Cognitive Process Instruction: Research on Teaching Thinking Skills

By Jack Lochhead; John Clement | Go to book overview

Mathematics and Learning: Roots of Epistemological Status
James J. Kaput
INTRODUCTION
This paper is concerned with certain vital aspects of the platonism-constructivism issue and how they are reflected in our everyday work of teaching and learning. It is suggested that we are confronted with a pair of universes: one being the stark, atemporal, formal universe of ideal knowledge; the other being the organic, interior, processual universe of human knowing. The former, Plato's, has monopolized status and power, and is responsible for the fundamental dominance of product over process, the values by which legitimacy of knowledge is conferred, and the rules, particularly linguistic rules, under which any inquiry occurs. This paper illustrates how the manifold consequences of this status dominance permeate our academic lives at all levels, concentrating mainly on its exclusionary function in mathematics and mathematics knowing/learning.The illustrations include discussion of the following phenomena:
The symbol system of mathematics denies the reality or importance of the knower/learner. Our use of mathematical equality systematically denies the process/product distinction, a distinction that is fundamental and real in the universe of human knowing. It also denies the various and distinct heuristic/ linguistic functions of equality.
The processual meaning of mathematical operations is achieved through an essential, yet covert and unacknowledged, act of anthropomorphism, a projection from our internal cognitive experience onto the timeless, abstract- structural mathematical operations. (This includes, for example, operations in arithmetic, algebra and calculus.) In some respects this anthropomorphism acts as a metaphorical "structure preserving mapping," a morphism.
The basic, irreducible and essential metaphoric nature of human thinking has only an accidental, unacknowledged, and denigrated role in mathematics. As it is in any circumstance, the metaphorizing process in mathematics is our primary means for creating and, especially, transferring meaning from one universe to the other. However in mathematics this process is forced into a smuggling and bootlegging role, and never acknowledged for its crucial function. For example, virtually all of basic calculus (the study of change) achieves its primary meaning through an absolutely essential collection of motion metaphors. These metaphors control the notation. Hence we write limit statements using arrows and use image-laden words such as "diverge," "converge," "increasing," "constant," and "transform." However, the formal mathematical definitions associated with these notations, being atemporal, are not connected to motion.

In all three illustrations we see the active human contribution to knowing

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