On Educational psychology: A cognitive view.

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Title: Finding your way in qualitative research 
Authors: E. Henning, W. van Rensburg and B. Smit 
Year: 2004 
Publisher: Van Schaik 
ISBN: 0-627-02545-5 (paperback, 179 pages) 
'It's like trying to navigate a dangerous, and mainly uncharted raging river, full of hidden currents and submerged rocks, only no-one will give you a boat that doesn't leak.' This quote, taken from the diary of a postgraduate student attempting to conduct qualitative research for the first time, succinctly and trenchantly summarises the frustration and confusion often faced by 'novice' qualitative researchers. 
Whilst the use of qualitative methods in conducting research within ..

Finding your way in qualitative research, E. Henning, W. van Rensburg and B. Smit : book review

The book “ Teaching and Learning” is a valuable resource for distance educators and trainers who want to create relevant and intellectually challenging online courses. Nichols aims to share the best educational practices from traditional or face-to-face teaching and from the field of distance education. It is an ambitious goal that gives the book a distinctive purpose and helps the author strive for a comprehensive view of the educational process. The book provides practical guidelines for distance educators that are based solid research and classroom experiences. The book is divided into three major sections: the teaching and learning process, universal principles of teaching and learning and the final section describes how educational principles can impact vital instructional issues.

Teaching for Learning.

It is wellknown that the majority of South African learners achieve extremely poorly in mathematics. Many claim that one of the causes of this poor attainment is teachers’ weak knowledge of mathematics, and propose that improving teachers’ mathematical knowledge would improve learner attainment. However, the evidence-base for this proposed solution is currently relatively weak. We report on a quasi-experimental study examining the learning gains of Grade 10 learners from five secondary schools in the Johannesburg area whose teachers participated in a year-long professional development course aimed at improving the teachers’ knowledge of mathematics for teaching. Statistical analyses of pre- and post-test results show that the intervention group of learners (N = 586) taught by teachers who had participated in the professional development (N = 14) outperformed a matched control group of learners (N = 217) taught by teachers in the same schools (N = 7). An effect size of =0.17 for the intervention group is equivalent to two months’ additional progress. While the learning gains are small, they are statistically significant. These findings provide empirical support for claims that attending to teachers’ mathematical knowledge can impact learners’ attainment. Suggestions are made regarding the form and substance of such professional development.

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Can improving teachers’ knowledge of mathematics lead to gains in learners’ attainment in Mathematics?

This article reports on an analysis of errors that were displayed by students who studied mathematics in Chemical Engineering in derivatives of mostly trigonometric functions. The poor performance of these students triggered this study. The researcher (lecturer) works in a mathematics support programme to enhance students’ understanding of mathematics. The purpose of this study was to identify errors and their origins when students did calculations in derivatives of trigonometric functions. The participants of this study were a group of thirty students who were registered for Mathematics in a university of technology in Western Cape, South Africa. The researcher used a qualitative case study approach and collected data from students’ written work. This study used Dubinsky’s (1991) APOS Theory (Actions, Processes, Objects, and Schemas) to classify errors into categories and analyse the data collected. Errors displayed by students were conceptual and procedural; there were also errors of interpretation and linear extrapolation. Conceptual errors showed a failure to grasp the concepts in a problem and a failure to appreciate the relationships in a problem. Procedural errors occurred when students failed to carry out manipulations or algorithms, even if concepts were understood. Interpretation errors occurred when students wrongly interpreted a concept due to over-generalisation of the existing schema. Linear extrapolation errors occurred when students over-generalised the property f(a + b) = f(a) + f(b), which applies only when f is a linear function, to the form f(a * b) = f(a) * f(b), where f is any function and * any operation. The findings revealed that the participants were not familiar with basic operational signs such as addition, subtraction, multiplication and division of trigonometric functions. The participants demonstrated poor ability to simplify once they had completed differentiation. This study recommends the strategy of focusing on elimination of errors to develop students’ understanding of derivatives of trigonometric functions. This can be done through learning activities that lead to identification and analyses of students’ errors in classroom discussions.

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Analysis of errors in derivatives of trigonometric functions

Many studies point to the problem of poor mathematics content knowledge of mathematics teachers in South Africa. The purpose of this study was to investigate teachers' knowledge of the mathematics they are themselves teaching. Data was generated from the teachers (n = 253) written responses to test that was a shortened form of a previous Grade 12 Mathematics Paper One examination. The sample of teachers were studying towards an Advanced Certificate in Education (an upgrading high school mathematics qualification) at the University of KwaZulu-Natal in South Africa. The findings revealed that the teachers in this sample obtained an average of 57% in the test. Using an APOS theory analysis it was found that many teachers who were working at an action level of a concept would require help and scaffolding to move to process or object levels of understanding of that concept. Furthermore it was found that on average teachers obtained 29% on questions which were at the problem solving level, raising concerns about how these teachers would mediate tasks that are set at high cognitive levels, with their Grade 12 learners.

https://openscholar.dut.ac.za/bitstream/10321/2306/1/brijall_et_at_2014.pdf

An exploration of the common content knowledge of high school mathematics teachers

This qualitative case study in a rural school in Umgungundlovu District in KwaZulu-Natal, South Africa, explored Grade 12 learners’ mental constructions of mathematical knowledge during engagement with optimisation problems. Ten Grade 12 learners who do pure Mathemat-ics participated, and data were collected through structured activity sheets and semi-structured interviews. Structured activity sheets with three tasks were given to learners; these tasks were done in groups, and the group leaders were interviewed. It was found that learners tended to do well with routine-type questions, implying that they were functioning at an action level. From the interviews it appeared that learners might have the correct answer, but lacked conceptual understanding. Exploring learners’ mental constructions via their responses to activity sheets and interviews enabled common errors and misconceptions to be identified. Themes that emerged were that learners: 1) lacked the understanding of notation dy/dx, 2) had not constructed the derivative and minima/maxima schema, 3) had some difficulty in modelling problems, 4) preferred rules and formulas, and 5) applied algebraic notions incorrectly. Inferences are drawn for curriculum developers and teachers. This study also formulated itemised genetic decompositions for particular tasks, which contribute to APOS theory. Keywords: APOS theory, Calculus, genetic decomposition, optimisation

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High school learners' mental construction during solving optimisation problems in Calculus: a South African case study

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 m...

Pre-service Teachers’ Mental Constructions of Concepts in Matrix Algebra

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

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.

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Deonarain Brijlall

Papers

An exploration of the common content knowledge of high school mathematics teachers

High school learners' mental construction during solving optimisation problems in Calculus: a South African case study

Pre-service Teachers’ Mental Constructions of Concepts in Matrix Algebra

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

Exploring support strategies for high school mathematics teachers from underachieving schools