Luis Radford

A Cultural-Historical Perspective on Mathematics Teaching and Learning

Eighty years ago, L. S. Vygotsky complained that psychology was misled in studying thought independent of emotion. This situation has not signifi cantly changed, as most learning scientists continue to study cognition independent of emotion. In this book, the authors use cultural-historical activity theory as a perspective to investigate cognition, emotion, learning, and teaching in mathematics. Drawing on data from a longitudinal research program about the teaching and learning of algebra in elementary schools, Roth and Radford show (a) how emotions are reproduced and transformed in and through activity and (b) that in assessments of students about their progress in the activity, cognitive and emotional dimensions cannot be separated. Three features are salient in the analyses: (a) the irreducible connection between emotion and cognition mediates teacher-student interactions; (b) the zone of proximal development is itself a historical and cultural emergent product of joint teacher-students activity; and (c) as an outcome of joint activity, the object/motive of activity emerges as the real outcome of the learning activity. The authors use these results to propose (a) a different conceptualization of the zone of proximal development, (b) activity theory as an alternative to learning as individual/social construction, and (c) a way of understanding the material/ideal nature of objects in activity.

Re/thinking the Zone of Proximal Development (Symmetrically)

The notion of zone of proximal development has come to be used widely to theorize learning and learning opportunities. Unfortunately, following a simplified reading of its original definition and primary sense in the quote that opens this text, the concept tends to be thought of in terms of the opposition of individuals. One of these individuals, a teacher or peer, is more capable than another individual, the learner. Somehow they engage in an “inter-mental” or “inter-psychological” plane from where the learner constructs knowledge from himor herself on an “intra-mental” or “intrapsychological” plane. That is, such conceptualizations convey a substantialist approach that thinks learning as knowledge assimilation and collectivity in terms of ensembles of individual actors interacting unproblematically. Their interaction is thematized through the dubious prism of the differences of what happens within the individual consciousness and what happens in collective consciousness—as if they could exist separately. Speaking is reduced to the individual, subjective intention of the speaker, who, in speaking, is considered to externalize ideas that have previously formed on the inside. The approach is substantialist in that it takes some prior situation, including the institutional positions of the participants in an interaction (i.e., teacher, student), and uses it to make causal attribution about the events that ensue. But such approaches are unsatisfactory given that there is insufficient attention to the co-constitutive nature of subjective consciousness and collective consciousness. More so, such approaches convey notions of verbal expressions

Algebraic thinking from a cultural semiotic perspective

In this article, I introduce a typology of forms of algebraic thinking. In the first part, I argue that the form and generality of algebraic thinking are characterised by the mathematical problem at hand and the embodied and other semiotic resources that are mobilised to tackle the problem in analytic ways. My claim is based not only on semiotic considerations but also on new theories of cognition that stress the fundamental role of the context, the body and the senses in the way in which we come to know. In the second part, I present some concrete examples from a longitudinal classroom research study through which the typology of forms of algebraic thinking is illustrated.

On the epistemological limits of language: Mathematical knowledge and social practice during the renaissance

An important characteristic ofcontemporary reflections in mathematicseducation is the attention given tolanguage and discourse. No longer viewed asonly a more or less useful tool to expressthought, language today appears to beinvested with unprecedented cognitive andepistemological possibilities. One wouldsay that the wall between language andthought has crumbled to the point that nowwe no longer know where one ends and theother begins. At any rate, the thesis thatthere is independence between theelaboration of a thought and itscodification is no longer acceptable. Theattention given to language cannot ignore,nevertheless, the question of itsepistemological limits. More precisely,can we ascribe to language and to thediscursive activity the force of creatingthe theoretical objects of the world ofindividuals? In this article, I suggestthat all efforts to understand theconceptual reality and the production ofknowledge cannot restrict themselves tolanguage and the discursive activity, butthat they equally need to include thesocial practices that underlie them. Thispoint is illustrated through the analysisof the relationship between mathematicalknowledge and the social practice of theRenaissance.

Philosophical, multicultural and interdisciplinary issues

School mathematics reflects the wider aspect of mathematics as a cultural activity. From the philosophical point of view, mathematics must be seen as a human activity both done within individual cultures and also standing outside any particular one. From the interdisciplinary point of view, students find their understanding both of mathematics and their other subjects enriched through the history of mathematics. From the cultural point of view, mathematical evolution comes from a sum of many contributions growing from different cultures.

Grade 2 students’ non-symbolic algebraic thinking

The learning of arithmetic, it has recently been argued, need not be a prerequisite for the learning of algebra. From this viewpoint, it is claimed that young students can be introduced to some elementary algebraic concepts in primary school. However, despite the increasing amount of experimental evidence, the idea of introducing algebra in the early years remains clouded by the lack of clear distinctions between what is arithmetic and what is algebraic. The goal of this chapter is twofold. First, at an epistemological level, it seeks to contribute to a better understanding of the relationship between arithmetic and algebraic thinking. Second, at a developmental level, it explores 7–8-years old students’ first encounter with some elementary algebraic concepts and inquires about the limits and possibilities of introducing algebra in primary school.

Early Algebraic Thinking: Epistemological, Semiotic, and Developmental Issues

In this article I present some findings of an ongoing 5-year longitudinal research program with young students. The chief goal of the research program is a careful and systematic investigation of the genesis of embodied, non-symbolic algebraic thinking and its progressive transition to culturally evolved forms of symbolic thinking. The investigation draws on a cultural-historical theory of teaching and learning—the theory of objectification—that emphasizes the sensible, embodied, social, and material dimension of human thinking and that articulates a cultural view of development as an unfolding dialectic process between culturally and historically constituted forms of mathematical knowing and semiotically mediated classroom activity.

Of Love, Frustration, and Mathematics: A Cultural-Historical Approach to Emotions in Mathematics Teaching and Learning

Emotions have traditionally been characterized as inner, subjective, and physiological experiences, usually of an irrational nature. Against this subjectivist and physiological position, drawing on cultural psychology and anthropological research, in this article I advocate for a cultural conception of emotions and their role in thinking in general and mathematical thinking in particular. I argue that, rather than momentarily subjective phenomena, emotions (for instance, anger, frustration, love) are historically constituted. Emotions, I contend, are not opposed to thinking, but are an integral part of it. Emotions are as ubiquitous as breathing. I illustrate these ideas through the analysis of Grade 4 students working on a mathematical problem.

Historical formation and student understanding of mathematics

The use of history of mathematies in the teaching and learning of mathematics requires didactical reflection. A crucial area to explore and analyse is the relation between how students achieve under standing in mathematics and the historical construction of mathematical thinking.

The Semiotics of the Schema

What is the relationship between our mental activity and the empirical objects of the world? Kant raised this question in the Critique of Pure Reason and attempted to answer it by arguing that between the realm of concepts and that of sensuous phenomena lies the schema. Piaget re-elaborated the Kantian concept of schema and since then it has been extensively used in constructivist and psychological accounts of the mind. In this article, I discuss Kant’s and Piaget’s concept of schema from a semiotic-cultural perspective. Attention is paid to the epistemological premises on which the Kantian and Piagetian theoretical elaborations of the concept of schema were based and the role that signs played therein. I contend that the schema and its genesis can be better conceptualized if we take into account linguistic and non-linguistic mediated actions embedded in the social processes of meaning production and knowledge objectification. My discussion interweaves epistemological concerns with the semiotic analysis of a group of Grade 11 students dealing with the mathematical understanding and description of a natural phenomenon — the movement of a body along a ramp in a technological environment.