Implications for teaching and learning design

Many instructional designs make either implicit or explicit commitments to a performance success focus (Clifford, 1979, 1984; Schmidt & Bjork, 1992). A focus on achieving performance success, therefore, clearly necessitates the provision of relevant support structures and scaffolds during problem solving. In designing for productive failure, the focus was more on students persisting in problem solving than on actually being able to solve the problem successfully. In contrast to a focus on achieving performance success, a focus on persistence does not necessitate a provision of support structures as long as the design of the problem allows students to make some inroads into exploring the problem and solution spaces without necessarily solving the problem successfully. An important implication for the design of problems and problem solving activities is that there is efficacy in persistence itself even though it may not lead to success in performance.

However, this only begs the question: How does one design for persistence? In productive failure, designing for persistence minimally involved five interconnected principles:

a. Designing complex, ill-structured tasks. Two ill-structured problems were designed such that they possessed many problem parameters with varying degrees of specificity and relevance, as can be seen in the problem scenario in Appendix A. Some of the parameters interacted with each other such that their effect could not be examined in isolation. As a result, the ill-structured problem scenarios were complex, possessed multiple solution paths leading to multiple solutions (as opposed to a single correct answer), and often required students to make and justify

Learning through Productive Failure in Mathematical Problem Solving 59

assumptions (Jonassen, 2000; Spiro et al., 1992; Voss, 1988). In contrast, well-structured problems commonly found in textbooks afford normative representations and methods for solving them, which often resulting in a single correct answer. In such cases, either a student is able to solve the problem quickly or simply gives up. Hence, well-structured problems often do not afford opportunities for students to persist in problem solving.

b. Designing collaborative activities. The activity structure of collaboration helps students persist in solving problems more than what they may do individually. Hence, the choice of having students engage in collaborative problem solving was critical towards maximizing the likelihood of persistence in problem solving.

c. Setting expectations for persistence. It is important that teachers set appropriate expectations to assure students that it is okay not to be able to solve the ill-structured problems as long as they try various ways of solving them, especially highlighting to them the fact that there were multiple solutions to the problems. This setting of expectation is important because the usual norm in most classrooms (though not all) is not one of persistence. Instead, it is getting to the correct answer, of which there is only one, in the most efficient manner. Therefore, designing for persistence requires substantial and constant effort on the part of the teacher to set the appropriate expectations throughout the series of lessons.

d. Withholding assistance. It is also important for teachers to get comfortable with the idea of withholding assistance or help when students ask for it, and instead get students to try working through the problem themselves first. Students are used to asking their teachers for help so much so that they do so even before trying to figure out an answer by themselves, be it individually or in groups. At the same time, teachers are just as used to offering help and assistance when it is asked for so much so that sometimes opportunities for students to persist in solving the problem are missed; opportunities that are critical for realizing productive failure. In many ways, the first three principles of

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designing ill-structured problems, collaboration, and setting appropriate expectations may come to naught if teachers do not withhold assistance during initial problem solving.

e. Iterative design. Finally, it is important to note that designing for persistence is not a one-off design effort. Usually, one does not get it right the first time around. Decisions around the above design principles are not made in isolation but as part of an iterative design process that involves other teachers and students so that the complexity of the ill- structured problem scenarios can be developmentally calibrated with the age, grade, and ability level of the students. Before classroom implementation, multiple pilot tests with two to three groups of students are used to provide insights into and help fine-tune the design decisions described above. Classroom implementation provides additional insights that lead to further iterations and fine tuning of the design.

The abovementioned five principles are but one set of principles for designing for persistence. They are surely not the only way of doing so. Needless to say, an emphasis on persistence comes with its own set of problems because of students have varying levels of persistence, not all students persist in problem-solving, the nature of their persistence varies, and relationship between the extent to which students persist and the nature of their persistence relates to learning remains an open and important question for future research.

6 Conclusion

In the classrooms that I have been working in, the conventional bias has typically been towards heavy structuring of instructional activities right from the start. The basic argument being - why waste time letting learners make mistakes when you could give them the correct understandings? This arguably makes for an efficient process but what productive failure suggests is that processes that may seem to be inefficient and divergent in the short term potentially have a hidden efficacy about them provided one could extract that efficacy. The

Learning through Productive Failure in Mathematical Problem Solving 61

implication being that by not overly structuring the early learning and problem solving experiences of learners and leaving them to persist and possibly fail can be a productive exercise in failure. I contend that the work described in this chapter opens up an exciting line of inquiry into the hidden efficacies in ill-structured, problem-solving activities. Perhaps one should resist the near-default rush to structure learning and problem- solving activities for it may well be more fruitful to first investigate conditions under which instructional designs lead to productive failure as opposed to just failure.

Acknowledgements

The research reported in this paper was funded by grants 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 particularly indebted to Leigh Dickson who was instrumental in coordinating the logistics and data collection efforts. I am also grateful to Professors Beaumie Kim, David Hung, Kate Anderson, Katerine Bielaczyc, Liam Rourke, Michael Jacobson, Sarah Davis, and Steven Zuiker for their insightful comments and suggestions on this manuscript. This chapter summarized findings from two studies on productive failure; fuller manuscripts of the first study have already been published elsewhere (Kapur et al., 2008; Kapur, in press), whereas the fuller manuscript of the second study is currently under review (Kapur, under review).

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Appendix A