Designing for productive failure in mathematical problem solving

Designing for Productive Failure in Mathematical Problem Solving Manu Kapur & June Lee National Institute of Education, Singapore solving problems without the provision of support structures. While this belief may well be grounded in empirical evidence, it is also possible that by engaging novices to persist and even fail at tasks that are beyond their skills and abilities can be a productive exercise in failure. Research reported in this paper explores this very possibility of designing for productive failure (Kapur, 2008). Abstract This paper describes two quasi-experimental studies of productive failure (Kapur, 2008) in Singapore public schools for the curricular unit on average speed. In the first study, seventh-grade mathematics students from intact classes experienced one of two conditions: a) productive failure, where students solved complex, ill-structured problems on average speed without any instructional support or scaffolds up until a teacher-led consolidation, or b) traditional lecture and practice. Despite seemingly failing in their collective and individual problem-solving efforts, students from the productive failure condition significantly outperformed their counterparts from the other two conditions on both the well- structured and higher-order application problems on the post- test. The second study, conducted in another school with significantly lower academic ability students, replicated the findings of the first study. Findings and implications of productive failure for theory, design of learning, and future research are discussed. Failure and Structure Introduction When and how to design structure in learning and problem- solving activities is a fundamental theoretical and design issue in education and the learning sciences. Structure can be operationalized in a variety of forms such as structuring the problem itself, scaffolding, instructional facilitation, provision of tools, content support, expert help, and so on (Puntambekar & Hubscher, 2005). Thus conceived, structure is designed to constrain or reduce the degrees of freedom in learning and problem solving activities; the lower the degree of freedom, the greater the structure (Woods, Bruner, & Ross, 1976). By doing so, structure increases the likelihood of novices achieving performance success during problem solving, which they might not otherwise be able to in the absence of support structures. Indeed, a vast body of research supports the efficacy of such an approach. This has led some to argue that instruction should be heavily guided especially at the start, for without it, little if any learning will take place (e.g., Kirschner, Sweller, & Clark, 2006). Further support for starting with greater structure in learning and problem solving activities with a gradual reduction (or fading) over time as learners gain expertise comes from several quarters (e.g., Puntambekar & Hubscher, 2005; Vygotsky, 1978; Woods et al., 1976). More often than not therefore, researchers have tended to focus on ways of structuring learning and problem-solving activities so as to achieve performance success, whereas the role of failure in learning and problem solving remains largely underdetermined and under-researched by comparison (Clifford, 1984; Schmidt & Bjork, 1992). What is perhaps more problematic is that an emphasis on achieving performance success has in turn led to a commonly-held belief that there is little efficacy in novices The role of failure in learning and problem solving is no doubt intuitively compelling. For example, research on impasse-driven learning (VanLehn et al., 2003) with college students in coached problem-solving situations provides strong evidence for the role of failure in learning. Successful learning of a principle (e.g., a concept, a Physical law) was associated with events when students reached an impasse during problem solving. Conversely, when students did not reach an impasse, learning was rare despite explicit tutor- explanations of the target principle. Instead of providing immediate structure, e.g., in the form of feedback, questions, or explanations, when the learner demonstrably makes an error or is “stuck,” VanLehn et al’s (2003) findings suggest that it may well be more productive to delay that structure up until the student reaches an impasse—a form of failure—and is subsequently unable to generate an adequate way forward. Echoing this delaying of structure in the context of text comprehension (also with college students), McNamara (2001) found that whereas low-knowledge undergraduate learners tended to benefit from high-coherence texts, high-knowledge undergraduate learners benefited from low-coherence texts, and especially more so when a low-coherence text preceded a high- coherence one. This, McNamara argues, suggests that reading low-coherence texts may force learners to engage in compensatory processing by using their prior knowledge to fill in the conceptual gaps in the target text, in turn, preparing them better to leverage a high-coherence text subsequently. Further evidence for such preparation for future learning (Schwartz & Bransford, 1998) can be found in the inventing to prepare for learning research by Schwartz and Martin (2004). In a sequence of design experiments on the teaching of descriptive statistics with intellectually gifted students, Schwartz and Martin (2004) demonstrated an existence proof for the hidden efficacy of invention activities when such activities preceded direct instruction, despite such activities failing to produce canonical conceptions and solutions during the invention phase. Clearly, the relationship between failure and structure forms a common thread through the abovementioned studies. It is reasonable to reinterpret their central findings collectively as an argument for a delay of structure in learning and problem-solving situations, be it in the form of