Search

In 1997, forty invited mathematics educators met Jean Lave at the University of Oxford for a day-long seminar during which the possible place of theories of situated cognition in mathematics education was thoroughly aired and explored. A follow-up seminar took place during 1998. This meeting gave rise to a collection of papers (Watson, 1998). Sadly, at the time, no commercial publisher could be found who would publish the collection as an authentic record of the work of the seminar, or within a reasonable time. During the intervening years, some of the papers (e.g. those by Adler, Lerman, Winbourne and Watson) became influential beyond what might be expected from a small print-run. In addition, concepts associated with a situated perspective are now taken-as-shared in mathematics education research. It is time to review the subfield by drawing together a collection of up-to-date work which could be said to have been influenced at some stage by ‘situated cognition'. Many of the authors in this volume participated in the original Oxford seminars and contributed to the collection of papers. In all cases their thinking has moved on and this new collection represents mature, critical, organic perspectives on aspects of mathematics education, framed by political, social and mathematical concerns. In March 2006 most of the authors met at a video-conference to discuss key theoretical issues, and this was followed up with lively electronic discussion. The chapters of this book, while clearly the work of the individual authors or authoring teams, have been peer-reviewed within the team, many of whom communicated with each other throughout the final stages of writing. [ABSTRACT FROM AUTHOR]

Three authors bring a range of different perspectives to bear on the experiences of one of them. The chapter explores the complexity of hearing voice and tackles the methodological difficulties of making and understanding personal accounts: of using mathematics in non-formal settings, of learning mathematics in formal educational settings, of planning for and understanding the learning of others. It presents multiple voices and a shared multivoice analysis of Sandra's changing mathematical identity over time and place. We found that - no matter where we looked - none of us could properly account for Sandra's persistence with mathematics education, despite a history peppered with stories of the sort that might reasonably make people despair of mathematics education. Key words: life history, voice, representation, adult education, mathematics, discourse, narrative methodology [ABSTRACT FROM AUTHOR]

between everyday and school mathematical practices. The analysis focuses on differences in the practices between the settings of workplace and school in particular. Finally issues to emerge from this study are discussed in relation to the wider context of policy and practice. These include issues of relevance, questions of purpose, learner confidence and approaches to assessment in mathematics. Key words: social practice, learning mathematically, workplace learning, vocational education, relevance of mathematics, purpose in mathematics, learner confidence, assessment [ABSTRACT FROM AUTHOR]

This chapter reports on a small-scale case study involving 15-16 year old secondary school students participating in a vocational module under the General National Vocational Qualification (GNVQ) scheme that operated in England during the late 1990s. The development was a pilot study involving experience in the workplace in a small-scale light engineering context. An initial aim of the study was to explore the potential of the setting for the development of numeracy practices. The theoretical framework adopted is based on a social perspective on learning and a view of learning mathematically as social practice. Of particular interest were the differences between everyday and school mathematical practices. The analysis focuses on differences in the practices between the settings of workplace and school in particular. Finally issues to emerge from this study are discussed in relation to the wider context of policy and practice. These include issues of relevance, questions of purpose, learner confidence and approaches to assessment in mathematics. Key words: social practice, learning mathematically, workplace learning, vocational education, relevance of mathematics, purpose in mathematics, learner confidence, assessment [ABSTRACT FROM AUTHOR]

We seek to understand the process of learning mathematical notions as forms of practice in the classrooms of two different technical school courses. More specifically, we investigate students' and teachers' experiences and use of the mathematical concept of tangent line in these different contexts. We use empirical data collected through non-participant observation, analysis of studentś written responses to a questionnaire, and semi-structured interviews with groups of six students from each course observed. We take a situated perspective of learning which enables us to see these classroom activities as genuinely mathematical, though distinct. Through our analysis, we describe aspects of what we see as the common direction of learning mathematics in the two vocational course lessons; this is found to be closer to ‘being mathematical' in work settings than in school mathematics classrooms. Key words: situated learning, local communities of practices, vocational school classrooms [ABSTRACT FROM AUTHOR]

This chapter examines Mechanical Engineering and Mathematics undergraduates' understanding of the derivative and addresses institutional issues in the social formation of knowledge. It summarises results from a study and examines student and lecturer data. Significant differences over the course of the first year are noted. It is claimed that these differences arise from their participation in different departments (institutions). The closing section examines students' developing conceptions of the derivative in institutional settings by addressing the question: what brought about these changes in students' conceptual development? Key words: derivative, cognition, institutional setting, departmental affiliation, engineering and mathematics students [ABSTRACT FROM AUTHOR]

This chapter explores connections between theories of tacit knowing and theories of situated learning and communities of practice aiming at a better understanding of school mathematics as a socio-cultural practice. By contrasting both school and other socio-cultural mathematical contexts, we discuss the usefulness of a perspective from which the identification and circulation of tacit knowing within school mathematics practice is a central concern. Empirical data are presented to illustrate our ideas. Key words: explicit-tacit knowing, competencies-in-activity, situated learning, school and out-of-school mathematical practices, mathematics education [ABSTRACT FROM AUTHOR]

This chapter aims to explore and discuss the notion of learning as participation with artefacts in social practices. It uses insights and evidence from the empirical field of ardinas' practice in Cape Verde to support a discussion that combines a situated learning approach with elements of activity theory. Two artefacts are analysed - the calculator and the record table used by the ardinas. We claim that the regulation of participation made possible by these artefacts does not come from the artefacts themselves but from the way they become present in the everyday and the functions they have in the practice. The artefacts do not represent something fixed and external to the practice; their usefulness is not revealed in the characteristics identified outside of its use in the practice. Artefacts are artefacts in the practice, though they have to be read in the interaction with the forms of use that practitioners put into action. Our final discussion goes into two key concepts in situated learning that we connect with the notion of artefact and resource: technology of practice and shared repertoire. The two concepts are complementary: giving visibility to particular aspects: firstly, to the process of construction; secondly, to the history. In both the key idea of participation is present, and it is through participation that one contributes to construction and has access to history. Key words: mathematics learning, communities of practice, activity theory, participation, artefact, shared repertoire, technology of practice [ABSTRACT FROM AUTHOR]

We report here an instructional method designed to address the cognitive gaps in children's mathematical development where operational conceptions give rise to structural conceptions, such as when the subtraction process leads to the negative number concept. The method involves the linking of process and object conceptions through semiotic activity with models which first record intuitive processes on objects in situations outside school mathematics - invoking situated intuition - and subsequently mediate new mathematical activity, with mathematical signs, in the mathematical voice. We ground this in teaching experiments focused on (i) the negative integers and (ii) algorithms for two-digit subtraction. We conceptualise modelling as the transformation of outside-school knowledge into school mathematics, and discuss the opportunities and difficulties involved. Key words: modelling (RME), situated intuition, primary pedagogy [ABSTRACT FROM AUTHOR]

This chapter draws on Wenger's (1998) account of communities of practice to provide insights into the relationship between home and school mathematics practices and identities. The chapter presents and analyses an interaction between a 9-year-old boy and his mother as she attempts to help him with a mathematics homework task, consisting of a sheet of two-digit subtraction problems. The analysis reveals considerable tension and conflict at the boundary between home and school practices, as the different identities of mother and child negotiate with and challenge each other. These conflicts are exemplified by arguments about the appropriate methods for carrying out the subtractions, in which both participants justify their positions in terms of power and legitimacy instead of the underlying mathematical principles. One implication is that schools need to reconceptualise their approach to homework and parents' role in supporting homework if such interactions are to be more supportive of children's mathematics learning. Key words: communities of practice, boundaries, identities, mathematics homework [ABSTRACT FROM AUTHOR]

In this chapter we address the question: are mathematical abstractions situated? We first consider empiricist accounts of abstraction which see abstraction as a development process from the concrete to the abstract achieved through the recognition of commonalties isolated in a large number of instances. We discuss difficulties involved in empiricist accounts and propose an alternative approach which we call a dialectical account of abstraction. In this approach, an undeveloped initial idea develops through the use of mediational means and social interaction. This development is not from the concrete to the abstract but, rather, a dialectical to and fro between the concrete and the abstract. Unlike empiricist views, our approach regards context, in the formation of mathematical abstractions, as paramount. Although the construct ‘context' is difficult to delineate precisely, we focus on the importance of students' personal mathematical histories, the tools and knowledge artefacts they work with, the people they work with and the tasks they work on. We exemplify the importance of these contextual factors through a study where two teenage girls worked collaboratively, with an interviewer assisting them, in completing tasks designed to generate abstractions in the field of graphs of linear absolute value functions. Key words: abstraction, absolute value, context, dialectics, social interaction [ABSTRACT FROM AUTHOR]

In this chapter I explore the implications for teaching of applying theories of situated cognition to the teaching and learning of mathematics in school. I use accounts of some learners' experiences of mathematics in schools to suggest that teaching, as planning for learning, might more usefully be conceptualised in terms of planning for the development of powerful, identity-changing practices than in terms of the achievement of a range of pre-specified mathematical objectives. Key words: alignment, becoming, identity, community of practice, predisposition [ABSTRACT FROM AUTHOR]

Case-studies conducted as part of a longitudinal study of mathematics teaching and learning in primary (elementary) classrooms were originally intended to shed light on children's mathematical understandings. It became clear, however, that mathematical understanding and attainment were inseparable from the social identities as learners of mathematics that children were able to adopt. Such identities are not ‘givens' but are situated and made possible through the affordances and constraints the classroom cultures. In the first half of the chapter I explore the emergence of social identities as learners of mathematics and in the second half examine how shifting our attention from identities to relations has implications for how classrooms might be organized to allow more children access to a social identity as a successful mathematician. Key words: primary (elementary) mathematics, social identities, situated learning relations, classroom culture [ABSTRACT FROM AUTHOR]

This chapter looks at intentional teaching in detail, drawing out significant distinctions in whole-class interaction sequences which may, at first glance, look similar. Such episodes are sometimes analysed only according to the amount of participation, or the patterns of participation, rather than the mathematical qualities of participation. We find the notions of affordance, constraint and attunement helpful in looking at classroom interaction in terms of how mathematical activity is structured in such interactive sequences. These ideas allow differences in mathematical learning to be understood within a situated perspective by asking ‘what are the specific mathematical practices engendered in this lesson?' As well as offering a powerful frame for ‘getting inside' interactive sequences, this approach gives insight into how learners' mathematical identity might develop in subtly different contexts. Key words: mathematics teaching, mathematical activity, mathematical practices, whole class teaching, classroom interaction, affordances, constraints, attunements [ABSTRACT FROM AUTHOR]

This chapter points out that one of the purposes of having mathematics as a school subject is that it can contribute directly to learners' development of higher psychological functions, and hence to the development of their identity as mature people. It draws attention to the dangers of too narrow an interpretation of situated learning, and makes the case for mathematics in the school context being seen as having a deeper psychological effect than that of acquiring mathematical instruments to solve problems close to life. Rather, activity theory, with its different levels of operations, tasks and complex activities, is shown to enable mathematics in school to be seen as potentially contributing to the development of thinking, motivation and identity. Key words: epistemology in mathematics education, situated learning, activity theory, ‘intellectualisation', cognitive development [ABSTRACT FROM AUTHOR]