Adults' Mathematical Thinking and Emotions

A Study of Numerate Practice

Jeff Evans

• Publisher: Routledge
• Year: 2002
• ISBN:
  9781135701901
• Edition: 1
• Topic: Education
• Subtopic: Education General

The crisis around teaching and learning of mathematics and its use in everyday life and work relate to a number of issues. These include: The doubtful transferability of school maths to real life contexts, the declining participation in A level and higher education maths courses, the apparent exclusion of some groups, such as women and the aversion of many people to maths. This book addresses these issues by considering a number of key problems in maths education and numeracy:
*differences among social groups, especially those related to gender and social class
*the inseparability of cognition and emotion in mathematical activity
*the understanding of maths anxiety in traditional psychological, psychoanalytical and feminist theories
*how adults' numerate thinking and performance must be understood in context.
The author's findings have practical applications in education and training, such as clarifying problems of the transfer of learning, and of countering maths anxiety.

1 Introduction: Mathematics, the Difficult Subject

There seems to be a need for a more technological bias in science teaching that will lead towards practical applications in industry, rather than towards academic studies. Or to take other examples, why is it that such a high proportion of girls abandon science before leaving school?

(Callaghan 1976: 5)

There is considerable evidence that those who give up mathematics do so because they perceive it as ‘hard’ or ‘boring’, that is having little significance for them.

(Brown 1995)

I began this study of adults’ mathematical thinking and emotions during an exciting period for mathematics education. The findings of an enquiry into mathematics teaching in the UK (Cockcroft Committee 1982) had recently reported, and there was a growing concern with adult ‘numeracy’. Many in the research communities were discussing exciting and impressive research on the ways that people’s mathematical thinking might be different in different settings; for example, when doing school problems, working in street markets, or shopping in supermarkets (e.g. Carraher et al. 1985, Lave et al. 1984). In addition, a few writers on both sides of the Atlantic were discussing mathematics anxiety, and other kinds of feelings about mathematics (e.g. Tobias 1978, Buxton 1981, Nimier 1978).

In this chapter, I outline the background to the study in terms of policy concerns and research emphases in play when I began. I indicate the most important aspects of the conceptual basis of my work, and briefly describe my use of quantitative and qualitative methodologies.

Perpetual Concerns about Mathematics Learning and Use

Concerns over the mathematics curriculum and mathematics teaching have been a continuing feature of debates in education throughout this century (see e.g. Howson 1983, McIntosh 1981). These issues occasionally flare up as crises, around several flashpoints, including:

• Assertions that students fail to apply their mathematics learning from school to the workplace, to other ‘everyday’ settings, or to other subjects: the problem of application or ‘transfer’.
• Decreases in already low levels of participation in A-level and higher education (HE) mathematics courses – and allegedly lower standards of preparation among those accepted to study mathematics in HE (London Mathematical Society 1995): the participation and preparation problems.
• The apparent under-representation of particular groups – especially females – in mathematics study: the inclusiveness problem.
• Evidence of the perception of mathematics as ‘hard’, ‘boring’, anxiety-provoking, even hateful: the affective problem.¹

These four areas of concern are interrelated. For example, both the inclusiveness and affective issues affect the participation problem, and affective issues are also likely to affect students’ commitment, and hence their preparation. Sometimes, action on one area undermines another; for example, the attempt to combat many students’ lack of confidence and alienation, by connecting mathematics with what appears to be ‘accessible’ – typically, based in everyday contexts – risks disconnecting it from its roots in science and technology, which some argue is what is genuinely useful (Noss 1997). In particular, I shall argue in this book that affective issues are implicated widely in the learning and doing of mathematics, including the ‘transfer’ or application of such learning in other contexts.

The first epigraph heading the chapter displays concerns about all of the first three issues raised. The then Prime Minister James Callaghan’s Ruskin College speech in October 1976 launched the ‘Great Debate’ in education, and arguably opened the way for the Conservative Party’s educational reforms of the 1980s and 1990s. These led to construction of a national curriculum and testing framework, which looks set to remain in place well beyond 2000.

Despite the time elapsed and the changes occurring, both in the educational world and outside, since the speech, many current commentators still appear to share a traditional view of ‘mathematical ability’. It is seen as involving a set of abstract cognitive ‘skills’, which can be applied to perform a range of tasks, in a variety of practical contexts. This is considered to take place through a relatively straightforward process of transfer. In any formal educational system, the issue of transfer of learning is clearly of major importance, whether in the ‘application’ of school knowledge in contexts outside the school, or in the ‘harnessing’ of outside learning to help with school aims. (Though the latter process has so far received less attention, it is bound to be increasingly crucial in systems where ‘mature students’ or ‘recurrent education’ are encouraged.) In this traditional view, performance is usually measured by the number of correct responses to a set of test items, of the type often used in schools. However, the possibility that correct performance could be produced merely by rote learning has led mathematics educators to stipulate that correct performance should be produced through ‘real understanding’ (Skemp 1976).²

In response to concerns expressed by the Prime Minister and others, in 1977 the Labour government announced its decision to establish an Inquiry into the teaching of mathematics ‘with particular regard to its effectiveness and intelligibility and to the match between the mathematical curriculum and the skills required in further education, employment and adult life generally’ (Cockcroft Committee 1982). The Committee’s report focused on a number of issues, including how to promote the application of school mathematics in practical everyday life, drawing on research which it had commissioned. This is not to say that the interest in ‘practical mathematics’ began with Cockcroft: there were already a number of projects with a similar focus.³ However, Cockcroft used the term ‘numeracy’ to mean the use of basic mathematical operations with confidence in practical everyday situations – and also to emphasise that the ‘transfer’ from abstract mathematics to everyday applications might not be as straightforward as the traditional view suggested.

Three major developments in further education and adult education during the 1980s and 1990s reinforced the interest in ‘numeracy’ – though, as we shall see, the meaning of this term varies across contexts:

1. the mushrooming of adult literacy and ‘basic skills’ courses
2. the development of ‘Access’ courses, providing adults with intensive preparation and a ‘second chance’ alternative entry route for higher education
3. the development of vocational qualifications, such as GNVQs and NVQs.

Thus numeracy became important in several areas of educational policy-making. In particular, the adult literacy movement led to an awareness, starting among tutors, that a lack of ‘numeracy’ existed as a widespread problem, somewhat independently of illiteracy. Public discussions resulted, concerning the handicaps, for individuals and for society, resulting from this lack of numeracy (Evans 1989a).

Besides signalling their avowedly more practical focus, calling courses ‘numeracy’ probably served to deal with many students’, and teachers’, negative feelings towards mathematics – lack of confidence, anxiety and dislike. Many students perceive there to be fairly strong boundaries between the mathematics that they have met at school or college, and the other activities that make up ‘real life’ (or indeed other academic disciplines), even when the latter make substantial use of quantitative, spatial or other ‘numerate’ ideas and strategies. One might suspect that both the sense of ‘boundaries’, and the negative feelings, will be implicated in any possibilities of transfer between academic mathematics and practical activities.

Research in mathematics education and psychology since the 1950s has often produced standard findings on social differences in performance in school mathematics: namely, gender differences in favour of males, and social class differences in favour of the middle classes. In the 1970s concern about gender differences increased markedly: most (but not all) researchers accepted their existence (at least by the early teenage years in the USA and UK). In the USA, many attributed them to gender differences in ‘participation’ or course-taking. These issues of subject choice in turn were explained by attitudes and affect, that is, by gender differences in feelings towards mathematics: of enjoyment/dislike, confidence/anxiety, beliefs about usefulness and difficulty, and so on. However, by the mid-1990s, gender researchers began to argue that the situation was changing, so that gender differences in school-level performance, and even at university, are disappearing (for example Keitel et al. 1996, but see also Fennema 1995).

Nevertheless, partly in order to explain gender differences in mathematics performance, much research on affect and attitudes has been carried out with schoolchildren since the mid-1970s, especially in the USA. Much of this research on affect has focused on ‘mathematics anxiety’. Anxiety as a concept has its basis in Freud’s work which places crucial emphasis on the possibility that anxiety may be ‘latent’ or unconscious. Psychology took on board these concepts, but the stress on conscious, observable phenomena in mainstream American psychology from the late 1940s onwards led to an emphasis on ‘manifest anxiety’, which was considered observable and quantifiable. Later research, using broadly the same methodology, studied college students (largely in the USA), especially with regard to gender and other differences in reported anxiety, and the relationship between mathematics anxiety and performance.

One of the reasons that ‘mathematics anxiety’ has received most attention among affective factors has been the continuing attention paid on both sides of the Atlantic to Sheila Tobias’s Overcoming Math Anxiety (1978) and Laurie Buxton’s Do You Panic about Maths? (1981). These and other researchers have shown how some of the ways that school and college mathematics have traditionally been taught lead to negative emotions: its emphasis on abstraction, individualism, and speed have spawned feelings of boredom, isolation, and anxiety. Widely-held myths like the ‘one right method’ have often led to humiliation and anxiety (lest one’s own, perhaps ‘illicit’, methods be found out); myths about the ‘one right order’ to learn mathematics and the possession – or lack – of a ‘mathematical mind’ encourage fantasies about ‘starting back at the beginning’, and/or a slavish dependence on teacher (or text).

Despite the wealth of research undertaken in recent years, there are still gaps in our conceptions and our knowledge. There remain confusion about the idea of transfer of mathematical (and other) learning from school to outside contexts, and disappointment about students’ ability to accomplish this. In any case, how should we specify the context of learning and thinking?

There is still relatively little research on the mathematical thinking of adults, especially in the UK. Further, much of the research that has been done on adults has not paid much attention to affect or emotion, though it does focus more nowadays on contexts outside the academic.

Therefore, the research reported here studies thinking and affect, and their interrelationship, among a group of adults who were involved in college mathematics and who also were experienced in a wide range of practical activities that might provide a context for the use of ideas recognisable as ‘mathematical’.

Developing Ideas and Methodologies for the Study

The main study reported in this book is about ‘mathematical’ thinking in various contexts – college and out-of-college – by adult students. Now what is ‘mathematical’ is not straightforward. A recent symposium characterised mathematics variously as:

• a discipline in its own right
• a collection of skills for wide application
• a set of ‘thinking tools’
• a set of principles and techniques for modelling
• a powerful language for sharing and systematising knowledge
• a part of our cultural heritage (Hoyles et al. 1999).

There are certainly differences of view on this issue, which cannot be discussed fully here.⁴ In this book, in describing mathematical thinking, I aim to emphasise the idea that it is thinking in context.

In this study, adults are understood as people of a range of ages, who:

• participate in a substantial range of activities and social relations, normally including some outside the home, school or college
• have at least the opportunity for paid or voluntary work
• are conscious of having social or political interests.

Thus, I would include many ‘adolescents’, and virtually all ‘mature students’, who in the UK are 21 or over, and usually have previous work or child-care experience.

Adults’ lack of numeracy cannot fully be explained by their weakness in school mathematics, as measured by school tests for several reasons:

1. Adults use numerate ideas in a variety of social contexts, which could not be anticipated by school mathematics teaching.
2. Adults may lose certain school mathematics skills through lack of use.
3. Society’s definition of a minimum adult level of competence may change.⁵
4. Lack of confidence may be as important a problem for adults as lack of knowledge.
5. The errors made by adults may differ from those made by children – sometimes because adults’ greater knowledge and experience may make simple problems more complex (Withnall et al. 1981).

I began my work with the broad problem: How do adults develop, or fail to develop, mathematical thinking, and confidence in it? I developed a simple model which aimed to explain this by several factors or processes:

• socialisation depending on the individual’s position in class, gender and other terms
• critical incidents in one’s history of learning mathematics
• affective characteristics, which I saw as representing an ‘internalisation’ of the experiences from the first two sets of determinants.

I aimed to attempt to produce (or gain access to) information that would allow me to describe the ‘level’ of numeracy in the population of adults in the UK, or a reasonably representative sub-population; and to relate this to respondents’ backgrounds, experiences, and feelings about mathematics. At the start, I was interested in the population of adults at large, though the practical constraints of doing empirical research soon led me to choose a population that would allow a compromise between representativeness and convenience (see Chapter 2).

I considered that using a methodology including both quantitative and qualitati...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Studies in Mathematics Education Series
• Figures
• Tables
• Series Editor’s Preface
• Author’s Preface and Acknowledgements
• 1: Introduction: Mathematics, the Difficult Subject
• 2: Mathematical Thinking in Context Among Adults
• 3: Mathematics Performance and Social Difference
• 4: Affect and Mathematics Anxiety
• 5: Reflections on the Study So Far
• 6: Rethinking the Context of Mathematical Thinking
• 7: Rethinking Mathematical Affect as Emotion
• 8: Developing A Complementary Qualitative Methodology
• 9: Reconsidering Mathematical Thinking and Emotion in Practice
• 10: The Learners’ Stories
• 11: Conclusions and Contributions
• Appendix 1: Questionnaire Design and Fieldwork
• Appendix 2: Interview Problems for Solution
• Notes
• Bibliography