# Productive failure in mathematical problem solving

## Abstract

This paper reports on a quasi-experimental study comparing a “productive failure” instructional design (Kapur in Cognition and Instruction 26(3):379–424, 2008) with a traditional “lecture and practice” instructional design for a 2-week curricular unit on rate and speed. Seventy-five, 7th-grade mathematics students from a mainstream secondary school in Singapore participated in the study. Students experienced either a traditional lecture and practice teaching cycle or a productive failure cycle, where they solved complex problems in small groups without the provision of any support or scaffolds up until a consolidation lecture by their teacher during the last lesson for the unit. Findings suggest that students from the productive failure condition produced a diversity of linked problem representations and methods for solving the problems but were ultimately unsuccessful in their efforts, be it in groups or individually. Expectedly, they reported low confidence in their solutions. Despite seemingly failing in their collective and individual problem-solving efforts, students from the productive failure condition significantly outperformed their counterparts from the lecture and practice condition on both well-structured and higher-order application problems on the post-tests. After the post-test, they also demonstrated significantly better performance in using structured-response scaffolds to solve problems on relative speed—a higher-level concept not even covered during instruction. Findings and implications of productive failure for instructional design and future research are discussed.

### Keywords

Ill-structured problems Failure in problem solving Persistence Classroom-based research Mathematical problem solving### References

- Amit, M., & Fried, M. N. (2005). Multiple representations in 8th grade algebra classrooms: Are learners really getting it? In H. L. Chick & J. L. Vincent (Eds.),
*Proceedings of the 29th conference of the international group for the psychology of mathematics education (Vol. 2, pp. 57–64)*. Melbourne: PME.Google Scholar - Anderson, J. R. (2000).
*Cognitive psychology and its implications*. New York: Worth.Google Scholar - Barron, B. (2003). When smart groups fail.
*Journal of the Learning Sciences,**12*(3), 307–359. doi:10.1207/S15327809JLS1203_1.CrossRefGoogle Scholar - Brown, A. L. (1992). Design experiments.
*Journal of the Learning Sciences,**2*(2), 141–178. doi:10.1207/s15327809jls0202_2.CrossRefGoogle Scholar - Brown, J. S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of learning.
*Educational Researcher,**18*(1), 32–42.Google Scholar - Bruner, J. S. (1985). Vygotsky: A historical and conceptual perspective. In J. V. Wertsch (Ed.),
*Culture, communication, and cognition: Vygotskian perspectives*(pp. 21–34). Cambridge: Cambridge University Press.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. doi:10.1207/s15516709cog0502_2.CrossRefGoogle Scholar - Chi, M. T. H., Glaser, R., & Farr, M. J. (1988).
*The nature of expertise*. Hillsdale: Erlbaum.Google Scholar - Cho, K. L., & Jonassen, D. H. (2002). The effects of argumentation scaffolds on argumentation and problem solving.
*Educational Technology Research and Development,**50*(3), 5–22. doi:10.1007/BF02505022.CrossRefGoogle Scholar - Clifford, M. M. (1984). Thoughts on a theory of constructive failure.
*Educational Psychologist,**19*(2), 108–120.Google Scholar - Cohen, E. G., Lotan, R. A., Abram, P. L., Scarloss, B. A., & Schultz, S. E. (2002). Can groups learn?
*Teachers College Record,**104*(6), 1045–1068. doi:10.1111/1467-9620.00196.CrossRefGoogle Scholar - Dillenbourg, P. (2002). Over-scripting CSCL: The risks of blending collaborative learning with instructional design. In P. A. Kirschner (Ed.),
*Three worlds of CSCL. Can we support CSCL*(pp. 61–91). Heerlen: Open Universiteit Nederland.Google Scholar - Dixon, J. A., & Bangert, A. S. (2004). On the spontaneous discovery of a mathematical relation during problem solving.
*Cognitive Science,**28*, 433–449.CrossRefGoogle Scholar - Even, R. (1998). Factors involved in linking representations of functions.
*The Journal of Mathematical Behavior,**17*(1), 105–121. doi:10.1016/S0732-3123(99)80063-7.CrossRefGoogle Scholar - Fishman, B., Marx, R., Blumenfeld, P., Krajcik, J. S., & Soloway, E. (2004). Creating a framework for research on systemic technology innovations.
*Journal of the Learning Sciences,**13*(1), 43–76. doi:10.1207/s15327809jls1301_3.CrossRefGoogle Scholar - Garner, W. R. (1974).
*The processing of information and structure*. Potomac: Erlbaum.Google Scholar - Ge, X., & Land, S. M. (2003). Scaffolding students’ problem-solving processes in an ill-structured task using question prompts and peer interactions.
*Educational Technology Research and Development,**51*(1), 21–38. doi:10.1007/BF02504515.CrossRefGoogle Scholar - Gibson, J. J., & Gibson, E. J. (1955). Perceptual learning: Differentiation or enrichment?
*Psychological Review,**62*, 32–41. doi:10.1037/h0048826.CrossRefGoogle Scholar - Goel, V., & Pirolli, P. (1992). The structure of design problem spaces.
*Cognitive Science,**16*, 395–429.CrossRefGoogle Scholar - Goldin, G. A. (2002). Representation in mathematical learning and problem solving. In L. D. English (Ed.),
*Handbook of international research in mathematics education*(pp. 197–218). Mahwah: Erlbaum.Google Scholar - Greeno, J. G., & Hall, R. P. (1997). Practicing representation: Learning with and about representational forms.
*Phi Delta Kappan,**78*(5), 361–367.Google Scholar - Greeno, J. G., Smith, D. R., & Moore, J. L. (1993). Transfer of situated learning. In D. K. Detterman & R. J. Sternberg (Eds.),
*Transfer on trial: Intelligence, cognition, and instruction*(pp. 99–167). Norwood: Ablex.Google Scholar - Hardiman, P. T., Dufresne, R., & Mestre, J. P. (1989). The relation between problem categorization and problem solving among experts and novices.
*Memory & Cognition,**17*(5), 627–638.Google Scholar - Hatano, G., & Inagaki, K. (1986). Two courses of expertise. In H. Stevenson, H. Azuma, & K. Hakuta (Eds.),
*Child development and education in Japan*(pp. 262–272). New York: Freeman.Google Scholar - Hmelo-Silver, C. E. (2004). Problem-based learning: What and how do students learn?
*Educational Psychology Review,**16*(3), 235–266. doi:10.1023/B:EDPR.0000034022.16470.f3.CrossRefGoogle Scholar - Janvier, C. (1987).
*Problems of representation in the teaching and learning of mathematics*. Hillsdale: Erlbaum.Google Scholar - Jonassen, D. H. (2000). Towards a design theory of problem solving.
*Educational Technology Research and Development,**48*(4), 63–85. doi:10.1007/BF02300500.CrossRefGoogle Scholar - Kapur, M. (2006). Productive failure. In S. Barab, K. Hay, & D. Hickey (Eds.),
*Proceedings of the international conference on the learning sciences*(pp. 307–313). Mahwah: Erlbaum.Google Scholar - Kapur, M. (2008). Productive failure.
*Cognition and Instruction,**26*(3), 379–424. doi:10.1080/07370000802212669.CrossRefGoogle Scholar - 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: Cognitive Science Society.Google 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 - Kaput, J. (1999). Representations, inscriptions, descriptions and learning: A kaleidoscope of windows.
*The Journal of Mathematical Behavior,**17*(2), 265–281. doi:10.1016/S0364-0213(99)80062-7.CrossRefGoogle Scholar - Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work.
*Educational Psychologist,**41*(2), 75–86. doi:10.1207/s15326985ep4102_1.CrossRefGoogle Scholar - Kohl, P. B., Rosengrant, D., & Finkelstein, N. D. (2007). Strongly and weakly directed approaches to teaching multiple representation use in physics.
*Physical Review Special Topics-Physics. Education Research,**3*, 1–10.Google Scholar - Lampert, M. (2001).
*Teaching problems and the problems of teaching*. New Haven: Yale University Press.Google Scholar - Lesh, R. R., & Doerr, H. M. (2003).
*Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching*. Mahwah: Erlbaum.Google Scholar - Lobato, J. (2003). How design experiments can inform a rethinking of transfer and vice versa.
*Educational Researcher,**32*(1), 17–20. doi:10.3102/0013189X032001017.CrossRefGoogle Scholar - Marton, F. (2007). Sameness and difference in transfer.
*Journal of the Learning Sciences,**15*(4), 499–535. doi:10.1207/s15327809jls1504_3.CrossRefGoogle Scholar - McNamara, D. S. (2001). Reading both high-coherence and low-coherence texts: Effects of text sequence and prior knowledge.
*Canadian Journal of Experimental Psychology,**55*(1), 51–62. doi:10.1037/h0087352.Google Scholar - Mestre, J. (2005).
*Transfer of learning from a modern multidisciplinary perspective*. Greenwich: Information Age.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. doi:10.1207/s15326985ep4001_1.CrossRefGoogle Scholar - Reiser, B. J. (2004). Scaffolding complex learning: The mechanisms of structuring and problematizing student work.
*Journal of the Learning Sciences,**13*(3), 423–451. doi:10.1207/s15327809jls1303_2.CrossRefGoogle Scholar - Scardamalia, M., & Bereiter, C. (2003). Knowledge building. In J. W. Guthrie (Ed.),
*Encyclopedia of education*. New York: Macmillan Reference.Google 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. doi:10.1111/j.1467-9280.1992.tb00029.x.CrossRefGoogle Scholar - Schwartz, D. L., & Bransford, J. D. (1998). A time for telling.
*Cognition and Instruction,**16*(4), 475–522. doi:10.1207/s1532690xci1604_4.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. doi:10.1207/s1532690xci2202_1.CrossRefGoogle Scholar - Spiro, R. J., Feltovich, R. P., Jacobson, M. J., & Coulson, R. L. (1992). Cognitive flexibility, constructivism, and hypertext. In T. M. Duffy & D. H. Jonassen (Eds.),
*Constructivism and the technology of instruction: A conversation*. NJ: Erlbaum.Google Scholar - Tatar, D., Roschelle, J., Knudsen, J., Shechtman, N., Kaput, J., & Hopkins, B. (2008). Scaling up innovative technology-based mathematics.
*Journal of the Learning Sciences,**17*, 248–286. doi:10.1080/10508400801986090.CrossRefGoogle Scholar - Tharp, R. G., & Gallimore, R. (1982). Inquiry processes in program development.
*Journal of Community Psychology,**10*, 103–118. doi:10.1002/1520-6629(198204)10:2<103::AID-JCOP2290100202>3.0.CO;2-9.CrossRefGoogle Scholar - 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. doi:10.1207/S1532690XCI2103_01.CrossRefGoogle Scholar - Voss, J. F. (1988). Problem solving and reasoning in ill-structured domains. In C. Antaki (Ed.),
*Analyzing everyday explanation: A casebook of methods*. London: Sage Publications.Google Scholar - Voss, J. F. (2005). Toulmin’s model and the solving of ill-structured problems.
*Argumentation,**19*, 321–329. doi:10.1007/s10503-005-4419-6.CrossRefGoogle Scholar - Vygotsky, L. S. (1978).
*Mind in society*. Cambridge: Harvard University Press.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. doi:10.1111/j.1469-7610.1976.tb00381.x.CrossRefGoogle Scholar - Zeitz, P. (1999).
*The art and craft of problem solving*. New York: John Wiley.Google Scholar