ADVENTURES with MATHEMATICS Ann Kitchen TO'wing Icebergs

Home , The Last Theorem

Sixth Form College. Just looking at in whichy is the elevation of the water what was in everyday use not so long surface above or below the average ago gives some amusement. Did you level, x is distance, t is time and C is know that one slug was 14,606 the velocity of the waves'. The analysis kilograms? (I know that those in my carries on from there. garden are rather smaller). Or that the Although the author's back density of air is 0.00233 slugs per ground is in mechanics and fluid cubic foot? Rather more worrying is dynamics, the examples given in the the fact that a respected academic in book range far more widely, from the States does not recognise that population statistics to geography and things have changed this side of the economics. Some chapters take a Atlantic. single theoretical background with a ADVENTURESWITH This aside, the book takes a variety of applications. Thus, falling number of everyday and more dominoes are linked to tidal surges MATHEMATICS unusual situations and explores them. and wave motion. Ann Kitchen What happens and why does it The format of each chapter is the happen? How can the theory be used same. So chapter 14 deals with in other related situations? His style is skipping ropes and wind turbines. light and in places perhaps he over Now, to you and I there might not be Towing icebergs. falling simplifies, but those willing to read much connection between the two, dominoes and other adventures with care will find a wealth of starting but Professor Banks takes us through in applied mathematics points suggested. the mathematics of pendulums, Robert B Banks The book starts with a descrip ellipses and elliptic integrals to the tion of the enormous meteorite that mathematics of the Troposkein, or to Professor Banks has been involved in hit Arizona over 20,000 years ago. the less initiated, the shape a skipping teaching and research in engineering This is used as a focus for explaining rope makes when in use. This leads over the past forty years. He looks on the term Mach number: how, in any smoothly to the Darrius wind mathematics as a beautiful and given medium, it is given by the ratio turbine, whose blade is designed to be intriguing language that can describe, of the speed of the body to the speed the same shape. The book is not accurately and concisely, the almost of sound in that medium. This leads designed to be read straight through. limitless list of phenomena that we seamlessly to the calculation of the Find a chapter that interests you see all around us. His joy and delight speed of sound in both air and water. particularly and read it. in these is evident to all who read the As a teacher I might well want to look So why buy and read the book? It book. His stated audience is mainly more deeply into this phenomenon. is a must for any school library. The those who have finished their formal How does the speed of sound vary as diversity of topics and the easy style education in mathematics. However, the density of the medium decreases? mean that it may well be the way to his book is equally accessible to those What affects the distance sound introduce students to modelling and who are still learning or using mathe travels in any medium? Answers to to the importance of modelling to matics. It is not a text book, more a questions such as these are not to be almost every field of human book of exploration of ideas and found. Professor Banks passes lightly endeavour. Equally, it may provide concepts linked together by mathe on. And yet, perhaps this is a good the germ of an idea for a new investi matical modelling. thing. It would be all too easy to go gation or a way that a well known What is the place of this book in deeper and deeper into the mathe investigation may be extended or teaching today? Firstly, it is not a matics until all but the most gifted altered. There are some suggestions book that one can pick up and use to mathematician had become lost. This for such projects but no detailed teach. Those who are looking for a set has not happened. Instead, the work answers. Those will be for the reader of finely worked out solutions to an both helps the reader to get a feeling to provide. However, there is a equally well-defined set of problems for the model and work out a simple comprehensive set of references to will be disappointed. But those who solution, and also gives ideas to those books and articles that may well give want to get ideas for explorations of who have the ability, knowledge and you the answers you need. their own, to find methods of contin inclination to go further. The purist, All in all, this is a book that is well uing and refining the models given or however, could rightly point out cases worth having. to enjoy looking at and trying to where the mathematics was too follow mathematics just beyond their simplistic. Certainly there is no TO'Wing icebergs,jalling dominoes own level of attainment will find a and other adventures in applied attempt to provide step-by-step wealth of material here. This is not to mathematics. Robert B. Banks, 1998, proof. So, when investigating waves in say that there are not some anomalies. Princeton University Press.ISBN 0-691- water, the analysis starts with the The author writes for the US market 05948-9, 328pp. £19.95. words 'We could analyse this problem and those studying in Britain will be by using the equations for the conser hard put to recognise the set of units Mechanics in Action Project, vation of mass and the conservation that he states as being in common University of Manchester of momentum. Doing so we would everyday use in Britain. Poundals and obtain the following equation: slugs have long since disappeared from the vocabulary of the average df

60 MTI68 SEPTEMBER 1999

Academic copyright permission does NOT extend to publishing on Internet or similar system. Provide link ONLY © ATM 2010 • No reproduction (including Internet) except for legitimate academic purposes • [email protected] for permissions.

reminder of much mathematics I had There are several reasons: ANOTHER LOOK AT forgotten I ever knew and gaining • because at its heart it is a series of THE LAST THEOREM new insights into things learned long papers about one teacher's class ago. The ten appendices range from a room; Jenny Murray proof of Pythagoras' theorem to an • because there is not enough example of proof by induction. You do known about what forms of talk not have to read them, but you best promote mathematical probably will want to. understanding, nor the processes Fermat's last theorem The human side of the story by which this is achieved; Simon Singh comes through particularly at the • because it is published in a series like many others, I want the books I end, when the flaw was found in the about situated learning and read to be both entertaining and of proof. The penultimate chapter reads cognition, and I wanted an more than passing interest. I like a more like a detective story or thriller, opportunity to grasp these ideas book which falls in the mean between as Andrew Wiles worked towards the for MT; 'once picked up, you can't put it final breakthrough. Perhaps this is a • because it raises some interesting down' and 'once put down, you can't point at which you would be unwilling questions about the relationship pick it up'. Fermat'slast theorem is to put the book down. between research and practice. just such a book: eminently readable, But this is only the penultimate but in no way tyrannical. It has chapter. So many amateur mathe One teacher's classroom enough of the human story to keep maticians have worked on Fermat's Magdalene Lampert taught in a the non-mathematician happy and last theorem. What could they do variety of different schools for twenty interested, and enough mathematics now? The last chapter gives many years before she moved into research. to have an erstwhile number theorist unsolved mathematical problems Towards the end of her teaching like me engaged in a delightful which, like Fermat's last theorem, are career she created video, audio and version of a revision course. Of easy to understand with a knowledge written records of her teaching with course, as the story unfolds, as the of elementary mathematics. Lots to one class for a whole year, realising mathematics develops, nearly all get those talented sixth-formers that they would be a rich resource for readers will join the ranks of the non working away on! future investigations. Four chapters of mathematicians (those who do not this book are based on analysis of her really know exactly what is going on), Fermat'slast theorem, Simon Singh, practice. There is no intention to say 1998, London: FourthEstate, ISBN 1- because the proof covers areas which 'this is an excellent teacher and we 85702 -669-1, 362pp, £6.99 'only half a dozen people in the world should all be copying her'; rather the could fully grasp'. Suffolk idea is to use her practice to highlight There are three threads twined some of the things that teachers do, together in the book. As the mathe that they may not be aware they do. matics involved and the history of By identifying features of her practice number theory are unfolded some the authors develop languages and biographical details of Andrew Wiles READING ABOUT structures which can be offered to are also presented. He first met TALKING ABOUT other teachers as possibilities. (They Fermat's last theorem in a public also raise further questions for other library book when he was ten years MATHEMATICS researchers, but that is not my old. Since then he kept the determi concern here.) AnneWatson nation to solve this problem which is Peggy Rittenhouse describes the three centuries old and which has teacher as 'stepping in' and 'stepping been the centre of so many false out'. She regards mathematics as claims. For seven years he kept Talking mathematics in school: represented by particular kinds of himself in professional isolation, away studies of teaching and learning discourse, and the pupils' task as from other mathematicians, working Magdalene Lampert and Merrie L. learning to use, and becoming fluent on it. Finally, he announced the Blunk (editors) in, the discourse. The teacher, who is proof, only to be presented with a This book, like many academic publi the expert, 'steps in' to use the flaw (which to many mathematicians discourse alongside her pupils, to cations, started life as a collection of seemed an almost inevitable conse model useful forms of talk, to conference papers written by quence). transform what they say into mathe American mathematics education The mathematics is presented in matical discourse and to scaffold their researchers. Papers are then taken a very human way. There are portraits efforts to express their mathematics away, rewritten and extended as a and stories of many mathematicians appropriately. The teacher also 'steps result of discussions with others. The from Pythagoras on. Many mathe out' of the mathematical conversation editors then force them into a suitable matical curiosities, such as 1t to 1500 in order to identify and name explic order, write overarching comments, figures, Euler's Konigsberg bridge itly different kinds of mathematical and publish. Such collections are problem, Penrose tiling and Sam talk. For example, she says: 'disagree seldom a complete view of the field, Loyd's 15 puzzle are met along the ing is very important to do in a math and can be patchy in their style and way. There is also a clear exposition of class; mathematicians do it all the content. Why should this book be 'serious' number theory and its time.' On another occasions she reviewed for Mathematics Teaching? offshoots. I delighted in finding a remarks: 'that's a generalisation'. This

MTl68 SEPTEMBER 1999 61

last comment reminds me of finding experiences which can be worked on seen as supporting the pupil's myself saying to a pupil who had just and developed within mathematics maturing partiCIpation by being an generalised: "there you are, that's lessons. She suggests teachers should expert guide and role-model. In many what generalisation feels like!", be asking: sodo-economic contexts, this guiding hoping that naming it as he did it Howcan I get pupils to spontaneously and modelling takes place in situa would enable him to do it again some engage in arguing, defining, speculat tions in which everyone is there for time. ing,chaJ1enging and modell ing? similar purposes. Ta king 'definitions' as a focus she The phrase 'situated cognition' is Talk and mathematical shows that various kinds of defining frequently used to describe the way understanding take place in mathematics as in that apprentices learn from experts, Very little has been written about the everyday life, and that teachers can alongside them, in the workplace, power of dialogue as a model for harness the experiences pupils have using formal and informal language, mathematical thinking. Plato, in the had with less rigorous defining in doing similar jobs, watching others at Dialogues, and Lakatos, in Proofs order to negotiate degrees of defini work and so on. The theories thus to and refutations, show how dialogue, tion in mathematics. It is not developed have been extended even imaginary dialogue (which could straightforward to expect pupils to apply to all developmental learning, be transformed into inner speech) can bring into classrooms reasoning skills so that children become 'apprentice' be at the heart of conceptual under developed elsewhere, and the adults, 'apprentice' shoppers, standing. Rittenhouse's chapter is one teacher's role is to build linguistic 'apprentice' pigeon-fanciers . . . In of few in the book which give new bridges in order to make the transfer mathematics education the theory does insight into a relationship between more obvious and simple. not apply itself so easily, for it is classroom talk and mathematical Another chapter which appeared unclear what the learners are learning thought. It also provides an accessible to be getting to grips with a concep to be. If they are learning to be math categorisation which could be a useful tual relationship between talk and ematicians there is often no expert tool for other teachers. It prompts the mathematics had the enticing title mathematician, doing maths questions: Disciptined perception. However, there alongside them, with whom they can Am I ever explicitabout usefulways to was a third feature of the relationship act like apprentices. If they are talk alxmtmathematics? in this case: some sort of visual focus learning to be numerate adults, the Couldbe? I Should be? I Howcan I be? for mathematical discussions. During expert is unlikely to be displaying the Another chapter which gets close one-to-one tutorials on straight line average competent numeracy of graphs a tutor had used various kinds to this relationship is by Kay McClain adulthood. In fact, the expert is acting and Paul Cobb. They write about the of grid, and had occasionally hidden like a teacher, not like a mathemati cian, so if the pupils are apprentices role of imagery and discourse in the parts of the diagram, in order to focus to her they are learning to become development of mathematical under the pupil's attention on particular standing, taking the view that what features of his attempts. It was found teachers! the teacher says can guide the learner that many of the pupil's 'errors' were In this book, the authors take a variety of views about this problem. in useful directions. For example, the due to features of the grids and teacher can remind the pupil, indi diagrams, so that causing him to In general, they accept that the focus on other features, visually and mathematics classroom is a social rectly through use of specific words, setting, with mathematical knowl to return to the particular context in verbally, helped him to decide what was more appropriate. The rest of the edge, skills, tools and goals, but the which a problem was set, or to teacher's role is not necessarily to act specific apparatus. The teacher's chapter described a mathematically similar situation in which highway like an expert mathematician. In the words can trigger a particular image Stevens and Hall chapter a version of which is more familiar or useful than engineers were coming to a shared Cockcroft's plea (in the UK in 1982) the one the pupil is currently express understanding of their plans. It would not be easy to shift from one-to-one for more involvement in workplace ing. Here a parallel between language seminars to general classrooms, and mathematics is repeated. This is more and image is being employed, but I likely to lead to artificial contexts was sad that this was restricted to the authors, Reed Stevens and Rogers unrealistically used in schools than to contextual mathematics: pupils were Hall, do not suggest that. Instead they suggest bringing more real technolog any effective shift in mathematical being referred back to cubes, or to ical experience into schools. discussion. In other chapters the sweets, rather than to previously-held teacher has a managerial role to concepts. However, the general Situated cognition and school organise discussions and interactions approach could prompt the question: so pupils can be experts for each Are there s, through 11!Y use of mathematics other. In other cases the teacher is language, I can relate this v.x:rrktothe All the chapters in the book are set images 11!Y pupils have of previous within an understanding of mathe required to display an expert's v.x:rrk? matics as a social practice, that is a knowledge of mathematical discourse, and occasionally to do Mary Catherine O'Connor's shared collection of knowledge, skills mathematics alongside pupils. chapter cheered me. She writes about and tools with a shared sodal goal. Some of the authors believe that pupils giving examples and counter The goal is to enable pupils to the teacher should be encouraging examples in mathematics, and how become fully participating members pupils to act like real mathematicians; they might do this quite naturally in of society with respect to their mathe others that pupils should be making other areas of their lives, giving pupils matical competence. The teacher is sense of a body of knowledge; all

62 MTl68 SEPTEMBER 1999

believe that mathematics is a social activity mediated through talk. Some questions which occur to me are: A4 paper and rhombuses inscribed in a rectangle W7uu are nrypupils learningto be? In An ordinary sheet of A4 paper is what reIo1ed areas am I, as a mathe an interesting rectangle. matics teacher, an expert? How do the According to the European norm pupilsaccess nryexpertise? its size is 297 by 210 millimetres. Just take a calculator and divide Research and practice the longer side by the shorter and A strength gained from researchers what do you get? 1.4142857 ... talking to each other is the develop Up to four decimal places it is ment of new ways of seeing \12. Is this coincidence acciden- . classrooms; a danger is that they will tal? No, it is not.Those who chose become too distanced from class this format wanted that, as a result rooms. The final chapter of this book of folding the paper, you get two is written by an educator who is an smaller sheets which are rectan expert on classroom discourse, not a gles similar to the larger one. mathematics educator. This choice Owing to this special property falls well into the danger area. Why of A4 (or A3, A2, ...) paper, these does a book by mathematics sheets can be used for interesting education researchers need summing spatial constructions. The most up by an outsider? Research is judged interesting shape you can get this by its format and its contribution to way is the rhombic dodecahedron. theory. Therefore researchers Its faces are rhombuses. The most constantly need their work to be obvious way to get such a rhombus from an A4 sheet is this. verified by other researchers, outside The diagonals of the rhombus are funding bodies, and accountability the axes of symmetry of the A4 sheet. authorities. Some of the work in this This is not the only way of getting such a book could make a contribution to rhombus from an A4 sheet. Here is practice, but the authors, being another way. First fold one corner of the American, are hardly likely to make sheet precisely onto the other. Then efforts to do so in the UK. make another fold through the middle of The power of systematic investi the crease at right angles to it. The gation is that it can show patterns and creases are the diagonals of a rhombus. frequencies of which teachers can be Fold to get a rhombus and two flaps. unaware. The power of seeing class rooms through a theoretical lens is that it can suggest relationships, structures and language which are not visible from the practitioner's viewpoint.You, the reader, if you are a

classroom teacher, can tell a story Most people prove these two about the relationship between rhombuses are similar by using Pythagoras research and practice as it strikes you to calculate the lengths of their diagonals. in this review. However, you can only Those who are mathematically rather do this through my interpretation, more stylish compare the angles. which is somewhat slanted. Is there In fact, there are more general results. anything here which you could not Theorem: The largest and smallest have found out by watching some rhombuses inscribed in a colleagues teach? Or by videoing your rectangle are similar. own lessons for a while? Or by doing Theorem: In any rectangle, allrhombuses inscribed are similar. these things and meeting with others Wszystkie romby wpisane w prostokat sa podobne. to discuss what you saw?

Talking mathematics in school: . This last theorem can be proved by studies of teaching and learning, using one of the circle theorems. edited by Magdalene Lampert and Merrie L. Blunk, 1998, Cambridge University Press, ISBN 0-521-62136-4, 260 pp, KrzysztofMostmvski, Department of £40.00. MathematicsEducation, Siedlce University of Oxford University, Poland.

MT168 SEPTEMBER 1999 63

Academic copyright permission does NOT extend to publishing on Internet or similar system. Provide link ONLY

The attached document has been downloaded or otherwise acquired from the website of the Association of Teachers of Mathematics (ATM) at www.atm.org.uk Legitimate uses of this document include printing of one copy for personal use, reasonable duplication for academic and educational purposes. It may not be used for any other purpose in any way that may be deleterious to the work, aims, principles or ends of ATM. Neither the original electronic or digital version nor this paper version, no matter by whom or in what form it is reproduced, may be re-published, transmitted electronically or digitally, projected or otherwise used outside the above standard copyright permissions. The electronic or digital version may not be uploaded to a website or other server. In addition to the evident watermark the files are digitally watermarked such that they can be found on the Internet wherever they may be posted. Any copies of this document MUST be accompanied by a copy of this page in its entirety. If you want to reproduce this document beyond the restricted permissions here, then application MUST be made for EXPRESS permission to [email protected]

The work that went into the research, production and preparation of this document has to be supported somehow. ATM receives its financing from only two principle sources: membership subscriptions and sales of books, software and other resources.

Membership of the ATM will help you through

• Six issues per year of a professional journal, which focus on the learning and teaching of maths. Ideas for the classroom, personal experiences and shared thoughts about developing learners’ understanding. • Professional development courses tailored to your needs. Agree the content with us and we do the rest. • Easter conference, which brings together teachers interested in learning and teaching mathematics, with excellent speakers and workshops and seminars led by experienced facilitators. • Regular e-newsletters keeping you up to date with developments in the learning and teaching of mathematics. • Generous discounts on a wide range of publications and software. • A network of mathematics educators around the United Kingdom to share good practice or ask advice. • Active campaigning. The ATM campaigns at all levels towards: encouraging increased understanding and enjoyment of mathematics; encouraging increased understanding of how people learn mathematics; encouraging the sharing and evaluation of teaching and learning strategies and practices; promoting the exploration of new ideas and possibilities and initiating and contributing to discussion of and developments in mathematics education at all levels. • Representation on national bodies helping to formulate policy in mathematics education. • Software demonstrations by arrangement.

Personal members get the following additional benefits:

• Access to a members only part of the popular ATM website giving you access to sample materials and up to date information. • Advice on resources, curriculum development and current research relating to mathematics education. • Optional membership of a working group being inspired by working with other colleagues on a specific project. • Special rates at the annual conference • Information about current legislation relating to your job. • Tax deductible personal subscription, making it even better value

Additional benefits

The ATM is constantly looking to improve the benefits for members. Please visit www.atm.org.uk regularly for new details.

LINK: www.atm.org.uk/join/index.html