This thesis describes an experiment using technology to develop conceptual understanding of functions through graphical representations. It examines the effects of including dynamic representations in a conceptual approach to the teaching of functions. The study was implemented over a 5-day period in a Grade 9 class in a small, generally working class, rural school in Eastern Massachusetts. Participating students were observed during class discussions and video analysis, and their written responses and created functions were analyzed. The procedure used in the experiment was based on the Theory of Didactic Situations and used the Didactic Engineering methodology. The structure and sequencing of the thesis is also based on these concepts. Conclusions are drawn regarding the effects of using multiple representations systems to deepen understanding of functional relationships and suggested improvements to the introduction of the function concept in high school instructional programs are given.

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<p>The purpose of the study was to investigate prospective &lsquo / A&rsquo / level mathematics teachers&rsquo / knowledge of the concept of a function. The study was a case study of six prospective Zimbabwean teachers who were majoring in mathematics with the intention of completing a programme leading to certification as secondary mathematics teachers. At the time of the study the six prospective teachers were in their final year of study. Prospective teachers&rsquo / knowledge of the concept of a function was assessed through task-based interviews and reflective interviews. These interviews, which were done over a period of three months, were structured to capture the prospective teachers&rsquo / subject matter knowledge and pedagogical content knowledge for teaching the concept of a function. The interviews were also meant to capture the prospective teachers&rsquo / underlining pedagogical reasons for their choices of the examples, representations and teaching approaches when planning to teach the concept.</p> <p>As part of the study a theoretical framework for understanding prospective teachers&rsquo / knowledge of the concept of a function was developed. The framework, which was developed, was used as an analytical tool in analyzing prospective teachers knowledge of the concept of a function. The results of the study indicated that the prospective teachers had a process conception of a function although some of them had given a set-theoretic definition of a function in which a function is perceived as a mathematical object. They also confined the notion of a function to sets of real numbers. Functions defined on other mathematical objects (for example, the differential operator and the determinant function) were not considered as functions by five of the six prospective teachers.</p>

This study set out to investigate how the use of a Smart Board impacts on the teaching and learning of algebraic functions. The research took place in a school equipped with Smart Boards in each Mathematics classroom. Data collection involved lesson observations in three classes over three lessons each. The teachers and learners were interviewed post observation and the data obtained were analysed according to Sfard‘s three-phase model framework to determine if the learners had a procedural or object view of a function after having been taught on a Smart Board. The findings show that by using a Smart Board learners had both procedural and object view of functions however, much of the teaching occurred in a way which would have been possible without the use of a Smart Board, indicating that teachers did not fully utilise the potential of such a technological tool. However, it emerged that visualisation played an important role in allowing learners to operate on functions as objects. So while the visualization that technology enables encouraged reification or allowed teachers and learners to operate on functions as a whole or even on families of functions, this appeared simply to 'speed up‘ the normal teaching-learning process rather than promote the explorative and investigative aspect of learning. Still, it must be acknowledged that this kind of practice is bound to strengthen these learners‘ function concepts as was evident in the ways they appeared to operate confidently on the objects as shown in the study. It must be acknowledged that teachers were extremely enthusiastic about the possibilities of the technology and were inspired to use technology more in their lessons to allow learners‘ visualisation of concepts. Positive comments made by learners showed that they too, were also motivated by the use of the Smart Board. / Thesis (M.Ed.)-University of KwaZulu-Natal, Pietermaritzburg, 2011.

<p>The purpose of the study was to investigate prospective &lsquo / A&rsquo / level mathematics teachers&rsquo / knowledge of the concept of a function. The study was a case study of six prospective Zimbabwean teachers who were majoring in mathematics with the intention of completing a programme leading to certification as secondary mathematics teachers. At the time of the study the six prospective teachers were in their final year of study. Prospective teachers&rsquo / knowledge of the concept of a function was assessed through task-based interviews and reflective interviews. These interviews, which were done over a period of three months, were structured to capture the prospective teachers&rsquo / subject matter knowledge and pedagogical content knowledge for teaching the concept of a function. The interviews were also meant to capture the prospective teachers&rsquo / underlining pedagogical reasons for their choices of the examples, representations and teaching approaches when planning to teach the concept.</p> <p>As part of the study a theoretical framework for understanding prospective teachers&rsquo / knowledge of the concept of a function was developed. The framework, which was developed, was used as an analytical tool in analyzing prospective teachers knowledge of the concept of a function. The results of the study indicated that the prospective teachers had a process conception of a function although some of them had given a set-theoretic definition of a function in which a function is perceived as a mathematical object. They also confined the notion of a function to sets of real numbers. Functions defined on other mathematical objects (for example, the differential operator and the determinant function) were not considered as functions by five of the six prospective teachers.</p>

The introduction of graphing calculators into senior secondary schools and mandating of their use in high stakes assessment makes student expertise in finding a complete graph of a function essential. This thesis investigated the cognitive, metacognitive, mathematical, and technological processes senior secondary students used in seeking a complete graph of a difficult cubic function. A pretest of function knowledge was administered to two mixed ability classes in their final two years of secondary school. Five pairs of experienced users of TI-83 or 82 graphing calculators from these classes were audio and videotaped solving a problem task. Protocols were constructed and subjected to intensive qualitative macroanalysis and microanalysis using tools developed by the researcher from Schoenfeld’s work. / The findings were: (1)all students demonstrated understanding of the local and global nature of functions and the synthesis of these in determining a complete graph; (2) a range of mathematical and graphing calculator knowledge was applied in seeking a global view of the function with their combined application being more efficient and effective; (3) an understanding of automatic range scaling features facilitated efficient finding of a global view; (4) all pairs demonstrated having a clear mental image of the function sought and the possible positions of the calculator output relative to this; (5) students were able to resolve situations involving unexpected views of the graph to determine a global view; (6) students displayed understanding of local linearity of a function; (7) when working in the graphical representation, students used the algebraic but not the numerical representation to facilitate and support their solution; (8) scale marks were used to produce more elegant solutions and facilitate identification of key function features to produce a sketch but some students misunderstood the effect of altering these; (9) pairs differed in the proportion of cognitive and metacognitive behaviours demonstrated with question asking during evaluation supporting decision making; (10) correct selection of xxi an extensive range of graphing calculator features and use of dedicated features facilitated efficient and accurate identification of coordinates of key function features.

Submitted by Filipe dos Santos (fsantos@pucsp.br) on 2017-12-04T12:05:44Z No. of bitstreams: 1 Hércules Nascimento Silva.pdf: 5439522 bytes, checksum: ddbe77eb19d9ab9cca85c3cbacd80cab (MD5) / Made available in DSpace on 2017-12-04T12:05:44Z (GMT). No. of bitstreams: 1 Hércules Nascimento Silva.pdf: 5439522 bytes, checksum: ddbe77eb19d9ab9cca85c3cbacd80cab (MD5) Previous issue date: 2017-10-02 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The research aimed to verify to what extent dynamic constructions in GeoGebra and applied in a sequence of activities make it possible to facilitate learning of function or the real function defined by sentences. Some constructs for the teaching and learning of function concept, present in the Imagiciel, a computational environment and five notebooks with activities developed by French researchers between the late 1980s and early 1990s, which don't run more in current computers, were rebuilt in GeoGebra. With such elaborate reconstructions in this free software with more features and accessibility, some proposals present in a notebook of the Imagiciel were organized in a sequence of activities inspired by the Dialectic Object-Tool and applied to a group of students in the first year of high school. As research methodology Didactic Engineering elements were used with a priori and a posteriori analysis. With the posteriori analysis of replies given by the students of the survey, by the interaction between them and with the buildings established by the GeoGebra applet shape’s, it was found that the activities allowed students to build the concept of function, in particular, the function defined by sentences in real intervals and so answer the question this research / A investigação relatada neste documento teve como objetivo verificar em que medida construções dinâmicas no GeoGebra e aplicadas em uma sequência de atividades possibilitam facilitar a aprendizagem de função, em especial, a função real definida por sentenças. Algumas construções referentes ao ensino e aprendizagem do conceito de função, presentes no Imagiciel, um ambiente computacional e cinco cadernos com atividades desenvolvidas por pesquisadores franceses entre o final da década de 1980 e início da década de 1990, que não rodam mais nos computadores atuais, foram reconstruídas no GeoGebra. Com tais reconstruções elaboradas nesse software livre com mais recursos e acessibilidade, algumas propostas presentes em um caderno do Imagiciel foram organizadas em uma sequência de atividades inspiradas na dialética ferramenta-objeto e aplicadas a um grupo de alunos do primeiro ano do Ensino Médio. Como metodologia de pesquisa foram utilizados elementos da Engenharia Didática que contou com análise a priori e a posteriori. Com a análise a posteriori das respostas dadas pelos sujeitos da pesquisa, pela interação entre os alunos e com as construções elaboradas pelo GeoGebra, em forma de applet’s, verificou-se que as atividades permitiram que os alunos construíssem o conceito de função, em especial, o de função definida por sentenças em intervalos reais e assim responder a questão dessa pesquisa

Orientador: Rita de Cassia Pavani Lamas / Banca: Tatiana Bertoldi Carlos / Banca: Jéfferson Luiz Rocha Bastos / Resumo: O ensino de Matemática anseia por mudanças de metodologias e estas estão previstas nos Parâmetros Curriculares Nacionais, em particular, a Metodologia de Resolução de Problemas introduzida por George Polya no início do século passado. Hoje esta metodologia é estudada por diversos pesquisadores e é indicada no Currículo do Estado de São Paulo. Os objetivos deste trabalho se resumem em fundamentar teoricamente as propriedades da função quadrática para obtenção do seu gráfico; apresentar aspectos da metodologia de Resolução de Problemas e os resultados de sua aplicação na terceira série do Ensino Médio para o ensino da função quadrática / Abstract: Teaching of Mathematics craves change methodologies and these are set out in National Curriculum Parameters, in particular, the Troubleshooting methodology introduced by George Polya early last century. Today this methodology is studied by many researchers and is indicated in the curriculum of the State of São Paulo. The objectives of this work are summarized in theory support the properties of quadratic function to obtain your chart; present aspects of the methodology Troubleshooting and results of its application in the third year of high school for teaching quadratic function / Mestre

Made available in DSpace on 2016-04-01T17:55:15Z (GMT). No. of bitstreams: 0 Previous issue date: 2015-06-26. Added 1 bitstream(s) on 2016-04-01T18:01:15Z : No. of bitstreams: 1 000859950.pdf: 425478 bytes, checksum: 657621c1147fa7a19791e4c14552ae24 (MD5) / O ensino de Matemática anseia por mudanças de metodologias e estas estão previstas nos Parâmetros Curriculares Nacionais, em particular, a Metodologia de Resolução de Problemas introduzida por George Polya no início do século passado. Hoje esta metodologia é estudada por diversos pesquisadores e é indicada no Currículo do Estado de São Paulo. Os objetivos deste trabalho se resumem em fundamentar teoricamente as propriedades da função quadrática para obtenção do seu gráfico; apresentar aspectos da metodologia de Resolução de Problemas e os resultados de sua aplicação na terceira série do Ensino Médio para o ensino da função quadrática / Teaching of Mathematics craves change methodologies and these are set out in National Curriculum Parameters, in particular, the Troubleshooting methodology introduced by George Polya early last century. Today this methodology is studied by many researchers and is indicated in the curriculum of the State of São Paulo. The objectives of this work are summarized in theory support the properties of quadratic function to obtain your chart; present aspects of the methodology Troubleshooting and results of its application in the third year of high school for teaching quadratic function

In the school mathematics curriculum functions play a pivotal role in accessing and mastering algebra and the whole of mathematics. The study investigated the extent to which pupils with little experience in algebra would develop the function concept and was motivated by the need to bring the current Zimbabwean mathematics curriculum in line with reform ideas that introduce functions early in the secondary school curriculum. An instrument developed from literature review was used to assess the extent to which the Form1/Grade 8 pupils developed the concept. The teaching experiment covered a total of 26 lessons, a period of about eight weeks spread over two terms starting in the second term of the Zimbabwean school calendar. The problem-centred teaching approach based on the socio-constructivist view of learning formed the background to facilitate pupils' individual and social construction of knowledge. Data was collected from the pupils' written work, audio taped discussions and interviews with selected pupils. The extent to which each pupil of the seven pupils developed the aspects of function, change, relationship, rule, representation and strategies, was assessed. The stages of development and thinking levels of functional reasoning at the beginning of the experiment, then during the learning phase and finally at the end of the experiment, were compared. The results showed that functions can be introduced at Form 1 and pupils progressed in the understanding of most of the aspects of a function. / Educational Studies / M. Ed. (Mathematics Education)