## Volume 37 (December 2005) Number 6## ZDM## Zentralblatt für Didaktik der Mathematik## International Reviews on Mathematical Education
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.
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:
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.
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.
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
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). |