Identifying Student Responsibilities While Watching Mathematics Instructional Videos

This study investigated the ways in which college algebra students watch mathematics instructional videos about completing the square with the goal of identifying student responsibilities within a particular video and across different videos. Guided by the theory of didactic situations that have defined implicit teacher and student responsibilities within the context of the face-to-face mathematics classroom, participants watched three different videos about completing the square and answered interview questions. Using categories previously identified by the didactic contract for the face-to-face classroom, this study expanded the types of student responsibilities identified specifically for video watching and found that participants, regardless of overall prior knowledge but who had prior knowledge of completing the square, held a responsibility to use the specific set of steps they were taught by their teacher to solve problems. Findings may be useful to both mathematics teachers and video creators.

O GeoGebra 3D no Estudo de Projeções Ortogonais Amparado pela Teoria das Situações Didáticas

ResumoEste trabalho apresenta uma situação didática voltada para o estudo de projeções ortogonais, auxiliado pelo GeoGebra 3D, em sua versão de aplicativo para smartphone. Para este estudo foi utilizada como base conceitual a Teoria das Situações Didáticas (TSD). O objetivo deste trabalho é mostrar como a visualização de projeções ortogonais por meio da construção geométrica 3D auxilia na compreensão e no desenvolvimento da visão espacial do aluno. Para tal, apresentam-se os resultados de uma aplicação para o ensino de Geometria Espacial, desenvolvendo a relação entre a visão 2D e 3D dos alunos, orientada e elaborada com base nas dialéticas da TSD – ação, formulação, validação e institucionalização –, realizada de forma remota, devido ao cenário de pandemia em que o mundo se encontra. Para sua estruturação foi selecionada uma questão do Exame Nacional do Ensino Médio (ENEM) sobre o assunto de projeções ortogonais, trabalhada de forma dinâmica e interativa em um encontro virtual na plataforma Google Meet. A aplicação ocorreu com alunos de 3º ano do Ensino Médio de uma escola pública estadual situada em Sobral – CE. Em síntese, aponta-se que a TSD associada ao GeoGebra 3D têm grande potencial para estimular a evolução do pensamento geométrico do aluno, atendendo o que se busca desenvolver nos currículos escolares. Palavras-chave: Projeções Ortogonais. GeoGebra 3D. Teoria das Situações Didáticas. AbstractThis work presents a didactic situation aimed at the study of orthogonal projections, aided by GeoGebra 3D, in its version of smartphone application. For this study, the Theory of Didactic Situations (TSD) was used as a conceptual basis. The objective of this work is to show how the visualization of orthogonal projections through the 3D geometric construction helps in understanding and developing the student's spatial vision. To this end, the results of an application for the teaching of Spatial Geometry are presented, developing the relationship between the students' 2D and 3D vision, oriented and elaborated based on the dialectic of the TSD - action, formulation, validation and institutionalization - carried out remotely, due to the pandemic scenario in which the world finds itself. For its structuring, a question from the National High School Examination (ENEM) was selected on the subject of orthogonal projections, worked in a dynamic and interactive way in a virtual meeting on the Google Meet platform. The application occurred with 3rd year high school students from a state public school located in Sobral - CE. In summary, it is pointed out that the TSD associated with GeoGebra 3D has great potential to stimulate the evolution of the student's geometric thinking, attending to what is sought to develop in school curricula. Keywords: Orthogonal projections. GeoGebra 3D. Theory of Didactic Situations.

Formação para a pesquisa científica: trajetórias híbridas, ativas e colaborativas em um Programa de Pós-Graduação em Educação MatemáticaEducation for scientific research: hybrid, active and collaborative trajectories in a Postgraduate Program in Mathematics Education

ResumoO presente artigo descreve uma investigação que ocorreu no âmbito de um grupo de mestrandos em Educação Matemática que cursavam a disciplina “Metodologia da Pesquisa” do Programa de Estudos Pós-Graduados em Educação Matemática da PUC-SP. A estratégia aplicada no desenvolvimento das aulas foi criada a partir de um planejamento que previa o uso de metodologias ativas, ensino híbrido e que tinha, como referência para as atividades propostas, a Teoria das Situação Didáticas. O principal objetivo era o de criar um contexto de discussões e um espaço de trabalho com interações presenciais e virtuais de modo a fornecer elementos para que os alunos iniciassem suas trajetórias como pesquisadores e começassem a escrever as respectivas dissertações, movimento classificado por eles como de grande dificuldade. O texto detalha as intervenções, em regime colaborativo, que os estudantes construíram, as discussões realizadas, os progressos alcançados, além de detalhes relativos à estrutura da disciplina que foram criados para incentivar o processo de formação pretendido.Palavras-chave: Colaboração, metodologias ativas, ensino híbrido, formação de pesquisadores, teoria das situações didáticas. AbstractThis article describes an investigation that occurred within a group of masters´ degree students in Mathematics Education that studied the discipline "Methodology of Research" of the Program of Postgraduate Studies in Mathematical Education of PUC-SP. The strategy applied in the development of classes was created from a planning that predicted the use of active learning methodologies, hybrid teaching and that had, as a reference for the proposed activities, the Theory of Didactic Situations. The main objective was to create a context of discussions and a workspace with face-to-face and virtual interactions in order to provide elements for the students to begin their trajectories as researchers as well as to begin to write the respective dissertations, a movement classified by them as very difficult. The text details the collaborative interventions the students have built, the discussions held, the progress made, as well as details of the discipline structure that were created to encourage the intended education process.Keywords: Collaboration, active learning methodologies, hybrid teaching, researchers’ education, didactical situations theory. ResumenEste artículo describe una investigación que se desarrolló en el ámbito de un grupo de estudiantes de maestría en Educación Matemática que cursaban la asignatura "Metodología de la Investigación" del Programa de Estudios de Postgrado en Educación Matemática de la PUC-SP. La estrategia aplicada en el desarrollo de las clases se creó a partir de una planificación que preveía el uso de metodologías activas, docencia híbrida y que tenía como referencia para las actividades propuestas, la Teoría de Situaciones Didácticas. El objetivo principal fue crear un contexto de discusiones y un espacio de trabajo con interacciones presenciales y virtuales con el fin de brindar elementos para que los estudiantes iniciasen sus trayectorias como investigadores y empezasen a escribir sus respectivas disertaciones, movimiento que calificaron como de gran dificultad. El texto detalla las intervenciones colaborativas que construyeron los estudiantes, las discusiones realizadas, los avances logrados, así como detalles relacionados con la estructura de la disciplina que se crearon para incentivar el proceso de formación pretendido.Palabras clave: Colaboración, metodologías activas, docencia híbrida, formación de investigadores, teoría de situaciones didácticas.

Didactical Engeering of second generation: a proposal of the design and a teaching resource with support of the GeoGebra software in Brazil

This article presents a partial excerpt from an ongoing master's research on the methodological theoretical aspects of problems related to the content of volumes, selected from the evaluations of the National High School Exam (ENEM). The Exam consists of a test proposed by the Brazilian federal government, which aims to assess the performance of students at the end of high school, in addition to being used as a selection criterion to enter higher education. In this way, a didactic situation is proposed in the context of ENEM structured in the Theory of Didactic Situations (TSD) and subsidized by the use of the GeoGebra software as a technological resource for the construction and resolution of the problem situation on the volume of a cylinder. The didactic session was consolidated in Didactic Training Engineering (EDF) and developed to design a didactic resource for teaching and training mathematics teachers. GeoGebra was used to carry out the didactic transposition of the problem, as it benefits the student to break the difficulties inherent in the understanding and appropriation of geometric concepts. This structure facilitates didactic mediation, in addition to giving a new meaning to the study and teaching of volumes in the context of ENEM assessments.

A Didactic Engineering in the Research Process of the Generalization of the Padovan Sequence: An Experience in a Pre-Service Teacher Training Course

Background: Obstacles are found during the epistemological construction of mathematical concepts research, aiming to contribute to the Didactics of Mathematics through a study of Padovan sequence. Objectives: describe elements of a systematic study, based on Didactic Engineering in conjunction with the Theory of Didactic Situations. I addition, referring to the generalization model of Padovan sequence and promoting a historical-evolutionary understanding and its mathematical properties. Design: it presents the most representative data of an investigation supported by the foundations of Didactic Engineering research design, in association with the Theory of Didactic Situations teaching methodology. Setting and Participants: the research was developed in 2019 and applied in a Pre-Service Mathematics Teacher Training Course in the History of Mathematics discipline, with the eight students enrolled. Data collection and analysis: data validation occurred internally due to the short period of the research. Results: it describes an investigation around the object of study, the Padovan sequence, focusing on the generalization process of this sequence and its properties. Thus, three problem situations are elaborated and analyzed based on the assumed research and teaching methodologies, seeking to examine their properties and the student's intuitive thinking, before the insertion of a historical-epistemological conception of this investigation. Conclusions: the research makes it possible to extract repercussions, suggest and promote research scripts aiming at the formation of teachers (initial) in the context of the teaching of History of Mathematics.

Propuesta de un proceso de estudio de clases para la formación inicial del profesorado de Educación Infantil desde el paradigma del cuestionamiento del mundoProposal of a class study process for the initial training of Early Childhood Education teachers from the paradigm of questioning the world

ResumenComo docentes universitarios a cargo de la formación inicial del profesorado de Educación Infantil, observamos claros síntomas del paradigma monumentalista (visita a algunas obras tanto de Matemáticas como de Didáctica de las Matemáticas) cuando esta formación se organiza según el esquema tradicional clase de teoría/clase de prácticas. En este trabajo pretendemos identificar con nitidez el reto que supone la formación profesional funcional de futuros profesores, formular este reto como un problema de investigación dentro de la TAD, y explorar la potencialidad del dispositivo del estudio de clases como herramienta para desarrollar el equipamiento praxeológico del profesorado como respuesta a cuestiones profesionales vivas y auténticas. Se describirá el diseño de este dispositivo, para el caso de la formación inicial de profesorado de Educación Infantil sobre la enseñanza de los primeros conocimientos numéricos.Palabras-clave: Teoría Antropológica de lo Didáctico, estudio de clases, Educación infantil, Formación inicial de profesorado, Teoría de las Situaciones Didácticas.AbstractAs teacher educators involved in the initial education of prospective Early Childhood Education teachers, we observe evident signs of the monumentalistic paradigm (visiting some pre-stablished works in Mathematics as well as in Didactics of Mathematics) when the education of teachers is structured following the traditional scheme lecture-practice. In this paper, we aim at clearly identifying the challenge of a functional education of prospective teachers, formulating it as a research problem within the ATD, and exploring the potential of the lesson study device as tool to develop prospective teachers’ praxeological equipment as responses to live and authentic professional questions. We will describe de design of such device, for the case of the initial education of prospective Early Childhood Education teachers around the teaching of numbers and numbering.Keywords: Anthropological Theory of Didactics, study of classes, Early childhood education, Initial teacher training, Theory of Didactic Situations.

Facteurs de décisions didactiques dans l’enseignement des mathématiques au secondaire en AndorreFactors of didactic decisions in secondary mathematics education in Andorra

RésuméCet article porte sur les décisions didactiques prises par les enseignants de mathématiques et les facteurs qui peuvent les influencer. Nous nous situons au niveau secondaire en Andorre, contexte caractérisé par une diversité éducative particulière. Nous utilisons le modèle de décisions didactiques inscrit dans le cadre de la théorie anthropologique du didactique (TAD) avec les niveaux de codétermination articulé avec la structuration du milieu de la théorie de situations didactiques (TSD). À partir du modèle de la transposition, nous centrons notre étude sur l’enseignement de l’algèbre et en particulier la résolution d’équations algébriques. Nous proposons un modèle de référence pour une analyse comparative sur l’enseignement des mathématiques dans le contexte étudié. Notre travail vise à étudier le facteur institutionnel dans le contexte particulier de l’enseignement des mathématiques en Andorre.Mots-clés : Facteur institutionnel de décisions didactiques, TAD, Transposition didactique.AbstractThis paper is about didactical decisions taken by teachers of mathematics and the factors that can influence them. We are on secondary school in Andorra, a context characterized by a particular educational diversity. We use the model of educational decisions in the framework of the anthropological theory of the didactic (ATD) with the levels of co-determination combined with the structuring of the educational environment in Theory of Didactic Situations (TSD). From the model of the didactic transposition, we focus our study on the teaching of algebra and in algebraic solving equations. We propose a reference model for a comparative analysis of mathematics education in the context under study. Our work aims to study the institutional factor in the context of mathematics education in Andorra.Keywords: Institutional factor of didactic decisions, TAD, Didactic transposition.

Analytical proofs of expected value and variance of the main functions of continuous probability distributions, with examples of their applications

The present work proposes the design of mathematical problems, which allow the adequate understanding of mathematical concepts of probability for its correct interpretation and later application in the resolution of probabilistic problems. For the development of this work we rely on the theory of didactic situations of Brousseau (1997) and Sadovsky (2005). We believe that new materials and didactic models of this type have great educational potential because they encourage the analysis and understanding of various probability problems (Panizza, 2003). Accurate communication between teachers and students in the approach, interpretation, resolution, and testing of probability problems is of vital importance. The software used for this purpose is the MATHEMATICA program, a tool that facilitates the writing of formulas and calculations, as well as the construction of graphs, through a friendly interface, facilitating the self-taught work of the student and encouraging the development of analysis skills and problem solving. We believe that these materials will contribute to the teaching and learning processes of probability at higher education levels.

A teoria das situações didáticas no desenvolvimento de atividades com robótica educacionalThe theory of didactic situations in the development of activities with educational robotics

A robótica educacional vem tomando força no desenvolvimento de atividades pedagógicas. Apresento nesse texto algumas características da robótica educacional, associando essas características com a teoria das situações didáticas desenvolvida por Brousseau (1986). Relato uma experiência ocorrida em sala de aula com alunos do ensino fundamental, de modo que as ações desenvolvidas ganharam nova ótica sob a observação pela teoria das situações didáticas. Alguns materiais da Empresa Zoom Education também foram analisados e apresentados para dar apoio às análises. O resultado desta experiência mostrou que o aprendizado pode acontecer em momentos de investigação, desenvolvendo o trabalho em equipe, a criatividade e a curiosidade dos alunos, afinal, as situações-problema, trazem a oportunidade de sentirem-se parte integrante do contexto estabelecido.

Comparação entre a teoria das Situações Didáticas de Brousseau e a Matemática no Contexto das Ciências de CamarenaComparison between Brousseau's Theory of Didactic Situations and Mathematics in the Context of Camarena Sciences

Apresentamos neste artigo uma primeira articulação entre duas abordagens teóricas da Didática da Matemática: a Teoria das Situações Didáticas (TSD) e a Matemática no Contexto das Ciências (MCC), com o objetivo de compará-las e contrastá-las. Detivemo-nos nos princípios gerais de tais referenciais, no planejamento de situações de aprendizagem e em aspectos do trabalho em sala de aula envolvendo o docente e o discente. Como uma convergência entre as duas teorias podemos citar a análise das interações entre os diferentes atores presentes no sistema educativo (aluno, conteúdo e professor). Ambas têm por base ideias construtivistas, no sentido de Piaget, sendo que a MCC foi inicialmente idealizada para o ensino superior, enquanto a TSD pode ser empregada para qualquer nível de ensino. Como principal ferramenta de trabalho em classe, na MCC temos os eventos contextualizados, construídos a partir de aplicações da Matemática em diferentes áreas do conhecimento. Na TSD trabalha-se com situações adidáticas, que são planejadas a partir de situações fundamentais relativas a um conceito matemático. Nas teorias em foco, parte-se do pressuposto de que o aluno deve construir seu próprio conhecimento e, para tal, o professor assume o papel de mediador. Ambas contemplam etapas de contextualização e de descontextualização do saber.