T. M. Duffy (Eds.),
*Constructivist instruction: Success or failure*(pp. 127–143). New York, NJ: Routledge.Google Scholar - Tobias, S., & Duffy, T. M. (2010).
*Constructivist instruction: Success or failure*. New York, NJ: Routledge.Google Scholar - Van Lehn, 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.CrossRefGoogle Scholar - Wadsworth, B. J. (1996).
*Piaget’s theory of cognitive and affective development*. White Plains, NY: Longman.Google Scholar - Wood, D., Bruner, J. S., & Ross, G. (1976). The role of tutoring in problem solving.
*Journal of Child Psychology and Psychiatry and Allied Disciplines,**17*, 89–100.CrossRefGoogle Scholar