Our overall objective in this paper is to share a few observations made and insights gained while conducting a recently completed teaching experiment. The experiment had a strong pragmatic emphasis in that we were responsible for the mathematics instruction of a second grade class (7 year-olds) for the entire school year. Thus, we had to accommodate a variety of institutionalized constraints. As an example, we agreed to address all of the school corporation’s objectives for second grade mathematics instruction. In addition, we were well aware that the school corporation administrators evaluated the project primarily in terms of mean gains on standardized achievement tests. Further, we had to be sensitive to parents’ concerns, particularly as their children’s participation in the project was entirely voluntary. Not surprising, these constraints profoundly influenced the ways in which we attempted to translate constructivism as a theory of knowing into practice. We were fortunate in that the classroom teacher, who had taught second grade mathematics “straight by the book” for the previous sixteen years, was a member of the project staff. Her practical wisdom and insights proved to be invaluable.
Excerpt: Following the seminal influence of Jean Piaget, constructivism is emerging as perhaps the major research paradigm in mathematics education. This is particularly the case for psychological research in mathematics education, However, rather than solving all of the problems for our field, this raises a number of new ones. Elsewhere 1 have explored the differences between the constructivism of Piaget and that of von Glasersfeld (Ernest, 1991b) and have suggested how social constructivism can be developed, and how it differs in its assumptions from radical constructivism (Ernest, 1990, 1991a). Here I wish to begin to consider further questions, including the following: What is constructivism, and what different varieties are there? In addition to the explicit principles on which its varieties are based, what underlying metaphors and epistemologies do they assume? What are the strengths and weaknesses of the different varieties? What do they offer as tools for researching the teaching and learning of mathematics? In particular, what does radical constructivism offer that is unique? And last but not least: What are the implications for the teaching of mathematics?
This paper traces the development of constructivism as a theory of epistemology and learning, and identies ten key principles of this “grand theory.” It identies the need to further develop bridging theories that more closely link to empirical evidence. Within these bridging theories, it identies primary themes: grounding in action, activity and tools, alternative perspectives, student reasoning patterns and developmental sequences, student-invented representations, socioconstructivist norms, etc., that are useful in linking theory and practice. Finally, it discusses how these ideas have been evolving into a view of modelling as an orientation to mathematics and science instruction, and identies this approach as a successor to constructivist theories.
In the context of theories of knowledge, the name “radical constructivism” refers to an orientation that breaks with the Western epistemological tradition. It is an unconventional way of looking and therefore requires conceptual change. In particular, radical constructivism requires the change of several deeply rooted notions, such as knowledge, truth, representation, and reality. Because the dismantling of traditional ideas is never popular, proponents of radical constructivism are sometimes considered to be dangerous heretics. Some of the critics persist in disregarding conceptual differences that have been explicitly stated and point to contradictions that arise from their attempt to assimilate the constructivist view to traditional epistemological assumptions. This is analogous to interpreting a quantum-theoretical physics text with the concepts of a 19th-century corpuscular theory. It may be useful, therefore, to reiterate some points of our “post-epistemological” approach,1 so that our discussion might have a better chance to start without misinterpretations.
Excerpt: Why, despite the many sources of difficulty, is successful communication of mathematics among individuals possible? Why is it that to apply mathematical ideas, one must inevitably choose one or more notations in which to materialize those ideas? And why the large variation in the ways that these notations support and/or constrain our thinking processes? In this chapter I will not presume to answer such questions, but rather will attempt to provide some means for others, hopefully including the reader, to gain insight into them. Central to this task will be to describe the twin mediating roles of notation systems, in mediating between what is normally regarded as “pure mathematics” and one’s experienced world, and in mediating communication processes among individuals. In so doing, I hope also to show how a representational framework for mathematical cognition and learning is consistent with constructivism.
A theoretical model is proposed that explicates the generation of conceptual structures from unitary sensory objects to abstract constructs that satisfy the criteria generally stipulated for concepts of “number”: independence from sensory properties, unity of composites consisting of units, and potential numerosity. The model is based on the assumption that attention operates not as a steady state but as a pulselike phenomenon that can, but need not, be focused on sensory signals in the central nervous system. Such a view of attention is compatible with recent findings in the neurophysiology of perception and provides, in conjunction with Piaget’s postulate of empirical and reflective abstraction, a novel approach to the analysis of concepts that seem indispensable for the development of numerical operations.