Multicultural MathematicsAuthor(s): Brian HudsonSource: Mathematics in School, Vol. 17, No. 3 (May, 1988), pp. 47-48Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214501 .

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vestigations which the reader is encouraged to explore and explain rather than simply find the right answer. The first investigation begins with the following problem; Two players A and B have a pack of 9 cards. They take turns to remove 1, 2 or 3 cards from the pack--but a player must never remove the same number of cards as the previous player. The winner is the one who either takes the last card or leaves the other player with no valid move. Who wins?

This is followed by a series of structured questions which guide the reader through the discovery process. After three additional investi- gations it is assumed that the reader has acquired some strategies in the way problems are studied and so can tackle the extended investigations in Part 2. Finally the reader is presented with six additional problems with a shortened outline approach and hints.

This is a mathematics book with a difference. It is not about content in that the problems, al- though carefully chosen, are a means to an end, and that end is to cultivate the art of doing mathematics. It is a book which is likely to provide many hours of fascinating study and insight for teachers and pupils alike.

DAVID M. NEAL

Letters to the Editors

M nemonics

Dear Editor, I am making a collection of the various mnemonics by which pupils were taught (still are?) to remember the trigonometric ratios of sine, cosine and tangent, and would be very grateful for any contributions from readers. My present list includes:

"SOHCAHTOA" " The Old Aunt Sat On Her Coat And Hat" "Saving Old Halfpennies Can Always Help Towards Old Age" "Sailors Often Have Curly Auburn Hair Till Old Age"

Stephen Lerman South Bank Polytechnic, London

Statistics and Probability by Douglas Quadling Cambridge, 0521336155, A13.50

This is a statistics book designated for courses leading to "A" level Further Mathematics which would also serve the needs of first year courses at University. It is well set out and organised so that students can follow the text on an individual basis.

The basic ideas of probability and statistics are covered but it is assumed that the reader has some experience of the fundamental ideas on which the theory is developed. Full use is made of the calculator so as to avoid unnecessary arithmetical slog and readers with a microcom- puter will find it extremely useful.

The content is designed as a teaching course with topics developed spirally throughout the text. There are questions throughout the text that encourage the student to pause for thought and, I would hope, cause teachers to think again about the objectives of their lessons. I was particularly impressed with the project exercises, especially "It always rains on Thursday" which should produce some interest- ing discussion and conclusions. Furthermore the "Answers, hints and comments on the exercises" are extremely useful and will hope- fully set a trend to be followed by other authors.

All the text and questions are clearly presented with definitions emphasised by a pair of black lines. The content is divided into three sections each ending with revision exercises. Probability, randomness, estimation and discrete distri- butions form the first section. The second in- volves continuous distributions and hypothesis testing with the final section developing the theory in greater depth. I particularly enjoyed the constructive manner in which the author dealt with "The problem of estimation" and "Testing the model".

The feeling of the author for the subject matter is evident throughout the text and it is a pleasure to be treated to many graphical and geometric demonstrations. I am sure that this book will be valued by those for whom it was designed but it should be an essential book for all sixth form libraries and for sixth form teachers.

RUTH E. QUINN

Mathematics in School, May 1988

Multicultural Mathematics

Dear Sirs and Madam, I was pleased to read Michael de Villiers' letter written in response to my article on "Multi- cultural Mathematics" in the September 1987 issue of "Mathematics in School" and to learn of his interest in the issues raised. I do however feel the need to respond to a number of points which he makes in response to my article and also to comment on a number of issues which he raises in his letter.

With regard to his assertion that the activity outlined in Fig. 8 creates the misconception that Blacks are "confined" to the homelands I think that this is a matter of opinion and is not the central issue in any case. I do not see how one can escape from the essential facts concerning the human rights of the vast majority of Blacks in South Africa. For these people, their only official rights are to reside in a Bantustan. The migrant labour system and the immediate needs of the economy may well result in a policy of "confine- ment" being unnecessary. However the central issue surely is that their rights are confined to the Bantustans. I would like to point out that it is my wish to avoid misconceptions of any kind and do not see such an activity providing any answers. Rather I see it as the beginning of a process of questioning which will hopefully result in a greater awareness of the issues and a desire to find out more. At the same time stu- dents will be using and applying their mathema- tics for a meaningful purpose.

I am also pleased that my "honest attempts at addressing social and economic inequalities re- lating to racism" are recognised. I have been careful to fully document all sources of statistics. However some questions are raised about the use of the statistics relating to income distribu- tion. I should point out that these examples are drawn from a package which covers a wide range of issues and that further information is available on each of the major countries of the world (127 in total), including South Africa. This information includes total population and child-population figures as well as GNP, life expectancy rates, infant mortality rates and other data. This data is provided as a catalyst for further research and hence I am only too pleased to consider the further dimension regarding the age distributions of the Black and White populations. It should go without saying (although I feel the need to say it!) that I would be only too pleased to encourage my students to follow up the implications of this. Given the figures provided and some crude but quick approximations I would estimate that the amended figures are as follows:

Percentage of income Percentage of adult

population

White Black African African

64 26

12 37

Hence the ratio of income is roughly 2.5 to 1 in favour of the White population whilst Black adults outnumber White adults by 3 to 1. This hardly presents a picture of justice and equality!

I agree that a comparison of educational qualifications would put the matter into further perspective but I feel that we would be unlikely to share the same perspective. For myself it would be likely to highlight the differences in educational opportunity between Black and White children and the underlying structural racism of the political system in South Africa.

I note with interest that no criticism is made of the statistics relating to life expectancy and infant mortality. For the Black population the infant mortality rate is almost 6 times that for the White population and the corresponding life expectancy is up to 20% less.

I do not propose at any stage the "superficial treatment of statistics" but rather the opposite. In fact i would hope that the package that I produced would go a long way to avoiding this, including as it does 20 items of data on 127 countries of the world together with approx- imately 50 problems which focus on a wide range of related issues and topics. I will also reiterate that I see this quite extensive resource as a springboard to further research on the part of the students (and teachers). This resource combined with such an investigatory approach should not lead to "simplistic solutions" nor are any offered. The process which is encouraged is one of questioning, some of these questions no doubt raising issues to do with justice and equality (or the lack of these) for the Black community in South Africa.

I do wish to respond to a number of issues which Michael de Villiers raises and in particular to his comments about the Black popu- lation growth rate which he proceeds to apply to the "Third World" in general. To suggest that the problems of injustice and inequality in South Africa and in other parts of the world are likely to be solved by a decrease in the popu- lation growth rate is a simplistic notion. It completely ignores the direct link between pov- erty and levels of population growth. The higher the rate of child mortality the greater the pressure to have more children in the hope that some will survive and become economically productive through their labour in the future.

Equally simplistic is the notion that global problems can be solved simply by economic growth which flies in the face of a previously stated aim of developing "an awareness of our dependence on our environment" which I whole-heartedly support. Surely such an aware- ness leads to questions about what the planet can sustain and withstand and what its limits are.

I also seriously question the stated aim of "convincing Blacks of the benefits of the 'free- enterprise' system". If the aim was to "convince" students of the merits of a Marxist as opposed to a Capitalist economic system one would be rightly accused of promoting "Left-wing propa- ganda". I could not subscribe to such an aim from any particular perspective as I believe it to be based upon a lack of respect for the inde- pendence of the learner and the integrity of the teacher.

In general I feel that the issues and questions raised by Michael de Villiers are at a purely materialistic level and that the moral issues to do with justice, equality and human rights in gen-

47

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eral are ignored. I would hope that by using such examples and resources teachers would allow their students to explore these important issues in the mathematics classroom. In doing so it is my belief based upon several years in the class- room that this would be a valid response to the real concerns of many young people today.

Brian Hudson

Mathematics Education Centre, Sheffield City Polytechnic

Maths View: Fishy Chips (Vol 16 no 5)

Dear Sirs and Madam, Tony Gardiner argues that an understanding of mathematics comes from painfully working through numerous problems on one's own, and that the arrival of the calculator has done no- thing to change this process. This is clearly true. In fact the blind faith which children have in the correctness of a calculator's display makes it more important that they build up an under- standing of the rough limits within which an answer should fall.

However, Mr Gardiner's attacks on other people's non sequiturs leave room for several of his own. This is most evident in his discussion of calculator use in the workplace and its impli- cations for schools. Thus he is quite uncon- cerned to find that in most workplaces "clumsy" (sic) procedures are generally used, and that, if a problem can be approached in several ways, the most elementary level of mathematics will be the one chosen. It doesn't matter, he suggests, because software packages increasingly incor- porate "all essential calculations". In other words, while schoolchildren need to understand what they are doing, adults do not?

Obviously, people in the workplace can get by using clumsy procedures, just as one could always add up and never multiply. What is curious is Mr Gardiner's conclusion that such a situation has no implications for education. Does he really want to array himself with those academics to whom productivity is a dirty word?

Mr Gardiner also provides a rather misleading impression of existing research on industrial practices. The studies which he cites set out to examine the uses to which calculators were being put, not, as he seems to believe, whether they were being used correctly. Thus they can hardly be used to argue that we can and should

leave everything to efficient employers. Our own department has recently been

working closely with the Manpower Services Commission, and with a wide range of YTS schemes, to help improve young people's math- ematics skills at work. Calculator errors are un- fortunately extremely common, and not only because of problems with data entry. It would be nice to receive A1, let alone A10, for every time one has seen 30 minutes entered as .3 of an hour!

It will always be difficult to contradict an assertion of the type: "management apparently (sic) accepts the responsibility for ensuring that (calculators) are used consistently and correct- ly". It is, unfortunately, very easy to document that few managers make any formal provision for ensuring that this is the case. In a country where many employers do not even make any formal provision for training the new computer pro- grammers that they hire, the schools can expect little in the way of on-the-job mathematics or calculator tuition. In a world where calculator use is the norm, I would argue the contrary case to Mr Gardiner's: that we can and should in- corporate calculators into the mathematics classroom.

Alison Wolf Institute of Education, University of London

48

Short Notices

Rubik's Cubic Compendium by Erno Rubik et al Oxford University Press, 0198532024, A14.95

A delightfully produced book which provides a comprehensive guide to the Rubik cube and the mathematics contained in the simple practical activity. The clarity of the text involving clearly labelled coloured diagrams linked to the sym- bolic representation make the book widely readable and so suitable for the secondary school library.

The book contains six chapters, In Play by Erno Rubik, The Art of Cubing by Tamas Varga, Restoration Methods and Tables of Processes by Gerzson Keri, Mathematics by Keri and Varga, The Universe of the Cube by Gyorgy Marx, The Psychology of the Cube by Tamas Vekerdy, plus an introduction and conclusion by David Singmaster.

Teaching and Learning Mathematics Parts 1 & 2 by Paul Ernest University of Exeter, 850680948, 850680956, A2 These two booklets form numbers 33 and 34 of the series called Perspectives which consists of papers on current educational topics written by the staff of the University of Exeter and outside contributors.

Part 1 begins with a short paper by David Burghes entitled "Mathematics for the 21st Century" and following five other papers concludes with one entitled "The Use of the History of Mathematics in Teaching" by Derek Stander.

Part 2 contains papers by John Mason, Kath Hart, Dietmar Kuchemann, Barbara Jaworski and Neville Bennett, each one concerned with the research perspectives on the teaching and learning of mathematics.

ITeMS Catalogue V by F. R. Watson University of Keele, Staffs ST5 5BG

Ideas in the Teaching of Mathematics and Science is a catalogue of occasional papers and information leaflets for those concerned with the teaching of mathematics who work at Univers- ities, Polytechnics, Schools, LEA Centres, Teacher Associations and individuals.

The current edition contains 143 pages of advertisements for aids from SMP Resource Packs, Mathematical Association Calculator Packs, various software to ideas from individual teachers.

Table Teasers Books 1-4 by Peter Patilla et al Macmillan, 0333439546, A1.95

An attractive book to help children aged 7+ to learn multiplication facts through pattern finding activities and problem solving. Each page con- tains a series of imaginative pictures which lead the children to the tables patterns.

Books 2, 3 and 4 follow a similar format for children in succeeding years. A great improve- ment on the Giant Sum Book.

The copyright forbids all copying of the pages and as the children are expected to write in the book the 32 pages may prove too expensive for schools. However it is a series to recommend to parents.

Contributions

Contributions and correspondence concerning editorial matters should be sent to the editors: Mathematics in School, Faculty of Education, City of Birmingham Polytechnic, Westbourne Road, Birmingham B15 3TN.

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Mathematics in School, May 1988

This content downloaded from 173.79.184.220 on Wed, 23 Apr 2014 17:53:23 PMAll use subject to JSTOR Terms and Conditions