There are two purposes to this study. The first was for me, as a teacher, to try something new in my instruction and grow from it. The second purpose of this study focused on the students. I wanted to see what level of performance in problem solving my students are at currently, and how the use of journaling and discourse affected the students' problem solving abilities. A problem-solving unit was taught heuristically in order to introduce students to the various strategies that could be used in problem solving. Math journals were also used for problem solving and reflection. Classroom discourse in discussion of problem solving situations was used as a means of identifying strategies used to solve the problem. Explanations and justifications were then used in writing and discourse to support students' solution and methods. An analytic problem-solving rubric was used to score the problems solved by the students. These scores, along with explanations and justifications, and discourse were used as data and analyzed for common themes. The results of this study demonstrate overall improvement in student performance in problem solving. Heuristic instruction the students received on strategies in problem solving helped to improve their ability to not only select an appropriate strategy, but also implement it. This unit, along with the problem solving prompts solved in the journals, helped to improve the students' performance in explanations. It was discourse combined with all the previous instruction that finally improved student performance in justification.

The aim of this study was to explore the effects of a problem-solving heuristic instructional method on Grade 6 learners’ achievements in algebra. Two main theories inspired the design of this teaching method, namely the modelling and modelling perspective, and action, process, object, schema (APOS) theory. Modelling and modelling perspectives guided the development of modelling-eliciting activities used in the teaching method and the APOS theory guided the sequence of activities used to develop Grade 6 learners’ conceptions in algebra. The impact of the problem-solving heuristic instructional method was investigated with 198 Grade 6 learners from four different primary schools in the Zululand district of Kwazulu-Natal that were conveniently sampled. A mixed-method approach was used in this study and a hypothesis was formulated to investigate the effects of the teaching method on the learners’ achievements in algebra. The qualitative component consisted of a pre-intervention class observation of mathematics lessons of all four mathematics educators in the schools used for this study. The design and implementation of the problem-solving heuristic instructional method and the quantitative component employed non-equivalent control group design with pre-test and post-test measure. The main instruments for data collection were an observation schedule to document sequence of events in the classroom during the class observation, a standardized achievement test in algebra used to measure effects of the problem-solving heuristic instructional method and modelling-eliciting activities used as a medium of interaction between learners and the researcher during the implementation of the problem-solving heuristic instructional method. Findings from the class observation indicated all four schools made use of comparable traditional methods of instruction. The implementation of the problem-solving instructional method gave insights into how a problem-solving heuristic instructional method can be developed and used in Grade 6 algebra lessons, and the factors that could influence learners’ conceptual development in algebra. The findings from the quantitative component supported the initial hypothesis that improved scores in algebra are achieved through participation in the problem-solving heuristic instructional method. Quantitative data was analysed using the t-test, analysis of covariance, Johnson-Neyman (J-N) technique and the effect size. / Mathematics Education / D. Phil. (Mathematics Education)