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.

The exploratory study sought to elicit and document mathematics teachers‟ views on how they enacted the process of transition between contexts and mathematics. The study pursued to understand teachers‟ beliefs and knowledge of mathematics. A mixed methods sequential explanatory research design was employed where a quantitative phase was followed by the connecting phase and concluded through a qualitative phase involving three case studies. A purposive sample of 165 practicing teachers who had registered for a professional advancement developmental course at a university participated in the study by voluntarily completing a survey questionnaire. From this sample, three cases of individual teachers were pursued. The first two cases involved conducting in-depth interviews with the teacher who had rated sentences in the questionnaire differently while the last case involved the recording of an interview of one individual using field notes. The questionnaire sought teachers‟ biographical details (section A), their views on contexts and mathematics (section B) and their rating of sentences in a passage with regard to the mathematics embedded in the sentences (section C). Semi-structured interviews were conducted in the qualitative phase to elicit in-depth views of the teachers‟ regarding the research problem. All the instruments were tested for validity and reliability. Quantitative data gathered was analysed using frequencies, percentages, cross tabulations, bar charts and pie charts as well as the calculation of Pearson chi-square tests (Cohen, Manion & Morrison, 2011). Descriptive and inferential statistics were used to collate teachers‟ views from which themes were drawn and related to make inferences. It was found that teachers‟ positive views about contexts and mathematics did not translate into them recognising mathematics in some mathematics potent contexts as captured in their ratings in section C of the questionnaire. Statistically significant associations were recorded to support this. The study also conceptualised a mathematical participation model (MP-model) as a tool to describe and analyse participation that involves the use of real world data in the teaching and learning of mathematics. The MP-model involves four components, viz. the community of practice (CoP), real world data, mathematics and a model in which members of the CoP tap into the real world data and mathematics to model their participation. The study recommends the MP-model as tool for description and enactment of full mathematical participation. / Mathematical Sciences / D.Litt.et. Phil. (Mathematics Education)

Mathematics education is a human activity that helps to develop mental processes in order to enhance logical and critical thinking which will contribute to one’s decision-making process and to solve problems (DBE, 2011c). For one to be able to do Mathematics, strategies should be generated in order to solve problems. The performance in Mathematics is very poor and educational researchers have identified various reasons for the poor performance in mathematics. Therefore, South Africa has developed an assessment tool known as the ANA, to determine the learners’ weaknesses in mathematics at national, provincial, district and school level (DBE, 2013). An approach research (in the South African context) was to explore Intermediate Phase Mathematics teacher’s awareness of their metacognitive strategy use. The main purpose of my research study was to understand, to what extent Intermediate Phase Mathematics teachers become aware of metacognitive strategies during an adapted lesson study process. To achieve this purpose, the study aims to investigate the teachers’ awareness of metacognitive strategies before and during an adapted lesson study process. Empirical qualitative research based on a design research approach took place within the interpretative paradigm. Descriptive data was generated by means of semi-structured focus group interviews and a reflective diary was held with double-medium participants who were selected. The data were analysed by means of content analyses which proceeded by using mind maps, where codes and themes were related to the literature. The results show that most of the teachers were aware of the metacognitive strategies, but it can be that they lack knowing when, where and how to use these metacognitive strategies as they do not plan their lessons on a regular basis. Teachers also feel more comfortable when planning lesson collaboratively as they feel that they learn from one another. In conclusion an adapted lesson study could be a positive plan of action to provide teachers with the opportunity to plan lessons collaboratively and reflect on one another’s’ ideas. Teachers can become more aware of their metacognitive strategies when planning lessons in order to implement these metacognitive strategies during their lessons. In this way learners could be empowered to become metacognitive (think about their thinking) and to reflect on their actions which might contribute to their performance of mathematics. / MEd (Mathematics Education), North-West University, Potchefstroom Campus, 2015

Mathematics education is a human activity that helps to develop mental processes in order to enhance logical and critical thinking which will contribute to one’s decision-making process and to solve problems (DBE, 2011c). For one to be able to do Mathematics, strategies should be generated in order to solve problems. The performance in Mathematics is very poor and educational researchers have identified various reasons for the poor performance in mathematics. Therefore, South Africa has developed an assessment tool known as the ANA, to determine the learners’ weaknesses in mathematics at national, provincial, district and school level (DBE, 2013). An approach research (in the South African context) was to explore Intermediate Phase Mathematics teacher’s awareness of their metacognitive strategy use. The main purpose of my research study was to understand, to what extent Intermediate Phase Mathematics teachers become aware of metacognitive strategies during an adapted lesson study process. To achieve this purpose, the study aims to investigate the teachers’ awareness of metacognitive strategies before and during an adapted lesson study process. Empirical qualitative research based on a design research approach took place within the interpretative paradigm. Descriptive data was generated by means of semi-structured focus group interviews and a reflective diary was held with double-medium participants who were selected. The data were analysed by means of content analyses which proceeded by using mind maps, where codes and themes were related to the literature. The results show that most of the teachers were aware of the metacognitive strategies, but it can be that they lack knowing when, where and how to use these metacognitive strategies as they do not plan their lessons on a regular basis. Teachers also feel more comfortable when planning lesson collaboratively as they feel that they learn from one another. In conclusion an adapted lesson study could be a positive plan of action to provide teachers with the opportunity to plan lessons collaboratively and reflect on one another’s’ ideas. Teachers can become more aware of their metacognitive strategies when planning lessons in order to implement these metacognitive strategies during their lessons. In this way learners could be empowered to become metacognitive (think about their thinking) and to reflect on their actions which might contribute to their performance of mathematics. / MEd (Mathematics Education), North-West University, Potchefstroom Campus, 2015

published_or_final_version / Education / Master / Master of Science in Information Technology in Education

Studien behandlar matematiklärarnas syn på de sju matematiska förmågorna och hur förmågorna kommer till uttryck i lärarnas undervisning. Syftet studeras med hjälp av intervjuer av matematiklärare i gymnasieskolan samt observationer av lärarnas lektioner. Resultatet av studien visar på att lärarna är medvetna om hur deras undervisning behandlar de matematiska förmågorna. Vidar visar studien att lösning av rutinuppgifter dominerar undervisningen, vid dessa tillfällen berörs till största del procedur- och begreppsförmågan. I studien framkom att lärarna anser att det finns oklarheter i de matematiska förmågornas definition och att arbetet med förmågorna är tidskrävand. Baserat på undersökningen kan det tänkas att förmågorna anses vara tidskrävande på grund av att det inte finns material som är anpassat för arbete med förmågorna samt att nya områden inom matematiken måste introduceras i ett högt tempo. Lärarna lyfter att en positiv aspekt med de matematiska förmågorna är att de bidrar till en varierad undervisning. / This study´s purpose is to show mathematics teachers view on the seven mathematical competencies and how the competencies reflect the teacher’s teaching. The study is based on interviews of mathematics teachers in secondary schools and observations of the teacher’s lessons. The result of the study shows that teachers are aware of how their teaching covers the mathematical competencies. The study also shows that solving routine tasks dominates teaching. In these occasions mainly the procedural and conceptual competency are involved. It appeared that the teachers are unhappy with uncertainties in the definitions of the mathematical competencies. The teachers highlight that the competencies are a time consuming process. This study reveals that it can depend on that there is no material that is adapted to the work with the competencies and that new areas within mathematics must be introduced frequently. The teachers believe the positive aspects of the mathematical competencies are that the competencies contribute to a variation in the teaching.

The problem of this study was to determine the extent to which certain characteristics of teachers of first-year algebra were related to a) their pupils' attitudes toward algebra and b) their pupils' achievements in algebra. An additional aspect of the study was to ascertain and study the differences between Anglo- and Latin-American pupils' attitudes and achievements in algebra.

The problem of this study was to determine if the teachers of Algebra I rated highest and lowest according to the Texas Teacher Appraisal System differed significantly in a measure of achievement. The analyses indicated that there were significant differences in achievement between the classes taught by the highest and lowest ranked teachers.

Implementation of Oregon’s Educational Reform Act (HB 3565 and HB 2991) provides the context for this inquiry as its emphasis on problem solving has impacted mathematics teaching and learning throughout the state. Even though all Oregon teachers are responding to the same policy, their goals in teaching problem solving vary. These goals and these practices are influenced by the way teachers view the role of problem solving in the curriculum. Further, their practice is influenced by their knowledge and beliefs about mathematics content, teaching, learning, and the reform policy. The questions addressed in this study are: (1) What do exemplary middle school math teachers do to engage students in mathematical problem solving? and (2) On what bases do these teachers make decisions about what to emphasize when teaching problem solving? how to teach problem solving?, and when to teach problem solving? This qualitative study provides a fuller description of Standards-based classroom practice than presently represented in the literature by offering both examples of problem solving practice and the related influences on that practice. It considers the influences of policy, curriculum, professional development, administrators, and colleagues on teachers’ developing practice. The study also grounds the implementation of the Principles and Standards for School Mathematics (National Council of Teachers of Mathematics [NCTM], 2000) in the work of practicing middle school teachers. Finally, the study shows how, for these teachers, their curriculum has played a significant role in developing their perspectives on learning, teaching, and the nature of math, which has in turn, influenced their knowledge, beliefs, and instructional practice. This study demonstrates that teachers are able to teach in ways consistent with the NCTM Standards when their knowledge and beliefs about practice align with the recommendations. Further, they teach in this manner when professional development experiences are geared toward understanding and developing Standards-based instructional practice, curriculum is consistent with this vision of practice, and administrators and school cultures are supportive of such practice. When these internal and external conditions exist within and for teachers, their students have the opportunity to learn to become “problem solvers,” not just “problem performers.”

Sammanfattning Studiens fokus var dels vilka kategorier av innehåll som framkom i lärarnas kommunikation dels vilka förändringar i innehållet som uttolkades i matematiklärares kommunikation i början och i slutet av ett kompetensutvecklingsprojekt. Kontexten i kompetensutvecklingsprojektet var i form av modellen learning study, där sju lärarlag, sammanlagt 20 lärare deltagit. Ramverket i studien var tematisk analys, analys av matematiska kompetenser samt funktionell textanalys. Studiens resultat visar på flera förändringar, till exempel att de fem uttolkade huvudkategorierna innefattade fler underkategorier i slutet av kompetensutvecklingsprojektet. När det gäller matematiskt innehåll så visar analysen att lärarna kommunicerade matematika kompetenser på ett mer mångfacetterat sätt i slutet av kompetensutvecklingsprojektet. Då lärarna kommunicerade matematikundervisning använde de sig till viss del av ett annat sätt att uttrycka sig i slutet än i början av kompetensutvecklingsprojektet. I början talade de till exempel om att höra vad eleverna uttrycker till att i slutet använda sig av ordet lyssna. Resultatet visade även förändringar avseende hur lärarna samtalade om laborativt material. Fokus har flyttats från det laborativa materialet till att lärarna funderade över vilken matematik som eleverna uttryckte med hjälp av det laborativa materialet. / The focus of this study was to find out which categories of content emerged from the teachers' communication as well as which content changes could be seen in the communication of mathematics teachers in the early and final stages of a skills development project. The context of the project was in the form of a learning study in which a total of 20 teachers, divided into seven teams, participated. The theoretical framework of this study was thematic analysis, mathematical competencies as well as mathematical content, and functional text analysis. The results of the study identified five main categories and found several communication changes. Near the end of the project several subcategories emerged. In terms of mathematical content, the analysis shows that the teachers communicated more aspects of mathematical competencies emerged later in the project rather than in the beginning. While communicating their mathematics teaching, the teachers changed their way of expressing themselves over the course of the project. In the beginning, for example, they talked about hearing what their students said while, by the end, they talked about listening. This study also identified changes regarding how the teachers conversed about common tools and educational materials. Focus shifted from the common tools and educational materials to a consideration of the mathematics the students expressed with the help of said common tools and educational materials