Exploring teachers' mathematical meanings for teaching trigonometric ratios and functions to grade 10 learners in the Tshwane South District of Gauteng Province

Abstract

This study aimed to explore the mathematical meanings possibly entertained by Grade 10 teachers on the particular trigonometric topics of ratios and functions. The quest to embark on this exploration was primarily triggered by the desire to enhance our understanding of and provide plausible explanations for learners’ observed responses in relation to trigonometric problem-solving tasks. The study began by first addressing the key and contested conceptual polarisation of teacher knowledge and teacher knowing, which we consider carrying explicative power in allowing us to unpack the notion of mathematical meanings. Our research suggests that previous and accurate knowledge of these concepts is eminently important in enabling researchers to map out the exploration and study of teachers’ so-called “mathematical meanings” concerning the teaching of trigonometry. Mason and Spence (1999, cited in Thompson, 2015) argue that “knowing is more useful for thinking about teaching and learning”. In this study, this would constitute an act of knowing (by the teacher) what to teach, and how to teach. Hence the researcher views knowing as connoting the instructional activities of the knower, while knowledge connotes the acquired facts of the knower. Furthermore, the notion of mathematical meaning is conceived in our study as a reference to the conceptual images associated with mathematical constructs that eventually inform their instruction, the learning activities as chosen by classroom instructors (teachers), and how these intuitive images come to be communicated to and eventually received by learners. To conduct this conceptual exploration systematically, the researcher followed a case study research approach that embraced elements of exploratory research design. This research design took inspiration from the qualitative research paradigm and was adopted to facilitate the researcher’s intention to work with a small number of teachers to whom a trigonometric task was assigned, and who were later selected to participate in semi-structured interviews. Using convenience sampling procedures, the current study sampled 12 Grade 10 teachers who taught mathematics in the Tshwane South District in the Gauteng Province of South Africa. All teacher participants were given a trigonometry task to reflect on their conceptualisation and understanding of its mathematical content, which we considered to be an important, initial step in the effort to access these experts’ implicit mathematical meanings. The assigned trigonometric task was drawn and adapted from Thompson’s Mathematical Meanings for Teaching secondary mathematics (MMTsm) (Thompson, 2015), which proposes useful and authentic constructs to study respondents’ mathematical meanings. The analysis of teachers’ responses to the task led to a purposive sampling of two teachers to take part in subsequent semi-structured interviews. The purpose of the interviews was to, (1) probe teachers further on their responses; (2) verify our understanding and interpretation of teachers’ responses; and, (3) attempt to understand the argument that teachers’ responses provided direct insight into their entertained mathematical meanings toward particular mathematics topics. The observations that emerged from this analysis seem to indicate that teachers will often entertain differing mathematical meanings toward a single mathematical topic, some of which meanings are neither coherent nor productive.