# Productive failure in mathematical problem solving

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## 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## Notes

### Acknowledgments

The research reported in this paper was funded by a grant from the Learning Sciences Lab of the National Institute of Education of Singapore. I would like to thank the students, teachers, the head of the department of mathematics, and the principal of the participating school for their support for this project. I am also grateful to Professors David Hung, Katerine Bielaczyc, Katherine Anderson, Liam Rourke, Michael Jacobson, and anonymous reviewers for their insightful comments and suggestions on earlier versions of this manuscript. Parts of this manuscript have also been presented at the 2008 meeting of the Cognitive Science Society.

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