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Richard Lesh and Marsha Landau (eds.), Acquisition o f Mathematics Concepts and Processes, Academic Press, New York and London, 1983. This book presents a range of research projects concerning the learning of mathematics, covering many aspects of the primary and secondary curriculum. The chapters cover young children's approaches to problems involving addition and subtraction, older pupils' understanding of multiplicative problems, including those requiring ratio; space and geometry treated from two different standpoints; and three approaches to the process of problem solving. The researches discussed use a variety of methodologies, but typically they take a substantial curriculum area, and attempt to describe pupils' primitive conceptions and their development under the influence of teaching. Because of the importance of the book and the variety of approaches in the different chapters, it has seemed necessary to devote separate attention to each chapter rather than give a more superficial review. The introductory Chapter 1 describes the focus of the work in general as consisting of 'idea analyses', which it places at a level of generality between (a) the analysis of particular tasks, and (b) pupils' cognitive characteristics, for example, their Piagetian levels of thinking. Chapter 2, by Carpenter and Moser, is on the acquisition of concepts of addition and subtraction, that is, not simply on the learning of the meaning of these operations at the minimal level necessary for the learning of addition and subtraction facts and of methods of calculating, but rather on the understanding and solution of a broad range of problems in everyday contexts which require these operations. These become accessible as soon as children are able to count and to apply numbers to sets of objects. Such problems are classified broadly as Change, Combine and Compare problems-and across these categories, according to whether the unknown is the 'result' set, or the first or second component set (in senses which differ slightly across the three broader types). For example, "Connie has 13 marbles. 5 are red and the rest are blue. How many are blue?" is a Combine problem, where the unknown is the second component set. There are substantial differences in difficulty among the types, which the researchers attempt to relate to differences in the pupils' perceptions of the problems, and the solution procedures they adopt. For example, Change problems involving joining to or separating from a given set, with the result being unknown, and Combine problems with the two components sets given were correctly solved by almost all children, even the youngest (aged about 5); but Combine problems, like the one quoted, where one of the component sets is unknown, were correctly answered by less than half of the children aged 5 and 6. Compare problems were in general harder still. Childrens' solution procedures were observed and related to the problem types. For example, solution by first-graders to the Combine, part unknown problem given above consisted predominantly of 'separating from', that is making a set of 13 cubes, separating 5, and counting the remainder. Solutions by second and third graders included a significant number of 'count up from given', that is counting 6, 7 . . . . up to 13, and reporting 8. This indicates an improved ability to represent the problem mentally, and to recognise the 5 both as a subset in itself and also as part of the whole set of 13. These were the observable features of the solution procedures. This information and that on problem difficulty were used to formulate theories of development. These propose

Educational Studies in Mathematics 15 (1984) 103.

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three levels. At the first level children c a n n o t represent mentally the data of the problem and the role of each item, so they are limited to modelling the situation with cubes and counting all the cubes in each set. At the second level children can mentally record the items o f data a n d their roles, a n d can c o u n t on, and so solve Change U n k n o w n problems. But they can only represent concrete data, n o t u n k n o w n sets, so cannot solve Start U n k n o w n problems. Third level children, it is suggested, have a ' p a r t - w h o l e ' schema which enables them to recognise any problem of this class and identify given and missing elements; and (on one theory) they also are able mentally to reverse joining and separating actions. Considering this work from the viewpoint of the practitioner, we note first that current teaching programmes generally aim at teaching the meaning of addition and subtraction as operations, with the main object of enabling children to c a l c u l a t e - t o obtain 5 as the answer to 1 3 - 8 , and 47 as the answer to 1 3 2 - 8 5 . T h e y a s s u m e that the recognition of the needed operation in a given problem is trivial. The first contribution of this research therefore lies not in its o u t c o m e s but at its starting-point, in particular in its prior posing of the question, "What knowledge do children need in order to u n d e r s t a n d and manipulate mentally the addition and subtraction situations in everyday experience?" The restriction to verbally stated problems, presented as simply and briefly as possible, with no extraneous data, makes the problems less than fully representative of the situations outside m a t h e m a t ics lessons in which these ideas will find application. This is necessary for research which compares problem structures; there is a need for research which examines the consequence o f this gap. To make full use o f this research, the teacher needs to have available suitably varied sets of problems graded according to the difficulty levels revealed by the research. She needs to k n o w what are the problem features causing extra difficulty; to have some idea of why, in terms of the theory of children's developing knowledge and ability in this field, as sketched above; and to have some teaching strategies available to assist the develo p m e n t . The work described in this chapter takes us several steps along this road, b u t does n o t specifically consider the design of instruction. For example, is it sufficient to offer the graded set of problems, and to help children with the more difficult ones? Should we aim specifically to teach the part-whole schema, or reversal? Chapters 3, 4 and 5 each consider the development o f understanding in a particular conceptual field. Chapter 3 deals with proportional reasoning; the tasks each consist of comparing the sweetness o f two recipes for lemonade using given n u m b e r s of spoonfuls of l e m o n juice a n d sugar. The problem set is designed to include those in which the ratio of spoonfuls within or between recipes is equivalent to an integer (e.g. 3 to 12), those in which both ratios are integral, and those in which neither is integral. The problems require recognising whether the two concentrations are equal or not, and, if unequal, the finding of the necessary a m o u n t to add to equalise them. Problem difficulty depended strongly on whether or n o t the ratios were integral; also, problems were easier if the integral ratio was within rather than between recipes. The main differences a m o n g pupils' approaches were between proportional and additive m e t h o d s , non-integer ratios o f t e n leading them to adopt additive instead of multiplicative methods. A theory is suggested which might represent a pupil's path through a typical problem. It involves testing first the within and then the between ratios to see if they are integral, then comparing with the other ratios. It seems to be suggested that progress consists o f being able to reduce first integral and t h e n n o n - i n t e g r a l ratios to unit ratios. However, this appears to neglect the various ad hoc approaches depending on the particular numbers; such as those in which a comparison of 4 : 1 2 with 1 0 : 2 4 is made by seeing that 4:12 doubles to give 8:24; also correct partially additive approaches, such as 4: 1 2 , 4 : 12, and 2 : 6 making 1 0 : 3 0 . A particular feature of this chapter is its relating of available solution procedures

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(correlationally) to crystallised and fluid intelligence, field dependence and other general cognitive measures of pupils. Some teaching e x p e r i m e n t s are also discussed. The traditional teaching of proportions by expressing t h e m as a pair of fractions, to be tested for equality by cross-multiplication, are s h o w n to be used meaningfully by only a very few early adolescents. The ' u n i t ratio' approach appeared more promising, b u t needed dealing with in varied c o n t e x t s a n d with progressively m o r e difficult ratios. Distinguishing between additive and multiplicative situations also needed attention. Chapter 4, on rational n u m b e r s , describes a substantial teaching e x p e r i m e n t c o n d u c t e d with 8 - 9 year-olds over a period of a b o u t a year. This was used as a data generating situation, several particular studies being extracted from it. The chapter presents an analysis o f the c o m p o n e n t s o f the rational n u m b e r c o n c e p t - m e a s u r e , ratio, q u o t i e n t and operator, a n d in this respect can be compared with Freudenthal's p h e n o m e n o l o g y of fractions, which distinguishes the fraction as 'fracturer' (wholes broken into parts), comparer (ratio) and operator, the latter getting close to the fraction as measurer. ('Quotient' is perhaps n o t a fraction concept, rather (1) a t h e o r e m - t h a t if a is divided into b parts, each part is of size a/b, and (2) an e x t e n d e d use of the bar notation to signify a division, as 3.2/4.18 for 3 . 2 + 4.18). This chapter, in comparison with Freudenthal's discussion, conceives rational n u m b e r s m o r e in terms of what is c o m m o n l y learned in school (adding fractions is o f t e n m e n t i o n e d ) ; Freudenthal's is m o r e rooted in everyday experience. The main research data presented in this chapter concerns the relative difficulty of problems of constructing 3/4, 2/3 a n d 5/3 in different representations (circular and rectangular regions, sets o f discrete objects, and on a n u m b e r line), under various levels of distraction by consistent or inconsistent division lines. (For example, show 3/4 of EDD). Difficulty gradients are great, a n d in the expected directions, with 9 year olds showing almost total success for 3/4 of a quartered circle to 1 0 - 2 0 % success on an unhelpfully divided n u m b e r line. An interesting individual protocol shows a pupil's inability to accept an undivided quarter circle as both 1/4 and 3/12, although he recognises an adjacent divided-up 3/12 as a quarter. There are s o m e other c o m m e n t s on pupil's errors, which would be more helpful if brought into a systematic scheme, perhaps trying to explain the levels of success in the large-scale data by h y p o t h e s e s a b o u t pupils' developing approaches, as is done for additive problems in Chapter 2. A conclusion drawn in this chapter is that the quantitative notion of rational n u m b e r (i.e. its size) is the key to the linking o f the various c o m p o n e n t s o f the concept. This ranks as an i m p o r t a n t 'didactic fact'. Comparing two fractions for size is a good problem for forcing consideration o f equivalence, which is the basic fraction property; but awareness o f 'size' is also i m p o r t a n t for a fraction used as operator, or measurer. Thus this chapter, like the previous ones, uses a core of data on relative difficulty of a cluster of problems, varying in a few controlled ways; b u t it does not carry this through so fas as, say Chapter 2, towards constructing a theory of pupils' development. O n the other h a n d , it refers to a wider range of learning tasks, using several different embodiments. There are some teaching suggestions at the e n d - t h a t 'translating' a problem from one e m b o d i m e n t to another is helpful, that initial confusion can lead to greater learning; and that the inverse relation between size of parts and n u m b e r of parts is a key step in understanding. (It is helpful to distinguish passive 'confusion', from 'conflict' in which there is the active seeking of a resolution.) These are n o t points established in any rigorous way by the research. But they are the p r o d u c t of observation and reflection by expert teacher researchers, in the c o n t e x t of a co-operative research project, in which varied experimental teaching is taking place and its success m o n i t o r e d n o t only by critical observation b u t also by periodic testing. Perhaps we should think of experimental work as aimed not so m u c h at proving propositions to the naive outsider's satisfaction, but as an aid to a critical inquiry being pursued in the minds of the experimenters. In chapter 5, Vergnaud describes French theoretical and experimental work on

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problems involving multiplication and division. He classifies these into three types: isomorphism of measures, product of measures and multiple proportions. The first category includes rate problems, such as those relating cost and weight, distance and time, and also partition and quotition problems (e.g. number of sweets and number in each share). Product of measures includes area and volumes, and also problems of multiple classification. (An example of multiple proportions would be a problem relating the amount of milk produced to the number of cows and the number of days.) Problems of isomorphism of measures are subdivided into multiplications, first-type divisions (corresponding to partition) and second-type divisions (corresponding to quotition). A distinction is not strongly made between cases where the two measures connected are numbers of objects, and those where they are dimensioned quantities like distance or speed. All are regarded as examples of linear function. One of the main experimental results quoted is that children prefer to deal with scalar relationships rather than functional ones, so that a problem about consumption of heating oil (for 7 days, 21 litres, how much for 84 days?) would be treated by some pupils via • 12 rather than • 3 in spite of the easier ratio. This result conflicts with that of Karplus where the ratio within a recipe (spoonfuls of juice to spoonfuls of sugar) was preferred to the juice ratio between recipes. Clearly the fact that these are not two different types of quantity is relevant, as is the fact that they are bound together by being in the same recipe. The present chapter describes some experimental work on volume, as one example of product of measures. The main conceptual obstacles found relate to comparing capacities of objects and containers, to the inverse relation between measure-number and size of unit, and to the trilinearity of volume. In this chapter, as in the previous ones, some of the most interesting and useful material lies not in the experimental outcomes but in the analysis of the conceptual field which forms the basis of the experimental work. It now becomes possible, therefore, to argue the validity of the analyses, against possible alternatives. For example, other analyses of problems in this field give greater prominence to the size and type of number in the problem (integer, decimal, greater or less than 1), and argue that this factor influences the pupil's attempt to assimilate the problem to concepts of repeated addition, partition and quotition. Chapters 6 and 7 concern space and geometry, a field in which research on learning and teaching has been very limited, compared with those discussed so far. Would it be fair to say that, while education in number-related mathematics (and probability) bear a close relationship to peoples' actual experience in these fields, curricula in geometry have dealt with only a part of our common spatial experience? Our ability to visualise situations, to transform mental images-for example, to look at a road map and decide whether a given turn will be left or right on the ground, or to imagine the shape of a hill from contour readings-these receive little or no attention in the mathematics curriculum, though they are sometimes treated more fully in art or geography. Chapter 6, by Bishop, reviews two strands of research. One is from developmental psychology, which has described pupils' understandings at various ages of, for example, bilateral symmetry, area, representations of 3D objects, coordination of viewpoints, concepts of the earth. The other strand is the factorial analysis of spatial abilities; these include the ability to make a mental transformation of a pictorially presented stimulus object, and to 'comprehend the arrangement of elements within a visual stimulus pattern a n d . . , to remain unconfused by changing orientation'. Bishop's own preferred distinction for educational purposes is between the ability for interpreting figural information (IFI), including graphs, diagrams, projections of all types, and visual processing (VP), which includes the visualisation of abstract relationships and other non-figural information, and the mental manipulation of visual representations and images. Research indicates that it is relatively easy to train IFI, by practice with the relevant types of diagram or graph, but that the trainability of VP is more questionable. A wide range of VP activities seems

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probably necessary; b u t more transfer and retention studies need to be performed. Some research on students in Papua New Guinea is interpreted as showing t h e m to be very weak in IFI, b u t relatively strong on VP, which certainly could be plausibly explained in terms of their background experiences. This chapter is very different from the earlier ones. It surveys a very wide field, y e t succeeds giving some glimpses into childrens' perceptions with regard to space, as do the previous chapters with regard to n u m b e r . It raises, by implication, i m p o r t a n t questions a b o u t the g e o m e t r y curriculum, and provides references to work which might act as a resource in building up experiences for a more broadly conceived curriculum. Chapter 7 by Hoffer, is entitled 'Van Hiele-based research'. A more informative title would be "'Levels of abstraction a n d o f p r o o f in the learning o f g e o m e t r y " . In terms of curriculum implications, this chapter is less radical than the previous one. In practical terms it describes moves to reorganise the traditional U.S. one year high school geometry course: instead of being a deductively based presentation of Euclid t h r o u g h o u t , it would begin with an informal s t u d y o f shapes and their properties, and become more deductive gradually, thus allowing a kind of mathematical activity m o r e in s y m p a t h y with the pupils' intellectual level. In the scheme described by Van Hiele, at Level 0 figures are recognised by their global appearance, at Level 1 properties are recognised and figures are compared with regard to t h e m . Levels 2 and 3 are characterised by recognising ' t h e o r e m s ' relating figures and their properties; by the use of short deductive arguments and then of fully organised deductive systems. Level 4 is that o f the Hilbertian axiom system. (However, there are discrepancies between this account and that o f Wirzup (1976), regarding whether short deductive chains belong to t h e middle level or the next higher one). The main reform implied here is that of expecting pupils to u n d e r s t a n d and share in the building up of geometrical knowledge and in its deductive organisation, rather than being presented with proof-arguments for results which they would themselves see as justified by m o r e empirical t e s t s - o r in some cases as being self-evident without proof. This is a m o s t i m p o r t a n t step to take and no d o u b t the projects c o n d u c t e d in Oregon, Brooklyn and Chicago, aimed at showing the existence of the Van Hiele levels in American high-school students, will by n o w have produced s o m e evidence to encourage adoption o f this more activity based curriculum. Perhaps an English reviewer m a y be permitted a little historical reflection at this point. A Mathematical Association Report on G e o m e t r y (1923) described stages A, B and C in the learning o f geometry corresponding roughly to the l s t / 2 n d , 3rd and 4 t h of the Van Hiele levels. Text-books in England gradually took this scheme into their design; in particular Siddon and Snell's New G e o m e t r y , published around 1954, contained t h e deductively organised set o f t h e o r e m s and proofs as its second part, t h e Ftrst part consisting of m o r e informal e x p e r i m e n t and short deductions. A m o n g m o r e m o d e r n texts, SMP Book Z (the last volume o f the main secondary course for abler pupils) contains a section on Proof in which the nature and purpose of the process of deductive organisation is discussed, and results acquired as u n s t r u c t u r e d informal knowledge in earlier years are organised and proved. This work receives no m e n t i o n in the present chapter; and it is n o t even in Dutch or Russian! Chapters 8 a n d 10 m a y be considered together, since they both take the same view of problem solving as a sequence of decisions and actions at different l e v e l s - t h e tactical level of the use of specific tasks, concepts or skills, the middle level of domain specific or m o r e general heuristics, and the managerial or strategic level of planning decisions governing the course of the activity. Examples of domain specific heuristics are 'look for a p a t t e r n in the n u m b e r ' or 'make a table', general heuristics include 'being s y s t e m a t i c ' and means-end analysis (Where do I want to get? What do I have to do to get there?); managerial decisions are o f the form "I'11 spend 10 m i n u t e s on trying to find the angle". Lester, in Chapter 8, m e n t i o n s briefly work on 'task factors' which affect problem

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difficulty, such as complexity, level of mathematical content, setting; and characteristics of solvers such as reasoning ability and field dependence. But he concentrates on some m o r e interesting teaching experiments. These were typically c o n d u c t e d with classes of children aged between 8 a n d 11, over several weeks, with two problem solving sessions per week with their regular teachers, covering some 1 0 - 1 5 problems. In these experiments there was no direct instruction in heuristics; pupils worked at the problems individually or in pairs, with individual help from the teacher which consisted mainly of encouraging the sharing of ideas, drawing attention when necessary to relevant information, and in the last resort giving hints to ease progress. Each session was m e a n t to end with a discussion of the pupils' solutions and h o w they had reached them, but in the event this was unsuccessful due to little experience on the part of the teachers of this kind of discussion. Nevertheless, problem solving behaviour improved dramatically. A second experiment included pairs o f problems similar with respect to structure, type o f information or size o f n u m b e r . This enabled observations to be m a d e which showed that pupils generally recognised the similarity and r e m e m b e r e d relevant solution strategies. (An interesting conflict with previous research suggesting that pupils tend to r e m e m b e r contextual rather than structural features?). Other studies compared 'practice only' groups with others which received direct problem solving instruction; t h o u g h the heuristics taught were successfully learned, there was no significant difference in problem solving success. The main danger appeared to be that pupils taught particular strategies tended to use t h e m inappropriately, when other strategies were required. This is an important result. Schoenfeld's Chapter 10 draws attention to the powerful effect of managerial decisions in problem solving. By analysing an actual protocol he shows h o w managerial assessments take place at transitions between episodes in the solving process; he discusses how t h e y are made, notes their consequences, and considers whether or not they prove to be good decisions. He t h u s hopes to give such further insights into this process as will help students to be more aware of their managerial decisions and to make t h e m more effectively. Chapter 9, by Lesh, Landau and Hamilton, is entitled Conceptual Models and Applied Problem Solving Research. The latter term m e a n s by 'average students, of realistic problems containing substantial mathematical c o n t e n t ' . The problems discussed in the chapter relate to the Rational N u m b e r Project reported in Chapter 4. The n o t i o n o f conceptual model is discussed, and then the results are reported o f interviews with 80 pupils (aged 9 to 13) on a set of 11 'realistic' problems involving adding or multiplying fractions. It is admitted that it is difficult to m a k e such problems genuinely real; for example if a boy eats 1/2 of a m u s h r o o m pizza followed by 2/3 of a t o m a t o pizza, the question 'How m u c h has he eaten altogether?' it's hardly fikely, in real life, to expect the answer 7/6 o f a pizza! The c o m m e n t s on solution m e t h o d s for these problems are informative. General conclusions were, a m o n g others, that small differences in a problem statement or in m o d e of presentation could cause big changes in r e s p o n s e - a n indication that recognising what are the salient features of a problem f r o m t h e pupil's point of view is not easy. Concrete problems, e,g. with "realistic pizzas", are n o t necessarily easier than verbally or symbolically presented ones; they are certainly different. Pupils m a y well m o v e between several r e p r e s e n t a t i o n s - c o n c r e t e , diagrammatic, s y m b o l i c - i n the course of solving a problem (and m a y even accept inconsistent results). There are some interesting reports by pupils of their imagining parts o f circles when calculating with fractions. A c o m m o n error arose from using two parts instead of part and whole to form a fraction. This chapter presents fully, with briefer c o m m e n t s , the results of an extensive written test of rational numbers. These results m a y all be useful, but they leave the reader with the problem of digesting the mass o f data and attempting to interpret it for himself. This reader would have appreciated a little m o r e help from the researchers in this task. This b o o k is a b o u t research, and it might be helpful to finish with some comparative c o m m e n t s a b o u t the methodologies used and their outcomes. It seems best to start from

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the questions "What have I learned from the book? How far has it depended on the research m e t h o d o l o g y ? " 1 t h i n k that what I have learned springs less from the actualresearch results than from the concepts which had to be developed in order to obtain them. Yet w i t h o u t the thrust to obtain definite results, the concepts used for describing observations would have been unrefined. The three chapters on problem solving lend themselves to comparison, their methodologies being very different. T h e main o u t c o m e (for me) of Lester's chapter was an inference by comparison of results from several teaching experiments: experience of problem solving, with sparing hints as necessary, and with a little s u b s e q u e n t reflection on the process, is probably as good as direct instruction in particular heuristic m e t h o d s . This reinforces m y belief from classroom experience that direct instruction in problem solving strategies often fails to make contact at any deep level with pupils, because it describes strategies in over-general terms which the pupils cannot apply to their awareness of their problem solving. Schoenfeld's chapter convinces me that an adequate description of problem solving requires the concept of managerial decisions. These have perhaps been neglected in protocol analysis because they are not normally spoken, and apply to large c h u n k s rather t h a n to detail. Their importance is 'proved' by Schoenfeld essentially by his giving an analysis of two chosen protocols, and asking us to agree that his description 'explains' the protocol as a whole better than another would. There is no random sampling, n o representative set o f protocols with measures o f adequacy of description showing a correlation between successful problem solving and the use or non-use of managerial decisions. Yet the work does add to m y (our?) understanding of the process. The chapter of Lesh et al. presents interview report data and test results which I feel contain insights into children's approaches to concrete problems a b o u t fractions. It reminds m e that they s o m e t i m e s compare part with part instead of part with whole, and that they s o m e t i m e s c o u n t unequal parts without discrimination. But this data contains, I suspect, m u c h still-hidden information. One might ask the general question whether, in the present state of knowledge a b o u t mathematical education, we should progress faster by collecting 'hard' data on small questions, or 'soft' data on major questions. It seems to me that only results related to fairly i m p o r t a n t practitioner questions are likely to become part o f an intelligible scheme o f knowledge. T h e developing theory o f m a t h e m a t i c a l learning and teaching m u s t be a refinement, an extension and a deepening of practitioner knowledge, n o t a separate growth. Specific results unrelated to major themes do n o t become part of c o m m u n a l knowledge. On t h e other hand 'soft' results o n major t h e m e s , if t h e y seem interesting and provocative to practitioners, get tested in the m y r i a d of tiny experiments which teachers perform everyday when they 'try s o m e t h i n g and see if it works'. T h r o u g h this process, some ideas fade away and others become important. But we also have to reckon with fashion. The history o f the last thirty years in mathematical education gives evidence of false ideas which gained acceptance and had major curriculum reforms based on them, in spite o f the fact that their insufficiency could have been recognised at the outset, and was, by some people. It would be nice to t h i n k that never again would we perpetuate errors like that o f teaching y o u n g children to deduce 7 • 8 = 56 from the laws of algebra. Our safeguard against this is through the development of better understanding of the nature of m a t h ematieal learning and of appropriate t e a c h i n g - a n d the spreading o f such understanding t h r o u g h o u t the vast and c o m p l e x system o f schools, teachers, advisers, writers and controlling authorities. The work described in this b o o k represents a significant contribution to this process. It addresses i m p o r t a n t learning questions, and deals with recognisable and sizeable areas of the m a t h e m a t i c s curriculum. It represents also a step towards establishing an international c o m m u n i t y of knowledge, t h o u g h in this respect one could wish far more evidence o f critical interaction a m o n g the various authors, (there

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is some but more would be better) and m o r e awareness of other relevant work. One such gap has been m e n t i o n e d already. A n o t h e r is the scarcity o f references to the work o f the CSMS Project (Concepts in Secondary Mathematics and Science) at Chelsea College, London. However, the authors are to be congratulated and t h a n k e d for their considerable efforts at making this body of work available. Two lines of extension suggest themselves. First, the adoption o f substantial curriculum areas as the target o f interest is a great advance b e y o n d the s t u d y of single tasks. What remains to be tackled is the question of the ability to call the relevant mathematical knowledge into play in real situations, outside the m a t h e m a t i c s classroom. We know that pupils' perceptions of problem in general are d o m i n a t e d by c o n t e x t rather than structure; what are the implications of this? Secondly, t h o u g h the information in this book, suitably interpreted, is certainly of great value to teachers, there is a place for more concerted research on h o w these insights m a y be incorporated into teaching programmes. Chapter 9 promises a further publication on the first question, from the Applied Problem Solving Project; work on both is also in progress at our own Centre.

Shell Centre for Mathematical Education, University of Nottingham

A. W. B E L L