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Bishop's Argument for a Multiplicity of Different Forms of Mathematics

As some of the above commentary suggests, the answer to the question of whether mathematics is culture free depends critically on how "mathematics" is defined. Does the "mathematics" practised by "mathematicians" in universities around the world define, totally, the scope and essence of mathematics? Alan Bishop (1988) has argued persuasively against such a point of view. For him, mathematics is a pan-cultural phenomenon, something which exists in all cultures; and "Western Mathematics" (which he calls Mathematics with a capital M), is a particular variant of mathematics which has been developed through the ages by various societies.

For Bishop (1988), in addition to "Mathematics"(with a capital M) there are many different mathematics (with a small m). There is "Chinese mathematics, Greek mathematics, Roman mathematics, African mathematics, Islamic mathematics, Indian mathematics and Neolithic mathematics" (Bishop, 1988, p. 56). Furthermore, as Joseph (1992) has argued strongly, much of Western mathematics as we know it today has non-European roots.

Although Mathematics (capital M) is an internationalised discipline, it is nonetheless a specific line of knowledge development that has been cultivated by certain culture groups until it has reached the particular form that we know today. It is reasonably well defined, for as John Stillwell (1988), an Australian mathematician, has pointed out, probably 99 percent of mathematicians are now in agreement over what is a number, what is a function, etc, and recent claims that Fermat's Last Theorem has finally been proved has demonstrated that mathematicians the world over agree on what does and what does not constitute a legitimate mathematical proof.

There is a tension between the notion of mathematics as a pan-cultural phenomenon and Mathematics as a discipline practised by mathematicians in universities. Elementary school educators should be concerned that the mathematics curriculum links with the personal worlds of young children, but should this still be important as the level of education increases? Should mathematics become less culturally bound as the level of education increases? Can we assume that an important aim of mathematics education is to produce students who know Mathematics with a capital M, and if so, what proportion of school leavers should be in this category? Is it the main aim of school mathematics to produce students who know enough small m and capital M mathematics to be able to cope with everyday situations which demand the application of mathematical skills, including the selection and use of appropriate problem-solving strategies?

An Internationalised Mathematics Curriculum

Although some mathematics education researchers have distinguished between capital M and small m mathematics, many people believe that the discipline of mathematics is more or less culture free, and that there is no good reason why the content of mathematics, and the methods by which it is taught, should vary significantly from region to region and from nation to nation. It is also commonly assumed that the ability to acquire scientific and technological skills is dependent on having an adequate grounding in capital M mathematics (see, for example, Briggs, 1987).

A consequence of the belief that mathematics is culture free is that many educators believe that it is legitimate for some central authority to prescribe the type, the level and the extent of mathematics that students should be asked to learn. Elizabeth Oldham (1989, p. 200), after analysing mathematics curricula in the 25 countries that participated in the Second International Mathematics Study (SIMS), concluded that in most national systems of education, mathematics curricula are centrally decreed - by, for example, ministries of education - and that even the theoretical freedom to choose within a prescribed curriculum is usually constrained by the overriding importance of national examinations. Oldham concluded that there is a considerable degree of cross-national commonality and, although there are some distinct patterns of diversity, "there is indeed an international mathematics curriculum" (Oldham, 1989, p. 212).

The view that ultimately the task of defining mathematics curricula is sufficiently difficult and important that it should not be left to individual teachers, or schools, or local education authorities, would appear to have gathered momentum in recent years with the introduction of national curricula and assessment in the United Kingdom (Noss, 1989), the development and acceptance of the "Standards for School Mathematics" document in the United States (Crosswhite, Dossey, & Frye, 1989; National Council of Teachers of Mathematics Commission on Standards for School Mathematics, 1989), and the move in Australia, supported by politicians and many educators, to define principles and mathematics entitlements for compulsory and post-compulsory education levels in Australia (Baxter & Brinkworth, 1989).

Another pointer towards the existence of an implicitly defined international mathematics curriculum is the increasingly widespread use of the distance mode of teaching mathematics, both at the school and tertiary levels (Briggs, 1987, p. 26; Ellerton & Clements, 1989b). In particular, the almost universal acceptance of the idea that quality mathematics materials (like textbooks, for example) which are prepared in one country can be readily used, without major changes, in another, suggests that education authorities everywhere believe that mathematics is more or less the same the world over (Ellerton & Clements, 1989b, p. 4).

The Challenge for Mathematics Curriculum Developers

In this paper it has been our intention to provide a research basis for discussion of curriculum issues for those directly concerned with the teaching and learning of mathematics, especially in the nations of Southeast Asia. The present paper provides research perspectives which demonstrate that mathematics is not a culture-free phenomenon. One of the challenges for Southeast Asian mathematics education researchers is to design and carry out further research which will make more explicit the implications of these perspectives for the revision of mathematics curricula in their own countries. Only then will teachers, curriculum developers, textbook writers, and education administrators be in a position to make more informed decisions about the learning environments they wish to create.

Bishop (1988) has maintained in his book, Mathematical Enculturation, that mathematics curricula should always take full account of local cultural influences, so that the content and methods of teaching will be consistent with both the nature of mathematics and with research findings on how mathematics is learnt If his argument is correct, then the acceptance of the notion of an international mathematics curriculum, the existence of national curricula and testing bodies in many countries (including most Asian countries), and the moves towards more centralised curricula in the United Kingdom, the United States of America, and Australia are questionable; so too, is the increasing move towards the implementation of distance modes of teaching mathematics that make use of curriculum materials developed by institutions set in different cultures from the learners.

The pertinence of Bishop's argument is underlined by a report in the July 1988 edition of Commonwealth Education News on the activities of the Commonwealth Association of Science, Technology and Mathematics Educators (CASTME). According to this report:

CASTME facilitates the exchange of information among science, mathematics and technology (STM) educators, keeping in mind the great diversity of cultures, customs and technologies across Commonwealth countries. It is particularly concerned with the social implications of STM education. These include the relevance of STM curricula to local needs and conditions, and to the impact of technology, industry and agriculture on a local community.

Despite these sentiments, a paper prepared for the first Board meeting of the Commonwealth of Learning pointed to the fact that many developing countries are particularly short of scientists, technologists and engineers, and argued that there are distance-teaching materials already in existence that can be used to provide effective education and training in science and engineering (Commonwealth of Learning, 1988, p.2). Similar arguments can be, and in fact are being, applied in the area of mathematics education. The review provided in this paper suggests that such arguments are shortsighted and likely to result in the teaching and learning of inappropriate mathematics.

In the sense that different cultures will have different forms of "small-m" mathematics, it should be expected that mathematics learning should not be immune from the influence of culture, but rather it ought to be as culturally bound as learning in any other domain (Stigler & Baranes, 1988, p. 258). These ideas are in line with the writings of contemporary education philosophers such as Evers and Walker (1983), who argue that mathematical knowledge is but one aspect of a seamless web of knowledge, and should not be taught as if it is an "out-there," objective yet mystical a priori form of knowledge.

Yet, despite the large acceptance of these relativist notions of mathematics among the mathematics education research community, for most curriculum developers the image of mathematics is still one of being at the pinnacle of human reason. For them, mathematics curricula should be hierarchical in nature, and mathematics teachers should stress the need for students to learn basic mathematical facts and skills, and to make correctly sequenced verbal and written statements (Ellerton & Clements, 1990).

The flowering of what has come to be known as the "constructivist movement," as well as the anthropological emphasis typified by Bishop's research, have led mathematics education researchers not only to take into account pertinent philosophical, sociological, and critical theory literatures, but also to develop new research paradigms on which they are basing their own research. Thus, there is much mathematics education research in the 1980s and 1990s with a philosophical, anthropological, sociological, and epistemological flavour, and this contrasts with the psychological research emphases of the 1970s. Since 1980, qualitative methods of research have been increasingly used by mathematics education research

New paradigms Constructivism

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