In our presentation we will show how technology enables us to improve the teaching and learning of linear functions at the middle school level. Through various classroom activities that involve technology such as dynamic geometry software, graphing calculators and Excel, students explore functions and discover basic facts about them on their own. Students then work with real life data and on real life problems to draw graphs and form linear models that correspond to given situations as well as draw inferences based on their models. Participants will receive complete classroom materials for the unit on linear functions.

Este trabalho traz uma proposta de atividades, a serem desenvolvidas em sala de aula, com o objetivo de introduzir o conceito de derivadas para os alunos da primeira série do Ensino Médio. Antes das atividades, estão presentes algumas breves pesquisas. O histórico da presença de tópicos do Cálculo Diferencial e Integral no Ensino Médio no Brasil, assim como a análise de alguns livros didáticos, serve para mostrar como o assunto já foi e está sendo tratado no país. Também são exibidos aspectos sobre o Ensino Médio na Alemanha e nos Estados Unidos, países onde o cálculo está presente na Escola Secundária, embora de formas bastante diferentes. Um capítulo sobre a preparação adequada para as aulas também foi incluído, uma vez que a simples inserção da derivada poderia causar problemas de tempo para o cumprimento do cronograma e não trazer os resultados esperados. São necessários algum grau de adequação dos conteúdos ministrados e de cooperação com professores de Física. As atividades visando o ensino dos conceitos iniciais de derivada são motivadas por um problema físico de movimento. O foco é dado na intuição e na visualização de gráficos, para que haja uma melhor compreensão dos conceitos envolvidos. A utilização de um software de geometria dinâmica é requerida em boa parte do tempo, como importante ferramenta de apoio pedagógico / This paper presents a proposal of activities to be developed in the classroom, with the goal of introducing the concept of derivative for students in the first grade of secondary school. Before the activities, some brief researches are presented. The historical presence of the topics of Differential and Integral Calculus in brazilian High Schools, as well as the analysis of some textbooks, serves to show how it has been and is being treated in the country. Aspects of the High School are also shown in Germany and the United States, countries where the calculus is present in High School, though in quite different ways. A chapter about the proper preparation for these classes was also included, since the simple insertion of the derivative could cause problems for meeting the schedule and could not bring the expected results. Some degree of adequacy of the contents and cooperation with Physics teachers are needed. The activities aiming at teaching the initial concepts of derivatives are motivated by a physical problem of motion. The focus is given on intuition and visualization of graphs, so there is a better understanding of the concepts involved. The use of a dynamic geometry software is required for much of the time, as an important tool for pedagogical support

Text in Afrikaans / Real empowerment of school learners requires preparing them for the age of technology. This empowerment can be achieved by developing their higher-order thinking skills. This is clearly the intention of the proposed South African FET National Curriculum Statements Grades 10 to 12 (Schools). This research shows that one method of developing higher-order thinking skills is to adopt an integrated curriculum approach. The research is based on the assumption that an integrated curriculum approach will produce learners with a more integrated knowledge structure which will help them to solve problems requiring higher-order thinking skills. These assumptions are realistic because the empirical results of several comparative research studies show that an integrated curriculum helps to improve learners' ability to use higher-order thinking skills in solving nonroutine problems. The curriculum mentions four kinds of integration, namely integration across different subject areas, integration of mathematics with the real world, integration of algebraic and geometric concepts, and integration into and the use of dynamic geometry software in the learning and teaching of geometry. This research shows that from a psychological, pedagogical, mathematical and historical perspective, the theme conic sections can be used as an integrating factor in the new proposed FET mathematics curriculum. Conics are a powerful tool for making the new proposed curriculum more integrated. Conics can be used as an integrating factor in the FET band by means of mathematical exploration, visualisation, relating learners' experiences of various parts of mathematics to one another, relating mathematics to the rest of the learners' experiences and also applying conics to solve real-life problems. / Mathematical Sciences / D.Phil. (Wiskundeonderwys)

Este trabalho traz uma proposta de atividades, a serem desenvolvidas em sala de aula, com o objetivo de introduzir o conceito de derivadas para os alunos da primeira série do Ensino Médio. Antes das atividades, estão presentes algumas breves pesquisas. O histórico da presença de tópicos do Cálculo Diferencial e Integral no Ensino Médio no Brasil, assim como a análise de alguns livros didáticos, serve para mostrar como o assunto já foi e está sendo tratado no país. Também são exibidos aspectos sobre o Ensino Médio na Alemanha e nos Estados Unidos, países onde o cálculo está presente na Escola Secundária, embora de formas bastante diferentes. Um capítulo sobre a preparação adequada para as aulas também foi incluído, uma vez que a simples inserção da derivada poderia causar problemas de tempo para o cumprimento do cronograma e não trazer os resultados esperados. São necessários algum grau de adequação dos conteúdos ministrados e de cooperação com professores de Física. As atividades visando o ensino dos conceitos iniciais de derivada são motivadas por um problema físico de movimento. O foco é dado na intuição e na visualização de gráficos, para que haja uma melhor compreensão dos conceitos envolvidos. A utilização de um software de geometria dinâmica é requerida em boa parte do tempo, como importante ferramenta de apoio pedagógico / This paper presents a proposal of activities to be developed in the classroom, with the goal of introducing the concept of derivative for students in the first grade of secondary school. Before the activities, some brief researches are presented. The historical presence of the topics of Differential and Integral Calculus in brazilian High Schools, as well as the analysis of some textbooks, serves to show how it has been and is being treated in the country. Aspects of the High School are also shown in Germany and the United States, countries where the calculus is present in High School, though in quite different ways. A chapter about the proper preparation for these classes was also included, since the simple insertion of the derivative could cause problems for meeting the schedule and could not bring the expected results. Some degree of adequacy of the contents and cooperation with Physics teachers are needed. The activities aiming at teaching the initial concepts of derivatives are motivated by a physical problem of motion. The focus is given on intuition and visualization of graphs, so there is a better understanding of the concepts involved. The use of a dynamic geometry software is required for much of the time, as an important tool for pedagogical support

Les outils technologiques sont omniprésents dans la société et leur utilisation est de plus en plus grande dans les salles de classe. Du côté de l'enseignement et de l'apprentissage des mathématiques, ces outils se sont vu attribuer des rôles qui ont évolué avec les années. Les rôles de soutien, de visualisation et d'enrichissement des contenus en sont des exemples. Une utilisation des outils technologiques dans l'enseignement s'accompagne d'apports pragmatiques et épistémiques potentiels, mais comporte également des limites et des risques. Il s’avère important d’examiner le rôle accordé à l’outil technologique dans les activités qui le mobilisent. Puisque le raisonnement mathématique fait partie d'une des compétences visées à l’école (MELS, 2006) et que les futurs enseignants semblent accorder moins d'importance à la validation et la preuve comme composantes de ce raisonnement (Mary, 1999), nous émettons l'hypothèse qu'une séquence d'activités montrant la complémentarité de la preuve et des explorations tirant parti de la technologie pourrait aider les futurs enseignants à mieux saisir ces enjeux. La présente recherche s’appuie sur l'ingénierie didactique pour développer et valider une séquence d'activités intégrant le logiciel GeoGebra. Cette séquence d'activités a été conçue dans les buts suivants : initier les futurs enseignants en mathématiques au secondaire à un logiciel de géométrie dynamique et leur donner l'occasion de voir des activités mathématiques utilisant la technologie et visant le développement du raisonnement, par l’articulation de l’exploration et de la preuve. Le cadre théorique sur lequel repose cette recherche intègre des éléments de l'approche anthropologique (Chevallard, 1992, 1998, 2003) et de l'approche instrumentale (Vérillon et Rabardel, 1995; Trouche, 2000, 2003, 2007; Guin et Trouche, 2002). Certaines idées sur les constructions robustes et molles (Soury-Lavergne, 2011), la distinction figure/dessin (Laborde et Capponi, 1994) et le réseau déductif (Tanguay, 2006) ont servi de repères dans la construction de la séquence d'activités. Cette recherche s'est déroulée au cours de l'hiver 2016 dans une université québécoise, dans le cadre d’un cours de didactique de la géométrie auprès de futurs enseignants au secondaire en mathématiques. Un questionnaire pré-expérimentation a été rempli par les participants afin de voir leurs connaissances préalables sur les programmes, les outils technologiques ainsi que leurs conceptions au sujet de l'enseignement et de l'apprentissage des mathématiques. Par la suite, les étudiants ont expérimenté la séquence d'activités et ont eu à se prononcer sur les connaissances mises en jeu dans chacune des activités, l’opportunité de son utilisation avec des élèves du secondaire, et les adaptations perçues nécessaires pour sa réalisation (s'il y a lieu). Des traces écrites de leur travail ont été conservées ainsi qu'un journal de bord au fur et à mesure du déroulement de la séquence. En triangulant les diverses données recueillies, il a été constaté que la séquence, tout en contribuant à l’instrumentation des participants au regard du logiciel utilisé, a eu chez certains d’entre eux un impact sur leur vision du développement du raisonnement mathématique dans l’enseignement des mathématiques au secondaire. L’analyse des données a mis en lumière la place accordée au raisonnement par les futurs enseignants, les raisonnements mobilisés par les étudiants dans les diverses activités ainsi que des indices sur les genèses instrumentales accompagnant ces raisonnements ou les induisant. Suite à l’analyse de ces données et aux constats qui en découlent, des modifications sont proposées pour améliorer la séquence d’activités. / Technological tools are ubiquitous in society and their use is growing in the classroom. In mathematics education, these tools have been assigned roles that have evolved over the years: support, visualization, content enrichment. The use of technological tools in education comes with potential pragmatic and epistemic contributions, but also has limitations and risks. We must therefore examine at the activity level the role technology should play. Mathematical reasoning is one of the competencies aimed by school (MELS, 2006) and future teachers seem to place less emphasis on validation and proving processes as components of this reasoning (Mary, 1999). We hypothesize that a sequence of activities showing the complementarity of the proving processes with explorations leveraging technology could help future teachers better understand these issues. This research is based on didactical engineering to develop and validate a sequence of activities with GeoGebra software. The sequence of activities has been designed to: introduce pre-service secondary mathematics teachers to dynamic geometry software and give them the opportunity to see mathematical activities using technology that aim at developing mathematical reasoning and proof. The theoretical framework on which this research is based integrates elements of the anthropological theory of the didactic (Chevallard, 1992, 1998, 2003) and of the instrumental approach (Vérillon and Rabardel, 1995; Trouche, 2000, 2003, 2007; Guin and Trouche, 2002). Some ideas on robust and soft constructions (Soury-Lavergne, 2011), the distinction between figure and drawing (Laborde and Capponi, 1994) and the deductive network (Tanguay, 2006) served as benchmarks in the construction of the sequence of activities. This research took place at a Quebec university during the winter of 2016, in a geometry didactics course for pre-service secondary mathematics teachers. A preliminary questionnaire was given to the participants to capture their prior knowledge of programs, technological tools and conceptions about mathematics teaching and learning. Subsequently, the students experienced the sequence of activities and had to decide on the knowledge involved in each activity, the relevance of its use with high school students, and the perceived adaptations necessary for its realization (if considered). Written traces of their work have been kept as well as a diary as the sequence unfolds. By triangulating the various data collected, it was found that the sequence, while contributing to the instrumentation of the participants with regard to the software used, had, for some of them, an impact on their vision of the development of mathematical reasoning in mathematics education at secondary level. The analysis of the data highlighted the place given to the reasoning by the future teachers, the reasonings mobilized by the students in the various activities and also signs of the instrumental geneses inducing these reasonings and accompanying them. Following the analysis of these data and the findings that follow, modifications are proposed to improve the sequence of activities.

<p>Nowadays, Dynamic Geometry Software (DGS) is widely accepted as a tool for creating and presenting visually rich interactive teaching and learning materials, called dynamic drawings. Dynamic drawings are specified by writing expressions in functional domain-specific languages. Due to wide acceptance of DGS, there has arisen a need for their extensibility, by adding new semantics and visual objects (visuals). We have developed a programming framework for the Dynamic Geometry Software, SLGeometry, with a genericized functional language and corresponding expression evaluator that act as a framework into which specific semantics is embedded in the form of code annotated with metadata. The framework transforms an ordinary expression tree evaluator into an object-oriented one, and provide guidelines and examples for creation of interactive objects with dynamic properties, which participate in evaluation optimization at run-time. Whereas other DGS are based on purely functional expression evaluators, our solution has advantages of being more general, easy to implement, and providing a natural way of specifying object properties in the user interface, minimizing typing and syntax errors.LGeometry is implemented in C# on the .NET Framework. Although attributes are a preferred mechanism to provide association of declarative information with C# code, they have certain restrictions which limit their application to representing complex structured metadata. By developing a metadata infrastructure which is independent of attributes, we were able to overcome these limitations. Our solution, presented in this&nbsp; dissertation, provides extensibility to simple and complex data types, unary and binary operations, type conversions, functions and visuals, thus enabling developers to seamlessly add new features to SLGeometry by implementing them as C# classes annotated with metadata. It also provides insight into the way a domain specific functional language of dynamic geometry software can be genericized and customized for specific needs by extending or restricting the set of types, operations, type conversions, functions and visuals.Furthermore, we have conducted&nbsp; experiments with several groups of students of mathematics and high school pupils, in order to test how our approach compares to the existing practice. The experimental subjects tested mathematical games using interactive visual controls (UI controls) and sequential behavior controllers. Finally, we present a new evaluation algorithm, which was compared to the usual approach employed in DGS and found to perform well, introducing advantages while maintaining the same level of performance.</p> / <p>U dana&scaron;nje vreme softver za dinamičku geometriju (DGS) je &scaron;iroko prihvaćen kao alat za kreiranje i prezentovanje vizuelno bogatih interaktivnih nastavnih materijala i materijala za samostalno učenje, nazvanih dinamičkim crtežima. Kako je raslo prihvatanje softvera za dinamičku geometriju, tako je i rasla potreba da se oni pro&scaron;iruju, dodajući im novu semantiku i vizualne objekte. Razvili smo programsko okruženje za softver za dinamičku geometriju, SLGeometry, sa generičkim&nbsp; funkcionalnim jezikom i odgovarajućim evaluatorom izraza koji čini okruženje u kom su ugrađene specifične semantike u obliku koda označenog metapodacima. Ovo okruženje pretvara uobičajen evaluator stabla izraza u objektno orijentiran, te daje uputstva i primere za stvaranje interaktivnih objekata sa dinamičkim osobinama, koji sudeluju u optimizaciji izvr&scaron;enja tokom izvođenja. Dok se drugi DGS-ovi temelje na čisto funkcionalnim evaluatorima izraza, na&scaron;e rje&scaron;enje ima prednosti jer je uop&scaron;tenije, lako za implementaciju i pruža prirodan način navođenja osobina objekta u korisničkom interfejsu, minimizirajući kucanje i sintaksne gre&scaron;ke. SLGeometry je implementirana u jeziku C# .NET Framework-a. Iako su atributi preferiran mehanizam, koji povezuje C# k&ocirc;d sa deklarativnim informacijama, oni imaju određena ograničenja koja limitiraju njihovu primenu za predstavljanje složenih strukturiranih metapodataka. Razvijanjem infrastrukture metapodataka koja je nezavisna od atributa, uspeli smo prevladati ta ograničenja. Na&scaron;e re&scaron;enje, predstavljeno u ovoj disertaciji, pruža pro&scaron;irivost: jednostavnim i složenim vrstama podataka, unarnim i binarnim operacijama, konverzijama tipova, funkcijama i vizuelnim objektima, omogućavajući&nbsp; time programerima da neprimetno dodaju nove osobine u SLGeometry&nbsp; implementirajući ih kao C# klase označene metapodacima.</p>

The widespread use of Dynamic Geometry Software (DGS) is raising many interesting questions and discussions as to the necessity, usefulness and meaning of proof in school mathematics. With these questions in mind, a didactical sequence on the topic “Conics” was developed in a teacher education course tailored for pre-service secondary math methods course. The idea of the didactical sequence is to introduce “Conics” using a concrete manipulative approach (paper folding) then an explorative DGS-based construction activity embedding the need for a proof. For that purpose, the DGS software serves as an intermediary tool, used to bridge the gap between the physical model and the formal symbolic system of proof. The paper will present an analysis of participants’ geometric thinking strategies, featuring proof as an embedded process in geometric construction situations.

In our presentation we will show how technology enables us to improve the teaching and learning of linear functions at the middle school level. Through various classroom activities that involve technology such as dynamic geometry software, graphing calculators and Excel, students explore functions and discover basic facts about them on their own. Students then work with real life data and on real life problems to draw graphs and form linear models that correspond to given situations as well as draw inferences based on their models. Participants will receive complete classroom materials for the unit on linear functions.

The subject of teacher competencies or knowledge has been a key issue in mathematics education reform. This study attempts to identify and analyze teacher competencies necessary in the orchestration of a problem-centred approach to dynamic geometry teaching and learning. The advent of dynamic geometry environments into classrooms has placed new demands and expectations on mathematics teachers. In this study the Teacher Development Experiment was used as the main method of investigation. Twenty third-year mathematics major teachers participated in workshop and microteaching sessions involving the use of the Geometer's Sketchpad dynamic geometry software in the teaching and learning of the geometry of triangles and quadrilaterals. Five intersecting categories of teacher competencies were identified: mathematical/geometrical competencies. pedagogical competencies. computer and software competences, language and assessment competencies. / Mathematical Sciences / M. Ed. (Mathematical Education)

The subject of teacher competences or knowledge has been a key issue in mathematics education reform. This study attempts to identify and analyze teacher competences necessary in the orchestration of a problem-centred approach to dynamic geometry teaching and learning. The advent of dynamic geometry environments into classrooms has placed new demands and expectations on mathematics teachers. In this study the Teacher Development Experiment was used as the main method of investigation. Twenty third-year mathematics major teachers participated in workshop and microteaching sessions involving the use of the Geometer’s Sketchpad dynamic geometry software in the teaching and learning of the geometry of triangles and quadrilaterals. Five intersecting categories of teacher competences were identified: mathematical/geometrical competences, pedagogical competences, computer and software competences, language and assessment competencies. / Mathematics Education / M. Ed. (Mathematics Education)