Important prerequisites to understanding the definition of limit

Specialeforsvar ved Freja Elbro

Censor: Prof. Jacob Hjelmborg, SDU

Vejleder: Prof. Carl Winsløw, KU

Abstract:

The transition from high school to university level mathematics has been reported to cause difficulties for many students. The transition has been described as a move from describing to defining and from arguing to constructing proofs based on definitions. The formal definition of limit is often one of the first concepts which is taught to students in a way that requires post-transitional thinking. For this reason (and many others), student understanding of limits has attracted a lot of research interest. This study aims to compare the importance of five competencies, which have previously been shown or hypothesised to be important to learning the formal definition of limits or transitioning to university level mathematics in general:

• Being able to distinguish between valid and invalid proofs
• Having an active meaning orientation to studying
•  Understanding the role and nature of definitions in mathematics
• Not being coloured by common misconceptions of limits
• Being able to solve inequalities involving absolute values and connect the result to an
  image of a subset of the number line.

154 students took part in our study. They were first tested in each of the competencies above. Then they were taught the formal definition of limits, and finally their understanding of the concept was measured. We found that all of the competencies on the list above which are specific to mathematics (that is all of them except orientation) had moderate to strong correlations with understanding of the formal definition of limit, and that this correlation was significant, even when we controlled for general mathematical competency (as measured by a grade in an unconnected subject within mathematics). We did not find any strong indications of which competencies out of the four were most important. Orientation to studying was found to correlate only very weakly to understanding of the formal definition of limits. Hence this research indicates that if teachers want to improve student understanding of the definition, it is more important that they focus on the mathematical prerequisites rather than trying to change the study orientation of the students. Further research is needed to determine the order in which the different competencies should be taught.