Book•
Teaching Mathematics: Toward a Sound Alternative
01 Mar 1996-
TL;DR: In this article, the authors present an approach to the teaching of mathematics that departs radically from conventional prescription-oriented and management-based methods, focusing on commonsense distinctions drawn between thought and action, subject and object, individual and collective, fact and fiction, teacher and student, and classroom tasks.
Abstract: This book presents an approach to the teaching of mathematics that departs radically from conventional prescription-oriented and management-based methods. It brings together recent developments in such diverse fields as continental and pragmatist philosophy, enactivist thought, critical discourses, cognitive theory, evolution, ecology, and mathematics, and challenges the assumptions that permeate much of mathematics teaching. The discussion focuses on the language used to frame the role of the teacher and is developed around the commonsense distinctions drawn between thought and action, subject and object, individual and collective, fact and fiction, teacher and student, and classroom tasks and real life. The discussion also addresses the question of how mathematics teaching can be reformed to better suit current academic and social climates. Making use of the theoretical framework of enactivism, the book explores the subject through an account of a middle school teacher's appreciation and understanding of her role. Teaching mathematics, as both the report of this teacher's experience and the discussion make clear, demands an embracing of ambiguity, uncertainty, complexity, and moral responsibility. Courses for Adoption Education: Mathematics for Elementary Teachers, Methods for Teaching Elementary Schools, Methods for Teaching Secondary Schools, Curriculum Studies, Critical Pedagogy Special Features *Elucidates the importance and relationship between theory and practice. Employs reflective teaching techniques to focus students on their own learning, knowledge, and understanding of mathematics.Details a collaborative venture that traces the development of new thinking and insights about math teaching and learning. *A fine blending of theory with practice.
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TL;DR: In this paper, a methodology for studying mathematics teacher development in the context of reform is presented, which is an alternative both to studies that focus on teachers' deficits and to teachers' own accounts of their practice.
Abstract: In this article we articulate a methodology for studying mathematics teacher development in the context of reform. The generation of accounts of teachers'practice, an adaptation of the case study, provides an approach to understanding teachers' current practice and to viewing their current practice in the context of development toward envisioned reforms. The methodology is an alternative both to studies that focus on teachers' deficits and to teachers' own accounts of their practice. Conceptual frameworks developed within the mathematics education research community are applied to the task of investigating the nature of practice developed by teachers in transition. We characterize this methodology as explicating the teacher's perspective from the researchers' perspectives.
148 citations
Cites background from "Teaching Mathematics: Toward a Soun..."
...We understand the conceptual framework to be an evolving set of concepts and values comprised of both formulated, or "stateable" (Davis, 1996), and unformulated elements....
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TL;DR: In this article, a mathematics letter exchange with grade 4 students provided an opportunity for prospective teachers to learn about students' mathematical thinking and examine their interpretive practices, revealing changes in the patterns of their interpretations.
Abstract: Listening to students' mathematical thinking is one of the trademarks of reform-minded visions of mathematics teaching. The questions of when, where, how, and what might help prospective teachers learn to do so, however, remain open. This study examines how a mathematics letter exchange with Grade 4 students provided an occasion for prospective teachers to learn about students' mathematical thinking and to examine their interpretive practices. Analysis of the interactions between students and prospective teachers, and of the reflective writing of the latter, revealed changes in the patterns of their interpretations. I characterized these as changes in the focus of interpretation, from correctness to meaning, and in the interpretive approach, from quick and conclusive to thoughtful and tentative. I also discuss factors associated with these interpretive turns. The idea of teachers listening to and understanding students' thinking has been widely promoted and supported in the education community. In the Professional Standards (National Council of Teachers of Mathematics (NCTM), 1991), the analysis of students' thinking is highlighted as one of the central tasks of mathematics teaching. In this report, the analysis of students' thinking is seen as a resource that can help teachers make informed decisions in their classrooms and improve their practice. Such a listening orientation towards teaching promotes a learning environment conducive to and respectful of students' own sense making and intellectual autonomy (Davis, 1996; Kamii, 1989). In contrast, when teachers do not listen to or do not understand their students' thinking they tend to dismiss it by imposing their own formalized constructions onto the students (Cobb, 1988; Maher & Davis, 1990). Although there are various ways in which teachers can listen to their students' mathematical ideas, Davis (1996) reminded us that not all forms of listening are conducive and respectful of students' thinking. He discussed three different orientations teachers might have towards listening in the mathematics classroom. Teachers with an evaluative orientation, according to Davis, tend to listen to students' ideas in order to diagnose and correct their mathematical misunderstandings. Teachers with an inter- pretive orientation, on the other hand, listen to students' ideas with the
145 citations
Cites background from "Teaching Mathematics: Toward a Soun..."
...Such a listening orientation towards teaching promotes a learning environment conducive to and respectful of students’ own sense making and intellectual autonomy (Davis, 1996; Kamii, 1989)....
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...Although there are various ways in which teachers can listen to their students’ mathematical ideas, Davis (1996) reminded us that not all forms of listening are conducive and respectful of students’ thinking....
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...Furthermore, the study builds upon Davis (1996) conceptual categories of teachers’ orientations to listening in mathematics classrooms by using and elaborating the categories of evaluative and interpretive listening as useful ways to characterize and analyze preservice teachers’ interpretations....
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...We need a more concerted strategy in order to help prospective teachers begin to seriously consider Davis’ (1996) vision of “teaching as listening,” where teachers see an open-ended inquiry approach towards their students’ mathematical work as a viable and desirable teaching practice....
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TL;DR: Meanings of meaning of mathematics are discussed in this article, where the concept of function as an example is used to illustrate the meaning of meaning in mathematics education and common sense.
Abstract: Meanings of Meaning of Mathematics.- "Meaning" and School Mathematics.- The Meaning of Conics: Historical and Didactical Dimensions.- Reconstruction of Meaning as a Didactical Task: The Concept of Function as an Example.- Meaning in Mathematics Education.- Collective Meaning and Common Sense.- Mathematics Education and Common Sense.- Communication and Construction of Meaning.- Making Mathematics and Sharing Mathematics: Two Paths to Co-Constructing Meaning?.- The Hidden Role of Diagrams in Students' Construction of Meaning in Geometry.- What's a Best Fit? Construction of Meaning in a Linear Algebra Session.- Discoursing Mathematics Away.- Meaning and Mathematics.
110 citations
TL;DR: This paper argued that current characterizations of and distinctions among research methodologies in education are potentially counterproductive, in large part because they tend to be defined against or in terms of principles and methods that have been rendered problematic within the sciences.
Abstract: This article represents an attempt to reconcile discussions of aspects of educational research with recent developments in complexity science. It is argued that current characterizations of and distinctions among research methodologies in education are potentially counterproductive, in large part because they tend to be defined against or in terms of principles and methods that have been rendered problematic within the sciences. To develop this point, the authors draw on several contemporary discourses: poststructuralist methods are used to foreground the Euclidean (plane) geometric roots of much of the vocabulary of educational research; fractal geometry is taken as a source of images and analogies to support alternative conceptions of knowledge, learning and teaching; informed by poststructuralist and fractal geometric notions, the authors turn to complexity science and argue that it is fitted to and offers important elaborations of current discussions of educational research methodologies. In the proce...
109 citations
Additional excerpts
...(See Davis, 1996, for a fuller discussion.)...
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