Anna Sfard

• University of Haifa

Once teaching-learning events are conceptualized as inter-discursive encounters, it becomes clear that mathematics classroom talk is rife with invisible pitfalls. There are many types of unacknowledged discursive gaps, some of them necessary for learning and some potentially harmful. Such gaps may exist also between the teacher’s intentions and her...

Once teaching-learning events are conceptualized as inter-discursive encounters, it becomes clear that mathematics classroom talk is rife with invisible pitfalls. There are many types of unacknowledged discursive gaps, some of them necessary for learning, and some potentially harmful. Such gaps may exist also between the teacher's intentions and he...

According to commognitive conceptualization, development of mathematical thinking, whether historical or ontogenetic, requires periodic transitions to mathematical discourse incommensurable with the one that has been practiced so far. In this new discourse, some familiar mathematical words will be used in a new way. Historically, such discursive tr...

Once teaching-learning events are conceptualised as inter-discursive encounters, it becomes clear that mathematics classroom talk is rife with invisible pitfalls. There are many types of unacknowledged discursive gaps, some of them necessary for learning, and some potentially harmful. Such gaps may exist also between the teacher’s intentions and he...

Ever since the beginning of the twentieth century, language, widely recognized as the main source of all things human, has been gradually becoming the principal object of study in all human sciences. Mathematics education is no exception. Since its inception, this research has already generated two language-related lines of study, one of them devot...

According to commognitive conceptualization, development of mathematical thinking, whether historical or ontogenetic, requires periodic transitions to mathematical discourse incommensurable with the one that has been practiced so far. In this new discourse, some familiar mathematical words will be used in a new way. Historically, such discursive tr...

In this commognitive study, we take a close look at the interactive problem-solving by two middle-school students’ dyads, one of which participated in research conducted in Montreal, Canada in 1992, and the other, 25 years later, was a part of a classroom investigation in Melbourne, Australia. The present study was inspired by the second author’s i...

Conceptualizing numbers as discursive constructs generated in, and for the sake of, communication, we investigated the development of the numerical discourse of Milo, a boy who was 2 years and 8 months old when we first met him and whom we then followed for 18 months. Our analyses of the child’s evolving responses to the question “Where is there mo...

In this paper, learning is conceptualized as a process of routinization of learners’ actions. We begin by operationalizing the keyword routine. Two categorizations of routines are then offered. First, we distinguish between practical and discursive routines, discuss the process of their historical co-development, and claim that over time, new disco...

This commentary combines the insights offered by the contributors to this special issue into one consistent whole, a single story of identity and the learning of mathematics. Its somewhat unusual format—the story is told in an interview granted by the “Ideal Mathematics Education Researcher” to a journalist—signals that the author addresses the nar...

Abstract In this paper, learning is conceptualized as a process of routinization of learners’ actions.We begin by operationalizing the keyword routine. Two categorizations of routines are then offered. First, we distinguish between practical and discursive routines, discuss the process of their historical co-development, and claim that over time, n...

In this chapter, research in mathematics education is defined as a special type of discourse in which potentially useful stories about learning and teaching mathematics are being told. A consistent collection of stories coming from a given discourse is known as a theory. A commognitive version of theory of mathematics learning, made distinct by its...

In this symposium, we ask what mathematical engagement looks like in the context of play, focusing on contexts designed to support mathematical thinking through open-ended activity, and looking at ages that are traditionally overlooked in studies of play. We heed Dewey’s admonition to look beyond sugar-coating: we do not seek to claim that “play is...

What is human learning, what is it that makes it distinct, and how can we account for its uniqueness? In this talk, after a brief review of the widely differing answers that have been given to these questions by generations of thinkers, I reflect on lessons that can be learned from the history of research on learning. I follow with some observation...

This is a commentary to Sandra’s Crespo opening pleanry, titled “En la lucha/In the Struggle”

This article focuses on the theoretical-methodological question of how to identify reform-induced changes in school mathematics. The issue arose in our project The Evolution of the Discourse of School Mathematics (EDSM), in which we studied transformations in high-stakes examinations taken by students in England at the end of compulsory schooling....

In this chapter, while analysing the event from Chapter 2 and trying to identify the opportunities for learning the teacher has been creating for his students, I am guided by the conceptualisation of learning called commognitive. According to this approach, mathematics is a form of discourse and learning mathematics means becoming capable of partic...

With the intention to let her research findings reported in Chapter 4 have some impact on the practice of teaching, the author writes a letter to the teacher, in which she explains the difference between two modes of doing mathematics, the ritualized and the explorative. After defining the two notions and illustrating them with examples, the author...

South Africa is not just one among many comparable settings for mathematics education research and Jill Adler is not an ordinary mathematics education researcher who, her special prominence in the field notwithstanding, does basically the same type of work as all other members of this worldwide community. For the last two decades, South Africa has...

Research for Educational Change presents ways in which educational research can fulfil its commitments to educational practice. Focussing its discussion within the context of mathematics education, it argues that while research-generated insights can have beneficial effects on learning and teaching, the question of how these effects are to be gener...

Mathematical discourse is often described as abstract and devoid of human presence, yet many school curricula espouse an aim to develop active, creative mathematical problem posers and solvers. The project The Evolution of the Discourse of School Mathematics (EDSM) developed an analytic scheme to investigate the nature of school mathematics discour...

When Oksapmin number system met Western arithmetic: Answering the question of how cognitive and cultural change induce each other.

Among the diverse domains of human knowing, mathematics stands out as a hothouse for insights about teaching and learning in general. This chapter summarizes the ways that research on mathematics learning contributes to our understanding of how people learn. We begin with a brief historical overview, which explains what it is about mathematics that...

What is the relation between individual cognition and culture? This question has always been on Geoff Saxe’s mind, serving as the compass for his lifelong professional travel. The quandary is not a small one, to be sure, and not a new one either. Over centuries, it was pursued by the best of philosophical and scientific minds. So why tackle the old...

This symposium brings together researchers interested in studying mathematical proficiency through a focus on students' dispositions toward mathematics-their ideas and affect about mathematics and their patterns of engagement with it. While dispositions are useful for connecting important aspects of students' proficiency, they are also broad and ch...

Today’s widespread concern with mathematical literacy indicates the educators’ commitment to the old principle, formulated more than three decades ago by Hans Freudenthal: “The child should be able to use in society what [he or she] has learned at school.” As obvious and uncontestable as this maxim seems to be, the question of how the term mathemat...

Taking as a point of departure the vision of school algebra as a formalized meta-discourse of arithmetic, we have been following six pairs of 7th-grade students (12-13 years old) as they gradually modify their spontaneous meta-arithmetic toward the “official” algebraic form of talk. In this paper we take a look at the very beginning of this process...

It may be a useful exercise to look at the story of Paul Cobb’s life-long quest for an ever better understanding of educational processes as a modern representative of the same genre as the ancient tales of Argonauts attempting to recover the Golden Fleece or of Sir Galahad pursuing the phantom of the Holy Grail. As I will be arguing in this epilog...

The chapter reprinted in this part of the book outlines an analytical approach for teachers develop and revise their instructional practices. The institutional setting of mathematics teaching as we conceptualize it encompasses district1 and school policies for mathematics instruction. It therefore includes both the adoption of curriculum materials...

I completed my dissertation studies with Les Steffe and Ernst von Glasersfeld at the University of Georgia in 1983 and then accepted a faculty position at Purdue University in Indiana. The first study that I conducted at Purdue University was built on my dissertation work and focused on the psychological contexts within which young children interpr...

My career as a mathematics education researcher began in 1978 when I entered the master’s program in mathematics education at the University of Georgia. Prior to moving to Georgia with my wife Jenny, I had completed my undergraduate degree in mathematics at Bristol University in England and was a secondary math teacher for 2 years. Our primary reas...

The emergent perspective described in Chapter 4 serves to orient the analysis of the teacher’s and students’ actions and interactions in particular mathematics classrooms. As noted in Chapter 4, this framework was developed while analyzing the data generated in the course of two year-long classroom design experiments (an initial experiment conducte...

The notion of a classroom mathematical practice was introduced in the reprinted chapter in the last part. It is fair to say in retrospect that my initial use of this construct was largely intuitive: I had not defined a classroom mathematical practice with any precision or clarified how other researchers might identify the mathematical practices int...

The study presented in this paper, which serves as a pilot study for a future comprehensive project, was to investigate how students deal with the concepts of infinity and limit. Based on the communicational approach to cognition, according to which mathematics is a kind of discourse, we tried to identify the characteristics of students’ discourse...

In this paper I address the question of how to talk about learning so as to be able to cope with at least some of the longstanding quandaries and to arrive at new insights. After a very brief historical review, I concentrate on two basic metaphors for learning in which current educational research seems to be grounded: the metaphors of learning‐as‐...

Testing and measurement have always been inextricable parts of the processes of teaching and learning, but never did they occupy as central a place in educational processes as they do today. This unprecedented assessment frenzy is just a particular manifestation of a more general phenomenon: of our present tendency for speaking in numbers about abs...

While reading the articles assembled in this volume, one cannot help asking Why gestures? What’s all the fuss about them? In the last few years, the fuss is, indeed, considerable, and not just here, in this special issue, but also in research on learning and teaching at large. What changed? After all, gestures have been around ever since the birth...

1. Posing the question: What is it that changes when one learns mathematics? In the field of mathematics education, the terms discourse and communication seem to be these days in everybody's mouth. They feature prominently in research papers, they can be heard in teacher preparation courses, and they appear time and again in variety of programmatic...

This book contributes to the current debate about how to think and talk about human thinking so as to resolve or bypass such time-honored quandaries as the controversy of nature vs. nurture, the body and mind problem, the question of learning transfer, and the conundrum of human consciousness. The author responds to the challenge by introducing her...

Los textos que conforman este libro surgen de dos preocupaciones profesionales muy persistentes: una de índole general y otra personal. Como muchos de mis colegas no he estado satisfecha con la investigación existente sobre aprendizaje, y como muchos otros decidí remediar el problema tratando de revisar los anteojos conceptuales existentes… El enfo...

Il existe une tension apparente entre l’idée d’apprenant autonome, promue par les tenants des classes « participatives », et l’argument participatif selon lequel l’apprentissage est un processus intrinsèquement collectif d’induction vers des formes d’action historiquement établies. Dans cet article, nous essayons de comprendre la nature de cette te...

The interpretive framework for the study of learning introduced in this article and called commognitive is grounded in the assumption that thinking is a form of communication and that learning mathematics is tantamount to modifying and extending one's discourse. These basic tenets lead to the conclusion that substantial discursive change, rather th...

Why do different people react differently to the same situations, and why, in spite of the difference, do their actions display distinct family resemblance? How do members of society coordinate their actions without any conscious effort to do so? In what sense can we claim that actions performed by an individual are a product of collective doing?

Based on close observations of two 4-year-old children responding to their parents' requests for quantitative comparisons, we offer a "participationist" account of the origins and development of numerical thinking, one that portrays numbers as a product rather than a pregiven object of human communication. In parallel, we propose a participationist...

In this article, the authors make an attempt to operationalize the notion of identity to justify the claim about its potential as an analytic tool for investigating learning. They define identity as a set of reifying, significant, endorsable stories about a person. These stories, even if individually told, are products of a collective storytelling....

To investigate mechanisms of failure in mathematics, we adopt the communicational approach to cognition, which describes thinking as an activity of communication and learning mathematics as an initiation to a certain type of discourse. In the search for factors that impede students' participation in arithmetic communication, we examine the arithmet...

These seem to be very special times for mathematics education. The public interest in the topic has never been greater. Probably the most prominent among the occurrences that occasioned this recent leap in popularity are international comparative studies such as TIMSS and PISA. The fact that, in spite of the ongoing efforts toward reform in mathema...

The purpose of the study presented in this paper is to investigate how students deal with the concepts of infinity and limit. Based on the communicational approach to cognition, according to which mathematics is a kind of discourse, we try to identify the characteristics of students' discourse on the topic. Four American and four Korean students we...

In the attempt to account for striking differences between learning activities of immigrant mathematics students from the former Soviet Union and of their native Israeli classmates, we introduce the notions of actual and designated identities. These identities are subsequently presented as important factors that mold learning and influence its effe...

Part 1 Constructivism and the Learning of Mathematics: A Radical Constructivist View of Basic Mathematical Constructs, Ernst Von Glaserfeld Interaction and Children's Mathematics, Leslie P. Steffe and Ron Tzur Articulating Theories of Mathematics Learning, Stephen Lerman. Part 2 Psychology, Epistemology and Hermeneutics: Another Psychology of Mathe...

This paper presents a rationale and a conceptual framework for a wider research project dealing with mathematical communication, and in particular with actions performed by interlocutors whenever they wish to clarify their use of a symbol, a word or an expression. The aim of such actions is often to repair a communicational breach resulting from di...

Traditional approaches to research into mathematical thinking, such as the study of misconceptions and tacit models, have brought significant insight into the teaching and learning of mathematics, but have also left many important problems unresolved. In this paper, after taking a close look at two episodes that give rise to a number of difficult q...

The analysis of the Batteries and AIDS episodes presented in this article is guided by the assumption that thinking can be conceptualized as an activity of communication, and learning can be regarded as modifying and extending one's discursive ways. Within this framework, 1 of the aims of mathematical learning is to become skillful in the discursiv...

The authors of this volume claim that mathematics can be usefully re-conceptualized as a special form of communication. As a result, the familiar discussion of mental schemes, misconceptions, and cognitive conflict is transformed into a consideration of activity, patterns of interaction, and communication failure. By equating thinking with communic...

Traditional approaches to research into mathematical thinking, such as the study of misconceptions and tacit models, have brought significant insight into the teaching and learning of mathematics, but have also left many important problems unresolved. In this paper, after taking a close look at two episodes that give rise to a number of difficult q...

Without Abstract

In this article we take a close look at the now popular claim that many school subjects, and mathematics among them, are best learned in an interactive way, through conversation with others. Two types of specially devised analytical tools are used to analyze the data coming from a two-month-long series of interactions between two 13-year-old boys l...

Bridging the individual and the social: Discursive approaches to research in mathematics education Guest editorial While looking at the papers collected in this volume one feels that in spite of their diverse themes, these seven studies have quite a lot in common and, as a collection, seem to be signaling the existence of a distinct, relatively new...

The main thesis of this chapter is that to prepare teachers to their new role in the reformed mathematics classroom, we must first understand better the mechanisms of the interactions supposed to take place in these classrooms. Of particular interest are the instances of students’ collective problem-solving. The student-to-student classroom interac...

In this article, I take a critical look at some popular ideas about teaching mathematics, which are forcefully promoted worldwide by the reform movement. The issue in fo- cus is the nature and limits of mathematical discourse. Whereas knowing mathemat- ics is conceptualized as an ability to participate in this discourse, special attention is given...

This study deals with students' construction of mathematical objects. The basic claim is that the need for communication--any attempt to evoke certain actions by others--is the primary driving force behind all human cognitive processes. Effectiveness of verbal communication is seen as a function of the quality of its focus. Material objects may ser...

Presents a teaching experiment to turn students from external observers into active participants in a game of algebra learning where students use graphs to build meaning for equivalence of algebraic expressions. Concludes that the graphic-functional approach seems to make the introduction to algebra much more meaningful for the learner. (ASK)