Mathematics Education

Exploring the Culture of Learning

Barbara Allen, Sue Johnston-Wilder

• Publisher: Routledge
• Year: 2003
• ISBN:
  9781134338832
• Edition: 1
• Topic: Education
• Subtopic: Education General

Mathematics Education identifies some of the most significant issues in mathematics education today. Pulling together relevant articles from authors well-known in their fields of study, the book addresses topical issues such as:

• gender
• equity
• attitude
• teacher belief and knowledge
• community of practice
• autonomy and agency
• assessment
• technology.

The subject is dealt with in three parts: culture of the mathematics classroom, communication in mathematics classrooms and pupils' and teachers' perceptions.
Students on postgraduate courses in mathematics education will find this book a valuable resource. Students on BEd and PGCE courses will also find this a useful source of reference as will teachers of mathematics, mentors and advisers.

Section 1

Culture of the mathematics classroom – including equity and social justice

Each of the authors included in Section 1 is arguing about the importance of the creation of a classroom culture that supports effective learning. Underlying their work is the recognition that the values of the teacher impact upon the classroom but they do not assume that this is a simple system of cause and effect. The authors all see mathematics as a personal construction but are not necessarily agreed on the nature of mathematics.

If a classroom has a culture that values learners creating their own mathematics and becoming authors of mathematics, then the learners are more likely to become positioned as successful learners of mathematics. For this to happen you need a community of learners working together collaboratively and creatively. There needs to be a shift in the way some teachers view the nature of mathematics and an examination of the value they place on assessment and target setting. For a community of practice to flourish learners need to develop personal autonomy and be able to recognise for themselves that they are creating and understanding mathematics.

The first chapter by Paul Ernest focuses on the public image of mathematics. He is concerned that the public image of mathematics as cold, abstract and inhuman has an impact on the recruitment of students into higher mathematics.

Ernest highlights the importance of changing the negative public image of mathematics and challenges the general acceptance of an ‘I can’t do maths’ culture. He looks at teacher philosophy and values and argues that it is the values that have most impact on the image of mathematics in the classroom. This image of mathematics also impacts on the way learners position themselves as successful or unsuccessful. In a classroom where a learner is expected to develop techniques and skills with single correct answers to questions it is not unusual for them to see themselves as an unsuccessful learner of mathematics or indeed to become mathephobic (Buxton, 1981).

He argues that school mathematics is not a subset of the discipline of mathematics but a different subject made up of number, algebra, measure and geometry and not studied for its own sake. But, even so, he believes mathematics should be humanised, for utilitarian and social reasons.

Andrew Pollard’s research (Chapter 2) was not carried out in mathematics classrooms but has been included here because the findings are relevant for mathematics teachers. It is common for research about pupils’ views to be carried out across subjects rather than in a particular subject. Pollard argues that researchers should cooperate across the disciplinary boundaries of psychology and sociology, in a joint effort to look at learning in schools. One of his concerns, like many others in this book, is that little attention has been given to the effect that the new curriculum in the UK has had on learners.

Pollard looks at the changes in research into effective teaching practice over 30 years. That interest has gone from looking at teaching styles, to examining opportunities to learn, to considering the quality of tasks. He is also interested in pupils’ coping strategies and looks at those in subsequent articles – the focus here being on identity and learning. He looks at the relationship between self and others and the importance of social context in the formation of meaning – that is all part of developing a model of learning and identity. The identity of the learner is formed when they have a view of themselves as able to do mathematics or not. He demonstrates the importance of the social context in which learning takes place.

The article by Hilary Povey and colleagues (Chapter 3) takes the reader beyond Pollard to look at people in terms of identity and their responses to the classroom situation. The writers explore the idea of learners author/ing their own learning and how they come to know mathematics.

The article builds on Povey’s work with mathematics teachers with the main thrust being about discursive practices and how they can liberate a learner. The authors argue that when thinking of mathematics as a narrative rather than a fixed form, a learner can create their own narrative in the same way you would a story. Thinking of mathematics in this way enables the learner to have ownership and author/ship over their own learning thus giving greater autonomy to the learners. But both teacher and learners need to create a supportive and collaborative classroom environment in order for this to happen. Many current classrooms do not encourage autonomy because pupils are required to produce responses that are authored by another and not themselves.

Anne Watson’s article (Chapter 4) is concerned with a particular aspect of classroom culture, that of teachers’ informal assessment of students’ mathematics. She believes that the sort of assessment used by teachers reflects their values and, like Ernest, believes this has an impact on the classroom culture. Watson’s research with 30 UK mathematics teachers resulted in the identification of some differences in their practices that could lead to inequity in the classroom. She concludes that the teachers’ practices showed six contrasting beliefs and perceptions about assessment and that teachers could be positioned differently within each of these. It is these different forms of assessment that Watson believes could result in social inequity and contribute to a discriminatory curriculum.

Cooper and Dunne (Chapter 5) are particularly interested in the effects of social class on pupils’ learning. In this article they are concerned with those tasks in the National Curriculum tests that are termed realistic. Cooper and Dunne found that social class and gender differences were greater when ‘realistic’ tasks were used. So they argue that pupils from lower social classes are more likely to get better results on a task that is not ‘realistic’ but is abstract. The reason for this is in part because they do not have the cultural experience or ‘linguistic habitus’ (Zevenbergen, Chapter 7) to understand the game of answering realistic questions. These questions are not part of the home experience and discourse of the lower social class pupils and therefore the middle class pupils are advantaged.

This is of concern at a time when some colleagues are arguing that there is a need for more realistic tasks in the National Curriculum tests.

Goos, Galbraith and Renshaw’s research programme (Chapter 6) is based on sociocultural theory in which they are looking at the interactive and communicative conditions for learning. For them the idea of community is central where gaining knowledge is seen as the process of coming to know mathematics. In this community everyone is seen as having a voice and learners are author of their own mathematics. Their research shows that the roles of both teacher and learners need to change if the notion of a ‘community of practice’ is to take hold effectively.

Goos and colleagues found Vygotsky’s notion of a Zone of Proximal Development (ZPD) was a part-useful idea to work on as it highlighted the way in which pupils support each other so they are not fully reliant on the teacher. However, they also found that a teacher who does not have a good grasp of mathematics cannot see the links in order to help scaffold the pupils’ learning. A combination of mathematics and pedagogic knowledge is needed by teachers in the form of long-term continuing professional development so that mathematics classrooms may become communities of learners.

Further reading

Buxton, L. (1981). Do You Panic About Maths? Heinemann, London.

Cooper, B. (1998). Using Bernstein and Bourdieu to understand children’s difficulties with ‘realistic’ mathematics testing: An exploratory study. International Journal of Qualitative Studies in Education, 11(4), 511–532.

Murphy, P. and Gipps, C. (eds) (1996). Equity in the Classroom: Towards an effective pedagogy for girls and boys, RoutledgeFalmer.

Nickson, M. (1992). The culture of the mathematics classroom: an unknown quantity. In D. A. Grouws (ed.) Handbook of Research on Mathematics Teaching and Learning, Macmillan, New York, 100–114.

1 Images of mathematics, values and gender

A philosophical perspective

Paul Ernest

Abstract

This paper describes the widespread public image of mathematics as cold, abstract and inhuman, and relates it to absolutist philosophies of mathematics. It is argued that this image is consistent with ‘separated’ values (Gilligan, 1982) which help to make mathematics a ‘critical filter’ denying access to many areas of study and to fulfilling professional occupations, especially for women in anglophone western countries. In contrast, an opposing humanised image of mathematics, consistent with ‘connected’ values, finds academic support in recent fallibilist philosophies of mathematics. It is argued that although these two philosophical positions have a major impact on the ethos of mathematics classrooms, there is no direct logical connection. It is concluded instead that the values realised in the classroom are probably the dominant factor in determining the learner’s image and appreciation of mathematics (and hence, indirectly, that of society).

A widespread public image of mathematics is that it is difficult, cold, abstract, theoretical, ultra-rational, but important and largely masculine. It also has the image of being remote and inaccessible to all but a few super-intelligent beings with ‘mathematical minds’. Many persons operating at high levels of competency in numeracy, graphicacy and computeracy in their professional life in the UK still say ‘I’m no good at mathematics, I never could do it’. In contrast to the shame associated with illiteracy, innumeracy is almost a matter of pride amongst educated persons in western anglophone countries.

In fact, many such persons are not innumerate at all, and it is school or academic mathematics, not everyday mathematics, that they feel they cannot do. Numeracy, contextual mathematics, even ethnomathematics are perceived to be quite distinct from school/academic mathematics, and the latter is understood to be ‘real’ mathematics. The popular image of mathematics sets it apart from daily concerns of the public, despite the many social applications of mathematics referred to daily in the mass media, from sports and weather to economic and social indicators. Thus the widespread public image of mathematics is largely a negative and remote one, alien to many persons’ professional and personal concerns and their self-perceived abilities.

For many people the image of mathematics is associated with anxiety and failure. When Brigid Sewell was gathering data on adult numeracy for the Cockcroft Inquiry (1982), she asked a sample of adults on the street if they would answer some questions. Half of them refused to answer further questions when they understood it was about mathematics, suggesting negative attitudes. Extremely negative attitudes such as ‘mathephobia’ (Maxwell, 1989) probably only occur in a small minority in western societies, and may not be significant at all in other countries. Nevertheless it is an important phenomenon, and I have never heard of an equivalent ‘literaphobia’, although literacy is at least as important as numeracy.

The public image of mathematics is an important issue of concern for mathematics education. It is particularly important because of its social significance. Mathematics serves as a ‘critical filter’ controlling access to many areas of advanced study and better paid and more fulfilling professional occupations (Sells, 1973). This particularly concerns those occupations involving scientific and technological skills, but also extends far beyond this domain to many other occupations, including education, the caring professions and financial services. In addition, many adults leaving full-time education have not been empowered by their mathematics education as mathematically-literate citizens who are able to exercise independent critical judgements with regard to the mathematical underpinnings of crucial social and political decision-making.

If the image of mathematics is an unnecessary obstacle which blocks popular access to it, as well as failing to enable full participation in modern democratic society, then it is a great social evil. Of course, changing the image alone does little to address the problem. Instead the nature of the populace’s encounters with mathematics needs to be changed, to be humanised. A semiotic analysis of mathematical language views much of it as coercive (Rotman, 1993). Traditional classroom tasks instruct the learner to carry out certain symbolic procedures; to do, not to think; to become an automaton, not an independent exerciser of critical judgement. This plays a key role in dehumanising mathematics and the learner. Resistance may involve the adoption of a negative stance towards mathematics. This analysis is the subject of my current research and I shall not develop it here. Instead, in this paper I explore how the conceptions or philosophies of mathematics wh...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Contents
• List of figures
• List of tables
• Sources
• Introduction: issues in researching mathematics learning
• Section 1 Culture of the mathematics classroom – including equity and social justice
• Section 2 Communication in mathematics classrooms
• Section 3 Pupils’ and teachers’ perceptions
• Index