Multicultural MathematicsAuthor(s): Brian HudsonSource: Mathematics in School, Vol. 16, No. 4 (Sep., 1987), pp. 34-38Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214374 .

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by Brian Hudson, Sheffield City Polytechnic

G

Issues

Human National Rights Issues

Local

Issues

Health PovertyPPoverty PovertyPPoverty Health oet Health Unemployment

Housing

South-- North

Housing South Unemployment North-

Africa South

International

commodity markets

In beginning to counter racism in the mathematics class- room and to develop a multicultural approach to mathema- tics teaching, it may often be necessary to overcome feelings of indifference and sometimes of hostility. I would therefore initially like to place such issues in a wider context by considering some findings from two relevant reports.

The consultative document Education in Schools (DES, 1977) states that "Our society is a multicultural, multiracial one ... We also live in a complex, interdependent world .... The curriculum should therefore reflect our need to know about and understand other countries". This view was shared by the Swan Committee in its report Education for All (HMSO, 1985) in which is stated: "... the development of syllabuses that are both multicultural in their content and global in their perspective would remain equally valid ... whether there were ethnic minority pupils in our schools or not".

The Cockcroft Report recognises the immediate response of many mathematics teachers to attempts to relate their subject to other subjects and wider curricular concerns, as many will see such issues as having little direct relevance to themselves. In paragraph 485 for instance it states that: "... because of the ways in which mathematics can be used as a means of communication, it can play an important role in the learning process in curricular areas which may seem to be far removed from mathematics".

Traditionally mathematics has had an image of abstract purity and this image has predominated despite the more recent developments which have taken place in mathematics education. The consideration of potentially controversial issues in the mathematics classroom would in fact be intentionally avoided by many mathematics teachers. However, I would argue that it is important to reflect the complexity and interdependence of the real world and also the multicultural and multiracial aspects of our own society

34

and that as a mathematics teacher I have the opportunity to do so. I believe that the Secretary of State's statement on the curriculum in 1984 supports such a view, in the following extract: "... the curriculum should be relevant to the real world and to the pupils' experience of it".

I want to consider multicultural issues at three different levels. Firstly there are the cultural differences between various groups, secondly there is the bias that exists in texts and classroom materials, together with strategies for overcoming this, and thirdly there are social and economic issues which relate to racism.

The cultural differences have been discussed by Derek Woodrow'. A significant issue is the conflict between the reading of written script from left to right and the different orientation of mathematics in which significant features often occur on the right. For example the size of the number 231 is established by reference to the right hand side whereas the left hand side of 0.231 establishes its size. Thus languages which have strong structural order may well influence the style of decoding information.

Mathematics teaching traditionally has made certain demands upon pupils which may favour certain cultural groups rather than others. For instance, pupils have tra- ditionally been required to concentrate, to be self disci- plined, to be accurate, to conform to rules, to be quiet and to use precise and sophisticated language. In examining cer- tain general characteristics of particular groups it can be seen how mathematics discriminates towards certain per- sonality traits which may in turn be strongly culturally influenced. The danger in attempting to analyse general characteristics, however, is that this may merely serve to reinforce stereotypes and hence the risk of being counter- productive is run.

In detecting bias in classroom materials, I found it useful to consider it under three main headings: bias by omission,

Mathematics in School, September 1987

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stereotyping and tokenism. These headings are not totally exclusive and in some circumstances classification was not easy. However I found it to be a helpful checklist around which to centre my thinking.

In carrying out such an evaluation in some examples of commonly used classroom materials the most obvious aspect is the almost complete absence of references to members of non-white, non-Western groups. Such an evaluation does also highlight the sexist bias that exists in most texts: of the 22 illustrations at the head of each chapter of the SMP Books, X, Y and Z, 20 may be designated male; in a random sample of 90 questions from SMG Book 1, there were 21 containing references to persons identifiable by sex - and of these 19 were male'. Although both samples are quite limited, similar results are obtained from both. Male to female representation in the ratio of 10 to 1 is, in my opinion, a fair estimation of the bias that does exist within most classroom materials. The extent of the bias due to race is undoubted and does not reflect the multiracial nature of our society in Britain today.

The most obvious example of racial stereotyping in my experience occurs in Chapter 11 of SMP Book E. The pupil is asked to consider the following:

"If you found yourself in the Kalahari desert without your set of mathematical drawing instruments and with- out a penny in your pocket, how would you show a Bushman what an approximate circle was?"

There follows an illustration of three black men, with benign smiles, two of whom are dressed in loin cloths, carrying spears. The third man, who is also black, is dressed in the old colonial style of white shirt and shorts. He is also wearing a cap and a wrist watch, and is demonstrating how a circle may be drawn with two sticks. This example is only the second representation in the series of a non-white racial group. The potential damage to images of others and to the self-image of some is very significant, in my opinion, from such an example.

Strategies for overcoming such bias are fairly obvious but are dependent upon a conscious decision being made based upon an awareness of the problem and an acceptance that something ought to be done about it. Indications that publishers are becoming aware of this issue are noticeable within the SMP 11-16 scheme. In particular we are intro- duced to Angela in Level 4 who is not only a female but also black and employed by a computer firm.

There is also a popular assumption that is promoted in

many texts that mathematics developed from the time of the Greeks to the present day, only interrupted for a few centuries during the Dark Ages. The contributions of Arab, Chinese, Indians, Mayans and Persians are either ignored or mentioned in passing. A great deal can be done to increase pupils' historical awareness of the development of mathematics. For instance, the opportunity can be taken when studying Pascal's triangle to point out that records of this exist in Chinese manuscripts which were written 300 years before the birth of Pascal himself. The origin of the word 'algebra' can likewise be a starting point for a greater appreciation of history, originating as it does from the first word of the book "Al-jabr w' al Mugabulu" which was written in AD 820 by an Arab mathematician. The study of alternative number systems, e.g. the Mayan and Vedic systems, can also allow for a useful contribution to this process. What is very clear from studying most mathema- tics texts is the very limited and ethnocentric view of history that is presented within them.

There is very little material, if any, in school mathemat- ical texts relating to the mathematical and scientific achieve- ments of the peoples of the Indian subcontinent. This is despite the fact that at the beginning of the European colonisation of this region in the eighteenth century, reports were made of considerable scientific and mathematical advances. Descriptions exist highlighting high levels of mathematical capability in the areas of plane geometry, trigonometry, advanced algebra and also indicating the possible existence of some form of calculus'.

That so little historical evidence of Indian new mathema- tics in the eighteenth century is available to us should come as no surprise, since it is unlikely that the promotion of the cultural traditions and achievements of subjugates was high on the list of priorities of the occupying power at that time. An example of how we might begin to arouse an awareness of such issues in our young people would be to base some of our lessons on the patterns found in Islamic decorative art. The fact that the representation of the human figure is forbidden by the Mohammedan religion has led Islamic artists to perfect this purely abstract art to an incomparable degree of elegance. Such patterns are a rich and valuable source of mathematics in themselves; however, such a strategy should not be tokenist but part of a broader range of initiatives in the mathematics classroom. An example of the sequence of stages involved in one possible construction is given in Figure 1: This basic pattern could now be repeated to cover any required area and a further dimension

Fig. 1 Islamic tiling patterns: a classroom activity.

Mathematics in School, September 1987 35

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introduced by the use of colour. Such an exercise might be used to reinforce work with tessellations and basic geo- metrical constructions. As such it would be most applicable at the upper junior or lower secondary stage.

Such patterns would link in a very creative way with work on tessellations, symmetry and plane transformation geometry, and in so doing would help to develop children's awareness of space and relations between spaces. For further ideas on this topic, I would refer interested readers to a Dover book4 which contains 190 plates exhibiting a whole range of Islamic geometrical art, together with 28 examples of actual applications such as ceilings, windows and inlaid pavements. An article in MT86 on do-it-yourself Islamic patterns' gives examples of how similar patterns can be generated from successive reflections in, for example, four axes of symmetry at right angles on a dot lattice. These can very rapidly become quite complex and the patterns can be accentuated by using colour to produce some very pleasing artistic designs.

In beginning to consider how to deal with social and economic issues relating to cultural and racial differences I found it helpful to consider each at three different levels - the global, national and local level. Figure 2 best sum- marises my approach to thinking about the relevant issues. In beginning to deal with some of these issues at a local level I have found that local council publications of statistics have been very useful sources of information'.

The graph in Figure 3 gives details of jobs in the Bradford District during the period 1961-1981.

The graph in Figure 4 gives details of fifth formers leaving school for a job during a four-year period in the Bradford district. (a) Read off the percentage of each group finding a job for

each of the given years and display your results in a table.

(b) By what factor has the proportion of black school leavers finding work dropped between 1979/80 and 1982/83?

(c) What is the corresponding figure for white school leavers?

Since the council estimates that it will need 16,000 new houses in the 12-year period up to 1996, one calculation might be to work out the shortfall, assuming that the 1983 rate of housebuilding does not fall any further. The oppor- tunities for further research into up-to-date figures are obvious.

Other 220

200

180 Other t,

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1961 1966 1971 1976 1981

Year

(Source: District Trends 1984, City of Bradford Council.) Fig. 3 Local - UK region unemployment.

Global Issues

Human National Rights Issues

Local

Issues

HealthHealth PovertyP

Povert Health Unemployment Poverty

Housing

South-

Housing North South Unemployment North- Africa South

International commodity

markets

Fig. 2 Relevant social and economic issues.

Black school leavers

White school leavers

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55-

50-

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40- o

- 35- cn

30- a)

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1979/80 1980/81 1981/82 1982/83

Year

(Source: District Trends 1984, City of Bradford Council.) Fig. 4 Unemployment - black/white school leavers.

36 Mathematics in School, September 1987

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Year Number of new housing starts (Private, council & housing association)

1978 2,450

1979 1,250

1980 900

1981 850

1982 1,000

1983 750

(Source: District Trends 1984, City of Bradford Council.)

Fig. 5 Details of new house building in the Bradford district over a six- year period.

At a global level one might consider aspects of world trade as detailed in Figure 6 which represents the way in which the cost of a single banana is divided up amongst the groups involved in its production, transportation and eventual sale to the public at a cost of 10p. One approach might be to divide the class up into groups representing each stage of the process and ask them to divide up the cost of the banana themselves. Each group could report back and negotiation take place to produce a class response before revealing the actual figures. This could them lead to a comparison of the two sets of figures and a discussion of the implications of perhaps a fairer distribution of the profits from the trade in this particular commodity. For instance, what would be the implications for the retailer? Would he/she be able to sell more bananas or would the overall cost have to increase? If the proportion for each other group was increased, with the amount for the retailer remaining constant, what would be the effect upon the overall price? What in turn would be the likely effect upon sales?

The discussion of the terms of international trade could be further developed by using information relating to the value of raw materials in comparison with that of manu- factured goods such as tractors. The relative decline in the value of such raw materials could be illustrated in a variety of ways and calculations carried out to determine the effect upon the price of a single banana if its value had been maintained relative to the 1950 figure (see Fig. 7).

The issue of apartheid and South Africa could be ap- proached in the context of a mathematics lesson through the activity shown in Figure 87.

This activity could be supported with examples involving statistics based upon social and economic indicators of life in South Africa. For instance, statistics relating to income distribution and health could be analysed in order to gain a fuller picture of life in South Africa, involving mathemat- ical activity in the process.

To picker/grower

To shipping / To wholesaler companys

lp

1.5p

.5p 10p1

To retailer To packaging To importing company

company

(Source: World Studies 8-13, 1984, Simon Fisher & David Hicks.)

Fig. 6 The world banana trade.

1950

1970

1982

Tons of bananas

(Source: World Studies 8-13, 1984, Simon Fisher and David Hicks.)

Fig. 7 Tractors and bananas.

Income Distribution/Health Figure 9 gives the distribution of income and population

in South Africa in 1977. (a) Show this information on a chart which clearly presents

the distribution of income in relation to population. (b) Figure 10 gives information on life expectancy and

infant mortality rates for the same groups. Display this information in a similar way and comment upon any observations that you make from both tables considered together.

(c) What conclusions can you draw from the evidence presented in Figures 9 and 10 about the distribution of income and standard of health for each of the groups listed. Discuss your conclusions.

The whole issue of global inequality could be explored whilst also involving meaningful mathematical activity. For instance the statistics contained in Figure 11 could be used as an example in work on graphical representation or for further background information in more extensive pieces of work.

In South Africa 13% of the territory has been set aside as homelands for 18 million blacks, while the remaining 87% is for the country's 4 million whites.

(a) What is the total population of South Africa?

(b) What percentage of the people are black?

(c) What percentage are white?

(d) How many people are in your class?

(e) What number in your class is equivalent to the percentage of blacks in South Africa? Use the percentage that you calculated in (b).

(f) What number in your class is equivalent to the number of whites? Round off your answers to (e) and (f) to the

nearest whole numbers. (g) Measure the length and width of your

classroom and hence calculate its area.

(h) Work out 13% of the area and divide this off from the rest.

Choose the required number of people in your class to represent the blacks and to occupy the 13% of the room, which has been divided off, leaving the whites in the remainder. Finally whilst remaining in your positions discuss your activity and your thoughts about it.

(Adapted from an activity devised by the Center for Teaching Intemrnational Relations, University of Denver).

Fig. 8 Apartheid- A class activity.

Percentage of income

Percentage of population

Income Distribution

White Black Coloured Indian African African

64% 7% 3% 26%

15% 8.5% 3% 73.5%

(Source: South African Institure of Race Relations, 1978.)

Fig. 9

Life expectancy in years (1969-71)

Infant mortality rate per 1,000 live births (1974)

Official Mortality Rates

White Coloured Indian Black

68.4 52.5 61.6 55

18.4 115.5 32.0 105

(Source: Official Year Book of RSA, 1980/81.)

Fig. 10

Mathematics in School, September 1987 37

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Share of rich Share of the industrial nations rest of the world

(28 incl. USSR) (113 countries)

Population

Wealth (G

Petrol consumption

Natural gas consumption

Grain consumption per annum

Protein con- sumption (% of world pro- duction per annum)

24% (Approx. 1,000 million)

78%

83%

92%

16% for humans 62% 46% for animals

80%

76% (Approx. 3,300 million)

22%

17%

8%

30% for humans 38% 8% for animals

20%

(Source: It's not fair!, 1984, British Council of Churches.)

Fig. 11 The North-South divide.

There is obviously no end to the statistics which help to demonstrate the complexity, interdependence and in- equality of our world. What is required is not so much a source of statistics and information but rather an awareness of the need to relate mathematics to the wider world the statistics and information will then follow.

However, if we are truly serious about making our mathematics teaching multicultural in its content, then much more work needs to be carried out at all levels of mathematics education in order to ensure that such an aim is achieved.

References 1. Woodrow, D. (1984) "Multicultural Issues", Times Educational Sup-

plement, 11 May. 2. Hudson, B., "Social Division or Adding up to Equality? Bias in

Mathematics Text Books and Computer Software", The World Studies Journal, Vol. 5, No. 4, WSTTC (University of York).

3. Hemmings, R. (1980) "Multi-ethnic mathematics, I & II", New Approaches in Multiracial Education, Vol. 8, Nos. 3 & 4.

4. Bourgoin, J. (1973) Arabic Geometrical Pattern and Design; Dover. 5. Mathematics Teaching, No. 86, The Association of Teachers of

Mathematics, Kings Chambers, Queen Street, Derby, DE1 3DA. 6. District Trends 1984, City of Bradford Metropolitan Council. 7. Hudson, B. "Apartheid - A Class Activity", The World Studies Journal,

Vol. 5, No. 4, WSTTC (University of York).

Postscript This article is partly the result of research carried out during 1984/85 at

the World Studies Teacher Training Centre (now Centre for Global Education) at the University of York. A package of materials was developed, trialled and evaluated for the teaching of mathematics from a global and multicultural perspective. The package consists of a database of statistics on 127 countries of the world which relate to demographic, social and economic issues and also supporting problems of a more conventional nature, focusing upon such issues at a local, national and international level. The database is designed to be used in conjunction with QUESTD on the BBC and RML microcomputer systems. The package is available from the Centre for Global Education, University of York, Heslington, York, YO1 5DD.

Short Notices

Will Mathematics Count? Edited by Derek Ball et al AUCBE, 094603950X, A2

Following the Pendley Manor Report published in Mathematics in School in March 1985 a group of experienced teachers, and others con- cerned with teaching, met for a seminar at Stanhope to consider the impact of computers on mathematical education in schools. This booklet is based on the outcome of that four day seminar.

It begins with an overview of the current situation regarding the place of the computer in mathematical education. It then looks at class- room learning styles in the teaching of math- ematics and examines the possible impact of software packages and then analyses possible and inevitable changes in the mathematics cur- riculum. The report then examines problems and possible solutions of teacher education within the new context and adds some rather general comments on equipment for the classroom.

The final recommendations highlight the need for continuing curriculum development to repro- duce materials and software, increased research into the possible impact of new technology on the mathematics curriculum, devising a mechan- ism for the review and dissemination of current information, the provision of appropriate teacher support and the provision of adequate classroom equipment.

An interesting and informative report at a general level which would have been enhanced with more specific examples. Copies are available from the Mathematical Association in Leicester price A2.

Geometry for Grades K-6 Edited by Jane Hill NCTM, 0873532376, A9.50

A set of articles taken from copies of the Arithmetic Teacher published during the past ten years. The Arithmetic Teacher is one of the journals published by the National Council of Teachers of Mathemtics in America. It is written for teachers of children in the age range 5 to 12.

Many of the articles in this book are taken from the 'Lets do It' section of the journal and feature practical activities in the study of geome- try. The activities involve children classifying shapes, creating shapes and patterns, building and investigating shapes and a whole range of interesting activities.

A rich source of classroom material offering a variety of ideas and methods of presentation. Available from Jonathon Press Colchester.

Understanding Pure Mathematics by A. J. Sadler & D. W S. Thorning Oxford University Press, A11.95

This book is designed to cover the pure math- ematics component of a single subject Advanced level Mathematics course of any examination board.

First impressions are of a clearly presented book with carefully graded questions in exercises spaced throughout each chapter, but little different from a number of other text-books currently on the market. Closer examination reveals some well thought out explanations. Particularly impressive were the chapters on curve sketching.

This book is worthy of consideration by a teacher looking for a new A-level Pure Math- ematics text.

Studies of Meaning Language and Change by David Wells Rain Publications

Published three times a year in March, July and November. The March 1987 edition contains sections on the language of problems and inves- tigations, general concepts and teaching, false simplicity and true simplicity and finally a sec- tion entitled 'Against Problem Solving'.

For those thinking seriously about the ad- vantages and problems of investigational activi- ties in school mathematics this should prove interesting reading. Available from Rain Publi- cations, 6 Carmarthen Road, Westbury on Trym, Bristol BS9 4DU price A2 for three issues.

Number Activities and Games by Roy Edwards NARE, 0906730260, A2.40

A second edition of a collection of mathematical activities designed to offer interesting methods of practising number facts, skills and concepts. Particular areas covered are counting, place value and the four number operations. The book is essentially a set of ideas for the teacher to develop with children of primary school age who are experiencing learning difficulties. Each activity is clearly explained and full details are given of materials required and how the ideas should be developed.

Many of the activities are in the form of games or simple investigative experiences and although some teachers may find many activities familiar they do offer a useful resource to supplement any published scheme.

Likely to prove good value for money to many primary school teachers. The book is available from the National Association for Remedial Education, 2 Lichfield Road, Stafford stl7 4JX.

38 Mathematics in School, September 1987

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