Basic components in the scienctific didactical training of the secondary school mathematics teachers

Secondary mathematics teacher training in Spain is currently the subject of a heated revision debate. The speed of social, cultural, scientific and economic changes have left a hundred years old teacher training model well behind. However, academical inertia and professional interests are impeding a real new training of the mathematics teacher as an autonomous mathematical educator. Teachers of Didactic of Mathematics and the Spanish Associations of mathematics teachers have recently been discussing the issue. Their conclusions are included here.

Didactique des grandeurs en mesure et élèves en difficulté d'apprentissage du 2e cycle du primaire

Le programme -Une école adaptée à tous ses élèves-, qui s'inscrit dans la réforme actuelle de l'éducation au Québec, nous a amenée à nous intéresser aux représentations dans les grandeurs en mesure en mathématiques des élèves en difficulté d'apprentissage. Nous nous sommes proposés de reconduire plusieurs paramètres de la recherche de Brousseau (1987, 1992) auprès de cette clientèle. La théorie des champs conceptuels (TCC) de Vergnaud (1991), appliquée aux structures additives, a été particulièrement utile pour l'analyse et l'interprétation de leurs représentations. Comme méthode de recherche, nous avons utilisé la théorie des situations didactiques en mathématiques (TSDM), réseau de concepts et de méthode de recherche appuyé sur l'ingénierie didactique qui permet une meilleure compréhension de l'articulation des contenus à enseigner. Grâce à la TSDM, nous avons observé les approches didactiques des enseignants avec leurs élèves. Notre recherche est de type exploratoire et qualitatif et les données recueillies auprès de 26 élèves de deux classes spéciales du deuxième cycle du primaire ont été traitées selon une méthode d'analyse de contenu. Deux conduites ont été adoptées par les élèves. La première, de type procédural a été utilisée par presque tous les élèves. Elle consiste à utiliser des systèmes de comptage plus ou moins sophistiqués, de la planification aux suites d'actions. La deuxième consiste à récupérer directement en mémoire à long terme le résultat associé à un couple donné et au contrôle de son exécution. L'observation des conduites révèle que les erreurs sont dues à une rupture du sens. Ainsi, les difficultés d'ordre conceptuel et de symbolisation nous sont apparues plus importantes lorsque l'activité d'échange demandait la compétence "utilisation" et renvoyait à la compréhension de la tâche, soit les tâches dans lesquelles ils doivent eux-mêmes découvrir les rapports entre les variables à travailler et à simuler les actions décrites dans les énoncés. En conséquence, les problèmes d'échanges se sont révélés difficiles à modéliser en actes et significativement plus ardus que les autres. L'étude des interactions enseignants et élèves a démontré que la parole a été presque uniquement le fait des enseignants qui ont utilisé l'approche du contrôle des actes ou du sens ou les deux stratégies pour aider des élèves en difficulté. Selon le type de situation à résoudre dans ces activités de mesurage de longueur et de masse, des mobilisations plurielles ont été mises en oeuvre par les élèves, telles que la manipulation d'un ou des étalon(s) par superposition, par reports successifs, par pliage ou par coupure lorsque l'étalon dépassait; par retrait ou ajout d'un peu de sable afin de stabiliser les plateaux. Nous avons également observé que bien que certains élèves aient utilisé leurs doigts pour se donner une perception globale extériorisée des quantités, plusieurs ont employé des procédures très diverses au cours de ces mêmes séances. Les résultats présentés étayent l'hypothèse selon laquelle les concepts de grandeur et de mesure prennent du sens à travers des situations problèmes liées à des situations vécues par les élèves, comme les comparaisons directes. Eles renforcent et relient les grandeurs, leurs propriétés et les connaissances numériques.

Erreurs arithmétiques des élèves et interventions de l'enseignant débutant : une analyse didactique en termes de schèmes

Ancrée dans le domaine de la didactique des mathématiques, notre thèse cible le « travail de l’erreur » effectué par trois enseignants dans leur première année de carrière. Libérés des contraintes associées au système de formation initiale, ces sujets assument pleinement leur nouveau rôle au sein de la classe ordinaire. Ils se chargent, entre autres, de l’enseignement de l’arithmétique et, plus précisément, de la division euclidienne. Parmi leurs responsabilités se trouvent le repérage et l’intervention sur les procédures erronées. Le « travail de l’erreur » constitue l’expression spécifique désignant cette double tâche (Portugais 1995). À partir d’un dispositif de recherche combinant les méthodes d’observation et d’entrevue, nous documentons des séances d’enseignement afin de dégager les situations où nos maîtres du primaire identifient des erreurs dans les procédures algorithmiques des élèves et déploient, subséquemment, des stratégies d’intervention. Nous montrons comment ces deux activités sont coordonnées en décrivant les choix, décisions et actions mises en œuvre par nos sujets. Il nous est alors possible d’exposer l’organisation de la conduite de ces jeunes enseignants en fonction du traitement effectif de l’erreur arithmétique. En prenant appui sur la théorie de champs conceptuels (Vergnaud 1991), nous révélons l’implicite des connaissances mobilisées par nos sujets et mettons en relief les mécanismes cognitifs qui sous-tendent cette activité professionnelle. Nous pouvons ainsi témoigner, du moins en partie, du travail de conceptualisation réalisé in situ. Ce travail analytique permet de proposer l’existence d’un schème du travail de l’erreur chez ces maîtres débutants, mais aussi de spécifier sa nature et son fonctionnement. En explorant le versant cognitif de l’activité enseignante, notre thèse aborde une nouvelle perspective associée au thème du repérage et de l’intervention sur l’erreur de calcul de divisions en colonne.

A etnomatemática em uma cerâmica da região do Seridó/RN

This work has as objective to describe mathematical knowledge used as tools in the manufacture and marketing of tiles of red ceramic by potters of the Currais Novos village/ RN, located 250 km from the capital of Rio Grande do Norte. For us to reach our objective, we rely on conceptions ambrosianas of Ethnomatematics, besides of the qualitative research in an ethnographic approach. In the empirical part of the research, that went it accomplishes in the period from 2009 to 2012 in the Currais Novos Village, we support the following tools for data collection, semi-structured interviews, field diary, photographs, audio recordings and participant observations. In the analysis of the collected data, we can conclude that there are mathematical knowledge in the management of manufacture and marketing of tiles, often different from the academic mathematics, mainly in the wood cube, on cube of the clays, in the handler with the measures time, the count method , in the arrangement of tiles, in the preparation of the ceramic mass and sale of tiles. Theses knowledge were described and analyzed in the light of the theoretical Ethnomatematics, also supported in official documents, such as Parameters Nacional Curriculares. The analyzes of these knowledge generated subsidies for elaboration of an educational product - a proposal of didactic sequence destined to the Teaching of Mathematics in Elementary and Middle levels for the community schools and region, this proposal is in the Appendix to this work

Dificuldades na construção de gráficos de funções

This study describes about graduation s students difficulties of to draw functions graph. Specifically, we intend to observe their abilities evolution, as well as their difficulties during Calculus I subject in engineering course. For that, we show them publications about the elaboration of graphs and its difficulties in obstacle terms and some researches witch contain this subject and that it was done during postgraduate studies in mathematical education. It shows by research methodology aspects related to French didatic s mathematic and some theories of cognitive psychology considering the high value between theoretical-methodological relation that was evidenced in both theoretical conceptions about ways to understand and teach mathematic. This methodology is based on didactic engineering purpose, that consist in preliminaries analysis, conception and didactic sequence analysis prior, trials by application followed analysis up and conclusion. We had also used pedagogicals actions and analysis of results achieved, to classify types of errors made by the 2005 s students during second semester, from conceptions related to the episthemologic and didactics obstacles

Contribution of the research component in internship reports to the development of didactic knowledge of mathematics

Monogr??fico con el t??tulo: " Formaci??n de profesores. Perspectivas de Brasil, Colombia, Espa??a y Portugal"

A Didactic Model of Computer Supported Education in Higher Mathematics

Report published in the Proceedings of the National Conference on "Education and Research in the Information Society", Plovdiv, May, 2016

CLIL materials in Secondary Education: focusing on the language of instruction in the subject area of Mathematics

[EN]Applying a CLIL methodological approach marks a shift in emphasis from language learning based on linguistic form and grammatical progression to a more ‘language acquisition’ one which takes account language functions. In this article we will study the elements of the “language of instruction” of the area of Maths in Secondary Education, by focusing on the analysis of the communicative functions, and the lexical and the cultural items present in the textbook in use. Our aim is to present the CLIL teacher with the linguistic and didactic implications that he or she should take into consideration when implementing the bilingual syllabuses with their students. In order to do that, we will present our conclusions emphasizing the need for coordination in different content areas, linguistic and communicative contents, between the foreign language teacher and the CLIL subject one.

Online games for the teaching of mathematics

In this article we present a didactic experience developed by the GIE (Group of Educational Innovation) “Pensamiento Matemático” of the Polytechnics University of Madrid (UPM), in order to bring secondary students and university students closer to Mathematics. It deals with the development of a virtual board game called Mate-trivial. The mechanics of the game is to win points by going around the board which consists of four types of squares identified by colours: “Statistics and Probability”, “Calculus and Analysis”, “Algebra and Geometry” and “Arithmetic and Number Theory ”. When landing on a square, a question of its category is set out: a correct answer wins 200 points, if wrong it loses 100 points, and not answering causes no effect on the points, but all the same, two minutes out of the 20 minutes that each game lasts are lost. For the game to be over it is necessary, before those 20 minutes run out, to reach the central square and succeed in the final task: four chained questions, one of each type, which must be all answered correctly. It is possible to choose between two levels to play: Level 1, for pre-university students and Level 2 for university students. A prototype of the game is available at the website “Aula de Pensamiento Matemático” developed by the GIE: http://innovacioneducativa.upm.es/pensamientomatematico/. This activity lies within a set of didactic actions which the GIE is developing in the framework of the project “Collaborative Strategies between University and Secondary School Education for the teaching and learning of Mathematics: An Application to solve problems while playing”, a transversal project financed by the UPM.

Cognitive-didactic Model for Making Testing Items in Physics

Report published in the Proceedings of the National Conference on "Education and Research in the Information Society", Plovdiv, May, 2015

Didactic transposition: an example in the educative system in Costa Rica

This paper includes some fundamental ideas about the theory of didactic transposition of Yve Chevallard (1980), from which it is made an analysis of the knowledge transformation suffers from the mathematical level to the scholastic level. To demonstrate this transformation, it is analyzed the theme of integers, which is included in the seventh- grade curriculum of the educational system of Costa Rica.

Humor in mathematics teaching

HUMOR: OUR VIEW FOR MATHEMATICS TEACHING Our assumptions and context. Process humor and be able to produce is clearly a sign of intelligence, revealing, when done well, complex reasoning. Humor has an important social role, assuming as a cognitive experience that as well as creating a sense of well-being, predisposes people to work and can improve the productivity of that work. Mathematics is a discipline in which the reasoning occupies a very prominent place, both as a science as a school area. At the same time, students' interest for mathematics is not always the same and some have initially not very favorable feelings (Toh, 2009; Wanzer, Frymier & Irwin, 2010). Recent curriculum changes to the teaching of mathematics have been, in most countries of the world, showing the need for students to develop skills of critical nature, such as communication, thinking and problem solving along with the acquisition of mathematical knowledge. Also in Portugal, it is claimed the importance of promoting learning that combine the construction of mathematical knowledge with its use, when performing mathematical tasks and communicating mathematical ideas and mathematical reasoning. In the early years of schooling, corresponding to primary education in many countries, the use of texts such as short stories or comics, from which we can develop challenging mathematical tasks, is reported in the literature as having potential to promote learning specified in curricular documents (Wanzer, Frymier., & Irwin, 2010). In particular, some texts focus on mathematical topics in a humorous way and to be understood, students must develop their mathematical competence. The development of mathematical tasks from stories and other humorous presents big challenges to teachers (Flores & Moreno, 2011). Our questions. In this context, we put some questions: Primary teachers use in their classes tasks or situations that present, in a humorous way, mathematical ideas? What resources do they use? Also: How to select, adapt or build texts and tasks which have, in a humorous way, mathematical ideas with didactic potential for education in the early years of schooling? If the resources for this purpose have been produced and if teachers have been sensitized for their use, are they able to integrate them in their classes? Our intentions. This research project seeks to address these questions, focused on: (i ) assessment of teachers’ practices and underlying knowledge, resources available for the use of texts with mathematical ideas presented in a humorous way; (ii) selection, adaptation and construction of mathematical tasks from texts that present, in a humorous way, mathematical ideas with didactic potential in education for the early years of schooling; and ( iii ) integration and use, by primary school teachers, of texts that present , in a humorous way, contexts for the teaching of mathematics. So, the project is organized into three tasks and as a methodological design that combines qualitative elements with quantitative elements, the first one prevailing.