One goal of researchers in collegiate mathematics education is to develop a series

of mathematical activities that will enable all students to understand advanced

mathematical concepts. I will discuss a series of studies that suggest this goal

may be misplaced. In particular, I will show that some undergraduates learn and

reason about mathematical concepts primarily by focussing on informal

representations of these concepts (graphs, diagrams, etc.) while other students

tend to focus primarily on logic and deduction while not considering such informal

representations. Students are generally consistent with their mode of reasoning --

that is, students who do consider informal representations of concepts for one

task tend to do so on other tasks. One implication from this finding is that

students learn mathematics in different ways and there likely is not a specific

series of instructional activities that will be effective for all students.

Specific pedagogical suggestions from these findings will be discussed.