Effective Learning and Teaching in Mathematics and Its Applications

Peter Kahn, Joseph Kyle

• Publisher: Routledge
• Year: 2003
• Print ISBN:
  9780749435691
• eBook ISBN:
  9781135726898
• Topic: Education
• Subtopic: Education General

The Effective Learning and Teaching in Higher Education series is packed with up-to-date advice, guidance and expert opinion on teaching in the key subjects in higher education today, and is backed up by the authority of the Institute for Learning and Teaching. This book covers all of the key issues surrounding the effective teaching of maths- a key subject in its own right, and one that forms an important part of many other disciplines. The book includes contributions from a wide range of experts in the field, and has a broad and international perspective.

Part A: Issues in learning and teaching

1: The transition to higher education

John Appleby and William Cox

Introduction

The transition from one educational stage to another is usually a difficult and uncertain process. Examples include the first day at school, the primary/secondary interface, GCSE to A level (in the UK), school to university and even embarking upon postgraduate research. This article considers the transition to university and how we might better influence students’ prior expectations and preparation for this new phase of their life. In addition, we look at the implications for the current curriculum of the common problems of the transition as well as discussing initial assessment of students as they enter higher education (HE).

In mathematics there has been a great deal of recent publicity about the issues around the transition to HE, especially in the UK. Notable among these are the report from the London Mathematical Society (1992) and the more recent publication by the Engineering Council (2000). Other commentators include Cox (1994), Lawson (1997) and Gardiner (1997). Mustoe (1992), Sutherland and Pozzi (1995) and others look at the problem as it affects engineering programmes. Developments that impact on these issues include: changes in school/pre-university curricula, widening access and participation, the wide range of degrees on offer in mathematical subjects, IT in schools and social factors. In the UK, problems arising in the transition are mentioned in many Quality Assurance Agency (QAA) Subject Review reports in Mathematics, Statistics and Operational Research (MSOR), and also in some engineering Subject Review reports.

Various reports, including those listed above, point to the changes in schools as the source of problems in the transition, and make recommendations as to how things could be put right there. Indeed, in response to wider concerns about literacy and numeracy, recent government initiatives within the UK have, perhaps, partially restored some of the skills that providers of numerate degrees need; and these might feed into HE in the next decade. As well as changes at primary level, there is much talk of extending these to the 11–13 age group. But notwithstanding such changes, it is doubtful that we will ever return to a situation where school qualifications are designed solely as a foundation for HE, ensuring that any chosen university entrant will be capable in all the skills required by any university programme with a mathematics component.

However, difficulties in transition are caused not just by changes in school curricula, but also by changes in the ability range, in student attitudes and expectations, student financial burdens, in the resourcing available in HE, and in the curricula we expect them to undertake. Some of these are not at all under our control, and the best we can do is to be aware of them and how they affect our students. Others, including curriculum design and, to an extent, pastoral support, may need attention if those who choose our courses are to have the best chance of success. We might also usefully consider what our students know about our courses when they choose them, and how they might prepare themselves a little before they come: in attitude as well as in knowledge. For example, Loughborough University sends new students a revision booklet before they come (see Croft, 2001; and also the discussion in Chapter 11). Since 50 per cent of MSOR providers within England have been criticized for poor progression rates in Subject Reviews, it is clear that there is much to be done.

We can make an analogy with the position of English as a Second Language provision in adult education, where students apparently equally well qualified have widely variable skills and needs. An important component of such provision is rigorous initial assessment to determine students’ current skills profiles before designing a realistic programme that will meet their needs. We are not used to this in HE because there are usually a large number of students and they are traditionally expected to be capable of catching up for themselves if need be, and we are used to the idea that they have already covered everything that we need. It is therefore likely that a system of initial assessment will form an important part of our degree programmes, especially in the transition phase (see the section on initial assessment below).

Understanding the transition to higher education

A recent Teaching and Learning Undergraduate Mathematics (TaLUM) symposium categorized the issues impacting on the HE transition in mathematics as follows:

• The nature of mathematics: does ‘mathematics’ mean the same thing to schoolteachers, mathematics lecturers, engineering lecturers, etc? Mathematics taught in the context of mathematics honours courses and in ‘service’ courses for students whose study focuses on other relevant disciplines may be seen as having quite different purposes, by lecturers as well as students.
• Social component. HE is more student centred, students are less biddable, there are financial, emotional and other conflicting pressures.
• Diversity. There is wider participation, wider access, more variable backgrounds and more variable requirements.
• University requirements. Expansion in HE has led to a wide range of not always clearly articulated aims and objectives, so that entry qualifications no longer specify adequately what students should know and be able to do; some attempts have been made to address this through the work of the QAA.
• School provision. This is no longer geared to feed a coherent spectrum of HE institutions, and university staff are sometimes ignorant of it. There have been changes in school curricula in other subjects such as physics, which used to provide useful support and motivation for students learning mathematics.
• University provision. There are changes in content and style of university teaching compared with school. This is partly due to the fact that many university teachers are not trained (based on an assumption of self-motivated, able students).

The TaLUM symposium further noted that the following were also needed:

• Representation. There is a need to expand fora for school and university teachers to air views and concerns, debate issues, etc; the Learning and Teaching Support Network (LTSN) subject centres may now contribute to this within the UK.
• Liaison. There is a need for closer liaison between schools and universities.

The preceding list focuses on changes in the transition to HE, attempting to list some of the main categories of reasons for the increased difficulties. It is also useful to consider the students and their perspective, the university including its staff and its courses, and the change that is required for the one to adapt to the other. In this section we explore the contrast between attitudes and characteristics typical of lecturers and those of students, together with our intentions and expectations, usually implicit, of the course and of the students. We examine the curriculum in more detail in the following section.

University lecturers were not typical students even at the time they were students, and, in many cases, that was 10, 20, or even 40 years ago. We liked our subject, worked adequately if not hard, and also perhaps had a style of learning that was atypical. Understanding what we were doing, finding out for ourselves, checking our results, as well as having a natural bent for mathematics, were likely to be our characteristics. This attitude continues (except that there is far more emphasis on research compared to teaching these days), and we probably most like teaching those who are interested and make an effort. Most of us probably had family members who had been to university: not true for many students for some years after any major expansion.

In addition, we may well have enjoyed reading as a pastime, perhaps did not have a telephone or television in our lodgings, and certainly did not have a mobile phone or access to the Internet. We probably did not have term-time work, and also spent less on clothes, entertainment and drink, etc. School teaching also followed a different style as well as different syllabuses, with more routine practice though perhaps with fewer topics and possibly no calculator. On the other hand, we were less used to visiting the library, interviewing people, project work and group work and more used to textbook problems little connected with everyday problems. Furthermore, we also did not have to address the issues associated with IT and transferable skills, both of which are discussed in other chapters.

Our current student group, as well as having a great variety of qualifications and ability, also have variety in social background, age, work experience and ambitions or intentions. Lastly, some students, especially mature ones, may have had some gap since they last formally studied mathematics.

Other factors not often mentioned are the changes in other areas of the school curriculum. In the UK the most obvious are that double mathematics A level, formerly common for applicants to mathematics, physics or engineering courses, is now rare, and the changes in physics. Far fewer sixth-formers now study physics, so much so that many engineering courses now accept students without this qualification, but also the nature of Physics A level has changed. Partly because GCSE Mathematics contains no calculus, A level Physics has become less quantitative, and topics like simple harmonic motion and dimensional analysis are missing. Apart from the actual topics, this may also contribute to a relatively ‘Pure’ view of mathematics (at A level) as a subject with its own laws, rather than as a tool for describing the real world. It may also be relevant that in English, as well as in mathematics, there has been a reduced emphasis on standard techniques and the need for accuracy, and increased emphasis on creative use.

In the past we have not always been explicit about what we expect from students, and indeed, what we are trying to achieve in our undergraduate courses. Clearly, knowledge of the subject has been pre-eminent as an aim, with the implicit aim of understanding. After all, if we generally seek to understand what we do, and want to learn more, then it is easy to assume that the same is, or should be, true for our students. Thus our assessment methods have focused on skill at solving standard types of problems, with the real aims pursued incidentally (Ball et al, 1998). A student like ourselves will, in achieving mastery, attempt a wide range of problems and try to understand why the methods work when they do. However, a typical student (and, to an extent, this was always true) will see the assessment task as the mastery of a restricted skill and, in focusing on this problem type alone, may fail to make the connections we expect and achieve even that mastery. Therefore, to address the needs of a typical student, we must address the curriculum, be prepared to change it if necessary, as well as to prepare students for it (see following section).

A further area worth consideration is that of the change of pattern of learning as the student comes to HE. Instead of a school pattern, in which the majority of work is supervised in class or closely directed, students find that the majority of their work is undirected and unsupervised. In addition, they are expected to make use of a wide range of resources—books (and finding their way round the library and the Web), notes, handouts, computer-based materials—and new patterns of work: choosing their own tasks, managing their own time, and group work of a different kind. In the section on initial assessment below we consider further how we might address some of these issues for the new student.

The curriculum

We have two aims in considering the curriculum: Is the curriculum appropriate? How can we help students to cope with it? Moreover, we might consider various aspects of the curriculum including: syllabuses, assessment, resources, time-tabling and scheduling, style versus content, knowledge versus understanding, mathematics honours versus service teaching, and progression in several of the above over the three or four years of the course.

In this chapter, we can only reflect on the broader issues of what constitutes an appropriate curriculum, and offer some suggestions about the second question: how can we best support students, especially in the early days of their course? However, our reflections on transition should feed into discussions about the course as a whole. To some extent, we have to accept that an appropriate curriculum is one that our students can cope with (see Chapters 7 and 8 for a fuller discussion of curriculum design and related issues).

It is now a major challenge to match the first and subsequent years’ curriculum to the skills profile of what might be a wide range of incoming students. So many assumptions made about their background are now out of date. For example, consider the case of the integral:

which might occur as part of solving a differential equation in the first year of a typical mathematics degree programme, taking mostly students with around A level B grade standard. Twenty years ago the typical lecturer might have written the bald steps on the board as:

with little explanation, and very little or no time actually studying the method of integration.

In fact, five distinct techniques are used, namely:

• difference of two squares;
• partial fractions;
• substitutions for 2x-1, 2x+1;
• integral of reciprocal;
• properties of logs.

For a class today of grade B students even the individual steps of the above argument could be beyond a significant proportion (greater than 50 per cent). Putting them together therefore represents a significant learning task. This is no longer a case of revision with a few reminders, and such students can no longer be sent off to sort the details out for themselves without help. Time and resources must be allowed in the curriculum for the students to consolidate their skills in areas such as this. For engineering students, more revision of these skills has been routine for longer, but there have been changes in the examination. As well as a simpler integrand, the method of integration will probably be given. Other examples of changes over time include specifying all steps in examples, rather than most, specifying methods of solution for differential equations, and giving intermediate results in problems, which reduces the requirement for analysis and for accuracy.

So a continual rethink of the curriculum is required if even good A level students are not to be continually bombarded by the need to call instantly on basic skills that they simply do not have. It is easy to view this as ‘dumbing down’, falling standards, etc, and to begrudge the extra time needed for catching up. But this is nothing new, and there is little point in wasting time on recriminations: we simply have to work with the students we have. Against this, we must avoid a boring curriculum; methods are more interesting, relevant and memorable if seen as part of a whole and in context rather than in isolation.

Defining the curriculum in terms of learning outcomes

Encouraged by the programme of subject reviews within the UK we have become more explicit about our aims and objectives, or aims and learning outcomes, as they are now formulated. Specifically, a learning outcome is a statement of a performance under certain conditions to a certain standard (see Gronlund, 1978; Mager, 1990). We do not claim that wh...

Table of contents

• Cover Page
• Title Page
• Copyright
• About the editors
• Acknowledgments
• Forewords
• Preface
• Part A: Issues in learning and teaching
• Part B: Learning and teaching in context
• Further reading