Visualisation processes in teaching patterns and their generalisation: perceptions and experiences from senior phase mathematics teachers

Abstract

This study investigated how senior phase mathematics teachers used visualisation approaches to teach patterns and their generalisation as a result of an intervention programme. This project is an integral component of the VIPRO MATHS Project and one of a number of post-graduate research projects which has a particular focus on visualisation processes in mathematics education in the Southern African region. This case study of Senior Phase mathematics teachers in the Chris Hani West District (CHW) in the Eastern Cape Province of South Africa used a social constructivist theory. The study is grounded within an interpretive paradigm explanatory qualitative method design. It surveyed thirty-seven Senior Phase mathematics teachers from schools in the CHW district to get an idea of how teachers teach patterns and their generalisation, prior to an intervention process. After the survey, nine teachers were purposively selected to take part in an intervention programme. They were then observed (teaching) and interviewed to get their perceptions about what factors enable and/or constrain the teaching of patterns and their generalisation when visualisation approaches were used. The data from the survey questionnaire responses, the lesson observations and the interviews were analysed qualitatively. The findings from the survey questionnaire responses revealed that teachers in CHW district value the use of visualisation approaches, although they used visuals to teach patterns in a way that lacked depth, skill and rigor prior to the intervention. Challenges such as difficulty in keeping abreast with the prescribed syllabus, the lack of resources, time constraints, and the limited knowledge of algebra learners have for successfully transferring pictorial representations to symbolic presentations, emerged from the survey data. This necessitated the need for an intervention which focused on the use of a variety of visual strategies and tools to teach patterns and their generalisation effectively. Data from lesson observations showed that all observed participant teachers used visuals to generate image- related questions and created platforms for lively classroom discussions. The teachers used the generated questions to build on learners' prior knowledge and to promote learner understanding of pattern dynamics that stem from pictorial representations. The data from the interviews revealed that teachers' perceptions had drastically changed after the intervention. This was because they were now equipped and empowered to effectively make use of visual strategies that enabled them to: build on learners' prior knowledge, to manipulate visuals for the purpose of prompting learning, develop cognitive perception in learners, cultivate learner innovation and originality and enhance problem solving techniques. This study concludes that the use of visuals enhanced conceptual teaching of patterns and their generalisation in the observed participants. This has the potential to provide mathematics education researchers and curriculum developers with a strong basis to include visualisation processes and strategies into the design of policy documents and to initiate further research efforts. The implications for the teachers are that learners need to be taught ways to create and interpret visual representations in order to understand the dynamics of transferring a pictorial pattern to its resultant symbolic representation expressing pattern regularity. It is also hoped that the results of this study will be utilised by mathematics subject advisers and teacher training institutions to improve the teaching of patterns and their generalisation.