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.
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Notes
All inter-rater reliabilities reported in this paper are Krippendorff’s alphas.
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Appendices
Appendix A: Student engagement survey
For the statements below, please use the five-point agreement scale and circle the number that most closely matches your opinion.
1. I enjoyed the lesson very much | 5 (SA) | 4 (A) | 3 (N) | 2 (D) | 1 (SD) |
2. I felt I was engaged in the lesson | 5 (SA) | 4 (A) | 3 (N) | 2 (D) | 1 (SD) |
3. I was attentive during the lesson | 5 (SA) | 4 (A) | 3 (N) | 2 (D) | 1 (SD) |
4. I participated in the lesson’s activities | 5 (SA) | 4 (A) | 3 (N) | 2 (D) | 1 (SD) |
5. I learned a lot in the lesson | 5 (SA) | 4 (A) | 3 (N) | 2 (D) | 1 (SD) |
Appendix B: Representational flexibility items
For examples of the well-structured and high-order application items, see Kapur (2009).
Tabular representation item: The property market has been on the rise for the past few years. In the newspaper, you find the following table with the growth rate over the past 5 years.
Year | % Growth |
|---|---|
2003 | 2 |
2004 | 7 |
2005 | 11 |
2006 | 14 |
2007 | 16 |
Some people are saying that the property market is growing. Other are saying that it is slowing down. Based on the table above, what do you think—is the property market growing or slowing down? Explain your answer with calculations.
Graphical representation item: Bob drove 140 miles in 2 h and then drove 150 miles in the next 3 h. Study the two speed-time graphs A and B carefully. Which graph—A, B, or both—can represent Bob’s journey? Show your working and explain your answer.
Note: This item was adapted from Stanford Research International’s (SRI) research program on SimCalc and the Math of Change.
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Kapur, M. A further study of productive failure in mathematical problem solving: unpacking the design components. Instr Sci 39, 561–579 (2011). https://doi.org/10.1007/s11251-010-9144-3
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DOI: https://doi.org/10.1007/s11251-010-9144-3