Proof is an important component of advanced mathematical activity. Nevertheless, undergraduates struggle to write valid proofs. Research identifies many of the struggles students experience with the logical nature and structure of proofs. Little research examines the role mathematical content knowledge plays in proof production. This study begins to fill this gap in the research by analyzing what role mathematical content knowledge plays in the success of a proof and how undergraduates use mathematical content knowledge during proofs. Four undergraduates participated in a series of task-based interviews wherein they completed several proofs. The interviews were analyzed to determine how the students used mathematical content knowledge and how mathematical content knowledge affected a proof’s validity. The results show that using mathematical content knowledge during a proof is nontrivial for students. Several of the proofs attempted by the students were unsuccessful due to issues with mathematical content knowledge. The data also show that students use mathematical content knowledge in a variety of ways. Some student use of mathematical content is productive and efficient, while other student practices are less efficient in formal proofs.

South African learners underachieve in mathematics. The many different factors that influence this underachievement include mathematics teachers' role in teaching mathematics with understanding. The question arises as to how teachers' mathematical content knowledge states can be transformed to positively impact learners' achievement in mathematics. In this study, different kinds of teachers' knowledge needed for teaching mathematics were discussed against the background of research in this area, which included the work of Shulman, Ma and Ball. From this study an important kind of knowledge, namely mathematical content knowledge for teaching (MCKfT), was identified and a teacher's ability to unpack mathematical knowledge and understanding was highlighted as a vital characteristic of MCKfT. To determine further characteristics of MCKfT, the study focussed on the nature of mathematics, different kinds of mathematical content knowledge (procedural and conceptual), cognitive processes (problem solving, reasoning, communication, connections and representations) involved in doing mathematics and the development of mathematical understanding (instrumental vs. relational understanding). The influence of understanding different problem contexts and teachers' ability to develop reflective practices in teaching and learning mathematics were discussed and connected to a teacher's ability to unpack mathematical knowledge and understanding. In this regard, the role of teachers' prior knowledge or current mathematical content knowledge states was discussed extensively. These theoretical investigations led to identifying the characteristics of MCKfT, which in turn resulted in theoretical criteria for the development of MCKfT. The theoretical study provided criteria with which teachers' current mathematical content knowledge states could be analysed. This prompted the development of a diagnostic instrument consisting of questions on proportional reasoning and functions. A qualitative study was undertaken in the form of a diagnostic content analysis on teachers' current mathematical content knowledge states. A group of secondary school mathematics teachers (N=128) involved in the Sediba Project formed the study population. The Sediba Project is an in-service teacher training program for mathematics teachers over a period of two years. These teachers were divided into three sub-groups according to the number of years they had been involved in the Sediba Project at that stage. The teachers' current mathematical content knowledge states were analysed with respect to the theoretically determined characteristics of and criteria for the development of MCKfT. These criteria led to a theoretical framework for assessing teachers' current mathematical content knowledge states. The first four attributes consisted of the steps involved in mathematical problem solving skills, namely conceptual knowledge (which implies a deep understanding of the problem), procedural knowledge (which is reflected in the correct choice of a procedure), the ability to correctly execute the procedure and the insight to give a valid interpretation of the answer. Attribute five constituted the completion of these four attributes. The final six attributes were an understanding of different representations, communication of understanding in writing, reasoning skills, recognition of connections among different mathematical ideas, the ability to unpack mathematical understanding and understanding the context a problem is set in. Quantitative analyses were done on the obtained results for the diagnostic content analysis to determine the reliability of the constructed diagnostic instrument and to search for statistically significant differences among the responses of the different sub-groups. Results seemed to indicate that those teachers involved in the Sediba Project for one or two years had benefited from the in-service teacher training program. However, the impact of this teachers' training program was clearly influenced by the teachers' prior knowledge of mathematics. It became clear that conceptual understanding of foundation, intermediate and senior phase school mathematics that should form a sound mathematical knowledge base for more advanced topics in the school curriculum, is for the most part procedurally based with little or no conceptual understanding. The conclusion was that these teachers' current mathematical content knowledge states did not correspond to the characteristics of MCKfT and therefore displayed a need for the development of teachers' current mathematical content knowledge states according to the proposed criteria and model for the development of MCKfT. The recommendations were based on the fact that the training that these teachers had been receiving with respect to the development of MCKfT is inadequate to prepare them to teach mathematics with understanding. Teachers' prior knowledge should be exposed so that training can focus on the transformation of current mathematical content knowledge states according to the characteristics of MCKfT. A model for the development of MCKfT was proposed. The innermost idea behind this model is that a habit of reflective practices should be developed with respect to the characteristics of MCKfT to enable a mathematics teacher to communicate and unpack mathematical knowledge and understanding and consequently solve mathematical problems and teach mathematics with understanding. Key words for indexing: school mathematics, teacher knowledge, mathematical content knowledge, mathematical content knowledge for teaching, mathematical knowledge acquisition, mathematics teacher education / Thesis (Ph.D. (Education))--North-West University, Potchefstroom Campus, 2007.

South African learners underachieve in mathematics. The many different factors that influence this underachievement include mathematics teachers' role in teaching mathematics with understanding. The question arises as to how teachers' mathematical content knowledge states can be transformed to positively impact learners' achievement in mathematics. In this study, different kinds of teachers' knowledge needed for teaching mathematics were discussed against the background of research in this area, which included the work of Shulman, Ma and Ball. From this study an important kind of knowledge, namely mathematical content knowledge for teaching (MCKfT), was identified and a teacher's ability to unpack mathematical knowledge and understanding was highlighted as a vital characteristic of MCKfT. To determine further characteristics of MCKfT, the study focussed on the nature of mathematics, different kinds of mathematical content knowledge (procedural and conceptual), cognitive processes (problem solving, reasoning, communication, connections and representations) involved in doing mathematics and the development of mathematical understanding (instrumental vs. relational understanding). The influence of understanding different problem contexts and teachers' ability to develop reflective practices in teaching and learning mathematics were discussed and connected to a teacher's ability to unpack mathematical knowledge and understanding. In this regard, the role of teachers' prior knowledge or current mathematical content knowledge states was discussed extensively. These theoretical investigations led to identifying the characteristics of MCKfT, which in turn resulted in theoretical criteria for the development of MCKfT. The theoretical study provided criteria with which teachers' current mathematical content knowledge states could be analysed. This prompted the development of a diagnostic instrument consisting of questions on proportional reasoning and functions. A qualitative study was undertaken in the form of a diagnostic content analysis on teachers' current mathematical content knowledge states. A group of secondary school mathematics teachers (N=128) involved in the Sediba Project formed the study population. The Sediba Project is an in-service teacher training program for mathematics teachers over a period of two years. These teachers were divided into three sub-groups according to the number of years they had been involved in the Sediba Project at that stage. The teachers' current mathematical content knowledge states were analysed with respect to the theoretically determined characteristics of and criteria for the development of MCKfT. These criteria led to a theoretical framework for assessing teachers' current mathematical content knowledge states. The first four attributes consisted of the steps involved in mathematical problem solving skills, namely conceptual knowledge (which implies a deep understanding of the problem), procedural knowledge (which is reflected in the correct choice of a procedure), the ability to correctly execute the procedure and the insight to give a valid interpretation of the answer. Attribute five constituted the completion of these four attributes. The final six attributes were an understanding of different representations, communication of understanding in writing, reasoning skills, recognition of connections among different mathematical ideas, the ability to unpack mathematical understanding and understanding the context a problem is set in. Quantitative analyses were done on the obtained results for the diagnostic content analysis to determine the reliability of the constructed diagnostic instrument and to search for statistically significant differences among the responses of the different sub-groups. Results seemed to indicate that those teachers involved in the Sediba Project for one or two years had benefited from the in-service teacher training program. However, the impact of this teachers' training program was clearly influenced by the teachers' prior knowledge of mathematics. It became clear that conceptual understanding of foundation, intermediate and senior phase school mathematics that should form a sound mathematical knowledge base for more advanced topics in the school curriculum, is for the most part procedurally based with little or no conceptual understanding. The conclusion was that these teachers' current mathematical content knowledge states did not correspond to the characteristics of MCKfT and therefore displayed a need for the development of teachers' current mathematical content knowledge states according to the proposed criteria and model for the development of MCKfT. The recommendations were based on the fact that the training that these teachers had been receiving with respect to the development of MCKfT is inadequate to prepare them to teach mathematics with understanding. Teachers' prior knowledge should be exposed so that training can focus on the transformation of current mathematical content knowledge states according to the characteristics of MCKfT. A model for the development of MCKfT was proposed. The innermost idea behind this model is that a habit of reflective practices should be developed with respect to the characteristics of MCKfT to enable a mathematics teacher to communicate and unpack mathematical knowledge and understanding and consequently solve mathematical problems and teach mathematics with understanding. Key words for indexing: school mathematics, teacher knowledge, mathematical content knowledge, mathematical content knowledge for teaching, mathematical knowledge acquisition, mathematics teacher education / Thesis (Ph.D. (Education))--North-West University, Potchefstroom Campus, 2007.

This study investigates to what extent educators’ pedagogical content knowledge affects learners’ achievement in quadratic functions. The components of pedagogical content knowledge (PCK) examined are: (i) mathematical content knowledge (MCK), (ii) knowledge of learners’ conceptions, and misconceptions, and (iii) knowledge of strategies. The participants were seventeen mathematics educators and ten learners from each educator’s class. The sample of educators was a convenient sample, while the sample of learners was selected by means of random sampling. A mixed method design was used to execute the study. Data about educators’ MCK, and knowledge of learners’ misconceptions were collected by means of a questionnaire. An interview was used to gather data about educators’ knowledge of strategies. Data on learners’ achievements and misconceptions was collected by means of a questionnaire. Descriptive statistics were used to describe the effect of each component of the educators’ PCK on learners’ achievements. The result indicates that the achievement of learners who are taught by educators who have strong PCK is higher than the achievement of learners who are taught by educators who have weak PCK. / Mathematical Sciences / M. Ed. (Mathematics Education)

This study investigates to what extent educators’ pedagogical content knowledge affects learners’ achievement in quadratic functions. The components of pedagogical content knowledge (PCK) examined are: (i) mathematical content knowledge (MCK), (ii) knowledge of learners’ conceptions, and misconceptions, and (iii) knowledge of strategies. The participants were seventeen mathematics educators and ten learners from each educator’s class. The sample of educators was a convenient sample, while the sample of learners was selected by means of random sampling. A mixed method design was used to execute the study. Data about educators’ MCK, and knowledge of learners’ misconceptions were collected by means of a questionnaire. An interview was used to gather data about educators’ knowledge of strategies. Data on learners’ achievements and misconceptions was collected by means of a questionnaire. Descriptive statistics were used to describe the effect of each component of the educators’ PCK on learners’ achievements. The result indicates that the achievement of learners who are taught by educators who have strong PCK is higher than the achievement of learners who are taught by educators who have weak PCK. / Mathematical Sciences / M. Ed. (Mathematics Education)

South Africa is the first country in the world to offer Mathematical Literacy as a school subject. This subject was introduced in 2006 as an alternative to Mathematics in the Further Education and Training band. The purpose of this subject is to provide learners with an awareness and understanding of the role that mathematics plays in the modern world, but also with opportunities to engage in real-life problems in different contexts. A problem is the beliefs some people in and outside the classroom have regarding this subject such as teachers believing ML is the dumping ground for mathematics underperformers (Mbekwa, 2007). Another problem is the belief of some principals that any nonmathematics teacher can teach ML. In practice there is Mathematics teachers who teach ML in the same way that they teach Mathematics; non-Mathematics teachers who in many cases lack the necessary mathematical content knowledge and skills to teach ML competently; and Mathematics teachers who adapted their practices to teach ML using different approaches than those required for teaching Mathematics. Limited in-depth research has been done on the ML teachers, what they believe and what knowledge is required to teach this subject effectively and proficiently. The purpose of this study is to investigate the way in which ML is taught in a limited number of classrooms with the view to exploring the relationship between ML teachers’ knowledge and beliefs and their instructional practices. According to Artzt, Armour-Thomas and Curcio (2008) the instructional practice of the teacher plays out in the classroom where teachers’ goals, knowledge and beliefs serve as the driving force behind their instructional efforts to guide and mentor learners in their search for knowledge. To accomplish this aim, an in-depth case study was conducted to explore the nature of teachers’ knowledge and beliefs about ML as manifested in their instructional practices. A qualitative research approach was used in which observations and interviews served as data collection techniques enabling me to interpret the reality as I became part of the lives of the teachers. My study revealed that there is a dynamic but complex relationship between ML teachers’ knowledge and beliefs and their instructional practices. The teachers’ knowledge, but not their stated beliefs were reflected in their instructional practices. Conversely, in one case, the teacher’s instructional practice also had a positive influence on her knowledge and beliefs. It was further revealed that mathematics teacher training and teaching experience played a significant role in the productivity of the teachers’ practices. The findings suggest that although mathematical content knowledge is required to develop PCK, it is teaching experience that plays a crucial role in the development of teachers’ PCK. Although the study’s results cannot be generalised due to the small sample, I believe that the findings concerning the value of teachers’ knowledge and the contradictions between their stated beliefs and practices could possibly contribute to teacher training. Curriculum decision-makers should realise that the teaching of ML requires specially trained, competent, dedicated teachers who value the subject. This exploratory study concludes with recommendations for further research. / Thesis (PhD)--University of Pretoria, 2011. / Science, Mathematics and Technology Education / unrestricted

Educators and policymakers have demonstrated interest in finding ways to better equip mathematics teachers so they can help students achieve at a higher level. Academic coaching has been identified as an effective professional development activity for teachers. The purpose of this study was to investigate the difference between students’ achievement levels before and after a mathematics initiative in a Northeast Tennessee school district. In this study I analyzed grades 3 – 6 students’ Tennessee Comprehensive Assessment Program or TCAP scores in the year prior to the hiring of a mathematics coach and their respective scores 2 years after the placement of the mathematics coach. All statistical analyses were analyzed at a .05 level of significance. All null hypotheses under both research questions were analyzed with a pairsampled t-test using repeated-measures design. The results indicate significant difference in students’ TCAP scores prior to and after specialist. Scores after specialist were significantly higher than scores before specialists. The difference was present for students who attended Title I schools as well as for students who attended non-Title I schools. School administrators and school district leaders can benefit from such a study because it presents academic coaching as a viable means to equip teachers so they can help students increase their achievement in mathematics.