Deonarain Brijlall

Exploring the Pedagogical Content Knowledge for Teaching Probability in Middle School: A South African Case Study

Abstract This paper reports on an exploration into in-service teachers’ pedagogical content knowledge required for the delivery of lessons in probability. The section on probability is taught in Mathematical Literacy to the middle school learners in South Africa. As a theoretical framework the work initiated in the nineteen eighties was sought. This paper adopted a refined framework into domains of pedagogical knowledge. The four domains are: Common Content Knowledge; Specialised Content Knowledge; Knowledge of Content and Students as well as Knowledge of Content and Teaching which are defined and applied to show their necessity in a teaching situation. Examples of tasks from an open ended questionnaire were discussed and identification of the pedagogical content knowledge on probability was made. Teachers’ written responses were collected and analyzed to verify or refute suggested strands of knowledge provided earlier. The written responses were from in-service teachers (n = 86) who were studying towards an Advanced Certificate in Education (an upgrading qualification) at a South African university.

Exploring support strategies for high school mathematics teachers from underachieving schools

Abstract Algebra, trigonometry and calculus concepts are dealt with in grades 10 to 12 of South African high school topics, as prescribed by the Curriculum and Assessment Policy Statement. This paper reports on a study exploring practicing teachers’ subject matter knowledge of certain topics they were expected to teach. The theoretical perspectives for the design and analysis of this study were based on Shulman’s conceptual framework. This study was a qualitative case study (n = 28). Practising teachers from the Pinetown district in KwaZulu-Natal participated in this mathematics’ teacher intervention. Focused tasks on algebra, trigonometry and calculus were designed and administered to those teachers. The researchers analysed the data from those tests and found that certain areas of their pedagogical content knowledge required attention.

Exploring Pre-service Teachers' Mental Constructions When Solving Problems Involving Infinite Sets

Abstract In this study, the researchers explore the meaning of the, is equal to symbol by exposing pre-service teachers (n = 38) to two tasks. The first task was baseequivalence of two countable infinite sets. The second task focused on the comparison of the two numbers 0,9 and 1.The study was carried out at a South African university, where activity sheets comprising the two tasks were completed by pre-service teachers specializing to become high school mathematics teachers. Their written responses were analyzed against the initial itemized genetic decompositions formulated in accordance with APOS (Action-Process-Object-Schema) Theory. The findings of the exploration led to modified itemized genetic decompositions. These modified genetic decompositions have pedagogical implications. They also contribute to the APOS theory.

Pre-service teachers' mental constructions of concepts in matrix algebra

This study is part of ongoing research in undergraduate mathematics education. The study was guided by the belief that understanding the mental constructions the pre-service teachers make when learning matrix algebra concepts leads to improved instructional methods. In this preliminary study the data was collected from 85 pre-service teachers through a structured activity sheet. The aim was to explore their mental constructions of matrix algebra concepts and how they concur with a preliminary genetic decomposition. The study is underpinned by APOS theory (action–process–object–schema). The findings revealed that the mental constructions made by pre-service teachers in most cases concur with the preliminary genetic decomposition. In terms of APOS theory, the responses revealed that most pre-service teachers were operating at action and process stages, with few operating at the object stage. It was observed that the schemata for basic algebra and real number are necessary for the conceptual development of matrix algebra, and that familiarity of the correct use of terminology and notations promotes the earning of matrix algebra. On the basis of the detailed response analysis, modified genetic decompositions are proposed for the matrix order concept, matrix transpose concept and for matrix operations.

Exploring Practical Work as a Sustainable Strategy in Rural Mathematics Classrooms: A Case of Addition of Fractions

Abstract Addition of fraction is dealt with from grade 6 onwards in South African schools topics as prescribed by the Curriculum and Assessment Policy Statement. This paper reports on a study exploring practical work as a strategy to sustain rural mathematics classrooms. The conceptual perspective for the design and analysis of this study was based on a learning ecology framework consolidated by socio-culturalism and situated learning. This study was a qualitative case study (n = 42). The participants in this study were from a rural school of the Umhlali ward, which is in the Ilembe district in KwaZulu-Natal, South Africa. The learners engaged in four activities in four lessons. They used either diagrams or fraction circles to complete the tasks in the four activities. The researcher analysed the written responses, observed their working in groups and carried out interviews. This case study found that practical work, in a rural setting, was a suitable strategy to sustain an effective mathematics learning ecology.

Pre-service mathematics teachers’ mental constructions of the determinant concept

Abstract This study aims to reveal the nature of mental constructions made by pre-service teachers whenlearning the determinant concepts and to contribute to APOS theory (Action, Process, Object and Schema) interms of instructional strategies. This is guided by the belief that understanding the mental constructions thestudents make when learning mathematical concepts leads to improved instructional methods. The data wascollected through a structured activity sheet and interviews. The findings revealed that the mental constructionsmade by pre-service teachers in most cases do concur with the preliminary genetic decompositions, and that manypre-service teachers are operating at an action/process stages, with the exception of the few operating at an objectstage. It also revealed that many students could carry procedures effectively, even if the meaning of the conceptis not constructed, which indicates that students mainly possess procedural knowledge of the determinant concept.

Improving Proficiency in Mathematics through Website-based Tasks: A Case of Basic Algebra

Abstract Understanding how university students make sense of mathematics is always of concern to their lecturers. A group of lecturers at a South African university studied written responses of first year students to explore what mental conflicts arose during their students’ formulations of solutions to assigned tasks. The mental conflicts as illustrated by their responses were categorized. One of the sections considered was basic algebra which is the case discussed in this paper. After detecting the common conflicts, the lecturers designed website-based tasks. The tasks were meant to target mental conflicts with the hope of removing these from students’ cognition. The website-based tasks were designed in accordance with the principle of scaffolding. The researchers analysed the various conflicts within the framework of Kilpatrick’s five strands of mathematical proficiency. In this paper the researchers discuss the design of the activities and interactive collaborations during tutorial sessions with students. We found that the website material seemed to have helped in removing common mathematical conflicts of some of our undergraduate mathematics students. Also it was found that non-mathematical terminology used could contribute to students’ mental conflicts.

Using Questionnaires in the Learning of Congruency of Triangles to Incite Formal and Informal Reasoning

ABSTRACT This qualitative study examined the influence learning material design has on the learning of geometry in the middle school curriculum. The middle school mathematics learners (n =82) involved were from two South African rural schools in the province of KwaZulu-Natal. The research was conceptualised in terms of Vygotsky’s educational theory and the process of scaffolding. Researchers from a South African university designed questionnaires based on scaffolding guidelines suggested by Zhao and Orey. The questionnaires comprised of a series of geometry tasks which spanned two weeks. These questionnaires were administered to the learners and their written responses were analyzed. After analysis of these responses interviews were carried out to verify or refute the views of the researcher. Data yielded by these research instruments confirmed certain assumptions and literature claims. The stu dy revealed that the intervention design effectively managed to integra te the two types of geomet ries to strengthen the concept of congruency of triangles. Many recommendations, for the teaching of congrue ncy of triangles, emanated from the findings.

Innovative Pedagogy: Implications of Genetic Decompositions for Problem Solving in Management Courses

Mathematics is a subject on which many business and management courses rely. In this paper the researcher reports on the processes involved in the design of itemised genetic decompositions for certain mathematics tasks. The manner by which this is done may be extended to teaching and learning in management courses in higher education. The researcher used two tasks, one in a high school context from a previously published study and another from a qualitative case study which explored the development of mental constructions of a group of fourth-year pre-service teachers during definition making of certain mathematical concepts in real analysis, at a South African higher education institution. Questionnaires based on two items involving the concept of infinity were administered to those participating pre-service teachers. The pre-service teachers were allowed to respond to these items without lecturer involvement so that a learner-centred learning environment was created. Their mental constructions of these concepts were analysed by using an APOS (Action-Process-Object-Schema) framework. Forty six pre-service teachers, specialising in the teaching of mathematics in the FET school curriculum, participated in the project. It was found that the APOS designed questionnaire generated thick data that led to a modification of the initial genetic decomposition formulated. This modified genetic decomposition has didactical implication for business and management courses taught in higher education. This is illustrated in the recommendations section of this paper.

Exploring the stages of Polya’s problem-solving model during collaborative learning : a case of fractions

Abstract This paper reports on a small-scale action research, which investigated the processes involved in problem-solving in a mathematics class. Grade ten learners (n = 47) at a South African middle school were involved in the study. The participants from two classes attempted the solution of tasks involving the fraction concept. In one class, the learners worked in groups and in the other class, the learners worked individually. A qualitative method was adopted for data capture and analysis. Social constructivism was adopted as a theoretical framework and the stages advocated by Polya were interrogated when analyzing the learner responses on their problem solutions. The results revealed that those learners working in groups demonstrated most of the stages of the Polya linear problem-solving model. The findings helped identify which stages of the model promote effective problemsolving and some recommendations are made for classroom practitioners engaging their learners in problemsolving.