Giving effective instructional explanations is one of the most important teacher competences. Recent di dactic literature provides, however, little insight on teacher explanations. In our previous work we devel- oped guidelines for designing comprehensible explanations in the field of business (teacher) education, which are along general lines transferable to other subject areas and target audiences. In this article, we first compare our guidelines to the state of research in general and mathematics didactics. We then in vestigate their applicability to teaching operations research at university level, based on interviews with professors of the international operations research community.

Student Number : 8710172X - MSc research report - School of Education - Faculty of Science / This study investigates mathematics for teaching, specifically in the case of functions at the grade 10 level. One teacher was studied to gain insights into the mathematical problem-solving a teacher does as he/she goes about his/her work. The analysis of data shows that the mathematical problems that this particular teacher confronts as he goes about his work of teaching can be classified as defining, explaining, representing and questioning. The resources that he draws on to sustain and drive this practice can be described as coming from aspects of mathematics, his own teaching experience and the curriculum with which he works. Of interest in this study are those features of mathematical problemsolving in teaching as intimated by other studies, particularly restructuring tasks and working with learners’ ideas; which are largely absent in this practice. This report argues that these latter aspects of mathematical problem-solving in teaching are aligned to a practice informed by the wider notion of mathematical proficiency. The report concludes with a discussion of why and how external intervention is needed to assist with shifting practices if mathematical proficiency is a desired outcome, as well as with reflections on the study and its methodology.

Resumé Studiens syfte var att se om det fanns möjligheter till att kunna anpassa undervisningen i ett klassrum för alla de individuella behov som ryms där. För att finna ett lämpligt svar på ovanstående fundering, beslutade författarna sig för att ta reda på vilka olika undervisningsätt det finns för de elever som lär kinestetiskt, alltså genom att få röra på sig i undervisningen. Detta gjordes genom att lärarna fick ett fiktivt case att utgå ifrån. Studien genomfördes med en kvalitativ inriktning mot ett fenomenografiskt angreppsätt. Författarna valde att göra kvalitativa intervjuer med lärare i skolan. Författarnas tolkningar av lärarnas svar är att de gärna skulle vilja arbeta mer i grupper inom klasserna för att kunna tillgodose alla elevers lärande. Detta gäller inte enbart det kinestetiska lärandet. Arbetssättet av lärarna skiljde sig inte nämnvärt åt, eleverna med det kinestetiska lärandet skulle ha tillgång till en stor mängd konkret material som både lockar till att bygga, fritt eller efter ritningar, samt till att konstruera mönster, vilket är matematikens bas. Lärarna försökte göra undervisningen så varierande som möjligt för att ta tillvara på elevernas förmågor. I olika teman vävdes matematiken in och användes även utomhus. Där kunde eleverna sortera föremål, leta siffror, uppskatta antal och leta efter geometriska former som var av vikt för de som lär kinestetiskt. Resultatet visar att det var läraren som måste använda en undervisnings stil som passade eleverna med det kinestetiska lärandet, det vill säga att undervisningen inte skulle ske på lärarens villkor utan på elevernas. Det var även viktigt att använda olika metoder som läraren själv trodde på, för annars tror inte eleverna att de kommer att lyckas i sitt lärande. / Abstract The purpose of the study was to study if there are some possibilities for adjust teaching in a classroom for all individual needs. To find a suitable answer for the above thoughts, the authors decided to find out more about which different teaching methods there are for student who learn kinesthetic, in other words the students who needs to budge in the teaching. This was done through a fictitious case which was left to the teachers to assume from. The study was done with a qualitative direction with a phenomenographics approach. The authors conducted qualitative interviews with some teachers in school. The authors’ interpretation from the teachers’ answer is that they want to work more within groups in the classes for satisfying all the students learning. These were pertaining not to only the student who learns kinesthetic according to the interviews. The way of teaching didn’t differ between the teachers, and, the students, who learn by the kinesthetic way, should have access to a big part of concrete material, entice to build, free or after plan, and they entice to design model, which is the mathematics basis. The teachers tried to do the teaching as rich of variation as possible to take care of the students’ abilities. In different themes the mathematics was interweaved and used outdoors. There the students could classify object, look for figure, estimate quantity and look for forms of geometry. This was very important for those who learn by the kinesthetic way. The result shows that it is the teachers who have to use teaching styles which fits the student, who learn by the kinesthetic way. It means that the teaching should concern more to the students’ conditions than the teachers. It is also important that the teachers use different methods which the teacher believes in, otherwise the student don’t believe neither and don’t success in their learning.

<p>Resumé</p><p>Studiens syfte var att se om det fanns möjligheter till att kunna anpassa undervisningen i ett klassrum för alla de individuella behov som ryms där. För att finna ett lämpligt svar på ovanstående fundering, beslutade författarna sig för att ta reda på vilka olika undervisningsätt det finns för de elever som lär kinestetiskt, alltså genom att få röra på sig i undervisningen. Detta gjordes genom att lärarna fick ett fiktivt case att utgå ifrån.</p><p>Studien genomfördes med en kvalitativ inriktning mot ett fenomenografiskt angreppsätt. Författarna valde att göra kvalitativa intervjuer med lärare i skolan.</p><p>Författarnas tolkningar av lärarnas svar är att de gärna skulle vilja arbeta mer i grupper inom klasserna för att kunna tillgodose alla elevers lärande. Detta gäller inte enbart det kinestetiska lärandet. Arbetssättet av lärarna skiljde sig inte nämnvärt åt, eleverna med det kinestetiska lärandet skulle ha tillgång till en stor mängd konkret material som både lockar till att bygga, fritt eller efter ritningar, samt till att konstruera mönster, vilket är matematikens bas. Lärarna försökte göra undervisningen så varierande som möjligt för att ta tillvara på elevernas förmågor. I olika teman vävdes matematiken in och användes även utomhus. Där kunde eleverna sortera föremål, leta siffror, uppskatta antal och leta efter geometriska former som var av vikt för de som lär kinestetiskt.</p><p>Resultatet visar att det var läraren som måste använda en undervisnings stil som passade eleverna med det kinestetiska lärandet, det vill säga att undervisningen inte skulle ske på lärarens villkor utan på elevernas. Det var även viktigt att använda olika metoder som läraren själv trodde på, för annars tror inte eleverna att de kommer att lyckas i sitt lärande.</p> / <p>Abstract</p><p>The purpose of the study was to study if there are some possibilities for adjust teaching in a classroom for all individual needs. To find a suitable answer for the above thoughts, the authors decided to find out more about which different teaching methods there are for student who learn kinesthetic, in other words the students who needs to budge in the teaching. This was done through a fictitious case which was left to the teachers to assume from.</p><p>The study was done with a qualitative direction with a phenomenographics approach. The authors conducted qualitative interviews with some teachers in school.</p><p>The authors’ interpretation from the teachers’ answer is that they want to work more within groups in the classes for satisfying all the students learning. These were pertaining not to only the student who learns kinesthetic according to the interviews. The way of teaching didn’t differ between the teachers, and, the students, who learn by the kinesthetic way, should have access to a big part of concrete material, entice to build, free or after plan, and they entice to design model, which is the mathematics basis. The teachers tried to do the teaching as rich of variation as possible to take care of the students’ abilities. In different themes the mathematics was interweaved and used outdoors. There the students could classify object, look for figure, estimate quantity and look for forms of geometry. This was very important for those who learn by the kinesthetic way.</p><p>The result shows that it is the teachers who have to use teaching styles which fits the student, who learn by the kinesthetic way. It means that the teaching should concern more to the students’ conditions than the teachers. It is also important that the teachers use different methods which the teacher believes in, otherwise the student don’t believe neither and don’t success in their learning.</p>

The main objective of this study was to investigate the effect of using a suggested mathematical word-problem training programme on Primary 4 pupils' performance in mathematical word-problems. The study had a pre-post control group design. A treatment and a no-treatment group were exposed to pre-post methods of gathering data (a mathematical word-problem achievement test and a mathematical word-problem attitude scale). The treatment group was given direct and explicit training on how to solve mathematical word-problems, while the pupils of the no-treatment group received no such training; they were taught the same material they study at school. A "t" test was used to compare the means of scores of the control group pupils and those of the experimental group in the pre-post measurements. Results of the study revealed a significant improvement in the experimental group pupils' performance in mathematical word-problems because they had attended the suggested programme. Results also revealed that experimental group subjects' attitudes towards mathematical word-problems underwent an exceptional change because they had attended the suggested programme. Their attitudes towards mathematical word-problems became more positive than before. In the light of the results of the study, some recommendations were made for improving mathematics teacher training programmes, for mathematics teaching, and for further research.

This study investigates the ways in which parents' and teachers' experiences and representations mediate their child's mathematics learning as they make the transition between home and school to either a multiethnic or mainly white school. In particular, it examines if the forms of mediation they adopt can shed light on the academic success of the child in school mathematics. The focus on mathematics learning has been chosen for the study because of its relative neglect, until recent times, to be seen as a subject influenced by cultural representations. Furthermore, there are significant implications in the relative neglect of understanding the achievement of ethnic minority pupils in mathematics. The research was framed by Vygotskian sociocultural theory and Wenger's (1998) communities of practice to explore the construction of meaning, identity and representations of practice. The amalgam of Wenger's communities of practice with sociocultural theory provided three key theoretical facets: (i) multiple levels of understanding in the form of meaning, practice and identity, (ii) the scope to explore the social and cultural worlds of the learner and (iii) understanding the ways that past experiences impact on current practice. Three different forms of qualitative data collection were used within the context of an ethnographic approach: (i) investigations in the form of classroom observations, (ii) in-depth semi-structured interviews and (iii) a child identity task. Twenty-two parents, eight teachers and fifty-eight children took part in the interviews, which form the main part ofthe data analysis. Out ofthese fifty-eight children, twenty-seven undertook the child identity task. The research took place in three schools with different ethnic make-up: a multicultural school, a mainly white school and a predominantly South Asian school. Two year groups were chosen, year 2 (ages 6/7 years) and year 6 (10/11 years), balancing high and low achievers. This study has provided data, which suggests that the way parents and teachers mediate the child's learning involves more than representations of mathematics. In making meaning of the mathematical, they draw on wider representations of the educational world, which include aspects like child development, notions of achievement, past experiences and the child's projected futures. This complex picture emerged from studying the highly interwoven aspects ofthe construction of meaning, identity and representations of practice. Representations of learning can be borrowed from both communities, providing the ethnic minority pupil with the potential to create hybrid representations of learning as they make the transition between home and school, which may be attributed a cultural status within the home. Each social actor has the potential to borrow from the home or school community to a greater or lesser degree. lfthe gap between the shared representations of the home and school are large, then this increases the likelihood of difficulties for the child in transition. However, the data suggests that even if the cultural representations of the home are very different from the school, the identification of high achievement and the engagement in mathematical activity at home can still provide success in learning. From the school community perspective, classrooms were represented by the teacher informants as 'cultureless' in both the multi ethnic and mainly white school. For example, in the multicultural school the teachers felt that there were so many ethnicities that differences were not visible. In the mainly white school, there were so few ethnic minority children that teachers also struggled to identify issues of culture. In the predominantly South Asian school, issues surrounding culture were brought to the forefront of the teacher discourse. However, in many ethnic minority homes, parents described how culture was influential in mediating representations ofleaming. This has implications in the educational arena with respect to the teachers' understanding of the transitional process that ethnic minority children undergo and the levels of visibility that culture and ethnicity is given in the school community.

Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references.

The thesis presents the analytical process and the findings of a study on: lecturers teaching practice with first year undergraduate mathematics modules; and lecturers knowledge for teaching with regard to students mathematical meaning making (understanding). Over three academic semesters, I observed and audio-recorded twenty-six lecturers teaching to a small group tutorial of two to eight first year students, and I discussed with the lecturers about their underlying considerations for teaching. The analysis of this thesis focuses on a characterisation of each of three (of the twenty-six) lecturers teaching, which I observed for more than one semester. I chose the teaching of three experienced lecturers, due to diversity in terms of ways of engaging the students with the mathematics, and due to my consideration of their commitment to teaching for students mathematical meaning making. The distinctive nature of the study is concerned with the conceptualisation of university mathematics teaching practice and knowledge within a Vygotskian perspective. In particular, I used for the characterisation of teaching practice and of teaching knowledge the notions tool-mediation and dialectic from Vygotskian theory. I also used a coding process grounded to the data and informed by existing research literature in mathematics education. I conceptualised teaching practice into tools for teaching and actions with tools for teaching (namely strategies). I then conceptualised teaching knowledge as the lecturers reflection on teaching practice. The thesis contributes to the research literature in mathematics education with an analytical framework of teaching knowledge which is revealed in practice, the Teaching Knowledge-in-Practice (TKiP). TKiP analyses specific kinds of lecturer s knowing for teaching: didactical knowing and pedagogical knowing. The framework includes emerging tools for teaching (e.g. graphical representation, rhetorical question, students faces) and emerging strategies for teaching (e.g. creating students positive feelings, explaining), which were common or different among the three lecturers teaching practice. Overall, TKiP is produced to offer a dynamic framework for researcher analysis of university mathematics teaching knowledge. Analysis of teaching knowledge is important for gaining insights into why teaching practice happens in certain ways. The findings of the thesis also suggest teaching strategies for the improvement of students mathematical meaning making in tutorials.

Recent international reports, for example TIMSS (2011 & 2015), point to serious challenges in South African learner performance in Mathematics and Science. Of greatest concern is that research findings (e.g. Graven, Venkat, Westaway and Tshesane 2013) suggest that many South African learners show signs of mathematical knowledge gaps in the lower grades. Hence, there is a need to address challenges of this nature very early in Foundation Phase. This study was undertaken with a view to contribute towards addressing mathematical challenges encountered by learners in Foundation Phase This empirical enquiry was undertaken under the auspices of the South African Numeracy Chair Project (SANCP) at Rhodes University whose mission is to develop sustainable ways of improving quality teaching and learning of Mathematics in South Africa. A relatively new SANCP programme called Pushing for Progression (PfP) run as part of the after-school Maths Clubs to develop the number sense and four Operations in learners was used to achieve the research aims of this study. Research participants were drawn from the Maths Clubs established by the researcher in a small rural town of Ottosdal in the North West Province of South Africa. This Study is grounded on the Vygotskian perspective and uses the interpretivist qualitative research method for data collection and analysis. Sampling was done opportunistically by enlisting participants (12 teachers and 117 learners) on the basis of their availability and willingness to participate. Pre- and post-assessment of learners’ proficiency on the four Basic Operations was conducted at the beginning and at the end of the research project, respectively. This was done to determine the impact of the project on learner performance. Data analysis was done thematically and through the comparison of learner results of the pre- and post-assessment. The findings point to the effectiveness of the PfP Programme in learner performance. This can be deduced from improved scores between pre- and post-assessment and the observations made by participant-teachers on their respective club learners’ mathematical proficiencies. Accordingly, based on the findings, this study recommends, inter alia, that since the PfP programme is still in its early stages, similar research be conducted elsewhere. Additionally, the Department of Basic Education could consider exploring the PfP programme as one of several other strategies to help improve learner proficiency in Mathematics.

It can be argued that the Namibian curriculum is largely influenced by a Western epistemology. However, many studies suggest that learners make powerful meanings of mathematical concepts they are learning if they visualise these and experience them in relation to what they already know from their own cultural backgrounds. According to the Namibian National Curriculum for Basic Education (NNCBE, 2010), it is not only important for learners to acquire mathematical knowledge and skills, but also to develop and grow their identities, cultures and values as individuals. The aim of this study was to explore and investigate how selected mathematics teachers employ visualisation as a teaching approach to re-contextualise indigenous knowledge (IK). The study set out to explore how conceptual understanding is enhanced by participation in an intervention programme. The use of visualisation is considered an important mediating and pedagogical tool in the mathematics classroom to enhance the mathematics learning of learners. The research is informed by a socio-cultural theory of learning and is located within an interpretive paradigm. The study was conducted at four schools in the Oshana region of Namibia and involved four mathematics teachers who were purposefully selected due to their willingness to use visualisation-IK approaches in their teaching, based on their responses to the survey. The methodologies used are qualitative and quantitative case study. To gather data, document analysis, a survey, lesson observations and focus group interviews were used. The study found that most teachers in the Oshana region have an understanding of the effective use of visualisation-IK approaches. However, the study revealed that visualisation- IK approaches are mostly used in grades 4 to 7 mathematics classrooms. It also discovered that most of the visualisation approaches that teachers employed aligned well with the curriculum and promoted conceptual understanding in the teaching of mathematics. The results of the study showed that teachers have experience of the enabling and constraining factors in using a visualisation-IK approach to teaching mathematics.