Le monde de l'éducation a été marqué au début de ce siècle par l'implantation, notamment dans les... more Le monde de l'éducation a été marqué au début de ce siècle par l'implantation, notamment dans les écoles québécoises et mexicaines, de compétences et plus particulièrement en mathématiques. Dans cette nouvelle vision de l'enseignement et de l'apprentissage, la modélisation mathématique est omniprésente et liée à la pensée divergente. Notre projet porte sur le développement de cette pensée chez les élèves lors de la généralisation algébrique pour négocier le passage de l'arithmétique à l'algèbre dans la transition primaire-secondaire. À travers de la proposition de conditions d'enseignement et d'apprentissage misant sur un travail collaboratif et l'utilisation de ressources technologiques, nous rapportons le développement chez les élèves québécois et mexicains de ce que nous avons nommé la pensée arithmético-algébrique sollicitée lors de la généralisation de suites figurées. Dans cette expérimentation, les élèves des deux pays, ont mobilisé une structure cognitive arithmético-algébrique qui s'est opérationnalisée à travers des procédures de prédiction, des conjectures, du processus de
The aim of this document is to address the articulation between arithmetic and algebra. The liter... more The aim of this document is to address the articulation between arithmetic and algebra. The literature reports a discussion of arithmetic or algebraic thinking, not mentioning at all the importance of a pre-algebraic thinking related to an arithmeticalgebraic thinking. Through activities designed for this purpose, we propose a visual approach, with the intention of promoting mathematical visualization processes, which in turn constitute cognitive structures on the control exercised by students in solving a mathematical task. Our design is linked to a theoretical framework for action, collaborative learning (ACODESA) and use of technology in the mathematics classroom. In this study, we present our research project between Québec and Mexico, but restrict ourselves to the results obtained with the Mexican population in relation to polygonal numbers in secondary school.
O conteúdo deste livro está licenciado sob uma Licença de Atribuição Creative Commons Atribuição-... more O conteúdo deste livro está licenciado sob uma Licença de Atribuição Creative Commons Atribuição-Não-Comercial NãoDerivativos 4.0 Internacional (CC BY-NC-ND 4.0). Direitos para esta edição cedidos à Editora Artemis pelos autores. Permitido o download da obra e o compartilhamento, desde que sejam atribuídos créditos aos autores, e sem a possibilidade de alterá-la de nenhuma forma ou utilizá-la para fins comerciais. A responsabilidade pelo conteúdo dos artigos e seus dados, em sua forma, correção e confiabilidade é exclusiva dos autores. A Editora Artemis, em seu compromisso de manter e aperfeiçoar a qualidade e confiabilidade dos trabalhos que publica, conduz a avaliação cega pelos pares de todos manuscritos publicados, com base em critérios de neutralidade e imparcialidade acadêmica.
El descubrimiento de diferentes infinitos en matemáticas trajo consigo una discusión sobre su exi... more El descubrimiento de diferentes infinitos en matemáticas trajo consigo una discusión sobre su existencia desde épocas muy tempranas. La filosofía éleática (siglo V a. de C.), a través de las paradojas de Zenón, intentaban mostrabar a filósofos-matemáticos que las concepciones que se tenían sobre el infinito llevaban a contradicciones. Aristóteles (384-322 a. de C.) quizo cerrar el capítulo argumentando que solamente existe un infinito en matemáticas (el infinito potencial) y que el infinito real no tenía cabida alguna. Una implicación de esta postura la podemos ver en el Axioma 8 de Euclides (325-265 a. de C.) : "El todo es mayor que la parte"; sin embargo, el querer contar con una matemática libre de contradicciones habría nuevamente la caja de Pandora… Muchos intentos se realizaron, pero se tuvo que esperar al trabajo de Kant (1790) en filosofía y de Bolzano (1817 y 1851) en matemáticas (sobre la continuidad de funciones y sobre las paradojas del infinito) para que la pr...
In the transition from arithmetic to algebra and in light of the disjunction between the natural ... more In the transition from arithmetic to algebra and in light of the disjunction between the natural and symbolic approach to algebra and the choice of a natural way of learning, this paper discusses the development of a cognitive control structure in pupils when they are faced with a mathematical task. Researchers sought to develop, in novice pupils in both Quebec (12–13 years old) and Mexico (14–15 years old), an arithmetic-algebraic thinking structure that would promote mathematics competencies in a method based on collaborative learning, scientific debate and self-reflection (ACODESA, acronym which comes from the French abbreviation of Apprentissage collaboratif, Debat scientifique, Autoreflexion), and immersed in an activity theory approach. This paper promotes the equal use of both paper and pencil and technology in order to solve a mathematical task in a sociocultural and technological environment.
El presente trabajo presenta los resultados de una investigacion donde se concibio e implemento u... more El presente trabajo presenta los resultados de una investigacion donde se concibio e implemento una secuencia didactica que contribuyera a la construccion y comprension del concepto de Identidad Trigonometrica en estudiantes de segundo semestre del nivel Medio Superior. Tomando como principal herramienta la visualizacion matematica, basada en la creacion de redes de representacion, se induce al estudiante en tratamiento, transito y conversion entre diversas representaciones del concepto. En esta labor el software GeoGebra es de gran ayuda ya que brinda las herramientas necesarias para el logro de nuestros objetivos. Con base en los resultados obtenidos, se realiza un analisis que permite identificar los niveles de comprension del concepto en cada estudiante; dicho logro esta determinado por el numero y la fuerza de las conexiones que se presentan entre las distintas representaciones.
In the proposed integration of the different branches of STEM (science, technology, engineering a... more In the proposed integration of the different branches of STEM (science, technology, engineering and mathematics) education, one area of concern for researchers studying the didactics of mathematics is that mathematical modeling should play more of a central role in
El profesor de matemáticas que desea utilizar la tecnología en el aula de matemáticas en forma ra... more El profesor de matemáticas que desea utilizar la tecnología en el aula de matemáticas en forma razonada, debe tomar en consideración una gran cantidad de variables que le permitan llegar a tener una visión amplia de los problemas de enseñanza y de aprendizaje de las matemáticas en ambientes tecnológicos. Si tomamos la famosa frase de Euclides (siglo II antes de Cristo) formulada al rey Ptolomeo: “No hay camino real para aprender geometría”, lo podríamos aplicar aquí: “No hay camino real para saber cómo utilizar la tecnología en el aula de matemáticas”. La elección de qué tecnología utilizar en el aula de matemáticas y por qué, debe tomar en consideración diferentes variables para una elección razonada. Las variables en juego pueden ser de diferente tipo, cognitivas (para responder al por qué), económicas (uso de paquetes de cómputo de uso libre o comercial), sociales (promover aprendizaje individualizado y/o aprendizaje en colaboración) o institucionales (ligadas por ejemplo al curr...
Training and Evolution of Functional-Spontaneous Representations through Sociocultural Learning. ... more Training and Evolution of Functional-Spontaneous Representations through Sociocultural Learning. The present research aims to understand the role of the students' functional-spontaneous representations through the study of the pupils' external spontaneous representations in the process of solving a problem research situation. Through a qualitative methodology, the spontaneous representations of secondary students in the learning of covariation between variables are analysed. A particular goal is to study how these representations are likely to evolve through internal communication of students and through communication and validation with their pairs. Hence the importance of having a method that promotes a social construction of learning. In our case, we have opted for a learning process in a sociocultural environment: ACODESA (Collaborative Learning, Scientific Debate, Self-reflection and Institutionalization). From this perspective, in this paper, we consider collaborative research that allows the researcher to acquire knowledge about the teacher's practice, and vice versa, the teacher acquiring research knowledge in the mathematics classroom through the evolution of pupils' representations. The results show that the functional-spontaneous representations are the engine of the learning process of mathematical concepts. Résumé. Cette recherche vise à comprendre le rôle des représentations fonctionnellesspontanées dans l'étude des représentations spontanées (externes) des élèves pendant le processus de résolution d'une situation d'investigation. Plus précisément, nous analysons, grâce à une méthode qualitative, les représentations spontanées des élèves du secondaire lors d'un apprentissage sur la covariation entre variables. Nous portons un intérêt particulier à la manière dont ces représentations sont susceptibles d'évoluer grâce à une réflexion personnelle des élèves suivie d'un travail de collaboration et validation par les pairs, d'où l'importance d'avoir une méthode qui favorise la construction sociale de l'apprentissage. Dans notre cas, nous avons opté pour un apprentissage dans un environnement d'enseignement socioculturel : l'ACODESA (apprentissage collaboratif, débat scientifique, autoréflexion et institutionnalisation). Dans cette perspective, nous avons choisi la recherche collaborative pour mener à bien notre expérimentation. En effet, celle-ci permet au chercheur d'acquérir des connaissances sur la pratique de l'enseignant et, réciproquement, l'enseignant acquiert des connaissances sur la recherche dans la classe de mathématiques sur l'évolution des représentations des élèves. Dans notre article, nous nous interrogeons sur le rôle des représentations fonctionnelles-spontanées comme moteur du processus d'apprentissage des concepts mathématiques. Mots-clés. Représentation fonctionnelle-spontanée, représentation socialement construite, apprentissage dans un milieu socioculturel, ACODESA.
Recently, it has been considered important to reflect on the coincidences between the mathematica... more Recently, it has been considered important to reflect on the coincidences between the mathematical thinking of the “School of Mathematics ” and the “Discipline of Mathematics”. It is widely accepted that a professional mathematician has naturally developed reasoning abilities that are essential to his practice. Such reasoning abilities are considered central to student F. HITT, F. BARRERA-MORA and M. CAMACHO-MACHÍN 94 learning at different educational levels. While the problem is not entirely
This article is divided into four parts. The first part presents some introductory remarks on the... more This article is divided into four parts. The first part presents some introductory remarks on the use of Computer Algebra System (CAS) technology in relation to the long-standing dichotomy in algebra between procedures and concepts. The second part explores the technical-conceptual interface in algebraic activity and discusses what is meant by conceptual (theoretical) understanding of algebraic technique – in other words, what it means to render conceptual the technical aspects of algebra. Examples to be touched upon include seeing through symbols, becoming aware of underlying forms, and conceptualizing the equivalence of the factored and expanded forms of algebraic expressions. The ways in which students learned to draw such conceptual aspects from their work with algebraic techniques in technology environments is the focus of the third part of the article. Research studies that have been carried out by my research group with a range of high school algebra students have found evide...
Mathematics Education Across Cultures: Proceedings of the 42nd Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 2020
This volume collects most recent work on the role of technology in mathematics education. It offe... more This volume collects most recent work on the role of technology in mathematics education. It offers fresh insight and understanding of the many ways in which technological resources can improve the teaching and learning of mathematics. The first section of the volume focuses on the question how a proposed mathematical task in a technological environment can influence the acquisition of knowledge and what elements are important to retain in the design of mathematical tasks in computing environments. The use of white smart boards, platforms as Moodle, tablets and smartphones have transformed the way we communicate both inside and outside the mathematics classroom. Therefore the second section discussed how to make efficient use of these resources in the classroom and beyond. The third section addresses how technology modifies the way information is transmitted and how mathematical education has to take into account the new ways of learning through connected networks as well as new way...
Mathematics Education Across Cultures: Proceedings of the 42nd Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 2020
The first calculus course in the province of Quebec (Canada) is taught in the first year of colle... more The first calculus course in the province of Quebec (Canada) is taught in the first year of college (17–18 year-old students) before university. Statistics show that this course is the most difficult one for students at the collegial level and that it prompts many to drop out of school. The literature has highlighted the cognitive problems students have with integrating concepts related to pre-calculus and their problems in learning calculus concepts related to infinity. In the current century, a new variable is added to the teaching of introductory calculus courses, namely, the introduction of situational problems as a way to generate new knowledge. This new approach to teaching is mainly related to modelling processes in the generation of knowledge. In this paper, we show the cognitive problems students have in learning calculus when solving situational problems. More precisely, we observe students’ mathematical activity as they solve an open-ended task related to speed. We analyze their use of different representations in the process of modelling the situation, the evolution of these representations when students work in a collaborative learning approach, and their construction of the concept of the derivative.
Le monde de l'éducation a été marqué au début de ce siècle par l'implantation, notamment dans les... more Le monde de l'éducation a été marqué au début de ce siècle par l'implantation, notamment dans les écoles québécoises et mexicaines, de compétences et plus particulièrement en mathématiques. Dans cette nouvelle vision de l'enseignement et de l'apprentissage, la modélisation mathématique est omniprésente et liée à la pensée divergente. Notre projet porte sur le développement de cette pensée chez les élèves lors de la généralisation algébrique pour négocier le passage de l'arithmétique à l'algèbre dans la transition primaire-secondaire. À travers de la proposition de conditions d'enseignement et d'apprentissage misant sur un travail collaboratif et l'utilisation de ressources technologiques, nous rapportons le développement chez les élèves québécois et mexicains de ce que nous avons nommé la pensée arithmético-algébrique sollicitée lors de la généralisation de suites figurées. Dans cette expérimentation, les élèves des deux pays, ont mobilisé une structure cognitive arithmético-algébrique qui s'est opérationnalisée à travers des procédures de prédiction, des conjectures, du processus de
The aim of this document is to address the articulation between arithmetic and algebra. The liter... more The aim of this document is to address the articulation between arithmetic and algebra. The literature reports a discussion of arithmetic or algebraic thinking, not mentioning at all the importance of a pre-algebraic thinking related to an arithmeticalgebraic thinking. Through activities designed for this purpose, we propose a visual approach, with the intention of promoting mathematical visualization processes, which in turn constitute cognitive structures on the control exercised by students in solving a mathematical task. Our design is linked to a theoretical framework for action, collaborative learning (ACODESA) and use of technology in the mathematics classroom. In this study, we present our research project between Québec and Mexico, but restrict ourselves to the results obtained with the Mexican population in relation to polygonal numbers in secondary school.
O conteúdo deste livro está licenciado sob uma Licença de Atribuição Creative Commons Atribuição-... more O conteúdo deste livro está licenciado sob uma Licença de Atribuição Creative Commons Atribuição-Não-Comercial NãoDerivativos 4.0 Internacional (CC BY-NC-ND 4.0). Direitos para esta edição cedidos à Editora Artemis pelos autores. Permitido o download da obra e o compartilhamento, desde que sejam atribuídos créditos aos autores, e sem a possibilidade de alterá-la de nenhuma forma ou utilizá-la para fins comerciais. A responsabilidade pelo conteúdo dos artigos e seus dados, em sua forma, correção e confiabilidade é exclusiva dos autores. A Editora Artemis, em seu compromisso de manter e aperfeiçoar a qualidade e confiabilidade dos trabalhos que publica, conduz a avaliação cega pelos pares de todos manuscritos publicados, com base em critérios de neutralidade e imparcialidade acadêmica.
El descubrimiento de diferentes infinitos en matemáticas trajo consigo una discusión sobre su exi... more El descubrimiento de diferentes infinitos en matemáticas trajo consigo una discusión sobre su existencia desde épocas muy tempranas. La filosofía éleática (siglo V a. de C.), a través de las paradojas de Zenón, intentaban mostrabar a filósofos-matemáticos que las concepciones que se tenían sobre el infinito llevaban a contradicciones. Aristóteles (384-322 a. de C.) quizo cerrar el capítulo argumentando que solamente existe un infinito en matemáticas (el infinito potencial) y que el infinito real no tenía cabida alguna. Una implicación de esta postura la podemos ver en el Axioma 8 de Euclides (325-265 a. de C.) : "El todo es mayor que la parte"; sin embargo, el querer contar con una matemática libre de contradicciones habría nuevamente la caja de Pandora… Muchos intentos se realizaron, pero se tuvo que esperar al trabajo de Kant (1790) en filosofía y de Bolzano (1817 y 1851) en matemáticas (sobre la continuidad de funciones y sobre las paradojas del infinito) para que la pr...
In the transition from arithmetic to algebra and in light of the disjunction between the natural ... more In the transition from arithmetic to algebra and in light of the disjunction between the natural and symbolic approach to algebra and the choice of a natural way of learning, this paper discusses the development of a cognitive control structure in pupils when they are faced with a mathematical task. Researchers sought to develop, in novice pupils in both Quebec (12–13 years old) and Mexico (14–15 years old), an arithmetic-algebraic thinking structure that would promote mathematics competencies in a method based on collaborative learning, scientific debate and self-reflection (ACODESA, acronym which comes from the French abbreviation of Apprentissage collaboratif, Debat scientifique, Autoreflexion), and immersed in an activity theory approach. This paper promotes the equal use of both paper and pencil and technology in order to solve a mathematical task in a sociocultural and technological environment.
El presente trabajo presenta los resultados de una investigacion donde se concibio e implemento u... more El presente trabajo presenta los resultados de una investigacion donde se concibio e implemento una secuencia didactica que contribuyera a la construccion y comprension del concepto de Identidad Trigonometrica en estudiantes de segundo semestre del nivel Medio Superior. Tomando como principal herramienta la visualizacion matematica, basada en la creacion de redes de representacion, se induce al estudiante en tratamiento, transito y conversion entre diversas representaciones del concepto. En esta labor el software GeoGebra es de gran ayuda ya que brinda las herramientas necesarias para el logro de nuestros objetivos. Con base en los resultados obtenidos, se realiza un analisis que permite identificar los niveles de comprension del concepto en cada estudiante; dicho logro esta determinado por el numero y la fuerza de las conexiones que se presentan entre las distintas representaciones.
In the proposed integration of the different branches of STEM (science, technology, engineering a... more In the proposed integration of the different branches of STEM (science, technology, engineering and mathematics) education, one area of concern for researchers studying the didactics of mathematics is that mathematical modeling should play more of a central role in
El profesor de matemáticas que desea utilizar la tecnología en el aula de matemáticas en forma ra... more El profesor de matemáticas que desea utilizar la tecnología en el aula de matemáticas en forma razonada, debe tomar en consideración una gran cantidad de variables que le permitan llegar a tener una visión amplia de los problemas de enseñanza y de aprendizaje de las matemáticas en ambientes tecnológicos. Si tomamos la famosa frase de Euclides (siglo II antes de Cristo) formulada al rey Ptolomeo: “No hay camino real para aprender geometría”, lo podríamos aplicar aquí: “No hay camino real para saber cómo utilizar la tecnología en el aula de matemáticas”. La elección de qué tecnología utilizar en el aula de matemáticas y por qué, debe tomar en consideración diferentes variables para una elección razonada. Las variables en juego pueden ser de diferente tipo, cognitivas (para responder al por qué), económicas (uso de paquetes de cómputo de uso libre o comercial), sociales (promover aprendizaje individualizado y/o aprendizaje en colaboración) o institucionales (ligadas por ejemplo al curr...
Training and Evolution of Functional-Spontaneous Representations through Sociocultural Learning. ... more Training and Evolution of Functional-Spontaneous Representations through Sociocultural Learning. The present research aims to understand the role of the students' functional-spontaneous representations through the study of the pupils' external spontaneous representations in the process of solving a problem research situation. Through a qualitative methodology, the spontaneous representations of secondary students in the learning of covariation between variables are analysed. A particular goal is to study how these representations are likely to evolve through internal communication of students and through communication and validation with their pairs. Hence the importance of having a method that promotes a social construction of learning. In our case, we have opted for a learning process in a sociocultural environment: ACODESA (Collaborative Learning, Scientific Debate, Self-reflection and Institutionalization). From this perspective, in this paper, we consider collaborative research that allows the researcher to acquire knowledge about the teacher's practice, and vice versa, the teacher acquiring research knowledge in the mathematics classroom through the evolution of pupils' representations. The results show that the functional-spontaneous representations are the engine of the learning process of mathematical concepts. Résumé. Cette recherche vise à comprendre le rôle des représentations fonctionnellesspontanées dans l'étude des représentations spontanées (externes) des élèves pendant le processus de résolution d'une situation d'investigation. Plus précisément, nous analysons, grâce à une méthode qualitative, les représentations spontanées des élèves du secondaire lors d'un apprentissage sur la covariation entre variables. Nous portons un intérêt particulier à la manière dont ces représentations sont susceptibles d'évoluer grâce à une réflexion personnelle des élèves suivie d'un travail de collaboration et validation par les pairs, d'où l'importance d'avoir une méthode qui favorise la construction sociale de l'apprentissage. Dans notre cas, nous avons opté pour un apprentissage dans un environnement d'enseignement socioculturel : l'ACODESA (apprentissage collaboratif, débat scientifique, autoréflexion et institutionnalisation). Dans cette perspective, nous avons choisi la recherche collaborative pour mener à bien notre expérimentation. En effet, celle-ci permet au chercheur d'acquérir des connaissances sur la pratique de l'enseignant et, réciproquement, l'enseignant acquiert des connaissances sur la recherche dans la classe de mathématiques sur l'évolution des représentations des élèves. Dans notre article, nous nous interrogeons sur le rôle des représentations fonctionnelles-spontanées comme moteur du processus d'apprentissage des concepts mathématiques. Mots-clés. Représentation fonctionnelle-spontanée, représentation socialement construite, apprentissage dans un milieu socioculturel, ACODESA.
Recently, it has been considered important to reflect on the coincidences between the mathematica... more Recently, it has been considered important to reflect on the coincidences between the mathematical thinking of the “School of Mathematics ” and the “Discipline of Mathematics”. It is widely accepted that a professional mathematician has naturally developed reasoning abilities that are essential to his practice. Such reasoning abilities are considered central to student F. HITT, F. BARRERA-MORA and M. CAMACHO-MACHÍN 94 learning at different educational levels. While the problem is not entirely
This article is divided into four parts. The first part presents some introductory remarks on the... more This article is divided into four parts. The first part presents some introductory remarks on the use of Computer Algebra System (CAS) technology in relation to the long-standing dichotomy in algebra between procedures and concepts. The second part explores the technical-conceptual interface in algebraic activity and discusses what is meant by conceptual (theoretical) understanding of algebraic technique – in other words, what it means to render conceptual the technical aspects of algebra. Examples to be touched upon include seeing through symbols, becoming aware of underlying forms, and conceptualizing the equivalence of the factored and expanded forms of algebraic expressions. The ways in which students learned to draw such conceptual aspects from their work with algebraic techniques in technology environments is the focus of the third part of the article. Research studies that have been carried out by my research group with a range of high school algebra students have found evide...
Mathematics Education Across Cultures: Proceedings of the 42nd Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 2020
This volume collects most recent work on the role of technology in mathematics education. It offe... more This volume collects most recent work on the role of technology in mathematics education. It offers fresh insight and understanding of the many ways in which technological resources can improve the teaching and learning of mathematics. The first section of the volume focuses on the question how a proposed mathematical task in a technological environment can influence the acquisition of knowledge and what elements are important to retain in the design of mathematical tasks in computing environments. The use of white smart boards, platforms as Moodle, tablets and smartphones have transformed the way we communicate both inside and outside the mathematics classroom. Therefore the second section discussed how to make efficient use of these resources in the classroom and beyond. The third section addresses how technology modifies the way information is transmitted and how mathematical education has to take into account the new ways of learning through connected networks as well as new way...
Mathematics Education Across Cultures: Proceedings of the 42nd Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 2020
The first calculus course in the province of Quebec (Canada) is taught in the first year of colle... more The first calculus course in the province of Quebec (Canada) is taught in the first year of college (17–18 year-old students) before university. Statistics show that this course is the most difficult one for students at the collegial level and that it prompts many to drop out of school. The literature has highlighted the cognitive problems students have with integrating concepts related to pre-calculus and their problems in learning calculus concepts related to infinity. In the current century, a new variable is added to the teaching of introductory calculus courses, namely, the introduction of situational problems as a way to generate new knowledge. This new approach to teaching is mainly related to modelling processes in the generation of knowledge. In this paper, we show the cognitive problems students have in learning calculus when solving situational problems. More precisely, we observe students’ mathematical activity as they solve an open-ended task related to speed. We analyze their use of different representations in the process of modelling the situation, the evolution of these representations when students work in a collaborative learning approach, and their construction of the concept of the derivative.
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