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The effectiveness of a diagram in problem solving is dependent on its utility as a cognitive tool. In order to develop students ’ ability to use diagrams as cognitive tools, teachers need to assess the quality of students ’ diagrams and provide them with the necessary support. However assessing the quality of diagrams is problematic. This paper discusses how theoretical prototypes and exemplars of level of performance provide a practical and effective avenue for assessing the quality of students ’ diagrams. Mathematics educators strongly advocate the use of the strategy draw a diagram for mathematical problem solving (e.g., National Council of Teachers of Mathematics [NCTM], 1989). This perspective is strongly grounded in the belief that generating a diagram facilitates the conceptualisation of the problem structure (van Essen & Hamaker, 1990). Although the use of a diagram as a tool of mathematics can empower primary students to deal with novelty (NCTM, 1989), effective problem solving depends on the quality of students ’ diagrams (Yancey, Thompson, & Yancey, 1989). Thus, the generation of high quality diagrams should be a goal in any instructional programme on diagram use. The salient question that emerges, which is the focus of this paper, is how to assess the quality of students ’ diagrams. THE USE OF THE DIAGRAM IN PROBLEM SOLVING The advantages of generating a diagram are related its utility as a cognitive tool (e.g.; Larkin & Simon, 1987; van Essen & Hamaker, 1990). For example, diagrams act as an external sketch pad where interconnected pieces of information can be chunked together in a holistic manner (van Essen & Hamaker, 1990). Thus, implicit information within a problem may become explicit to the solver on a diagram (Larkin & Simon, 1987). However not all diagrams have the potential to be cognitive tools. For example, some students generate diagrams that focus on the surface (literal) features of the problem at the expense of representing the problem structure (e.g., Dufoir-Janvier, Bednarz &

Year: 2008

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