Understanding mathematical concepts : the case of the logarithmic function

by (Signe Elizabeth), 1963- Kastberg

The purpose of this study was to describe students’ understanding of the logarithmic function, changes in their understanding, and ways of knowing they use to investigate problems involving the logarithmic function. Understanding is defined as a student’s beliefs about a mathematical concept. Four categories of evidence (conception, representation, connection, and application) were used to make conjectures about the students’ beliefs over three instructional phases (preinstruction, instruction, and postinstruction). Nine interviews were conducted over a two-month period with students (3 female, 1 male) from two college algebra classes at a rural southeastern two-year college. Case studies were developed based on evidence gathered from phenomenological interviews, clinical interviews, participant observation, student constructed maps, and drawings. The students’ understanding contained a central theme: the logarithmic function as a collection of problems to do. Four categories of beliefs (level of difficulty, problem types, tools, character of the function) associated with the theme were identified in all three phases of the study. The static nature of the categories suggests students’ understanding of a mathematical concept is influenced by their beliefs about mathematics and understanding. The changes in the content of the beliefs were the result of instruction and the reconstructive nature of memory. A modified theory of understanding using beliefs about mathematics and understanding and four categories of evidence is suggested for further research. Four ways of knowing (number patterns, successive approximation, More A – More B, and responses to inconsistencies) were used by the students to investigate problems involving the logarithmic function. These ways of knowing are suggested as a starting point for the teaching of logarithmic functions.