Making Links Between Solutions to an Unstructured Problem: The Role of Pre-Written, Designed Student Responses

Introduction

When solving an unstructured problem, novice problem solvers are not readily reminded of similarly structured problems from the past and can find it challenging to re-cycle ‘old’ knowledge in new situations (Gick & Holyoak, 1983). They often set vague, unstructured goals or their goals are flawed (Juwah et al., 2004). Instead, they rely on naïve, inefficient strategies such as ‘trial and improvement’ (Evans & Swan, 2014). These strategies they pursue relentlessly, without pausing to review their validity or consider alternative strategies (Schoenfeld, 1992). Failure to effectively monitor their emerging solution may be because they are uncertain of the criteria to judge the quality of their work (Bell, Philips, Shannon, & Swan, 1997), other than checking the correctness of the answer. Or when monitoring, they simply cannot think of a replacement strategy. On the other hand, expert problem solvers recognize structural similarities between a new problem-situation and past problems and are able to retrieve relevant knowledge to construct effective strategies. They spend time setting hierarchical goals and as their solution unfolds they carefully monitor and regulate their progress against their goals (Schunk & Zimmerman, 2006).

This contrasting behavior can be for a number of reasons. It is widely recognized that insufficient original learning of mathematical concepts can prevent students transferring knowledge to new situations (e.g. Lee & Pennington, 1993). In particular, when solving an unstructured problem, lack of knowledge of how a concept can be represented in different ways (Rittle-Johnson & Star, 2007) together with inadequate metacognitive strategies (Schoenfeld, 1989) can restrict students’ capacity to flexibly and usefully adapt their original approach. The research indicates linking different solutions to the same problem can broaden students’ understanding of a specific problem-situation, and more generally develop their conceptual understandings and metacognitive strategies (Brousseau, 1997; Chazan & Ball, 1999; Lampert, 2001; Stein, Eagle, Smith, & Hughes, 2008). As Kaput (1989, pp. 179–180) proposed: “the cognitive linking of representations creates a whole that is more than the sum of its parts...it enables us to see complex ideas in a new way and apply them more effectively.” However, while there is much research into students’ working with multiple solutions to a problem (Thompson, 1994), there are only a few studies that explore the process of students’ linking these solutions (Wilmot, Schoenfeld, Wilson, Champney, & Zahner, 2011).

This chapter forms part of a design research study in which resources are specifically created to develop students’ capacity to solve unstructured problems. Central to this design intention is the provision of opportunities for students to link different solutions to a problem. As an illustrative focus for the chapter, data from one lesson is explored. The chapter begins by considering the research that shaped both the resources and the analytical framework. Then, to explicate the design researcher’s intentions, the resources are described in detailed. This is followed by an analysis of the data. The analysis is framed by questions such as ‘When do students make links between different solutions, ‘Why do students make links?’ and ‘What properties do students link?’. The findings are then discussed. Finally, the author reflects on how the materials could be refined to provide further opportunities for both students’ link-making and robust investigations into their link-making.