Given the more extensive experience of the authors in relation to pure mathematics, this section draws heavily on views expressed by others in writing and at conferences, particularly the work of Hibberd (2002). Mathematics graduates, whether they embark upon postgraduate study or enter a career outside academic life, are expected to possess a range of abilities and skills embracing subject-specific mathematical knowledge and the use of mathematical and computational techniques. They are also expected to have acquired other less subject-specific skills such as communication and teamworking skills.
For most mathematics degree programmes within the UK, the acquisition of subject- specific knowledge, essential IT skills, the use of mathematical and statistical software and subject-specific problem-solving skills are well embedded in the curriculum (QAA, 2001).
Typically these are delivered through formal lectures supported by a mixture of tutorials, seminars, problem classes and practical workshop sessions. As noted earlier, assessment is often traditional, making much use of examinations. Increasingly it is recognised that these approaches do not provide students with the non-mathematical skills that are much valued by employers. This has prompted a search for some variety of learning and teaching vehicles to help students develop both subject-specific and transferable skills.
We introduce one such approach in Case study 1, which is based on experience at the University of Nottingham.
A typical goal in implementing a modelling element is to stimulate student motivation in mathematical studies through ‘applying mathematics’ and to demonstrate the associated problem-solving capabilities. It also offers the chance to provide a synoptic element that brings together mathematical ideas and techniques from differing areas of undergraduate studies that students often meet only within individual modules. This in itself can lead students into a more active approach to learning mathematics and an appreciation and acquisition of associated key skills.
The underlying premise in this type of course can be accommodated through activities loosely grouped as ‘mathematical modelling’. Associated assessments and feedback designed around project-based work, whether as more exten- sive coursework assignments or as substantial reports, can allow students to demonstrate their understanding and problem-solving abilities and enhance both mathematical and key skills. Often quoted attributes gained by graduates are the subject-specific, personal and transferable skills gained through a mathematics- rich degree.
Increasingly, students are selecting their choice of degree to meet the flexible demands of a changing workplace, and well-designed MSOR programmes have the potential to develop a profile of the knowledge, skills, abilities and personal attributes integrated alongside the more traditional subject-specific education.
A mathematical model is typically defined as a formulation of a real-world problem phrased in mathematical terms. Application is often embedded in a typical mathematics course through well-defined mathematical models that can enhance learning and understanding within individual theory-based modules through adding reality and interest. A common example is in analysing predator–
prey scenarios as motivation for studying the complex nonlinear nature of solutions to coupled equations within a course on ordinary-differential equations;
this may also extend to obtaining numerical solutions as the basis of coursework assignments. Such a model is useful in demonstrating and investigating the nature of real-world problems by giving quantitative insight, evaluation and predictive capabilities.
Other embedded applications of mathematical modelling, particularly within applied mathematics, are based around the formal development of continuum models such as those found, for instance, in fluid mechanics, electromagnetism, plasma dynamics or relativity. A marked success in MSOR within recent years has been the integration of mathematics into other less traditional discipline areas of application, particularly in research, and this has naturally led to an integration of such work into the modern mathematics curriculum through the development 256 ❘ Teaching in the disciplines
Case study 1: Mathematical modelling
Mathematics and statistics ❘ 257
of mathematics models. Applied mathematics has always been a strong part of engineering and physical sciences but now extends to modelling processes in biology, medicine, economics, financial services and many more.
The difficulty often inherent in practical ‘real-world’ problems requires some preselection or guidance on initial problems to enable students to gain a threshold level of expertise. Once some expertise is gained, exposure to a wider range of difficulties provides students with a greater and more realistic challenge.
In general, modelling is best viewed as an open-ended, iterative exercise. This can be guided by a framework for developing the skills and expertise required, together with the general principle ‘solve the simple problems first’, which requires some reflection on the student’s own mathematical skills and competencies to identify a ‘simple problem’.
As with most learning and teaching activities, the implementation of modelling can be at a variety of levels and ideally as an integrated activity through a degree programme. The learning outcomes that can be associated with an extensive modelling provision include:
• knowledge and understanding;
• analysis;
• problem-solving;
• creativity/originality;
• communication and presentation;
• evaluation;
• planning and organisation;
• interactive and group skills.
In practice most will only be achieved through a planned programme of activities.
(The Authors)
According to Hibberd (2002), mathematical modelling is the process of:
• translating a real-world problem into a mathematically formulated representation;
• solving this mathematical formulation; and then
• interpreting the mathematical solution in a real-world context.
The principal processes are about how to apply mathematics and how to communicate the findings. There are, however, many difficulties that can arise. For those with an interest in exploring the ideas offered in Case study 1, the article by Hibberd (2002) contains further case studies illustrating the application of these principles; while Townend et al.
(1995) and Haines and Dunthorne (1996) offer further practical materials.