NORMA

SCHOOL

MATHEMATICS

C. PRESMEG

IN CULTURE-CONFLICT

SITUATIONS Towards a Mathematics Curriculum for Mutual Understanding when Diverse Cultures Come Together in the Same Classroom’ ABSTRACT. In times of cultural change, education plays an especially important role. The writer suggests that even mathematics curricula, which have traditionally been considered culture-free, have a role to play in fostering mutual understanding amongst members of different cultures, after a period of cultural upheaval. Anthropological and educational sources are used to suggest points of relevance when a mathematical curriculum is designed for multi-cultural classrooms.

As indicated by the word “towards” in the subtitle, the writer does not regard this paper as providing ultimate solutions to problems which are extremely complex. Writing, as she is, about South African cultures and sub-cultures all of which are in a state of intense ferment at this time, she is aware of complexities and contradictions which are implicit in some of the issues (and some of which are not confined to South African society). The analysis of elements which may contribute to a mathematics curriculum for mutual understanding is offered as a first approximation to an answer to the question of how pupils from diverse cultures can best learn mathematics together after a period of cultural change.

A PERIOD

OF

INTENSE

CULTURAL

FERMENT

IN

SOUTH

AFRICA

University of Durban-Westville (U D-W) is an institution which, in line with the trend at all South African universities, is increasingly opening its doors to students of all races. At present there are approximately 6000 students, the majority (86%) of whom still are Indians. The remaining 14% are predominantly black students. 40% of the academic staff of 425 are Indians, almost all of the remainder being White. Of the total staff complement of more than 1000, about 70% are Indians. “Dr Presmeg, what must I do?” Yusuf, a fourth-year mathematics education student, stood in the doorway of the writer’s office at U D-W with a piece of paper in each hand. One piece of paper was a police notice prohibiting unlawful gatherings on the campus of U D-W in terms of the state of emergency regulations; the other was a notice from the Students’ Representative Council calling all students Educational Studies in Mathematics 19 (1988) 0 1988 by Kluwer Academic Publishers.

163-177.

164

NORMA

C. PRESMEG

to a mass protest meeting. The students were newly returned to campus after previous boycotts and disruptions had resulted in the early closing of the university for the July holiday. (The academic year ends in December.) Some student leaders were still in detention. Yusuf did not really have a choice. He attended the meeting, which in spite of a heavy security force presence did not end in violence this time. As another student informed the writer, at a previous mass meeting to decide whether the students would boycott lectures, one student spoke against the boycott. A student leader said, “Don’t listen to him. We’ll deal with him. Let’s get on with it!” No student spoke against the boycott after that. Voting was by a show of hands. Students who attended lectures that afternoon (some of whom were part-timers who did not know of the decision) were forcibly removed from some lecture rooms by intimidators. Yusuf’s dilemma illustrates the conflict situation in which not only students but black pupils in many of the townships find themselves. Burning and boycott of schools which are seen as symbols of white domination are actions which are encapsulated in the slogan, “Liberation first, education later!” But the vocabulary is changing as fast as it emerges (Gangat, 1986), and this slogan, popular in 1985, has already been superceded by “People’s education for people’s power!” The term “people’s education” also seems to have superceded the term “alternative education”, which has been used in the last few years to describe forms of oppositional or counter-hegemonic educational programmes or approaches (Walters, 1986). It is beyond the scope of this paper to document all the social, political, economic and ideological forces at work in South African society today, but the foregoing serves to illustrate three points, as follows. (1) The social, cultural and ideological changes which are taking place amount to an ongoing revolution. (2) Education is seen as “a prime catalyst in and towards change” (Gandat, 1986). (3) The emergent cultures as well as those they are replacing, are fraught with contradictions. The events on the campus of U D-W illustrate a phenomenon which is widespread in counter-hegemonic strategies in South Africa at present. In the protest against lack of true democracy, methods are employed which in fact perpetuate this lack of democracy. It appears that cultural change is not a rational, logical process. Indeed, some of its elements may not be conscious at all. Apple (1982, p. 24) wrote of the “contradictions, conflicts, mediations and especially resistances” which he found in American school sub-cultures. Negotiation and conflict are aspects of cultural hegamony. Ironically, in the

CULTURE-CONFLICT

165

SITUATIONS

rejection of mental labour by working class boys, Apple found that “The seeds of reproduction lie in this very rejection” (ibid., p. 99). It is a danger, then, in all cultural change, that unwanted elements of a former hegemonic ideology may be reproduced unwittingly. One master may simply be exchanged for another. However, the formation of ideologies is not a simple act of imposition. “It is produced by concrete actors and embedded in lived experiences that may resist, alter or mediate these social messages” (ibid., p. 159). It may be asked what role mathematics education can, or should, play in the reproduction of ideologies. The political, social and economic environment in South Africa today is highly sensitive (Natal Teachers’ Society, 1986). It seems to the writer that mathematics education in South Africa in the times ahead can play a positive role towards healing rifts and bitternesses and in promoting understanding and tolerance of cultural differences. Writing in and about Brazil during a time of social transition, Freire wrote: The time of transition involves a rapid movement in search of new themes and tasks. In such a phase, man needs more than ever to be integrated with his reality. If he lacks the capacity to perceive the ‘mystery’ of the changes, he will be a mere pawn at their mercy (Walters, 1986, p. 2).

The need to be “integrated with his reality”, particularly during cultural dislocations or periods of intense social change, is discussed in a general anthropological context in the next section, while ways of achieving this integration in the context of mathematics education are suggested in the final section. With Bishop (1985e) and Berry (1985), the writer is convinced that a mathematics curriculum which is experienced as real by a pupil can be developed only by adults who belong to the same cultural group as the pupil. This aspect is complicated by the multiplicity of cultures in South Africa, but the writer believes that this is a goal which is capable of realisation. However, the necessary understanding and tolerance cannot be learned in separate educational systems, or indeed in separate classrooms. To gain this understanding and tolerance, children from all cultural groups will need to come together in the same classrooms in the future. Under these conditions, a mathematics curriculum designed by a group of people representative of all cultures involved would have a positive role to play in promoting understanding and tolerance. SOME

PROBLEMS

OF

ACCULTURATION:

LIVING

IN

TWO

WORLDS

In this section are discussed some relevant aspects of the dislocation which may occur when a “Western” school culture does not resonate with the home culture of pupils. This problem is well documented (Spindler, 1974),

166

NORMA

C. PRESMEG

and has several implications for a mathematics curriculum for mutual understanding, which are listed at the end of the section. As Singleton (1974, p. 28) pointed out, culture encompasses “patterns of meaning, reality, values, actions and decision-making that are shared by and within social collectivities”. All these patterns are relevant in the learning of mathematics (Bishop, 1985a,c,d). Cultural transmission includes both the transmission of tradition from one generation to the next and the transmission of new knowledge and cultural patterns from anyone who “knows” to anyone who does not. This distinction underlies that between enculturation, “the process of generational continuity” and acculturation, “the process of individual and group change, caused by contact with various cultural systems” (Singleton, op. cit., p. 28). It is the dynamic aspects of acculturation which are relevant in multi-cultural classrooms. In the English public school tradition, Hilton College in Natal is a private, boys’ residential school which usually provides finalists in the national Mathematics Olympiads. Hilton admits boys of all races. One of the mathematics teachers at Hilton told the writer that black pupils leave by train at the start of the school holidays wearing their school uniforms, but change their attire before arriving at their destinations. This action is symbolic of the “living in two worlds” which may be experienced by such pupils as cultural dislocation-but not necessarily. In the following case studies of school acculturation, elements are identified which shed light on the question of why, and under what conditions, dislocation is not inevitable. The establishment of Western schools, especially boarding schools, and curricula in non-Western societies is likely to constitute an extreme type of cultural discontinuity and may do much to force ‘either-or’ choices on their learners (DuBois in Sindell, 1974, p. 333).

This extreme type of cultural discontinuity was experienced by Mistassini Cree children who left their homes at age five or six to attend the residential school at La Tuque in Canada. Sindell (1974) described many aspects of the cultural dislocation of these children. In the field of interpersonal relations, they learned that dependent behaviour such as crying was effective in gaining an adult’s attention. At home, self-reliance and independence were valued and therefore crying was ignored from an early age and children learned not to cry. In school the children also learned that the smallest scratch would elicit concern, thus contradicting their early training in silently enduring pain. In the case of the Mistassini Cree, preschool children learned behavioural patterns and values which were highly functional for participating as adults in the traditional hunting-trapping life of their parents.

CULTURE-CONFLICT

SITUATIONS

167

Because they must go to school, their development into trappers or wives of trappers is arrested. Prolonged residential school experience makes it difficult if not impossible for children to participate effectively in the hunting-trapping life of their parents. Not only do they fail to learn the necessary technical skills, but they acquire new needs and aspirations which cannot be satisfied on the trapline. Yet most Mistassini parents want their children to return to the bush. It remains to be seen how the students will resolve their dilemma (ibid., pp. 340-I).

The extreme cultural dislocation and “either-or” choices facing Mistassini Cree children have also been documented for children in roughly similar circumstances in Papua New Guinea (Lancy, 1983), and for Hopi children during certain periods of their history (Eggan, 1974). However, discontinuity is experienced in a less extreme form, if at all, if the schooling is seen as relevant in the pupils’ future without either-or decisions having to be made. This point is illustrated in the self-perceptions of Sisala pupils in Northern Ghana (Grindal, 1974). The distinction between the “traditional” and “modern” sectors of African life need not be perceived by the actors as a dichotomy of two worlds if they can simultaneously embody the continuity and values of the traditional society and the changes brought by colonization and modernization. This conception is illustrated in the apparent ease with which African tribal leaders such as Chief Mangosuthu Buthelezi of the Zulus switch back and forth from traditional roles amongst their people, to various roles at national and international levels as representatives of their people. One significant aspect of the problem of “two worlds” is implied in the following exchange between two new young teachers in charge of a village school among the Ngoni of Malawi, and a senior chief: The teachers bent one knee as they gave him the customary greeting, waiting in silence until he spoke. ‘How is your school? ‘The classes are full and the children are learning well, Inkosi.’ ‘How do they behave? ‘Like Ingoni children, Inkosi.’ ‘What do they learn? ‘They learn reading, writing, arithmetic, scripture, geography and drill, Inkosi.’ ‘Is that education? ‘It is education, Inkosi.’ ‘No! No! No! Education is very broad, very deep. It is not only in books, it is learning how to live. I am an old man now. When I was a boy I went with the Ngoni army against the Bemba. Then the mission came and I went to school. I became a teacher. Then I was chief. Then the government came. I have seen our country change, and now there are many schools and many young men go away to work to lind money. I tell you that Ngoni children must learn how to live and how to build up our land, not only to work and earn money. Do you hear? ‘Yebo, Inkosi’ (Yes, 0 Chief) (Spindler, 1974, p. 308).

In this conversation the chief also pointed to the general shape of the solution to the problem of “two worlds”, in his reference to education as learning how to live.

168

NORMA

C. PRESMEG

Margaret Mead made an important point when she wrote that children need the stability of the cultural heritage, especially when their society is marked by rapid change (Nash, 1974). At such a time, adults may also learn from their children (and not only children from adults) in what Mead called prefigurative enculturation. These points are illustrated in one more case study, set in Schonhausen, an urbanising German village in the Remstal. Incidentally, in its ethos, the Grundschule in Schbnhausen, in the Federal Republic of Germany, bears a striking resemblance to Izotsha Primary School in Natal as it was a few decades ago, even to the nature excursions which took place (“Wanderungen”), disciplinary use of the “Ohrfeige” (ct.@, and use of the German language (now an alternative medium of instruction only in the first two years of schooling at Izotsha). The Izotsha school was established more than a century ago by German settlers in Natal, the descendants of whom are predominantly sugar cane farmers in the area. In the years between 1945 and 1968, Schdnhausen almost doubled its population (from 1300 to 2500) as migrants arrived from what was the east zone or from the outlying prewar German minorities, or from other parts of Western Germany. From being a region almost entirely devoted to the cultivation of wine grapes and subsistence farming, the whole area moved rapidly towards urbanisation and industrialisation. The importance of the case study of the four-year Grundschule in Schonhausen for this paper lies in the fact that although the divergencies in backgrounds between natives and newcomers set the scene for potentially explosive confrontations, these confrontations did not occur. Spindler (1974, p. 233) wrote as follows: Particularly intriguing is the fact that on the whole this great influx of manic) population was assimilated without any apparent disturbance. crime, suicide and juvenile delinquency suggests that there has been no the social and psychological ills that often accompany rapid urbanisation. area around it give every appearance of social and economic health.

diverse (though GerThe low incidence of substantial increase in Schiinhausen and the

According to Spindler’s ethnography, Heimatkunde (learning about the homeland) and Naturkunde (learning about the land and nature) were important components of the Grundschule curriculum. In an atmosphere of freedom and exploration, during a six-and-a-half-hour Wanderung the pupils covered a distance of nearly eighteen kilometres and discussed elements of the forests, meadows and waterways as well as the new apartment buildings and industries plainly identifiable in the valley. On another occasion, the children were taught about the history of the four great bells in the tower of the local Protestant church, after which they walked across to the church to experience the bells first-hand. The Wanderung and the bells illustrate the role that knowledge of local culture and

CULTURE-CONFLICT

169

SITUATIONS

history can play in fostering stability and an understanding of change. As the Schonhausen children grew older they increasingly chose to follow the “easier” life of modern apartments and hxed salaries; but one gains the impression that the changes did not usually involve rejection, but ongoing appreciation of the old culture. From the foregoing, several relevant points emerge. For a mathematics curriculum for mutual understanding when diverse cultures come together, the following points appear to be of importance: (1) Children need the stability of their cultural heritage, especially during periods of rapid social change. (2) The mathematics curriculum should incorporate elements of the cultural histories of all the people of the region. (3) The mathematics curriculum should be experienced as “real” by all children, and should resonate, as far as possible, with diverse home cultures. (4) The mathematics curriculum should be seen by pupils as relevant to their future lives. CHANGING

VIEWS

OF

THE

MATHEMATICS

CURRICULUM

As Howson, Keitel and Kilpatrick (1982) point out, curriculum means more than the syllabus: it must encompass aims, content, methods and assessment procedures - and curriculum reform is never completed. Amongst the pressures that serve to initiate curriculum development they list societal and political pressures, mathematical and educational pressures. In common with most Western countries, pressures in South Africa for curricular reform in mathematics resulted in the introduction of “modern mathematics” in the 1960s and more recent curriculum reforms which are still in the process of being implemented. In the future, however, the curriculum changes will be far more fundamental and far-reaching as first and third worlds attempt to come to terms with mathematics in the same classrooms. The magnitude of the issues is realised by some mathematics educators in South Africa at present. At a workshop open to all to discuss the subject, “People’s Education and the Role of Mathematics”, it was stated that “underlying all [the] debates are questions concerning the kind of society envisaged in the medium and long term future, and the role of mathematics education in achieving these goals. These are enormous issues, and the most that a workshop of this nature can achieve is to raise some of the main questions” (Taylor et al., 1986). The workshop was seen as an early step in a long process. The political aspects of the “democratisation of knowledge”

170

NORMA

C. PRESMEG

are focal issues at present, but the writer is looking beyond these issues (important as these are) to what she sees as an essential stage, inevitable if growth and stability are goals, in which healing of prejudices, rifts and bitternesses must take place. It has been pointed out (Howson et al., 1982) that it is possible for teachers to teach a new curriculum as they taught the old, thereby undermining the intentions of the curriculum developers. The actions of mathematics teachers in the classroom are predicated on their beliefs about the nature of mathematics and the teaching and learning of mathematics (Cooney, 1984; Cooney et al., 1985). In-service and pre-service courses for mathematics teachers would therefore be important components in the implementation of any radically new mathematics curriculum. Furthermore, in all societies prejudiced beliefs about different cultures may be deep-seated or unconscious (Reynolds and Reynolds, 1974). However, studies of anti-Semitic prejudice following World War II provided research evidence that if people can be induced to change their actions, then changes in their belief-systems in line with these actions are likely to follow (Selltiz et al., 1963). In order to reify the values of tolerance and mutual understanding, these values would necessarily be conscious goals on the part of people of all cultures in the classroom. It is in the philosophical assumptions underlying perceptions by mathematics educators of the nature of mathematics, that changes in world thinking have been taking place. It might be asked to what extent it is meaningful to speak of cultural mathematics. Several writers have argued convincingly that mathematics is not culture-free, but culture-bound (Bishop, 1985b,e; Breen, 1986; D’Ambrosio, 1984; Fasheh, 1982; Gerdes, 1985). The traditional view is that the propositions of mathematics are absolute and transcend questions of culture. After all, six plus six must equal twelve in any culture! But even this statement may be called into question. In Papua New Guinea there are more than 700 different languages (not dialects) and the counting systems may be classified into four main types only one of which includes our familiar base ten system (Lancy, 1977). If one’s counting system goes 1, 2, 3,4, many, or numerals are attached to various parts of the body, or the system changes according to what one is counting, then 6 + 6 = 12 as an absolute truth, becomes meaningless. However, the view that mathematics is culture-bound is probably too limited if stated baldly, like this, without qualification. Mathematicians from diverse cultural backgrounds have no difhculty in understanding each other. Indeed, this point is illustrated strikingly in the account of G. H. Hardy’s discovery of the mathematical genius of Ramanujan. Ramanujan was a poor

CULTURE-CONFLICT

171

SITUATIONS

clerk from Madras who had not been able to enter Madras University because “he could not matriculate in English” (Hardy, 1979, p. 35). He was brought to Cambridge by Hardy, who recognised from his manuscripts that Ramanujan was, in terms of natural mathematical genius, in the class of Gauss and Euler. In his 1933 Spencer lecture at Oxford, Einstein said, Experience be deduced mathematical p. 274).

may suggest the appropriate mathematical concepts, but they most certainly cannot from it. Experience remains, of course, the sole criterion of the physical utility of a construction. But the creative principle resides in mathematics (Einstein, 1973,

There is a universal element in the principle to which Einstein refers here. This element is present, too, in the curtailment, generalisation and logical economy evident in the mathematical thinking of Krutetskii’s (1976) “capable” pupils. The universal or absolute aspect of mathematics must be available to those pupils of all cultures who can master it. In plural South African society it must be possible for future Ramanujans to learn the mathematics which will enable them to make their contributions. Perhaps the schools of the future will teach such mathematics in special groups to those pupils who desire it or who require it for their chosen vocations. The rigour of mathematical concepts needed in what Bishop (1985e) has called mathematico-technological (MT) culture could be developed gradually from the concepts of a cultural mathematics curriculum. For all pupils, cultural mathematics as a basic groundwork could provide a core curriculum which is meaningful in their reality. Some of the problems and possibilities of such a core curriculum are addressed in the following section. ELEMENTS

OF

A CURRICULUM

FOR

MUTUAL

UNDERSTANDING

By virtue of the fact that they would be sharing a common school experience (first section), not all aspects of the realities of pupils from diverse cultural backgrounds would be different. Lawton (1975, p. 5) wrote appositely in this regard: One view is that a common curriculum must be derived from a common culture. But this in turn raises other difficult issues. What is meant by a common culture? Is it meaningful to talk of a common culture in a pluralistic society?

Lawton came to the conclusion that there were no convincing against the existence of some elements of a common culture society - sufficient at least as a basis for a common curriculum.

arguments in English

172

NORMA

C. PRESMEG

In the much more diverse South African context, it might be useful to ask, further, to what extent it is possible for cultures to combine. An anecdote is appropriate in this regard. The writer’s children, who live on a small farm in Natal, frequently play with the black farmworker’s children who live on the property. One day when the children were playing together, they found in the veld (grassland) a stray domestic cat, which had obviously been living wild. The black children indicated that this find was “ukudla, nyama” (food, meat); the white children indicated, no, and demonstrated that the cat was to be petted and stroked. The cat, tough from the ways of the wild, was named “Tiger” and became a family pet. It is suggested that individual elements of cultures cannot combine: one either pets the cat or one eats it. But the writer also suggests that social interchange permits and promotes understanding and tolerance of ways that are different. If these differences can be aired naturally in mathematics classrooms, in a non-evaluative atmosphere, mutual understanding may be facilitated especially among children, whose patterns of thought are more pliable than those of their parents (Spindler, 1974). It is inevitable that new cultural forms will evolve from the old in this “melting pot” experience, and in this sense cultures will grow closer to one another. It is possible to observe this growing together of cultures in language changes in South African townships. Nundkumar (1985) described graphically the “changing face of English” in the Indian township of Phoenix, near Durban. He quoted a poem written in “Township lingo”, a verse of which is as follows: This is poetry, brazo, and if you think I’m ‘g’-ing you, read it, Twaai it hacker, I come from the township mei brew, i An’ I can’t skryf or choon anything except in the township st!yle. Don’t worry about the lingo bhai, the poem is more important.

English, Afrikaans,

Hindi and American terms appear in the extract.

Vital influences on ‘Township lingo’ are television and films, especially those depicting Black Americans. At the moment there is a craze for imitating Black Americans - their style of dress, the way they walk and talk. The ‘Township lingo’ is not restricted to the Indian Township. There is a constant diffusion of words and phrases from the other communities, especially the Black communities. When Blacks talk of a friend they can him ‘bra’ (short for brother). In the Indian township people talk of a ‘bra’ but use the term ‘brazo’ more frequently (ibid., p. 69).

It is likely that the future lingua franca of South Africa will be English, because this is the only South African language which has international currency. The issue of language is important in the learning of mathematics because it is possible that the structure of the learner’s mother tongue has a strong influence on mathematical cognitive processes such as classification and recognition of equivalences and relationships (Gay and Cole, 1967;

CULTURE-CONFLICT

SITUATIONS

173

Glick, 1974; Bishop, 1983, Lancy, 1983; Berry, 1985). In his illuminating article, Berry ( 1985) classified language-related mathematical learning problems into two types. Type A problems occur typically when the language of instruction is not the student’s mother tongue. “Remedial strategies are linguistic, not mathematical-the treatment is to improve the learner’s fluency in the instructional language” (ibid., p. 20). Type B problems result from the “distance” between the cognitive structures natural to the learner and implicit in his mother tongue and culture, and those assumed by the teacher or curriculum designer. Type B problems may occur among unilinguals being taught in their mother tongue, as Berry illustrated in a Botswana context in which the mathematics textbooks were English books simply translated into Setswana. The remedy for type B problems is to modify the curriculum and methodology to build on the learner’s natural modes of cognition. Two points emerge. Once again the importance of a multicultured development team is stressed, so that each member of this team is au fait with the cognitive structures of at least one of the cultural groups involved, and each group is represented on the team. The second point is that analysis of the cognitive structures implicit in the home languages of the pupils for whom the curriculum is intended might provide a useful groundwork for the development of this curriculum. With regard to this second point, the work of Pinxten et al. in developing a Navajo geometry, is excitingly relevant (Pinxten, 1984; Pinxten et al., 1983). The fundamental principle used in the development of a Navajo geometry for the primary school was that the mathematical ideas must be based upon, and develop from, Navajo pupils’ frame of reference. The model must derive from the Navajo child’s own world. Analysis of the Navajo language revealed a preponderance of verbs and verb forms: their thinking was characteristically dynamic. Objects in their culture were typically viewed in terms of actions which could be performed. It was found that intuitive, action-based notions formed a basis for many standard geometrical concepts, e.g. parallelism may be understood in terms of two people running side by side, whose paths do not diverge or approach each other. However, no attempt was made to make the curriculum conform to preconceived ideas of Euclidean geometry. Rather, major elements in the Navajo child’s world were studied in the classroom from a mathematical, but action-based, point of view, in a way which represented the authentic thinking of the Navajo child. Thus projects which were modelled and studied included the rodeo, the hooghan (traditional Navajo house), herding sheep, weaving and the school compound. Evolving concepts of continuity,

174

NORMA

C.

PRESMEG

circularity, size, projections of a three-dimensional model in two-dimensional representations, and many others, were developed naturally from the models. The emphasis was on action by the children in the making and studying of the models. Although it would of necessity differ from the development of mathematics for Navajo children (because the cultures are different), the Zulu input, for instance, in the development of a common core curriculum for the primary school in Natal, might adopt the same principles as those followed in the Navajo project. Thorough study of the language to identify characteristic elements of thinking, constructs and world views, would need to be followed by identification of mathematical concepts which are already embedded in Zulu culture. Projects based on the world of Zuln children (which would be done by all children in the common core curriculum), could then be used to develop and broaden these intuitive mathematical concepts. For instance, the traditional Zulu village is based on a pattern of concentric circles moving out from the circular cattle kraal in the centre, through the circle of huts arranged in a particular way, to the enclosing outer fence. Krige ( 1965, p. 39) wrote that “The village everywhere is built on the same plan with few variations, and even these are very slight, never disturbing the customary arrangements of the huts, cattle-kraal, etc.” The arrangement of the huts is based on the status of different wives, the indlunkulu or hut occupied by the chief wife being situated at the top end of the kraal exactly opposite the main entrance. Although a decreasing number of Zulu children actually live in traditional villages, as discussed earlier it is important that all children be made aware of their cultural heritage, especially when social change is rapid. Furthermore it is likely that elements of cultural life are reflected in their language and therefore in their cognitive structures. Children from non-Zulu cultures would grow in understanding, through exposure to elements of Zulu tribal life in their learning of mathematics. Each culture would have its turn as activities or elements from that culture came to the fore in aspects of the mathematics curriculum. Interestingly, Bishop (1985a) suggested that a focus on activities rather than on lessons entailing the learning of a fixed piece of mathematical knowledge, would promote the negotiation of mathematical meaning in the classroom. Excellent work has been done by Gerdes (1985 and 1986) in analysing mathematical elements in traditional Mozambiquan life and culture, e.g. in weaving, fishing and building. Some of his writings are openly propagandist, however (only the hegemonic ideology being represented), and in a mathematical curriculum designed for mutual understanding of all cultures involved in a society, a more balanced representation would be essential.

CULTURE-CONFLICT

175

SITUATIONS

Inevitably, the success of a curriculum such as the one envisaged in this paper (and indeed whether its development could be realised at all), must depend at least partly on the kind of society which emerges in South Africa. Professor A. J. Thembela, Vice-Rector of the University of Zululand, expressed the view that political change must precede the finding of solutions to some of the problems experienced in black education in South Africa today (Thembela, 1986). In its consideration of some aspects of a common-core mathematics curriculum for diverse cultures, this paper has done little more than scratch the surface of a problem which may require a great deal of time, and which will certainly require the dedication of a large number of South African mathematics educators. Even some mathematical features which are common to all cultures, such as the need to measure, may admit of cultural interpretations and outcomes. This point was illustrated by Bishop (1985d, p. 2) in the following conversation which he had with a university student in Papua New Guinea. I asked him how he would find the area of a rectangular piece of paper. He replied: ‘Multiply the length by the width.’ ‘You have gardens in your village. How do your people judge the area of their gardens? ‘By adding the length and the width.’ ‘Is that difficult to understand? ‘No, at home I add, at school I multiply.’ ‘But they both refer to area.’ ‘Yes, but one is about the area of a piece of paper and the other is about a garden.’ So I drew two (rectangular) gardens on the paper, one bigger than the other. ‘If these two were gardens, which would you rather have? ‘It depends on many things, I cannot say. The soil, the shade. . .’ I was then about to ask the next question, ‘Yes, but if they had the same soil, shade. .‘, when I realised how silly that would sound in that context.”

What we are not aiming for in the schools of the future is the cognitive compartmentalisation epitomised in the comment, “At home I add, at school I multiply”, even if a pupil is capable of embodying both worlds in an integrated personality structure (as discussed earlier). Cultures are not static, and if the various South African cultures are to move towards each other in the future, a sound understanding of each other’s cultures will be a precondition. This understanding might be promoted, inter alia, by a common core mathematics curriculum, along the lines suggested in this paper.

NOTE ’ The substance

Dynamics.

of this paper

is the same as that

in an article

written

for the journal,

Cultural

176

NORMA

C. PRESMEG

REFERENCES Apple, M. W.: 1982, Education and Power, Routledge & Kegan Paul, Boston. Berry, J. W.: 1985, ‘Learning mathematics in a second language: Some cross-cultural issues’. For the Learning of Mathematics 5(2), 18-23. Bishop, A. J.: 1983, ‘Space and geometry’, in Lesh, R. and Landau, M. (eds.), Acquisition of Mathematics Concepts and Processes, Academic Press, New York. Bishop, A. J.: 1985a, ‘The social construction of meaning-a significant development for mathematics education?, For the Learning of Mathematics 5(l), 24-28. Bishop, A. J.: 1985b, The Social Dimension of Research into Mathematics Education, Plenary paper presented at the research pre-session, N.C.T.M. Annual Conference, San Antonio, 1617 April, 1985. Bishop, A. J.: 1985c, The Social Dynamics of the Mathematics Classroom, Talk to Austrian teacher educators in Klagenfurt, Austria, 22 May, 1985. Bishop, A. J.: 1985d, The Social Psychology of Mathematics Education, Plenary paper presented at the Ninth P.M.E. conference, Noordwijkerhout, Holland, July, 1985. Bishop, A. J.: 1985e, A Cultural Perspective on Mathematics Education, Paper presented at a conference of mathematics educators, Alice Springs, Australia, September, 1985. Breen, C. J.: 1986, Alternative Mathematics Programmes, Proceedings of the Eighth National Congress of Mathematical Association of South Africa, Stellenbosch, July, 1986. Cooney, T. J.: 1984, Investigating Mathematics Teachers’ Beliefs: The Pursuit of Perceptions, Paper prepared. for ‘short communications’ at the Fifth International Congress on Mathematical Education, Adelaide, August, 1984. Cooney, T., F. Goffree, M. Stephens and M. Nickson: 1985, ‘The Professional Life of Teachers’, For the Learning

of Mathematics

5(2),

24-30.

D’Ambrosio, U.: 1984, Socio-Cultural Bases for Mathematical Education, Plenary paper presented at ICME V, Adelaide, August, 1984. Eggan, D.: 1974, ‘Instruction and affect in Hopi cultural continuity’, in Spindler, G. D. (ed.), Education and Cultural Process, Holt, Rinehart and Winston, New York. Einstein, A.: 1973, Ideas and Opinions, Souvenir Press, London. Fasheh, M.: 1982, ‘Mathematics, culture and authority’, For the Learning of Mathematics 3(2), 2-8. Gangat, F.: 1986, Education: A Prime Catalyst in and towards Change, Open lecture, Faculty of Education, University of Durban-Westville, Oct., 1986. Gay, J. and M. Cole: 1967, The New Mathematics and an Old Culture: A Study of Learning among the Kpelle of Liberia, Holt, Rinehart and Winston, London. Gerdes, P.: 1985, ‘Conditions and strategies for emancipatory mathematics education in underdeveloped countries’, For the Learning of Mathematics 5(l), 15-20. Gerdes, P.: 1986, ‘How to recognize hidden geometrical thinking: A contribution to the development of anthropological mathematics’, For the Learning of Mathematics 6(2), l&12. Glick, J.: 1974, ‘Culture and cognition: Some theoretical and methodological concerns’, in Spindler, G. D. (ed.), Education and Cultural Process, Holt, Rinehart and Winston, New York. Grindal, B. T.: 1974, ‘Students’ self-perceptions among the Sisala of northern Ghana: A study in continuity and change’, in Spindler, G. D. (ed.), Education and Cultural Process, Holt, Rinehart and Winston, New York. Hardy, G. H.: 1979, A Mathematician’s Apology, Cambridge University Press, Cambridge. Howson, G., C. Keitel and J. Kilpatrick: 1982, Curriculum Development in Mathematics, Cambridge University Press, Cambridge. Krige, E. J.: 1965, The Social System of the Zulus, Shuter & Shooter, Pietermaritzburg. Krutetskii, V. A.: 1976, The Psychology of Mathematical Abilities in Schoolchildren, University of Chicago Press, Chicago.

CULTURE-CONFLICT

177

SITUATIONS

Lancy, D. F.: 1911, The Indigenous Mathematics Project: A Progress Report, Address to the Linguistic Society of Papua New Guinea, Lae, September, 1977. Lancy, D. F.: 1983, Cross-Cultural Studies in Cognition and Mathematics, Academic Press, New York. Lawton, D.: 1975, Class, Culture and the Curriculum, Routledge and Kegan Paul, London. Nash, R.: 1974, ‘The convergence of anthropology and education’, in Spindler, G. D. (ed.), Education and Cultural Process, Holt, Rinehart and Winston, New York. Natal Teachers’ Society (NTS): 1986, NTS Strategyfor Maintaining Relevance and Impact in the Changing South African Context, Document presented at the 71st Annual Conference of the NTS (theme “Education for Liberation”), June, 1986. Nickson, M.: 1984, Perceptions of Mathematics and the Social Context of the Mathematics Classroom, Paper presented at the Fifth International Congress on Mathematical Education, Adelaide, August, 1984. Nundkumar, R.: 1985, ‘The changing face of English in the township of Phoenix’, University of Durban-

Westville:

Faculty

of Education

Journal

2(4),

6s-72.

Pinxten, R., I. Van Dooren and F. Harvey, 1983, Anthropology Natural

Philosophy

and

Semantics

of Navajo

Indians,

of Space:

Explorations

Philadelphia. Pinxten, R.: 1984, ‘Navajo spatial representation and Navajo geometry’, Communication mation

V&23),

into the

University of Pennsylvania Press, Infor-

266-289.

Reynolds, D. A. T. and N. T. Reynolds: 1974, ‘The roots of prejudice: California Indian history in school textbooks’, in Spindler, G. D. (ed.), Education and Cultural Process, Holt, Rinehart and Winston, New York. Selltiz, C., M. Jahoda, M. Deutsch and S. W. Cook: 1963, Research Methods in Social Relations, Holt, Rinehart and Winston, New York. Sindell, P. S.: 1974, ‘Some discontinuities in the enculturation of Mistassini Cree children’, in Spindler, G. D. (ed.), Education and Cultural Process, Holt, Rinehart and Winston, New York. Singleton, J.: 1974, ‘Implications of education as cultural transmission’, in Spindler, G. D. (ed.), Education and Cultural Process, Holt, Rinehart and Winston, New York. Spindler, G. D.: 1974, ‘Schooling in Schonhausen: A study in cultural transmission and instrumental adaptation in an urbanizing German village’, in Spindler, G. D. (ed.), Education and Cultural Process, Holt, Rinehart and Winston, New York. Spindler, G. D. (ed.): 1974, Education and Cultural Process, Holt, Rinehart and Winston, New York. Taylor, N., J. Adler, F. Mazibuko and L. Magadla: 1986, People’s Education and the Role of Mathematics, Workshop at a conference held by the Department of Education, University of the Witwatersrand, Kenton-on-the-Jukskei, 31 Oct.-3 Nov., 1986. Thembela, A. J.: 1986, Black Education, Paper presented at the 71st Annual Conference of the Natal Teachers’ Society, Durban, June, 1986. Walters, S.: 1986, PeopIe’s Education: A Conceptual Frameworkfor Analysis, Paper presented at a conference at the Centre for Adult and Continuing Education, University of the Western Cape, July, 1986.

Faculty of Education, University of Durban- Westville, Private Bag X.54001, Durban, 4000 Rep. of South Africa.