Threshold concepts and troublesome knowledge in a secondlevel mathematics course

Abstract

The term threshold concept derives from education theory to denote concepts that are essential to knowledge and understanding within particular disciplines. Threshold concepts act like doorways that once crossed enable students to comprehend a topic not previously understood. In turn this enables the learner to progress to higher levels of learning. This process can be immediate or drawn out over long periods of time. Grasping the threshold concept transforms the students’ perception of the subject area, and they are better able to relate the topic to wider fields of study (Meyer and Land 2003). However the transformative nature of this process can at times be troublesome and challenging for students (Perkins 1999). In mathematics, once a threshold concept is grasped, the student will see the calculations they are working on in a different light. They will gain insight into what the calculations are doing and how they work. This enables the student to translate the threshold concept to different and more difficult problems. The student is also able to see the relevance of this form of mathematical thinking to other areas of mathematics and to applications beyond the field of mathematics. Very little work has been undertaken to identify the threshold concepts in higher-level mathematics and what is troublesome for students to learn. This paper reports on a study carried out in a large second-level mathematics course at the University of Queensland. The course is taken by students of mathematics, engineering and physical sciences, and covers topics such as advanced ordinary differential equations, vector calculus and linear algebra. Data was collected from course documents and interviews with tutors, and surveys and quizzes completed by students. Analysis of this data identified potential threshold concepts for this content area, and areas of troublesome knowledge experienced by students. This paper reports on the findings from this study, and on the implications of these findings for enhancing learning and teaching of mathematics.