This paper reports the results of research investigating what high-performing college algebra students know about major components of the function concept. Quantitative data was gathered by administering a written exam to thirty students at the completion of the course. Follow-up interviews were conducted on five select inidviduals. Analysis of quantitative and qualitative data provided insights concerning the mathematical knowlege and mathematical habits motivating specific student responses.
Successful college algebra students possess a narrow view of functions, believing that all functions are definable by a single algebraic formula and all functions must be continuous. They do not understand the language of functions and are unable to represent "real world" relationships using function notation. In addition to their conceptual misunderstandings, the interview transcripts reveal that high-performing college algebra students possess very little persistence in pursuing solutions to problems. They do not appear to trust the mathematics that they know when solving problems, nor do they engage in activities which demonstrate an expectation of "sense making" when constructing their responses. High performing college algebra students demonstrate difficulties with much of the information which was explicitly taught during their course.
Marilyn P. Carlson, Ph.D.
Director of First Year Mathematics
Arizona State University