The effect of instruction on children's use of diagrams in novel problem solving

The research reported in this thesis investigated a theory of instruction that was developed to teach the use of the strategy draw a diagram in novel problem solving.Consistent with an explanatory case study design, the theory was derived from the literature and tested through the implementation of an instructional programme informed by that theory. This theory predicted that if the instructional programme addressed specific content related to diagram use and adopted a particular instructional model then the instructional programme would be effective.

Explanatory case study designs utilise how and why questions to determine support for theories and enable theories to be refined. Two research questions were posed. The first research question - How will instruction in diagram use affect children's problem solving peiformance on novel problems? - focused on the outcomes of instruction. The second research question - "Why will instruction in diagram use affect children's problem solving peifonnance on novel problems? - explored the relationship between instruction and changes in children's use of diagrams in novel problem solving.

The content of the theory prescribed essential knowledge about diagrams and the sequence of exploration of that knowledge in an instructional programme. For example, the concept of a diagram and symbolic representations were features of the knowledge component. As understanding the concept of a diagram provides the foundation for using symbolic representations, the concept of a diagram should be explored first in the instructional sequence. The content of the instructional theory is crucial to children's understanding about diagram use because children experience a range of knowledge-related difficulties in using diagrams as tools for problem solving.

The instructional model comprised six components that prescribed ways to support learning about diagram use in problem solving. These components were: (a) the learner, (b) the teacher, (c) the instructional tasks, (d) the classroom interaction, (e) participant structures, and (f) management issues. Attention to these components was predicted to support learning in two ways. First, these components focused on the creation of supportive conditions for learning, such as the importance of considering the prior experiences of the learner and the need for cognitively challenging tasks.

Second, these components addressed issues that are specifically related to learning about diagram use, such as the learner's preference for a visual method of solution and the need to present tasks according to their problem structures. Because the instructional model influences children's learning about diagrams, it is considered an essential part of the theory of instruction in diagram use. However, many aspects of the instructional model are relevant for teaching other mathematical topics.

The instructional programme consisted of twelve half-hour lessons on general purpose diagrams (networks, hierarchies, matrices, and part-whole diagrams). The goals of the instruction were for the students to: (a) employ the strategy draw a diagram, (b) generate networks, matrices, hierarchies, and part-whole diagrams, where appropriate, (c) reason appropriately with diagrams in the solution process, and ( d) use the diagram to produce a successful solution to a problem.

The researcher implemented the programme with a class of Grade 5 students. Twelve of these students were participants in a single case study to test the theory. The mean age of the participants was 10 years 3 months (range from 10 years 8 months to 9 years 8 months). To ensure a cross section of participants, three students were purposefully selected for each of four different profiles of performance and frequency of diagram use based on a novel problem solving test. The profiles were: (a) a high performance score and a high frequency score; (b) a high performance score and a low frequency score; (c) a low performance score and a low frequency score; and (d) a low performance score and a high frequency score. The classroom teacher was also a participant in the study, in that, he provided contextual data about the participants, their mathematical experience and capabilities. The student participants were interviewed individually on five novel tasks prior to, and at the conclusion of the instruction. The tasks had problem structures that could be represented with general purpose diagrams. The tasks in the pre- and post-instruction interviews were isomorphic.

The effectiveness of the instruction was ascertained by testing a series of assertions related to the goals of instruction. The criteria used to test the assertions were: (a) the frequency and autonomy of diagram use, (b) the quality of the diagram that was generated, (c) the appropriateness of the reasoning with the diagram, and (d) the success rate for tasks in which a diagram had been used. In order to compare the preand post-instruction diagrams that were generated and the associated reasoning, a series of performance levels were developed for each of the pairs of isomorphic tasks. Analysis of the data provided support for each of the assertions. Hence, the response to the question - How will instruction in diagram use affect children's problem solving performance on novel problems? - was that children achieved each of the instructional goals associated with the successful use of the diagram in problem solving. Explanations for children's learning are provided by the theory of instruction. Thus, the response to the question - 'Why will instruction in diagram use affect children's problem solving performance on novel problems? - was that the instructional programme addressed essential content and provided appropriate conditions for learning about diagram use.

In testing the theory, some unexpected results emerged necessitating the refinement of the preliminary theory of instruction. One novel technique for investigating these results was to represent the interview data visually on data maps, which provided an overview of the interviews, and facilitated the wholistic analysis of data. One of the refinements to the content was the need to teach students about tracking strategies. Some students had difficulty locating their position after moving about on a diagram. Refinements were also made to the instructional model. For example, the learners' use of a diagram was influenced by their beliefs about the advantages of using a diagram and by their level of confidence with diagrams. The refined theory provides the basis for future instructional programmes.

The problem solving strategy draw a diagram is advocated in many curriculum documents. However, the successful use of diagrams may not occur spontaneously. The conclusion of this study is that instruction can improve children's use of diagrams in novel problem solving by developing the appropriate knowledge of diagrams as a problem solving tool. The literature base on instruction in diagram use has been limited. This study contributes to the field by providing a theoretical framework to inform effective instructional programming and curriculum development.

The major implication for teachers and curriculum developers is that the mathematics curriculum should include specific instruction about the use of diagrams in problem solving. As teachers may be ill-prepared to provide instruction in diagram use, appropriate curriculum guidance is necessary. To ensure that students become mathematically literate citizens, instruction in diagram use in problem solving should be included in the reform agenda for mathematics education for the 21st century. Furthermore, effective use of diagrams in problem solving involves visual literacy or graphicacy. Hence, the scope of literacy in the classroom needs to extend beyond numeracy, oracy, and written literacy to include literacy with various forms of visual representation, which includes diagrams.