1.1. From the Publisher.

Degree students of mathematics are often daunted by the mass of definitions and theorems with which they must familiarise themselves. In the fields algebra and analysis this burden will now be reduced because in A Handbook of Terms they will find sufficient explanations of the terms and the symbolism that they are likely to come across in their university courses. Rather than being like an alphabetical dictionary, the order and division of the sections correspond to the way in which mathematics can be developed. This arrangement, together with the numerous notes and examples that are interspersed with the text, will give students some feeling for the underlying mathematics. Many of the terms are explained in several sections of the book, and alternative definitions are given. Theorems, too, are frequently stated at alternative levels of generality. Where possible, attention is drawn to those occasions where various authors ascribe different meanings to the same term. The handbook will be extremely useful to students for revision purposes. It is also an excellent source of reference for professional mathematicians, lecturers and teachers.

1.2. From the Preface.

The enormous increase in mathematical activity and knowledge during the past half century has not been achieved without a corresponding increase in the number of terms which mathematicians use. Not only have names had to be given to new concepts and objects, for example categories and functors, but there has also been a need to attach new and/or more precise meanings to certain terms such as 'function' which have been used, in some sense or other, for centuries. Again, deeper insight into mathematical structure has often yielded an alternative way of approaching such well-established ideas as that of a derived function. This creation of new definitions and rewriting of old ones has not made life any the easier for the reader of mathematics. Certainly, an explanation of any of the terms can be found somewhere, but finding the correct source can be time consuming. The attractions of a reference book of definitions are, therefore, obvious.

Alas, the difficulties of compiling such a work are no less obvious! As soon as such a book were to appear it would be months out of date, for each issue of every mathematical journal can be expected to introduce at least one new term to the vocabulary of mathematics and frequently a new symbol - or an alternative usage of an old one - to accompany it. Moreover, talents equal to those of the troops of Bourbaki would be required to produce a comprehensive and authoritative work.

The objectives of this handbook, then, must be somewhat circumscribed. It is, for example, intended to meet the needs of the undergraduate and the school teacher rather than the university lecturer or the research student. It is concerned only with algebra and analysis. Nevertheless, I hope that within these limits the handbook will prove of value and that by its arrangement into sections, its examples and its notes, it will give some indication of, and some feeling for, the mathematics that has given rise to the definitions listed. Some cynics have asserted that 'modern mathematics' is 'all definitions', and it cannot be denied that it does contain a great number of them. It must be stressed, therefore, that the terms defined here are only the 'words' of mathematics and that a mathematician is interested not only in learning new 'words' but in 'creative writing' and in appreciating the subject's 'literary heritage'.

Presenting a definition out of context is not a simple matter. 'How do I get to Newcastle from here?' is a question which has many answers. Is the questioner travelling on foot or on horseback, by bicycle or by car, or, come to that, by boat ? Does he want the fastest route or the least congested? Would he like to see the new motorway, or travel on minor roads past some fascinating historical monuments? These questions all have their mathematical analogues - as does the apocryphal Irishman's answer that 'If I wanted to go to Newcastle, I shouldn't start from here.' There is always the chance, too, that the questioner really wanted to go to Newcastle-under-Lyme and has been directed in error to Newcastle-upon-Tyne. In an attempt to meet some of the analogous mathematical problems, many terms have been defined in this handbook in alternative ways; for example, 'continuity' is discussed in three different sections, namely, Metric spaces, Topological spaces, and Real-valued functions of a real variable. Theorems too have often been stated at alternative levels of generality. I have also tried to draw attention to words and phrases which are used in different senses by different authors, for example 'ring' and 'set of natural numbers'.

It would, of course, be difficult to answer the traveller without using numbers, yet no one is likely to preface his reply by an account of how the number system is constructed. In a similar manner, one cannot define terms in mathematics without using some words or symbols taken from logic. For this reason, the first section of the handbook describes those words and symbols which we shall wish to borrow from logic, without, however, concerning itself with the foundations of that subject.

The use of symbols in mathematical literature is, if anything, even more bewildering than that of defined terms. Only the ' + ' sign in the context of the addition of real numbers springs to mind as a symbol used by all writers in an unambiguous manner. (Lest readers press the case of the companion ' - ' sign, I was informed by a Scottish Inspector of Schools that he visited one primary school where the teacher taught the class that ' - ' was to be translated as 'from' and that the children were, therefore, to write equations such as 2 - 8 = 6. The teacher explained to the inspector that the children experienced less difficulty with this convention!) I have tried, therefore, when introducing symbols to list any alternatives which are widely used. In particular, in the sections on differentiation I have presented definitions, results and examples in a variety of notations - both 'ancient and modern' - in the hope that the opportunities thus provided for comparing the different notations will more than compensate for any possible loss in clarity.

Whilst producing this book I have received help and encouragement from many sources. As far as content is concerned, I am indebted to a host of mathematicians ranged alphabetically from Abel to Zorn and chronologically from Pythagoras to Cohen! Several colleagues or former colleagues at Southampton have been kind enough to read sections of the book and to offer most valuable advice. In particular, I should like to acknowledge my especial indebtedness to Professor T A A Broadbent and to Dr Keith Hirst who read the final manuscript and whose observations led to the removal of several errors and, I hope, to a more readable book. I am also most grateful to Jennifer, my wife, for - amongst many other things - her assistance in the compilation of the index to this book.

1.3. Review by: Editors.
Mathematical Reviews MR0349289 (50 #1783).

This is an interesting book: it is essentially a very carefully organised textbook without the proofs. The material is arranged in 38 short chapters (starting with "Some mathematical language" and "Sets and functions" and ending with "Measure and Lebesgue integration" and "Fourier series") which, if worked through in sequence, would form an integrated and coherent course. The level is roughly that of a first course in abstract algebra and a second course in calculus (excluding differential equations but including contour integration), with an appetiser of real-variable theory. The material is well indexed and cross-referenced and an appendix on some "named" theorems and properties lists a substantial number of well-known mathematicians with their dates.

1.4. Review by: Norman Schaumberger.
The Mathematics Teacher 66 (4) (1973), 350.

A valuable reference work for teachers and workers in mathematics and related fields. Most of the definitions of basic terms, theorems, and properties that arise in the broad areas of algebra and analysis are included. This compact book is about as readable and interesting as a reference of this type can be. A wealth of illustrative examples and notes are used at appropriate points in an effort to clarify meanings and give the reader a feeling for the concept under discussion. A copy belongs in every school and college mathematics library.

1.5. Review by: D A Quadling.
The Mathematical Gazette 57 (399) (1973), 76-77.

How is one to review a book such as this? Certainly not, as I first tried, by beginning at p. 1 and hoping to reach the end. But as I struggled on, it began to dawn that I had been in this situation before. Might not the volume have been entitled "A Dictionary of Modern Mathematical Usage"? Was not Howson Cambridge's answer to Oxford's Fowler ? Heartened by this analogy, I opened the pages (as it were at random) ...

... and so it goes on, through a wide range of undergraduate pure mathematics. The publishers (who are to be congratulated on the handling of this complex production) have thoughtfully produced a paperback version within the compass of a student's grant or a teacher's salary; and one hopes that its use will not be confined to the night before 'finals', but that it will be dipped into from time to time for the reassurance that this kind of enlightened overview of mathematics can provide. Of course, Fowler is not English Literature, nor is Howson Mathematics. But I intend to keep both far closer to hand than the bound volumes of the classics (in either field) which grace my bookshelves.

2.1. From the Publisher.

This book is about mathematics and curricula, and therefore of interest both to mathematicians and to those with a wider interest in the curriculum. It pioneers the bringing together of both practicing mathematics teachers and curriculum developers in discussions about the curriculum in a crucial area which affects every school, college and university.

The book looks at the forces, theoretical, social and practical, which have shaped the curriculum in the past, which are affecting it now and the considerations which should shape the curriculum in the future. The main purpose of the book is to make readers aware of what is being taught, why and what it hopes to achieve.

2.2. Review by: Edith Robinson.
The Arithmetic Teacher 22 (3) (1975), 219.

The setting is essentially British, but the theme is universal: the curriculum is inherited. For any given country, that inheritance has evolved within a specific social and political history. Moreover, intervention by professionals, whether individuals, committees, or commissions, has not been independent of those same forces.

The authors offer a penetrating and insightful account of the evolution of the mathematics curriculum in Great Britain, and in so doing, highlight its strengths and weaknesses. In addition, they explore the nature of mathematics according to several schools of thought, and point out associated curricular implications. Finally, some examples of innovative projects and areas of needed research are identified.

Our inheritance is somewhat different; we do not, for example, have the external examination. Nevertheless, there are points of similarity; and it may also be true here, as conjectured by the authors, that we can "define middle-class ... as ability to profit from the educational system!"

Today we in the United States are assailed by criticisms of our educational system in general, and mathematics in particular. As a result changes are taking place. History will tell us whether these are improvements, but there is less chance of making only superficial changes if we see the curriculum in its true perspective. Griffiths and Howson have presented a perspective that can be compared and contrasted to our own. Their book merits careful reading by all concerned with curricular reform.

2.3. Review by: H Neill.
The Mathematical Gazette 59 (408) (1975), 115-116.

This book on mathematical education developed from a third year course on Mathematical Curriculum Studies given by the authors at Southampton University. It is, as far as I know, the only book of its type to have been published in Britain so it seems useful to describe it in more detail than usual before giving an opinion about it.

The first part briefly discusses the nature of mathematical education before considering in more detail the historical development of the teaching of mathematics. Because examinations have very close links with the mathematics which is taught and the manner in which it is taught, the development of the English examination system finds a place here. The second and third parts are concerned with how changes in mathematics teaching have been caused by forces outside mathematics such as social pressures, educational research, educational technology and the availability of money, and how changes have been inspired by developments within mathematics itself and by increases in the areas of application of mathematics. Part 4 develops the mechanism of these changes, showing how these developments in mathematics teaching have been brought about in schools by professional bodies and curriculum development projects. Parts 5 and 6 are concerned with the mathematics curriculum, Part 5 being called The curriculum in the large and Part 6 being The curriculum in the small. In Part 5 we see an account of the development of the curriculum through to the 'modern' courses of the main school while Part 6 considers the mathematics in greater detail and the problems of teaching it in the classroom. Finally, Part 7 takes a closer look at examinations.

For me the core of the book was certainly the work on the curriculum, with the work preceding it serving to set the scene. Much of the material of the first four parts will be familiar to teachers of mathematics who are members of this Association, but it is useful to see it assembled in this way where it throws light on the situation in schools now. I kept finding myself saying "yes, that's it" when an idea that I held but had not formulated explicitly was put forward.

The part of the book dedicated to curriculum development is concerned mainly with the content of courses although it is recognised that content and methods of instruction cannot be separated. One of the courses discussed is that of the naval cadets at Dartmouth at around 1900. This is fascinating reading, consisting of a syllabus plus a commentary from the head of department justifying the inclusion of material and also suggesting methods of teaching and viewpoints on it. What a model for a present day department head to copy! Other more modern syllabuses are also discussed, one of them, naturally enough, being the SMP syllabus, but I found myself particularly interested by the objectives and reasoning behind the authors' curriculum studies course at Southampton.

The mathematical part of the book is concerned with the problems of communication between mathematicians and teachers, between mathematicians and children and between teachers and children. The mathematical educator (not a phrase I like) has to attempt to construct approaches in a school which lead naturally into the mathematics that the professional mathematician recognises, uses and approves. The authors have discussed this problem by considering geometry, the number system and applied mathematics.

The part of the book concerned with examinations discusses briefly the objectives of examinations and the way in which certain questions in specific examinations succeed or do not succeed in achieving these objectives. The book finally closes with some specimen examination papers.

Throughout the book, and to my mind an important part of it, are many exercises which are challenging and provocative. It would certainly not be possible for a reader to do all these exercises, but thinking seriously about at least some of them is a valuable part of assimilating the book. There is also a light touch running throughout the book; who could fail to be charmed by the idea of the Grand Peano Project? In short, this is an important book and would be profitable reading for anyone concerned with the teaching of mathematics or the teaching of teachers of mathematics. It could, if it is taken seriously, have a lasting effect on the development of mathematics teaching by bringing into focus questions concerning mathematical curriculum development which are in the background but not asked sufficiently often.

2.4. Review by: P Reynolds.
Mathematics in School 3 (6) (1974), 33.

A unique and scholarly reference book, on mathematical education, which grew out of the distinctive undergraduate course "Mathematical Curriculum Studies" which has been available at Southampton University since 1966. For many years, Professor Griffiths and Dr Howson, who both teach mathematics at Southampton University, have shown a real interest in school mathematics. About twelve years ago Dr Howson was the editor establishing the very new SMP textbooks, whilst Professor Griffiths (with Professor Hilton) was giving a Wednesday afternoon in-service course in Birmingham to awaken teachers (including me!) to modern mathematics - a series of lectures made permanent in Classical Mathematics (Van Nostrand Reinhold). Since then, both authors have continued to make substantial contributions to mathematical education.

The particular merit of the Southampton course is that students, who are capable mathematicians, are guided to investigate, with a mathematical approach, school mathematics. The book represents a textbook, a basic course, for such students. It informs them of the essential facts, provides plenty of references and sets exercises designed to stimulate further thought. It does this job very well.

However, its value to the ordinary classroom teacher is problematical. Clearly the teacher in training (or involved in further study) will find it useful, but its sheer size and level of discussion may mean that for everyday practical matters its contents are too inaccessible for speedy reference: its nature is reflective not pragmatic. Nevertheless, every secondary school mathematics department should have one for reference; every head of department planning a new syllabus or style of working would be better informed by consulting this book.

2.5. Review by: Philip Peak.
The Mathematics Teacher 68 (1) (1975), 52.

The basic source of information on which this book is based is the British system of mathematics instruction at all levels; however, it does draw comparisons and illustrations from the U.S. In the study of mathematics curricula, this book provides an excellent resource. It generalises on the problems of curricular change, the outside forces at work, and the internal built-in biases. The authors consider pupils, teachers, money, research, media, applications, professional associations, and developmental projects as factors of change and also consider each one's contribution. They close the text with a look at curricula in general, with a review of examinations as instruments of evaluation.

Each section includes a set of exercises that raise significant questions for the reader's consideration. There are 50 pages of specimen examination papers, a very complete bibliography, and a scope and sequence chart from A.D. 2000 to the present as visualised through English eyes. If you do not offer a course in mathematics curricula, this should at least be a resource for any teaching of a mathematics course.

3.1. From the Introduction.

Modern linguistics shares the delusion - the accurate term, I believe - that modern 'behavioural sciences' have in some essential respect achieved a transition from 'speculation' to 'science'. ... Obviously any rational person will favour rigorous analysis and careful experiment; but to a considerable degree, I feel, the 'behavioural sciences' are merely mimicking the surface features of the natural sciences; much of their scientific character has been achieved by a restriction of subject matter and a concentration on rather peripheral issues.

Chomsky, Language and Mind, 1968.

This study began as an attempt to compile an annotated bibliography of writings on the interaction of language and mathematical education. Interest in this topic has greatly increased in recent years and, in particular, conferences on this theme have been organised by UNESCO (Nairobi, 1974), the Commonwealth Association for Science and Mathematics Education (Accra, 1975) and the Ministry of Education of the Ivory Coast (Abidjan, 1978). However, in the course of our work, doubts similar to those expressed in the above quotation from Chomsky began to arise. Would a bibliography concentrating on research activities to date adequately reflect the ways in which language and mathematical education interact, or might it offer only a restricted view in which peripheral issues dominated? We felt that there was a real danger that the latter would indeed be the case. For that reason we have chosen to preface the bibliography by an essay. In this we indicate possible areas for investigation, and draw attention to those regions in which research activity has already commenced.

The book, then, differs from a survey paper within the field of mathematics. The latter aims to provide an exhaustive survey of the state of our knowledge: too often the present book can only indicate the extent of our ignorance. For that reason much of the writing may appear superficial. If, however, it helps to provide an incentive for less shallow work, then there is no cause for concern.

3.2. Review by: M J Cahill.
Mathematics in School 9 (5) (1980), 34.

This study has been reprinted from Educational Studies on Mathematics 10 (1979) and consists of an annotated bibliography of writings on the interaction of language and mathematical education, prefaced by an essay in which the authors indicate possible areas for investigation and draw attention to those regions in which research activity has already commenced.

They point out that studying this interaction between mathematical education and language is, in fact, viewing the whole of mathematical education from one particular vantage point. The discussion is structured via a framework consisting of the language of the learner, the language of the teacher and, finally, the language(s) of mathematics. The result is an extremely readable informative essay which gives plenty of encouragement to mathematics teachers to pay some attention to linguistics. Sections are included on language and concept formation, readability, language in the classroom, mathematical symbolism and linguistic structure and the whole essay runs to some 20 pages. About 240 articles or reports are included in the bibliography and these are drawn from writers on linguistics as well as from the pages of this and other journals.

4.1. From the Publisher.

In the mid-1970s the curriculum development boom in mathematics was to end almost as rapidly as it had begun. In this book the authors, who come from countries with differing educational traditions and patterns, consider these developments in their historical, social and educational context. They give not only a descriptive account of developmental work in a variety of countries, its aims and the patterns of management utilised, but also attempt to identify trends and characteristics and thus provide a theoretical base for criticism and analysis. The reader will find numerous case studies, including extracts from such renowned authors as Bruner, Dieudonné and Piaget.

4.2. Review by: David M Neal.
Mathematics in School 10 (5) (1981), 38-39.

Teachers who, like me, still find Mathematics: Society and Curricula a valuable resource book will be eager readers of this new scholarly analysis of curriculum development in mathematics.

The rapid curriculum change of the past two decades has given way to a more reflective stage of curriculum evaluation. The authors of this new text have provided a base for analysis and criticism by setting curriculum development in mathematics in "both a historical and a more general social and educational context".

The book begins with a study of the components of curriculum development, the historical influences and then examines three curriculum projects in mathematics; the projects being chosen to demonstrate the variety of approaches which have been adopted. It continues with an examination of the practice of the management of curriculum development in schools and how this management reflects the various theories of curriculum change. It then takes a retrospective look at the periods of reform in both Great Britain and the USA before analysing in some detail the problem of curriculum evaluation. It concludes with some pertinent thoughts and questions about the lessons for today and tomorrow.

A book which will be appreciated most by those teachers and students who are involved in post-graduate, in-service and other courses of mathematical education where detailed study will be made of the questions raised at the end of each chapter of the book and the extensive bibliography. It should also prove a valuable resource for all concerned in curriculum development in mathematics.

4.3. Review by: Morris Kline.
The American Mathematical Monthly 91 (2) (1984), 150-151.

The failings - one could say the failure - of mathematics education on all levels are so well known that substantiation of this fact is hardly necessary. Intelligent people, many of whom have done superb work in non-mathematical areas, fail to acquire even a modicum of mathematical knowledge and indeed are repelled by mathematics of any sort. Surely these people would not deny the importance of mathematics not only for science and engineering but also for its impact on all branches of our culture including the very branches in which these intelligent people have made their mark. Such an anomalous situation calls for an explanation and remedy. The role of mathematics in our civilisation is too fundamental and vital to allow such a situation to be tolerated.

The explanation involves what is certainly the major feature of mathematics education, the curriculum. The curriculum is fashioned by mathematicians. But mathematicians, especially in our time, make assumptions which vitiate what they seek to achieve. One assumption is that mathematics proper is intrinsically attractive and should interest all students. Indeed in the past twenty or so years we have had curriculum developments that not only concentrate on mathematics proper but stress many fine points - rigour - that eluded the best mathematicians from Greek times to about 1900. This approach adds insult to injury. One such development, popularly known as the new math, fortunately fell flat on its face, and now there is no need to whip a dead horse. But a return to the older, somewhat more intuitive curriculum is also not the answer. This type of curriculum was a failure for many generations before the new math was fashioned.

Should we then abandon the teaching of all mathematics beyond the practical elementary arithmetic except perhaps to future scientists and engineers and those few "perverted" young minds who for some queer reason take a liking to mathematics? Of course not. Mathematicians generally know the importance and relevance of the subject even though they fail to convey it in the curricula. This is the major failure that must be remedied.

This failing has been emphasised by no less a man than Hermann Weyl. In his Monthly article of October 1951, Weyl said:

One may say that mathematics talks about things which are of no concern at all to man. Mathematics has the inhuman quality of starlight, brilliant and sharp but cold. But it seems an irony of creation that man's mind knows how to handle things the better the farther removed they are from the centre of his existence. Thus we are cleverest where knowledge matters least: in mathematics, especially in number theory.

The proper approach to mathematics education was urged by Alfred North Whitehead as far back as 1912. In his essay "The Aims of Education," Whitehead said:

In scientific training the first thing to do with an idea is to prove it. But allow me for one moment to extend the meaning of prove; I mean - to prove its worth. ... The solution which I am urging is to eradicate the fatal disconnection of subjects which kills the vitality of our modem curriculum. There is only one subject matter for education and that is Life in all its manifestations. Instead of this single unity, we offer children Algebra, from which nothing follows; Geometry, from which nothing follows.... Our course of instruction should be planned to illustrate simply a succession of ideas of obvious importance.

In some of his essays in his book Essays in Science and Philosophy, Whitehead elaborates on the above theme.

In substantiation of Whitehead's suggestion I might note that I have asked any number of high school teachers why anyone should learn how to solve quadratic equations or trigonometric identities. Did they have any occasion to use this knowledge outside the classroom? The answer was an abashed silence.

The answer to curriculum deficiencies is in broad terms relevance or application. These may vary from games and puzzles to physical and social science applications depending upon the ages of the students. Such motivations are available. But they must not be pieces of candy thrown out occasionally to cheer up the students. They must be an integral part of the curriculum and they must be inserted in the texts in the proper places where they will be most effective.

To many professional mathematicians this motivation may be judged as desecration or a waste of time that could be better devoted to more mathematics. But even God's words must be interpreted for the layman.

If the contention that mathematics courses must contain motivations drawn from outside of mathematics is correct, then it is a corollary that teachers must be trained to present such material as carefully and as assiduously as they now present mathematics proper.

If any additional evidence is needed to support the contention that mathematics courses must teach not only what mathematics is but also what mathematics does, the book under review supplies it at least indirectly. The essence of the book is a review of the many new curricula fashioned during the last twenty or so years not only in the United States but also in several foreign countries. It is a valuable record even though it is a record of failures to improve mathematics education.

The authors state that the boom in curricula has about ended. Has work on new curricula ended? No! Now many teachers are on new binges: problem-solving, back to basics, developmental mathematics, and the New York State Department of Education three-year high school curriculum.

Anyone planning a new curriculum should certainly study this history of the various factors that motivated and in the end defeated the objectives of the many movements. The authors are experienced and competent.

4.4. Review by: John K Backhouse.
The Mathematical Gazette 66 (435) (1982), 70-71.

This book is a timely one. After the flurry of curriculum development in the 1960s and 1970s, it is possible to look back during a period of comparative calm when people are reappraising the changes that have taken place. The meaning of "curriculum" has been extended in recent years to include aims, teaching methods and assessment procedures as well as content, so the authors have a wide field to cover.

From the authors names, many readers will deduce that the book crosses national boundaries. This means that the authors have been faced with a major problem of selection while they simultaneously have the opportunity to take examples from any country which provides suitable instances of curriculum development. They also faced the problem of readers' greatly different experiences working in national systems of education. They have therefore provided a considerable number of lengthy quotations to fill the gaps in readers' knowledge, and this has been done very effectively. Another device they have used is to outline three case studies to which they refer back later in the book. International collaboration is not easy but the authors have attempted a unified text rather than chapters written by named individuals. One may guess at the authorship of sections from their examples but the flow of ideas is not impeded by this.

Curriculum development is placed by the authors with a quick sketch in an historical background; they are more thorough and successful in placing it in its social context. The emergence and nature of a project depend on various factors: society's aims and traditions, social and geographical factors, the professional status and training of teachers, the organisation of the educational system, the place of textbooks and publishers and, of course, the way the project is managed - in particular the relation between the 'centre' and the 'periphery'. One aspect of the work which the reviewer found most helpful was the allocation of projects to types depending on their theoretical approach: behaviourist, new-math, structuralist, formative, and integrated-teaching. While any such analysis must inevitably simplify, it helps to make the mass of information about projects both more digestible and understandable. The chapter following this analysis applies it first to the American based projects and then to those with an English base. This chapter is well provided with illuminating examples from project texts.

While there is no generally agreed theoretical approach for curriculum developers to adopt, perhaps evaluation provides the thorniest issue in this book. The chapter on this topic is well provided with quotations and accounts are given of the evaluation of projects with different theoretical approaches. There is no necessity for the evaluator to use the same theoretical approach as the initiators of the project and this led to some interesting situations.

By the end of the book the reviewer was receiving a very clear message that the success of a project in producing a lasting change in curriculum largely depends on the way in which teachers are involved in it. It is all too easy for them to turn project materials to their own ends instead of those of the project developers. Since every teacher is involved in curriculum development, the authors write, there are obvious reasons why he should know as much as possible about its construction and be able to examine it critically. Clearly with the book priced as it is, not every teacher will read it. The publishers blurb on the jacket commends the work to all those seriously concerned with the development of mathematical curricula especially students of mathematics education or curriculum development. This is a far more realistic readership and to such the reviewer commends it, particularly as it fills a gap in the literature. Such students and their tutors will find in the text plenty of thought-provoking exercises intended to form the starting point for discussion, further reading, dissertations, theses or general research work. Nevertheless it would be a pity if the readership were so confined.

5.1. From the Publisher.

The teaching of mathematics has a history stretching back some hundreds of years. From its infancy through to its adolescence institutions at which mathematics was taught were thinly and somewhat haphazardly spread over the country and so individuals were extremely influential in developing curricula and methods of teaching. Indeed this has continued to be a feature of English mathematics education. In this authoritative account Geoffrey Howson follows the history and development of mathematics teaching by looking at the careers of some of these individuals in detail.

5.2. From the Preface.

To my knowledge this is the first book to be published which attempts to tell the story of the development of mathematics education in England. That this should be so is rather surprising; for one can turn to histories of the teaching of science and to a history of mathematics teaching in Scotland. Any attempt to fill such a gap is, therefore , fraught with difficulties, for the 'only book' is likely to be invested with an authority it may not deserve. Extra problems may also arise as a result of my having chosen to present the material through the medium of biographies, Emerson's claim that 'there is properly no history; only biography" could be used to justify this decision. The truth is, however, more mundane; for I abandoned a 'chronological' account thinking it would have little appeal for the general reader as opposed to the serious student. I believe also that a biographical account, even though it requires frequent scene-settings and 'flashbacks', better demonstrates the great part which individuals have always played in the advancement of mathematics education in England. The book, however, is not to be compared with Macfarlane's Ten British Mathematicians of the Nineteenth Century, for that author set out to identify the ten 'greatest' mathematicians of that age. My basis for selection has been different, for I have chosen subjects from various periods and traditions whose own mathematical education and whose contributions to mathematics teaching provided a framework around which I could construct a representative story. Often there were competing candidates: why, for example, Godfrey, rather than Perry, Nunn or Siddons! The answer lies not in the weight of their contributions to mathematics education, but in their attraction to me as persons and in a number of diverse associations. The reader is warned, therefore, as he should be when reading all histories, that the work has a subjective air. Yet, although the choice of descriptions and quotations is bound to reflect my own interests and beliefs, I have attempted to refrain from explicitly drawing parallels and linking the past to the present. Nevertheless, I hope the reader will think throughout about implications for today, for I have always aimed to show that history can not only help us to understand the past, but, more importantly, that it can lead us better to comprehend the present.

5.3. Review by: John K Backhouse.
British Journal of Educational Studies 32 (1) (1984), 95-97.

This book has an unusual format. Howson abandoned a chronological account, thinking it would have little appeal to the general reader and he has boldly presented his material in nine biographies: Robert Recorde, Samuel Pepys, Philip Doddridge, Charles Hutton, Augustus De Morgan, Thomas Tate, James Wilson, Charles Godfrey and Elizabeth Williams. These names indicate the period covered. The first account starts in the 16th century and the last brings the reader up to 1960. These biographies are sandwiched between a Prelude which refers to the coming of St Augustine in 597 and a Postlude of seven pages, concluding with the Cockcroft report. This is followed by an appendix containing a 'selection of examination papers, syllabuses, etc.' which illuminate the text in a way that is only possible by the perusal of specific questions and syllabuses.

The list of nine people comes as a shock. Was Pepys a mathematics educator? '... without having any pretensions to being a mathematician, Pepys was to have considerable influence on the development of mathematics education in England ..' So Howson has no intention of selecting the nine most important mathematics educators since the mid-16th century. Rather, he has chosen people 'whose own mathematical education and whose contributions to mathematics teaching provided a framework around which I could construct a representative story.' In some cases the choice was one of personal preference, as others from the same period could have done equally well. What he has done is to weave a story around these nine people. So he has had to provide a considerable amount of scene-setting and 'flashbacks'. But the result of the nine periods of history with their human touches is a most readable account of the development of mathematics education in England. There were times when I thought that comparisons with other centuries would have been illuminating (there were some references) but the scope of the book is so wide, covering as it does university to infant school mathematics, that this would not have been feasible.

The book contains about 640 notes. The character of each is indicated to the reader in three ways: parentheses for sources of further reading, square brackets for biographical notes and curly brackets for those of a more general expository nature. This I found to be a very helpful device in view of the very large number of notes. These are most rewarding. There are many containing information which, while it would distract from the flow of the text, adds to the interest of the body of material presented. In this way the cover of the period is made more complete; indeed, a considerable number of names in the index are to be found only in the notes. There are two indexes, for names and subject matter, but no bibliography. This may not matter to those who read the book for pleasure, but it could be very irritating for someone using it for reference. For instance, it occurred to me to check whether reference was made to W W Westaway's Craftsmanship in the teaching of elementary mathematics; Westaway is not mentioned in the index although his book is referred to in what I take to be an extract from an interview with Mrs Williams.

As the first book on the history of mathematics education in England, and an authoritative one which may be expected to deserve its place on bookshelves for many years to come, it is to be welcomed as a valuable addition to the bibliography of mathematics education, and as a contribution to scholarship. But it also deserves a wider readership since any curious person with a lively mind will find it natural to inquire how we have arrived at our present state of affairs. So on these grounds the book may be commended to the considerable number of people who are involved in teaching mathematics at all levels in England, with the proviso that the reader cannot expect to find very much on post-1960 developments.

The reader who takes up this book may expect to be provoked to reflect on our present-day problems in mathematics education. While there have undoubtedly been considerable changes in syllabus content, several problems are shown not to be new. The danger of rote learning is mentioned in Recorde's Ground, which first appeared in 1543. Rote learning is presumably due to a shortage of good mathematics teachers. We find that in 1713, because of their shortage, the Act to Prevent the Growth of Schism specifically excused teachers of mathematics, navigation or mechanical arts from subscribing to the Thirty-nine Articles of the Church of England. (This is particularly topical in view of ACSET's recent recommendation to terminate exemption from training.) Then one is forced to think about the pace of curriculum change on reading that Priestley in 1768 was advocating as essential subjects exactly the same subjects that Howson took in his School Certificate in the 1940s: Latin, English, French, mathematics, physics, chemistry, history and geography. The domination of the curriculum by examinations is not new either. About 1911, Godfrey wrote, 'English education is dominated by examinations. Examiners cannot test outlook and they cab test understanding only by testing manipulation ...'
...

Certainly Howson shows us that there have been some changes, besides syllabus over the period covered by the book. His final biography is that of a woman and is clear that girls and women are now far more commonly studying mathematics on a par with boys and men. That there should also have been changes in views on psychology comes as a matter of little surprise.

Looking back at the past we can see that there have been periods of development in mathematics education, and times of stagnation. The excellent Mathematical Association report The teaching of geometry in schools (1923) effectively crystallised geometry teaching. Are we now in a period of stagnation? Have the developments of the 1960s and the current economic recession left teachers of mathematics in a state where they simply want to survive? The Cockcroft Report has stimulated much interested talk; which of its ideas, few of which are new, will be implemented in the next ten, twenty, or three hundred years?

5.4. Review by: I Grattan-Guinness.
The British Journal for the History of Science 17 (1) (1984), 97-98.

It is a shame to start a review of such a useful book on a note of caution, but the first problem arises with the title. Howson provides a history of mathematical educators, nine figures from Recorde of the 16th century to Williams of our own time via Pepys, Hutton and de Morgan, among others. While he exercises great skill in placing appropriate developments in mathematics education at appropriate places in several chapters, the role of other figures does not come out sufficiently, and the biography as such is interrupted.

In his preface Howson alludes to this difficulty, but omits another one - the level of the mathematics education being described. Most of the account deals with school mathematics, but at times (in the de Morgan chapter, for example) university education takes a larger role. The overall impression is incoherent from this point of view, however, and several important university-level developments are passed over. The Cambridge reform of the 1820s, for example, gave special places to mathematics and logic, especially through the influence of Whewell, who receives very little space in this book. Connections between mathematics, logic and psychology were common in 19th-century England, and they were taken into education too. Boole is a particularly good example; but only his widow, who was actually proselytising his ideas on education, appears here.

One of the reasons why university-level developments need study on the history of school mathematics is that, for better or usually worse, university demands have influenced school curricula. This point emerges several times in the book, with a good discussion provided on pp. 163-166 of the International Commission for Mathematics Education, which was very active in the early decades of this century and produced some valuable studies, especially in Germany. Howson quotes Godfrey, his biographee of the moment, that 'in Germany the freest use is made of intuition, and the sphere for rigorous mathematics is admitted to be the university'. The opinion should be qualified, however, for the prominence given to intuition was a view of Gottingen's Klein, very active in the Commission; it was derided by the Berlin purists. A most important chapter in the history of mathematics education lies here - and not only for Germany but also in England, when Hardy and Littlewood began to prosecute Berlin-style mathematical analysis, with important influences acting in turn on school calculus.

Another feature of the story which comes over well is how small a role has been assigned by mathematics education to the history of mathematics. Although a 1919 report gave as its (last) recommendation 'That portraits of the great mathematicians should be hung in the mathematics classrooms, and that reference to their lives and investigations should frequently be made by the teacher in his lessons', the suggestion has almost always been ignored. Thus the history of mathematics has remained a backwater, and the history of mathematics education with it.

This collection of biographies constitutes a valuable contribution to a neglected theme. A notable feature is the end-notes, which contain much valuable material and are helpfully keyed by the author in his text by three different kinds of bracket around the note number, to discriminate between the kinds of information buried away at the end of the book by the publisher.

5.5. Review by: Richard C Weimer.
The Mathematics Teacher 76 (6) (1983), 446.

This historical development of mathematical education in England is based on biographical accounts of the following nine individuals: Robert Recorde, Samuel Pepys, Philip Doddridge, Charles Hutton, Augustus De Morgan, Thomas Tate, James Wilson, Charles Godfrey, and Elizabeth Williams. These nine essays represent different ages, traditions, interests, and attitudes covering a period of approximately 450 years. An appendix contains a selection of syllabi and examination papers dating from 1800 to the present, twenty-one pages of notes elaborating on details, an index of names, and a subject index.

5.6. Review by: James G O'Hara.
Isis 75 (3) (1984), 575.

The author, a university reader in mathematical curriculum studies, traces the history and development of mathematics education in England proper from the Middle Ages to the present by looking at the careers of individuals who were influential in developing curricula and methods of teaching. The book provides biographical essays in all on nine characters representing different ages, traditions, interests, and attitudes; the mathematical education of each character and his or her contribution to mathematics education are described and set in a national context; besides the nine a great deal of information is provided about others who influenced the course of English mathematics education. The work includes an appendix containing a selection of syllabuses, reports, and examination papers (e.g., Cambridge Tripos, University of London Matriculation, Teacher Training College and local scholarship examinations).

The author prefers biographies to a chronological account, partly to cater to the general reader but also because he believes that biographies best demonstrate the influential roles of individuals in developing curricula and methods of teaching. The nine chosen to provide a framework for are presentative story are Robert Recorde (1510-1558), Samuel Pepys (1633-1703), Philip Doddridge (1702-1751), Charles Hutton (1737-1823), Augustus De Morgan(1806-1871), Thomas Tate (1807-1888), James Wilson (1836-1931), Charles Godfrey (1873-1924), and Elizabeth Williams (1895-1986). The subjects were chosen according to the author's personal preference rather than for the weight of their contributions to mathematics education. Far from being the greatest mathematicians of their respective centuries, the chosen individuals were generally of secondary importance to mathematics proper, De Morgan being perhaps the only one of the nine who was also an outstanding mathematician. To be sure, the greatest mathematicians were often poor teachers or had little involvement with education or curriculum development.

This history of mathematics education in England has been written especially for teachers, lecturers, and students of education as well as for amateur mathematicians; it assumes no prior knowledge on the part of the reader. Few historians of science and technology are likely to read it from cover to cover; however, it does provide some interesting reading for the serious scholar in the field. There is considerable overlapping with such fields as history of institutions, organisation of science, science and education policy, and philosophy of education. To sum up, the book provides a good overall view of the main developments and movements in English mathematics education, and the author, though not a professional historian, shows a thorough acquaintance with the literature and methods of historical research in his field.

5.7. Review by: Bryan Lang.
The Mathematical Gazette 67 (442) (1983), 309-310.

Perhaps it is a sign of the reviewer's advancing years that a selling price of £25 suggests a volume of coffee table proportions, lavishly illustrated and beautifully prepared! In this case an outlay of £25 has a return of some 200 pages of text supplemented by 41 pages of detailed textual notes, with virtually no diagrams or other illustration; all of which does not pre-dispose a reviewer to be charitable despite the immaculate presentation and the evident care which the production must have entailed. Having made that point, it must be said at once that the book has absolutely no need of charity to proclaim its importance for mathematical educators. The author begins his preface with the claim that 'this is the first book to be published which attempts to tell the story of the development of mathematics education in England' and even a preliminary glance at the text assures the reader that any such prior material would not have escaped the scrutiny of so thorough a researcher.

In a country such as England where there is no history of centralised control over curricula, so that in any given period each school has responded to its perception of needs in its own individual way, a history of development has to wrestle with many complexities. This presents a considerable challenge to any writer who sets out to clarify the progression over almost 500 years, distinguishing major events from minor and pointing up significant influences at each stage of development. The author meets and overcomes this challenge admirably.

The device which he has chosen is a biographical one, with each of the nine chapters devoted to one person who played some part at a critical stage of the development of mathematics education in England. As some of the names might surprise prospective readers, they are worth listing here: Roberte Recorde, Samuel Pepys, Philip Doddridge, Charles Hutton, Augustus De Morgan, Thomas Tate, James Wilson, Charles Godfrey and Elizabeth Williams. Those who do not have central importance serve as catalysts for discussion of the major issues of mathematics education in their time and all have relevance for the special character of their own education in mathematics. Extensive notes on the main text are collected in later pages and each foot-note number is enclosed by brackets of a particular sort to distinguish for the reader the character of the notes, either further readings, biographical notes or general, expository commentary. The main chapters are bounded by prelude and postlude, and there is an appendix of selected examination papers and syllabuses from the years 1802 to 1944.

A major source of enjoyment for readers is the way in which contemporary issues of mathematics education can be detected in the writings of mathematicians for up to 500 years. Recorde wrote that '... you muste prove yourselfe to do some thynges that you were never taught, or els you shall not be able to doo any more then you were taught, and were rather to learne by rote (as they cal it) than by reason'. So the rote-learning debate began in the sixteenth century! De Morgan pioneered in-service training of mathematics teachers in the early part of the nineteenth century. Steps towards a process model of the curriculum were first taken, also by De Morgan, who wrote that 'the actual quantity of mathematics acquired ... is ... of little importance when compared to the manner in which it has been studied'. In the mid-nineteenth century Tate offered principles of teaching which will be immediately recognised by followers of Bruner and Paiget as being central to their models of educational psychology. This book is full of such quotations and illuminating commentaries which put present-day problems into a developmental context. They give a perspective to the criticisms of the Cockcroft Report that 'much of it was being said twenty years ago and nothing happened, so why expect it to do so now?'; the evidence from history is that even the best of ideas takes a long time to become widely accepted and has to be continually re-iterated.

In the Postlude developments are brought forward from 1960 as far as the 1982 publication of the Cockcroft Report. The author discusses a number of contemporary concerns such as the considerable pressures on the teaching of mathematics from outside the profession, increasing attempts to control the curriculum by politicians and society at large and difficulties of differentiation of students and curricula in comprehensive education. In his view, the recently established research in mathematics education has so far produced little which can be interpreted into classroom practice and, in general, solutions to these problems have not yet been found. He concludes with the sentence;

It is essential then that careful consideration should be given to those processes of change and I hope that this book will contribute to a greater understanding of them.

The book does contribute to our understanding in large measure. Its style makes it accessible to the casual reader, yet its accurate and careful identification of sources are all that a serious student of the subject could wish. It contains considerable food for thought for everyone involved in mathematics education and deserves to be widely read.

5.8. Review by: Morris Kline.
Mathematical Reviews MR0683878 (84b:01057).

This book is devoted to what the title describes and covers the period from about 1500 to 1960. A Prelude sketches the history of mathematics education in England up to 1500. The individual chapters of the book proper are each headed by and largely concerned with the work of one man, such as Robert Recorde, Samuel Pepys, and about ten others. However, the chapter titles are somewhat misleading in that each chapter also describes the work of educators whose work was closely related to the principal figure of that chapter.

The account deals with the rise of mathematics education at all levels from grammar school to the universities. What is surprising is the slow spread of that education and the opposition to it from various quarters such as the Anglican Church. Mathematics in particular does not contribute to the spread of Christianity and so was not, to say the least, favoured. The low state of mathematics education in the 17th century, to say nothing of earlier and later ones, makes one wonder how England could have produced Newton.

Beyond what the book has to say about the rise of mathematics education, it has much historical material about the rise of mathematical research and biography that would justify a broader title. In particular the chapter devoted to Augustus De Morgan and the partial contents of other chapters are as significant for the history of mathematics proper as for mathematics education.

There are some interesting quotations which are echoed in the arguments one hears or reads about education in the United States: There are no bad pupils; there are only bad teachers.

There is an interesting Appendix on the contents of English examination papers used in the past few centuries.

6.1. From the Abstract.

This third volume of a review prepared for the Cockcroft Committee of Inquiry into the Teaching of Mathematics in Schools in Great Britain concerns research with relevance to the mathematics curriculum. The role of history is discussed briefly in the introduction, and has an impact throughout the report. The 13 chapters consider: (1) introduction, (2) early curriculum development and theories, (3) curriculum development in England in the early 20th century, (4) the Consultative Committee and mathematics, (5) later development of curriculum theories, (6) strands of curriculum development, (7) lessons to be learned, (B) evaluation in Britain, (9) evaluation in the United States, (10) the management of curriculum development, (11) centres and networks for curriculum development in mathematics, (12) common problems, and (13) the teacher's role in curriculum development. A bibliography is found at the end of the report.

6.2. From the Preface.

This review is in three parts, differing to some extent in subject, in style and in format. In particular, in the section on the Social Context of Mathematics Teaching the main discussion of the material comes first and is followed by Conclusions and Recommendations. In the section on Learning and Teaching, the main conclusions to which the work leads are stated in the Introduction, in a similar way, the introductions to the chapters are used to place the material in perspective and to relate it to current practice.

Part C is essentially as it was presented to the Cockcroft Committee, but new references have been added where they are of particular interest and reflect more recent developments.

6.3. From the Introduction.

A 1973 paper by D F Walker bears the title 'What curriculum research?' It is a significant question, for, indeed, there are few research findings relating to the curriculum which can be quoted. Goodlad (1969) summed up the position when he spoke of 'the paucity of ordered "findings" from curriculum research - findings in the sense either of scientific conclusions from cumulative inquiry or of tested guidelines for curriculum decisions.' The position has not changed significantly in the years that have elapsed since the two papers quoted (and which referred to education in general, and not mathematics education in particular) were written.

The reasons are not hard to find. Although small-scale research into particular constituents of a curriculum can be readily carried out, investigations relating to the entire curriculum are much more expensive and difficult to mount. Such as have been mounted, and to which we shall refer later (e.g., NLSMA, IMU) have not been particularly successful in illuminating general issues. The construction of appropriate research models has not proved easy, for what was to be investigated was something exceedingly complex and the initial situation could often never be replicated.

Yet it would be foolish to ignore the developments of recent years simply because of the scarcity of 'objective' research findings. What the world has seen has been the greatest classroom experiment ever in the field of mathematics education. That experiment has taken a variety of forms and the outcomes have been many. Any attempt to summarise and to select must therefore reflect personal bias and the reader must take this into account.

In a recent book, the American historian J H Hexter divided historians into two camps - the 'lumpers' who attempted to identify movements and themes, and the 'sifters' who always have a counter-example to hand to deny any general statement. The reader is warned that, on this occasion, the writer identifies himself with the 'lumpers', whilst recognising that a 'sifter' might well tell a different story.

The reference to history is not merely a passing one, for if one is to understand the events of the past twenty years it is essential to see what preceded them. As Goodlad indicates, the best a curriculum developer can hope for is guidance and theory arising from cumulative inquiry and experience. It is essential then, when considering recent curriculum development and methods of dissemination and implementation, to have knowledge of the way in which such processes were traditionally carried out. Only when considered in a historical context can the achievements of the past twenty years be best evaluated.

6.4. Review by: Michael C Hynes.
The Arithmetic Teacher 32 (7) (1985), 57.

Howson begins the text by emphasising the scarcity of research findings related to the mathematics curriculum. Rather than relating isolated evidence from research that one curriculum design is superior to another, he tells the story of curricular development in mathematics from a British point of view. The historical perspective offered in the book is concise and insightful, and the information about curricular projects and theories is well formulated and complete. The British perspective is dominant, with references to the influence of American educators and curricular projects. (One chapter is devoted to evaluation studies conducted in the U.S.)

The book would be a useful reference for mathematics educators who are interested in curricular development and reform. It could be helpful as a companion to NCTM's History of Mathematics Education in the United States and Canada in a graduate-level course dealing with the mathematics curriculum.

7.1. From the Publisher.

This 1987 study seeks to identify key issues and basic questions within mathematics education, to propose and comment upon alternative strategies, and to provide a stimulus for more detailed, less general discussions, within more limited geographical and social contexts. The text is based upon an international symposium held in Kuwait in February, 1986 and attended by selected mathematics educators drawn from all parts of the world.

7.2. From the Foreword by Jean-Pierre Kahane.

The International Commission on Mathematical Instruction is planning a series of studies on topics of current interest within mathematics education. The first study was on the impact of computers and informatics on mathematics and its teaching at university and senior high school level. That study had as its centre-point an international symposium held in Strasbourg, France in March, 1985 and attended by some seventy participants drawn from thirteen different countries.

The second study, which gave rise to this volume, has taken a slightly different form. First a discussion document, School Mathematics in the 1990s by A G Howson, B F Nebres and B J Wilson, was sent to all National Representatives of ICMI and circulated widely in the original English and in translation. A small, closed international seminar was then held in Kuwait in February, 1986 at which an invited group of mathematics educators, named on the title page of this book, considered issues raised in the discussion document, points made by those who had responded to that paper, and, of course, attempted to remedy its many omissions. This book is based on those discussions and has been prepared by Geoffrey Howson and Bryan Wilson. It is, as such, a compilation of views and certainly would not have the effect of drawing discussion amongst those who participated in Kuwait to an end. Its aim, however, is not to terminate discussions, but rather to provoke and stimulate them. Further serious and detailed debate will be required before sound responses to problems can be formulated. It is ICMI's hope that this book will facilitate such debate and decision-taking.

The subject is a vital one, for there are few issues which are of such concern to all countries throughout the world. Mathematics education faces a vast variety of challenges as we move towards the 1990s and it is by no means certain what responses to some particularly crucial questions will prove most appropriate. There are few certainties. Perhaps one of the few things of which we can be certain is that ICMI's approach to the problems must be based on a broader appreciation of mathematics education than is suggested by its somewhat outdated name. 'Instruction' or 'teaching' will always remain a key issue for us, but, as is emphasised in this volume, 'learning' demands equal consideration, and, what is still insufficiently recognised, this must include consideration of what mathematics students learn, and of the mathematical activities in which they engage, outside, as well as in, the classroom.

We hope also that this book will prove timely. In many countries there is a growing lack of confidence in our ability to teach mathematics successfully. Education is poorly regarded. Such a crisis of confidence must be overcome, for there is no doubt that well-informed, active citizens in the 1990s will require more mathematics and a greater comprehension of mathematics. We must react against any tendencies to see the problems of mathematics education as intractable.

Finally, I should like to acknowledge with gratitude the leading role which Geoffrey Howson has played at all stages of this study.

7.3. Review by: John Leamy.
The Mathematics Teacher 81 (1) (1988), 76-77.

The last several years have seen considerable discussion about the future of mathematics education in the United States. This volume, the result of an international conference held in Kuwait in February 1986, considers the question from an international perspective. The participants, from thirteen different countries, ask questions about what mathematics should be taught to whom in the next decade and what techniques could be used. Specific topics discussed include the place and aims of mathematics in the schools; mathematics and general education goals; the content of the school mathematics curriculum; and teachers and classrooms in the 1990s. The authors do not propose solutions. After a brief but complete discussion of a topic, options are presented in an alternative-possible-consequences format. The intent is to promote discussion and debate, and the authors stress that different countries may find varying solutions to the questions posed.

A distinguishing feature of all the discussions is the recognition of the role that political considerations have played and will continue to play in many curricular decisions.

This book should be required reading for anyone involved in curriculum planning or teacher training; it is a valuable resource to anyone involved in the teaching of mathematics. A very good bibliography is provided for anyone wishing to do further study.

7.4. Review by: Bonnie H Litwiller.
The Arithmetic Teacher 35 (4) (1987), 42.

This book is the second of a series published by the International Commission on Mathematics Instruction (ICMI). The series addresses topics of current interest within mathematics education; the first in the series concerned the impact of computers and informatics on mathematics and its teaching.

The development of the book is as follows. First a discussion document, School Mathematics in the 1990s, was sent to all national representatives of ICMI. Then a small international seminar was held in Kuwait in February 1986 at which an invited group of mathematics educators considered issues raised in the discussion document and points made by those who responded to the paper. This book is based on the discussions. Its aim is to provoke and stimulate further discussions. The nine areas of discussion are (1) mathematics in a technological society, (2) mathematics and general educational goals, (3) the place and aims of mathematics in school, (4) the content of the school mathematics curriculum, (5) particular content issues, (6) classrooms and teachers in the 1990s, (7) research, (8) the processes of change, and (9) the way ahead.

This book is recommended for those mathematics educators who are interested in the mathematics curriculum at the international level. This well-written book will stimulate considerable discussion.

8.1. Review by: John W Vander Beek.
The Mathematics Teacher 85 (2) (1992), 152.

In the light of another round of falling mathematics SAT scores and poor ranking at the international level, mathematics educators of the United States may do well to look at how other countries cope. This volume offers a European's summary of the European Community's (plus Hungary's and Japan's) national curricula in mathematics. By the author's own admission, little of what is intended by those who draw up national curricula is ever implemented, but those curricula do carry a message about the system - its aims, aspirations, and shortcomings. Thus this book can be valuable reading for those who are truly concerned about the state of mathematics education.

The author summarises the national curricula of thirteen countries and the United Kingdom. These summaries are necessarily brief outlines of very extensive documents in an effort to encourage educators to examine and compare these curricula in light of the various social and cultural traditions from which they arise. The summaries are preceded by the author's comments on national curricula in general, observations concerning their divergence, and his personal view.

The first part of the book is written in an easy style and becomes necessarily encyclopaedic when the summaries are presented. This book can be a good stimulus for reflective reading by mathematics educators everywhere.

9.1. From the Publisher.

Mathematics Textbooks: A Comparative Study of Grade 8 Texts is the third in a series of monographs that is being produced by the Third International Mathematics and Science Study (TIMSS), an international comparative study coordinated from the University of British Columbia. In Mathematics Textbooks: A Comparative Study of Grade 8 Texts. Geoffrey Howson, Emeritus Professor of Mathematical Curriculum Studies at the University of Southampton, England, examines eight mathematics textbooks for 13-year-olds for their pedagogical and philosophical similarities and differences. Texts from the United States, the Netherlands, the United Kingdom, Norway, Spain, France, Switzerland, and Japan are carefully studied. Professor Howson has published extensively in the fields of mathematics, education, and mathematics education, and has been closely involved in curriculum development for over thirty years. He has visited educational institutions in over forty countries.

9.2. Review by: Diane Hoemeke.
The Mathematics Teacher 89 (3) (1996), 258.

One product of the Third International Mathematics and Science Study is Mathematical Text books: A Comparative Study of Grade 8 Texts by Geoffrey Howson. This monograph is an analysis of textbooks used by thirteen year-old students in six European countries, Japan, and the United States. Howson hopes that this international comparative study of classroom textbooks will help future authors and promote further research in the field.

Howson recognises that although many different materials are used in the classroom, the textbook continues to be "the main published aid to learning and teaching." He presents a critical discussion of the historical and appropriate roles of textbooks in the mathematics classroom.

Howson's methodical analysis of the content of the selected textbooks is most useful. Readers will find his comparative charts - with country-by-country listings of content - a real highlight. It is interesting that most countries teach only a few topics at the eighth-grade level, with the notable exception of the approach in the United States, which seems to take on an "encyclopaedic nature." In addition to content, Howson looks at how application, presentation, and integration with other disciplines are managed. He does caution that what actually transpires in the classroom cannot be determined solely on the basis of the textbooks.

One chapter is devoted to a brief but interesting history of mathematics textbooks. Another focuses on philosophical and pedagogical considerations. Concern about motivation, the ordering of mathematical topics, the use of rote learning, and the application of mathematical knowledge are among many mentioned topics that have challenged educators for years in diverse parts of the world.

This monograph should certainly be a source of encouragement for researchers to investigate further the nature and role of mathematics textbooks. Educators not involved in research will find it a moderately useful textbook-evaluation tool.

9.3. Review by: Carol E Malloy.
Mathematics Teaching in the Middle School 1 (10) (1996), 842.

This book summarises a supplementary study that is part of a broader study of mathematics textbooks being undertaken by the Third International Mathematics and Science Study. It examines and compares mathematics textbooks for students in grade 8 from England, France, Japan, the Netherlands, Norway, Spain, Switzerland, and the United States. The author's aim is not to show the merit of the different textbooks; he investigates discernible trends and differences in the presentation of mathematics and in aims and expectations and questions the direction of further research. Textbooks were reviewed for pedagogy, philosophy of mathematics, treatment of algebra and geometry, and the use and application of mathematics.

9.4. Review by: Derek Foxman.
Mathematics textbooks across the world (National Foundation for Educational Research, 1999), 3-4.

In an analysis of Grade 8 textbooks from eight countries, Howson distinguishes between their roles in the classroom and in the educational system of a country. In the classroom, he notes the following uses a textbook may provide:

- a source of problems and exercises;

- a reference book setting out 'kernels': theorems, rules, definitions, procedures, notations and conventions which have to be learned as knowledge;

- 'explanations' - not themselves kernels - which prepare the students for the kernels.

Textbook series were seen by Howson as attempting to meet one or more of the goals listed below. Which ones would depend on the nature of the system; for example whether or not the curriculum was centralised.

1. Fleshing out a centrally prescribed curriculum.

2. Attempting to update pedagogy within a centrally prescribed curriculum.

3. Responding to new, non-statutory proposals on pedagogy, for example National Council of Teachers of Mathematics (NCTM) Standards in the United States or the Cockcroft Report in Britain.

4. Helping to define a new curriculum.

He suggested that the aims of a country's texts are determined by the nature of the country's educational system. Japan's texts can only aim at 1 and 2. All of its series attempt to put flesh on the new skeletal national curriculum, but, in addition, can attempt to increase pupils' motivation, to proceed from the concrete to the abstract, to make greater use of contextualised examples and problems. The French text he analysed contained innovations directed at 2, that is, a greater use of technological aids. The US text aimed primarily at 3 but also at 4. In England, where School Mathematics Project (SMP) texts were used by over 40 per cent of the Population 2 pupils in the year of the TIMSS surveys, Howson proposed that a planned total rewrite of SMP 11-16 materials (Cambridge University Press, 1983, 1992) should seek to satisfy 1 and 2, whilst still hoping to achieve some success with 4.

10.1. School Mathematics Project, Book T (1964), edited by A G Howson.

10.2. Developing a new curriculum (1970), edited by A G Howson.

10.3. Developments in Mathematical Education: Proceedings of the Second International Congress on Mathematical Education (1973), edited by A G Howson.

10.4. The influence of computers and informatics on mathematics and its teaching (1986), edited by A G Howson and J P Kahane.

10.5. Perspectives on Mathematics Education (1986), edited by B Christiansen, A G Howson, and M Otte.

10.6. Mathematics as a service subject (1988), edited by A G Howson, J P Kahane, P Lauginie and E de Turckheim.

10.7. The Popularization of Mathematics (1990), edited by A G Howson and J P Kahane.