This research study investigated the ways in which a mathematics module, informed by an enactivist philosophy, enabled pre-service teachers to unpack the reality of their teaching practice in terms of proficient teaching. Given the generally poor mathematics results in South Africa it is not enough for teachers to be merely proficient in Mathematics. They also need to be in a position to explain important mathematical concepts to children in a manner that will encourage and develop an understanding of the basic mathematical concepts. It was my intention with this study to determine whether a mathematics education module, that embraced the underlying themes of enactivism as part of its teaching pedagogy, could have the potential to develop and increase the skills of pre-service teachers’ teaching for proficiency in Mathematics. The mathematics module was underpinned by five themes of enactivism namely: autonomy, embodiment, emergence, sense-making and experience and was designed to supplement the pre-service teachers’ basic skills in Mathematics in the Intermediate Phase. This mathematics module was offered to fourth year pre-service teachers completing a B.Ed. in the Foundation Phase at a private institute specialising in the training of teachers. The theoretical framework was informed by enactivism and how the themes of enactivism could be used as a vehicle to develop teaching proficiency. The study was qualitative in nature and situated within an interpretivist paradigm. The specific perspectives of interpretivism that were used were hermeneutics, phenomenology and reflexivity. The research design was a case study that contained elements of action research and encompassed three phases of data collection. The first phase focused on the pre-service teachers’ approach to teaching Mathematics and what this brought forth in terms of the reality of their teaching practice and the problems they encountered. The second phase undertook to determine what growth and development of teaching proficiency in Mathematics had emerged over the research period. The final phase was undertaken after the pre-service teachers had graduated and were employed as full time teachers in the Intermediate Phase. The analytical framework and lens through which the data was analysed was that of Kilpatrick, Swafford and Findell’s (2001) strands of mathematical proficiency. The argument that I present is that the themes of enactivism did contribute to the growth of the pre-service teachers’ teaching for mathematical proficiency. The themes of embodiment and experience were major contributions in revealing that this was a reality for the pre-service teachers from a practical perspective and was what they would be able to take away with them. However the theme of emergence stood out as the principle that generated the most awareness and growth and which, in turn, affected the participants’ autonomy.

This investigation is primarily concerned with the In-Service Education and Training (INSET) needs of primary school mathematics educators. The research is grounded in the proposition that in order for senior primary mathematics educators to keep abreast with the current knowledge explosion and rapid rate of technological growth, they are under serious obligation to improve their expertise, knowledge and skills in mathematics teaching and learning through INSET. Pre-Service Education and Training (PRESET) serves only as preparation for entry into the teaching profession and cannot last the whole teaching career. INSET is thus necessary for the senior primary mathematics educator’s continuing education. In this investigation an attempt is made at establishing a framework for INSET and mathematics educators. It is hoped that these theoretical frameworks will enable mathematics educators to cope with the changing needs of senior primary mathematics education. There is no doubt that the developments of senior primary mathematics education and INSET in the United Kingdom can have a profound effect on the senior primary mathematics education in South Africa. The extent to which the developments in the United Kingdom influences the educational initiatives presently being undertaken in South Africa will depend upon those who teach senior primary mathematics and those who are responsible for the provision of INSET of senior primary mathematics educators. The past South African discriminatory policies led to numerous iniquities in the provision of mathematics materials and development of mathematical human resources. Consequently, there is a large number of unqualified and under-qualified mathematics educators especially amongst Blacks in South Africa. It is through INSET that this condition can be rectified. A questionnaire survey revealed that senior primary mathematics educators are fully conscious of the importance and significance of INSET. The recommendations made for the INSET of primary school mathematics educators based on the literature survey and the empirical investigation are: the need to establish a national and provincial policy for the INSET of primary school mathematics educators; there must be a concerted effort to establish school focused INSET; many of the methods of INSET courses emphasizing the participative approach needs to be explored; the value of teachers’ centres as exciting brokers for new ideas and as networks for personnel proves invaluable and finally, pivotal to the INSET of primary school mathematics educators is the need for them to update their in-depth knowledge of mathematics. There is no doubt that the INSET of primary school mathematics educators is a crucial factor in KwaZulu-Natal (KZN). It is clear from this research that INSET for both academic and professional upgrading of the senior primary mathematics educators and the improvement of mathematics teaching and learning I the primary school is only limited by one’s imagination. / Dissertation (MEd)--University of Pretoria, 2006. / Humanities Education / MEd / Unrestricted

In the majority of the schools in the Eastern Cape, South Africa, teaching and learning takes place in the second language, English, of both teachers and learners. The purpose of this research was to elicit the perceptions of teachers in multilingual mathematics classes about language issues that they encounter and to ascertain whether they could experientially learn the theory of dialogic teaching through an intervention in order to introduce dialogue in practice in their classes. The effect of the intervention on teacher practices was qualitatively observed and the effect of the teacher practices on learner reasoning competence, numeracy competence and English language competence was quantitatively tested by using validated pre- and post-tests. The study follows a mixed method concurrent triangulation design with both quantitative and qualitative results. Two cohorts of students/teachers studying for qualifications at Nelson Mandela Metropolitan University centres throughout the Eastern Cape expressed their opinions about language challenges and solutions through questionnaires, reflective writing and poetry. A cohort of BEd Honours (Mathematics and Science) students experienced a semester long intervention on the theory and practice of dialogic teaching, particularly exploratory talk, and were tasked to introduce the practice into their multilingual mathematics classes in the form of reported action research. The next phase of the study focussed on the practices of three teachers and their grade seven multilingual mathematics learners who were observed and tested over a period of nine months. The following year the observations and testing were repeated with one teacher and his grade seven learners to ascertain whether the intervention would result in similar findings. iv The results enhance the validity of the Vygotskian claim concerning the relationship between language use, social interaction and reasoning development. In classes where there was evidence of dialogic practices the learners collaborated in groups using code-switching and their main language. Their reasoning, numeracy and English skills test scores improved statistically significantly. Teachers were able to give voice to their deep-felt emotions through poetry. They felt that the devaluing of isiXhosa had resulted in the loss of learners’ main language literacy competencies and consequent loss of cultural capital; however they considered it necessary to develop English competence in the learners, even if it was at the expense of developing mathematical competence. The introduction of exploratory talk in their home languages served the dual purpose of promoting the value of isiXhosa in an academic environment as well as enhancing mathematical reasoning. It appears that when teachers focus on developing language as a tool for reasoning, significant improvements in learners’ problem solving competences occur. When the language used is the main language of both teachers and learners both mathematical understanding and cultural identity are enhanced. The study concludes with a suggestion for a model for future interventions to train teachers to introduce dialogic practices in multilingual mathematics classes.

The Department of Education was tasked by Government and the Department of Labour to develop learning programmes which would provide skills to learners. The National Certificate Vocational (NCV) programmes were developed, which provided an alternative to completing a National Senior Certificate (NSC). The NC(V) programmes consist of seven subjects of which Mathematical Literacy is offered as a fundamental subject. The NC(V) programmes were officially implemented in 2007 using the FET College sector as a vehicle. FET College educators had to be skilled and re-skilled to teach the various new subjects. One of the new subjects at the time was Mathematical Literacy. Selected educators were provided with a short course to prepare themselves for the implementation of Mathematical Literacy. This study is aimed at investigating the pedagogical experiences of educators who were, and are still part, of the implementation of Mathematical Literacy in the FET College sector. A phenomenological approach was followed in order to capture the lived experiences of the educators. Three educators were selected from different FET colleges within the Eastern Cape Province. A qualitative research was done, making use of interviews. The research found that educators have divergent pedagogical experiences. They make use of different strategies to implement teaching and learning within their classrooms. Though there are good experiences, the research has managed to point out that there are some frustrations too. Recommendations are made with regard to teaching and learning strategies, as well as the emerging trends that surfaced during the research.

No description available.

Thesis (M. Ed. (Mathematics Education)) --University of Limpopo, 2017 / Sarton (1936) stated that mathematics has grown so large for a single mind to grasp. Mack (1961) attributes that phenomenon by claiming that mathematics differs from science in that it keeps on adding new concepts to existing ones, whereas in science there is reduction of concepts. This continuing growth makes it impossible for an individual to study mathematics as a whole (Krantz, 2010). Van Bendegem (2009, p. 137) calls the mathematics world a “mad world”. Recently, Ellerton (2014) compared mathematics to a growing tree. A number of challenges arise out of the observations made above. Is the mathematics that is taught in secondary schools an appropriate reflection of the mathematics that is out there today? Is an individual an appropriate embodiment of a secondary mathematics teacher? In the mist of these and many other questions, this study locates itself in the second question and investigated the notion of an embodiment of a secondary mathematics teacher. The main research question that was pursued was ‘How adequate is an individual as an embodiment of a secondary mathematics teacher?’ This question should be understood and interrogated in the context of Festinger’s (1962) dissonance cognitive theory that also serves as the theoretical framework for the study. The expectations of a secondary mathematics teacher do not fit in with an individual’s capacity to embody those. Grounded theory (Glaser, Strauss & Beer, 1967) was used to generate and develop what Elliot and Higgins (2012) called a substantive theory. This was a desktop grounded theory study and data was collected from existing literature of published journals and books. Since the use of documents is recommended as one of the qualitative data collection methods in grounded theory (Strauss & Corbin, 1990), the documents served as primary data where only a few that were relevant to the issues discussed were selected (Breckenridge & Jones, 2009). Content and thematic analyses procedures were used. Content analysis assisted to organise data according to various eras, tracing the growth in mathematics education and mathematics content, comparing them to a mathematics teacher of different eras, which assisted in bringing the answer to the research question posed (Bowen, 2009). Thematic analysis was used to identify commonalities and differences with regard to the notion of a teacher in various eras (Fereday & Muir-Cochrane, 2006). The findings revealed that the notion of a secondary mathematics teacher of the current era is completely not a suitable embodiment of a secondary mathematics teacher. The current notion of an embodiment of a secondary mathematics teacher is seriously challenged by this ever growing subject. Secondary mathematics is so large for an individual to acclimatise with (Sarton, 1936), and there seems to be a need for more than an individual to ensure that mathematics is well taught and learned by learners. It is recommended that other studies should be undertaken to determine as to how many individuals can constitute a composite suitable to embody the requirements of an ideal secondary mathematics teacher.

This study focused on the evaluative listening practices of four teachers who participated in an algebra professional development involving lesson study. This instrumental case study operationalizes the enactment of teacher listening followed by teacher questions and responses to define listening-to-question. Also, questioning-to-listen is operationalized as the enactment of purposefully posing questions to posture oneself to listen to students' mathematical thinking. Because of the tacit aspect of teacher listening and the visibility of teacher questioning, interrelating listening and questioning affords teachers an accessible point of entry into developing listening practices. In response to participants wondering as to when evaluative listening is appropriate in the mathematics classroom, this study discusses six instances of teaching excerpts along a continuum of listening orientations from directive to observational to responsive. The results indicate positive aspects of evaluative listening towards an observational and responsive listening stance. Results of the study also confirm a reliance on low-order gathering information questions as the predominant type of teacher question posed in mathematics teaching. This study reveals the necessity of contextualizing teacher questions to inform appropriate uses of evaluative listening. Future professional development should consider emphasizing positive aspects of evaluative listening in mathematics teaching.

This research paper is a description and analysis of a test devised to measure knowledge of the four mathematical operations at the 6th grade level.

The controversy around the inclusion of zero in the number system has been widely documented. Influential mathematicians in various ancient cultures did not accept zero as a number. The idea of the empty set was too abstract and they could not conceptualise division by zero. Surprisingly, understanding of the concept is still a matter of concern today. In spite of expansive reports on and recommendations for developing conceptualisation of the concept, learners and teachers still experience problems similar to those that ancient mathematicians struggled with. The study was initiated by an observation of Grade 7 learners' inability to solve the problems 4 × 0 and 0 ÷ 7 effectively or at all. I investigated why Grade 3 to 6 learners and mathematics teachers on a BEd (in-service) course and an accredited ACE course experience problems with the concept of zero. I was especially interested in the understanding of multiplication and division by zero. I investigated teachers' knowledge of zero's characteristics as a number, the history of zero and how they teach the concept, in order to support my assumptions. The data production process was performed over a period of two years. It involved a multi-case opportunity sample approach embedded in the empirical field that formed the backdrop of my involvement as mathematics education specialist in schools in the Western and Eastern Cape. The interpretative orientation of the study allowed me to conduct inquiries that served to confirm or challenge my assumptions and enabled me to construct generalisations that depict learners' and teachers' knowledge construction. The qualitative data analysis informed the presentation and discussion of the findings. The single most important message conveyed to readers of this study is that the value of zero as a number, its importance in the number system, its properties and its behaviour in calculations, should not be underrated. Teaching of this abstract concept requires competent teachers who are able to mediate understanding in the most effective and innovative manner. Professional development programmes should orchestrate this competence and curriculum developers and textbook authors should acknowledge the significance of learning and teaching the concept of zero.

This is a study of secondary mathematics teachers' knowledge of the dynamics of learning and teaching mathematics in multilingual classrooms in South Africa. It probes teachers' articulated and tacit knowledge through a qualitative methodology that includes In-depth interviews, classroom observations, and reflective workshops. The sample is purposive and theoretical, comprising SIX teachers drawn from three different multilingual school contexts. Categories of description and analytic narrative vignettes enable a qualitative, layered analysis of what the teachers said and how they acted.