The major goals of this study were to conceptualize a framework for analyzing student conceptions of mathematics, gather baseline information about the conceptions of mathematics held by mathematically talented students and by average high school students, and begin to generate hypotheses about how they are related to the nature of student learning. The central dimensions used to examine student conceptions of mathematics are presented after a brief review of the related literature. A study comparing mathematically talented high school students' conceptions of mathematics to those of typical high school algebra students is reported and discussed. Three dimensions were used to characterize the nature of mathematical knowledge themes: composition of mathematical knowledge, structure of mathematics knowledge, and status of mathematical knowledge. Both the composition dimension and the structure dimension demonstrated consistent differences in response patterns for mathematically talented high school students and typical high school algebra students. Contains 23 references. (MKR)