ABSTRACT This study investigates what and how mathematics (for teaching) is constituted in classroom practice. Specifically mathematical knowledge for teaching fractions in Grade 7. One teacher was studied to gain insight into the mathematical problemsolving the teacher does and the dilemmas he faces as he goes about his work. The analysis of the data show that the mathematical problem-solving that this particular teacher engaged in can be classified as demonstrating, encouraging and working with learner ideas. He appealed to mathematics (rules & empirical), experience (everyday) and the curriculum (tests and exams) to fix meaning. The mathematical problem solving and appeals he made threw up dilemmas of representing the content, competing goals and student thinking. This aided in providing a description of what mathematics for teaching is in this practice. The report concludes with a discussion of what teachers need to know or study in order to become better mathematics teachers and where do they find these courses to accommodate their need to improve as mathematics teachers.

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.

Thesis (Ph.D.)--University of the Witwatersrand, Faculty of Humanities, School of Education, 2013 / Mathematics-for-teaching (MfT) is complex, multi-faceted and topic-specific. In this study, a Financial Mathematics course for pre-service secondary mathematics teachers provides a revelatory case for investigating MfT. The course was designed and taught by the author to a class of forty-two students at a university in South Africa. Eight students, forming a purposive sample, participated as members of two focus tutorial groups and took part in individual and group interviews. As an instance of insider research, the study makes use of a qualitative methodology that draws on a variety of data sources including lecture sessions and group tutorials, group and individual interviews, students’ journals, a test and a questionnaire. The thesis is structured in two parts. The first part explores revisiting of school mathematics with particular focus on compound interest and the related aspects of percentage change and exponential growth. Four cases are presented, in the form of analytic narrative vignettes which structure the analysis and provide insight into opportunities for learning MfT of compound interest. The evidence shows that opportunities may be provided to learn a range of aspects of MfT through revisiting school mathematics. The second part focuses on obstacles experienced by students in learning annuities, their time-related talk, as well as their use of mathematical resources such as timelines and spreadsheets. A range of obstacles are identified. Evidence shows that students use timelines in a range of non-standard ways but that this does not necessarily determine or reflect their success in solving annuities problems. Students’ use of spreadsheets reveals that spreadsheets are a powerful tool for working with annuities. A key finding with regard to teachers’ mathematical knowledge, and which cuts across both parts of the thesis, is the importance of being able to move between compressed and decompressed forms of mathematics. The study makes three key contributions. Firstly, a framework for MfT is proposed, building on existing frameworks in the literature. This framework is used as a conceptual tool to frame the study, and as an analytic tool to explore opportunities to learn MfT as well as the obstacles experienced by. A second contribution is the theoretical and empirical elaboration of the notion of revisiting. Thirdly, a range of theoretical constructs related to teaching and learning introductory financial mathematics are introduced. These include separate reference landscapes for the concepts of compound interest and annuities

Submitted by Roberta D´Angela Menduni Bortoloti (robertamenduni@yahoo.com.br) on 2016-07-20T14:39:35Z No. of bitstreams: 1 tese_FIM_Roberta.pdf: 154195498 bytes, checksum: 7914e88c9de15de25d87beef14a3d99f (MD5) / Approved for entry into archive by Maria Auxiliadora da Silva Lopes (silopes@ufba.br) on 2016-07-21T14:35:10Z (GMT) No. of bitstreams: 1 tese_FIM_Roberta.pdf: 154195498 bytes, checksum: 7914e88c9de15de25d87beef14a3d99f (MD5) / Made available in DSpace on 2016-07-21T14:35:10Z (GMT). No. of bitstreams: 1 tese_FIM_Roberta.pdf: 154195498 bytes, checksum: 7914e88c9de15de25d87beef14a3d99f (MD5) / UESB / Apresentamos uma matemática para o ensino como um modelo para o ensino do conceito de proporcionalidade. Este modelo permite reunir uma variabilidade de formas de comunicar o conceito de proporcionalidade e (re)apresentá-la por meio de uma estrutura teórica que organiza seus modos de ocorrência. O objetivo geral da pesquisa foi a construção de um modelo de uma matemática para o ensino do conceito de proporcionalidade, no qual identificamos diferentes modos de comunicar o conceito em questão, utilizando três fontes: artigos científicos, um grupo de professores e livros didáticos de matemática. Três objetivos específicos foram propostos para que se alcançasse o objetivo geral. O primeiro consistiu em construir uma matemática para o ensino do conceito de proporcionalidade a partir de uma revisão sistemática de literatura, identificando e sintetizando estudos. Fundamentamos os dois outros objetivos no método qualitativo, sendo o segundo o de construir uma matemática para o ensino do conceito de proporcionalidade a partir de um grupo com professores da educação básica e, o terceiro objetivo construir uma matemática para o ensino do conceito de proporcionalidade a partir de livros didáticos de matemática da educação básica. A justificativa para a escolha do método qualitativo encontra-se na construção do modelo por meio do que é comunicado como proporcionalidade, seja por professores da educação básica ou autores de livros didáticos de matemática. Inspirados em Brent Davis, recorremos ao Estudo do Conceito como dispositivo investigativo para a produção dos diferentes usos do conceito de proporcionalidade. A apropriação que fizemos desse dispositivo, entrelaçado às definições teóricas dos trabalhos desenvolvidos pela pesquisadora Anna Sfard, se constituiu em instrumento de análise e estratégia de modelagem teórica. Os resultados mostraram uma diversidade de realizações do conceito de proporcionalidade, distribuída em três cenários, formando, assim, um modelo teórico para o ensino do conceito de proporcionalidade. No primeiro cenário, o conceito de proporcionalidade foi relatado como razão e realizou-se como taxa, escala, divisão, probabilidade, razão trigonométrica, porcentagem, divisão e quotização proporcionais, vetor e intervalos musicais. No segundo, ele foi descrito pela igualdade entre razões a partir do uso da regra de três, da divisão proporcional de segmentos e da porcentagem. No último cenário, esse conceito foi apresentado como taxa de variação de uma função, podendo ser identificada como uma constante de proporcionalidade, um fator-escala, um coeficiente angular ou uma declividade. / ABSTRACT We present Mathematics for the teaching as a model for the teaching of the proportionality concept. This model allows to gather a variability of ways of communicating the proportionality concept and (re) introduce it through a theoretical structure that organizes its ways of occurrence. The general objective of the study was the building of a model of Mathematics for the teaching of the proportionality concept. We have identified three different ways to communicate this concept, through the use of three sources: scientific papers, a group of teachers and mathematics textbooks. There were proposed three specific objectives in order to achieve the general objective. The first one was to build Mathematics for the teaching of the proportionality concept from a systematic review of literature, through the identification and syntheses of the studies. We have founded the two other objectives in the qualitative method, being the second one to build Mathematics for the teaching of the proportionality concept through a group with Elementary School teachers, and the third one to build Mathematics for the teaching of the proportionality concept through textbooks of Mathematics in Elementary School. The reason for the choice of the qualitative method can be found in the building of the model through the way of what has been taught as proportionality, has it been done by Elementary School teachers or authors of mathematics textbooks. Being inspired by Brent Davis, we used the Concept Study as an investigative tool for the production of the different uses of the proportionality concept. The appropriation that we made of this tool, together with the theoretical definitions of the work by the researcher Anna Sfard, were used in the analysis and strategy of theoretical modeling. The results showed diversity for the proportionality concept that had been distributed in three different landscapes and, this way, creating a theoretical model for the teaching of the proportionality concept. In the first landscape, the proportionality concept was related as ratio and it was hold as rate, scale, division, probability, trigonometric ratio, percentage, proportional division and partition, vector and music intervals. In the second one, it was described through the equality between ratios through the use of the rule of three, the proportional division of segments and percentage. In the last landscape, this concept was presented as a rate of variation of a function and it can be identified as a constant of proportionality, a scale factor, an angular coefficient or a declivity.

Submitted by Jean Coutinho (jeanlbiko@hotmail.com) on 2015-12-16T19:31:10Z No. of bitstreams: 1 MatematicaparaoEnsinoAC.pdf: 2609857 bytes, checksum: aae033891c7c90a4c130cf4f93a30ae3 (MD5) / Approved for entry into archive by Maria Auxiliadora da Silva Lopes (silopes@ufba.br) on 2015-12-17T18:56:16Z (GMT) No. of bitstreams: 1 MatematicaparaoEnsinoAC.pdf: 2609857 bytes, checksum: aae033891c7c90a4c130cf4f93a30ae3 (MD5) / Made available in DSpace on 2015-12-17T18:56:16Z (GMT). No. of bitstreams: 1 MatematicaparaoEnsinoAC.pdf: 2609857 bytes, checksum: aae033891c7c90a4c130cf4f93a30ae3 (MD5) / O objetivo deste estudo foi modelar uma Matemática para o Ensino do conceito de combinação simples em Análise Combinatória. Os materiais de análise utilizados nesta pesquisa foram observados em duas fontes: produções científicas a partir de uma Revisão Sistemática e um estudo com professores. A estrutura de análise proposta foi o Estudo do Conceito e suas ênfases: realizações, panoramas e vinculações. Para tal propósito, foi analisado um corpus de dez artigos publicados em periódicos brasileiros, nas áreas de Educação e Ensino, avaliados pelo sistema WebQualis da CAPES como A1, A2, B1 e B2. Além disso, foi organizado um estudo coletivo cujos integrantes foram seis professores atuantes nos níveis fundamental, médio e/ou superior que possuíam experiência no ensino de Análise Combinatória. Como resultado, foi apresentado um modelo de Matemática para o Ensino de combinação simples, estruturado em quatro panoramas: formalista, instrumental, ilustrativo e comparativo, que sugerem implicações para o fazer do professor que ensina combinação simples e desdobramentos da pesquisa. / ABSTRACT The aim of this study was to model a Mathematics for Teaching the concept of simple combination in Combinatory Analysis. Materials observed in this investigation came from two sources: a systematic review of scientific production and a study with teachers. The proposed structure for the analysis was a Concept Study in its emphases: realizations, landscapes and entailments. In favor of that, a corpus of ten articles published in Brazilian journals in the areas of Education and Teaching was analyzed, all of them evaluated by CAPES’ system WebQualis as A1, A2, B1 and B2. In addition, there was a collective study with six teachers acting in primary, secondary and/or higher education who had experience in teaching Combinatory Analysis. As a result, presented a model of Mathematics for Teaching the concept of simple combination, structured in four landscapes: formalist, instrumental, illustrative, and comparative, which suggest implications for the actions of the teacher that teaches simple combination, and for possible outspread of research.

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