The Activity of Defining (original) (raw)
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3—349 the Activity of Defining
1993
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 differences in the interlocutors ’ uses of words. As was found in our study, only some of the defining actions would result in texts known as mathematical definitions. The point of departure of our research project is that the effectiveness of the defining actions is as much a function of the action itself as of its contexts. Our focus in this research is thus broader than in the past studies on definitions, and includes the when and why of defining along with their how. Today, it is a common belief that learning with peers in small groups has many advantages over frontal learning, where the teacher is often the only speaker. And yet, such f...
The Codevelopment of Mathematical Concepts and the Practice of Defining
Journal for Research in Mathematics Education
We examined the codevelopment of mathematical concepts and the mathematical practice of defining within a sixth-grade class investigating space and geometry. Drawing upon existing literature, we present a framework for describing forms of participation in defining, what we term aspects of definitional practice. Analysis of classroom interactions during 16 episodes spanning earlier and later phases of instruction illustrate how student participation in aspects of definitional practice influenced their emerging conceptions of the geometry of shape and form and how emerging conceptions of shape and form provided opportunities to develop and elaborate aspects of definitional practice. Several forms of teacher discourse appeared to support students' participation and students' increasing agency over time. These included: (a) requesting that members of the class participate in various aspects of practice, (b) asking questions that serve to expand the mathematical system, (c) model...
Exploring Mathematical Definition Construction Processes
Educational Studies in Mathematics, 2006
The definition of 'definition' cannot be taken for granted. The problem has been treated from various angles in different journals. Among other questions raised on the subject we find: the notions of concept definition and concept image, conceptions of mathematical definitions, redefinitions, and from a more axiomatic point of view, how to construct definitions. This paper will deal with 'definition construction processes' and aims more specifically at proposing a new approach to the study of the formation of mathematical concepts. I shall demonstrate that the study of the defining and concept formation processes demands the setting up of a general theoretical framework. I shall propose such a tool characterizing classical points of view of mathematical definitions as well as analyzing the dialectic involving definition construction and concept formation. In that perspective, a didactical exemplification will also be presented.
Definitions and Concepts in Discourses of Mathematics , Teaching and Learning
2009
However, the only specific aspect of language identified is “vocabulary” – in fact, mathematical language appears to be identified with its vocabulary. The title of the book, its format (mainly consisting of lists of words) and the repeated emphasis on vocabulary, terminology and words (see the extract in Appendix 1 of the introductory article) construct an image of mathematical language as a collection of discrete terms. Although there are suggestions of language activities such as discussing, hypothesising, reading or writing instructions that hint at the complex functions of language in mathematics, these are presented only as “opportunities to develop [children‟s] mathematical vocabulary” (p.3) rather than as development of a more broadly conceptualised mathematical language.
Construction of mathematical definitions: an epistemological and didactical study
Proceedings of the 28th Conference of the …, 2004
The definition-construction process is central to mathematics. The aim of this paper is to propose a few Situations of Definition-Construction (called SDC) and to study them. Our main objectives are to describe the definition-construction process and to design SDC for classroom. A SDC on "discrete straight line" and its mathematical and didactical analysis (with students' productions) will be presented too.
2005
We have chosen to put together this book for two reasons. First, we acknowledge dial an increased focus on communication has influenced and shaped work in a range of perspectives and areas in mathematics education research and practice. Over the past decade and a half, we have witnessed more and more studies that-although not directly concerned with analyses of communication-are considering communication as an integral part of pedagogy and didactics in mathematics classrooms, mathematics education curricula, and broader educational structures. The notion of communication opens up to embrace not only what is happening and what is being said by the participants in a classroom setting, but also conveys the CluilleiigingPerspectives on Mathematics Classwom Communication, pages 3-45 All rights of reproduction in any form reserved. 3 4 A. CHRONAKI and I.M. CHRISTIANSEN
Clarifiable Ambiguity in Classroom Mathematics Discourse
Investigations in mathematics learning, 2019
Ambiguity is a natural part of communication in a mathematics classroom. In this paper, a particular subset of ambiguity is characterized as clarifiable. Clarifiable ambiguity in classroom mathematics discourse is common, frequently goes unaddressed, and unnecessarily hinders in-the-moment communication because it likely could be made more clear in a relatively straightforward way if it were attended to. We argue for deliberate attention to clarifiable ambiguity as a critical aspect of attending to meaning and as a necessary precursor to productive use of student mathematical thinking. We illustrate clarifiable ambiguity that occurs in mathematics classrooms and consider ramifications of not addressing it. We conclude the paper with a discussion about addressing clarifiable ambiguity through seeking focused clarification. KEYWORDS Discourse; ambiguity Researchers who look closely at the complexities of communicating in mathematics classrooms see one particular aspect of communicationsometimes referred to as ambiguity (e.g., Barnett-Clarke & Ramirez, 2004; Barwell, 2003; Foster, 2011)as both inherent (and thus unavoidable) and as providing opportunities for learning. Broadly one might define ambiguity as involving "a single situation or idea that is perceived in two self-consistent but mutually incompatible frames of reference" (Byers, 2007, p. 28). As Barwell (2003) stated, "it is the potential for ambiguity inherent in all language that allows students to investigate what it is possible to do with mathematical language, and so to explore mathematics itself" (p. 5). In fact, as Byers (2007) argued, "the power of ideas resides in their ambiguity. Thus, any project that would eliminate ambiguity from mathematics would destroy mathematics" (p. 24). Whenever students are placed in a sensemaking situation, they are working with ideas they do not fully understand and, as a result, their current vocabulary is insufficient. Thus, ambiguity is a natural part of learning and an essential aspect of mathematics. There are, however, instances of ambiguity where students are capable of clarifying what they said (although not necessarily what they meant). To illustrate such instances of ambiguity, consider the following example from the junior high school mathematics classroom of an award-winning teacher. While studying data about a group of bikers on a multi-day trip, students were examining a graph where distance was measured by the distance from a given city (see Figure 1). In a discussion about the graph in Figure 1, the class interpreted the plotted points at times 1.5 and 2 as an indication that the bikers were stopped on the interval between 1.5 and 2 hours. A student then volunteered, "And then they went up." The student statement, "And then they went up," is ambiguous for a couple of reasons. First of all, it is unclear to what the student is referring by they. In this context they likely refers to either the bikers on the road or the dots on the graph, and these two interpretations have very different
Introduction: Developing mathematical discourse—Some insights from communicational research
International Journal of Educational Research, 2012
Although all the papers in this special issue speak about mathematics and learning, they may appear too diverse in their foci to be bound together in a single volume. At first glance, such issues as the development of algebraic or geometrical thinking, dependence of mathematics on language, the fidelity of curricular implementation, interactions between children trying to learn mathematics in collaborative groups, and the impact of emotions on mathematics learning do not seem to have much in common; all the more so that these topics, when taken together, cover at least three separate research areas, those of cognition, affect, and social interactions. Traditionally, these three types of study differ from each other not just in their themes, but also in their foundational assumptions and methods. Coming from such diverse frameworks, the papers in this volume may appear as targeting several different audiences, with none of these audiences interested in, or even able to access, all seven of them. It is not by accident, however, that the penultimate sentence of the last paragraph has been qualified with the word ''traditionally''. One of the main aims of this special issue is to break out from the grip of the separatist tradition, the tradition of using different, often unbridgeable discourses for dealing with different aspects of learning. The importance of the project of bridging and unifying can hardly be overestimated. If the collective effort of those who study learning-teaching processes is to result in a picture of the proverbial elephant rather than in a collection of possibly misleading partial images, researchers need to build on each other's work; to be able to do so, they have to communicate with one another; and in order to communicate, they need a common discourse, one in which cognitive and affective, as well as intra-personal and interpersonal (or individual and social) aspects of teaching-learning processes would all be seen as members of the same ontological category, to be studied with an integrated system of tools, grounded in a single set of foundational assumptions. A unified discourse of research is what brings the different parts of this special issue together. Between the covers of this volume, the foundational and methodological diversity disappears. Thanks to this discursive unity, the relation between stories told in the different papers is that of continuity and complementarity. When put one next to another, these seven texts combine into a single narrative about mathematics and its development through teaching and learning and, to some extent, also in the course of history. The picture that emerges from this collective tale is more than a sum of the parts. This synergetic effect is made possible by the fact that the seven teams of authors, dissimilar as they are in their interests, are nevertheless united by their ways of talking and their sense of understanding one another. Their common language and the shared set of basic tenets and of research methods make all their papers equally accessible to anybody who is prepared to become a participant in their discourse.
Modes of algebraic communication: moving from spreadsheets to standard notation
2006
Many students seem to find it difficult to learn to express generalisations in standard algebraic notations. One approach to making the acquisition of these skills easier for students is to use the environment of the spreadsheet as a means of semiotic mediation for them in forming meanings for algebraic symbolism.