Learning Mathematics

From Hierarchies to Networks

Prof Leone Burton, Leone Burton

• Publisher: Routledge
• Year: 2012
• Print ISBN: 9780750710084
• eBook ISBN: 9781135698478
• Topic: Education
• Subtopic: Education General

Learning Mathematics brings together a collection of interrelated and forward-looking chapters by internationally recognized experts that explores changes in the theories and practices of learning (and teaching) mathematics.
The authors reject a traditional, transmission view of the teaching of mathematics which has proved so ineffective for learning. In its place they offer information gathered from research and from practice about effects on the learners seeking to create and negotiate meaning. Learners are presented as actively attempting to make sense of the mathematics they encounter, and learners, teachers and researchers are offered examples of ho such sense-making activities, incorporated into mathematics classrooms, impact on coming to know.
The book celebrates both diversity, in the range of different perspectives, contributions and topics, and unity, in the linking chapters and themes, It will be fascinating reading for those mathematics educators who are eager to engage with a socio-cultural perspective in order to better understand the complexity of learning mathematics.

Section Three

Teaching and Learning Mathematics

In the five chapters in this section, the authors look closely at teaching and learning from the points of view of students and their teachers. The focus continues to be on the making of meaning and on respect for those engaged in this enterprise. In Chapter 9, Barbara Jaworski relates theories to teaching practices and she does not avoid the problematics of trying to do so. The issues that she raises about interpreting, questioning, inculcating or eliciting knowledge and classroom style are central to the concerns of most teachers. Terry Wood and Tammy Turner-Vorbeck focus, in Chapter 10, on how differences in the patterns of interaction and discourse influence what is learned in mathematics classrooms. Labels, for John Mason, in Chapter 11 are a key device that enables reference to, and consequently learning from, experience. To establish networks of meaning demands, therefore, attending to the use of, and meaning given to, labels. In Chapter 12, the group of authors from Canada develop further their theory about the growth of mathematical understanding as dynamic and requiring what they call ‘folding back’. They point to the implications of taking a non-hierarchical view of growth in understanding both for the learning and for the mathematics. Finally, Hilary Povey and Leone Burton explore authoring in Chapter 13. Authoring is, for them, a way of moving towards control of new learning, author/ity. To this end, they discuss what it means to be a learner/ author, how that affects classroom behaviours and what discourses and practices encourage or discourage these. They engage in dialogue with two teachers about their experiences of trying to move towards a classroom that respects all those within it.

9 Tensions in Teachers’ Conceptualizations of Mathematics and of Teaching

Barbara Jaworski

I feel in my head I have a system of mathematics. I don't know what it looks like but it's there, and whenever I learn a new bit of mathematics I have to find somewhere that that fits in. It might not just fit in one place, it might actually connect up a lot of places as well. When I share things it's very difficult because I can't actually share my mathematical model or whatever you want to call it, because that's special to me. It's special to me because of my experiences. So, I suppose I'm not a giver of knowledge because I like to let people fit their knowledge into their model because only then does it make sense to them. Maybe that's why if you actually say, ‘Well probability is easy. It's just this over this’, it doesn't make sense because it's got nowhere to fit. That's what I feel didactic teaching is a lot about, isn't it? Giving this knowledge, sharing your knowledge with people, which is not possible? (Ben, in Jaworski, 1994a, p. 157)

These are the spontaneous words of a teacher, Ben, talking to a researcher, myself, between lessons at which I was present as participant observer. Over a period of 9–12 months I explored with Ben his beliefs and theories, the motivations guiding his mathematics teaching and their rationalization with his classroom practice. This took place as part of a four-year classroom-based study of six teachers who engaged in an investigative approach to their teaching of mathematics at secondary level.

This chapter addresses dilemmas in the practice of the six teachers from a perspective of alternative paradigms of knowledge growth and, through this, relationships between theory and practice in the growth of knowledge about learning and teaching mathematics. These relationships reflect the complex interconnections between growth of knowledge of mathematics and growth of knowledge of teaching mathematics; between personal theories of the human beings involved and established theories of the research community; and between theory and practice in both teachers’ interpretations of theory in their classroom work, and researcher conceptualizations of the theory-practice interface as manifested in these classrooms.

Interpretation and Construction

It is important to recognize that this research was interpretive at many levels. The teachers in the study interpreted an investigative approach in their classrooms. This involved interpretations of teachers’ own theoretical perspectives relative to the various social/cultural worlds impinging on their classrooms: school and classroom culture, the culture of mathematics, the demands of schooling and assessment, home, family and ethnic groupings, and so on. As a researcher, representing what I saw to occur or understood teachers to think, I offer interpretations arising from my own perspectives. Many of these were fed back to the teachers for their comments. However, since they are written from my own theoretical positions, and research paradigms in which the research is embedded, they often differ in emphasis from the teachers’ perspectives. I therefore need to recognize the total situation and context of analytical outcomes, including research(er) decisions, to ensure rigour in the research. These alternative interpretations might be seen as narratives that jointly illuminate knowledge growth for students, teachers and researchers relative to their domain of experience (e.g. Bruner, 1986; Burton, Chapter 2 in this book); or, following Mason (Chapter 11), they might be seen as fragments from which insights into teaching and teachers’ thinking can emerge.

The early years of fieldwork in this research coincided with the emergence of radical constructivism (von Glasersfeld, 1984) as a paradigm through which to view growth of knowledge in mathematics and mathematics education. The thinking of several teachers was seen, initially, to fit with a radical constructivist perspective, and later one that is better described as social constructivist. I thus interpreted events through a constructivist lens, and an important part of my theoretical synthesis involved a justification of interpretations relative to this position.

There was no intent in this research to label the teachers as constructivist. Indeed, I question whether such a label has any meaning at all, since constructivism is not about pedagogy. In fact, constructivism was never mentioned between the teachers and myself during the classroom study. Despite this, I argue that teachers could be seen as developing their teaching from a constructivist perspective. One teacher, reading my work much later, commented ‘so I was constructivist before I knew what one was. Does that mean I constructed constructivism?’ (Mike, in Jaworski, 1994a, p. 132). The quotation from Ben, with which this chapter began, is paradigmatic of the position of teachers in this study. Analysis of this position will be left until after some discussion of the theoretical perspectives involved.

In order to explain, illuminate and critically situate issues arising from the classroom situations, some reference to established theories of the growth of knowledge is necessary. This is not to reduce the classroom richness and complexity to exemplification of particular theoretical perspectives, but rather to engage in a dialectical relationship between given theories and interpretations of practice. Demonstrating validity in interpretive research leads to a positing of theoretical perspectives, which unsurprisingly intersects with theories in the public domain. John Shotter (1995), in a critique of a number of theoretical positions, questions relationships between theory and practice:

If practice is not learned by first learning theory—and theory is not merely an accurate representation of a state of affairs—then what do we academics have to say that is of any worth to practitioners? And if we do have anything, how best should it be communicated to them if not as a theoretical representation? (p. 42)

My perspective here is to use theory in the public domain, together with its associated questions, issues and dichotomies, as a lens or lenses into practice, to, reflexively, inform and critique perceptions of knowledge and its growth from both theoretical and practical positions. Inevitably of course the discourse remains theoretical, and its relationship to its practical manifestations a matter of philosophical distinction.

Theoretical Perspectives

Radical Constructivism

Throughout the 1980s constructivism in mathematics education developed in a very theoretical way. Largely through the writings of Ernst von Glasersfeld, drawing strongly on the work of Piaget, a philosophy, epistemology or ideology, relating to mathematical cognition and cognitive processing was promulgated. Von Glasersfeld, supported by other scholars, presented a persuasive view of ‘coming to know’ in mathematics, which he referred to as Radical Constructivism. His definition of two principles is now very well known:

1	Knowledge is not passively received either through the senses or by way of communication. Knowledge is actively built up by the cognizing subject.

2	a.	The function of cognition is adaptive, in the biological sense of the term, tending towards fit or viability;
	b.	Cognition serves the subjects’ organization of the experiential world, not the discovery of an objective ontological reality (von Glasersfeld, 1990, p. 22).

It is the second of these principles that is radical in that it breaks away from a traditional metaphysical epistemology. It requires a recognition of the adaptive nature of cognition and the relative position of knowledge. From a radical constructivist perspective it is impossible to talk about the status of knowledge in absolute terms. Rather, knowledge needs to be related to its (historical) situation and context, which often means that of the individual knower. Contrary to some criticisms of radical constructivism, this position does not reduce to solipsism (Gergen, 1995; Lerman, 1989, 1996a). The very recognition of the relative nature of knowledge forces a critical rationalization with experience, as well as with socially constituted bodies of knowledge and other knowers (which has profound implications for research). Kenneth Gergen's claim (1995, p. 28) that this implies ‘that there is a real world that is separate from one's experiences of it, thus reasserting the dualist assumption’, is not one that I accept. Radical constructivism says no more than that if there is such a world we cannot know it except through our experiences.

The profound implications for research in a constructivist paradigm are highlighted by Martin Hammersley, who, referring to the ‘cultural relativism of constructivism’, suggests it is unclear how constructivism differs from ‘fiction or ideology’, in that ‘research reports...cannot be judged in terms of their validity, in the sense of how accurately they represent the events of the world’ (Hammersley, 1993, p. 6). Essentially, what is problematic with this statement is its assumption that validity rests with ‘how accurately [research reports] represent the events of the world’, since it assumes we can have objective knowledge of these events. In a constructivist paradigm, validity must be related to making sense of judgements in terms of their full situation and context. Other paradigms take a related stance, for example the narrative paradigm mentioned earlier and the discipline of noticing articulated by John Mason (Chapter 11). Hammersley, however, speaks from an objectivist paradigm incommensurable with constructivism, since it depends on acceptance of ontological reality beyond the experience of the knower. It is worth drawing attention to this as relevant to the validity of the research in focus, but also since the issue is at the root of teaching dilemmas that are discussed later in this paper.

The Social Dimension

The social constitution of knowledge is central to a social constructivist orientation. This orientation is not an ‘add-on’ to a radical position, but rather the result of placing a magnifying lens onto certain aspects of human experience that foster rationalization, that is interactions and communication between human beings. To emphasize the importance of the social perspective, Taylor and Campbell-Williams (1993, p. 135) have offered a ‘third principle’ to the two from von Glasersfeld:

The third principle derives from the sociology of knowledge, and acknowledges that reality is constructed intersubjactively, that is it is socially negotiated between significant others who are able to share meanings and social perspectives of a common lifeworld (Berger and Luckmann, 1966). This principle acknowledges the sociocultural and socioemotional contexts of learning, highlights the central role of language in learning, and identifies the learner as an interactive coconstructor of knowledge.

While this articulation of the social position is helpful, it must be recognized that it is not an ‘extra’ to the other two principles, but rather a qualification of the second. It emphasizes the importance of socio-cultural settings in influencing cognition, and of cooperation and negotiation (in the sense of debate, not compromise—see also Wood, Chapter 10) with others in offering alternative perspectives and challenging constructions. What the authors do not do, is account for intersubjective construction of reality, and the roles of socio-cultural and socio-emotional contexts in knowledge growth. We need to turn to a debate about Piagetian and Vygotskian developmental frames to see why this might be problematic.

The Individual and the Social: Piaget and Vygotsky

Although radical constructivism can be seen to develop from the work of Jean Piaget (e.g. von Glasersfeld, 1983), particularly from the point of view of accommodation of experience and reflective abstraction of concepts, social constructivism has been aligned with Vygotskian theory (Ernest, 1991). Lev Vygotsky emphasized the fundamental role of social influences on learning, particularly the role of language. Thus, it might seem seductive to seek a conjunction between aspects of Piagetian and Vygotskian world-views to elaborate the social dimension of constructivism. However, it has been pointed out that this leads to an inconsistent absorption of Piaget into Vygotsky or vice versa (Confrey, 1995; Lerman, 1996a).

Vygotsky wrote that ‘Human learning presupposes a special social nature by which children grow into the intellectual life of those around them’ (1978, p. 88). This might seem in striking contrast to a Piagetian view (in Jerome Bruner's terms ‘a paradigm of a lone organism pitted against nature’; 1985, p. 25) focusing largely on individual cognition and developing the well known and much criticized stage theory. Yet Piaget not only recognized the importance of the social domain, he articulated explicitly its position with regard to his theory of intellectual development (Piaget, 1950...

Table of contents

• Front Cover
• Half Title
• Studies in Mathematics Education Series
• Title Page
• Copyright
• Contents
• List of Figures and Tables
• Series Editor’s Preface
• Foreword
• Section One Abandoning Hierarchies, Abandoning Dichotomies
• Section Two Mathematics as a Socio-cultural Artefact
• Section Three Teaching and Learning Mathematics
• Notes on Contributors
• Index