Learning mathematical symbolization: conceptual challenges and instructional strategies in secondary schools

This paper investigates South African 12th Grade students’ conceptual challenges with mathematical symbolization and instructional strategies that teachers use to mitigate mathematical symbolization. The study is motivated by the students’ failure to connect representations between symbolic and mathematical ideas to understand concepts and procedures. The study attempts to gain insight into mathematical symbols as potential barriers to students’ understanding of mathematical concepts and processes. The study consists of 120 randomly selected 12th Grade students and 15 purposefully selected mathematics teachers from Sekhukhune district of Limpopo Province, South Africa. Data was collected through questionnaires and focus group interviews. A mixed-method sequential explanatory design was employed. An SPSS cluster analysis of data produced three (3) clusters consisting of 50 (41.6%), 47 (39.3%) and 23 (19.1%) students with severe, mild, and minor challenges with mathematical symbols. Two themes emerged from the students’ difficulties with mathematical symbols. Firstly, students lack symbol sense for mathematical concepts and algebraic insight for problem-solving. Secondly, students disregard conceptual and contextual uses of symbols. The study therefore suggests that students’ negotiation of discourse between the mathematical symbol and the mathematical concept or procedure is crucial developing symbolic meaning. Therefore, teachers need to use appropriate strategies to engage students in processes that allow them to make meanings of mathematical symbols. The study recommends that concepts should be understood before symbolised

Mathematical symbolisation: challenges and instructional strategies for Limpopo Province secondary school learners

This study reports on an investigation into the manner in which mathematical symbols influence learners’ understanding of mathematical concepts. The study was conducted in Greater Sekhukhune and Capricorn districts of Limpopo Province, South Africa. Multistage sampling (for the district), simple random sampling (for the schools), purposive sampling (for the teachers) and stratified random sampling with proportional allocation (for the learners) were used. The study was conducted in six schools randomly selected from rural, semi-urban and urban settings. A sample of 565 FET learners and 15 FET band mathematics teachers participated in the study. This study is guided by four interrelated constructivist theories: symbol sense, algebraic insight, APOS and procept theories. The research instruments for the study consist of questionnaires and interviews. A mixed method approach that was predominantly qualitative was employed. An analysis of learners’ difficulties with mathematical symbols produced three (3) clusters. The main cluster consists of 236 (41.6%) learners who indicate that they experience severe challenges with mathematical symbols compared to 108 (19.1%) learners who indicated that they could confidently handle and manipulate mathematical symbols with understanding. Six (6) categories of challenges with mathematical symbols emerged from learners’ encounters with mathematical symbols: reading mathematical text and symbols, prior knowledge, time allocated for mathematical classes and activities, lack of symbol sense and problem contexts and pedagogical approaches to mathematical symbolisation. Two sets of theme classes related to learners’ difficulties with mathematical symbols and instructional strategies emerged. Learners lack symbol sense for mathematical concepts and algebraic insight for problem solving. Learners stick to procedurally driven symbols at the expense of conceptual and contextual understanding. From a pedagogical perspective teachers indicated that they face the following difficulties when teaching: the challenge of introducing unfamiliar notation in a new topic; reading, writing and verbalising symbols; signifier and signified connections; and teaching both symbolisation and conceptual understanding simultaneously. The study recommends teachers to use strategies such as informed choice of subject matter and a pedagogical approach in which concepts are understood before they are symbolised.Mathematics, Science and Technology EducationD. Phil. (Mathematics, Science and Technology Education

Exploring Grade 11 Learners’ Mathematical Connections when Solving Trigonometric Equations

In this paper, we explored the intra-mathematical connections that grade 11 learners make when solving trigonometric equations. The study was guided by Mowat’s theory of mathematical connections in which nodes and links are used to connect mathematical concepts and topics. We used a qualitative case study design within an interpretive paradigm to explore the intra-mathematical connections learners make as they solved trigonometric equations. The study was conducted in a high school in Mankweng Circuit, Limpopo Province, South Africa. Convenience sampling was used to select 30 learners who participated in the study. Data was collected using documents and task-based interviews. Data were analysed using inductive thematic analysis. The findings showed that learners made were able to make algebraic connections when solving trigonometric equations. They, however, were unable to make connections within trigonometry itself. This study, therefore, recommends that teachers stress the importance of connections when teaching trigonometry so that learners will not learn trigonometric concepts in isolation. In addition, it is recommended that further research be conducted on teaching strategies to improve learners’ mathematical connection skills when solving trigonometric equations.In this paper, we explored the intra-mathematical connections that grade 11 learners make when solving trigonometric equations. The study was guided by Mowat’s theory of mathematical connections in which nodes and links are used to connect mathematical concepts and topics. We used a qualitative case study design within an interpretive paradigm to explore the intra-mathematical connections learners make as they solved trigonometric equations. The study was conducted in a high school in Mankweng Circuit, Limpopo Province, South Africa. Convenience sampling was used to select 30 learners who participated in the study. Data was collected using documents and task-based interviews. Data were analysed using inductive thematic analysis. The findings showed that learners made were able to make algebraic connections when solving trigonometric equations. They, however, were unable to make connections within trigonometry itself. This study, therefore, recommends that teachers stress the importance of connections when teaching trigonometry so that learners will not learn trigonometric concepts in isolation. In addition, it is recommended that further research be conducted on teaching strategies to improve learners’ mathematical connection skills when solving trigonometric equations

Exploring Grade 11 Learners’ Mathematical Connections when Solving Trigonometric Equations

In this paper, we explored the intra-mathematical connections that grade 11 learners make when solving trigonometric equations. The study was guided by Mowat’s theory of mathematical connections in which nodes and links are used to connect mathematical concepts and topics. We used a qualitative case study design within an interpretive paradigm to explore the intra-mathematical connections learners make as they solved trigonometric equations. The study was conducted in a high school in Mankweng Circuit, Limpopo Province, South Africa. Convenience sampling was used to select 30 learners who participated in the study. Data was collected using documents and task-based interviews. Data were analysed using inductive thematic analysis. The findings showed that learners made were able to make algebraic connections when solving trigonometric equations. They, however, were unable to make connections within trigonometry itself. This study, therefore, recommends that teachers stress the importance of connections when teaching trigonometry so that learners will not learn trigonometric concepts in isolation. In addition, it is recommended that further research be conducted on teaching strategies to improve learners’ mathematical connection skills when solving trigonometric equations.In this paper, we explored the intra-mathematical connections that grade 11 learners make when solving trigonometric equations. The study was guided by Mowat’s theory of mathematical connections in which nodes and links are used to connect mathematical concepts and topics. We used a qualitative case study design within an interpretive paradigm to explore the intra-mathematical connections learners make as they solved trigonometric equations. The study was conducted in a high school in Mankweng Circuit, Limpopo Province, South Africa. Convenience sampling was used to select 30 learners who participated in the study. Data was collected using documents and task-based interviews. Data were analysed using inductive thematic analysis. The findings showed that learners made were able to make algebraic connections when solving trigonometric equations. They, however, were unable to make connections within trigonometry itself. This study, therefore, recommends that teachers stress the importance of connections when teaching trigonometry so that learners will not learn trigonometric concepts in isolation. In addition, it is recommended that further research be conducted on teaching strategies to improve learners’ mathematical connection skills when solving trigonometric equations

Learners’ Graphical Efficacy When Solving Trigonometric Problems

This study explored grade 12 learners’ graphical efficacy when solving problems involving trigonometric graphs. A structured test consisting of five trigonometric problems, with variations in context and structure, was administered to a purposefully selected group of 25 Grade 12 learners from the Sekhukhune District in South Africa. Insights into learners' graphing efficacy were obtained through task-based interviews. Data were analysed using direct interpretation which involved deductive thematic analysis of the task-based interviews and content analysis of the test scripts to match learners’ responses to the themes drawn from the Meta-Representational Competence (MRC) framework. The results showed that most learners lack invention and functioning, critiquing and reflection efficacies and hence this affected their drawing and interpretation of the graphs and consequently lead to incorrect solutions.  Furthermore, the results show most learners have critiquing efficacy. This indicates that learners lack graphical efficacy for solving trigonometric problems involving trigonometric functions. This finding has pedagogic implications: the apparent lack of graphical efficacy in graphical solutions may suggest inadequate mastery of the concept. Therefore, this study recommends that the teaching and learning of trigonometric graphs should consider the development of invention, functioning, critiquing and reflection efficacies