Mathematical question spaces

It is uncontroversial to assert that learning mathematics is only effective when it is an active process on the part of the learner. Setting questions is a ubiquitous technique to engage students, and answering such questions constitutes a large proportion of the activity they undertake. Indeed, asking students questions is a central part of all theories of learning. This paper examines in detail the process of randomly generating versions of mathematical questions for CAA. In doing this we examine not only a single mathematical question, but how such questions are linked together into coherent structured schemes. Two important pragmatic reasons are often cited by colleagues for wishing to generate a random sequence of questions. • Randomly generated questions may reduce plagiarism • Distinct but equivalent questions may be used for practice Even if giving each student a distinct problem sequence reduces plagiarism, professional experience unfortunately demonstrates it is not eliminated. However, some students are well aware of the potential benefits of collaborative learning, possibilities for which are traditionally hard to provide in the mathematics classroom. As one student commented in their feedback evaluations: "The questions are of the same style and want the same things but they are subtly different which means you can talk to a friend about a certain question but they cannot do it for you. You have to work it all out for yourself which is good." Notice here the student voices the opinion that the questions "want the same things but they are subtly different". In this paper we address exactly this issue, by examining equivalent mathematical problems in some detail.