The effects of a framework for procedural understanding on college algebra students' procedural skill and understanding

Abstract

This dissertation examined the effectiveness of an instructional treatment consisting of lecture content, homework tasks, and quiz assessments built around a common Framework for Procedural Understanding. The study addressed concerns about increasing numbers of students enrolling in remedial mathematics courses because they did not develop sufficient understanding in previous courses. The Framework-oriented instruction was designed to help students develop deep and well-connected knowledge of procedures, which has been shown to facilitate recall and promote future learning. Data collection spanned the Fall 2005 semester at a western land-grant university. In the quasi-experimental design, instructors from six intact sections of college algebra were matched into pairs based on prior teaching experience, and the treatment condition was assigned to one member of each pair. Equivalence of treatment and control groups was established by comparing ACT / SAT scores for the 85% of students for whom those scores were available. Data collection consisted of classroom observations, homework samples, common hour exams scores, procedural understanding assessments, supplemental course evaluations, and a final interview with treatment instructors. Analysis of covariance was the primary statistical tool used to compare treatment and control group performances while controlling for attendance rates and pre-requisite mathematical knowledge. Treatment group students scored significantly higher than control group students on the procedural understanding assessments. Moreover, although treatment students were assigned 18% fewer drill questions than controls and 8% fewer problems overall, the gains in procedural understanding were realized without declines in procedural skill. The relationship between understanding and skill was also examined, and students with greater procedural understanding tended to score higher on the skills-oriented final exam regardless of which treatment condition was assigned to them. Finally, the interview with the treatment instructors provided insight into the implementation issues surrounding the treatment. They expressed concerns about time constraints and reported initial discomfort with, but eventual appreciation for, using the Framework for Procedural Understanding to guide instruction. The Framework-oriented treatment was found to be effective at helping students develop deeper procedural understanding without declines in procedural skill. Additional implications and recommendations for future research are also discussed.