Rethinking School Mathematics

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Andrew Noyes

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    Acknowledgements

    Many individuals have been influential in the writing of this book but particular mention needs to be made of my mathematics teacher education colleagues at the University of Nottingham: Peter Gates, Kieran Murphy, Stef Sullivan and Mark Simmons. They have been supportive throughout and willing to read parts of the manuscript at various stages of completion. That is not to say that they would agree with everything that is written herein, but their openness to considering the complexities of mathematics education is sustaining. Hilary Povey and Linda Haggarty are due special mention for their helpful comments on drafts of the manuscript. Many other mathematics educators and scholars have been influential in shaping the ideas of this volume, not least because I am aware of not being alone in thinking that there must be a better way to do mathematics education in schools. Some of these people I know well and others I know only through their writing. Nevertheless I am pleased to be part of what is a global debate about the purposes of school mathematics.

    Secondly I want to thank the four beginning teachers who read the manuscript and were so positive in response: Katia, Anneke, Joe and James. I hope that you are able to make use of the ideas in the book and that you will keep questioning the status quo and, when necessary, go against the flow of tradition and current trend in pursuit of a more socially just, relevant and engaging mathematics education.

    Finally, thanks to Helen Fairlie at Sage who saw something in this book proposal and who has been encouraging and patient throughout.

    Introduction

    This book explores some big questions concerning mathematics education – questions that I think every teacher of mathematics should consider from the outset of their career. These include:

    • Why teach mathematics?
    • Of what use is the mathematics curriculum to different groups of learners?
    • Why are popular views of mathematics often so negative and what might teachers do in an attempt to challenge these?
    • Where has mathematics education come from and where is it going? Who decides?
    • How does/can mathematics contribute to general education; young people's personal, social, spiritual, moral and cultural development?

    The issues with school mathematics education explored herein cannot be resolved once and for all. In our changing world, school mathematics, perhaps more than any other curriculum area, is subject to the competing perspectives and controlling influences of various groups in society. These perspectives are not only intellectual, but ideological, political and historical, and as you read this book you will be invited to reflect upon your own position in relation to these themes. If you are someone concerned about mathematics teaching and learning in schools, you probably have fairly strong beliefs about the subject and how best it should be taught and learnt; though you might not be fully aware of them. This book will help you to explore those beliefs and practices and enable you to recognize that your position is not neutral.

    This is not an impersonal textbook but is intended to provoke a response in you; to make you think about the complex issues of teaching school mathematics. Provoking people can be a risky business but my aim is to make you think, to see the work of teaching mathematics as intellectual, political, cultural and social. Perhaps the book should therefore come with a health warning, namely, that I would not expect every reader to agree with every argument and position. Moreover the classroom activities and ideas in Part II might not interest you. Such difference is good as the world would be rather dull place if we all thought the same things and agreed on everything. Although this is one person's account of where we are now at and where mathematics teaching might go, you will find many references to literature that you can read in order to explore these ideas further.

    When I moved from teaching secondary school mathematics into higher education, initially to work with those learning to become teachers of mathematics, I had some nagging concerns about the nature of what I had been doing as a teacher of mathematics and the impact that it had had upon learners. At that time I had not formulated those concerns into meaningful questions, let alone managed to answer them in a convincing way. On the whole I enjoyed my own schooldays and, being in the top groups for mathematics, had also enjoyed the privilege (and unfairness) of being taught by those considered to be the best teachers in the school. After an uninspiring undergraduate degree, I started to teach in a school with a rich tradition of curriculum innovation. One of the authors of the South Notts. Project (Bell, Rookey and Wigley, 1979) had been a previous head of department and an investigative, creative tradition had continued until the early 1990s when I joined the school. It was in that school and departmental culture that I learned to teach mathematics. I am indebted to my colleagues, and in particular to Sue Pope, my first head of department, who inspired me to keep thinking about innovative pedagogy, ethnomathematics (though I did not know it was that at the time), the value of collaborative learning, rich learning environments, and so on. Many children in that school learned their mathematics in all-ability groups, but this was not to last for long as the 1990s brought huge changes to the culture of UK schools. First the National Curriculum, then national Key Stage tests, Office for Standards in Education (Ofsted) inspections, school league tables and, by the time I left that school, Curriculum 2000 had happened – with disastrous effect for mathematics – and Key Skills (including Application of Number) had come … and nearly gone. That department had in many ways been transformed by the new performance and audit culture of the education market.

    I am not suggesting that this department was perfect by any means, but rather I want to signal that during those transition years, with all of the changes of staff and policy developments, something was lost. The way that mathematics was being experienced in school had shifted and the root cause of this was the so called ‘standards’ agenda. So it was that I began to understand the political nature of knowledge, including mathematical knowledge. I also came to realize that even the liberal mathematics learning culture that I had known was not explicitly addressing relationships between mathematics, society, culture and power; this will be a central theme of this book. So, for the last five years I have been trying to make sense of this, not as an interested observer standing apart from the transitional turbulence of curriculum change and policy experimentation, but as someone still in the mix, caught up in mathematics education at a crucial time in its development, albeit with a particular historical view. I cannot claim that this book is impartial, but then neither can any of us when it comes to understanding our beliefs and practices. However, in exploring possible alternatives and missing ingredients in current school mathematics lessons, I hope that you will be encouraged to understand more fully your own views on mathematics education and the implications that this might have for classroom teaching and learning, and for learners themselves. This reflective journeying will include you mapping out your own part of the terrain in which you have learned and are now teaching mathematics. You will recognize certain freedoms and constraints that you have and are now enjoying, and understanding these features is important in your ongoing development as a teacher of mathematics.

    This book is written primarily for teachers of mathematics but also those who are concerned with the challenges of mathematics education. It is grounded in the context of English school mathematics education and explores popular, ongoing national antipathy to things mathematical. However, the issues considered might well be of interest to mathematics educators in other countries who share similar concerns about the social role of mathematics education.

    Part I explores questions about the nature of school mathematics, with brief recourse to history, references to culture and an overview of the changing national policy context for school mathematics. International comparisons (for example, TIMSS, PISA) have resulted in ‘reform’ education policies in many countries, and so it is not only mathematics educators in England who are questioning the nature of teaching and learning mathematics. Since the late 1980s, a series of policies have sought to improve the teaching and learning of school mathematics. Academics, politicians and education commentators differ in their views on how successful these initiatives have been. Such political work is far from over, as during the writing of this book we will have moved from three- to two-tier General Certificate of Secondary Education (GCSE) system; new pathways through 14–19 mathematics will have been devised and published; students will be learning ‘functional mathematics’ alongside their traditional exam syllabuses. Moreover, we now have a National Centre for Excellence in the Teaching of Mathematics (with a budget of £15 million for its first three years!). Why then, with all of this effort and money expended on improving the quality of school mathematics education, do so many still love to hate mathematics?

    The chapters in Part I will develop your broader understanding of mathematics education and provide some rationale for exploring alternative (but complementary) classroom activities. Part II extends the consideration of the purpose of school mathematics and its relevance for the twenty-first-century citizen by bringing together a collection of classroom approaches and resources that develop some of the themes from Part I. There are plenty of resources that support the development of high-quality teaching of mathematics. Numerous existing books offer plenty of food for thought and the Internet – combined with increasing interactive capability – allows you access to the best ideas (and the worst!) from across the world. However, this overwhelming volume of material can distract from asking the big questions about what we teach in mathematics classrooms, why and how we teach it, and what the outcomes are for the learner. So Part II will focus on ways of thinking about classroom mathematics that take account of social, cultural, political and historical contexts in which mathematics has developed and is now used. Of course these ideas are not new but, hopefully, you will find some of them brought together in a way that might be helpful when exploring alternatives to the current trends of mathematics teaching.

    Although you will be encouraged to develop a broader view of mathematics learning in school you should not expect this to be an alternative textbook. Critically appraising, selecting and redesigning learning tasks should always be a fundamental aspect of teachers' work, so this book is not intended as a ‘how to do’, but rather a ‘how to think’ about the teaching and learning of mathematics. This is an important point but, in an age where mathematical instruction and rule-following takes precedence over education and exploratory learning, this is not always welcomed. For example, David, a beginning teacher said of his Postgraduate Certificate of Education (PGCE) course: ‘I thought that we would learn to teach, or we would be taught to teach mathematics, whereas we're taught to think about teaching mathematics. And one of the problems that I can see with that, is that, basically, that I will teach things the way that I was taught.’ I do not think that the first point made by David automatically leads to the second but the principle that he hints at is important, namely, that how you experienced mathematics has a strong influence over how you might teach it (Noyes, 2004b). What he means by ‘being taught’ is being instructed or trained, which is not what he got. Rather he was being educated to think for himself, to learn the craft of teaching which is intellectual, experimental, creative and always developing. Stuart, another beginning teacher who had struggled with his own mathematics education, described his experiences as follows:

    I don't blame myself only for struggling. I think it were the teaching. You worked from the SMP Y series, the yellow books. You basically walked in and sat down and ‘right, get on with where you left off’. In my time at school I very rarely remember the maths teacher up at the board. I remember the maths teacher I had for my GCSEs, she were just sat at the front desk: all lesson, every lesson. ‘If you've got any problems just come and sit next to me and see me’.

    Stuart found the whole-class interactive components of teaching difficult to develop and although I do not want to suggest a direct causal relationship here it is worth you thinking about how your own learning story is influencing your teaching. One of my hopes for David, Stuart and for you as a reader of this book is that you might develop critical dispositions towards teaching. Brookfield (1987, cited in Scott, 2000: 3) described four dimensions of criticality as:

    • identifying and challenging assumptions
    • challenging the importance of context
    • imagining and exploring alternatives
    • developing reflective scepticism.

    These four dimensions give a good idea of what you might expect to be engaged with through the remainder of this book. First, I hope to challenge your assumptions about the teaching and learning of mathematics and to get you involved in what Ulrich Beck (1994) terms ‘self-confrontation’. That is not to say that you will necessarily change your position but, hopefully, you will be more aware of it and be able to justify your views. You will be thinking about the historical and political context of mathematics education and will be invited to imagine some alternatives. Finally, I hope that you develop that sense of reflective scepticism about the value and effectiveness of school mathematics. This should not mean a wholly negative view of what happens in classrooms, but rather a more realistic, critically informed view, which might inspire you to play a small part in rethinking school mathematics.

    The many excellent resources concerned with the principles and practicalities of constructing high-quality mathematics learning experiences have been referred to above. A large part of mathematics education research has been concerned with understanding mathematical cognition and how teaching can be designed to improve learning. Given this wealth of knowledge, it is interesting that many young people do not have positive experiences of learning mathematics and I want to explore the possible reasons for this. Furthermore, I doubt whether (1) the application of current knowledge is sufficient to affect the kind of transformation in mathematics education that so many people are calling for, and (2) this is the right way to move. A change in direction, or shift in emphasis, is required. That is not to suggest that good teaching and learning (however one defines it) that exists already should be overlooked. On the contrary, as a community of mathematics educators we need to strive for increased quality of mathematics learning for all students in schools. However, the message of this book is that currently there are missing threads in school mathematics and these need to be woven into the fabric of existing quality teaching.

    Indirectly, this is a book written for all those children who have not been served well by traditional mathematics teaching and curriculum; those who have found their mathematics education uninspiring and irrelevant. Whether or not it will serve to help any of these students remains to be seen. That is not to say that my vision for mathematics education is for one group over another, but rather there should be developments that will benefit all young people. One of the arguments you will be considering is that mathematics education in secondary schools should reflect the broader aims of general education. This was Hans Werner Heymann's (2003) argument regarding German mathematics education, which provoked considerable discussion about many of the same issues that you will be thinking about when reading this book. For his context he cautions that ‘the path to instruction oriented more strongly toward general education cannot be enforced from external sources … but can only consist of small steps involving many participants for whom these steps make good sense’ (Heymann, 2003: 84). So this book is my attempt to mark a pathway to a more engaging, equitable, democratic mathematics education and to invite you to consider walking this way – if it makes ‘good sense’.

    When Nel Noddings (2004) briefly outlined a possible alternative direction for US mathematics curricula at the 2004 American Education Research Association's annual conference, she described her vision as ‘an exercise of the imagination’. Hopefully, this book will exercise you in the same way. My aim is to get you thinking about some alternative directions for school mathematics education that might be better suited to the needs of all citizens, particularly those of the majority: school students who will not proceed to study advanced mathematics courses beyond their compulsory schooling. In so doing, this book will hopefully provoke discussion amongst teachers and other mathematics educators about the best way to steer a course towards a mathematics education fit for the young learners of the twenty-first century. And if you do feel provoked but do not agree with what I am arguing for, then I would encourage you to develop your own arguments and position on these matters.

    Finally, some thoughts on how you might engage with the book. Parts I and II are quite different in their aims and style. You will be able to dip into the chapters of Part II or read them through as a whole. They contain ideas for classroom activities. However, they are not tasks that can be lifted straight into your own classrooms; rather they are starting points and processes that you might tailor to your own classroom contexts. The earlier chapters require a more traditional reading but do have a number of questions and reflection points that should help to get you thinking about the issues. I would recommend that you take some time to record something in response to these questions for it is all too easy to skip over them. This need not be in any particular form; diagrams, concept maps, lists would all be fine. You might have emotional responses to the questions or feel that you are not in a position to answer them. All this is fine but, in keeping with the critical focus that I am asking you to adopt, do try and make some sense of these various responses. Most people who read this book will be pre- or in-service teachers of mathematics, which means that the range of questions and tasks are aimed at teachers at quite different points in their careers. So use them as you see fit. Knowing that I never read a book like this when I was a teacher presents me with a certain challenge – to make you want to read on whilst at the same time not oversimplifying the issues. I congratulate you for having got this far!

    I firmly believe that the work of teaching is deeply intellectual and so this book does try to engage you with some big ideas, research and theory. However, I hope that you will find what follows both intellectually stimulating and helpful for critical reflection upon, and the development of, both your teaching and students' learning of mathematics. I would be very happy for you to let me know whether or not I have managed to do this.

    andrew.noyes@nottingham.ac.ukNovember 2006
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