Topics and Tasks. Having discussed the roles of the teacher and the student in the problem-solving process, we now focus on the third most important aspect, namely the problems themselves and the tasks and skills required to solve them. By accident of birth, our Project is aimed specifically at problem-solving in the context of pure mathematics. (Another project, also funded by the National HE STEM Programme, is aimed at mathematical modelling and applied problem-solving.) For this reason, five of our six case-studies are in pure mathematics, the exception being Manchester, which is also involved with the modelling project. We freely acknowledge the central importance of mathematical modelling and believe it to be an indispensable part of any mathematics degree. No student should graduate without knowing how to translate an amenable real-world problem into a mathematical form, and to learn solve it, to test its validity as a model, to refine it and to analyse its limitations. We focus on the ‘solve it’ stage as an indispensable part of the modelling process. Moreover, almost all topics taught in a typical first degree in mathematics—pure or applied—are amenable to a problem-solving approach When setting up a problem-solving module for the first time, we strongly advise against trying to author original problems, not least because of the amount of work required. Writing tailor-made problems that are of sufficiently high-quality to give students the best experience can be left until the module has run once or twice. Xxxxxxx, for example, began teaching analysis using Burn’s Steps into Analysis (1992) before original workbooks were written in its second year of operation. Birmingham’s geometry class used Xxxxxxxxxxx and Xxxxxxxx’x Lines and Curves (2004) for several years before Xx. Xxxxxxx wrote new problems covering Apollonian circles, and Leicester’s graph theory class used Xxxxx-Method resources from the Journal of Inquiry-Based Learning in Mathematics. After a problem-solving module is established, it is advisable to review the problems in the light of student and staff feedback and student performance, and to consider writing a fresh set, or at least revising the existing problems, to improve their effectiveness and suitability. If there is a risk of students colluding from one year to the next, having several sets of problems can help to avoid it. A further recommendation is to restrict the content of the problems in a module to a particular area or topic in mathemati...
Topics and Tasks. The problems handed out should be commensurate with each student’s abilities, pre- senting them with sufficient challenge while offering a fair chance of success within a reasonable time. • Choosing problems from a specific area of mathematics has advantages: the students learn the techniques that work in that area and acquire a deeper understanding of that particular body of knowledge. There is also something to be said for exposing students to problems that bring together ideas for different areas so that they come to appreciate the inter-relatedness and essential unity of mathematics. • There is value in creating sequences of problems, where each problem builds on those before and the progression leads to a significant result at the end. Students become familiar with the circle of ideas and techniques involved and derive satisfaction from the sense of having undertaken a significant mathematical journey. • Creating a body of suitable problems from scratch is hard. When setting up a problem- solving module, it is advisable to borrow tried-and-tested problems from a variety of published sources. Only when the module has become established should the leader be tempted to write original problems to suit their style, the subject matter, and the students’ backgrounds Assessment • Timed exams are better for assessing knowledge than problem-solving skills, which function more effectively in a relaxed study environment. • Continuous assessment can evaluate the various skills in action and also give students an incentive to attend the classes. • Course credit can be offered for a variety of activities: original solutions, presentations, rubrics, portfolios, fair copies of solutions, posters and other types of record. • Credit can be awarded for group work, but carries the risk of weaker students becoming free riders. General • Teaching problem-solving is important and effective. It is not a substitution for learning mathematics but a key component of the process. • Setting up and running a problem-solving module calls for a significant, but worthwhile, investment of resources. Beginning with an optional module on a modest scale will enable a department to analyse its benefits before making a full-scale commitment.
