Date of Award


Degree Name

Doctor of Philosophy



First Advisor

Dr. Christian R. Hirsch


This study focuses on the fostering of reflective thinking in students in a reform calculus course through completion of homework assignments incorporating reflective tasks, and the effect of these assignments on student understandings of calculus and conceptions of mathematics. The study, conducted in Fall 1997, involved two sections of first-semester calculus at a large midwestern university and used quantitative (n = 25, 18) and qualitative (n = 7) techniques.

Homework assignments incorporating reflective tasks included asking students to compare and contrast textbook ideas; to write about how obstacles were overcome as they attempted exercises; to develop concept maps organizing and relating course material; and to explain, in writing, strategies regarding specified tasks. Analysis of covariance with pretest achievement scores as covariate was used to analyze student performance on four examinations by section. Student responses at the beginning and end of the semester to an inventory of mathematical conceptions were analyzed by section using a two-sample t-test. Audiotaped "think aloud" problem sessions were conducted with selected treatment section students and analyzed by category of thought using time-line graphs, which provided detail on reflective thinking used during problem solving unavailable from in-class examinations.

No significant differences in adjusted means were determined on the four examinations. Inspection of regression lines of examination scores and intersection points revealed an interaction between treatment and precalculus achievement. Students scoring at the 12th percentile had better achievement on Exam 1 than control students. Students had better achievement than the control students at the 28th percentile for Exam 2, at the 32nd percentile for Exam 3, and at the 44th percentile for the Final examination. As the semester progressed, an increasing number of students appeared to benefit from the treatment.

The "think aloud" problem sessions supported this benefit of treatment. By the end of the semester, students exhibited categories of reflective thinking, such as Direction of Thinking, which were virtually absent at the beginning of the semester and exhibited more repetition and variety in their categories of thought.

Access Setting

Dissertation-Open Access