La créativité est considérée comme une compétence cruciale pour le monde contemporain. La recherche décrite dans cette thèse a eu comme contexte principal le projet MC Squared. Réalisé entre octobre 2013 et septembre 2016. L'objectif du projet était de développer une plate-forme numérique pour le développement de C-books destinés à l'enseignement des mathématiques de manière à développer la pensée mathématique créative chez les étudiants et les auteurs. Cette thèse propose une analyse de la conception, du développement, de la mise en oeuvre et du test des activités numériques et non numériques dans le but d'améliorer et d'encourager la pensée mathématique créative ayant des fonctions et des algorithmes comme objets mathématiques à analyser. Les questions de recherche suivantes ont été soulevées à partir du problème: -Comment opérationnaliser et réviser les définitions existantes de la pensée mathématique créative? -Quels sont les composants nécessaires d'une situation ou d'un artefact permettant un processus de pensée mathématique créative? -Comment pouvons-nous évaluer l'avancement d'un processus impliquant la pensée mathématique créative?-Le modèle "Diamant de la créativité" est-il un outil d'analyse utile pour cartographier le cheminement du processus créatif? Pour répondre à ces questions, la recherche a suivi une méthodologie basée sur une recherche agile basée sur le design. Quatre activités ont été développées de manière cyclique. Le premier, appelé Function Hero, est un jeu numérique qui utilise les mouvements du corps du joueur pour évaluer la capacité de reconnaissance des fonctions. Trois autres activités appelées Binary Code, Fake Binary Code et Op'Art, visant au développement de la pensée computationnelle. Le modèle principal de cette thèse est le modèle "Diamond de créativité" pour cartographier le processus de résolution des problèmes rencontrés dans chaque activité, en évaluant le processus et les produits dérivés du travail des étudiants.Pour valider les hypothèses de recherche, nous avons collecté des données pour chaque activité et les avons analysées quantitativement et qualitativement. Les résultats montrent que les activités développées ont éveillé et engagé les étudiants dans la résolution de problèmes et que le modèle "Diamond of Creativity" peut aider à identifier et à identifier des points spécifiques du processus de création / Creativity is considered as a crucial skill for the contemporary world. The research described in this thesis had the Project MC Squared as the main context. Carried out between October 2013 and September 2016. The objective of the project was to develop a digital platform for the development of C-books for teaching mathematics in a way that develops Creative Mathematical Thinking both in the students and the authors. This thesis, entitled: Didactic Situations for the Development of Creative Mathematical Thinking proposes an analysis of the design, development, implementation, and testing of digital and non-digital activities with the aim of improving and fostering Creative Mathematical Thinking having Functions and Algorithms as mathematical objects to analyze. The following research questions raised from the problem: • How to operationalize and revise existing definitions of Creative Mathematical Thinking? • How can we assess the progress of a process involving Creative Mathematical Thinking? • How the "Diamond of Creativity" model is an useful analytic tool to map the Creative Process path? To answer such questions, the research followed a methodology based on an agile Design-Based Research. Four activities were cyclically developed. The first one, called: "Function Hero," is a digital game that uses body movements of the player to evaluate recognizability of functions. Three other activities called "Binary Code," "FakeBinary Code" and "Op’Art", aimed at the development of Computational Thinking. The main constructs of this thesis are: (a) the "Diamond of Creativity" model to map the process of solving problems found in each activity, evaluating the process and the products derived from the students’ work. (b) The digital game: "Function Hero". To validate the research hypotheses, we collected data from each activity and analyzed them quantitatively and qualitatively. The results show that developed activities have awakened and engaged students into problem-solving and that the "Diamond of Creativity" model can help in identifying and labeling specific points in the creative process

This study investigates how five secondary students think mathematically and process information in a technology-supported environment while solving mathematics problems. In the study, students were given open-ended problems to explore in an online dynamic learning environment and to solve the problems in computer environments. Given that all the work was done in the computer environments, both online and offline, students’ work was recorded by using screen capturing software. A new method, the frame analysis method, was used to describe and analyze students’ thinking processes while they were interacting with mathematical objects in the dynamic learning environment and solving mathematics problems. The frame analysis method is a microgenetic method based on information processing theory and is developed to analyze students’ work done in computer environments. Two reasons make the analysis method used in this study unique: (a) collecting data with minimized disturbance of the students and (b) analyzing students’ artefacts through researcher’s (teacher) perspective, meaning that integrates teachers within the analysis process. The frame analysis method consists of multiple steps to observe, describe, interpret, and analyze students’ mathematical thinking processes when they are solving mathematics problems. I described each step in detail to explain how the frame analysis method was used to monitor students’ mathematical thinking and to track their use of technology while solving problems. The data emerged from this study illustrates the importance of using dynamic learning environments in mathematics and the potential for transformation of mathematical representational systems from symbolic to visual. Moreover, data suggest that visual representation systems and linked multi-representational systems encourage students to interact with mathematical concepts and advance their mathematical understanding. Rather than dealing with the grammar of algebra only, students may benefit from direct interaction with the visually represented mathematical concepts. It appears that recording students’ problem-solving processes may engage teachers and mathematics educators to seek opportunities for implementing process-oriented assessment into their curriculum activities. Furthermore, students may benefit from sharing their work through peer collaboration, either online or offline, and metacognition and self-assessment. Suggestions for further studies include using audio and video recording in the frame analysis method.

Problems with students’ performance in Mathematics at tertiary level are common in South Africa − as it is worldwide. Pass rates at the university of technology where the researcher is a lecturer, are only about 50%. At many universities it has become common practice to refer students who do not have a reasonable chance to succeed at university level, for additional support to try to rectify this situation. However, the question is which students need such support? Because the Grade 12 marks are often not perceived as dependable, it has become common practice at universities to re-test students by way of an entrance exam or the "National Benchmark Test"- project. The question arises whether such re-testing is necessary, since it costs time and money and practical issues make it difficult to complete timeously. Many factors have an influence on performance in Mathematics. School-level factors include articulation of the curriculum at different levels, insufficiently qualified teachers, not enough teaching time and language problems. However, these factors also affect performance in most other subjects, but it is Mathematics and other subjects based on Mathematics that are generally more problematic. Therefore this study focused on the unique properties of the subject Mathematics. The determining role of prior knowledge, the step-by-step development of mathematical thinking, and conative factors such as motivation and perseverance were explored. Based on the belief that these factors would already have been reflected sufficiently in the Grade 12 marks, the correlation between the marks for Mathematics in Grade 12 and the Mathematics marks at tertiary level was investigated to assess whether it was strong enough for the marks in Grade 12 Mathematics to be used as a reliable predictor of success or failure at university level. It was found that the correlation between the marks for Mathematics Grade 12 and Mathematics I especially, was strong (r = 0,61). The Mathematics marks for Grade 12 and those for Mathematics II produced a correlation coefficient of rs = 0,52. It also became apparent that failure in particular could be predicted fairly accurately on the basis of the Grade 12 marks for Mathematics. No student with a Grade 12 Mathematics mark below 60% succeeded in completing Mathematics I and II in the prescribed two semesters, and only about 11% successfully completed it after one repetition. The conclusion was that the reliability of the prediction based on the marks for Grade 12 Mathematics was sufficient to refer students with a mark of less than 60% to receive some form of additional support. / MEd, Learning and Teaching, North-West University, Vaal Triangle Campus, 2011

Esta dissertação apresenta o desenvolvimento de uma pesquisa referente ao ensino da Análise Combinatória, por meio da Resolução de Problemas, em uma turma de 6º ano do Ensino Fundamental. Para isso, elaborou-se uma sequência didática que buscava proporcionar aos educandos um contato com esse conteúdo antes do Ensino Médio. A partir dessa sequência analisou-se como a Resolução de Problemas, segundo Onuchic e Allevato, auxiliou os alunos a compreender os conceitos iniciais de Análise Combinatória, buscando também como referencial teórico o estudo referente ao Pensamento Matemático, de David Tall. Concluímos que a Resolução de Problemas auxiliou a expandir e modificar as Imagens dos Conceitos que os alunos possuíam com relação à Análise Combinatória. / This dissertation shows the development of research related to teaching Combinatorics, through Problem Solving, at a 6th grade level. A lesson plan was prepared and aimed to confront students of middle school with problems involving Combinatorics, allowing them to work with such concepts before high school. Based on this lesson plan, our intent was to verify how Problem Solving, according to Onuchic e Allevato, helped the students to understand initial concepts of Combinatorics. Also, using David Tall’s studies about Mathematical Thinking as reference. We could verify that the Problem Solving Theory helped the students to expand and modify their Concept Images related to Combinatorics.

Problems with students’ performance in Mathematics at tertiary level are common in South Africa − as it is worldwide. Pass rates at the university of technology where the researcher is a lecturer, are only about 50%. At many universities it has become common practice to refer students who do not have a reasonable chance to succeed at university level, for additional support to try to rectify this situation. However, the question is which students need such support? Because the Grade 12 marks are often not perceived as dependable, it has become common practice at universities to re-test students by way of an entrance exam or the "National Benchmark Test"- project. The question arises whether such re-testing is necessary, since it costs time and money and practical issues make it difficult to complete timeously. Many factors have an influence on performance in Mathematics. School-level factors include articulation of the curriculum at different levels, insufficiently qualified teachers, not enough teaching time and language problems. However, these factors also affect performance in most other subjects, but it is Mathematics and other subjects based on Mathematics that are generally more problematic. Therefore this study focused on the unique properties of the subject Mathematics. The determining role of prior knowledge, the step-by-step development of mathematical thinking, and conative factors such as motivation and perseverance were explored. Based on the belief that these factors would already have been reflected sufficiently in the Grade 12 marks, the correlation between the marks for Mathematics in Grade 12 and the Mathematics marks at tertiary level was investigated to assess whether it was strong enough for the marks in Grade 12 Mathematics to be used as a reliable predictor of success or failure at university level. It was found that the correlation between the marks for Mathematics Grade 12 and Mathematics I especially, was strong (r = 0,61). The Mathematics marks for Grade 12 and those for Mathematics II produced a correlation coefficient of rs = 0,52. It also became apparent that failure in particular could be predicted fairly accurately on the basis of the Grade 12 marks for Mathematics. No student with a Grade 12 Mathematics mark below 60% succeeded in completing Mathematics I and II in the prescribed two semesters, and only about 11% successfully completed it after one repetition. The conclusion was that the reliability of the prediction based on the marks for Grade 12 Mathematics was sufficient to refer students with a mark of less than 60% to receive some form of additional support. / MEd, Learning and Teaching, North-West University, Vaal Triangle Campus, 2011

This study investigates how five secondary students think mathematically and process information in a technology-supported environment while solving mathematics problems. In the study, students were given open-ended problems to explore in an online dynamic learning environment and to solve the problems in computer environments. Given that all the work was done in the computer environments, both online and offline, students’ work was recorded by using screen capturing software. A new method, the frame analysis method, was used to describe and analyze students’ thinking processes while they were interacting with mathematical objects in the dynamic learning environment and solving mathematics problems. The frame analysis method is a microgenetic method based on information processing theory and is developed to analyze students’ work done in computer environments. Two reasons make the analysis method used in this study unique: (a) collecting data with minimized disturbance of the students and (b) analyzing students’ artefacts through researcher’s (teacher) perspective, meaning that integrates teachers within the analysis process. The frame analysis method consists of multiple steps to observe, describe, interpret, and analyze students’ mathematical thinking processes when they are solving mathematics problems. I described each step in detail to explain how the frame analysis method was used to monitor students’ mathematical thinking and to track their use of technology while solving problems. The data emerged from this study illustrates the importance of using dynamic learning environments in mathematics and the potential for transformation of mathematical representational systems from symbolic to visual. Moreover, data suggest that visual representation systems and linked multi-representational systems encourage students to interact with mathematical concepts and advance their mathematical understanding. Rather than dealing with the grammar of algebra only, students may benefit from direct interaction with the visually represented mathematical concepts. It appears that recording students’ problem-solving processes may engage teachers and mathematics educators to seek opportunities for implementing process-oriented assessment into their curriculum activities. Furthermore, students may benefit from sharing their work through peer collaboration, either online or offline, and metacognition and self-assessment. Suggestions for further studies include using audio and video recording in the frame analysis method.

This paper aims to present a model of teaching and learning mathematics in China. The model is “Two Basic”, basic knowledge and basic ability. Also, the paper will analyze some of the background of the model and why it is efficient in mathematics education. The model is described by a framework of “slab” and based on a model of learning cycle, allow students to work with mathematical thinking. Though the model looks of demonstration and practice looks very traditional, the explanation behind allows us to understand why Chinese students achieved well in many international studies in mathematics. The innovation of the model is the teacher intervention during the learning process. Such interventions include repeated practice, and working on group of selected related questions so that abstraction of the learning process is possible and student can link up mathematical expression and process. Examples used in class are included and the model can be applied in teaching advanced mathematics, which is usually not the case in some many other existing theories or framework.

Esta dissertação apresenta o desenvolvimento de uma pesquisa referente ao ensino da Análise Combinatória, por meio da Resolução de Problemas, em uma turma de 6º ano do Ensino Fundamental. Para isso, elaborou-se uma sequência didática que buscava proporcionar aos educandos um contato com esse conteúdo antes do Ensino Médio. A partir dessa sequência analisou-se como a Resolução de Problemas, segundo Onuchic e Allevato, auxiliou os alunos a compreender os conceitos iniciais de Análise Combinatória, buscando também como referencial teórico o estudo referente ao Pensamento Matemático, de David Tall. Concluímos que a Resolução de Problemas auxiliou a expandir e modificar as Imagens dos Conceitos que os alunos possuíam com relação à Análise Combinatória. / This dissertation shows the development of research related to teaching Combinatorics, through Problem Solving, at a 6th grade level. A lesson plan was prepared and aimed to confront students of middle school with problems involving Combinatorics, allowing them to work with such concepts before high school. Based on this lesson plan, our intent was to verify how Problem Solving, according to Onuchic e Allevato, helped the students to understand initial concepts of Combinatorics. Also, using David Tall’s studies about Mathematical Thinking as reference. We could verify that the Problem Solving Theory helped the students to expand and modify their Concept Images related to Combinatorics.

Submitted by EDUARDO CARDOSO DE SOUZA null (eduardoc@fc.unesp.br) on 2016-04-07T19:46:41Z No. of bitstreams: 1 DISSERTAÇÃO_VERSÃO_FINAL.pdf: 4673018 bytes, checksum: 18ab7c434cd38e291de6086e815d206a (MD5) / Approved for entry into archive by Felipe Augusto Arakaki (arakaki@reitoria.unesp.br) on 2016-04-08T12:52:50Z (GMT) No. of bitstreams: 1 souza_ec_me_bauru.pdf: 4673018 bytes, checksum: 18ab7c434cd38e291de6086e815d206a (MD5) / Made available in DSpace on 2016-04-08T12:52:50Z (GMT). No. of bitstreams: 1 souza_ec_me_bauru.pdf: 4673018 bytes, checksum: 18ab7c434cd38e291de6086e815d206a (MD5) Previous issue date: 2016-02-25 / Não recebi financiamento / O baixo desempenho em matemática obtido pelos alunos brasileiros nas avaliações externas revela que o ensino da matemática é um grande desafio tanto para as nações marcadas pela desigualdade social, quanto para as nações mais desenvolvidas. Em busca da superação dos baixos índices de rendimento dos estudantes em matemática, este trabalho de cunho qualitativo, investigou por meio da aprendizagem situada numa comunidade de prática de programadores as formas pelas quais os alunos com baixo desempenho em matemática se relacionam com a mesma durante e após participarem de oficinas de programação. A pesquisa contou com oito oficinas de programação utilizando a ferramenta Processing 2. Foi possível delinear seis categorias de análise que, conforme os objetivos da pesquisa, sinalizam arranjos da aprendizagem situada da matemática numa comunidade de prática de programadores - não linearidade, ênfase no saber fazer, informal, construção do conhecimento a partir da necessidade, interatividade e engajamento e tentou-se buscar subsídios para responder perguntas como: Quais são as vantagens de aulas de matemática utilizando ferramentas de programação? Quais são as dificuldades ao desenvolver aulas de matemática utilizando ambiente de programação? Que tipos de mudança o uso de ferramentas de programação provocam na dinâmica das aulas de matemática? Para a realização da pesquisa foi escolhida uma escola da rede municipal de ensino fundamental do município de Santa Cruz do Rio Pardo – SP, com seis alunos, na faixa etária de 12 anos que apresentavam baixo rendimento na disciplina. A programação de computadores como instrumento didático-pedagógico, no emprego da ferramenta Processing 2 aponta avanços na aprendizagem dos alunos, no tocante a promoção de um trabalho mais cooperativo, que coloca o erro como elemento natural no processo de aprendizagem, e torna a aprendizagem mais interativa, contribuindo para um feedback imediato, e avança sobretudo na apropriação de conceitos matemáticos adjacentes da atividade de programação. Foi possível evidenciar que no transcorrer das oficinas os alunos passaram a perceber a necessidade e importância da matemática enquanto constroem programas que representam seus anseios e desejos. Assim o conhecimento parte da necessidade pontual e, com isso, explorar uma aprendizagem mais ativa. No desencadeamento das oficinas os estudantes deixam de ser meramente receptores de informações, e se engajam de maneira ativa nas práticas da comunidade, o que contribui para uma aprendizagem ativa. Um dos indicativos de sucesso para a nova alfabetização do século XXI está na interatividade e no tempo rápido de respostas conforme se evidenciou neste trabalho. Enquanto os alunos trabalhavam no desenvolvimento dos projetos, criando as sequências de comandos, eles estavam a aprender sobre o processo de construção, do processo de como formalizar uma ideia e transformá-la num projeto completo e funcional. Assim, estavam a “programar para aprender”. À medida que se avança nas oficinas, os alunos aumentam sua atitude reflexiva, de modo que a ferramenta passou a ser apenas um suporte. Neste processo encontravam seus erros, procuravam corrigi-los, testavam e aprendiam os conceitos envolvidos na solução dos problemas. / The poor performance in mathematics obtained by Brazilian students in the external evaluation shows that mathematics teaching is a major challenge for both nations marked by social inequality, and for the most developed nations. In search of overcoming low levels of student performance in mathematics, the qualitative nature of work, investigated through learning located in a developer community of practice the ways in which students with low math performance relate to the same during and after participating in scheduling workshops. The research included eight workshops programming using Processing tool 2. It was possible to outline six categories of analysis that, as the research objectives, signal arrangements situated learning of mathematics in a developer community of practice - nonlinearity, emphasis on know-how, informal, knowledge building from the need, interactivity and engagement and we tried to get subsidies to answer questions such as: What are the advantages of math classes using programming tools? What are the difficulties in developing math classes using programming environment? What kinds of changes using programming tools cause the dynamics of math classes? For the research was chosen a school elementary school municipal system of the city of Santa Cruz do Rio Pardo - SP, with six students, aged 12 who had low performance in the discipline. The programming computers as didactic and pedagogical tool, the use of Processing 2 tool shows progress in student learning, regarding the promotion of a more cooperative work, which puts the error as a natural element in the learning process, and makes learning more interactive, contributing to an immediate feedback, and advancing above all in the ownership of adjacent mathematical concepts of programming activity. It became clear that in the course of the workshops students have come to realize the need and importance of mathematics as they build programs that represent their wishes and desires. Thus the knowledge of the specific need and, therefore, explore a more active learning. In triggering the workshops students are no longer merely information receptors, and to engage actively in community practices, which contributes to an active learning. One of the success indicative for the new literacy of the twenty-first century is on interactivity and quick response time as was evident in this work. While students worked in project development, creating the command sequences, they were to learn about the construction process, as the process of formalizing an idea and turn it into a complete and functional design. So were "programmed to learn." As we advance in the workshops, students increase their reflexive attitude, so that the tool just happened to be a support. In this process they found his mistakes, sought to correct them, were testing and learned the concepts involved in solving problems.

Esta dissertação apresenta o desenvolvimento de uma pesquisa referente ao ensino da Análise Combinatória, por meio da Resolução de Problemas, em uma turma de 6º ano do Ensino Fundamental. Para isso, elaborou-se uma sequência didática que buscava proporcionar aos educandos um contato com esse conteúdo antes do Ensino Médio. A partir dessa sequência analisou-se como a Resolução de Problemas, segundo Onuchic e Allevato, auxiliou os alunos a compreender os conceitos iniciais de Análise Combinatória, buscando também como referencial teórico o estudo referente ao Pensamento Matemático, de David Tall. Concluímos que a Resolução de Problemas auxiliou a expandir e modificar as Imagens dos Conceitos que os alunos possuíam com relação à Análise Combinatória. / This dissertation shows the development of research related to teaching Combinatorics, through Problem Solving, at a 6th grade level. A lesson plan was prepared and aimed to confront students of middle school with problems involving Combinatorics, allowing them to work with such concepts before high school. Based on this lesson plan, our intent was to verify how Problem Solving, according to Onuchic e Allevato, helped the students to understand initial concepts of Combinatorics. Also, using David Tall’s studies about Mathematical Thinking as reference. We could verify that the Problem Solving Theory helped the students to expand and modify their Concept Images related to Combinatorics.