Dynamic Geometry Sofware

Les nouveaux programmes du cycle 3 (MEN 2015) associent l'enseignement de la geometrie a une initiation a la programmation. Tout comme la geometrie dynamique a apporte un point de vue nouveau sur la geometrie (Assude et Gelis, 2002), qu'en est-il de la programmation ? En nous appuyant sur des travaux de didactique de la geometrie (Perrin-Glorian & Godin, 2014, Petitfour 2015), nous interrogeons les apports et les limites de cette approche de la geometrie a travers la programmation par la comparaison de la mise en oeuvre d'une tâche de reproduction d'une figure geometrique dans differents environnements.

El objetivo principal de este proyecto es la creación de un videojuego para dispositivos móviles utilizando realidad virtual, con el fin de utilizar los conocimientos adquiridos en el Máster para entretener y divertir a la gente, y además involucrar a más personas en el tema de los videojuegos con realidad virtual que actualmente están en auge. Con el objetivo de determinar el tipo de aplicación a desarrollar, el primer paso fue el interés que me despertaba la realidad virtual vista en el Máster de Desarrollo de Software para Dispositivos Móviles. Además, considerando el interés del público por los videojuegos, así como la expansión de las tecnologías móviles, establecen la base para llevar a cabo este proyecto. En este caso, para un videojuego que utilice realidad virtual, y puesto que soy aficionado al fútbol, decidí realizar un videojuego de fútbol en el que podemos simular un terreno de juego e interactuar de forma virtual a través de nuestro dispositivo móvil. En el videojuego propuesto el usuario será el portero y tendrá que evitar encajar un gol. El rol de portero permitirá utilizar realidad mixta, es decir, realidad virtual para simular el entorno, y realidad aumentada con la intención de utilizar las manos del usuario para interactuar con el balón. También se puede considerar la opción de que el usuario maneje al jugador para intentar marcar gol. A partir de ahora a este videojuego le llamaremos Footer 3D, que combina dos términos: 'Football' y 'Shooter', es decir, goleador de fútbol.

This is to certify that Mr. Dipesh Prajapati has completed his thesis entitled Effectiveness of GeoGebra in Teaching and Learning Geometry under my supervision during the period prescribed by the rules and regulation of Tribhuvan University, Nepal. The study embodies the result of investigation conducting during the period under the

O objetivo do estudo é de analisar a gestão de pessoas no agronegócio, a relevância da evolução do recurso humano para produtividade agropecuária. Na cultura da zona rural em geral para viabilizar a produção e o manejo é através de bons profissionais, assim como todo o processo e procedimentos do agronegócio são realizados por capital humano para atrair bons funcionários e serem atualizados, necessitam ser regidos por organização e coordenação, que envolve uma série de técnica, método, treinamentos, tática entre outros meios que garanta o progresso e eficiência no setor. Através do levantamento bibliográfico o estudo fundamenta a importância de gestão de pessoas para o agronegócio, pessoas profissionais, motivados e com habilidades para obter uma alta eficiência e garantir ganhos altamente desejáveis no setor rural. Constatou a importância de evolução do recurso humano na gestão de pessoas para a utilização de tecnologia e implantação de técnicas adequadas, na formação de excelentes profissionais admissíveis que componham sua preparação, formação e seja motivado para exercer seu trabalho no setor do agronegócio.

Palavras-Chave: Recursos Humanos. Organização e Coordenação. Produtividade. Profissionais. Treinamento.

I would like to acknowledge and extend my heartfelt gratitude to the following persons who have made the completion of this research a reality. Considering that it was my first time of doing a research, it would not be possible without the support of my two supervisors, professor Maria Luiza Cestari and professor Per Henrik Hogstad. Maria Luiza has been my didactic and theoretical guide. She has helped me to develop ideas, to find the relevant theory and provided guidance during the writing process of this research. Per Henrik has helped me with his expertise as a programmer and his experience to develop the program for use as aids in mathematics. Given that it was my first time to conduct research, it would not have been possible without the support of my two supervisors: Professor Maria Luiza Cestari and Professor Per Henrik Hogstad. Maria Luiza has been my didactic and theoretical guide. She helped me to develop ideas and to find the relevant theory. She also provided guidance during the writing process of this research. Per Henrik helped me with his expertise as a programmer and his experience in developing the programs for use as teaching and learning aids in mathematics. Special thanks go to my friends and colleagues, especially Suela Kacerja, for her feedback and valuable advice, and Bebwa Isingoma, for his willingness to help me to correct the grammar and to improve my study. I would also like to thank the teachers in Albania, who supported me in carrying out this research in their school, especially the headmaster, Professor Violeta Volumi, who kindly received me and appreciated my initiative and work. Finally, I would like to thank my family for their moral support and their encouragement.

Para ello se diseñó e implementó una cartilla virtual de actividades bajo el tema de teselados donde se incorpora el GeoGebra, haciendo uso de la plataforma Moodle. Se contempló el estudio de elementos básicos de geometría plana, transformaciones geométricas en el plano, y su relación con el arte, tomando como elemento motivador los diseños formados por combinaciones de figuras que permiten realizar un teselado.

This study investigates a crucial didactical challenge for mathematics teachers when they integrate 3D dynamic geometry environments (3D DGEs) into upper secondary classrooms. The challenge concerns teachers' regulation of the interaction of two processesempirical perception (often intertwined with actions) and theoretical deduction-in student learning. We establish a framework to characterize teachers' behaviour regarding the two processes of students, referring to it as the "coordination behaviour" framework. The framework is then applied to a case study involving an experienced French mathematics teacher. The analysis identifies three distinct patterns in the teacher's coordination behaviour, revealing that despite her goal of fostering students' theoretical deduction and empirical perception-action concurrently, she sometimes has to make compromises due to contextual and personal factors, leading to a diminished focus on one of these two processes. The results highlight the intricacy of teachers' coordination in upper secondary classrooms integrating 3D DGEs. We therefore call for the development of teacher education programmes that specifically prepare teachers to address this challenge.

This case study describes and investigates the instructional practices of Grades 1 to 5 teachers and the levels of geometry thinking of the learners, according to the Van Hiele model, with a view to determining whether there is a match between the instructional practice and the learners' level of thinking. The instructional practices of the teachers were observed and analysed, and their learners' levels of geometry thinking were accessed through a Van Hiele test. The results suggest that there is not a simple relationship between the phases of learning, as described by Crowley in 1987, and geometric development in terms of the Van Hiele levels. It is, however, possible to explain the geometric development to a limited extent in terms of the Van Hiele levels of the observed teaching activities. Although the presence of activities on an appropriate level does not guarantee growth in terms of the Van Hiele model, the absence thereof results in stagnation. The instructional practices in primary schools in all Grades should span geometry experiences on all the levels, because the previsualisation level and Van Hiele Level 1 thinking are still evident up to Grade 5.

This case study describes and investigates the instructional practices of Grades 1 to 5 teachers and the levels of geometry thinking of the learners, according to the Van Hiele model, with a view to determining whether there is a match between the instructional practice and the learners' level of thinking. The instructional practices of the teachers were observed and analysed, and their learners' levels of geometry thinking were accessed through a Van Hiele test. The results suggest that there is not a simple relationship between the phases of learning, as described by Crowley in 1987, and geometric development in terms of the Van Hiele levels. It is, however, possible to explain the geometric development to a limited extent in terms of the Van Hiele levels of the observed teaching activities. Although the presence of activities on an appropriate level does not guarantee growth in terms of the Van Hiele model, the absence thereof results in stagnation. The instructional practices in primary schools in all Grades should span geometry experiences on all the levels, because the previsualisation level and Van Hiele Level 1 thinking are still evident up to Grade 5.

Abstract: This study investigated the attitude towards, as well as the level of understanding of Euclidean geometry in pre-service mathematics education (PME) students. In order to do so, a case study was undertaken within which a one group pre-post-test procedure was ...

Este artículo plantea el desarrollo de un entorno virtual de la tabla periódica con el propósito de adquirir una herramienta que apoye el aprendizaje de la química, usando técnicas de realidad virtual. Para el desarrollo de esta propuesta, primero, se realizó la caracterización de los diferentes conceptos de la tabla periódica y lo concerniente a Realidad Virtual; luego, se diseñaron los componentes de la arquitectura, se modeló la base de datos y se realizaron los mockups; a partir de esto, se implementó el prototipo de la tabla periódica a través de técnicas de realidad virtual y modelado 3D y finalmente, se validaron los resultados del entorno virtual mediante un grupo de control de estudiantes de la materia química orgánica del Politécnico Jaime Isaza Cadavid, profesores y personas externas y se hizo la retroalimentación del entorno, con base en la encuestas realizadas.

This study examined the effects of an open-ended approach on scientific argumentation and creative scientific thinking among senior high school science students. Conducted with 40 Grade 12 STEM learners from Laguna State Polytechnic University-San Pablo Campus during the Second Semester of the Academic Year 2024-2025, the research aimed to compare students' pretest and post-test performances in these areas. Two main tools were used: a researcher-developed instrument based on the Claim-Evidence-Reasoning (C-E-R) framework to measure scientific argumentation and an adapted version of the Scientific Structure Creativity Model to assess creative scientific ability. Results showed significant improvements following the implementation of the open-ended approach. Students advanced notably in all components of scientific argumentation, particularly in reasoning, where 95% of them moved to higher proficiency levels. Likewise, students demonstrated substantial growth in creative scientific ability, particularly in flexibility, where the percentage of high-performing students increased from 2.5% to 45%. Statistical analysis using paired samples t-tests confirmed that all gains were significant (p < .001). Overall, the open-ended approach proved to be effective in promoting critical thinking and enhancing students' capacity to reason with evidence and scientific creativity. The study recommends its integration into science education to foster scientifically literate and creative learners.

The aim of this study was to investigate the relationship between level of geometrical thinking and classification skills of polygons of 7 th grade students. Relational survey model was used in this study. The sample of the study consists of 318 7 th grade students from a public school in a city in Central Anatolian region. Data were collected by Geometrical Thinking Levels Determination Test, developed by Özcan (2012) and Polygon Perception and Classification Scale, developed by Ergün (2010). The obtained data were analyzed by using nonparametric tests as Kruskall Wallis-H, Mann Whitney-U and Spearman Brown correlation analysis. According to results of the study, it was found that 7 th grade students' geometrical thinking levels are mainly 1st level; more than half of the students are in the 2nd level and upper level. Students' classification skills of polygons were at medium level. It was stated that there is a positive and meaningful relationship between classification skills of polygon and geometrical thinking levels of students. It was seen that the students with high level of geometrical thinking were successful in classifying of polygons and finding relations between polygons.

Translating Euclid reports on an effort to transform geometry for students from a stylus-and-claytablet corpus of historical theorems to a stimulating computer-supported collaborative-learning inquiry experience. The origin of geometry was a turning point in the pre-history of informatics, literacy, and rational thought. Yet, this triumph of human intellect became ossified through historic layers of systematization, beginning with Euclid's organization of the Elements of geometry. Often taught by memorization of procedures, theorems, and proofs, geometry in schooling rarely conveys its underlying intellectual excitement. The recent development of dynamic-geometry software offers an opportunity to translate the study of geometry into a contemporary vernacular. However, this involves transformations along multiple dimensions of the conceptual and practical context of learning. Translating Euclid steps through the multiple challenges involved in redesigning geometry education to take advantage of computer support. Networked computers portend an interactive approach to exploring dynamic geometry as well as broadened prospects for collaboration. The proposed conception of geometry emphasizes the central role of the construction of dependencies as a design activity, integrating human creation and mathematical discovery to form a human-centered approach to mathematics. This book chronicles an iterative effort to adapt technology, theory, pedagogy and practice to support this vision of collaborative dynamic geometry and to evolve the approach through on-going cycles of trial with students and refinement of resources. It thereby provides a case study of a design-based research effort in computer-supported collaborative learning from a human-centered informatics perspective.

Las nuevas tecnologias aportan al campo de la educacion aspectos innovadores que suponen una mejora cualitativa en las formas de ensenar y aprender. Su introduccion no solo reduce el coste efectivo de aplicacion de teorias y principios de intervencion pedagogica sino que abre las posibilidades de exploracion de modelos procedentes de diferentes campos, facilitando su compatibilizacion y permitiendo ofrecer una vision que los englobe. En este sentido la Realidad Aumentada se puede considerar un concepto tecnologico relativamente nuevo, a pesar de que sus primeros conceptos fueron concebidos a principios de los anos 1990 (Tom Caudell). Es en la actualidad en donde estos conceptos se pueden aplicar de manera practica dado el enorme avance que ha tenido la tecnologia en los ultimos anos. La Realidad Aumentada comprende aquella tecnologia capaz de complementar la percepcion e interaccion con el mundo real, brindando al usuario un escenario real, aumentado con informacion adicional genera...

Chère (s) lectrice (s).. cher (s) lecteur(s)..
Rien que le progrès…Progressez-vous donc en mathématiques !!!... Faites-le avec méthode!!!
      C’est le quatrième numéro de votre Collection Méthodos, intitulé
«Géométrie et grandeurs (2)». Ce numéro de la série "Maths méthodiques au collège"  est susceptible de vous fournir un outil méthodologique précieux qui vous permettra de savoir comment analyser des problèmes de mathématiques de manière simple mais efficace.
      Les exercices proposés, dont les solutions détaillées sont présentées immédiatement après les exercices de méthodes de chaque chapitre, sont organisés selon un degré de difficultés progressif, allant du simple au complexe, et du général au particulier.
      Étant donné que c’est justement la méthodologie qui compte, ne vous lancez jamais, et ne vous précipitez jamais, à voir, sans faire de véritables efforts, les solutions des exercices. Ce n’est qu’au terme d’un travail bien structuré que vous avez droit à comparer ce produit avec la solution proposée.             
        Par ailleurs, exprimer des contenus scientifiques, oralement et par écrit,  en une langue claire et correcte, c’est l’objectif que je vous conseil fermement de se fixer en permanence et durant toute votre carrière d’apprenant, maintenant en secondaire collégial, demain en secondaire qualifiant, après demain à l’université... et le processus d’apprentissage n’a pas de limite..
NOUREDDINE  HALI

Chère (s) lectrice (s).. cher (s) lecteur(s)..
Rien que le progrès…Progressez-vous donc en mathématiques !!!... Faites-le avec méthode!!!
      C’est le deuxième numéro de votre Collection Méthodos, intitulé
« Géométrie et grandeurs (1)». Ce numéro de la série "Maths méthodiques au collège"  est susceptible de vous fournir un outil méthodologique précieux qui vous permettra de savoir comment analyser des problèmes de mathématiques de manière simple mais efficace.
      Les exercices proposés, dont les solutions détaillées sont présentées immédiatement après les exercices de méthodes de chaque chapitre, sont organisés selon un degré de difficultés progressif, allant du simple au complexe, et du général au particulier.
      Étant donné que c’est justement la méthodologie qui compte, ne vous lancez jamais, et ne vous précipitez jamais, à voir, sans faire de véritables efforts, les solutions des exercices. Ce n’est qu’au terme d’un travail bien structuré que vous avez droit à comparer ce produit avec la solution proposée.             
        Par ailleurs, exprimer des contenus scientifiques, oralement et par écrit,  en une langue claire et correcte, c’est l’objectif que je vous conseil fermement de se fixer en permanence et durant toute votre carrière d’apprenant, maintenant en secondaire collégial, demain en secondaire qualifiant, après demain à l’université... et le processus d’apprentissage n’a pas de limite..
NOUREDDINE  HALI

En este artículo se presenta parte del trabajo de grado que estoy desarrollando para optar por el título de maestría en Docencia de las Matemáticas. Específicamente, se recuentan detalles de una experiencia de aula en la que el objetivo es formular una conjetura en la clase de geometría con estudiantes de grado octavo, y a partir de tal recuento se realiza un análisis parcial del rol del profesor, bajo el modelo de la Mediación Semiótica. Dicha experiencia está centrada en el uso de un software de geometría dinámica como artefacto, y basada en un ciclo didáctico fundamentado en la aproximación metodológica para la enseñanza del grupo AE • G.

Sozlukler, milletlerin tarihteki varliklarinin sembollerinden birisi olan ses bayraklarinin yani dillerinin sahip oldugu kelimeleri bunyesinde toplayarak degerlerini emanet ettigi kelimeleri korumakta ve kelimelerin zaman icerisinde kazandigi anlamlar hakkinda kullanicilarina bilgi vermektedir. Ayrica kullanicilarinin kelime servetlerine cesitli kelime ve kavramlari katarak kelime hazinelerinin genislemesini saglamakta ve dusunmenin temeli olan anlam kaliplarinin zenginlesmesine yardimci olmaktadir. Tutumlar davranislarin kazanilmasinda onemli bir yere sahiptir. Bundan dolayi ortaogretim ogrencilerinin sozluklere yonelik olumlu veya olumsuz tutumlarinin sozluk kullanma aliskanliklari uzerinde etkili bir yeri bulunmaktadir. Ogrencilerin sozluklere yonelik olumlu tutumlari, sozluklerden daha etkili bir sekilde yararlanmalarini ve sozluk kullanmayi aliskanlik hâline getirmelerini saglayacagi gibi sozluklere yonelik olumsuz tutumlari da sozluklerden sogumalarina, onu kullanmak istememel...

The paper examines the quality of digitized tasks designed by 10 (small) groups of prospective upper secondary school teachers as part of a geometry course assignment. The results indicate that a small instructional intervention, addressing the planning and implementation of tasks in digitized task environments as well as how to stimulate students to make mathematical generalizations, led to a relatively high proportion (8 out of 10) of high-quality tasks designed by the prospective teachers.

Bu araştırma, dördüncü sınıf fen bilimleri ders kitaplarında yer alan beslenme temalı etkinliklerin tür bağlamında incelenmesini amaçlamaktadır. Araştırma, nitel araştırma modeliyle desenlenmiş olup, doküman incelemesi tekniğiyle gerçekleştirilmiştir. Araştırmanın veri kaynağını ise Millî Eğitim Bakanlığı Talim ve Terbiye Kurulu Başkanlığı tarafından 2023 yılında onaylanan ve iki farklı dördüncü sınıf fen bilimleri ders kitabı oluşturmuştur. İncelenen ders kitaplarında yer alan beslenme temalı etkinlikler Doyle (1988) ile Swan (2007)'nin etkinlik sınıflamaları çerçevesinde betimsel analiz yöntemine tabi tutulmuştur. İncelenen ders kitapları ile gerçekleştirilen analizler sonucu elde edilen veriler tanımlanarak temalaştırılarak kodlanmıştır. Elde edilen bulgular, FBDK 1'deki etkinliklerin genellikle "anlama, yorumlama ve tahmin" türüne odaklandığını, FBDK 2'de ise daha çok "hatırlama" ve "standart algoritma kullanma" türlerinde yoğunlaştığını göstermiştir. Ayrıca, etkinliklerin üst düzey bilişsel süreçleri desteklemek yerine sınırlı türlerde yoğunlaşması, ders kitaplarında çeşitlilik eksikliğine işaret etmektedir. Araştırma, beslenme etkinliklerinin üst düzey bilişsel becerilere yönelik olarak çeşitlendirilmesi gerektiğini vurgulamakta ve etkinlik tasarımında pedagojik yaklaşımların daha dengeli bir şekilde uygulanmasını önermektedir. Araştırmanın sonuçları, beslenme etkinliklerinin tür çeşitliliğini artırarak, öğrencilere daha zengin öğrenme deneyimleri sunulabileceğini göstermektedir.

From a grading list of 28 of the highest ranked mathematics education journals, the six highest ranked journals were chosen, and a systematic search for inquiry-based mathematics education and related keywords was conducted. This led to five important theme/issues for inquiry-based learning in mathematics: 1) communication in the mathematics classroom, 2) mathematical competence, 3) moving in and out of the mathematical domain, 4) tools and resources for planning and implementing inquiry-based learning, and 5) professional development and collaboration. From these five issues, three principles were developed to determine which implications were important for the didactical intervention of the design in the Quality in the subjects Danish and Mathematics (KiDM) project.

Geometry environment. The study was made through a teaching experiment, and shows the development of two of the participant teachers regarding their argumentative schemes and argumentative practices. In the teaching experiment there was an inadequate use of a deductive thinking. At the beginning of the teaching experiment, teachers use symbolic and factual argumentative schemes, and then they adopt, in most argumentative practices, analytical schemes.

In this paper we report on a research aimed to identify and characterize secondary school students' reasoning and proof abilities when working with 3-dimensional geometric solids. We analyze students' answers to two problems asking them to prove certain properties of prisms. As results of this analysis, we get, on the one side, a characterization of students' answers in terms of Van Hiele levels of reasoning and, on the other side, a classification of these answers in different types of proofs. Results from this research give directions to grade and organize secondary school instruction on 3-dimensional geometry.

Définition : Une symétrie axiale correspond à un pliage sur une droite, appelée axe de symétrie. Définition : Dire que deux figures F et F' sont symétriques par rapport à une droite (d) signifie qu'elles se superposent par pliage le long de cette droite (d). Le segment [𝑴'𝑵'] est le symétrique du segment [𝑴𝑵] par rapport à la droite (∆). 𝑰' est le symétrique de 𝑰 par rapport à (∆), avec 𝑰 milieu de [𝑴𝑵] donc 𝑰' milieu de [𝑴'𝑵']. Propriété : Le symétrique d'un segment par rapport à une droite est un segment de même longueur. Remarques : (∆) est la médiatrice des segments [𝑴𝑴'] et [𝑵𝑵']. Si on place un point 𝑪 tel que 𝑪 ∈ (∆) alors son symétrique par rapport à (∆) est lui-même.

In this paper, we report on how a group of UK 12-13 year old students work with hierarchical defining and classifying quadrilaterals, an area which students find very difficult to understand. We implemented a geometry test to 9 groups of students. Quantitative data suggest that the students found very difficult to undertaking hierarchical defining and classifying quadrilaterals even in collaborative learning situations. Our qualitative video-recorded data from the four groups suggest that we find that even in collaborative learning settings prototypical images of geometrical figures strongly influence students’ ways of hierarchically defining and classifying quadrilaterals. In addition, groups often had opportunities to examine their ideas, but they did not explore these opportunities because each member did not see what others were saying ‘as if through the eyes of another’

Besides offering an aesthetic attraction, the use of proofs, which rely on simple geometrical figures suggest intuitive ideas. Two such proofs are presented here. One is a recreation with a stair-like decomposition of a rectangle. The second is a simple game that proves Pythagoras Theorem.

Las matemáticas, como cuerpo de conocimiento científico, están conformadas no sólo por un conjunto de resultados que aparecen en libros y revistas, sino también por métodos empleados para descubrir, construir, añadir esos resultados y validar los resultados. Este último proceso tiene como herramienta la llamada demostración matemática, la cual vale la pena ser incluida en el proceso educativo. Para la realización del estudio se aplicaron dos pruebas en las que debían justificar sus resultados a una muestra aleatoria estratificada. Sin embargo, los alumnos de bachillerato (15-18 años) incluidos en el estudio no emplearon de manera natural la demostración. Es por ello que la Universidad Autónoma de Querétaro (UAQ), a fin de buscar medios para el aprendizaje de la validación matemática ha llevado a cabo un estudio diagnóstico en su Escuela de Bachilleres para detectar las estrategias utilizadas por los alumnos y así poder proponer a futuro medios didácticos para disminuir la distancia entre lo que hacen y lo que se pretende que conozcan. El resultado fundamental es la descripción de la práctica argumentativa de elección preferente por los estudiantes y la identificación que los esquemas en que se agrupan.

La validación del conocimiento construido en Matemáticas es una parte epistemológicamente importante en el proceso metodológico que se conoce como demostración matemática. Su enseñanza incluye procesos complejos y obstáculos que aparecen a lo largo del desarrollo cognitivo del individuo. Con el uso de Software de Geometría Dinámica (SGD), es posible diseñar actividades orientadas a promover la producción de conjeturas y justificaciones en ambientes geométricos, y que facilitarían su aprendizaje. En este trabajo presentamos los tipos de esquemas de argumentación de alumnos de Secundaria (14-15 años) cuando trabajan en el desarrollo de justificaciones matemáticas a partir de exploraciones. Se muestra cómo los alumnos centran su atención en características figurales que resultan irrelevantes en procesos de demostración deductiva. Asimismo, se discute el tipo de propuestas didácticas que deben diseñarse e implementarse para facilitar en los alumnos el desarrollo de esquemas de argumentación analíticos que implican deducciones. Palabras clave. Geometría dinámica, justificaciones geométricas, aprendizaje de la demostración. A validação do conhecimento construído em matemática é uma parte epistemologicamente importante no processo metodológico e é conhecida como a prova matemática. Seu ensino inclui processos complexos e obstáculos que aparecem ao longo do desenvolvimento cognitivo do indivíduo. Usando software de geometria dinâmica (SGD), é possível projetar attividades para promover a produção de conjecturas e justificaçãos em atividades ambientes geométricos e que facilitam a sua aprendizagem. Neste artigo, apresentamos os tipos de esquemas de argumentação de estudantes do ensino médio (14-15 anos), quando trabalham no desenvolvimento de justificativas matemáticas das verificações. Ele mostra como os alunos centrar a sua atenção sobre características figurativas que são irrelevantes no processo da prova dedutiva. Além disso, o tipo de propostas educacionais de ser concebido e implementado para facilitar os alunos no desenvolvimento de esquemas de raciocínio analítico envolvendo deduções é discutido. Palavras chave. Geometria dinámica, justificaçãos geométricas, aprendizagem de prova.

Explanatory videos play an important role in the flipped classroom (FC) approach, which has become highly important especially since the COVID-19 crisis. In this qualitative study, we explore the potential of using explanatory videos (EVs) in mathematical modelling education as important component of FC and assess its merits and deficiencies. We focus on the perspectives and experiences of 44 preservice mathematics teachers (PSTs) attending a German university in their master phase. Considering EVs in an interactive FC context, we employed a multifaceted approach to data collection that included interviews, online surveys with open-ended questions, EVs generated by the PSTs, and written documents resulting from collaborative group activities. The findings show that the EVs used in FCs in modelling can enhance understanding and simplifying modelling content and real-world situations. Moreover, PSTs report that using EVs in flipped modelling seminars can offer personalized learning, optimize teaching hours, decrease anxiety, and increase learners' attention. PSTs also improved their skills in developing EVs on modelling and evaluating their quality in terms of structure, content, and focus. The experienced challenges include time-consuming content development, workload, technical hurdles, and feasibility of FC implementations. Although these challenges are important, they are not content-specific issues, and we conclude that the notable benefits of using EVs in FCs have the potential to promote technology-enhanced modelling education.

La experiencia en aula que se presenta en este documento fue desarrollada en el marco de la Practica de Integracion Profesional a la Escuela de la Licenciatura en Matematicas de la Universidad Pedagogica Nacional en el periodo 2014-I. La practica consistio en realizar una propuesta didactica para la ensenanza de movimientos en el plano: rotacion, reflexion y translacion; mediante la construccion de teselados.

In this article we present a case story of a teaching episode that takes place in an accelerated middle school geometry class (N = 14) using Shape Makers, an interactive geometric curriculum. We use Tawfeeq's bidirectional discourse model and a modified version of Bonnstetter's (1998) Inquiry Continuum to frame (1) the nature of the mathematical discourse, (2) the instructional decisions by the teacher, and (3) progressive conceptualizations of the students' geometric ideas elicited by a non-traditional representation of a kite in an interactive geometric curriculum.

Our engagement with this ICMI Study Conference began when we accepted invitations from the ICMI President, Frederick Leung, to co-chair the Study 26: Advances in Geometry Education. Although we have both worked internationally in geometry education for decades, we had not previously met one another. A briefing meeting with the President, and the Secretary General, Jean-Luc Dorier, provided context and scope for the Study. As with all ICMI studies, the central aims are to consider the topic (in our case geometry education) with respect to theory, research, practice, and policy; and to suggest new directions of research that take into account contextual, cultural, national, and political dimensions of the field. We then worked collaboratively with Professors Leung, Dorier, and other members of the ICMI Executive, to establish an International Program Committee (IPC) with expertise in geometry education. The IPC members have invested substantial time and effort to be part of the IPC, over an extended period. We are grateful for their collective wisdom and technical expertise in framing the core work of the Study. We are especially appreciative of Professor Fabien Emprin's commitment to the Study by taking on the responsibility of Conference Convenor. Fabien's leadership and organisational skills became on integral part of the Study's success. We would also like to acknowledge the effort that some participants from economically disadvantaged countries have made to attend the conference, and the economic support we have received from ICMI and Trent University (Canada) to support their attendance.

Bien que possédant des caractéristiques permettant d'envisager qu'il y soit d'un apport spécifique, une intervention réussie de l'ordinateur dans l'apprentissage ne va pas de soi. Le but auquel nous nous attachons est donc de poser et proposer des solutions à quelques questions soulevées par l'élaboration et l'utilisation d'un logiciel qui fournisse effectivement une aide dans un domaine d'enseignement. meilleure appréhension par l'élève de ces concepts. La deuxième étape, décrite dans un deuxième chapitre, s'attache plus particulièrement aux problèmes de l'intégration de l'ordinateur dans le système didactique. Il s'agit d'effectuer le choix justifié d'un type d'intervention de l'ordinateur dans ce système et de caractériser les moyens logiciels qui permettront cette intervention. Ce choix s'appuie sur l'étude du système didactique et de son fonctionnement dans le cadre de situations d'apprentissage s'appuyant sur la résolution de situations problématiques. intervention dans le système élève-maître-savoir, n'est pas seulement le résultat d'un calcul. En fait, le logiciel, dont nous décrivons la conception dans ce chapitre, est aussi, comme beaucoup d'autres, le fruit de la mise en oeuvre d'idées originales élaborées au cours de pratiques d'enseignement. L'analyse du système permet d'expliciter la prise de conscience de difficultés et d'affiner les stratégies permettant d'y remédier. Trouver les raisons profondes ayant motivé ces idées et leur mise en oeuvre au sein de l'apprentissage est l'occasion d'en renforcer la pertinence. Les réponses apportées à ces questions préliminaires fournissent les bases d'un cahier des charges du logiciel et permettent d'engager sa réalisation. Dans celle-ci se mêlent autant de problèmes conceptuels d'élaboration de structures de données que de problèmes techniques de calcul ou de tracé à l'écran d'objets géométriques. Ces problèmes et les choix qui ont permis d'en fournir une solution sont décrits dans un troisième chapitre. La réalisation proprement dite doit aussi trouver les moyens permettant de rapprocher la géométrie de l'écran qui s'appuie sur un ensemble fini de nombres discrets de la géométrie analytique qui s'appuie sur les réels. La réalisation doit encore élaborer les moyens permettant une communication entre l'apprenant et l'ordinateur répondant aux concepts de l'ergonomie dont la pertinents du point de vue de l'enseignement. l'intervention de l'ordinateur dans le système didactique. De façon plus générale, la mise en oeuvre du logiciel dans une séquence d'enseignement est l'occasion d'évaluer l'intérêt que présente le logiciel et d'observer les rapports que les élèves établissent avec lui. Pour encore augmenter nos chances de réussite dans cette entreprise, nous avons constitué une équipe de spécialistes de différents domaines : mathématiciens, informaticiens, (Milhaud, 1908, p.420) Pour être convainquant, ces discours doivent être construits de manière logique. Mais, il est aussi nécessaire d'utiliser des notions de base : postulats ou prémisses dont le Comment se sert-on de Droite ? Ce qu'on voit apparaître : Comment se sert-on de Cercle ? Ce qu'on voit apparaître : Comment se sert-on de Segment def.2pts ? Ce qu'on voit apparaître : Comment se sert-on de Droite def. 2pts ? Ce qu'on voit apparaître : Comment se sert-on de Triangle def. 3pts? Ce qu'on voit apparaître : Comment se sert-on de Cercle def. 2pts? Ce qu'on voit apparaître : 271 Annexe 2 Comment se sert-on de Milieu ? Ce qu'on voit apparaître : Comment se sert-on de Droite parallèle ? Ce qu'on voit apparaître : Comment se sert-on de Droite orthogonale ? Ce qu'on voit apparaître : Comment se sert-on de Centre d'un cercle ? Ce qu'on voit apparaître : Comment se sert-on de Point sur objet ? Ce qu'on voit apparaître : Comment se sert-on de Intersection de 2 objets ? Ce qu'on voit apparaître :

While representations of 3D shapes are used in the teaching of geometry in lower secondary school, it is known that such representations can provide difficulties for students. In this paper, we report findings from a classroom experiment in which Grade 7 students (aged 12-13) tackled a problem in 3D geometry that was, for them, quite challenging. To analyse students' reasoning about 3D shapes, we constructed a framework of levels of 3D geometrical thinking. We found that students at a lower level of 3D thinking could not manipulate representations effectively, while students operating at a higher level of 3D thinking controlled representations well and could reason correctly.

The purpose of the research was to learn about students' attitudes toward and evaluation of the pedagogical procedure based on integrating dynamic geometry software (DGS) in dealing with construction problems in geometry. The research population consisted of mathematical education students. It was observed that the students preferred to use DGS when dealing with construction problems and appreciated the dynamic features of DGS to a large extent, and that the developed pedagogical procedure improved the students' mode of usage of the computerized environment. We suggest that the explored teaching procedure enables students to construct and assimilate more complete and wide-scale knowledge. Over the few last decades technology has become an important component of mathematical education. Numerous recent studies have indicated that integration of technology into math classrooms has a significant impact on teaching mathematics by enhancing the acquisition of mathematical learning Hollerbands, 2007). Unfortunately, despite these benefits, in practice such integration meets serious difficulties on the part of teachers as well as students .