1.1. From the Foreword.

It will be well to indicate at the outset, in the words of the author himself concerning the hypothesis here explored, which is based on correlation tables, that the statements that are going to be made, whether of topological or of quantitative character, will be statistical statements referring to groups of individuals and not to single individuals. This caveat cannot be entered too often in the discussion of educational and psychological researches employing statistical techniques. When it is ignored we find statistical concepts and parameters, whose legitimacy in reference to populations is ascertainable, used in arguments concerning characteristics of the individual, and we are heading straight into fallacy.

Since this investigation has important bearings on the teaching of mathematics we must bear in mind that classes of children are sample populations and hence that, with proper precautions, it is perfectly legitimate to apply statistical hypotheses to the collective responses of classes. Statistics, in its practical aspects, is essentially an administrative instrument. For the administrator has both to predict and to organise the behaviour of populations . He knows, if he is honest, that he cannot avoid errors but he wants to minimise these. On the positive side he wants to maximise the distribution of whatever benefits his work confers on the population. The teacher is a kind of administrator. He organises the learning-behaviour of populations and predicts progress . The make-up of his population shifts by discrete amounts year by year. In any one year it may amount, in total, to over a hundred different individuals and in a life-time of teaching, say 40 years, over 2,000 different individuals may each have had two or three years' instruction at his hands.

1.2 Review by: P E Vernon.
The British Journal of Sociology 11 (2) (1960), 190-191.

Dr Dienes's brief monograph is primarily of interest to psychologists, though it would have some important implications for education, and sociology, if its findings were confirmed and extended. It is a careful study of the formation of abstract concepts or generalisations among 10-year-old children, the author believing that such formation is not merely a matter of amount of general intelligence, but that various qualitatively different types or patterns can be observed which probably relate to the thinker's personality. For example some are more flexible or rigid, some more synthetic, others analytic, etc. He therefore presented individually two novel and complex practical problems and, by giving a standard series of hints, recorded the stages in their solution. He also obtained ratings from teachers on certain characteristics of the children's thinking and applied intelligence tests and Raven's Controlled Projection test which is supposed to reveal personality variables. Correlational analysis did indeed yield some suggestive relationships between thinking and such personality tendencies as anxiety, introversion, conformism, etc., which differed as between the two sexes. However the results cannot be considered as more than exploratory, since no clear hypotheses were set up beforehand, and the instruments employed were far too unreliable to provide stable measures of the thinking and personality traits.

The monograph is pretty hard going, since it draws not only on Piaget's work on intellectual development and mathematical logic, but also on Lewin's so-called topological psychology, the Californian investigations of the authoritarian personality and Eysenck's factorial studies of personality; and it makes use of advanced tools like canonical correlations. Curiously the author is unaware of the overlapping between his approach (and findings) and those of the German typologists who used, for example, the Rorschach test. But he is certainly fertile in ideas which deserve fuller investigation.

2.1. From the Introduction.

The purpose of this article is to show that there are ways in which mathematical concepts can be caused to develop in children so that the techniques they learn are preceded by an understanding of the corresponding mathematical structures.

There have been a number of attempts in recent years to do this, notably those of Cuisenaire and Stern. However, without minimising the importance of these initiatives, we hope to show that a great deal remains to be done, if insightful mathematics learning is to be achieved throughout children's mathematical careers. The full development of some of the more difficult concepts means that some experiences must be assimilated perhaps several years before the concepts are actually formed. It follows too that whereas a variety of experiences may be suitable for forming an early concept, only some of them will be fully suitable as preliminary experiences for later concepts. A full count of mathematical experiences, therefore, must be planned, both according to a carefully thought-out mathematical progression, and according to a definite theory of concept formation in children.

The mathematical progression chosen will, to some extent, depend on our theory of mathematics, as well as on how mathematical experiences can be dove-tailed to our theory. Here we may remark that it is perhaps not generally realised by teachers that different and opposing theories of mathematics are held by professional mathematicians themselves and that to such differences may well correspond differences in the ways in which children understand mathematics. We have, therefore, to allow for such differences in children and in mathematical theory so as to minimise their effect on learning. This means practically that the learning experiences we give should be varied enough to allow children to form their own concepts in their own way, instead, as we now tend to do, of trying to teach them in ours.

Even to a casual observer it is clear that different people think differently. If they did not, there would never be any arguments and there would be no problems of communication and of understanding. We are not here particularly concerned with how these differences arise, whether they are innate or acquired, but only with the fact that they exist.

2.2. Review by: H Martyn Cundy.
The Mathematical Gazette 44 (350) (1960), 301-303.

Dr Dienes sets himself in this article to elucidate his view of what Dr Biggs means by "meaningful" teaching; his answer is the "structured game." (The word "game" is extended to include experiments of all kinds). This is an activity designed to establish a new concept; e.g. hanging weights on a beam at different distances from the pivot in order to discover conditions of balance, so leading to the concept of "moment". This, when grasped, is confirmed by "practice games", which would presumably include routine paper work. Dr Dienes applies these principles to two cases; the use of similar figures of various shapes to establish the concept of a "square number" - how many identical triangles must be used to build a triangle three times as much each way?-and even to establish such results as $(a + b)^{2} = a^{2} + 2ab + b^{2}$; and the use of "multibase blocks" to establish positional notation, not merely to base ten, but to a wide variety of bases. He claims that this is not difficult for children - though it may confuse an adult with preconceptions! But when he goes on to use similar material to establish trinomial factors and logarithms, we must ask (what unfortunately is not stated) what age of child is envisaged!

3.1. From the Cover.

Written in the hope of encouraging a greater love of mathematics among children and teachers alike.

3.2. From the Preface.

In a book on Education Through Art which was first published in 1943, I suggested that the method of education I advocated - a method that is formally and fundamentally aesthetic - should be applicable to all subjects in the curriculum, and not merely to the teaching of the arts. The integral education which I conceive is relatively indifferent to the fate if individual subjects, since its underlying assumption is that the purpose of education is to develop generic qualities of insight and sensibility, which qualities are fundamental even in mathematics and geography. Dr Dienes, the author of the present book, might be justly critical of that little word 'even', for it is his contention that essentially in mathematics such qualities are required, and he would go so far as to say that if mathematics is to be effectively taught to children, the means must be artistic. My own mathematical capacities were effectively stifled in childhood and I have therefore always hesitated to venture into a realm in which nevertheless I always felt that an aesthetic approach was relevant. I rejoice, therefore, in this scientific demonstration of my intuitive conviction. As Dr Dienes says, the simple psychological fact that construction must precede judgement or analysis has for a long time been conveniently forgotten, with disastrous effects on teaching methods in mathematics (and, as I would contend, in all subjects involving the learning of abstractions).

The alternative method of teaching, explained with beautiful clarity in Dr Dienes's second chapter, depends on the appreciation of certain psychological processes that enable insights to be generated spontaneously in children. The whole learning process, and particularly the essential steps of concept-formation, is still a mystery, but one on which much light has been thrown by the researches of Piaget, Bruner and Bartlett. Dr Dienes has carried these researches a step further in the particular field of mathematics and in the further chapters of his book he shows how the Dynamic Principle he has established can be given practical application in the classroom (and the playroom). Concept formation becomes a natural result of perpetual experience. It may seem to the conventional teacher that mathematics is being turned into a game, or a number of games, and indeed it is. But there is plenty of evidence to suggest that to the best mathematicians it has never been anything else. And yet mathematics is the indispensable instrument of all scientific progress. Hence the extraordinary importance of this book. If our scientific progress has been retarded by wrong teaching methods (and certainly our leading scientists seem to be anxious about the matter) then the method advocated by Dr Dienes at least deserves the urgent consideration of all who are in authority in the world of education.

3.3. Review by: H Martyn Cundy.
The Mathematical Gazette 45 (352) (1961), 147-148.

Dr Dienes is a reformer impelled by iconoclastic zeal. In an area so wide as that of mathematical teaching in this country it is obviously possible to find many targets for criticism, but Dr Dienes goes further. He suggests, or at least the reviewer understands him to suggest, that the whole of traditional mathematical education is faulty. Thorough-going rethinking and reconstruction of the methods and aims of the curriculum are called for.

The author begins by criticising existing aims. These, he says, are as follows. (a) To give skill in computation needed in everyday life. But apparently very little of this is really needed; all the rest will be done by automatic computers. (b) To train the mind to think logically. "But what could be more illogical than to perform a great number of quite unintelligible acrobatics with symbols for the simple reason that you would get a detention if you did not?" (c) To provide basic techniques for scientific development. But - and here the argument does not seem very clear - to do this we must teach understanding of processes, to make them generally applicable. Only a few pupils have the capacity to understand fully enough to make them expert in the application of mathematics, so that this aim leads to greater specialisation: teaching more mathematics to fewer people. "The mathematically fit survive by natural selection; the rest get gradually relegated to the mathematical lumber-room as second-class citizens unfit for initiation into the mysteries." (d) To build up personality. But for children do not enjoy mathematics the punishment-reward system is disintegrating, while for those who do, operating within closed sets of techniques can have little integrating effect.

By this time most practising teachers will be thinking that they would like to introduce Dr Dienes to some of their pupils in their lower streams. What remedy does the author propose? Basing his theory on the work of Piaget, Bruner, and Bartlett, Dr Dienes outlines the process of learning mathematics as a cyclic scheme in which concepts are learnt and then used as the basis for further concepts. Each concept must first be experienced in preliminary play, then recognized as the common element in a wide variety of structured games, then used and practised until in turn it can be made the basis of games leading to recognition of a further concept. "It will be clear", says the author," that the kind of mathematics-learning described here is not one that very often takes place in the conventional class-lesson." Instead he recommends work (or play) in groups of two or three, using assignments and much concrete material. Detailed consideration is given in the chapters that follow to the materials used for formation of various concepts: place-value in arithmetic (Multibase Arithmetic Blocks); distributive laws in algebra (balance with hooks and rings); equations (pegs, strips and squares); factorisation (squares and triangles, strips and pegs); exponentiation and logarithms (Blocks again); functions (charts and graphs).

It is quite clear that this is all going to be great fun for somebody; is equally clear that it will need an exceptionally skilled teacher to ensure that very much is learnt from it. The thought of what could happen in a classroom littered with blocks of all shapes and sizes removed from their " radix " boxes will make many teachers hesitate. Most will suspect that the huge private joke which has evidently come off in illustration $V$ has nothing whatever to do with factorising $A^{2} - 5A + 6$. But it would be wrong to dismiss the book as visionary. Too many of us are conscious that, in spite of our best efforts, mathematics to many of our pupils is a game played with meaningless marks on paper. This book may help us to find out some of the reasons why. But, in the reviewer's opinion at least, a wholehearted reformation might lead to the equal danger of mathematics becoming a meaningless game played with blocks, balances, pegs and rings, if carried out exclusively on the lines advocated in this book.

3.4. Review by: Herbert F Miller.
The Mathematics Teacher 11 (2) (1964) 125-127.

Acquaintance with this book is a must for all who wish to be informed on developments in mathematics education on elementary levels because the book indicates, at least partially, what is happening in British thought on a topic which parallels the educational phenomenon in the U.S. called "new mathematics" Dienes develops a "theory of mathematics-learning," based on an analysis of the status of mathematics education in Britain and the tentative conclusions indicated by the researches of such men as Piaget and Bruner in concept formation and abstraction. Separate chapters are devoted to the application of this theory to arithmetic, to algebra, and to the learning of the mathematical concept of function.

In his introduction, the author states his belief that some of the difficulties in learning abstractions stem from an educational climate strongly influenced by the mathematical philosophy which is predominantly formalistic. His reaction is expressed in a learning theory which is constructive rather than analytic, inductive rather than deductive. Hence, his title is Building Up Mathematics. Not only is he concerned with creating for the learner the concepts and abstractions which "build up" the structure of mathematics, but also with the effects that such learning experiences have on the personality of the learner. He points out that authoritarian teaching and lack of opportunity for the learner to discover and February 1964 create can hardly be expected to lead to an integrated personality for the learner. He concludes that, to achieve the most desirable effects upon the personality development of the learner, "we [the teachers] need to create mathematical learning-situations, partly as if we were practising an art-form, and partly as if we were devising an original research-situation."

3.5. Review (of 2nd edition 1965) by: A G Sillitto.
The Mathematical Gazette 49 (369) (1965), 328.

This is a second edition of a book first published in 1960 and reviewed in the Gazette of May 1961. A 29-page appendix, "Scalars, Vectors and Matrices" has been added, "to deal with the introduction of directed numbers". Dienesian principles lead to the introduction of several dimensions at once ... "structures like that of the complex algebra should make their appearance far earlier than is normally the case ... experience shows the soundness of this view. Primary school children in Hawaii, South Australia, Paris, Massachusetts and Geneva have shown that multi-dimensional vector-spaces, complex and hypercomplex algebras, matrices etc. are well within their grasp, given suitable experiences ...".

This "revised" edition refers still to a "perceptual variability principle" which the author in "The Power of Mathematics" (1964) has renamed. The appendix bears the marks of hasty writing. Page 129 becomes clearer when one reaches p. 131 and finds the relevant diagram. It is not clear just how much of the work here described has been carried through with children; we are told about many things that "can" be done or "could" be used. Great inventiveness is shown in devising situations which exhibit the mathematical structures the author wants the children to encounter; whether the education of children in and through Mathematics really requires the running of such elaborate mazes is, one supposes, for discussion. Teachers of Mathematics will certainly find interest and stimulus here.

4.1. Contents.

Preface / Jerome S Bruner.

Foreword.
Introduction.
1. On the function of play in mathematical thinking.
2. Abstraction and construction.
3. Generalisation and analysis.
4. Symbolism and interpretation.
5. Educational implications.
Further reading.

Appendix I. Description of the experimental groups.

Appendix II. Summary of research on mathematical-learning in an Italian university.

4.2. Review by: John Biggs.
International Review of Education / Internationale Zeitschrift für Erziehungswissenschaft / Revue Internationale de l'Education 13 (1) (1967), 93-96.

The aim of An Experimental Study of Mathematics Learning is to "observe mathematics learning in situ" so that a learning theory applicable to mathematics learning may eventually be constructed. Accordingly, five small groups of children, drawn from elementary grades two to six, were given different sets of materials and games all with a mathematical structure, and their activities were observed closely. The rich data thus provided are used to suggest modifications to Dienes's initial hypotheses about learning and to help to generate new ones. The emphasis here is upon psychology rather than upon pedagogy and it is to be appraised accordingly.

Play is regarded as the basis of all good learning. At this point Dienes makes much of a rather peculiar distinction between "natural" and "artificial" learning, which leads him to such statements as "fortunately it is impossible to learn skating or riding a bicycle from a book or many people would have a try." Why "fortunately?" And why shouldn't people have a try; indeed, don't they already? If there is a distinction to be made here it is between the acquisition of sensori-motor and formal schemata. But even then his statement is just not true. Cross-modal transfer of information is a well established phenomenon, although it is true that children may find it much more difficult than adults to convert symbolic input into sensori-motor output. At other times the "natural-artificial" distinction seems to refer not to the nature of the data being learned but to whether the behaviour is intrinsically or extrinsically motivated. The moralistic, even messianic, tone of much of Dienes's writing is at best irritating and irrelevant, but here it downright misleading.
...
The best summary of Dienes's contribution is given by Bruner in his Preface: "Dienes's investiveness lies in translating mathematical ides into demonstrable embodiments that are in the grasp of children." As such, anyone who is in any way concerned with the teaching of mathematics at primary or secondary levels cannot afford to miss reading this book.

4.3. Review by: W O Storer.
The Mathematical Gazette 50 (372) (1966), 194-197.

An Experimental Study of Mathematics-Learning is a report of a "Mathematics-Learning Project " carried out by the author in collaboration with Professor J Bruner and others during a year spent at the Harvard University Center for Cognitive Studies. In the investigation Dr Dienes set out from his views of mathematics-learning and used his apparatus and story-methods to study the function of play and the incipient processes of abstraction and construction, generalisation and analysis, and symbolisation and interpretation in children's progress in mathematical thinking. He regarded his work as "largely exploratory, ... undertaken more to help research workers ask relevant questions than to answer them with any certainty".

A full understanding of this report is made easier by a knowledge of his other books and also by practical experience with his apparatus, namely Multibase Arithmetic Blocks and Algebraic Experience Materials, but an Appendix describes some of this briefly. The children involved consisted of a whole class of children in grade 2 (aged 7 years) and smaller numbers in higher grades up to grade 6 (aged 11). The Multibase Arithmetic Blocks and Algebraic Experience Materials were used for a wide variety of games and tasks leading to such results as the importance of place value, factorisation of a quadratic expression, indices and logarithms and the summation of geometrical series. The older children were given prepared games with a rule-structure exemplifying a vector-space, a group, complex algebra or some other mathematical system. The structures were presented and made interesting by stories of the kind mentioned above, and much ingenuity of design is in evidence throughout.

It is refreshing because unusual to be able to read reports of what children said and did in their exploration of mathematical situations, as the best substitute for observing them oneself. The printed word no doubt makes them a little more formal than they actually were, but the reports seem very realistic and sometimes amusing if perhaps mystifying ...

5.1. From the Preface.

The teaching of mathematics in schools is in a state of ferment today. Every country in the world is finding itself short of the scientists, technicians and other specialists needed to carry on a technological civilisation. The bottleneck clearly lies in the education of young people. The basic skill underlying all scientific and technological skills is control of the tools of mathematical structures, and not enough young people even become aware of their existence. The majority of them go through school regarding mathematics as an arduous conditioning process which enables them to pass qualifying examinations for entry into the professions. A fundamental re-thinking of the role of mathematics in schools is therefore being undertaken in many parts of the world, coupled in some centres with actual experimental work in the classroom. In this way it is hoped to establish that radical changes are practicable, as well as economically desirable. It is certainly being found that exploration of the 'difficult' underlying mathematical  structures, in place of the customary rote-learning of rules, delights rather than repels children. Any child, unless spoilt by long conditioning through extrinsic punishment-reward systems, will respond to a challenge that awakens his natural curiosity.

5.2. Review by: John Biggs.
International Review of Education / Internationale Zeitschrift für Erziehungswissenschaft / Revue Internationale de l'Education 13 (1) (1967), 93-96.

The Power of Mathematics attempts to do for algebra teaching what Dienes's earlier Building Up Mathematics did for arithmetic teaching. As in the latter, Dienes bases his approach on the proposition that "one of the greatest sources of difficulty (in traditional mathematics learning) is the rash use of symbolism before adequate experience has been enjoyed of that which is symbolised".

After an exposition of the nature of learning, the book consists largely of showing how, by manipulating their experiences, children may reach a certain conceptual sophistication, with each concept being frozen by symbolisation so that it may be operated upon efficiently. The conceptual structures treated include powers, roots, logarithms, matrix algebra, vector spaces and irrational numbers. The emphasis throughout is on the basic mathematical laws such as associativity, commutativity, etc. and their application to these fields, rather than on the systematic development of these areas themselves.

Many of these concepts are traditionally not taught until tertiary level; and bearing in mind the inevitable lack of background mathematical knowledge on the part of the greater majority of his readers, one's first reaction is that 176 pages can provide nowhere near an adequate treatment of the material, and one wonders therefore just how practicable the book is. One is dazzled with Dienes's obvious virtuosity - and it seems the children's too - but what about the poor old teacher himself? He is given little hint of how to relate these fascinating and evidently most erudite activities to the conventional mathematics that he himself knows and applies to real problems. This is a criticism of the teachers' education rather of Dienes's own books: but it is an essential part of the problem that he has set himself to solve and he should therefore recognise it. Thus, it is doubly unfortunate that Dienes writes in a sweeping style that so easily induces a false now-I-know-it-all confidence in the reader.

In short, this in not an adequate teaching manual qua manual, neither is it an attempt to show the teacher how this mathematics of groups and matrices fits into the everyday scheme of things. But if Dienes' intention is to what the reader's appetite for more - to show him how it may be done, given a lot more blood, toil, tears and sweat than this book provides for - then Dienes must be regarded as having been remarkably successful. However, it is a fatal omission that he does not answer the question "Where do we go from here?"

5.3. Review by: W O Storer.
The Mathematical Gazette 50 (372) (1966), 194-197.

In his earlier book Building Up Mathematics Dr Dienes expounded convincingly his view of the development and learning of mathematical concepts and described materials and apparatus he used in accordance with his four principles: the dynamic principle of preliminary, structured and practice games; and the principles of constructivity, mathematical variability and perceptual variability. In The Power of Mathematics he now extends his treatment from the constructive phase of mathematical development to the analytic phase. While still emphasising the need for his dynamic principle at all levels, he insists also on the pupils' deeper study of the structure of mathematical systems than has been customary. He aims to achieve this partly by the use of apparatus and also by investigations presented in the guise of story-situations designed to exemplify different mathematical structures. A great deal of ingenuity is shown in both these sections of the work, and many valuable ideas are obtainable from them at many different levels. The topic of indices and logarithms, for example, begins with the use of his Logs and Indices Blocks (to bases 2, 3, 4 and 5) and continues with the story of a magician whose hocus-pocus and abracadabra accomplish the operations of multiplication and raising to a power. The factorising of a quadratic expression similarly receives both concrete and fictional representation. The topics of vector spaces, real and complex algebras and groups are covered mainly by stories, but some practical exemplifications are also given. A dance-floor story, a boating-lake story and a farm story are devised with different rules for this purpose.

At a first reading many of these stories appear too elaborate and artificial to attract the interest and stimulate the mental activity of pupils. But we can accept as something more than a sanguine hope that "it is certainly being found that exploration of the 'difficult' underlying mathematical structures, in place of the customary rote-learning of rules, delights rather than repels children."

6.1. Review by: Stephen S Willoughby.
The Mathematics Teacher 60 (7) (1967), 793-794.

In this book, the results of an experiment carried out by Dienes and Jeeves are reported and analysed. The experiment was meant to help determine how people organise apparently chaotic situations. The first five chapters of the book are largely taken up with a description of the experiment, and the last two chapters consist of the authors' conclusions regarding the meaning of their results for the teacher.

Perhaps the most interesting conclusion the authors draw from their experiments is that of two similar tasks, one of which is more complex than the other, the more complex should be considered first rather than second, as is usually done. The systems considered in the experiments are somewhat analogous to the complex and real number systems, though much simpler (the systems have two and four elements). Thus, the authors infer, it may be better to teach children the complex number system before the real number system. Or, for similar reasons, it might be better to start with the rational number system before the natural number system. The fact that this reasoning can be carried to a somewhat ludicrous conclusion does not negate the potential benefits that may accrue from limited and reasonable use of the principle - as the authors suggest, more experimentation should probably be carried out along these lines (Dienes has al ready made some progress in this direction involving the complex and real number systems).

The authors also suggest that the sort of test they used (the subject is to determine a structure from apparently chaotic facts - and is encouraged to make shrewd guesses regarding the structure) may be a better test of children's intellectual abilities than is the conventional IQ test. Although there is little evidence to substantiate this claim (unless one accepts the hypothesis that ability to play this sort of game well is a mark of high intelligence) it is an interesting idea, and may be substantiated with more work. In any case, the test does seem to rely less heavily on cultural background than does the standard IQ test.
...
Taken as a whole, I would say that the experiments have made a substantial contribution to thinking about the teaching of mathematics, and may lead to some significant improvements in the teaching of mathematics. However, I wish the authors had chosen and described their samples more carefully, and had spent more time editing the book itself so as to make it easier for a teacher to read.

7.1. From the Preface.

This booklet of instructions in how to induce young children to learn "modern" mathematics is presented to the public in the hope of persuading at least some people in the educational world that the present spring cleaning in the mathematics classroom should start way down in the Kindergarten, where it will have the greatest effect, and at the same time create the greatest amount of enjoyment and therefore love of and learning of mathematics. At no point is there any mathematical cheating; rather suggestions are put forward as to the ways of presenting the "modern" mathematics which are tailored to the type of thinking that particular age-groups are now known to be capable of.

Reference is made to the work of William Hull, who has pioneered 81 the use of the logical blocks, to Paul Rosenbloom and Patrick Suppes, who have pioneered the introduction of sets to young children, and to the present author's work on the explicit use of powers with young children, involving variable systems of numeration.

7.2. Review by: Julia Adkins.
The Arithmetic Teacher 13 (6) (1966), 509.

This booklet describes the introductory teaching of certain mathematical concepts to children of ages from five to eight years. The suggested methods and visual materials are an attempted synthesis of research studies by William Hull, Paul Rosenbloom, Patrick Suppes, and the author of this booklet, Z P Dienes. "The aim of the methods suggested is complete understanding of every detail of mathematical activity by every child in the school." In working toward the achievement of this aim, the author indicates that there are three stages through which the child progresses: (1) a preliminary groping stage, a time of playful exploring activity; (2) a structured stage, a time of insight; and (3) an application stage, a time of analysis and practice.

8.1. Review by: Shirley Hill.
The Arithmetic Teacher 15 (8) (1968), 740-742.

Were it not for one minor drawback, I would not hesitate to recommend these two paperbacks [the other is Learning Logic, Logical Games] for the library of any teacher of children in the approximate age range of four to eight years. They provide a large number of interesting activities, rich in their potential for mathematical concept development. For the most part, the ideas represented in these activities are mathematically sound and basic; exposition is clear and to the point; and the games are motivating and often quite ingenious.

The drawback is that many of the best activities require materials that are not readily available in and around most classrooms. I am thinking particularly of the "attribute blocks" which are critical to most of the logical games described. The "attribute blocks" are a set of blocks of varying shapes, sizes, and colours. They are an excellent device for providing early experiences in classification and developing basic logical ideas of conjunction, disjunction, set union, and set intersection. They are available commercially from several sources and could, I suppose, be easily and inexpensively constructed by someone handy with saw and wood. But it is so often the very situations in which children are most in need of these concrete experiences that resources for nonstandard materials are least available.

On the other hand, many of the games and activities in Sets, Numbers and Powers can be carried on utilising only familiar and easily secured objects.

9.1. Review by: Shirley Hill.
The Arithmetic Teacher 15 (8) (1968), 740-742.

Were it not for one minor drawback, I would not hesitate to recommend these two paperbacks [the other is Sets, Numbers and Powers] for the library of any teacher of children in the approximate age range of four to eight years. They provide a large number of interesting activities, rich in their potential for mathematical concept development. For the most part, the ideas represented in these activities are mathematically sound and basic; exposition is clear and to the point; and the games are motivating and often quite ingenious.

The drawback is that many of the best activities require materials that are not readily available in and around most classrooms. I am thinking particularly of the "attribute blocks" which are critical to most of the logical games described. The "attribute blocks" are a set of blocks of varying shapes, sizes, and colours. They are an excellent device for providing early experiences in classification and developing basic logical ideas of conjunction, disjunction, set union, and set intersection. They are available commercially from several sources and could, I suppose, be easily and inexpensively constructed by someone handy with saw and wood. But it is so often the very situations in which children are most in need of these concrete experiences that resources for nonstandard materials are least available.
...
It seems to me that Learning Logic, Logical Games provides many more unique and original ideas. Both books will be invaluable to teachers as sources of good ideas for classroom activities, but Learning Logic, Logical Games provides an excellent basis for development of some important concepts not likely to be provided by the standard mathematical textbooks.

10.1. Review by: Richard B Skemp.
Research in Education 0 (6) (1971), 91-92.

Since Professor Dienes is professionally qualified both as a psychologist and as a mathematician, his contributions in the field of mathematics learning are of particular importance. This book is the second of two written in collaboration with Professor Jeeves, a psychologist. being the outcome of their joint research when both were at the University of Adelaide. We need to consider it from three points of view: psychological, mathematical and educational.

Even the most tough-minded experimental psychologist is unlikely to find fault with the experimental design and technique. Here is a good example of the use of 'hardware' to make possible the recording of every response of the subjects on IBM punch cards. By feeding these into a computer the data could be rapidly analysed in an assortment of ways. This is a development of what Bruner, Goodnow and Austin (1956) describe as the externalisation of the cognitive processes of a subject while engaged in solving a problem.

The tasks were in this case much harder, being the learning of a large number of ordered trios of stimuli. The experiment was designed to indicate whether the learning, when achieved, could best be explained by an S-R model or by one which asserts that structured learning was taking place. Analysis of the results gave support to the latter view and it is an indication of the hard-line experimental approach that information from the subjects about how they tackled the job was not used, nor even (apparently) requested.

The pattern built into the machine which presented the learning task to the subjects, and recorded their responses, was based on a mathematical structure known as a group, and chapter 2 is given over to an explanation of this for (they say) the non-mathematical reader. Such a reader, with no previous acquaintance with groups, might well find it hard going. But someone who knows just a little about them would learn a great deal more, Groups are a topic popular among exponents of 'modern mathematics' because they have a clearly defined, fairly simple structure, of which it is possible to find many different embodiments. Piaget also considers that a certain stage of development of children's thinking shows this structure; so it is worth learning about from both psychological and mathematical points of view.

10.2. Review by: Michael Holt.
The Mathematical Gazette 55 (394) (1971), 471-473.

As part of an on-going research project now active in Sherbrooke, Canada, this important monograph is a record of how [Professors Dienes and Jeeves] used mathematical group structures to externalise modes of thinking and, in the process, to test various models of thinking. In this book the relative merits of two such models - a Stimulus-Response (S-R) model, and a role model are compared.

The S-R model which Koestler has tellingly called "flat-earth ", invokes associative thinking - "Hastings" and "1066" or "Swan" and "Edgar ", for example; the role of structural model calls for structural relations -"Claudius" - "Hamlet" and " Richard III " - "the Princes in the Tower", both being examples of the "uncle-nephew" relationship. The stimulus for this piece of research, as for that recounted in their earlier book Thinking in Structures, stems from the seminal work of Sir Frederick Bartlett. The general lines of the first enquiry I summarised in an article for the Gazette: "Maths for Tomorrow's Children" (May 1970). The aim was to externalise the subject's responses to thinking problems. A machine has now replaced the flesh-and-blood experimenter and the algebraic structures have been augmented. The following algebraic groups were used: the 3-by-3 group, Klein 4-group, the 5-group, the 6-group and the two forms of the nine-element group, the cyclic and the 3-by-3 variants.

The experimental subject sat before a console of panels and push-buttons tagged by the symbols, triangle, circle and square arranged as for the leading rows of a group table. The machine would light up a panel showing one of the symbols and the subject would press a button of his own choosing corresponding to one of the three symbols. The subject's task was then to predict the outcome of combining these two symbols, from a knowledge of the rules of the game. Once he had grasped the basic table of combinations - that is the group table-he had to solve various equations again based on the group table, as a check on his learning. Adults and children took part in the experiment - psychology undergraduates with, one gathers, little mathematics, and eleven-year-olds. Both groups were equally bright - though it is not clear why the children's mean IQ should be quoted to the degree of accuracy of 114.06. The sample one infers was of the order of 60 although nowhere is it explicitly stated. This has a bearing on the needlessly high degrees of accuracy quoted for the standard deviations in the numerous tables of extrapolations from the raw data.

The experimenters set out to discover how well or badly subjects transferred the rules of one algebraic group to that of an analogous group. Their subjects were, in effect, learning isomorphisms - either by heart, that is by associating Stimulus-Response-Outcome triads from one task to another - or by insight, that is, by discovering the role of operators. :Paired group tables were ingeniously labelled to indicate the kinds of learning transfer needed between the two tasks. Thus subjects learning the 3-group after the 4-group probably saw the elements as split into "the neutral element" (common to both groups) and "the others", all of which in this particular pair of tasks required relearning. Roles were further classified into "new", "extrapolation" and, as one might expect from Dienes, "generalisation". The order of presentation seemingly has a marked effect on the performance of adult and child subjects - though not, perhaps expectedly, in the same way.

Typical findings: adults perform better than children doing the 3-group before the 5-group and the 3- before the 6-group. And, in line with the authors' earlier work, children find generalisation harder than adults do.

11.1. Review by: Philip Peak.
The Mathematics Teacher 65 (5) (1972), 440-441.

The content of this book is based on the theoretical and practical research by the authors and others in the teaching of mathematics. It uses the principles of learning, the processes of communication and the results of experimentation and research to formulate ways in which learning mathematics can be implemented. This book illustrates the need to develop the theoretical base if continued success is to be achieved in the teaching of mathematics.

12.1. Review by: Philip Peak.
The Mathematics Teacher 65 (5) (1972), 442.

A textbook in mathematics for prospective elementary teachers to enable them to handle modern mathematics as teachers in the elementary school. The development is logical and clear, starting with the concept of set and going to vector space. There is one chapter on geo metric transformation and one on logic. The illustrations are very helpful for understanding and the big ideas stand out. The discourse carries the student through the development and its implications rather than having him work a series of exercises.

13.1. Review by: Julia Matthews.
Mathematics in School 2 (2) (1973), 35.

It is by now well-established that the framework of mathematics is sets and relations, that many experiences (sorting, matching, ordering, ...) are needed before there can be understanding of numbers and operations on them and that in the past spatial relations were neglected in favour of numerical ones. Young children learn by doing and by playing with real objects and not just imagining them, being told of them or even just shown drawings of them. Is there, then, still a place for expendable work-books for infants? The series under review consists of eight of these, two "Elephant" books on sorting, matching and counting; two "Dog" books on joining sets and pictures for sorting; two "Cat" books on relations and two "Panda" books on space. The books are attractive and thoughtfully designed and could well play a useful part in supplementing practical activities. An interesting feature is that they are totally non-verbal, though this is not an unmixed blessing as motivation provided by mathematics can often help the children to learn to read. Some children in my school enjoyed working through these books and felt a sense of achievement, but on the whole, they needed guidance at every fresh step and so made demands on the busy teacher. The teacher's book is well written and vital to the series but again demanding on the teacher's time. The series can, however, be recommended as supplementary materials if the capitation will stretch.

14.1. Summary of the 'Six Stages' by Z P Dienes.
https://zoltandienes.com/academic-articles/zoltan-dienes-six-stage-theory-of-learning-mathematics/

Stage 1.
Most people, when confronted with a situation which they are not sure how to handle, will engage in what is usually described as "trial and error" activity. What they are doing is to freely interact with the situation presented to them. In trying to solve a puzzle, most people will randomly try this and that and the other until some form of regularity in the situation begins to emerge, after which a more systematic problem solving behaviour becomes possible. This stage is the FREE PLAY, which is or should be, the beginning of all learning. This is how the would-be learner becomes familiar with the situation with which he or she is confronted.

Stage 2.
After some free experimenting, it usually happens that regularities appear in the situation, which can be formulated as "rules of a game". Once it is realised that interesting activities can be brought into play by means of rules, it is a small step towards inventing the rules in order to create a "game". Every game has some rules, which need to be observed in order to pass from a starting state of things to the end of the game, which is determined by certain conditions being satisfied. It is an extremely useful educational "trick" to invent games with rules which match the rules that are inherent in some piece of mathematics which the educator wishes the learners to learn. This can be or should be the essential aspect of this part of the learning cycle. We could call this stage learning to play by the rules, as opposed to the free learning characteristic of stage one.

Stage 3.
Once we have got children to play a number of mathematical games, there comes a moment when these games can be discussed, compared with each other. It is good to teach several games with very similar rule structures, but using different materials, so that it should become apparent that there is a common core to a number of different looking games, which can later be identified as the mathematical content of those games that are similar to each other in structure, even though they might be totally different from the point of view of the elements used for playing them. It is even desirable, at one point, to establish "dictionaries" between games that have the same structure, so to each element and to each operation in one game, should correspond a unique element or operation in the other game. This will encourage learners to realise that the external material used for playing the games is less important than the rule structure which each material embodies. So learners will be encouraged to take the first halting steps towards abstraction, which is of course becoming aware of that which is common to all the games with the same rule structure, while the actual physical "playthings" can gradually become "noise". This stage could be called the comparison stage.

Stage 4.
There comes a time when the learner has identified the abstract content of a number of different games and is practically crying out for some sort of picture by means of which to represent that which has been gleaned as the common core of the various activities. At this point it is time to suggest some diagrammatic representation such as an arrow diagram, table, a coordinate system or any other vehicle which would help fix in the learner s mind what this common core is. We cannot ever hope to see an abstraction, as such things do not exist in the real world of objects and events, but we can invent a representation which would in some succinct way give the learner a snapshot of the essence that he has extracted or abstracted through the various game activities. Each one of the learned games can then be "mapped" on to this representation, which will pinpoint the communality of the games. This stage can be called the representation stage.

Stage 5.
It will now be possible to study the representation or "map" and glean some properties that all the games naturally must have. For example it could be checked whether a certain series of operations yields the same result as another series of operations. Such a "discovery" could then be checked by playing it out in one or more of the games whose representation yielded the "discovery". An elementary language can then be developed to described such properties of the map. Such a language can approximate to the conventional symbolic language conventionally used by mathematicians or freedom can be exercised in inventing quite new and different symbol systems. Be it one way or another, a symbol system can now be developed which can be used to describe the properties of the system being learned, as the information is gathered by studying the map. This stage can be called the symbolisation stage.

Stage 6.
The descriptions of the symbolisation stage can get very lengthy and often quite redundant. There comes a time when it becomes desirable to establish some order in the maze of descriptions. This is the time to suggest that possibly just a few initial descriptions would suffice, as long as we appended ways of deducing other properties of the map, determining certain definite rules that would be allowed to be used in such "deductions". In such a case we are making the first steps towards realising that the first few descriptions can be our AXIOMS, and the other properties that we have deduced can be our THEOREMS, the ways of getting from the initial axioms to the theorems being the PROOFS. This stage could be called the formalisation stage.

14.1. Review by: D St John Jesson.
The Mathematical Gazette 58 (405) (1974), 225.

In this slim volume Dienes extends and summarises his theoretical position which starts from the assumption that "the child may best construct his store of knowledge from a rich environment", and that 'plunging the child in at the deep end' "often facilitates the process of learning - of abstraction, generalisation and finally of transfer."

Most of the book is taken up with discussing three examples in logic, group theory and order relations which illuminate and exemplify the six stages outlined in the first chapter. So we find detailed descriptions of an environment enriched by provision of materials for children to 'play' with and secondly a strategy for directing children's investigations in order to highlight the essential common structure involved. What is valuable here is the recognition of the sameness of the patterns arising in seemingly diverse situations. It is this structure which is at the heart of the mathematics Dienes would have children learn.

In place of a detailed description of the six stages of the title, note Dienes' contention that the normal methods of classroom instruction in mathematics reverse the natural process of learning. "In traditional pedagogy... a formal system is introduced by means of symbols. One perceives that often the child does not really understand such a system and, consequently, audio/visual means are used to make him understand. Thus in starting from symbolism one proceeds to representation. Then it is found that the child is not really applying the concepts... so it is necessary to show him real life applications. So finally the child reaches the real life situation, which is where he should have started."

If we catch a reflection of our own experience here, then perhaps we may be the more prepared to consider the revolutionary impact for secondary schools of what Dienes has to say. And if after all the reforms of the past fifteen years it remains largely true that the teaching of mathematics is ineffective and inefficient for the majority of pupils, is it not about time that a more constructive scheme of curriculum renewal took place? Then perhaps the process by which mathematics is learned may become more like the process by which it is taught.

15.1. Review by: C A R Bailey.
The Mathematical Gazette 60 (413) (1976), 227.

Professor Dienes is one of the most eminent living mathematical educationists, and a new book from him deserves careful notice. Relations and functions has been edited on his behalf by Mr Seaborne, who is perhaps the leading protagonist of Dienes' ideas in this country; the book is clearly written, and sets out step by step, with the minimum technical notation, the ideas which so often appear indigestibly in the first few pages of Chapter 0 of more advanced mathematical texts. The non-mathematical reader should gain from this book an understanding of such terms as equivalence, order, succession, function, transitive, periodic - a simple and incomplete list which does not do justice to the careful development which is a feature of the book.

Presumably the book is intended for teachers and others interested in the mathematics learned by children of primary school age. Much of the mathematics is illustrated by reference to various kinds of structural material, with the strong implication that children themselves should learn the same mathematics by using the same materials. The viability of this process is vouched for by the experience of Dienes and his colleagues in their own work with children, but for a statement of the philosophical basis for the belief that the structure of logical relationships is a proper study for such children, and that the way children best learn that structure is through apparatus carefully devised to embody such relationships, we must go back to some of Dienes' earlier books. The work under review can stand on its own as a simple introduction to the subject-matter of its title; seen as part of a larger corpus, it can also prompt us to re-examine some of our fundamental purposes.

15.2. Review by: D J Winteridge.
Mathematics in School 6 (1) (1977), 36.

Dienes considers mathematics to be the study of relations. This book aims to explain what is meant by relations and functions and to suggest ways in which their study may be made accessible to children. However, mathematics is to do with ideas and abstractions. To help young children deal with these abstractions, Dienes suggests that they should be presented in the form of concrete representations. Thus he advocates ample use of Logiblocs, People Logic Set, Trimath, Multibase Arithmetic Blocks and Cusenaire Rods. In his words "we shall need to become familiar with relations through handling them, thinking about them and doing things with them".

The book deals with various types of relations, paying particular attention to equivalence and order relations. The properties of different types of relations are introduced through considering their graphs. The concept of a function is introduced as a special type of relation and the last part of the book deals with various types of function, including periodic ones.

As usual, Dienes makes a number of interesting and enlightening points and his approach is original and often ingenious. However, it is an unsatisfactory book in a number of respects. Dienes explains the games and activities for the children very laboriously. Often they are very difficult to follow and the point of the task is not always very clear. His diagrams are much more helpful and sometimes make the rather lengthy explanation redundant. The age-range for which the programme is designed is not clear. Some of the activities suggested in the first part of the book could be carried out profitably with quite young children, but the underlying concepts of the work on functions are likely to be too sophisticated for Junior children, even if it is possible for them to complete the tasks.

The book does contain many valuable suggestions for activities which should provide insights into aspects of relations and functions. The difficulty for the reader will be in finding them. The book lacks an obvious structure, so it is not easy to "dip into". Nor is it an easy book to read as an introduction to the topic, but, for those already having a basic understanding of relations and functions, it does provide useful teaching suggestions.

16.1. From the Publisher.

Dr Zoltan Dienes is a world-famous theorist and tireless practitioner of the 'new mathematics' - an approach to mathematics learning which uses games, songs and dance to make it more appealing to children. Holder of numerous honorary degrees, Dr Dienes has had a long and fruitful career, breaking new ground and gaining many followers with his revolutionary ideas of learning often complex mathematical concepts in such fun ways that children are often unaware that they are learning anything. This is an honest account of an academic radical, covering his sometimes unconventional childhood in Hungary, France, Germany and Britain, his peripatetic academic career, his successes and failures and his personal affairs. Occasionally sad or moving, frequently amusing and always fascinating, this autobiography shares some of the intelligence, spirit and humanity that have made Dr Dienes such a landmark figure in mathematics education. A 'must-read' for anyone with a professional interest in the field, this is also an absorbing and frank book for anyone interested in the life of a man of ideas who was not afraid to take on the might of the traditionalist educational establishment.

16.2. Review by: John L Wisthoff.
The Mathematics Teacher 94 (7) (2001), 616.

Zoltan Dienes contributed greatly to the "new mathematics," which was developed in reaction to the Soviet launch of Sputnik in 1957. His principal interest was in the early learning of mathematical concepts, and he can justifiably be viewed as the father of the extensive use of manipulatives and play in teaching those concepts. His genius lies in designing experiences that can expose children to higher-order mathematical concepts.

If the readers of this book are interested in the intellectual de tails of his approach to teaching mathematical ideas, they will be disappointed. This memoir is about a life's journey from Hungary to Vienna, Paris, and England. It is a "grand tour" of mathematics education venues in Germany, Italy, Australia, and Canada. The book celebrates the life experiences of a mathematics educator. It is about a boy who can speak four languages by the time he is eight years old. It is about a naive academic who takes jobs without knowing their tenure. It is a detailed description of houses and apartments occupied, transportation used, and even individual meals eaten. Dienes describes his life in a very readable and enjoyable manner.

Within this personal memoir are descriptions of lessons given to children in places from Italy to Papua New Guinea; lessons devised by a mathematician with the singular ability to explore concepts through directed movement of the body or manipulatives. The book challenges one to read the author's considerable writings of a more technical nature?

16.3. Review by: B H Neumann.
The Mathematical Gazette 84 (500) (2000), 348-350.

A maverick, originally 'a calf or yearling found without an owner's brand' is also according to the Oxford English Dictionary 'a masterless person: one who is roving and casual'. Professor Dienes, known to his friends as Zed or Zeddie, fits this description well: he is himself a master of his trade, and he has travelled widely all his life. A mathematician according to the same source, is 'one who is skilled or learned in mathematics' and this fits him too.

The foreword by Paul Ernest starts 'I first met Zoltan Dienes in the summer of 1985'. I have known members of the Dienes family, including Zoltan and his wife Tessa, since 1962 when they first settled in Australia, and had known of Z P Dienes as a mathematician (and of his father, who was in his time a well-known mathematician in Great Britain) for quite a few years before that. In fact it must have been during the 1950s that I met Dr Dienes on a trip to Leicester.

The autobiography makes delightful reading. Zeddie was born in Hungary, and his young years were spent mainly in Budapest, but also Vienna, Nice, Paris and the Oberammergau: so he naturally became fluent in Hungarian, French and German and later learnt to speak several more languages fluently, especially English and Italian. His parents were divorced, and his father remarried and accepted an academic appointment in Wales (and later in London). So, eventually Zeddie also migrated to England, and went to school and university there. However this bald statement does not do justice to the many interesting trips Zeddie went on with much walking in Scotland, Italy, and Corsica and, of course, England. In fact he became an inveterate walker and explorer, helped by the many languages he acquired. He worked for and took his PhD in mathematics, and he married Tessa, a girl he had known for years, remains his wife and companion after more than 61 years.

During the Second World War, Dr Dienes, as he was by then, taught at a number of schools, mostly in the south of England. This did not last long. He was appointed to a lecturing job at Southampton University College, one of the many University Colleges that the University of London nursed to full university status. He spent two years there, two years at the University of Sheffield, two years at the University of Manchester (before I arrived at that University), and finally went to Leicester University College where he remained for more than twelve years. This bare recital does not do justice to the many charming anecdotes that fill these, as indeed all, chapters of the book. During the time at Leicester, Zeddie, and then Tessa, became members of the Society of Friends, commonly known as the Quakers.

Dr Dienes' interests shifted from pure mathematics to mathematics education, educational psychology, and educational philosophy. He designed materials such as wooden blocks or picture cards, with which he taught quite young children some interesting mathematical structures. In fact the first chapter of the book describes how he began to teach village children in somewhat inaccessible places in what was then the (Australian administered) Territory of Papua New Guinea. The description of what he did there comes some 15 chapters later. In 1961 he and his family, by now five children - some already grown-up, moved to Australia where he had been offered a Readership at the University of Adelaide. The post was not in mathematics but in psychology, which in his case meant educational psychology. Although he had met and worked with Jean Piaget, and admired him (as does this reviewer), his theories went well past Piaget's.

The time in Australia lasted only some four years during which be not only lectured at the University of Adelaide, but also introduced many of his educational ideas into schools in Australia, in Papua New Guinea, and on many visits to other countries. In 1966 Zeddie was then enticed to found and direct his own institute at the University of Sherbrooke, in Canada. He describes the years there as the zenith of his professional life: he was in great demand all over the world to advise on childhood mathematics education, to demonstrate his methods and to lecture in some of the many languages he commanded by then. The book is full of his travels, in which Tessa usually accompanied him, to very many places in very many countries, some professional, some recreational but most of them a combination of both. Their adventures on these travels make fascinating reading.

The wonderful Sherbrooke experience eventually went sour because of professional jealousy (my interpretation, not the author's, who put it down to his own lack of diplomacy). He and Tessa moved to Winnipeg, where Brandon University had a centre at which Zeddie then held a full professorship. But this also went sour with a change of dean of the faculty. So Zeddie and Tessa returned to Europe where they divided their time between Devonshire and Italy, of course again with many trips to other places. They eventually returned permanently to Canada, where they now live in quite active retirement - in his 84th year Zeddie is still professionally alive and well, in spite of some health problems that seem natural to his age.

17.1. From the publisher.

This collection of poems describes events that have taken place during the author's life. The first group of autobiographical poems includes the author's experiences as a refugee in Austria and Hungary, his move to England, and his schooldays and holidays. The second section provides some insights into the author's feelings for his beloved wife, Tessa. There are a number of poems that are intended to make the reader laugh at some of the absurd things that we take for granted. A few poems refer to the difficulties that some children have in trying to understand mathematics because of the way that it is taught, and give examples of fun ways of teaching it. Finally, there are poems that aim to make the reader think more deeply about the world in which we live; the beauty of the ever-changing seasons; reason and spirit; the meaning of life; hearing God's voice; and walking with God.

17.2. Review by: B H Neumann.
The Mathematical Gazette 85 (504) (2001), 534.

This is a collection of poems, but Dr Dienes is no poet: many of the rhymes are artificial, and the scanning is often awkward. However, the 'poems' are worth reading for their contents. They add some interesting detail to the author's earlier book Memoirs of a Maverick Mathematician, especially in Part One, 'Autobiographical', which occupies about half the book. His, and his wife Tessa', joining of the Society of Friends is convincingly and sensitively described. Part Three 'Mathematically inclined' need not frighten anybody not so inclined. The final Part Five, 'To make you think', is naturally informed by his religion as a Quaker. The book forms an interesting foil to his Memoir.

18.1. From the Michael O J Thomas.

This book contains a collection of the most recent ideas for learning mathematics from Zoltan Paul Dienes, one of the outstanding contributors to the field of mathematics education over the last 50 years. He writes in the introduction to this new book "I have been thinking that it is time to put together some of the ideas that I have evolved about mathematics and about how it is best approached. I thought that one way of doing this would be to put together some of my writings of the past decade or so - I don't really think you need to have any pre-knowledge of mathematics to read my various contributions. ... All my contributions are full of practical things that you can do, indeed that you would be advised to do, if you want to get the best out of what I have provided - I think I have managed to leave behind, as a swan song, a more complete account of what I have considered important, even exciting, for living this short life of ours! I hope you will find some excitement, some beauty and perhaps even some mathematics in what follows. Good hunting!"

18.2. From the Publisher.

Zoltan Dienes (born 1916) is a Hungarian mathematician whose ideas on the education (especially of small children) have been popular in some countries. He is a world-famous theorist and tireless practitioner of the "new mathematics" - an approach to mathematics learning that uses games, songs and dance to make it more appealing to children. One of the best known examples of the application of these principles is Dienes' multibase blocks, which have been used in learning by many around the world to embody the concepts of place values. Mike Thomas is an Associate Professor at the Mathematics Education Unit in the Department of Mathematics at the University of Auckland, and has been in regular contact over the last ten years with Zoltan Dienes whose manuscripts have appeared regularly in the New Zealand Mathematics Magazine. Mike is confident that all mathematic educators will find something valuable in this publication as will their students.

18.3. Contents.

Foreword.
Introduction.
1.     Mathematics Everywhere! - A Poem.
2.     Mathematical Fun Without Numbers, Letters, Formulae or Equations - Part I.
3.     Mathematical Fun - Part II.
4.     Mathematical Fun - Part III.
5.     Mathematical Fun - Part IV Five Colours.
6.     Mathematical Fun - Part V Two-Colour Sliding.
7.     Mathematical Fun - Part VI Sequences Using Three Strips.
8.     Strips for a Two Dimensional Space Based on the Eight and the Nine Element Fields.
9.     Circular Villages Part One.
10.   Circular Villages Part Two.
11.   Magic Polygons.
12.   How to Make Infinity Intelligible.
13.   Six Stages with Rational Numbers.
14.   Points, Lines and Spaces.
15.   Playful Logic.
16.   Games to Amuse.
17.   Cube Cards.
18.   Introducing the Tetrahedron.
19.   What Can You Do With Odd Numbers?
20.   Cyclic Nightmare.
21.   Measure.
22.   Some Problems With Logic Blocks.
23.   Using Algebra to Solve a Problem.
24.   Some Reflections on Order and Density - A Child's Path to the Bolzano-Weierstrass Theorem.
25.   A Learning Game.
26.   Mathematics as an Art Form: An Essay About Communicating Mathematics to Non-Mathematicians.
27.   Collected Poems.
28.   Peace Poems.
29.   Conversations with God.