Assessment and Contract-Like Relationships in Undergraduate Mathematics Education

“Well, I let him pass, but I am pretty sure that if I had given my questions a twist, his lack of understanding would have been revealed.” At traditional oral or written examinations, this kind of impression is not unusual among examiners. Knowing that the student has dutifully met the course requirements, the only reasonable verdict is “pass.” Yet it is given with an uneasy feeling that the student has not really been put to the test. However, it seems that the circumstances and traditions governing university mathematics teaching make it difficult to assess more than the use of standard techniques or the passive knowledge of textbook material.

It is commonly acknowledged that assessment procedures influence university students' activities in many ways. At a very global level this influence can be seen as simultaneously inevitable, necessary and regrettable. Inevitable because universities need to grant a credible diploma and hence must assess and declare their holders' competencies in depth; the fact that this declaration may be decisive for the students' future career opportunities is, at a global level, an important reason for students to direct their study activity towards maximizing their “declared competency” (or, more modestly, to just get their diploma). It is necessary in the sense that (outside paradise) explicit reward is sometimes necessary to get people and hence society to work. This may be true not least for studies with many technical and difficult parts located more or less necessarily at the beginning of the curriculum: the necessity to do these parts (in order to achieve the target diploma) acts as a default motivation for students who do not themselves acknowledge their attraction or necessity. Finally it is regrettable because the existence of such a default incentive to study may seem to suspend the need for other rationalities for teaching and study, and hence reduce academic teaching and studies to something highly un-academic: work based solely on control and rule following. Obviously one would like to minimize the regrettable effects of assessment, while retaining a visible incitement for students to meet necessary work requirements as well as a credible declaration of the results of this work.