The tension and the balance between one mathematical concept and students' constructions of it: The case of solution to a differential equation.

摘要

One of the developing areas of research on undergraduate mathematics education is student understanding of concepts related to differential equations (DE). A concept that exemplifies the basic purpose of a course in elementary differential equations is that of a solution to a DE. Our main intent is to capture students' thought processes following the introduction of the concept of solution to a first order DE and study how the initial concept image is reconstructed as they face higher order DEs, systems of DEs and the numerous methods of finding solutions to DEs. A second focus is to scan student responses to find emerging patterns of processes of construction.; Toward that end we present and analyze the data collected from five in-depth semi-structured clinical interviews of six undergraduate students of an introductory course in elementary differential equations. The supplemental data is provided by classroom observations and notes, student quizzes and exams, and a questionnaire completed by the students after the completion of the course.; Our first theoretical contribution is the refinement of the current framework of concept image and definition into a nested hierarchy of entity-object-concept. This refinement served well to explore particular aspects of solution, and can be adapted to explain aspects of other concepts that can be separated into two tiers of entity and object; such as functions or differential equations.; Our second theoretical contribution is the adaptation, refinement and the patterns of construction that emerged from the data collected. The present tri-level framework was augmented by a fourth level, and each existing level was further refined to adequately incorporate additional tendencies that our students exhibited. These tendencies are neither unique to these students, nor to the concept of solution to a DE, but to the stages of accommodation and assimilation that follow the introduction of any new concept. We are confident that this extended learning theory framework can serve well as a tool to explain cognition processes.; In addition, we offer a categorization of the students as learners into three levels of sophistication (learners, potential learners and pseudo-learners) as determined by their approaches towards learning as a connected activity. We show snapshots of their different approaches leading to either assimilation or compartmentalization of new knowledge.; In this thesis we have (1) presented and analyzed the evolution of students' concept images of (a) solution as an entity, (b) the relationship of solution with the DE, (c) uniqueness of solution to an initial value problem and the role of the Existence and Uniqueness theorem; (2) compared and contrasted the expectations and the effects of the definition of solution and the processes used to obtain it on students' concept images of solution; (3) identified one cognitive root and located the possible cognitive obstacles creating gaps between the concept definitions and the concept images; (4) discussed the general tendencies that the students exhibited in the accommodation stages of knowledge construction as well as the tendencies that led to assimilation of new knowledge; (5) suggested probable pedagogical remedies and future directions in research to improve student understanding of solution to a DE.