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Keyword: ‘Luis Radford’

“Three Key Concepts of the Theory of Objectification: Knowledge, Knowing, and Learning”: Luis Radford

16/10/2013 Deja un comentario

Abstract

In this article I sketch three key concepts of a cultural-historical theory of mathematics teaching and learning—the theory of objectification. The concepts are: knowledge, knowing and learning. The philosophical underpinning of the theory revolves around the work of Georg W. F. Hegel and its further development in the philosophical works of K. Marx and the dialectic tradition (including Vygotsky and Leont’ev). Knowledge, I argue, is movement. More specifically, knowledge is a historically and culturally codified fluid form of thinking and doing. Knowledge is pure possibility and can only acquire reality through activity—the activity that mediates knowledge and knowing. The inherent mediated nature of knowing requires learning, which I theorize as social, sensuous and material processes of objectification. The ideas are illustrated through a detailed classroom example with 9–1 0-year-old students.

Keywords: objectification; knowledge, knowing, learning, consciousness.

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“Working With Cultural-Historical Activity Theory”: Wolff-Michael Roth, Luis Radford & Lionel LaCroix

14/01/2013 Deja un comentario

Abstract: This article focuses on the experiences of two researchers, Wolff-Michael ROTH and Luis RADFORD, using cultural-historical activity theory in mathematics education. The aim is to provide insights into the ways these researchers see and engage with activity theory, how they have come to adopt and expand it, and some of the challenges and concerns that they have had using it. These questions are not usually addressed within typical scientific papers. Yet, they are important for understanding both the dynamics of research and the practical use of cultural-historical activity theory. Since the format of research report papers is not necessarily well suited to convey personal experiences and thinking, the present article takes the form of a conversation, which provides an effective vehicle for exploring and articulating these matters. This provides a basis for understanding more deeply the underlying assumptions of this theory; its dynamics and how it is applied in research of mathematics practice, thinking, and learning; and insights into the manner in which experienced researchers grapple with the theoretical dimensions of their research.

Key words: cultural-historical activity theory; dialectical thinking; Leont’ev; Vygotsky; mathematics education; objectification; subjectification

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“Elementos de una teoría cultural de la objetivación”: Luis Radford

19/12/2012 Deja un comentario

RESUMEN
En este artículo se presentan los lineamientos generales de una teoría cultural de la objetivación –una teoría de la enseñanza y el aprendizaje de las matemáticas que se inspira de escuelas antropológicas e histórico-culturales del conocimiento. Dicha teoría se apoya en una epistemología y una ontología no racionalistas que dan lugar, por un lado, a una concepción antropológica del pensamiento y, por el otro, a una concepción esencialmente social del aprendizaje. De acuerdo con la teoría, lo que caracteriza al pensamiento no es solamente su naturaleza semióticamente mediatizada sino sobre todo su modo de ser en tanto que praxis reflexiva. El aprendizaje de las matemáticas es tematizado como la adquisición comunitaria de una forma de reflexión del mundo guiada por modos epistémico-culturales históricamente formados.

PALABRAS CLAVE: Objetivación, pensamiento matemático, semiótica, sentido, significado, significación cultural, signos.

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“Bakhtin, Alterity, and Ideology”: Luis Radford

12/11/2012 Deja un comentario

Commentary on the Chapter by Richard
Barwell, “Heteroglossia in Multilingual
Mathematics Classrooms”

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.

1 Language in the Mathematics Classroom

One way or another, for one reason or another, since the time of Babylonian schools, institutional educations have always faced the question of linguistic diversity. However, the manner in which this diversity has been addressed and understood has not always been the same. Contemporary schools seem to be led to address this diversity along the lines of contemporary concerns about equity and social justice. These concerns, of course, are a token of social and political interests in coming to grips with cultural diversity, brought forward by unprecedented migratory movements of a global scale.

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“On the Cognitive, Epistemic, and Ontological Roles of Artifacts”: Luis Radford

20/08/2012 Deja un comentario

1 Introduction

Galileo opens his Discourses and Mathematical Demonstrations Relating to Two New Sciences with a remark about the famous 16th century Venetian arsenal, which he praises for its impressive amount of instruments and machines; this arsenal, he says, offers an opportunity to wonder and think. With their unprecedented variety of tools and artifacts, contemporary classrooms may have looked like the Venetian arsenal to Galileo. True, some of the artifacts that are part of our educational settings have been there for a long time now – for example, textbooks. Others, however, made their appearance with the digital technological progress during the 20th century. And, like the instruments and machines of the Venetian arsenal, they offer new possibilities for thinking and learning.

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“On Signs and Representations. A Cultural Account”: Luis Radford

31/12/2011 Deja un comentario

“l’écriture, la lettre, l’inscription sensible ont toujours été considérées par la tradition occidentale comme le corps et la matière extérieurs à l’esprit, au souffle, au verbe et au logos”.
J. Derrida, De la grammatologie, 1967, p. 52.

1. The ghost of the metaphysics of pre-sence

One of the oldest discussions on the relationships between language and ideas is found in Plato’s dialogue Cratylus.

In this dialogue, Plato deals with several conceptions of language. One of them is defended by Hermogenes, the poor brother of the rich Callias, who claims that names and language are merely conventional —like, he says, the names of slaves, that may be given and changed at pleasure. Another conception of language is held by Cratylus, who maintains that there is a perfect match between the things and their names. The name or the sign of a thing, according to Cratylus, discloses or uncovers the true nature of the thing. In fact, Cratylus goes further, for, in a certain passage of the dialogue, he affirms that all truth and knowledge derive from language and names. Then, Socrates, with his usual subtle spirit of controversy, replies that if knowledge comes from names, then the names must have preceded the things. “But”, he adds conclusively, “how could he [who gives names to things] have learned or discovered things from names if the primitive names were not yet given?” (Cratylus, 347b).

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“La razón desnaturalizada: ensayo de epistemología antropológica”: Luis Radford

12/12/2011 Deja un comentario

Resumen:
La epistemología tradicional, desde Kant, ha planteado la investigación del conocimiento en términos de los fundamentos y de las representaciones que la Razón se hace de las cosas fuera de la mente. Dicha agenda de investigación supone, empero, una concepción de la Razón y de sus razones. ¿Cuáles son las razones de la Razón? En este ensayo proponemos que la epistemología tradicional, en cuyo marco el pensamiento moderno encontró un sostén decisivo, elaboró un concepto de Razón cuya raíz ha de buscarse en el movimiento de la Ilustración, por un lado, y en lo que Max Weber llamó la “razón instrumental”, por el otro. Este ensayo relata dos de los momentos centrales que han conducido a un cuestionamiento profundo de las bases en que reposa la racionalidad moderna y a su ineludible consecuencia: la desnaturalización de la razón. Se aboga, al final, por una concepción más amplia de racionalidad que, prestando más atención a las diferentes voces de los alumnos, nos permita proponer prácticas educativas más aptas para la enseñanza de las matemáticas.

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“Sujeto, objeto, cultura y la formación del conocimiento”: Luis Radford

16/11/2011 Deja un comentario

Resumen: La relación sujeto-objeto ha sido reconocida tradicionalmente como un elemento clave en las diferentes teorías del conocimiento. Mientras que, a partir de Kant, la relación se plantea en términos de un sujeto que construye el objeto, con Hegel y luego el materialismo dialéctico, la relación mencionada es vista de tal forma que el objeto de conocimiento es inseparable de la actividad de los individuos Ambas aproximaciones sirvieron de punto de apoyo a elaboraciones teóricas posteriores en ramas como la psicología, la epistemología, la filosofía y la sociología, interesadas en la comprensión del desarrollo del conocimiento. La intención de este artículo es contribuir a la identificación de ciertas diferencias teóricas en corrientes contemporáneas en la Educación Matemática –diferencias que han conducido a lo que Sfard (1999) ha sugestivamente llamado ‘la guerra de los paradigmas’.

 

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“The cultural-epistemological conditions of the emergence of algebraic symbolism”: Luis Radford

25/10/2011 Deja un comentario

ABSTRACT
The main thesis of this paper is that algebraic symbolism emerged in the Renaissance as part of a new type of thinking − a new type of thinking shaped by the socioeconomic activities that arose progressively in the late Middle-Ages. In its shortest formulation, algebraic symbolism emerged as a semiotic way of knowledge representation inspired by a world substantially transformed by the use of artefacts and machines. Algebraic symbolism, I argue, is a metaphoric machine itself encompassed by a new general abstract form of representation and by the Renaissance technological concept of efficiency. To answer the question of the conditions which made possible the emergence of algebraic symbolism, I enquire about the cultural modes of representation of knowledge and human experience and look for the historical changes which took place in cognitive and social forms of signification.

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“Cerebro, cognición y matemáticas”: Luis Radford y Mélanie André

08/10/2011 Deja un comentario

RESUMEN. En este artículo abordamos el problema de la relación entre el cerebro, la cognición y las matemáticas. En la primera parte discutimos algunos elementos de la anatomía y crecimiento del cerebro; a partir de esos elementos y de resultados recientes de investigaciones en neurociencias, en la segunda parte presentamos un esbozo de las regiones cerebrales que generalmente están asociadas al pensamiento aritmético. Aquí, ponemos una particular atención a las áreas cerebrales que se activan en el pasaje del pensamiento aritmético perceptual (común en varias especies) al simbólico calculatorio (específico del humano).

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“Semiótica cultural y cognición”: Luis Radford

02/10/2011 Deja un comentario

La psicología debe proceder necesariamente del hecho que entre la conciencia individual y la realidad objetiva existe el “enlace mediador” de la cultura históricamente formada, la cual actúa como prerrequisito y condición de la actividad individual mental.

Evald Ilyenkov, The Concept of the Ideal.

Introducción

La semiótica cultural que será abordada en este trabajo plantea el problema de la cognición humana desde un punto de vista antropológico. A la pregunta general que formula la antropología acerca de la relación entre cognición y cultura, la semiótica cultural responde en dos etapas.

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