Volume 37 (December 2005) Number 6


Zentralblatt für Didaktik der Mathematik

International Reviews on Mathematical  Education

Articles • Electronic-Only Publication • ISSN 1615-679X

(Full texts are available to subscribers only)

Theories of mathematics education: A global survey of theoretical frameworks/trends in mathematics education research
Bharath Sriraman, The University of Montana
Lyn D. English ,  Queensland University of Technology

In this article we survey the history of research on theories in mathematics education. We also briefly examine the origins of this line of inquiry, the contribution of Hans-Georg Steiner, the activities of various international topics groups and current discussions of theories in mathematics education research.  We conclude by outlining current positions and questions addressed by mathematics education researchers in the research forum on theories at the 2005 PME meeting in Melbourne, Australia.
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On the theoretical, conceptual, and philosophical foundations for research in mathematics education 
Frank K. Lester, Jr., Indiana University

The current infatuation in the U.S. with “what works” studies seems to leave education researchers with less latitude to conduct studies to advance theoretical and model-building goals and they are expected to adopt philosophical perspectives that often run counter to their own. Three basic questions are addressed in this article:  What is the role of theory in education research?  How does one’s philosophical stance influence the sort of research one does? And, What should be the goals of mathematics education research? Special attention is paid to the importance of having a conceptual framework to guide one’s research and to the value of acknowledging one’s philosophical stance in considering what counts as evidence.
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The fundamental cycle of concept construction underlying various theoretical frameworks
John Pegg, The University of New England
David Tall, University of Warwick

In this paper, the development of mathematical concepts over time is considered. Particular reference is given to the shifting of attention from step-by-step procedures that are performed in time, to symbolism that can be manipulated as mental entities on paper and in the mind. The development is analysed using different theoretical perspectives, including the SOLO model and various theories of concept construction to reveal a fundamental cycle underlying the building of concepts that features widely in different ways of thinking that occurs throughout mathematical learning.
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The articulation of symbol and mediation in mathematics education
Luis Moreno-Armella, CINVESTAV-IPN
Bharath Sriraman, The University of Montana

In this paper we include topics which we consider are relevant building blocks to design a theory of mathematics education. In doing so, we introduce a pre-theory consisting of a set of interdisciplinary ideas which lead to an understanding of what occurs in the “central nervous system” – our metaphor for the classroom and eventually in more global settings. In particular we highlight the crucial role of representations, symbols viewed from an evolutionary perspective and mathematics as symbolic technology in which representations are embedded and executable.
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Trends in the evolution of models & modeling perspectives on mathematical learning and problem solving
Richard Lesh, Indiana University  
Lyn D. English, Queensland University of Technology

In this paper we briefly outline the models and modelling (M&M) perspective of mathematical thinking and learning relevant for the 21st century.Models and modeling (M&M) research often investigates the nature of understandings and abilities that are needed in order for students to be able to use what they have (presumably) learned in the classroom in “real life” situations beyond school.  Nonetheless, M&M perspectives evolved out of research on concept development more than research on problem solving; and, rather than being preoccupied with the kind of word problems emphasized in textbooks and standardized tests, we focus on (simulations of) problem solving “in the wild.” Also, we give special attention to the fact that, in a technology-based age of information, significant changes are occurring in the kinds of “mathematical thinking” that is coming to be needed in the everyday lives of ordinary people in the 21st century – as well as in the lives of productive people in future-oriented fields that are heavy users of mathematics, science, and technology.
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Mathematics education as a design science
Richard Lesh , Indiana University
Bharath Sriraman, The University of Montana

We propose re-conceptualizing the field of mathematics education research as that of a design science akin to engineering and other emerging interdisciplinary fields which involve the interaction of “subjects”, conceptual systems and technology influenced by social constraints and affordances. Numerous examples from the history and philosophy of science and mathematics and ongoing findings of M& M research are drawn to illustrate our notion of mathematics education research as a design science. Our ideas are intended as a framework and do not constitute a “grand” theory (see Lester, 2005, this issue). That is, we provide a framework (i.e., a system of thinking together with accompanying concepts, language, methodologies, tools, and so on) that provides structure to help mathematics education researchers develop both models and theories, which encourage diversity and emphasize Darwinian processes such as: (a) selection (rigorous testing), (b) communication (so that productive ways of thinking spread throughout relevant communities), and (c) accumulation (so that productive ways of thinking are not lost and get integrated into future developments).
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