# Bibliography Of Maths Project On Statistics

## Mathematics Education Bibliography

The following bibliography consists of a collection of books and other resources which may be useful references for those studying or otherwise interested in mathematics education. The descriptions of the books are taken, in the main, from that provided on the book cover. The bibliography does not contain details of text-books or other specifically classroom resources. If you feel other sources should be included, you are invited to contact the compiler. Peter Gates School of Education The University of Nottingham Nottingham NG7 2RD, UK peter.gates@nottingham.ac.uk September 1997

Pages for some publishers are listed below:

In addition you can see another bibliography on Semiotics run by Adam Vile at South Bank University, London, at http://www.scism.sbu.ac.uk/~pme/bibliography_of_semiotics.html

### Janet Ainley, Enriching Primary Mathemtics with IT, Hodder and Stoughton, 1996

### Margaret Alic, Hypatia's Heritage, Women's Press, 1986

The author has discovered an entire and fascinating lost heritage of female scientific achievement which had been stolen, plagiarised, simply ignored, or published under male pseudonyms to prevent public opprobrium. Unique and highly acclaimed, the book draws on wealth of biographical and scientific evidence to provide a fascinating and overdue insight into the life and times of outstanding women scientists from prehistory to the late 19th century.

### Michael Armstrong, Closely Observed Children. The Diary of a Primary Classroom, Writers and Readers, 1980

This book is a meticulous description of a year Michael Armstrong spent in Leicestershire in an 'informal' primary school, documenting the intellectual growth and development of children. Going beyond the conventional researcher's role, he made extensive daily notes, and worked with the class teacher to analyse in detail the children's learning. The book draws heavily on the children's own work as well as the records of teachers and of the school.

### Sue Atkinson, Developing a Scheme of Work for Primary Mathematics, Hodder and Stoughton, 1996.

### Carol Aubrey, Mathematics Teaching in the Early Years. An Investigation of Teachers' Subject Knowledge, Falmer Press 1996

This book picks up and examines issues of teachers' subject knowledge - issues which are often trivialised by the media and politicians. It offers a detailed investigation of the construction of early years mathematics, in school and out, and brings together and synthesises a number of fields, conceptual and empirical and provides a springboard for a wide-ranging exploration of pedagogical subject knowledge. Reviewing current research on the learning and teaching of early number and experimental curricula across the world, this book contributes to the current debate concerning the teaching of number and baseline assessment. It carries important implications for initial teacher education and the early years curriculum.

### John Backhouse, Linda Haggarty, Susan Pirie and Jude Stratton, Improving the Learning of Mathematics, Cassell, 1992

At the heart of this book is the belief that teaching mathematics should start with learners as individuals. The authors show how teachers can encourage a positive attitude to mathematics and a capacity in the pupils to take responsibility for their own progress.

### Alan Bell, John Costello and Dietmar Kücheman, A Review of Research in Mathematical Education. Part A: Research on Learning and Teaching, NFER-Nelson, 1983

This is the first of three comprehensive reviews of research commissioned by the Cockcroft Committee into different aspects of mathematics education. It draws together for the first time the 'hard evidence' in debates on teaching effectiveness and learning difficulties, pointing to those changes and developments in school mathematics which should take priority.

### Nadine Bednarz, Carolyn Kieran and Lesley Lee, Approaches to Algebra. Perspectives for Research and Teaching, Kluwer Academic Publishers, 1996.

In the international research community, the teaching and learning of algebra have received a great deal of interest. The difficulties encountered by students in school algebra show the misunderstandings that arise in learning at different school levels and raise important questions concerning the functioning of algebraic reasoning, its characteristics, and the situations conducive to its favorable development. This book looks more closely at some options that aim at giving meaning to algebra, and which are considered in contemporary research : generalization, problem solving, modeling, and functions. Salient research on these four perspectives addressed the question of the mergence and development of algebraic thinking by a dual focus on epistemological (via the history of the development of algebra) and didactic concerns. Through the theoretical issues raised and discussed, and the indication of given situations which can promote the development of algebraic thinking, Approaches to Algebra will be of interest and value to researchers and teachers in the field of mathematics education.

### David Bird and Michael Hiscox, Mathematics in School and Employment: a study of liaison activities, Methuen Educational, 1981

This is a study of the Schools Council project, Mathematics and the Young Entrant to Employment, which looked at the work of thirty-three liaison groups. It concentrates on the processes of the work of the groups, rather than the outcomes. It brings out the methodological issues, illustrating them with many examples.

### Marion Bird, Mathematics for Young Children: an Active Thinking Approach, Routledge, 1991

This book shows how children as young as four and five of all abilities can be encouraged to carry out their own mathematical explorations whilst at the same time covering the content of a prescribed curriculum. It includes a section of case studies from the author's own work with young children and a wide range of examples of children's work.

### Alan Bishop and Marilyn Nickson, A Review of Research in Mathematical Education. Part B: Research on the Social Context of Mathematics Education, NFER-Nelson, 1983

This is the second of three comprehensive reviews of research commissioned by the Cockcroft Committee into different aspects of mathematics education. It considers the affect of social factors on the teaching and learning of mathematics and looks at the conclusions of research into issues such as teacher perceptions and pupil attitudes.

### Alan Bishop, Mathematical Enculturation; A Cultural Perspective on Mathematics Education, Kluwer Academic Publishers, 1988

This book breaks new ground in the area of Mathematics Education by taking as its focus the idea of Mathematics as a cultural product. This idea is both simple and profound. It is simple because common sense tells us that all knowledge must be culturally produced. But anthropological and cross-cultural research in the last 20 years have generated a wealth of data which do not just substantiate this common sense view, they also enable us to really understand the meaning and importance of the idea. The book demonstrates the substantiation and the meaning of Mathematics as a cultural product. It draws on a wide variety of sources and references, and integrates this literature into a new conceptual schema. Equally the idea is a profound one because of its potential for development within Mathematics Education. The author is not content with merely surveying the 'cultural' field, he also explores the significant aspects of this development. He presents a new curriculum structure, which generates new ideas as well as supporting the relevant ideas and procedures that already exist. But enculturation is not just about curriculum content, it is also about process, and the mathematical enculturation process is well explored in the book. Finally, and naturally, there are several important implications for mathematics teacher preparation and for the whole process of teacher education, which are discussed in the final chapter.

### Jo Boaler, Experiencing School Mathematics. Teaching Styles, Sex and Setting, Open University Press, 1977.

This book is the first of its kind to provide direct evidence for the effectiveness of ÔtraditionalÕ and ÔprogressiveÕ teaching methods. It reports upon extensive case studies of two schools which taught mathematics in totally different ways. Three hundred students were followed over three years and the interviews that are reported in this book give compelling insights into what it meant to be a student in the classrooms of the two schools. The different school approaches are compared and analysed using student interviews, lesson observations, questionnaires given to students and staff and a range of different assessments including GCSE examinations. Questions are raised about (a) the effectiveness of different teaching methods in preparing students for the demands of the Ôreal worldÕ and the 21st Century; (b) the impact of setted and mixed ability teaching upon student attitude and achievement and (c) gender and learning styles. This book draws radical new conclusions about the ways that traditional teaching methods lead to limited forms of knowledge that are ineffective for non-school settings.

### Rafaella Borasi, Learning Mathematics Through Inquiry, Cassell Education, 1994.

This book relates the experience of the author working with two secondary school girls who did not like mathematics, and how they began to see themselves as 'real mathematicians' engaged in original mathematical explorations .She raises issues about the teaching and learning of mathematics.

### Tom Brissenden, Mathematics Teaching. Theory and Practice, Harper and Row, 1980.

This book offers a programme of self-evaluation and self-improvement for teachers who are studying alone or in a group. Lesson extracts and examples are drawn from actual experience and are used to develop a theory of classroom interaction in teaching mathematics. The exposition is accompanied by numerous suggestions for practical activities and attention is given to exercise lessons and methods for stimulating mathematical discussion. Lesson plans, schemes of work and mathematical situations are also discussed.

### Tom Brissenden, Talking About Mathematics, Basil Blackwell, 1988.

This book offers guidance to teachers looking for alternatives to the traditional teacher-led question and answer pattern. It offers a programme of self-development for primary teachers in the skills needed for discussion-based learning of mathematics; supported and illustrated by practical ideas, activities and case studies. The book offers suggestions for developing and conducting small group activities and planning suitable mathematical situations, including work with calculators and microcomputers. Later chapters examine how staff can share ideas, classroom organization, parental involvement and the contribution of mathematics to a school language policy.

### Tony Brown, Mathematics Education and Language. Interpreting Hermeneutics and Post-structuralism, Kluwer Academic Publishers, 1997

Contemporary thinking on philosophy and the social sciences has been dominated by analyses that emphasise the importance of language in understanding societies and individuals functioning within them; important developments which have been under-utilised by researchers in mathematics education. This book reaches out to contemporary work in these broader fields; drawing on original sources in key areas such as Gadamer and Ricoeur's development of hermeneutics, Habermas' work in critical social theory, Schutz's social phenomenology, Saussure's linguistics and the post-structuralist analysis of Derrida Foucault an Barthes. Through examining the writings of these major thinkers it is shown how language is necessarily instrumental in developing mathematical understanding; but a language that is in a permanent state of becoming, resisting stable connections to the ideas it locates. The analysis offered extends from children doing mathematics to teachers inspecting and developing their own professional practices.

### Leone Burton, Thinking Things Through, Basil Blackwell, 1988.

A practical guide for teachers to problem solving and investigational work in mathematics. As well as providing assistance and advice, 35 problems are presented with notes on problem resolution and extension.

### Leone Burton (Ed), Girls Into Maths Can Go, Holt, Reinhart and Winston, 1986.

The significance of gender for mathematics education is examined from a number of different perspectives by a wide range of contributors. Section one, 'What is the problem?', discusses the influence of socialisation and learning and teaching styles on girls' attitudes and attainment in mathematics. The second section, 'What can be done?', offers some examples of effective practice in classrooms and schools that encourage girls' participation in mathematics education. An appendix, 'Herstory of mathematics', offers a brief review of women's contribution to the subject. Throughout the aim is to heighten educators' awareness of the issues and of the strategies that can be used at all levels of education. The Book accompanies the Open University pack 'Girls into Maths' (PM647)

### Leone Burton (Ed), Gender and Mathematics: an International Perspective, Cassells Educational, 1990.

This collection of papers, from the work of the International Organisation of Women and Mathematics Education, contains original, unpublished papers which reflect the range of recent studies on the links between gender and the learning of mathematics; the book presents new and provocative material for students and teachers of mathematics. It also provides insights into methodologies and questions that are relevant to different cultures. The book is divided into four sections - gender and classroom practice, gender and the curriculum, gender and achievement and women's presence. In identifying trends that are replicated as well as those that appear to be contradicted, it draws attention to the complexity of the issues involved in providing equal opportunities in mathematics education.

### Leone Burton and Barbara Jaworski (eds), Technology in Mathematics Teaching - a bridge between teaching and learning, Chartwell-Bratt, 1995.

Blackboard and chalk are no longer the sole technology available to the mathematics teacher. In many classrooms, students might be expected to learn through using computers, calculators, film, television, video and other media. However, very often, technology is introduced into mathematics classrooms without due consideration of its role, or its impact on learning and teaching. This book addresses issues raised by the introduction of technology into the teaching and learning of mathematics. It uses the metaphor of technology acting as a bridge between the teacher's planning and the learner's developing understanding. Chapters address the learning of mathematiccs at every level, primary to tertiary. The technologies discussed are substantially those of the computer and calculator but reference is made to others including video and film.

### Leone Burton (ed), Who Counts? Assessing Mathematics in Europe, Trentham Books, 1994.

This is a 'state-of-the-art' book on assessment in mathematics in ten European countries, with a focus on gender differences in achievement. Few of the countries officially differentiate their assessment by gender despite the findings from those that do, indicating that attainment is indeed affected by gendered responses to learning and assessment styles. No country differentiated by race, again despite indications from the USA that results are racially differentiated. The editor sets the cross-cultural context comparing the ten different European systems, and their assessment styles, with what is known about differentiated outcomes. She outlines the data collection procedures needed so that differentiation in mathematics attainment can be effectively monitored and she speculates on the researchable questions that would bring European education closer to a social justice approach to assessment in mathematics.

### Guy Claxton, Live and Learn. An Introduction to the Psychology of Growth and Change in Everyday life, Harper and Row, 1984.

Although not specifically about mathematics Education, this book is an introduction to the psychology of learning which takes a fresh approach. It sees learning in the context of everyday experience, and looks at the kinds of things people learn, how they learn them and what makes it easier or more difficult for learning to take place. It points to the opportunities that arise for learning in the course of everyday existence, and shows how people deal with these opportunities or get in the way of their own learning. The book tells a coherent story of learning in which the psychology of Skinner, Rogers, Bruner, Kelly and Piaget each has its place. Rather than recapitulate there theories, the book aims to show how they enhance, illuminate and complement each other.

### David and Wendy Clemson, Mathematics in the Early Years, Routledge, 1994.

This clear and readable book takes the reader from debates about how children learn and what children know and can do when they start school; through to a discussion of how mathematics can be managed, assessed and evaluated in the school and classroom. Linking these two parts of the book is a section on the subject of mathematics itself, from which the non-specialist reader can gain a view of what mathematics is, what needs to be thought about in planning and offering a curriculum and the special dilemmas faced in teaching and learning mathematics as a subject. A bank of case studies offers an opportunity to see mathematics in action in a variety of classrooms.

### Paul Cobb (ed), Learning Mathematics. Constructivist and Interactionist Theories of Mathematical Development, Kluwer Academic Press, 1994.

The common theme linking the six contributions to this book is the emphasis on students' inferred mathematical experiences as the starting point of the theory building process. The focus of the first five chapters is primarily cognitive and addresses the process by which students construct increasingly sophisticated mathematical ways of knowing. The conceptual constructions addresses include multiplicative notions, fractions, algebra, and the fundamental theorem of calculus. The primary goal in each of these chapters is to account for meaningful mathematical learning - learning that involves the construction of experientially-real mathematical objects. The theoretical constructs that emerge from the authors' analyses of sudents' mathematical activity can be used to anticipate problems which might arise in learning-teaching situations, and to plan solutions to them. The issues discussed include the crucial role of language and symbols, and the importance of dynamic imagery. The remaining chapter complements the others by bringing intothe social dimensions of mathematical development. It focuses on the negotiation of mathematical meaning, thereby locating students in ongoing classroom interactions and the classroom micro-culture. Mathematical learning can therefore be seen to be both an individual and a collective process.

### Barry Cooper, Renegotiating Secondary School Mathematics, Falmer Press, 1985.

Over the 1970s and 1980s there has been increased interest in the issues surrounding curriculum control not only by specialists but by curriculum sociologists and political scientists. This prompted this book which investigates the ways in which different educational systems have promoted and reacted to curriculum innovation since the early 1960s. In the late 1950s and early 1960s a major reform of the content of selective secondary school mathematics was carried out throughout Europe and the USA. In the climate of crisis which was believed to exist within the subject elements of 'modern mathematics' were introduced into new textbooks and syllabuses and much of the content of existing mathematics seemed destined to disappear,. Recently, a similar perception of crisis developed and various individuals and groups have again successfully argued that another phase of redefinition of mathematics is necessary. Each new wave of redefinition of mathematics has failed to adequately recognize the importance of previous reconstructions and struggles over the definition of the subject. In such a situation it is vitally important that the history of these subject events is recovered so that new initiatives can be viewed in the light of previous experience. This book focusses on some of the key aspects of the historical reconstruction of school mathematics in the post-war period. It should be noted though that the criteria for selectivity in the account are primarily sociological and the concern above all theoretical. Hence the account does not set out to be a detailed history of school mathematics and should not be judged as such. However this work is important as a broadly conceived notion of curriculum history herein.

### Michael Cornelius (ed), Teaching Mathematics, Croom Helm, 1982.

Mathematics is indisputably an essential part of the curriculum. It is an important subject also both in terms of the number of teachers involved in teaching it and the amount of time devoted to it in the school timetable. However, at a time when there is a shortage of qualified teachers of mathematics, there is no general agreement on crucial issues of how the subject should be taught, and they are bombarded with a bewildering array of materials and ideas which they have little time to assimilate. This book provides a survey of the problems which face teachers of mathematics in the classroom. Issues discussed include teaching at primary and secondary levels, learning difficulties and the place of calculators and games in the classroom. Each chapter reviews a key problem and offers practical guidance in helping both the student and established teacher to teach the subject in a professional and intelligent way.

### John Costello, Teaching and Learning Mathematics 11-16, Routledge, 1991.

The teaching of mathematics in secondary schools has undergone major changes in the last five years, as has the education of secondary mathematics teachers. This book analyses the effects of these changes. It focusses on the demands of the National Curriculum, and also explores developments such as individualised learning, investigations, microcomputers and the GCSE. It discusses social issues such as multi-cultural education, gender and provision for children with special needs, and shows how they concern the mathematics teacher.

### Harry Daniels and Julia Anghileiri, Secondary School Mathematics and Special Needs, Cassell Education, 1994.

This is a book for mainstream mathematics teachers, particularly those with responsibility for "whole school" policies. It will also be valuable for special needs teachers in mainstream and special schools. The book falls into three main sections: Part 1 looks at different justifications for teaching mathematics to children with special needs. In part 2 the authors consider the control of the teaching and learning of mathematics. Part 3 examines issues in the learning of mathematics, including issues in psychology.

### Andrew Davis and Deirdre Pettitt, Developing Understanding in Primary Mathematics, Falmer Press, 1994.

This is not, on the whole, a book for educational theories. It is addressed to the reflective primary teacher who wants learning mathematics to go beyond the acquisition of rules and facts; to avoid a predominance of solitary written exercises; and who wishes to relate mathematics in motivating and educationally appropriate ways to other curriculum areas. The contributors see mathematics as a subject which owes its existence to people interacting in groups and societies, and which is constantly being developed, reinvented or reconstructed from one generation to the next. The book contains a wide variety of practical classroom applications of this view, including extensive discussion of the potential role of story, drama, pattern and children's drawing in developing mathematical understanding, given appropriate teacher roles.

### Philip J Davis & Reuben Hersh, The Mathematical Experience, Harvester Press, 1981.

The authors offer their view of science and demonstrate that 'mathematical truth, like other kinds of truth is fallible and corrigible'. They have created a work of history, reflection, exposition, and philosophy which presents both the complexities and the beauty of a subject which is one of the most pervasive and esoteric of human endeavours.

### Brenda Denvir, Chris Stolz and Margaret Brown, Low Attainers in Mathematics 5-16; Policies and Practices in Schools, Methuen Educational, 1982.

This report of the Schools Council Low Attainers in Mathematics Project is concerned primarily with those pupils in the 5-16 age range who fall, for whatever reason, into the lowest 20 per cent of mathematical attainment. In examining the various causes and types of low attainment, and the overall problem of teaching these pupils, the project team consulted schools, working groups research projects and advisers. Each chapter deals with a different aspect of the problem and ends with points for discussion to assist authorities, schools and teachers to consider how to help their low attaining pupils.

### Charles Desforges and Anne Cockburn, Understanding the Mathematics Teacher: A Study of Practice in First Schools, Falmer Press, 1987.

This book investigates the mathematics teaching in a selection of First Schools. The authors set out to discover why there is such a huge gap between what mathematics education experts say should be taught, and what really is taught The authors have made a detailed analysis of the mathematics teaching of seven experienced infant teachers. They found that all of them used commercial published schemes and that this lead to the teaching of procedures rather than the exploration of mathematics. In trying to understand why this happened they concluded that the difficulties that teachers laboured under are not sufficiently considered by education experts. (From a review by Sheila Ebbutt which appeared in the Times Educational Supplement)

### Linda Dickson, Margaret Brown and Olwen Gibson, Children Learning Mathematics: A Teacher's Guide to Recent Research, Holt Educational.

In this book, the authors try to acquaint primary and secondary teachers with recent knowledge concerning how children come to learn or fail to learn mathematics. Relevant research in the UK, the US and other countries is summarised. The book includes quotes and examples of children's responses which give valuable insight into complexity of the learning process. It is divided into 4 sections spatial awareness, measuring, number and language.

### Margaret Donaldson, Children's Minds, Fontana, 1987.

This book has become a classic inquiry into the nature of human thought. Margaret Donaldson shows how thought and language originally depend upon the interpersonal contexts within which they develop, and how, given the support of such contexts, children are already skilled thinkers and language users by the time they come to school. However, when school begins, success depends on the development of new modes of thinking. Margaret Donaldson suggests what these entail and analyses the difficulties which they present. She claims that we have not fully understood the nature of these difficulties and so have not known how to best help children with them. She suggests a range of strategies that can be used arguing that the way in which reading is taught is even more important that we have supposed.

### Paul Dowling and Richard Noss, Mathematics Versus the National Curriculum, Falmer Press, 1990.

This book outlines fundamental theories and practical objections to the national curriculum, with mathematics as its focus. It looks at the NC's inherent curricular shortcommings and the role teachers have been assigned in its development. The NC is perceived as the mechanism by which market forces are to be introduced to education, and rather than offering teachers room to manoeuvre, the NC is seen as a process of deskilling and a means of removing any remaining autonomy.

### Kevin Durkin and Beatrice Shire, Language in Mathematical Education. Research and Practice, Open University Press, 1991.

Mathematical knowledge is transmitted, explored and advanced via language: face-to-face, in printed text and in other media. Children and their teachers write, read, talk and listen mathematics. In the diverse contexts of mathematical communication and instruction, language can help the pupil to gain insight, yet it can serve also to obscure and impede the learning process. This book presents current perspectives and reviews the key issues that are attracting the attention worldwide of researchers and educators involved in mathematical development and teaching. It brings together in one volume important work from an international array of specialists in this rapidly growing field. Authors provide concise and accessible accounts of their own and others' work, and each chapter draws out the practical implications in clear terms which will inform and guide teachers. Following introductory reviews, there are sections exploring language and early experiences of number; language and meanings in mathematical education; word problems; the uses of discussion; language disabilities and mathematics; and cross-linguistic issues. Mathematics education begins and proceeds in language, it advances or stumbles because of language and its outcomes are often assessed in language. This book examines how these processes occur, and provides illuminating analyses of the benefits, ambiguities and complexities of language in the maths curriculum.

### Suzanne Edwards, Managing Effective Teaching of Mathematics 3 - 8, Paul Chapman Publishing, 1998.

This book, written for teachers and student teachers working with children 3-8, presents a comprehensive study of mathematics curriculum management in nursery and infant schools. It aims to highlight and clarify the different elements that have been identified in recent reports such as �good practice� in managing the teaching and learning of mathematics in the early years of schooling. The book includes the formulation of a mathematics policy, the role of the mathematics co-ordinator, resource management, partnerships with parents, planning process, classroom and lesson management, assessment, managing differentiation record-keeping and reporting.

### Derek Edwards and Neil Mercer, Common Knowledge; The Development of Shared Understanding in the Classroom, Open University Press, 1987.

This book is based on a study of video-recorded lessons with groups of 8- to 10- year-olds. It suggests that classroom communication takes place against a background of implicit understanding, some of which is never made explicit to the pupil, whilst there develops during the lessons a context of assumed common knowledge between pupil and teacher about what has been said, done or understood. The authors present a considerable amount of classroom dialogue which they subject to considerable intensive scrutiny.

### Paul Ernest (Ed), Mathematics Teaching - The State of the Art, Falmer Press, 1989.

Mathematics teaching is in a state of ferment and change. New technologies like the microcomputer and new assessments such as the G.C.S.E. examination demand radically different patterns of teaching. At no time in the history of mathematics teaching have there been changes on so broad a front or so rapid a growth of knowledge as today. This book offers an overview of these changes from those at the cutting edge of development and a state of the art account of research in mathematics education.

### Paul Ernest, The Philosophy of Mathematics Education, Falmer Press, 1991.

Although many agree that all teaching rests on a theory of knowledge, there has been no wide-ranging in-depth exploration of mathematics for education. Building on the work of Lakatos and Wittgenstein amongst others, this book challenges the notion that mathematical knowledge is certain, absolute and neutral, and offers instead an account of mathematics as a fallible social construction. One of the outcomes of this book is a model of five educational ideologies each with its own epistemology, values, aims, history and social group of adherents. An analysis of the impact of these groups on the British National Curriculum offers a critique of the NC revealing the questionable assumptions and interests upon which it rests. The book finishes on an optimistic note, arguing that an appropriate pedagogy allows the achievement of the radfical aims of educating confident problem posers and solvers who are able to critically evaluate the social uses of Mathematics.

### Paul Ernest (ed), Mathematics, Education and Philosophy, Falmer Press, 1994.

This book addresses the central problem of the philosophy of mathematics education: the impact of the conceptions of mathematics on educational practice. It also embodies a far reaching interdisciplinary enquiry into philosophical and reflective aspects of mathematics and mathematics education. It combines falibilist and social philosophies of mathematics with exciting new analyses from post-structuralist and post-modernist theories, offering both a reconceptualisiation and a critique of mathematics and mathematics education. The outcome is a set of new perspectives which bring out the human face of mathematics as well as acknowleging its social responsibility.

### Paul Ernest (ed), Constructing Mathematical Knowledge: Epistemology and Mathematical Knowledge, Falmer Press, 1994.

The emphasis in this book is on epistemological issues encompassing multiple perspectives on the learning of mathematics, as well as broader philosophical reflections on the genesis of knowledge. It explores constructivist and social theories of learning and discusses the role of the computer in the light of these theories. It brings new analyses from psychoanalysis, hermeneutics and other perspectives to bear on the issues of mathematics and learning. It enquires into the role of language. Finally, it relates the history of mathematics to its teaching and learning.

### Elizabeth Fennema and Gilah Leder, Mathematics and Gender, Teachers College Press, New York, 1990.

In this collection, written by researchers with a long standing comittment to the field, the authors report on various studies that have increased understanding of why females learn different kinds and amounts of mathematics. The longitudinal nature of a number of these studies and the cross-cultural perspective provided by the inclusion of American and Australian research make this edition noteworthy. In addition to presenting a background to the debate on gender differences, this book also provides insight into the interactions between internal beliefs and external influences, and how they in turn affect the learning of mathematics.

### David Fielker, Extending Mathematical Ability, Hodder and Stoughton, 1996.

This book helps teachers work with more advanced mathematical ideas to the most challenging pupils by promoting discussion through interactive whole class teaching, enhancing the experience of mathematics for all pupils. Strategies are given for developing suitable activities, which cover all aspects of mathematics, and these are presented in an accessible way that provides the understanding necessary to teach them effectively.

### Ann Floyd (Ed), Developing Mathematical Thinking, Addison-Wesley, London, 1981.

This is the reader which accompanies the Open University course Developing Mathematical Thinking. It is aimed at principally the non-specialist teacher of mathematics. It will be of value to students of education courses and practicing teachers who are responsible for teaching children between the ages of five and fourteen. It contains selected articles on the theme of how children learn to investigate and solve mathematical problems.

### Marilyn Frankenstein, Relearning Mathematics. A Different Third R - Radical Maths, Free Association Books, London, 1989.

Why do so many people have problems with maths? How can teachers help students to deal with 'maths anxiety'. Intended for students who have been made to feel a failure at maths, this textbook overcomes learning obstacles by developing methods that help students. Using examples from other disciplines such as politics and history, the text encourages students to reflect on their own learning, to analyse their work for error patterns, and to look for hidden messages in maths exercises. They learn to approach problems more carefully by first understanding, comparing and rounding numbers, before solving problems involving addition, subtraction, multiplication or addition. At a time when government demands are narrowing educational approaches this book provides an alternative - a critical maths 'literacy'. It sets maths problems in the context of issues around race, gender and class.

### Hans Freudenthal, Weeding and Sowing, D. Reidel Publishing Co.,1978.

After raising and answering fundamental quesitons on science and on education, the author goes on to question the scientific character of what presents itself, and what is accepted as educational theory today. He claims that a science of teaching must start with a science of teaching something and, trusting historical analogies, he believes that the first subject to develop a science of education will be mathematics. He explains anew his previous ideas on levels and discontinuities in learning processes and adds new ideas which are illuminated by a wealth of examples.

### Hans Freudenthal, Didactical Phenomenology of Mathematical Structures, Kluwer Academic Publishers, 1983.

### Hans Freudenthal, Mathematics as an Educational Task, D Reidel Publishing Company, 1973.

This book is a comprehensive philosophy of mathematics education that analyses a large part of mathematics teaching in a way which integrates the scientific and the teaching aspects. The teaching theory of levels in the learning process has never been applied to mathematics learning in such depth as in this book. The applications and social implications of mathematics are also deal with. The author places mathematics into its historical, developmental and social context, and mathematics education into the context of general education, as it has developed and still continues to develop. At the summit of scientific production it is generally acknowledged that mathematics is an activity rather than a well-established stock of knowledge. The author's teaching philosophy says that this applies to all levels of the learning process. These levels have been analysed in numerous examples. This general theory is followed by an analysis of several crucial concepts and domains. In contrast with the modern tendency to over-stress the cardinal aspects, the number concept is scrutinised from a wide spectrum of logical and developmental aspects.he author shows what the uses of set theory are, while at the same time warning against its misuse. Geometry is no longer regarded as being the the summit of deductivity; its task is now seen as that of exploring the space we live in, and of offering a field of mathematical activity.

### Hans Freudenthal, Revisiting Mathematics Education: China Lectures, Kluwer Academic Publishers, 1991.

Hans Freudenthal died on 13 October 1990 and this book represents his last major contribution to the field of mathematics education. In it Freudenthal revisits and reanalyses some of his earlier ideas, with the aim of comprehensiveness being achieved, as he says, "not by compiling, but selecting and streamlining, by including essentials and by eliminating contradictions". This book is therefore the definitive Freudenthal. The book is in three parts, which cover the three foci of his academic interests. "Mathematics phenomenologically" describes and analyses the mathematical context within which the "didactical principles" of the second part are systematically developed. Finally, and now equiped with well-shaped language and constructs, the reader is guided through "The landscape of mathematics education".

### Martin Gardiner, AHA! Insight, Scientific American, 1978.

This book presents a considerable number of problems from various branches of mathematics; geometry, number, logic. It illustrates how insight and a creative approach may shed light on their solution.

### Ernst von Glasersfeld (ed), Radical Constructivism in Mathematics Education, Kluwer Academic Publishers, 1991.

All too frequently, the present ways of teaching mathematics generate in the student a lasting aversion against numbers, rather than an understanding of the useful and sometimes enchanting things one can do with them. Parents, teachers, and researchers in the field of education are well aware of this dismal situation, but their views about what causes the widespread failure and what steps should be taken to correct it have so far not come anywhere near a practicable consensus. The authors of the chapters in this book have all had extensive experience in teaching as well as in educational research. They approach the problems they have isolated from their own individual perspectives. Yet, they share both an over-all goal and a specific fundamental conviction that characterized the efforts about which they write here. The common goal is to find a better way to teach mathematics. The common conviction is that knowledge cannot simply be transferred ready-made from parent to child or from teacher to student, but has to be actively built up by each learner in his or her own mind. This book is intended to clarify how the didactic attitude changes when the constructivist theory of knowing is put into practice, and what results have been attained in doing so. Rather than forecasts of what might be achieved in the future, the papers assembled in this book report on experiments and implementations that have actually been carried out.

### Ernst von Glasersfeld, Radical Constructivism: A Way of Knowing and Learning, Falmer Press, 1994.

In the past decade or two, the most important theoretical perspective to emerge in mathematics education has been that of constructivism. This burst onto the international scene at the controversial 11th International Conference on the Psychology of Mathematics Education in Montreal in the summer of 1987. No one there will forget von Glasersfeld's authoritative plenary presentation on radical constructivism, and his replies to critics. Ironically, the conference, at which attacks on radical constructivism were perhaps intended to expose fatally its weaknesses, served as a platform from which the theory was launched to widespread international acceptance and approbation. Radical constructivism is a theory of knowing that provides a pragmatic approach to questions about reality, truth, language and human understanding. It breaks with the philosophical tradition and proposes a conception of knowledge that focuses on experiential fit rather than metaphysical truth. It claims to be a useful approach, not the revelation of a timeless world. The ten chapters of this book present different facets in an elegantly written and thoroughly argued account of this epistemological position, providing a profound analysis of its central concepts.

### Maria Goulding, Learning to Teach Mathematics, David Fulton Publishers, 1997.

This is a text book for student and newly qualified teachers of secondary mathematics and their school-based mentors. It aims to link the practical experience gained in school placements with the theoretical background available from reading and research. Guidance is drawn from accounts of experiences in classrooms, giving students and newly qualified teachers practical ideas for planning and evaluating pupils' learning, and insights into their own development as new teachers.

### Alan Graham, Help your Child with Maths; A Guide for Parents, Fontana, 1985.

As the name suggests this book is directed at parents who want to become involved in their children's education; for any parent keen to help their child with maths, but not sure how to go about it. It does however have much that would commend it to teachers of pupils in the 0-12 age range. It provides a clear and logical explanation of basic maths, including numbers, decimals, fractions and percentages. It highlights the particular difficulties children have in learning maths and gives numerous examples of the sort of conversations that can be held with children in the kitchen, the supermarket or on a long journey which will help to improve their mathematical awareness and understanding.

### Linda Haggarty, New Ideas for Teacher Education, Cassell, 1995.

The author of this book takes the opportunity to look at ways of moving forward in teacher education within the current climate of change. Research evidence already available points to a new model of teacher education that requires schools and universities to work together to plan an integrated and coherent set of experiences for student teachers. Within this model, university tutors and school mentors have distinctive roles in helping students to learn about teaching in an active and questioning way. The context for discussion throughout the book is that of mathematics. The first part of the book explores the literature relating particularly to mathematics; mathematics mentors and beginning teachers provide the focus of study in the second part; and a mathematics programme is developed in part 3. However, the model, the findings and suggestions have universal relevance for everyone involved in initial teacher education

### Gila Hanna (ed), Towards Gender Equity in Mathematics Education - An ICMI Study, Kluwer Academic Publishers, 1996.

This book consists of original articles that develop further issues pertaining to gender equity in mathematics education. The premise - that there is no physical or intellectual barrier to the participation of women in mathematics, science, and technology - provides the starting point for analyses and discussion. The authors explore the attitudinal and societal/structural reasons for the gender imbalance in these fields and look at foci for change, including curriculum and assessment practices, classroom and school cultures, and teacher education programs. A major part of the book comprises a series of detailed descriptive studies of education systems across the world from the perspective of mathematics and gender equity issues.

### Mary Harris (ed), Schools, Mathematics and Work, Falmer Press, 1991.

This book appears at a time when the relationship between school and work mathematics is a live and controversial issue, with the values of industry tending to override those of education. It challenges simplistic and narrow views of school mathematics education and offers evidence for more positive and substantial ways forward. The book illustrates the wide range of issues that`should affect any consideration of the relationship between mathematics education and the use of mathematics for practical and work purposes. The concept of ethnomathematics is discussed, and biases, including those of gender, in the context of mathematics and mathematics education are exposed. The views of researchers, users, training boards and employing organisations are presented, and international contributions provide a broader perspective through which to assess existing practices and to suggest alternatives.

### Kath Hart (Ed), Children's Understanding of Mathematics: 11-16, John Murray, 1981.

This book makes available the results of five years' research by the mathematics team of the programme "Concepts in Secondary Mathematics and Science" (CSMS). It is aimed at establishing hierarchies of understanding for eleven major topics which appear in secondary school mathematics curriculum. Written both for teachers and for developers of mathematics curicula, it gives insights into the difficulties children face when attempting to solve mathematical problems. Ten of the chapters deal with particular mathematical topics. Each chapter describes the methods used by children to solve problems, errors made and the resulting levels of understanding in the hierarchy, before going on to point out suggestions for teaching that topic.

### Kath Hart (Ed), Children's Mathematical Frameworks. 8 - 13: A Study of Classroom Teaching, NFER-Nelson, 1989.

An early grasp of fundamental mathematical ideas - such as place value, fractions, measurement, equations and ratio - is essential to children's later mathematical development; all of these topics are identified in the National curriculum. This book provides primary and secondary mathematics advisers, teachers and student teachers with comprehensive information on how mathematical concepts are commonly presented and interpreted, and the problems children encounter in attempting to understand and apply them. This book also provides assistance in assessing levels of performance and can serve as an aid in determining programmes of study for Key Stages 2 and 3 in the National Curriculum. The authors describe in detail children aged between 8 and 13 years as they learn and generalize specific mathematical relationships. Interviews with children are used to illustrate important aspects of their thinking during key stages of learning and observations made during the lessons are noted. As well as highlighting areas of difficulty, these descriptions enable the reader to perceive the essence of how mathematics is understood by children as they move from introductory activities to more formal representations.

### Derek Haylock and Anne Cockburn, Understanding Mathematics in the Lower Primary Years, Paul Chapman Publishing, 1997.

Written for teachers and student teachers working with children 4-8, this book provides them with a clearer view of the mathematical material they deal with in the classroom. To develop the understanding, he book examines fundamental mathematical ideas such as number, and number operations, division, transformation and equivalence, comparison and measurement, pattern and generalisation and concepts associated with shape and space. Particular attention is given to the use of symbols in mathematics and the importance of networks of connections between symbols, concrete experiences, language and pictures.

### Derek Haylock, Mathematics Explained for Primary Teachers, Paul Chapman Publishing, 1995.

### Derek Haylock and Anne Cockburn, Understanding Early Years Mathematics, Paul Chapman Publishing, 1989.

This book is for those who teach mathematics to children in Infant or Primary schools and who wish to have a clearer understanding of the mathematical material they deal with in the classroom. Teachers concerned about implementing the National Curriculum proposals for mathematics will recognise the importance of a thorough understanding of the ideas which underlie the attainment targets and programmes of study. This book aims to develop this understanding by examining such fundamental mathematical ideas as number, addition, subtraction, multiplication, division, transformation and equivalence, measurement and concepts associated with shape and space. Particular attention is given to the use of symbols in mathematics and the importance of building connections between symbols, concrete experience, language and pictures.

### Derek Haylock, Teaching Mathematics to Low Attainers, 8 - 12, Paul Chapman Publishers, London, 1991.

This is a book for teachers and students concerned about the problems of those children in primary and middle schools whose attainment compares unfavourably with that of their contemporaries. Drawing on his classroom-based research with low attaining children in the age range 8 - 12 years, the author discusses the factors associated with their low attainment and proposes a strategy for dealing with them. The author shows that many low-attaining pupils will surprise their teachers with what they can do when they are put in situations where they have to use mathematics to make something happen. The discussion of appropriate objectives for low-attaining pupils in aspects of basic numeracy, such as place value, measurement and confidence with number focussed especially on understanding and the development of language. The book is practical with numerous examples and suggestions for appropriate activities to be used with low-attaining pupils. Many of these demonstrate how calculators can be used effectively to develop numeracy. A central objective for numeracy, analysed in detail, is that the pupil should know what to enter on a calculator in all the practical situations which might be encountered in everyday life where calculations are required. The final chapter demonstrates how the principles and practical suggestions contained in the book might be embodied in a scheme of work on a specific topic.

### Rita Headington, Supporting Numeracy, David Fulton Publishers, 1997.

This handbook gives sound but simple explanations of mathematics in the early years and puts them into action through practical activities. It aims to reduce the 'fear factor' of mathematics and increase confidence by providing those who assist in early years settings with knowledge of mathematics learning, language and equipment. This is an ideal resource for classroom assistants and volunteer parents working in schools and a useful handbook for trainees and trainers on Specialist Teacher Assistant (STA), BTEC and NNEB courses and for headteachers and teachers in early years settings.

### Douglas Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid, Penguin Books, 1979.

Linking the music of J S Bach, the graphic art of M C Escher and the mathematical theorems of Gödel, as well as ideas drawn from logic, biology, psychology, physics and linguistics, the author illuminates one of the greatest mysteries of modern science: the nature of human thought processes. His book has much in common with the works of Lewis Carroll, drawing together an astonishing range of ideas. It is 'an entertainment, a literary achievement and a triumph of the imagination'.

### Peter Holmes, Ramesh Kapadia and Neil Rubra, Statistics in Schools 11-16: a Review, Methuen Educational, 1981.

This is the report of the Schools Council Statistical Education Project set up at the University of Sheffield in 1975. It is a summary of project papers assessing the prevailing situation in statistical education. It gives an account of the project's secondary-school survey and considers statistics taught in mathematics and science courses and its use in the humanities and social sciences. It analyses the statistics content and approach met in texts used in class work, and examination syllabuses and includes detailed lists of resources for teachers.

### John Holt, How Children Fail, Penguin, 1965.

This book records a teacher's search for an answer to the question of why children fail. It developed from the journal which John Holt kept whilst observing children in class. He analyses the strategies children use to meet or dodge the demands which the adult world makes on them, the effect of fear and failure on children, the distinction between real and apparent learning and the ways in which he feels schools fail to meet the needs of children. A lot of time in this book is spent looking at young children attempting to do mathematics.

### John Holt, How Children Learn, Penguin, 1967.

John Holt's claim in this book is that left to themselves, young children are capable of grasping new ideas faster than most adults give them credit for (and this included teachers and psychologists as well as parents). But they have their own way of understanding, of working things out; and in most cases this fresh, natural style of thinking is destroyed when the child goes to school and encounters formal methods of learning.

### Christine Hopkins, Sue Gifford and Sandy Pepperall (eds), Mathematics in the Primary School, David Fulton Publishers, 1996.

Mathematics in the Primary School is designed to provide student teachers, NQT's and practising teachers with the background needed to teach mathematics to children aged 3-11. It aims to blend idealism and pragmatism by: identifying the important principles which underpin mathematics teaching; providing a sense of direction for teaches as they guide and help children; providing a published mathematics scheme to be used by teachers as a resource to help them structure their teaching.

### Geoffrey Howson, A Review of Research in Mathematical Education. Part C: Curriculum Development and Curriculum Research, NFER-Nelson, 1983.

This is the third of three comprehensive reviews of research commisioned by the Cockcroft Committee into different aspects of mathematics education. It looks at ideas and research which have contributed to the development of the mathematics curriculum and puts British developments into an international perspective.

### Geoffrey Howson, National Curricula in Mathematics, Mathematical Association, 1991.

The imposition of National Curricula in the U. K. and the approach of 1992, with increased European collaboration, have prompted greater U. K. interest in the teaching of mathematics in other countries. This growing interest in comparative studies also appears to be shared by educators elsewhere. Increasing attention is being focused on comparisons of the structure of educational systems and on expectations, attainment and emphases. What is being done to cope with pupils of different abilities, interests and aspirations? how is 'mathematics for all' being interpreted? What provisions are being made for pre-school children? How many hours are devoted to teaching mathematics and for how many years is its study compulsory? To what extent are curricula and teaching methods influenced by calculators and micros? Which new topics are finding their way into mathematics curricula and which are being dropped? These are some of the questions around on this issue. This book considers the place of national curicula, comments on points of divergence to be found in the curricula of different countries, and looks at some of the lessons to be learned from current practices. Descriptions of the school systems and national curicula of the E.C. countries, Hungary and Japan are included.

### Geoffrey Howson, Christine Keitel and Jeremy Kilpatrick, Curriculum Development in Mathematics, Cambridge University Press, 1981.

The ideas of mathematics can be understood through the techniques needed to solve problems which arise in everyday life. Conversley, these problems can illustrate how mathematics develops 'naturally'. The essence of this informal book is to motivate mathematics by examining models of situationsa and problems that occur in the real world. Each chapter deals with a specific mathematical topic and each topic is introduced at different levels to provide motivation for students of varying mathematical maturity. The authors emphasise that although applications provide motivation they are not trying to supply aids for teaching applied mathematics. It is mathematical notions that are important and how they can be developed. We have thus an amusing inversion of aims: physics, biology, linguistics etc., are here applied to the teaching of mathematics.

### Geoffrey Howson & Jean-Pierre Kahane (et al) (eds), The Influence of Computers and Informatics on Mathematics and its Teaching, ICMI Study Series, Cambridge University Press, 1986.

This first ICMI study is concerned with the influence of computers and computer science on mathematics and its teaching in the last years of school and at tertiary level. In particular, it explores the way in which the computer has influenced mathematics itself and the way mathematicians work, likely influences on the curriculum of high school and undergraduate students and the way in which the computer can be used to improve mathematics teaching and learning.

### Geoffrey Howson and Bryan Wilson (eds), School Mathematics in the 1990s, ICMI Study Series, Cambridge University Press, 1986.

Courses throughout the world are faced with the problem of adjusting school mathematics curricula in an attempt to match rapid changes in society, technology and educational systems. This ICMI study is intended to help those who wish to form a vision of what school mathematics miught be in the 1990s and who would want to work towards the fulfilment of specific goals. In doing this it draws on the experience of the past thirty years which have taught us that what is desirable might not be attainable; and that goals must be set which acknowledge the existence of constraints. This study identifies key issues and basic wquestions within mathematics education, proposes and comments upon alternative strategies, and provided a stimulus for more detailed, less general discussions within more limited geographical and social constraints.

### Geoffrey Howson & Jean-Pierre Kahane (eds), The Popularisation of Mathematics, ICMI Study Series, Cambridge University Press, 1990.

This book presents papers arising from the ICMI study seminar on the popularisation of Mathematics held in 1989. The Seminar consisted of three plenary talks followed by sessions discussing the problems faced in the popularisation through the mass media. The image of mathematicians, TV and films and mathematics in different cultures were al included in the discussions.

### Celia Hoyles and Rosamund Sutherland, Logo Mathematics and the Classroom, Routledge, 1989.

The use of computers in the classroom has brought iwth it fresh challenges and ways of thinking for pupils and teachers alike. In particular, the iuse of Logo in the context of the mathematics National Curriculum provides new opportunities for pupils to engage in mathematical thinking and mathematical discussion. Inevitably it also raises questions about children's mathematical learning and the teacher's role in this process. In this book, the authors use extense case study material of pupils aged from 11 to 14 to tackle these questions. Originally published in 1989, the book has been updated to take account of new developments in the national Curriculum, and revised to make it more suitable for teachers.

### Celia Hoyles and Richard Noss (eds), Learning Mathematics and Logo,

### Martin Hughes, Children and Number, Basil Blackwell, 1986.

Martin Hughes proposes a new perspective on children's early attempts to understand mathematics. He describes the surprisingly substantial knowledge about number which children acquire naturally before they start school, and contrasts this with the difficulties presented by the formal written symbolism of mathematics in the classroom. He argues that children need to build links between their informal and their formal understanding of number, and shows what happens when these links are not made. In the book he describes many novel ways in which young children can be helped to learn about number. He shows that the written symbols children often invent for themselves are more meaningful to them than the conventional symbols they are taught. He presents simple number games for introducing children to mathematical symbols in ways they can appreciate and understand. He also describes how LOGO can be adapted for young children, and shows the dramatic effect it can have on their mathematical understanding.

### Zelda Isaacson, Teaching GCSE Mathematics, Hodder and Stoughton, 1987.

This book provides a complete guide to the new syllabuses for the practicing teacher. It details new features, their implications for the classroom, and suggests a wide range of teaching strategies directly applicable to the work of a mathematics teaching program.

### Barbara Jaworski, Investigating Mathematics Teaching: A Constructivist Enquiry, Falmer Press, 1994.

This book is a personal account of a research study into the nature of an investigative approach to mathematics teaching at secondary level. The study, consisting of close observations of three teachers, is reported in detailed case studies featuring significant episodes in the teaching and its analysis, so highlighting issues and tensions which were central to their practice. The teaching was seen to result in high level mathematical thinking from studetns in the classroom. The research involves a weaving of theory and practice in which constructivism was central to both learning and teaching and to the research process itself. Both teachers and researcher were seen as reflective practitioners whose critical analysis of the teaching and research process contributed to the rigour of the research.

### David Johnson and Alison Millett (eds), Implementing the National Curriculum. Policy. Politics and Practice, Paul Chapman Publishers, 1996.

This book gives a broad view of developing policy in the years leading up to the National Curriculum with a close focus on the intricacies of policy implementation and review �friction at the interface�. It dealt with a two-year research study evaluating the implementation of the National Curriculum Mathematics Key stages 1, 2 and 3. The focus of the study was on primary and secondary teachers and their perceptions and actions in a period of major change bought about by the move to a statutory National Curriculum. This book is an examination of that research study. Its genesis and influence on subsequent events. The book begins by tracing the evolution of the content from which the authors� research into the implementation of the National Curriculum was conceived. The central chapters focus on the use to which teachers make use of commercial schemes; the professional development of teachers concerned with achieving �ownership� of the curriculum and perceptions of �Using and Applying Mathematics� arising form teacher readings of National curriculum documents. This book ends by considering the impact of the research on event in 1993-94 a period that included the Dearing review.

### George Joseph, The Crest of a Peacock, I. B. Taurus, 1991. (Also Pengiun Books, 1993)

Cultural arrogance has led many Westerners to ignore the magnificent heritage of non-Western mathematics, and to regard the development of mathematics as Europe-centred and Europe-led. This pioneering book reinstates the fundamental contribution to mathematics made by non-Western peoples. To the traditional European mind mathematics was supposedly 'invented' by the Greeks, stored by the Arabs of Moorish Spain and rediscovered and further developed in Europe from the Renaissance onwards. The belief in a 'Greek miracle', followed by a thousand-year period of cultural stagnation conveniently labelled the 'Dark Ages', was a forceful expression of European intellectual self-confidence and feelings of superiority over other races. By considering mathematical developments in the ancient civilisations of Mesopotamia and Egypt, in China and pre-Columbian America, in India, the Arab world and Africa, and continuing the discussion to developments in the early modern period, George Gheverghese Joseph questions, and successfully demolishes, many familiar assumptions and enlarges our sense of what we mean by mathematics. A celebration of some of the most fascinating and neglected achievements in intellectual history, the book is accessible even to those with a basic school maths background.

### Imre Lakatos, Proofs and Refutations (The Logic of Mathematical Discovery), Cambridge University Press, 1976.

Much of this book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. The author is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts.

### L.A.M.P., Better Mathematics, H.M.S.O., 1987.

This book is the report of the Low Attainers in Mathematics Project, which was aimed at developing and encouraging 'good practice' in mathematics teaching. Throughout it there are many recommendations for action and it is an invitation to those concerned with education at all levels to experiment, discuss, debate, strengthen and refute as a result of their experiences.

### Tony Larcombe, Mathematical Learning Difficulties in the Secondary School; Pupil Needs and Teacher Role, Open University Press, 1985.

The Author explores the inter-relationship of learner, teacher, mathematics and schooling. His starting point is the needs of pupils, particularly the needs of low attaining, disaffected or reluctant learners. He indicates the relevance of their feelings and attitudes to mathematics, paying close attention to notions of perceived failure and worth, and how they affect pupil motivation. Taking a detailed look at concept development, problem-solving and assessment, Tony Larcombe develops a rationale for effective classroom practice. Further he translates why into how by using telling examples of what teachers can do, particularly in the context of older pupils' special needs.

### Anna Lewis, Discovering Mathematics with 4 - 7 Year-olds, Hodder and Stoughton, 1996.

### Pamela Liebeck, How Children Learn Mathematics; A Guide for Parents and Teachers, Penguin Books, 1984.

Aimed at teachers and parents who are concerned with children from infancy through the Primary School years. It takes the reader through the development of abstract thought in very young children and describes how children's experiences can be used to help them grasp the principles underlying such processes as addition, subtraction, multiplication, division and measurement. It then goes on to discuss fractions and decimals, calculators and computers, some theories of learning and the vexed problems of why some children fail at mathematics.

### Stephen Lerman, (ed), Cultural Perspectives on the Mathematics Classroom, Kluwer Academic Publishers, 1994

Mathematics teaching and learning have been dominated by a concern for the intellectual readiness of the child, debates over rote learning versus understanding and, recently, mathematical prosesses and thinking. The gaze onto today's mathematics classroom is firmly focussed on the individual learner. Recently, however, studies of mathematics in social practices, including the market place and the home, have initiated a shift of focus. Culture has become identified as a key to understanding the basis on which the learner appropriates meaning. The chapters in this book attempt to engage with this shift of focus and offer original contributions to the debate about mathematics teaching and learning. The chapters adopt theoretical perspectives while drawing on the classroom as both the source of investigation and the site of potential change and development.

### D. S. Macnab and J. A. Cummine, Teaching Mathematics 11-16; A Difficulty-Centred Approach, Blackwell, 1986.

This book examines the reasons for failure in mathematics and provides ideas for enhancing mathematical attainments of pupils. It considers the aims of and attitudes to the teaching and learning of mathematics and presents a brief description of some of the main themes of which have engaged the attention of learning theorists and mathematics educators. These include Piaget's developmental approach, the structural approach of Bruner and Dienes as well as Constructivism.

### Helen Mansfield (ed), Mathematics for Tomorrow's Young Children International Perspectives on Curriculum, Kluwer Academic Publishers, 1996.

The contributions in this book, intended for mathematics education professionals and teachers of mathematics, come from a broad range of countries and cultures; they are representative of different theoretical perspectives and classroom experiences. All contributors are concerned with helping teachers explore ways to develop children's mathematical understanding appropriate for the new millennium. The authors present complex ideas about mathematical understanding and provide readers with powerful classroom examples. Recommendations for changing the curriculum for young children are also suggested. The book documents four years of development in the field. Among the emergent developments described is the importance of context to mathematical development - it is not only the physical context, but also the social context of the classroom and school that stimulate conceptual growth. The book also locates current theoretical perspectives in a broad framework. Finally the book is organized around four interconnected themes all related directly to teaching and learning mathematics.

### John Mason, Learning and Doing Mathematics, Macmillan Educational, 1988.

By recognizing that whilst learning mathematics, students do get 'stuck', this book encourages the adoption of a clearly defined set of strategies for coping with it. There are a wide variety of interesting and entertaining examples used to illustrate the possible strategies. Readers are also urged not only to work through the examples, but also to reflect on and develop their own approaches to mathematical problems.

### John Mason, Kaye Stacey and Leone Burton, Thinking Mathematically, Addison Wesley, 1982.

Demonstrates how to encourage, develop and foster the processes which seem to lie at the heart of doing mathematics. Intensely practical, it demands that the reader participates in each question posed, so that the subsequent discussion speaks to immediate experience. In this was, a deep seated awareness of the nature of mathematical thinking can grow.

### Mathematical Association, Maths Talk, Stanley Thornes, 1987.

The skills of spoken language are as important in mathematics as they are in other areas of the primary curriculum. This book offers some interesting suggestions in answer to many questions this raises. It looks at the part discussion plays in the development of mathematical thinking and the benefits talking and listening can bring to pupils and teachers. It also looks at the teacher's role in language based mathematics and the implications this approach may have on the forms of classroom management and organization.

### Mathematical Association, Sharing Maths With Children, Stanley Thornes, 1987.

There is an ever growing need for education to be seen as a shared task for schools and parents. With mathematics in particular, a major obstacle to progress may be the parents' own past experiences. The suggestions in this book show how teachers at all levels can plan school-based activities as a means of overcoming this problem.

### Mathematical Association, Mental Methods in Mathematics: A First Resort, The Mathematical Association, 1992.

Try asking some children or a group of teachers to do a few simple calculations in their heads and then ask them to describe the methods they used. You may be surprised at the great variety of approaches adopted, most of them very different from the standard procedures. Effective mental calculation requires a flexible approach where the method employed is adapted to suit the particular numbers involved. The problem for the teacher is to devise ways of encouraging children to develop their own strategies and to use them confidently and effectively. The aim of this book is to encourage teachers to explore for themselves the rich variety of mental methods that children and adults commonly use and to suggest classroom activities which might stimulate and enhance children's powers of mental imagery and mental calculation from 5 - 16.

### Steig Melin-Olsen, The Politics of Mathematics Education, Laurence Erlbaum, 1984.

In this book, the author analyses mathematics education from a political standpoint, arguing that pupils' failure to learn mathematics is attributable to their failure to recognise the full significance of the subject in, for instance, administration, economy and industry. Key concepts are 'production of culture', 'oppression' and 'resistance'. The author benefits here from recent developments in third world education. In addition he uses among others, Bateson's communication theory, Freud's theory of oppression and resistance, and early Soviet activity theory to construct a basis for a comprehensive mathematics education theory. The book includes a series of examples and case studies which illustrate the politicisation of mathematics in education.

### Ruth Mertens & Jeff Vass, Sharing Maths Cultures IMPACT: Inventing Maths for Parents and Children and Teachers, Falmer Press, 1990.

This book tells the story of IMPACT, one of the largest projects ever launched to involve parents routinely in the primary school curriculum. Mathematics is a source of anxiety for both parents and children; IMPACT has designed curriculum materials that are valid across the different cultures in and out of school that a child inhabits. The book describes and promotes educational settings in which the child is not the 'centre' or 'object' of teaching, but is the initiator and tutor. It explains how any teacher or school can set up and run a similar initiative. Detailed examples are given of the processes involved and the specific methods used for effective parental involvement. It provides and discusses sample materials for such a scheme, and outlines a radically new approach to evaluation.

### Ruth Mertens & Jeff Vass, Partnership in Maths. Parents and Maths, Falmer Press, 1993.

This book provides a series of accounts of the work of parents in support of the teaching of mathematics in schools. Tsaking the IMPACT Project as their focus, parents, teachers, advisors and inspectors, LEA officers and academkcs write about the ways in which an effective collaboration between parents, teachers and children may be achieved through the medium of shared mathematics tasks in the home.

### Tim Miles and Elaine Miles, Dyslexia and Mathematics, Routledge, 1991.

Dyslexia is seen primarily as a limitation in the ability to deal with symbolic material. As far as the symbols of mathematics are concerned, therefore, special teaching techniques are needed, just as they are for the teaching of reading and spelling. In this book it is argued that with suitable help dyslexics have every chance of being successful at mathematics. This book contains material on individual cases and on children of different ages. Two central themes are discussed: first that dyslexics need to carry out the operations of addition, dividing and so on before being introduced to the symbolism; and second they need to be shown the many regularities and patterns which can be found in the number system.

### Bob Moon, The New Maths Curriculum Controversy, Falmer Press, 1986.

This book has the value of developing comparative insights into a significant curriculum reform movement. Above all the work reminds us that the general features of a national system of education offer limited help for an analysis of how the curriculum is produced at any particular point in time. It warns us that the outcome of educational change reflects a much more complex interaction between interest groups than the unified concepts of control of the curriculum would imply. But if the focus is mainly on the interest groups and networks involved in curriculum negotiations this book also alerts us to the continuing significance of school textbooks. Here the author is following the work of Kuhn and more recently of Apple. Kuhn was clear that textbooks were a major socializing device and the study of textbooks a vital element in studies of curriculum change.

### Ann Montague-Smith, Mathematics in Nursery Education, David Fulton Publishers, 1997.

With the introduction of nursery curriculum guidelines for mathematics, this book aims to encourage adults working or studying to work with children under five, to have increased understanding of mathematics and to become more confident in planning, teaching and assessing mathematical learning. In an accessible style, the book brings together a wealth of research evidence and current professional good practice; outlines the key concepts for the mathematical topics; offers suggestions for language development, considers planning, assessment and managing the parts played by other adults, including assistants; suggests ways in which to enhance partnership with parents. Mathematics in Nursery Education is intended for anyone who works with children in the early years: teachers, nursery nurses, classroom assistants, playgroup leaders and workers, child-minders and day care staff. It will also be a suitable course text for BA QTS and early childhood degrees or diplomas

### David Nelson, George Joseph and Julian Williams, Multicultural Mathematics. Teaching Mathematics from a Global Perspective, Oxford University Press, 1993.

Mathematics is a truly international language and field of study that knows no barrier between race, culture or creed. How can we exploit its rich cultural heritage to improve the teaching of mathematics and educate our children for life in a multicultural society? The history of mathematics is one of creation and discovery in many parts of the world, and yet few people are aware that Pythagoras' Theorem was known to the Babylonians a thousand years before the Greeks. Similarly, Pascal's Triangle of 1645 was in use much earlier in China. There is a rich field of African, Middle Eastern, Chinese, and Indian mathematics that is often ignored in the present teaching of the subject. This book is the first to explore ways of helping school children understand the universality of mathematics, in that it offers suggestions and guidance for a multicultural approach.

### Perla Nesher and Jeremy Kilpatrick, Mathematics and Cognition, ICMI Study Series, Cambridge University Press, 1990.

This book is aimed at teachers, mathematics educators and general readers who are interested in mathematics education from a psychological point of view. It describes research findings thatg shed light on the learning of mathematics from early arithmetic to high levels of geometry and algebra. The book is the collaborative effort of sa number of members of the International Group for the Psychology of Mathematics Education and primarily describes their work whilst at the same time covering many issues that interest researchers in mathematics education.

### Marilyn Nickson and Stephen Lerman (eds), The Social Context of Mathematics Education: Theory and Practice, Southbank Press, 1992.

This book contains papers from the first three years of conferences of the Group for Research into Social Perspectives of Mathematics Education. The group's focus is the mathematics classroom as a socio-cultural setting and the papers reflect the growing interest in the area within the mathematics education community. Contributors include those drawn from the fields of mathematics, mathematics education, sociology and philosophy. The four sections of the book are : the National Curriculum in mathematics - a cultural perspective; the social context of mathematics education; theoretical frameworks and current issues, and political action through mathematics education. The contributions include the following topics; ethnomathematics;discourse analysis; the sociology of the mathematics classroom; gender issues; cultural origins of mathematics; problem posing; multicultural mathematics; political implications of the national Curriculum; 'barefoot statisticians".

### Mogens Niss (ed), Cases of Assessment in Mathematics Education - An ICMI Study, Kluwer Academic Publishers, 1993.

This book is one of the first to present a variety of carefully selected cases to describe and analyze in depth and considerable detail assessment in mathematics education in various interesting places in the world. The book is based on work presented at an invited international ICMI seminar and includes contributions from first rate scholars from Europe, North America, the Caribbean, Asia and Oceania, and the Middle East. The cases presented range from thorough reviews of the state of assessment in mathematics education in selected countries, each possessing `archetypical' characteristics of assessment, to innovative or experimental small or large scale assessment initiatives. All the cases presented have been implemented in actual practice. The book will of interest to mathematics educators - at all levels - who are concerned with the innovation of assessment modes in mathematics education, as well as everybody working in the field of mathematics education: in research and development, in curriculum planning, assessment institutions and agencies, mathematics specialists in ministries, teacher trainers, textbook authors, frontline teachers.

### Mogen Niss (ed), Investigations into Assessment in Mathematics Education. An ICMI Study, Kluwer Academic, Press, 1993.

This book attempts a systematic in-depth analysis of assessment in mathematics education. It deals with assessment in mathematics education from historical, psychological, sociological, epistemological, ideological and political perspectives. It is based on work presented at an international conference and includes chapters by a team of prominent scholars in the field. Based on the observation of an increasing mismatch between goals and accomplishment of mathematics education and prevalent assessment modes, the book aims to assess assessment in mathematics education and its effects. In so doing it pays particular attention to the need for and possibilities of assessing a much wider range of abilities than before including understanding, problem solving and posing and modelling and creativity.

### Richard Noss and Celia Hoyles, Windows on Mathematical Meanings, Kluwer Academic Publishers, 1996.

Why are mathematical ideas so hard? Is mathematics an unassailable peak, which only the few can ever hope to conquer? Or can mathematics be broadened to be accessible to the many? The authors have written a book which challenges some of the conventional wisdoms on the learning of mathematics. They use the computer as a window onto mathematical meaning-making, drawing together the threads of their individual and collaborative research over more than a decade. the pivot of their theory is the idea of webbing which explains how someone struggling with a new mathematical idea can draw on supportive knowledge, and reconciles the individual's role in mathematical learning with the part played by epistemological, social and cultural forces.

### Terezinha Nunes and Peter Bryant, Children Doing Mathematics, Blackwell, 1996

This book provides a review of the substantial recent work in children's mathematical understanding. The authors also present important new research on children's understanding of number, measurement, arithmetic operations and fractions both in and out of school. The central theme of this book is that there are crucial conditions for children's mathematical learning. First, Children have to come to grips with conventional mathematical systems. Equally importantly, they have to be able to represent mathematical knowledge in a way that is solving problems. The book also diseases how mathematical activities and knowledge involve much more than is currently viewed as mathematics in the school curriculum, illustrating how children can be successful in mathematical activities outside school when they fail in similar activities in the classroom.

### Terezinha Nunes, Analucia Dias, and David Carraher, Street Mathematics and School Mathematics, Cambridge University Press, 1993.

This book provides a systematic comparison of mathematics used in schools and out of school, describing the two forms of activity as different cultural practices that are based upon the same mathematical principles. Many philosophers and psychologists have recognised that reasoning abut number and space is part of people's everyday experience as well as part of the formal discipline of mathematics. However discussions of everyday mathematical reasoning have been speculative because until the work described here little systematic research had been carried out comparing mathematical knowledge developed in and outside of school. This book illustrates the advantages and disadvantages of the two practices as they are now observed, pointing out the trade-off between preservation of meaning and potential for generalisation in mathematical knowledge. The empirical findings are analysed within a broad framework in the concluding chapter, which discusses the educational implications of these findings and presents a case for realistic mathematics education - a form of teaching that builds upon formal mathematical knowledge on the foundations of street mathematics.

### Antony Orton, Learning Mathematics, Cassell, 1987.

A comprehensive introduction to a major strand of mathematics education. It provides a much needed overview of research in psychology and education that explores children's comprehension of mathematical concepts. Written from the viewpoint of the mathematics teacher, the text examines issues of continuing concern, the theories that surround then and the role in teaching practice. Why, for example do some pupils achieve more than others? Must we wait until a child is 'ready'? Can children discover mathematics for themselves? Does language interfere with mathematics learning? Each chapter explores a particular issue, illustrating the interaction between theory and practice. Suggestions for further reading and study are provided together with a comprehensive reference list and points for discussion.

### Antony Orton and Geoff Wain (eds), Issues in Mathematics Education, Cassell Educational Press, 1993.

This book brings together a collection of essays by British authors considering current issues in mathematics education. Different chapters address: social issues, language, new developments and the place of a national curricula.

### Lynn Osen, Women in Mathematics, M.I.T. Press, 1974

This is an interesting and useful book to put the women in mathematics in proper perspective. From Hypatia in 300 B.C. to Emmy Noether in 1900, we see how the women who rose to the top in mathematics were clearly superior to mathematicians in general. It also illustrates well the inherent motivations at work in the real scholar. The eight biographies of women mathematicians parallel the growth of mathematics and provide a new approach to the history of mathematics. It is also clear that these women scholars were also very human people. It would help everyone if [(s)he] were to read the final chapter, 'The Feminine Mathtique'. The colourful lives of these women, who often travelled in the most avant-garde circles of their day, are presented in fascinating detail. The obstacles and censures that were also a part of their lives are a sobering reminder of the bias against women still present in this and other fields of academic endeavour. Mathematicians, science historians, and general readers will find this book a lively history; women will find it a reminder of a proud tradition and a challenge to take their rightful place in academic life today.

### Seymour Papert, Mindstorms (Children, Computers and Powerful Ideas), Harvester Press, 1980.

Enabling children to program computers - mastering a powerful technology and coming into contact with some of the deepest ideas from science, mathematics and model making - this book shows how the fundamental concepts of mathematics can be understood and mastered by young children through techniques of Logo programming. The author aims to show how mathematics phobia is created in the classroom and how it can be overcome with the help of computers.

### Neil Pateman, Teaching Mathematics - a Tantalising Enterprise. On the Nature of Mathematics and Mathematics Teaching, Deakin University Press, Australia, 1989.

This Book forms part of a course entitled 'Teaching and Learning Mathematics', at Deakin University. It begins by looking at mathematics teaching today and at what we may learn from classrooms. It follows this up with studies of the main branches of the philosophy of mathematics educational,. Including, empiricism, absolutism, logicism, intuitionism and radical constructivism and activity theory. It concludes with a chapter on the sociology of mathematics and considers mathematics teaching in the future.

### John Allen Paulos, Innumeracy, Penguin Books, 1988.

Poor education, psychological blocks and romantic misconceptions about mathematics have made much of the population innumerate. Yet if we are to resist the false claims of advertisers and politicians, quack doctors and pseudo-scientists, we urgently need a bit of health scepticism about statistics. In this book John Allen Paulos brings together many intriguing and practical examples to make accessible the beauty and sheer power of mathematics - and put the fun back into facts and figures.

### Terry Perl, Math Equals, Addison Wesley, 1978.

An eminently readable account of nine women mathematicians, their lives and their work, with mathematics activities which relate to the work of each. Ranging from Hypatia in 4th century Alexandria to the 20th century Emmy Noether of Germany, these women of greatly varying social and educational backgrounds had one thing in common: they excelled in a field usually thought to be reserved for men. The activities in this book are accessible to anyone with elementary school mathematics and geometry, yet touch a broad range of areas in mathematics.

### David Pimm, Speaking Mathematically, Routledge and Kegan Paul, 1987.

David Pimm examines the school teaching of mathematics from the perspective of mathematics as a language. His claim is that seeing mathematics and its teaching in linguistic terms can illuminate many events which occur daily in mathematics lessons, and can allow important questions to be asked. The first half is concerned with verbal interactions, looking at pupil talk in many different settings as well as the structure of teacher/pupil interactions. The second half examines aspects of pupil written mathematics, particularly in the Secondary School with the increased move towards symbolization.

### David Pimm (Ed), Mathematics, Teachers and Children, Hodder and Stoughton, 1988.

This book is the reader for the Open University Course Using Mathematical Thinking. It comprises a collection of articles and readings designed for all those concerned with the teaching of mathematics in the primary and secondary curriculum. Through it, key issues of problem solving, problem posing, assessment and evaluation are all addressed.

### David Pimm (Ed), Symbols and Meanings in School Mathematics, Routledge 1995.

This book explores the various uses and aspects of symbols in school mathematics, and also examines the notion of mathematical meaning. It is concerned with the power of language which enables us to do mathematics, giving us the ability to name and rename, to transform names and to use names and descriptions to conjure, communicate and control our images. It is in the interplay between language, image and object that mathematics is created and can be communicated to others. One theme which runs throughout the book is the core metaphor of manipulation. How does the omnipresence of this term, describing the doing of mathematics, connect to the two fundamental metaphors in English for understanding (touch and sight), and how is manipulation implicated in the common tension in mathematics teaching between helping to develop fluency and understanding? The book also addresses a set of questions of particular relevance to the last decade of the 20th century, which arise due to the proliferation of machines offering mathematical functionings.

### David Pimm and Eric Love (eds), Teaching and Learning School Mathematics, Hodder and Stoughton, 1991.

This collection of articles explores key issues facing all of those concerned with the teaching of mathematics at both primary and secondary levels. The authors are writing in response to a variety of emerging educational challenges and constraints, including imposed national curicula and the changing political and technological arenas for education. The chapters are arranged in four sections. The first focusses on the practices of teachers and the experiences of learners, the next offers new ways of thinking about the mathematics taught in schools. Then several writers examine the major constraints on what is taught from within the mathematics curriculum and in the final section there are six pieces which explore issues which are present behind any mathematics teacher's action. The reader is one component of the Open University course "Learning and Teaching Mathematics".

### George Polya, How To Solve It, Princetown University Press, 1957, (Reprinted 1991, Penguin).

This book involves the study of the methods and rules of discovery and invention. The author uses specific examples taken largely from geometry. His principle aim is to teach a method which can be applied to the solution of other problems. The particular solution of a particular problem is, for his purposes, of minor importance. The approach is constant regardless of its subject and can be expressed in simple but incisive questions: "What is the unknown? What are the data? What is the condition? Do you know a related problem". Deftly the teacher shows us how to strip away the irrelevancies which clutter our thinking and guides us toward a clear and productive habit of mind.

### Michael Preston (Ed), Mathematics in Primary Education, Falmer Press, 1987.

This book brings together a series of contributions from practising teachers, advisers and teacher trainers and focuses on key issues which face the development of Primary Mathematics to the end of this century. Each chapter stands on its own and sets out to stimulate the reader to think about the questions raised. This book is not a textbook of what should be done but a challenge to all those concerned with providing mathematical education to primary children.

### Sal Restivo, Mathematics in Society and History, Kluwer Academic Press,

### Sal Restivo, Jean Paul van Bendegem & Roland Fischer (Eds), Maths Worlds. Philosophical and Social Studies of Mathemnatics Education, State University of New York,.

An international group of authors brings a variety of resources to bear on the major issues in the study and teaching of mathematics, and on the problem of understanding mathematics as a cultural and social phenomenon. All are guided by the notion that our current understanding of mathematical knowledge must be grounded in and reflect the realities of mathematical practice. Chapters on the philosophy of mathematics illustrate the growing influence of a pragmatic view in a field traditionally dominated by platonic perspectives. In a section on mathematics, politics and pedagogy, the emphasis is on politics and values in mathematics education. Issues addressed include gender and mathematics, applied mathematics and social concerns, and the reflective and dialogical nature of mathematical knowledge. The concluding section deals with the history and sociology of mathematics and social change.

### Jim Ridgway, Assessing Mathematical Attainment, NFER-Nelson, 1988.

The study focused on fairly stable implementations of a first-edition *Standards*-based high school mathematics curriculum that was used by all students in each of three schools.

It involved students who experienced up to seven years of *Standards*-based mathematics curricula and instruction in middle school and high school.

It monitored students' mathematical achievement, beliefs, and attitudes for four years of high school and one year after graduation.

Prior to the study, many of the teachers had one or more years of experience teaching the *Standards*-based curriculum and/or professional development focusing on how to implement the curriculum well.

In the study, variations in levels of implementation of the curriculum are described and related to student outcomes and teacher behavior variables.

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