Overcoming the Technical Nature in Learning Translation: Cabri Geometry, a Tool to Sustain Conceptual Comprehension?

This article presents the results of a didactic experimentation conducted with two elementary pupils (11-12 years of age) in a Quebec school in the of learning geometric translation with a mathematical software, called Cabri Geometry. The teaching-learning sequence experimented with Barth’s (2001) process of conceptualization. The working strategies of pupils were analyzed in relationship to this process. Their comprehension of the concept of translation was considered before and after the teaching-learning sequence using the Herscovics and Bergeron (1988) model of comprehension of conceptual schemata. Results indicated that by following the teachinglearning sequence, the logical mathematical comprehension of the concept of translation evolved in the pupils, while the logical- physical comprehension of this concept remained inaccurate. Moreover, the results revealed different utilization procedures of Cabri Geometry software. These utilization procedures likely affected the process of conceptualization and, consequently, the comprehension of pupils

Contribution of the Analysis of the Mathematical Concordance to Understand the Teachers' KTMT

UIDB/04647/2020 UIDP/04647/2020Technology is recognized by its potential to promote mathematical learning. However, achieving this potential requires the teachers to have the knowledge to integrate it properly into their practices. Several authors have intended to characterize the teachers' knowledge and developed several models, but this approach has often been criticized by its static approach, not attending neither valuing the teachers' practice. In this study we adopt the KTMT - Knowledge for Teaching Mathematics with Technology model, assuming the teachers' practice as the main scenario of analysis. We focus on the options guiding the teachers' decisions when confronted with a situation of lack of mathematical concordance while teaching functions. The situations of lack of mathematical concordance (i.e., situations where the mathematics addressed by the students is different from the one intended by the teacher) are assumed as rich and encapsulating the potential to reveal significant aspects of the teachers' KTMT. The main goal of the study is to understand what domains of the teachers' KTMT are highlighted in these circumstances. A qualitative methodology is adopted and one episode of one 10th grade teacher's practice is analyzed, based on the KTMT model. The conclusions reached show the relevance of different knowledge domains, but emphasize the Mathematics and Technology Knowledge (MTK). They also raise questions about the impact of the specific technology being used on the teachers' KTMT.publishersversionpublishe

Bridging knowing and proving in mathematics An essay from a didactical perspective

Text of a talk at the conference "Explanation and Proof in Mathematics: Philosophical and Educational Perspective" held in Essen in November 2006International audienceThe learning of mathematics starts early but remains far from any theoretical considerations: pupils' mathematical knowledge is first rooted in pragmatic evidence or conforms to procedures taught. However, learners develop a knowledge which they can apply in significant problem situations, and which is amenable to falsification and argumentation. They can validate what they claim to be true but using means generally not conforming to mathematical standards. Here, I analyze how this situation underlies the epistemological and didactical complexities of teaching mathematical proof. I show that the evolution of the learners' understanding of what counts as proof in mathematics implies an evolution of their knowing of mathematical concepts. The key didactical point is not to persuade learners to accept a new formalism but to have them understand how mathematical proof and statements are tightly related within a common framework; that is, a mathematical theory. I address this aim by modeling the learners' way of knowing in terms of a dynamic, homeostatic system. I discuss the roles of different semiotic systems, of the types of actions the learners perform and of the controls they implement in constructing or validating knowledge. Particularly with modern technological aids, this model provides a basis designing didactical situations to help learners bridge the gap between pragmatics and theory

Pour dépasser le technicisme de l’apprentissage de la translation: Cabri-géomètre, un outil de soutien à la compréhension conceptuelle?

Cet article présente les résultats d’une expérimentation didactique réalisée avec deux élèves du primaire (11-12 ans) dans une école du Québec au sujet de l’apprentissage de la translation avec le logiciel Cabrigéomètre. La séquence d’enseignement-apprentissage expérimentée a été conçue de façon analogue à la démarche d’enseignement-apprentissage proposée par Barth (2001) sur le processus de conceptualisation. Les stratégies des élèves en lien avec ce processus ont été analysées. La compréhension de ces élèves du concept de translation a été examinée avant et après la séquence d’enseignement-apprentissage en fonction du modèle de compréhension de schèmes conceptuels de Herscovics et Bergeron (1988). Les résultats indiquent que, suite à la séquence d’enseignement/apprentissage, la compréhension logico-mathématique du concept de translation a évoluée chez ces élèves, alors que leur compréhension logico-physique de ce concept est demeurée inexacte. Par ailleurs, les résultats font état de procédures d’utilisations différentes du logiciel Cabri-géomètre. Elles ont possiblement influencé le processus de conceptualisation et, par conséquent, la compréhension des élèves

A symbolic dance: the interplay between movement, notation, and mathematics on a journey toward solving equations

This article analyzes the use of the software Grid Algebra with a mixed ability class of 21 nine-to-ten-year-old students who worked with complex formal notation involving all four arithmetic operations. Unlike many other models to support learning, Grid Algebra has formal notation ever present and allows students to "look through" that notation and interpret it either in terms of physical journeys on a grid or in terms of mathematical operations. A dynamic fluidity was found between the formal notation, imagery of movements on a grid, and the process of mathematical operations. This fluidity is interpreted as a "dance" between these three. The significant way in which this dynamic took place reflects the scaffolding and fading offered by the software, which was crucial to the students' fluency with formal notation well beyond what has been reported from students of that age. © 2014 Taylor and Francis Group, LLC

Ciências da Educação, Escolarização e Abjeção: diferença e construção da desigualdade.

Neste artigo, traço uma história das ciências dirigidas à escola navirada do século XX e hoje, focando primordialmente os EUA, e exploro os gestosdúbios de esperança e medo como uma história do presente. Considero os temas desalvação da educação progressista americana na pedagogia escolar, no cosmopolitismoda criança que envolve seu gesto de oposto da abjeção e da exclusão nas novas sociologiasda comunidade e da vida da familiar urbana. Retorno aos temas da salvação, daeducação e da comunidade urbana nas reformas contemporâneas num cosmopolitismodo presente que dirige a atenção às qualidades da criança associadas à aprendizagem aolongo da vida que existe em relação aos medos daquela criança que vive fora do espaçocultural desse cosmopolitismo

PISA: numbers, standardizing conduct, and the alchemy of school subjects

Este artículo examina la red de prácticas que confieren inteligibilidad a la organización del conocimiento de las materias escolares que hace PISA. El programa PISA es tratado como un acontecimiento histórico. Su estudio consiste en hacer visible los principios que ordenan y clasifican los objetos “vistos” y que actúan sobre el “conocimiento práctico” de las materias escolares. La política de PISA, a mi juicio, son los principios que rigen lo que los chicos deberían conocer, cómo ese conocimiento se hace posible, y el estudio de la inclusión y exclusión incorporadas en estas prácticas. La primera parte analiza históricamente la transformación de los números en “hechos”, una presunción que posibilita las comparaciones de PISA. En la segunda parte, dirige la atención a los principios de las materias escolares que ordenan y clasifican los “hechos” de las mediciones de PISA.This article examines the grid of practices that give intelligibility to PISA’s organizing the knowledge of school subjects. PISA is treated as an historical event. Its study is to make visible the principles that order and classify the objects “seen” and acted on the “practical knowledge” of school subjects. The politics of PISA, I argue, are in the principles that order what children should know, how that knowing is made possible, and issues of inclusion and exclusion embodied in these practices. The first section historically traces the making of numbers as “facts”, a presumption that makes the comparisons of PISA possible. In the second section, focuses the attention to the principles of school subjects that order and classify the “facts” of PISA’s measurements.Grupo FORCE (HUM-386). Departamento de Didáctica y Organización Escolar de la Universidad de Granada

Using the computer as a tool for constructivist teaching: a case study of Grade 7 students developing representations and interpretations of mathematical notation when using the software Grid Algebra

The aim of this research was to investigate how I engaged in constructivist teaching (CT) when helping a group of low-performing Grade 7 students to develop new meanings of notation as they started to learn formal algebra. Data was collected over a period of one scholastic year, in which I explored the teacher-student dynamics during my mathematics lessons, where students learnt new representations and interpretations of notation with the help of the computer software Grid Algebra. Analysing video recordings of my lessons, I observed myself continuously changing my teaching purpose as I negotiated between the mathematics I intended to teach and the mathematics being constructed by my students. These shifts of focus and purpose were used to develop a conceptual framework called Mathematics-Negotiation-Learner (M-N-L). Besides serving as a CT model, the M-N-L framework was found useful to determine the extent to which I managed to engage in CT during the lessons and also to identify moments where I lost my sensitivity to students constructions of knowledge. The effectiveness of my CT was investigated by focusing on students learning, for which reason I developed the analytical framework called CAPS (Concept-Action-Picture-Symbol). The CAPS framework helped me to analyse how students developed notions about properties of operational notation, the structure and order of operations in numerical and algebraic expressions, and the relational property of the equals sign. Grid Algebra was found to be a useful tool in helping students to enrich their repertoire of representations and to develop new interpretations of notation through what I defined as informal- and formal-algebraic activities. All students managed to transfer these representations and interpretations of notation to pen-and-paper problems, where they successfully worked out traditionally set substitution-and-evaluation tasks

Implementation, use and analysis of open source learning management system “Moodle” and e-learning for the deaf in Jordan

When learning mathematics, deaf children of primary school age experience difficulties due to their disability. In Jordan, little research has been undertaken to understand the problems facing deaf children and their teachers. Frequently, children are educated in special schools for the deaf; the majority of deaf children tend not to be integrated into mainstream education although efforts are made to incorporate them into the system. Teachers in the main stream education system rarely have knowledge and experience to enable deaf students to reach their full potential. The methodological approach used in this research is a mixed one consisting of action research and Human Computer interaction (HCI) research. The target group was deaf children aged nine years (at the third grade) and their teachers in Jordanian schools. Mathematics was chosen as the main focus of this study because it is a universal subject with its own concepts and rules and at this level the teachers in the school have sufficient knowledge and experience to teach mathematics topics competently. In order to obtain a better understanding of the problems faced by teachers and the deaf children in learning mathematics, semi-structured interviews were undertaken and questionnaires distributed to teachers. The main aim at that stage of research was to explore the current use and status of the e-learning environment and LMS within the Jordanian schools for the deaf in Jordan. In later stages of this research, semi-structured interviews and questionnaires were used again to ascertain the effectiveness, usability and readiness of the adopted e-learning environment “Moodle. Finally pre-tests and post-tests used to assess the effectiveness of the e-learning environment and LMS. It is important to note that it was not intended to work with the children directly but were used as test subjects. Based on the requirements and recommendations of the teachers of the deaf, a key requirements scheme was developed. Four open source e-learning environments and LMS evaluated against the developed key requirements. The evaluation was based on a software engineering approache. The outcome of that evaluation was the adoption of an open source e-learning environment and LMS called “Moodle”. Moodle was presented to the teachers for the purpose of testing it. It was found it is the most suitable e-learning environment and LMS to be adapted for use by deaf children in Jordan based on the teachers requirements. Then Moodle was presented to the deaf children’s to use during this research. After use, the activities of the deaf and their teachers were used and analysed in terms of Human Computer Interaction (HCI) analysis. The analysis includes the readiness, usability, user satisfaction, ease of use, learnability, outcome/future use, content, collaboration & communication tools and functionality