In this paper, we turn to the notion of networking theories with the aim of contrasting two theor... more In this paper, we turn to the notion of networking theories with the aim of contrasting two theoretical mathematics education perspectives inspired by Vygotsky's work, namely, the Theory of Objectification and the Documentational Approach to Didactics. We are interested in comparing/contrasting these theories in accordance with the following three main questions: (a) the role that the theories ascribe to language and resources; (b) the conceptions that the theories bring forward concerning the teacher, and (c) the understandings they offer of the mathematics classroom. In the first part of the paper, some basic concepts of each perspective are presented. The second part includes some episodes from a lesson on the teaching and learning of algebra in a Grade 1 class (6-7-year-old students). The episodes serve as background to carry out, in the third part of the paper, a dialogue between proponents of the theoretical perspectives around the identified main questions. The dialogue shows some theoretical complementarities and differences and reveals, in particular, different conceptions of the teacher and the limits and possibilities that language affords in teaching-learning mathematics.
The overwhelming presence of a procedural meaning of equality and equations reported in previous ... more The overwhelming presence of a procedural meaning of equality and equations reported in previous research has led to a call for suitable pedagogical interventions to nurture a relational meaning of these concepts. This paper is a response to that call. Drawing on the theory of objectification, the first part deals with the configuration of a Grade 3 (8-9-year-old students) teaching-learning activity that seeks to create the classroom conditions for the formation of the mathematical operations and operation-based rules that underpin the algebraic simplification of linear equations. Instead of using problems involving abstract open arithmetic sentences or alphanumeric equations (e.g., 5 + __ = 16; 2 + 3 = 11), the teaching-learning activity resorts to story-problems. Two visual semiotic systems serving to model and solve the story-problems were devised. The story-problems were framed in narratives that allowed the teacher and the students to infuse equations, their equating parts, and the mathematical operations with contextual meanings. The first part of the paper includes the theoretical assumptions about the teachinglearning activity and its configuration, and a rationale behind the devisal of the semiotic systems. The second part presents a Vygotskian multimodal genetic analysis of the teaching-learning activity; that is, an analysis that shows the formation of concepts in motion, in the process of their genesis. The genetic analysis sheds some light on the way students, in their work with the teacher, encountered and refined the cultural-historical algebraic meanings of the equal sign and equations, and the concepts required in solving equations. Keywords Algebra • multimodal semiotics • Vygotsky • gestures • algebraic operations • storyproblems
Teaching and Learning Secondary School Mathematics, 2018
In this chapter, we discuss some ideas of a cultural-historical theory of the teaching and learni... more In this chapter, we discuss some ideas of a cultural-historical theory of the teaching and learning of mathematics. The basic ideas emerged from, and have evolved during, an ongoing long-term collaboration between researchers and teachers. Since its inception, this long-term collaboration has sought to offer an alternative to child-centred individualist educational perspectives. It endeavours to understand and foster mathematics thinking, teaching, and learning conceived of as cultural-historical phenomena. This collaboration has led to what has been termed the theory of objectification. We illustrate the basic ideas through the discussion of a classroom episode where what is at stake is the production and understanding of graphs in a grade 10 mathematics class.
En la presentacion de la obra, Bruno D’Amore y Luis Radford ponen en evidencia, a la luz de nuevo... more En la presentacion de la obra, Bruno D’Amore y Luis Radford ponen en evidencia, a la luz de nuevos enfoques –sobre todo socioculturales–, que progresivamente se han venido imponiendo en el campo de la educacion matematica, la necesidad de repensar hoy en dia algunas nociones centrales de la didactica, como aquellas del saber, del conocimiento y del aprendizaje. Los dos autores se refieren con frecuencia a estas nociones a lo largo de la coleccion de textos.
The Second Handbook of Research on the Psychology of Mathematics Education, 2016
In the past few decades, language has become an active focus of investigation in educational rese... more In the past few decades, language has become an active focus of investigation in educational research, including research in mathematics education. Such a focus is a symptom of a relatively recent paradigmatic shift whose chief characteristics are a new understanding of the student and an increasing awareness of the complexities of learning contexts, such as, notably, the complexities arising from cultural and linguistic diversity.
... L. Radford Fig. B While drawing Figure 5, Erica goes back to Figure 4 to count the squares on... more ... L. Radford Fig. B While drawing Figure 5, Erica goes back to Figure 4 to count the squares on the top row. The finger helps her to see and count ... Aintervened and anticipated the key aspects of the figure. Carl said: 1. Carl: We do 6 plus 6 equals 12, plus 1 2. Erica: Yes... No... ...
This article deals with the question of the development of algebraic thinking in young students. ... more This article deals with the question of the development of algebraic thinking in young students. In contrast to mental approaches to cognition, we argue that thinking is made up of material and ideational components such as (inner and outer) speech, forms of sensuous imagination, gestures, tactility, and actual actions with signs and cultural artifacts. Drawing on data from a longitudinal classroom-based research program where 8-year old students were followed as they moved from Grade 2 to Grade 3 to Grade 4, our developmental research question is investigated in terms of the manner in which new relationships between embodiment, perception, and symbol use emerge and evolve as students engage in patterning activities.</p> <p>Este artículo aborda la cuestión del desarrollo del pensamiento algebraico en estudiantes jóvenes. En contraste con los enfoques mentales de la cognición, sostenemos que el pensamiento está compuesto por componentes materiales y del mundo de las ideas tales como el discurso (interior y exterior), formas de imaginación sensitiva, gestos, tacto y acciones reales con signos y artefactos culturales. Con base en datos obtenidos de un programa de investigación longitudinal basado en el aula en el que se siguió el paso de estudiantes de 8 años de segundo grado a tercero y a cuarto, nuestra pregunta de investigación acerca del desarrollo es investigada en términos de la forma en que surgen y evolucionan nuevas relaciones entre el cuerpo, la percepción y el inicio del uso de símbolos a medida que los estudiantes participan en actividades sobre patrones.
In this article we describe and analyze a teaching sequence whose goal was to introduce students ... more In this article we describe and analyze a teaching sequence whose goal was to introduce students to the operation of the unknown and to the related elementary algebraic techniques to solve linear equations. The design of the activity and the analysis of the students' reasoning were conducted within a semiotic-cultural theoretical framework. A discourse analysis of the students' interaction in the classroom suggests that the overcoming of the well-known difficulties that novice students usually encounter in operating on the unknown, as found in traditional approaches to algebra, was accomplished here through the students' complex co-ordination of different sign systems (gestures, natural language, drawings, etc.) and their successful mastery and articulation of levels of abstraction (going from manual industrious actions to intellectual and sign- based actions).
Mathematics classrooms are sites of encounter for different voices, perspectives, and ideas. Thos... more Mathematics classrooms are sites of encounter for different voices, perspectives, and ideas. Those differences become even more visible when the object of difference is language. In his chapter, Barwell draws on Bakhtin's concept of heteroglossia to explore the tensions that underpin multilingual classrooms. He enquires about how those tensions influence the teaching and learning of mathematics and the implications that they may have for equity in mathematics teaching. In my comments, I would like to dwell upon the question of language in the mathematics classroom and on some issues about equity.
The purpose of this article, which is part of a longitudinal classroom research about students&#x... more The purpose of this article, which is part of a longitudinal classroom research about students' algebraic symbolizations, is twofold: (1) to investigate the way students use signs and endow them with meaning in their very first encounter with the algebraic generalization of patterns and(2) to provide accounts about the students' emergent algebraic thinking. The research draws from Vygotsky's historical-cultural school
Atti del Convegno di didattica della matematica, 2004
Puisque lors d'une généralisation, un fait ne peut pas renvoyer à lui-même, tout acte de générali... more Puisque lors d'une généralisation, un fait ne peut pas renvoyer à lui-même, tout acte de généralisation suppose le recours à quelque chose d'autre. Cette « autre chose » relève du domaine de la représentation. Mots, gestes et symboles constituent des couches sémiotiques différentes offrant chacune des possibilités et des limites pour exprimer et objectiver ce qu'il y a à généraliser. Le passage à des généralisations symboliques complexes pose des difficultés précises aux élèves qui commencent à apprendre l'algèbre. Ces difficultés peuvent être mieux comprises si on compare les généralisations symboliques aux généralisations présymboliques que ces mêmes élèves effectuent. En nous situant dans le cadre d'une approche anthropologique culturelle qui s'inspire des travaux de Vygotsky, de la phénoménologie de Husserl et de l'épistémologie kantienne, nous comparerons, dans cette présentation, les généralisations que nous avons appelées « factuelles » et « contextuelles » à la généralisation algébrique symbolique. Nos résultats suggèrent que le passage à la généralisation symbolique nécessite la mise en place d'un système de signification qui se trouve en opposition aux significations qui dérivent du langage naturel et d'une gamme de gestes ostensifs et iconiques qui, opérant ensemble, permettent de situer l'expérience mathématique de l'élève dans le temps et dans l'espace. La mise en place du nouveau système de signification oblige alors l'élève à se situer à l'extérieur de ses repères spatio-temporels créant ainsi un vide « désubjectivé », c'est-à-dire un vide dont il est lui-même exclu comme sujet, et d'où va naître la possibilité du discours mathématique objectif.
In this paper, we turn to the notion of networking theories with the aim of contrasting two theor... more In this paper, we turn to the notion of networking theories with the aim of contrasting two theoretical mathematics education perspectives inspired by Vygotsky&amp;#39;s work, namely, the Theory of Objectification and the Documentational Approach to Didactics. We are interested in comparing/contrasting these theories in accordance with the following three main questions: (a) the role that the theories ascribe to language and resources; (b) the conceptions that the theories bring forward concerning the teacher, and (c) the understandings they offer of the mathematics classroom. In the first part of the paper, some basic concepts of each perspective are presented. The second part includes some episodes from a lesson on the teaching and learning of algebra in a Grade 1 class (6-7-year-old students). The episodes serve as background to carry out, in the third part of the paper, a dialogue between proponents of the theoretical perspectives around the identified main questions. The dialogue shows some theoretical complementarities and differences and reveals, in particular, different conceptions of the teacher and the limits and possibilities that language affords in teaching-learning mathematics.
The overwhelming presence of a procedural meaning of equality and equations reported in previous ... more The overwhelming presence of a procedural meaning of equality and equations reported in previous research has led to a call for suitable pedagogical interventions to nurture a relational meaning of these concepts. This paper is a response to that call. Drawing on the theory of objectification, the first part deals with the configuration of a Grade 3 (8-9-year-old students) teaching-learning activity that seeks to create the classroom conditions for the formation of the mathematical operations and operation-based rules that underpin the algebraic simplification of linear equations. Instead of using problems involving abstract open arithmetic sentences or alphanumeric equations (e.g., 5 + __ = 16; 2 + 3 = 11), the teaching-learning activity resorts to story-problems. Two visual semiotic systems serving to model and solve the story-problems were devised. The story-problems were framed in narratives that allowed the teacher and the students to infuse equations, their equating parts, and the mathematical operations with contextual meanings. The first part of the paper includes the theoretical assumptions about the teachinglearning activity and its configuration, and a rationale behind the devisal of the semiotic systems. The second part presents a Vygotskian multimodal genetic analysis of the teaching-learning activity; that is, an analysis that shows the formation of concepts in motion, in the process of their genesis. The genetic analysis sheds some light on the way students, in their work with the teacher, encountered and refined the cultural-historical algebraic meanings of the equal sign and equations, and the concepts required in solving equations. Keywords Algebra • multimodal semiotics • Vygotsky • gestures • algebraic operations • storyproblems
Teaching and Learning Secondary School Mathematics, 2018
In this chapter, we discuss some ideas of a cultural-historical theory of the teaching and learni... more In this chapter, we discuss some ideas of a cultural-historical theory of the teaching and learning of mathematics. The basic ideas emerged from, and have evolved during, an ongoing long-term collaboration between researchers and teachers. Since its inception, this long-term collaboration has sought to offer an alternative to child-centred individualist educational perspectives. It endeavours to understand and foster mathematics thinking, teaching, and learning conceived of as cultural-historical phenomena. This collaboration has led to what has been termed the theory of objectification. We illustrate the basic ideas through the discussion of a classroom episode where what is at stake is the production and understanding of graphs in a grade 10 mathematics class.
En la presentacion de la obra, Bruno D’Amore y Luis Radford ponen en evidencia, a la luz de nuevo... more En la presentacion de la obra, Bruno D’Amore y Luis Radford ponen en evidencia, a la luz de nuevos enfoques –sobre todo socioculturales–, que progresivamente se han venido imponiendo en el campo de la educacion matematica, la necesidad de repensar hoy en dia algunas nociones centrales de la didactica, como aquellas del saber, del conocimiento y del aprendizaje. Los dos autores se refieren con frecuencia a estas nociones a lo largo de la coleccion de textos.
The Second Handbook of Research on the Psychology of Mathematics Education, 2016
In the past few decades, language has become an active focus of investigation in educational rese... more In the past few decades, language has become an active focus of investigation in educational research, including research in mathematics education. Such a focus is a symptom of a relatively recent paradigmatic shift whose chief characteristics are a new understanding of the student and an increasing awareness of the complexities of learning contexts, such as, notably, the complexities arising from cultural and linguistic diversity.
... L. Radford Fig. B While drawing Figure 5, Erica goes back to Figure 4 to count the squares on... more ... L. Radford Fig. B While drawing Figure 5, Erica goes back to Figure 4 to count the squares on the top row. The finger helps her to see and count ... Aintervened and anticipated the key aspects of the figure. Carl said: 1. Carl: We do 6 plus 6 equals 12, plus 1 2. Erica: Yes... No... ...
This article deals with the question of the development of algebraic thinking in young students. ... more This article deals with the question of the development of algebraic thinking in young students. In contrast to mental approaches to cognition, we argue that thinking is made up of material and ideational components such as (inner and outer) speech, forms of sensuous imagination, gestures, tactility, and actual actions with signs and cultural artifacts. Drawing on data from a longitudinal classroom-based research program where 8-year old students were followed as they moved from Grade 2 to Grade 3 to Grade 4, our developmental research question is investigated in terms of the manner in which new relationships between embodiment, perception, and symbol use emerge and evolve as students engage in patterning activities.</p> <p>Este artículo aborda la cuestión del desarrollo del pensamiento algebraico en estudiantes jóvenes. En contraste con los enfoques mentales de la cognición, sostenemos que el pensamiento está compuesto por componentes materiales y del mundo de las ideas tales como el discurso (interior y exterior), formas de imaginación sensitiva, gestos, tacto y acciones reales con signos y artefactos culturales. Con base en datos obtenidos de un programa de investigación longitudinal basado en el aula en el que se siguió el paso de estudiantes de 8 años de segundo grado a tercero y a cuarto, nuestra pregunta de investigación acerca del desarrollo es investigada en términos de la forma en que surgen y evolucionan nuevas relaciones entre el cuerpo, la percepción y el inicio del uso de símbolos a medida que los estudiantes participan en actividades sobre patrones.
In this article we describe and analyze a teaching sequence whose goal was to introduce students ... more In this article we describe and analyze a teaching sequence whose goal was to introduce students to the operation of the unknown and to the related elementary algebraic techniques to solve linear equations. The design of the activity and the analysis of the students' reasoning were conducted within a semiotic-cultural theoretical framework. A discourse analysis of the students' interaction in the classroom suggests that the overcoming of the well-known difficulties that novice students usually encounter in operating on the unknown, as found in traditional approaches to algebra, was accomplished here through the students' complex co-ordination of different sign systems (gestures, natural language, drawings, etc.) and their successful mastery and articulation of levels of abstraction (going from manual industrious actions to intellectual and sign- based actions).
Mathematics classrooms are sites of encounter for different voices, perspectives, and ideas. Thos... more Mathematics classrooms are sites of encounter for different voices, perspectives, and ideas. Those differences become even more visible when the object of difference is language. In his chapter, Barwell draws on Bakhtin's concept of heteroglossia to explore the tensions that underpin multilingual classrooms. He enquires about how those tensions influence the teaching and learning of mathematics and the implications that they may have for equity in mathematics teaching. In my comments, I would like to dwell upon the question of language in the mathematics classroom and on some issues about equity.
The purpose of this article, which is part of a longitudinal classroom research about students&#x... more The purpose of this article, which is part of a longitudinal classroom research about students' algebraic symbolizations, is twofold: (1) to investigate the way students use signs and endow them with meaning in their very first encounter with the algebraic generalization of patterns and(2) to provide accounts about the students' emergent algebraic thinking. The research draws from Vygotsky's historical-cultural school
Atti del Convegno di didattica della matematica, 2004
Puisque lors d'une généralisation, un fait ne peut pas renvoyer à lui-même, tout acte de générali... more Puisque lors d'une généralisation, un fait ne peut pas renvoyer à lui-même, tout acte de généralisation suppose le recours à quelque chose d'autre. Cette « autre chose » relève du domaine de la représentation. Mots, gestes et symboles constituent des couches sémiotiques différentes offrant chacune des possibilités et des limites pour exprimer et objectiver ce qu'il y a à généraliser. Le passage à des généralisations symboliques complexes pose des difficultés précises aux élèves qui commencent à apprendre l'algèbre. Ces difficultés peuvent être mieux comprises si on compare les généralisations symboliques aux généralisations présymboliques que ces mêmes élèves effectuent. En nous situant dans le cadre d'une approche anthropologique culturelle qui s'inspire des travaux de Vygotsky, de la phénoménologie de Husserl et de l'épistémologie kantienne, nous comparerons, dans cette présentation, les généralisations que nous avons appelées « factuelles » et « contextuelles » à la généralisation algébrique symbolique. Nos résultats suggèrent que le passage à la généralisation symbolique nécessite la mise en place d'un système de signification qui se trouve en opposition aux significations qui dérivent du langage naturel et d'une gamme de gestes ostensifs et iconiques qui, opérant ensemble, permettent de situer l'expérience mathématique de l'élève dans le temps et dans l'espace. La mise en place du nouveau système de signification oblige alors l'élève à se situer à l'extérieur de ses repères spatio-temporels créant ainsi un vide « désubjectivé », c'est-à-dire un vide dont il est lui-même exclu comme sujet, et d'où va naître la possibilité du discours mathématique objectif.
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