A dialectical invariant for research in mathematics education

Abstract

IntroductionA central issue in the discipline is the answer to the question: what is mathematics education? Many answers to these questions have traditionally been, and continue to be, advanced. Standard reasons include the need to produce another generation of scholars to continue developing the discipline of mathematics, the supply of a cadre of scientists and others such as engineers who need strong mathematical competence, as training in logical thinking and problem solving, as exposure to what is as much a part of cultural heritage as literature or music. All of these, and more, are valid, but a deeper analysis is required3.Carlos Vasco models mathematics education as an octagon in which mathematics (which he classifies as research, scholars and daily life) is located on the inside and on the outside, forming the sides, are eight disciplines: philosophy, logic, computer science, linguistics, neurology, psychology, anthropology, history of mathematics and epistemology4.We believe that mathematics education integrates dissimilar disciplines as they are represented in Figure 1 which clearly illustrates the complexity on which we base the considerations that follow.Other important reflections are contained in the following sentences: So mathematics education, is fashioned to provide appropriate mathematical knowledge, understanding, and skills to diverse student populations5. In this sense we can say that mathematics education is a special type of teaching, engineering in the sense that it. is the personalization of the basic mathematical principles to meet the needs of teachers and students6, some even claim that it is a branch, of applied mathematics'7.Mathematics education today is a science that at a glance reveals two conflicting features: on the one hand sublimation and on the other a multiplicity of practical problems that must be resolved by the teacher in the classroom. Of course, this two-faced trait manifests itself in research carried out in different regions of the world, hi some cases theoretical centers immersed in general issues are at the center of the research being done, and in others there are institutions dealing with the real problems that teachers face in the classroom.A principal point must be made clear: what is a research in mathematics education?Dialectic theory delimits knowledge formation as an active, complex, ongoing process of organizing and reorganizing conceptual structures rather than an accumulation of fixed truths. Furthermore, in dialectic theory contradiction assumes a central role in the process of change and reorganization that the theory presumes to explain. Whether in the field of cognitive development or in the broader realm of psychology, a dialectical view also assumes that developmental processes are socially and culturally shaped and defined, and that concepts and meanings -whether mathematical or not - evolve in an emergent process of what Vygotsky8 and Leontiev9 called a collective activity system. The latter is understood to operate through the emergence of cognitive conflict within the conceptual system, leading to the ongoing resolution of that conflict in a dialectical manner- which is to say through the recognition and articulation of contiadictions and inconsistencies, and then: mediation in the context of a collective activity10.The teleology of the processes of structural cognitive development, as defined by the major western theorist of cognitive change, Piaget, is understood not only as a continuous movement from no balance towards reequilibration, but as progressively directed movement towards increasing equilibrations, which necessarily require a correspondingly higher organization of cognitive structure. Although it was never affirmed by Piaget, several theorists* 11 understand his psychological theory of cognitive development to be fundamentally dialectical. We can identity'' many parallels between Piaget and the other major theorist of the twentieth century, Vygotsky, at least on the level of the basic, conceptual mechanisms of cognitive development. …