Learning to Listen: A Student’s Understanding of Powers of Ten

ion (Piaget, 1971). These intellectual processes are, for the constructivist, the source and the content of knowledge. Knowledge is not the accumulation of information; it is the construction of cognitive structures that are enabling, generative and proven successful in problem solving. Thus, problem solving becomes an essential intellectual act not an enrichment, an occasional pastime. Figure 2 provides a simple model to describe this process of construction. The objectification of problems, seen in these terms, reveals a human tendency which is itself a part of the problem solving process. We act as though problems (and their solutions) are out there. Michael Polanyi in Personal Knowledge (1958) emphasized that a person (a student or a creative mathematician) needs to believe a problem capable of solution and be “looking for it [the solution] as if it were there, pre-existent” in order to consider it a problem. Thus, inherent in our establishment of a problematic is a belief that it is capable of being solved. We may not possess the confidence or arrogance to presume that we will be the successful ones what we are expressing is our belief that mathematics. If they do not carry “content”, why are they so compelling? Most LEARNING TO LISTEN 119 we will recognize the solution if we were to see it, though we may not be able to articulate anything about it. Our assurance is only in its possibility of solution an assurance which easily leads us to talk as though problems were out there. His characterization of the emotional dimensions of problem solving explain further how our investment in the process encourages us to confer on the problem an independent existence. Our belief that “we can’t create something from nothing” makes us wish to assume the problem solution came from outside. However, with a revised perspective on knowledge, we accept the recursive quality of knowledge construction and the emotional intensity helps to explain why problem solving is so successful in generating knowledge. As Polanyi put it, “obsession with one’s problem is, in fact, the mainspring of all inventive power .... And the intensity of our preoccupation with a problem generates also our power for reorganizing our thoughts successfully . . . (p. 127). Polanyi summed up the power of the problem solving activity in his definition of a problem: “A problem is an intellectual desire ... and like every desire it postulates the existence of something that can satisfy it .... As all desire stimulates the imagination to dwell on the means of satisfying it, and is stirred up in its turn by the play of the imagination it has fostered, so also, by taking interest in a problem we start speculating about its possible solution and in doing so, become further engrossed in the problem.” Asssumption Three: Problems serve a crucial role in the construction of knowledge. Problems reside in the mind of the student not in textbooks or in the mathematics. Problems are felt discrepancies, roadblocks to where a student wishes to be and therefore catalysts for action. To accept a problematic an individual must believe that it is capable of being solved and act as though the problem and solution were preexistent. The cycle of identifying (noticing) problematics, acting and operating on them and then reflecting on the This leads to a third assumpton for the constructivist position: