Robert L. Gray
Instructional strategies in physics have over the past decade been influenced by the developmental theory of Jean Piaget. In general, curriculum revision and design have been directed toward the incorporation of concrete materials in laboratory settings requiring active involvement on the part of students. Given the nature of Piaget's research methodology, it is no surprise to find exploratory manipulation the cornerstone of these newer instructional strategies. From kindergarten through freshman and sophomore college classes one finds example after example of action-oriented instruction ( Arons, 1973) in which concreteness is rigorously pursued. The theoretical rationale for this emphasis upon concrete objects derives from the unique and crucial role played by the object prior to the stage of formal operations ( Piaget, 1969). The interplay between sense data derived from objects and overt actions performed on them forms the intellect in the Piagetian model.
The educational community ought now to seek a theoretical base by which to guide the growth of formal operations. That part of an overall theory which connects the manipulation of real objects to the formation of concrete operations has fulfilled its educational promise. There seems, however, to be little evidence supporting the notion that formal operations grow solely out of the debris of paper, glue, test tubes, inclined planes, litmus paper or any other laboratory apparatus. The metaphor of a proportionality algorithm (a favorite Piagetian task indicative of formal operations) succinctly states the issue urgently requiring attention.
"Objects are to concrete operations as __________ is/are to formal operations."
The metaphor needs somehow to be completed. Piaget might insert "the performance of operations upon operations." ( Nodine, 1971) But what does that mean in an instructional setting?
The experiences of the author over the past few years in physics courses for both science and nonscience majors suggest the need for an instructionally useful, articulated distinction between reasoning at the concrete stage and reasoning at the formal stage. The notion of abstraction is in this paper the distinguishing feature, and the relationship between abstraction and concreteness is considered in the context of instructionally assisted promotion from concrete to formal operations.