Type of Document Dissertation Author Eraslan, Ali Author's Email Address email@example.com URN etd-11102005-162034 Title A Qualitative Study: Algebra Honor Students' Cognitive Obstacles as They Explore Concepts of Quadratic Functions Degree Doctor of Philosophy Department Middle and Secondary Education, Department of Advisory Committee
Advisor Name Title Leslie Aspinwall Committee Chair Elizabeth Jakubowski Committee Member Emanuel I. Shargel Committee Member Maria L. Fernandez Committee Member Keywords
- Algebra honor students
- Cognitive obstacles
- Secondary mathematics education
- Quadratic functions
Date of Defense 2005-10-24 Availability unrestricted AbstractWith the paradigm shift from a behavioral to a constructivist perspective in teaching and learning of mathematics, students’ thought processes have become a major focus for learning and students’ learning of the specific subject matter has been analyzed and approached more qualitatively. In parallel to this development, the present study attempted to describe two algebra-honor students’ cognitive obstacles in the learning of quadratic functions. In particular, along with students’ concept image and definition for the quadratic function (Tall & Vinner, 1981), five other aspects of quadratic functions were examined to identify students’ cognitive obstacles surrounding quadratic functions. These five aspects, adapted by Wilson (1994) who identified the most important aspects of the function concept for deep understanding, were as follows: translating, determining, interpreting, solving quadratic equations, and using quadratic models.
A multiple case study involving two algebra honor students was designed and implemented. Two honor students under the pseudonyms of Richard and Colin were purposely chosen and voluntarily participated in this study. Data were obtained from one-on-one clinical interviews, students’ written work (a test, quiz, and questionnaire), and classroom observations. The analysis particularly focused on identifying students’ cognitive processes as they worked on quadratic tasks during the interviews. The whole data were analyzed through the lens of an integrated framework using Schoenfeld’s (1989) level of mathematical analysis and structure and Tall and Vinner’s (1981) framework of concept image and concept definition.
The study revealed the cognitive obstacles that Richard and Colin encountered
during the study of quadratic functions. In light of these obstacles, the following four
assertions were made in this study: (1) one of the obstacles arises from a lack of making
and investigating mathematical connections between algebraic and graphical aspects of the concepts, (2) another cognitive obstacle arises from the need to make an unfamiliar idea more familiar, (3) a third cognitive obstacle arises from the disequilibrium between
algebraic and graphical thinking, and (4) the image of the quadratic formula or absolute value function has a potential to create an obstacle to mathematical learning.
This study has important applications for classroom teaching. By identifying the students’ cognitive obstacles based on the six aspects of quadratic functions, the study indicates which obstacles are associated with certain aspects of quadratic functions.
Moreover, in light of these obstacles, it emphasizes the interrelation and complementary aspect of algebraic and graphical thinking in an ongoing back-and-forth process in learning and teaching of quadratic functions.
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