This is a qualitative research that draws on Fairclough’s Critical Discourse Analysis methodology to analyze the discourse practices of the mathematics teacher educators in initial teacher training colleges in Malawi. The study involved four mathematics teacher educators in two teacher training colleges located in two different regions of Malawi. Specifically the study explored the following questions: 1) What are the discourse practices that mathematics teacher educators display in their descriptions of multilingual mathematics classrooms? 2) a) What are the discourse practices that mathematics teacher educators display in a college mathematics classroom? b) How do they make available the discourse practices for the student teachers to draw on? Data was collected through pre-observation interviews, classroom observations, reflective interviews and focus group discussions with the mathematics teacher educators. This study has shown that while there are some disconnections between the discourse practices produced in a school multilingual mathematics classroom and a college mathematics classroom, some of the discourse practices that mathematics teachers produced in a college mathematics classroom reinforces the common discourse practices being produced in multilingual mathematics classroom. There are three common discourse practices that were displayed in a college mathematics classroom. These discourse practices are: Initial-Response-Evaluation (Pimm, 1987), traditional lecturing and group discussions. I observed that the IRE and traditional lecturing discourse practices were accompanied by directive discourses for procedural control, and the procedural discourse was the prevalent discourse in all the discourse practices produced. iv Three major themes have emerged from the data analysis. Firstly, the research findings indicate that the mathematics teacher educators regard multilingualism and the language practices that come with it such as code-switching more as a problem rather than a resource for teaching and learning. Secondly, code-switching in college mathematics classroom is not as spontaneous as is research shows it to be in schools; rather it is very much controlled and restricted. Thirdly, the dilemmas of code-switching as discussed by Adler (1998, 2001) are more acute in teacher training colleges, mainly because of the mismatch in the Language-in-Education Policy (LiEP) in schools and tertiary level.

Made available in DSpace on 2016-04-27T16:57:53Z (GMT). No. of bitstreams: 1 dissertacao_eduardo_sad_costa.pdf: 3568903 bytes, checksum: 4e09f1b15f7714b64ad56708b0bd9974 (MD5) Previous issue date: 2007-05-21 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This work involves a qualitative study about whether and how mathematics High-School teachers work with their students the trouble-situations regarding linear Diophantine equations. The study was performed by means of analyzing semi-structured interviews applied on six mathematics teachers from the states of São Paulo and Minas Gerais, teaching at high-school level. The Numbers Elementary Theory has been treated by several researchers on Mathematical Education, as Campbell e Zazkis (2002), Resende (2007), as an adequate subject for the introduction and development of fundamental Mathematical ideas in High- School. However, the results of such investigation show that, although the interviewed teachers affirmed that they did work with problems of discreet mathematics that can be modeled through linear Diophantine equations, none of them seemed to work with their students using the knowledge of these equations properties in order to decide whether they have solution, and what these solutions would be / Neste trabalho apresento um estudo qualitativo sobre se, e como, professores de Matemática do Ensino Médio trabalham com seus alunos situações-problema que recaem em equações diofantinas lineares. O estudo foi feito por meio da análise de entrevistas semi-estruturadas realizadas com seis professores de Matemática dos estados de São Paulo e Minas Gerais que lecionam no Ensino Médio. A Teoria Elementar dos Números vem sendo tratada por diversos pesquisadores de Educação Matemática, como Campbell & Zazkis (2002), Resende (2007), como assunto propício para a introdução e desenvolvimento de idéias Matemáticas fundamentais no Ensino Básico. No entanto os resultados desta investigação indicam que embora os professores entrevistados afirmassem trabalhar com problemas de matemática discreta modeláveis via equação diofantina linear, nenhum deles deu indícios de trabalhar com seus alunos utilizando conhecimentos das propriedades dessas equações para decidir se as mesmas tem solução e quais seriam essas soluções

A presente pesquisa teve por objetivo analisar os modos de interação e as relações de alunos jovens e adultos com o conhecimento matemático dentro e fora da escola, bem como as possibilidades de aproximação entre conhecimento matemático escolar e não escolar. As referências teóricas compõem-se da concepção de Bernard Charlot (2001) sobre as interações do jovem com o saber; da noção de aprendizagem situada desenvolvida por Jean Lave e Etienne Wenger (1991); e da análise da matemática como cultura feita por Alan Bishop (1999). O desenvolvimento do trabalho apoia-se em análise de bibliografia sobre a temática aqui questão e em dados levantados por meio de acompanhamento de aulas e de entrevistas realizadas com alunos e um professor de duas classes de Educação de Jovens e Adultos de uma escola pública da cidade de São Paulo. Entre os principais resultados do trabalho, podem-se destacar a possibilidade de diálogo entre o conhecimento matemático escolar e o conhecimento matemático adquirido pelos alunos em diferentes contextos não escolares, bem como a possibilidade de relação entre contexto e aprendizagem de modo que cada ambiente crie situações e artefatos próprios para enriquecer momentos de aprendizagem. / This research aimed to analyze the modes of interaction and relationships of young and adult students with mathematical knowledge, inside and outside school, as well as possibilities of approach between mathematical knowledge school and non-school. The theoretical references consist of the conception of Bernard Charlot (2001) on the relationship of youth with knowledge; the idea of situated learning of Jean Lave & Etienne Wenger (1991); and the analysis made by Alan Bishop (1999) of mathematics as a culture. The work development is based on analysis of bibliography on the topic and data collected through monitoring classes and interviews with students and teacher of two classes of youth and adults in a public school in the city of São Paulo. Among the highlight results of the study, its present the possibility of dialogue between the school mathematical knowledge and mathematical knowledge acquired by students in different non-school contexts, as well as the relationship between context and learning, so that each environment creates situations and artifacts to enrich learning moments.

Made available in DSpace on 2016-04-27T16:58:29Z (GMT). No. of bitstreams: 1 Paulo Rogerio Salomao.pdf: 1515343 bytes, checksum: e8054aa96548e5060d5d9c759a64a899 (MD5) Previous issue date: 2007-10-02 / In the first term of 2005, I joined the Professional Master s degree on Mathematics Teaching at PUC/SP. In this same year, the research project AProvaME, whose goals are: investigating concepts about argumentation and proofs of teenager students at schools from São Paulo state; structuring groups composed by teachers and researchers in order to elaborate activities involving students in the building process of knowledge, arguments and proofs in Mathematics, the use of technology and the investigating the teacher s role as the mediator of this process. As a part of this project, I will structure my dissertation in order to investigate two situations. The first one to verify to what extent, by the teacher s mediation and by the activities proposed, it is possible to engage students in argument, justification and proof of conjectures about Arithmetical Progressions. On the second one, investigating if the use of technology can favor the building of arguments, justification and proofs in Arithmetical Progressions by the students. Oriented by these questions, I tried to raise some observations of how the teacher s mediation should be done, using activities related to Arithmetical Progressions to engage the students in argument, justifying and proof situations, as well as which type and how to use the technologies available: first of all, I realized the need for the teacher s mediation after each ending of a group of activities, making a closure, or else, proposing to the students that they needed to confront and discuss, giving arguments, justifying their answers, so that everyone could proceed to the following activities without compromising their conjectures; subsequently; I verified that the use of technology is an incentive to the performing of activities in any area of knowledge, because the students feel motivated to build geometrical figures in the computer to solve the Mathematics exercises, concluding, with relation to the use of technology, I noticed that in the activities of this essay the usage of one more computational tool for the validation of students answers, as the Excel software, could complement the results obtained. This essay was based, mainly on the nine types of tasks extracted from Balacheff et al. text (2001). The methodology used was the teaching experiment, always looking for an improvement, not only in the activity, but also in the teacher-studenttechnology interaction. The research involved 10th graders from the evening shift of a State public network school / No primeiro semestre de 2005, ingressei no curso de Mestrado Profissional em Ensino de Matemática na PUC/SP. Neste mesmo ano, iniciava-se o projeto de pesquisa AProvaME, cujos objetivos são: investigar concepções sobre argumentação e prova de alunos adolescentes em escolas do Estado de São Paulo; formar grupos compostos por professores e pesquisadores para elaboração de atividades envolvendo alunos em processos de construção de conhecimento, argumentos e provas em Matemática e o uso de tecnologia e investigar o papel do professor como mediador neste processo. Por fazer parte deste projeto, estruturarei minha dissertação para investigar duas situações. A primeira para verificar em que medida, por meio da mediação do professor e das atividades propostas, é possível engajar os alunos em situações de argumentar, justificar e provar conjecturas sobre Progressões Aritméticas. Na segunda, investigar se o uso de tecnologia pode favorecer a construção de argumentos, justificativas e provas em Progressões Aritméticas pelos alunos. Orientado por essas questões, procurei levantar algumas observações de como deve ser feita a mediação do professor, utilizando atividades de Progressões Aritméticas para engajar os alunos em situações de argumentações, justificativas e provas, bem como qual tipo e como usar as tecnologias disponíveis: em primeiro lugar, percebi a necessidade da mediação do professor a cada término de atividade ou a cada final de um grupo de atividades, fazendo um fechamento, ou seja, propondo que os alunos confrontassem e discutissem, argumentando e justificando suas respostas, para que todos pudessem prosseguir com as atividades seguintes sem comprometimento de suas conjecturas; em seguida, verifiquei que o uso de tecnologia é um incentivo para a realização de atividades em qualquer área do conhecimento, pois os alunos sentem-se motivados por construir figuras geométricas no computador para a resolução de exercícios de Matemática; ao finalizar, com relação ao uso da tecnologia, constatei que nas atividades deste trabalho a utilização de mais uma ferramenta computacional para validação das respostas dos alunos, como o software Excel, poderia complementar os resultados obtidos. Este trabalho fundamentou-se, sobretudo nos nove tipos de tarefas extraídos do texto de Balacheff et al. (2001). A metodologia utilizada foi o experimento de ensino, objetivando sempre um aperfeiçoamento, tanto das atividades, como da interação professor aluno tecnologia. A pesquisa envolveu oito alunos da 1ª série do Ensino Médio do período noturno de uma escola da rede pública estadual

Syftet med detta examensarbete är att undersöka undervisning och lärande under en matematiklektion som präglas av traditionell skolmatematik. Metoden för undersökningen var en deltagande observation av en matematiklektion i åk 3 på gymnasiet. Med hjälp av begreppen matematikens lärandeobjekt, matematiska resurser, eleven som lärande aktör och sociomatematiska normer har jag tolkat de resultat som genererats från observationen. Två slutsatser som kan dras av undersökningen är att eleverna stimuleras till att bli oberoende lärande aktörer i undervisningen av traditionell skolmatematik samt att det i första hand är läraren som synliggör potentiella matematiska resurser för eleverna. Medvetenheten om elevernas användande av matematiska resurser skulle kunna påverka elevernas lärande genom att läraren synliggör matematiska resurser på ett mer medvetet sätt. / The aim with this degree project is to examine teaching and learning during a mathlesson characterized by traditional school mathematics. The method of the study was aparticipant observation of a mathematics lesson in year 3 in upper secondary school. Using the concepts of mathematical learning objects, mathematical resources, and pupil as an active learner in combination with socio-mathematical norms, I have interpreted the results generated from the observation. Two main conclusions can be drawn from the study. Firstly, pupils are encouraged to become independent as active learners in the teaching of traditional school mathematics. Secondly, it is primarily the teacher who makes potential mathematical resources visible and available for the pupils. With an increasing awareness of pupils’ use of mathematical resources, teachers can affect pupils’ learning by making potential mathematical resources explicit in a more conscious way.

In 2004, the Transition Mathematics Project (TMP), funded by the state of Washington and The Bill and Melinda Gates Foundation, was established to create projects to help high school students gain the necessary skills to become college and work-ready. Aligned to TMP's College Readiness Mathematics Standards, a fourth-year capstone mathematics course was developed and implemented, titled Applications in Mathematical Reasoning (AMR), a rigorous course option for students to take during their senior year of high school. The purpose of this study was to explore any relationship between taking the AMR course and preparation for college level mathematics. Using causal-comparative study design and matching participants in the sample, variables were examined based on the number of precollege courses taken; college level math course completed and grade earned; and placement test results for students who took the AMR course compared to those students who took no mathematics during their high school senior year. Though findings for precollege and college level course-taking were inconclusive, mathematics placement test scores were found to be significantly higher for those students who completed the AMR course. The placement test findings supported other research that links rigorous mathematics courses taken in high school with improved college placement and persistence. Based on the research examined and the study findings, there was support to consider the following: (a) creating alternate but rigorous math course offerings for the high school senior year; (b) striving toward a four-years of mathematics graduation requirement for all high schools; (c) enacting mandatory placement at the community college for students placing into precollege courses; and (d) reducing barriers to successful transition between high schools and post secondary institutions by fostering K-16 communication, aligning standards, and improving course alignment. / Graduation date: 2012

The author shares some of the findings of the research he conducted in 2007 on grade 11 mathematics learners in two schools, one experimental and the other one control. In his study, the author claims that an open-ended approach towards teaching and learning of mathematics enhances understanding of mathematics by the learners. The outcomes of the study can be summarised as follows: 1. In the experimental school, where the author intervened by introducing an open-ended approach to teaching mathematics (by means of giving the learners an open-ended approach compliant worksheet to work on throughout the intervention period), the performance of the learners in the post-test was better than that of the learners from the control school. Both schools were of similar performance in the pre-test. The two schools wrote the same pre-test and same post-test. Both schools were following common work schedule. 2. Within the experimental school, post-test performance of the learners in the class where the intervention was monitored throughout the intervention period (thus ensuring compliance of the teacher to the open-ended approach) out-performed those in which monitoring was less frequent. 3. There was no significant difference in performance between learners from the unmonitored experimental class and those from the control class.

Text in Afrikaans, abstract in Afrikaans and English / Hierdie studie is daarop gemik om te bepaal tot welke mate wiskundeleerlinge op skool en onderwysstudente in wiskunde, onderrig in logika ontvang as agtergrond vir strenge bewysvoering. Die formele aspek van wiskunde op hoerskool en tersiere vlak is besonder belangrik. Leerlinge en studente kom onvermydelik met hipotetiese argumente in aanraking. Hulle leer ook om die kontrapositief te gebruik in bewysvoering. Hulle maak onder andere gebruik van bewyse uit die ongerymde. Verder word nodige en voldoende voorwaardes met stellings en hulle omgekeerdes in verband gebring. Dit is dus duidelik dat 'n studie van logika reeds op hoerskool nodig is om aanvaarbare wiskunde te beoefen. Om seker te maak dat aanvaarbare wiskunde beoefen word, is dit nodig om te let op die gebrek aan beheer in die ontwikkeling van 'n taal, waar woorde meer as een betekenis het. 'n Kunsmatige taal moet gebruik word om interpretasies van uitdrukkings eenduidig te maak. In so 'n kunsmatige taal word die moontlikheid van foutiewe redenering uitgeskakel. Die eersteordepredikaatlogika, is so 'n taal, wat ryk genoeg is om die wiskunde te akkommodeer. Binne die konteks van hierdie kunsmatige taal, kan wiskundige toeriee geformaliseer word. Verskillende bewystegnieke uit die eersteordepredikaatlogika word geidentifiseer, gekategoriseer en op 'n redelik eenvoudige wyse verduidelik. Uit 'n ontleding van die wiskundesillabusse van die Departement van Onderwys, en 'n onderwysersopleidingsinstansie, volg dit dat leerlinge en studente hierdie bewystegnieke moet gebruik. Volgens hierdie sillabusse moet die leerlinge en studente vertroud wees met logiese argumente. Uit die gevolgtrekkings waartoe gekom word, blyk dit dat die leerlinge en studente se agtergrond in logika geheel en al gebrekkig en ontoereikend is. Dit het tot gevolg dat hulle nie 'n volledige begrip oor bewysvoering het nie, en 'n gebrekkige insig ontwikkel oor wat wiskunde presies behels. Die aanbevelings om hierdie ernstige leemtes in die onderrig van wiskunde aan te spreek, asook verdere navorsingsprojekte word in die laaste hoofstuk verwoord. / The aim of this study is to determine to which extent pupils taking Mathematics at school level and student teachers of Mathematics receive instruction in logic as a grounding for rigorous proof. The formal aspect of Mathematics at secondary school and tertiary levels is extremely important. It is inevitable that pupils and students become involved with hypothetical arguments. They also learn to use the contrapositive in proof. They use, among others, proofs by contradiction. Futhermore, necessary and sufficient conditions are related to theorems and their converses. It is therefore apparent that the study of logic is necessary already at secondary school level in order to practice Mathematics satisfactorily. To ensure that acceptable Mathematics is practised, it is necessary to take cognizance of the lack of control over language development, where words can have more than one meaning. For this reason an artificial language must be used so that interpretations can have one meaning. Faulty interpretations are ruled out in such an artificial language. A language which is rich enough to accommodate Mathematics is the first-order predicate logic. Mathematical theories can be formalised within the context of this artificial language. Different techniques of proof from the first-order logic are identified, categorized and explained in fairly simple terms. An analysis of Mathematics syllabuses of the Department of Education and an institution for teacher training has indicated that pupils should use these techniques of proof. According to these syllabuses pupils should be familiar with logical arguments. The conclusion which is reached, gives evidence that pupils' and students' background in logic is completely lacking and inadequate. As a result they cannot cope adequately with argumentation and this causes a poor perception of what Mathematics exactly entails. Recommendations to bridge these serious problems in the instruction of Mathematics, as well as further research projects are discussed in the final chapter. / Curriculum and Institutional Studies / D. Phil. (Wiskundeonderwys)

Over the past several decades there has been an emphasis on educational research pertaining to learners’ performance in Mathematics and on finding methods to improve learner performance in this subject. In South Africa, Grade 12 learners’ results in Mathematics from 2010 to 2013 were unsatisfactory as shown in DBE, 2013a. The teachers are challenged to find new teaching methods that will make the subject more interesting and appealing to the learners (Oliver & Makar, 2010 in Goos, 2010). The purpose of this study was to investigate the effect of using computers in the teaching and learning of Mathematics with special reference to the topic of linear functions in order to improve learner performance. The literature reviewed shows that the use of computers not only improves the learners’ performance but also changes their attitude towards Mathematics (Bester & Brand, 2013). The quantitative research approach was used to gather the data, namely the quasi- experimental, non-equivalent control group pre-test-post-test design. Two intact classes formed part of the research study, that is an experimental group (n=50) and control group (n=50). The experimental group learnt the concept of linear function using GeoGebra software. The control group learnt the same concept through the traditional pen and paper method. The data were analysed using the SPSS on ANOVA. The results indicated that there was a significant difference between the mean scores of the experimental group (μ=70.5) and the control group (μ=47.5). From the results it was evident that the use of computers had a positive effect on learners understanding of linear functions as reflected in their performance and on their attitude towards Mathematics, as seen in the questionnaire responses. / Mathematics Education / M. Ed. (Mathematics Education)

A presente pesquisa teve por objetivo analisar os modos de interação e as relações de alunos jovens e adultos com o conhecimento matemático dentro e fora da escola, bem como as possibilidades de aproximação entre conhecimento matemático escolar e não escolar. As referências teóricas compõem-se da concepção de Bernard Charlot (2001) sobre as interações do jovem com o saber; da noção de aprendizagem situada desenvolvida por Jean Lave e Etienne Wenger (1991); e da análise da matemática como cultura feita por Alan Bishop (1999). O desenvolvimento do trabalho apoia-se em análise de bibliografia sobre a temática aqui questão e em dados levantados por meio de acompanhamento de aulas e de entrevistas realizadas com alunos e um professor de duas classes de Educação de Jovens e Adultos de uma escola pública da cidade de São Paulo. Entre os principais resultados do trabalho, podem-se destacar a possibilidade de diálogo entre o conhecimento matemático escolar e o conhecimento matemático adquirido pelos alunos em diferentes contextos não escolares, bem como a possibilidade de relação entre contexto e aprendizagem de modo que cada ambiente crie situações e artefatos próprios para enriquecer momentos de aprendizagem. / This research aimed to analyze the modes of interaction and relationships of young and adult students with mathematical knowledge, inside and outside school, as well as possibilities of approach between mathematical knowledge school and non-school. The theoretical references consist of the conception of Bernard Charlot (2001) on the relationship of youth with knowledge; the idea of situated learning of Jean Lave & Etienne Wenger (1991); and the analysis made by Alan Bishop (1999) of mathematics as a culture. The work development is based on analysis of bibliography on the topic and data collected through monitoring classes and interviews with students and teacher of two classes of youth and adults in a public school in the city of São Paulo. Among the highlight results of the study, its present the possibility of dialogue between the school mathematical knowledge and mathematical knowledge acquired by students in different non-school contexts, as well as the relationship between context and learning, so that each environment creates situations and artifacts to enrich learning moments.