Cognitive Incoherence of Students Regarding the Establishment of Universality of Propositions through Experimentation/Measurement

Abstract

The purpose of this research is to construct a conceptual framework for use in capturing students’ unstable perception regarding the establishment of the universality of propositions through experimentation/measurement in school geometry. As a conceptual framework, this research uses a quadrangular–pyramid model comprised of five representative aspects: idealism/teleology, pessimism, optimism, actualism, and naïve pre-established harmony. In constructing the framework, I decided on the following viewpoints: (1) the students take as a criterion of the universality of propositions whether the proposition and the result of experimentation/measurement match or not; (2) the students assume that the result will always match the proposition; and (3) the students consider improving the method of experimentation/measurement based on whether the result matches or not. To illustrate the phenomenon of students’ unstable perception, I focus on the intermediate aspects between actualism and naïve pre-established harmony, in which the cognitive incoherence of students is conspicuous in that it is easy-to-manifest. I discovered two types of unstable perception through analyzing two 8th graders’ responses to questionnaires: (1) one student understood asymptotic accessibility to the match by improving the measurement, although the student was skeptical about reaching the final stage; and (2) another student understood the necessity of the match in order to guarantee her geometrical activity, although she was confused when confronted with the result in that it did not match, even when measured with high-precision equipment.