Stimulating Students To Construct Boundary Examples

We address three common difficulties encountered by students: not appreciating the necessity of conditions in a theorem before using it; using non-generic examples as if they were generic; and ignoring counter-examples as pathologies. We propose the conjecture that students would be more likely to remember and to appreciate the importance of conditions, if they were stimulated to construct examples for themselves which show why each of the conditions is necessary. Furthermore, constructing their own examples is likely to prompt them to explore the space of possibilities admitted by definitions, and hence to appreciate both the range of situations encompassed by the definition, and the force of both definition and theorem. We illustrate our conjectures with some particular but generic tasks for students, and we use these to consider what is involved in constructing examples, leading to ways to support students in learning how to construct them for themselves. 1. Method Our method of enquiry is to identify phenomena we wish to study, and to seek examples within our own experience. We then construct task-exercises to offer to others to see if they recognise what we find ourselves noticing. Through refinement and adjustment of task-exercises in the light of experience and of reading relevant literature, we both extend our own awarenesses, and offer others experiences which may highlight or even awaken sensitivities and awarenesses for them. These sensitivities and awarenesses may inform their future practice. As task-exercises are developed and shared, actions which exploit what is noticed become part of regular teaching for the benefit of students. Our method does not attempt to capture or cover the experience of readers. Rather it aims to make contact with that experience, perhaps challenging interpretations, perhaps pointing to features not previously noticed. The data of this method are the experiences generated, the sensitivities to notice which are enhanced. If you recognise at least something of what we are talking about as a result of having worked on these problems, you may be stimulated to look out for similar experiences in the future, and over time, begin to act upon what you notice. Validity in this method lies in you finding your actions being informed in the future, not in what we say. The task-exercises which follow are intended to bring to the surface various features about the construction of examples to meet constraints, starting from the premise that if you simply ask students out of the blue to construct a mathematical object, they are likely to find it very difficult if not impossible. This may lead a tutor to lose confidence and to conclude that 'students can't do this sort of task'. The effect is a move from impoverished past experience (students are not aware of the construction of objects) to a continuation of impoverished experience (students are not called upon to construct examples, because 'it is too hard'), and so the cycle continues. The aim of this paper is to locate ways of breaking out of this cycle. 2. Task-Exercises 1 A Routine Problem: solve the differential equation f''(x) + b f'(x) + c f(x) = 0.