A Handbook for Teaching and Learning in Higher Education Enhancing academic and Practice

Mathematics and statistics
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Teaching from the microcosm
If students are to see mathematics as a creative activity it will help to focus on the different
strategies they themselves can employ to help make sense of mathematical ideas or
problems. It may also help to focus on the process of learning mathematics more broadly.
Specific teaching approaches, though, are required to realise this. Palmer (1998: 120), for
instance, argues that every problem or issue can become an opportunity to illustrate the
internal logic of a discipline. We can take a straightforward example of this. If you do not
understand an initial concept, it will be virtually impossible to understand a more
advanced concept that builds on the initial concept: to understand the formal definition
of a group, for instance, you first need to understand the concept of a variable (as well as
other concepts). But students cannot rely on the tutor always identifying these prior
concepts for them. They themselves need to be able to take a look at an advanced concept
and identify contributory concepts, so that they can then make sure they understand
them. The same applies to other strategies, as Kahn (2001) further explores, whether
generating one’s own examples, visualising, connecting ideas or unpacking symbols. In
terms of concrete teaching strategies, the tutor can model these strategies alongside a
systematic presentation of some mathematics or require students to engage in such
strategies as a part of the assessment process. We thus move away from an exclusive focus
on the content, to more direct consideration of the process by which we might come to
understand that content.
ISSUES PARTICULAR TO APPLIED MATHEMATICS
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 subjectspecific
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.