Ethnomathematics is a programme which looks into the generation, transmission, institutionalisation and diffusion of knowledge with emphasis on the socio-cultural environment. By drawing on the cultural experiences and practices of individuals and of communities, ethnomathematics allows for an easier flow of scientific ideas with children, reducing the effects of cultural blocks. [D’Ambrosio, 1990, p. 369]
The above description of ethnomathematics directly contrasts with the current views held by many of the modern and traditional philosophers and mathematicians, particularly those who strongly advocate the absolutist view of mathematical knowledge noted in chapter 3. However, despite these strongly held views, the notion of ethnomathematics, like that of modern technology (i.e. calculators and computers), is inevitably here to stay, particularly within the community of mathematics educators. In particular today, the need to produce great mathematicians at the expense of the majority of the citizens is strongly challenged by the need to produce mathematically literate citizens who are able to manage their affairs more successfully in a so-called ‘modern technological’ society. This subsequently leads to another concern which is of even greater importance to this particular literature survey. This relates to the question of how best to run the business of education, in particular, the teaching-learning process within mathematics as a means of enabling young people to not only become mathematically literate citizens, but also become successful everyday problem solvers. Thus in educational terms, this notion is well-expressed in the comments made by Bishop (1991b, p. 3) that:
Educating people mathematically consists of much more than just teaching them some mathematics. It is much more difficult to do, and the problems and issues are much more challenging. It requires a fundamental awareness of the values which underlie mathematics and a recognition of the complexity of educating children about those values. It is not enough merely to teach them mathematics, we need also to educate them about mathematics, to educate them through mathematics, and to educate them with mathematics.
Seen in the above context, ethnomathematics is therefore a notion generated as a result of what others might call a ‘self-defeating prophecy’ of the current practices of mathematics education. This simply means that not only have these practices been unsuccessful in fulfilling the educational needs of the majority of the learners, but in doing so, they have also not fulfilled their societal obligations in producing a mathematically literate society.
It is in the light of the above existing problems concerning the teaching-learning process that this particular chapter explores the notion of ethnomathematics in anticipation of incorporating the type of educational environment it advocates into the mathematics classrooms of today. This is envisaged not only as means to make mathematics learning more meaningful to the learners, but in doing so enable these learners to realise and to apply the power of mathematics in solving many of the everyday unfamiliar problem situations in their environment. This is above all, in this author’s view, the prime objective of the whole exercise of education in general and mathematics education in particular.
4.1 Background and definition
For many thousands of years, all societies one way or another have developed mathematical practices that are considered most appropriate to their daily lives and cultures, an area of mathematics known as ethnomathematics (Bishop, 1991a; Borba, 1990; D’Ambrosio, 1990a; 1991). In contrast to the more commonly known ‘academic mathematics’, that is mathematics taught and learned in schools, ethnomathematics is the “mathematics which is practised among identifiable cultural groups, such as national-tribal societies, labour groups, children of a certain age bracket, professional classes and so on, with their jargons, codes, symbols, myths and even specific ways of reasoning and inferring” (D’Ambrosio, 1991, p.18). Furthermore, ethnomathematics can also be seen as a very broad range of human activities, which throughout history have been replaced by equivalent practices through formalising and codifying and eventually incorporated into the so-called ‘academic’ mathematics found within the formal schooling systems (Bishop, 1991a; Borba, 1990; D’Ambrosio, 1990a, 1991; Gerdes, 1994). On the other hand, ethnomathematics is still alive in many of these culturally identified groups, subsequently forming an integral part of many of the routines in their practices (Lancy, 1983; Saxe, 1981). In this context, the notion of ethnomathematics therefore lies on the borderline between the history of mathematics and cultural anthropology (D’Ambrosio, 1990a, 1991; Gerdes, 1994; Pinxten, 1994).
4.1.2 Mathematics as a neutral and value-free knowledge
It was not until very recently that the notion of ethnomathematics started gaining popularity, probably because of the commonly held view of the “universality of mathematics” (D’Ambrosio, 1991, p. 15) which implied that mathematics is both culture- and value-free (Bishop, 1991a; Gerdes, 1994; Julie, 1993). As noted in the National Statement on Mathematics for Australian Schools:
The universal nature of many mathematical ideas and the extensive use of symbolic notation to portray abstract ideas ... lead many people to the mistaken view that mathematics is culturally neutral and value-free. (Australian Educational Council, 1991, p. 15)
The absolutist view of mathematical knowledge (i.e. formalism, logicism) has for the most part of the last two thousand years dominated much of mathematics. This view of mathematical knowledge is still the one currently held by many of the modern and traditional philosophers (Ernest, 1991; Pinxten, 1994). According to this view, mathematics is a “body of infallible and objective truth, far removed from the affairs and values of humanity” (Ernest, 1991, p. xi). As Bloor (1973), in support of this view further notes:
Mathematics and logic are seen as being about a body of truth, which exist in their own right independently of whether anyone believes them or knows about them. On this view even if there were no human beings mathematical truths would still be true. (cited in Stigler & Baranes, 1988, p. 257)
Hence, the implication for mathematics education according to this view is that teaching and learning mathematics consist of discovering the already existing truths of formal logic, thereby logically initiating the pupil in the deductive and basically decontextualised knowledge of the mathematical field (Pinxten, 1994). This view suggests that mathematics teaching involves teaching the students to view the quantities and their manipulations as contextless and divorced from meaning, that is to say, independent of human activities (Ascher & D’Ambrosio, 1994).
4.1.3 Mathematics as a socially constructed knowledge
The above view of mathematical knowledge is however challenged not only by mathematics educators because of the growing dissatisfaction with the current teaching programs based on the absolutist philosophy of mathematics (Ernest, 1991; Pinxten, 1994; Julie, 1993), but also by a growing number of philosophers and mathematics educators such as Abraham and Bibby (1988), Ascher and D’Ambrosio (1994), D’Ambrosio (1990a), Ernest (1991) and Bishop (1991a, 1991b) who affirm that mathematics is fallible, changing, and like any other body of knowledge, the product of human inventiveness. As argued further by Ascher and D’Ambrosio (1994):
The absolute suppression of context, and the quantification of values for comparison, in order to value something more than the other, is ... a sign of philosophical damage done to modern thought; it leads to a world deprived of human values, even of human feelings. (p. 40)
In addition, the absolutist view of mathematical knowledge is also challenged by recent research evidence from both anthropological and cross-cultural studies (e.g. Bishop, 1979; Gerdes, 1985; Lancy, 1983; Saxe, 1981, 1982; Zaslavsky, 1994) which not only support the idea that mathematics has a cultural history, but also that from these cultural histories have come what can only be described as different mathematics (Bishop, 1991a, 1991b). As D’Ambrosio (1991) further noted:
... recent research ... shows evidence of practices which are typically mathematical, such as counting, ordering, sorting, measuring and weighing, done in radically different ways to those which are commonly taught in the school system. (p. 15)
Mathematical practices were developed as a result of every single culture, tribe, community and individual trying to cope with everyday needs, problems and challenges for their survival in direct relationship with the environment and fellow human beings. It can be viewed as a means of trying to understand what is going on, to explain what is seen and felt, thereby contributing either directly or indirectly to the building up of knowledge (Ascher & D’Ambrosio; Bishop, 1979, 1991a; D’Ambrosio, 1990a, 1991; Stigler & Baranes, 1988). In other words, mathematical practices and ideas arose out of the real needs and interests of human beings (Zaslavsky, 1994). This view is also shared by Abraham and Bibby (1988) who, in their effort to generate discussion about the potential role of a Mathematics and Society curriculum in formal education, contend that “mathematics cannot be understood without some understanding of the social institution of mathematics - that is socially organised mathematical activity” (p. 4). This view implies that in order to have some understanding of the role mathematics plays in structuring our experiences and judgements, one needs to have some understanding of the human actions that give rise to such major developments in mathematics.
In the light of the above views, it is obvious that various cultural mathematics, referred to in this paper as ‘ethnomathematics’, existed in all societies in one form or another from the most simple everyday practical activities to more complex abstractions. Moreover, these views also suggest that new developments in mathematics can only occur when previously established knowledge is challenged by the need to solve unfamiliar problem situations occurring in everyday contexts. Seen in this context, ethnomathematics is therefore best summarised by D’Ambrosio (1990a, p. 369) that:
mathema [are] the actions of explaining and understanding in order to transcend and of managing and coping with reality in order to survive. Throughout all our own life histories and throughout the history of mankind, technes (or tics) of mathema have been developed in very different and diversified cultural environments, i.e. in the diverse ethnos. So, in order to satisfy the drives towards survival and transcendence, human beings have developed and continue to develop, in every new experience and in diverse cultural environments, their ethno-mathema-tics. These are communicated vertically and horizontally in time and for the reason of being more or less effective, more or less potent and sometimes even for political reasons, these various tics have either lasted and spread (e.g. counting, measuring) or confined themselves to restricted groups and even disappeared.
In other words, ethnomathematics refers to what Nunes (1992) called “forms of mathematics that vary as a consequence of being embedded in cultural activities whose purpose is other than doing mathematics” (p. 557). These include among other things, building houses, exchanging money, weighing products, and calculating proportions for a recipe which involve numbers, calculations, and precise geometrical patterns.
4.2 Ethnomathematics and the Development of Mathematics.
4.2.1 Mathematics as a cultural knowledge
In his philosophy of mathematics, Wittgenstein (cited in Stigler & Baranes, 1988), while simultaneously questioning the notion that mathematics is a system that exists outside the practice of doing mathematics, argued that mathematics is “social in nature, and inseparable from the social realm in which it is used” (p. 257). Thus, from Wittgenstein’s perspective, the formalisation of mathematics was not a discovery of the foundations of mathematics, but rather a theoretical enterprise that came after the development of mathematical practices. For Wittgenstein, mathematics is not a system of unique formal logic that exists on its own but rather a collection of techniques or games, each with its own consistent rules that are not necessarily connected to other games (Stigler & Baranes, 1988). As Bishop (1991a, p. 30) further elaborates: “The thesis is therefore that mathematics must now be understood as a kind of cultural knowledge, which all cultures generate but which need not necessarily ‘look’ the same from one cultural group to another”. Accordingly, he further argues that just as all human cultures generate language, religious beliefs, rituals, food-techniques etc., so it seems do all human cultures generate mathematics. Thus, in his view, mathematics is a pan-human phenomenon. Moreover, if the above perspective is to be taken as it is, it means that just as each cultural group generates its own language, religious beliefs etc., so it logically seems that each cultural group is also capable of generating its own mathematics. Obviously this kind of thinking has some fundamental implications for teaching and learning of mathematics in terms of re-examining many of our traditional beliefs about the theory and practice of mathematics education (Bishop, 1991a).
4.2.2 The universal mathematical activities
In the light of the description in the above section, it is worthwhile to point out that while there are specific mathematical practices or activities that differ within and among different cultural groups, in general, there are some practices that Bishop (1991a) called “universal” across different cultural groups. According to his analyses, Bishop (1991a) argued that there are six fundamental activities which are both universal, in that they appear to be carried out by every culture group ever studied, and also necessary and sufficient for the development of mathematical knowledge. These, as he described them, are as follows:
Counting : the use of a systematic way to compare and order discrete objects. It may involve body or finger counting, tallying or using objects or strings to record, or special number names. Calculation can also be done with the numbers, with magical and predictive properties associated with some of them.
Locating : exploring one’s spatial environment and conceptualising and symbolising that environment, with models, maps, drawing and other devices. This is the aspect of geometry where orientation, navigation, astronomy and geography play a strong role.
Measuring : quantifying qualities like length and weight for the purposes of comparing and ordering objects. Usually measuring is used where phenomena cannot be counted (e.g. water, rice) but money is also a unit of economic growth.
Designing : creating a shape or a design for an object or for any part of one’s spatial environment. It may involve making the object, as a copyable ‘template’, or drawing it in some conventional way. The object can be for technological or spiritual use and ‘shape’ is a fundamental geometrical concept.
Playing : devising and engaging in games and pastimes with more or less formalised rules that all players must abide by. Games frequently model a significant aspect of social reality, and often involve hypothetical reasoning.
Explaining : finding ways to represent the relationship between phenomena. In particular, exploring the ‘patterns’ of number, location, measure and design, which create an ‘inner world’ or mathematical relationships which model and thereby explain the outer world of reality. (Bishop, 1991a, p. 32-33)
Thus, mathematics as a cultural knowledge derives from humans engaging in the above six fundamental activities, as Bishop (1991a, p. 33) put it, “in a sustained and conscious manner”. Accordingly, he argued that the performance of these activities by members of the respective cultural groups can either be in a mutually exclusive way or perhaps, more significantly, by interacting together as in ‘playing with numbers’, examples of which are number patterns and magic squares which arguably contributed to the development of algebra. Taken in this context, Bishop (1991a) is also supported by many other writers (D’Ambrosio, 1991; Ernest, 1991; Gerdes, 1994; Pinxten, 1994) who argue that many of the highly significant ideas found in the current so-called ‘Western mathematics’ taught in schools are themselves the results of human engagement in activities such as those described above. As Bishop (1991a, p. 34), on the basis of the above six fundamental activities, further elaborated:
From these basic notions, the rest of ‘Western’ mathematical knowledge can be derived, while in this structure can also be located the evidence of the ‘other mathematics’ developed by other cultures. Indeed we ought to re-examine labels such as ‘Western Mathematics’ since we know that many different cultures contributed to the knowledge encapsulated by that particular label.
For example, Nelson (1993) in this respect provides some significant historical examples of mathematical developments such as the use of 360 degrees for complete revolution tracing it back to the Babylonian sexagesimal (60-based) number system, the notation for the angle to the Greeks, the sign for equality to Robert Recorde (1557), the notation to Descartes (1637), and both the decimal notation and the concept of zero to the Indians and Chinese, just to name a few. These examples illustrate the fact that the study of historical developments in mathematics can be of rich learning experience for the learners through the use of projects which can be investigation-based thereby simultaneously providing the opportunity for the learners to involve parents or other members of the local community who are the bearers of various mathematical practices embedded in cultural activities.
4.2.3 Mathematics as a cultural phenomenon
If the current experiences are taken to be any indication, there is another aspect of ethnomathematics which is usually left out by many of the school systems during the teaching of formal mathematics. Unlike the more obvious observable mathematical practices described so far, this aspect of ethnomathematics relates to the notion of what Bishop (1991a) called “mathematics as a cultural phenomenon” (p. 32). In other words, if ethnomathematics is to be taken as a cultural product, according to White (cited in Bishop, 1991a) this simply means that the functions of culture, more than anything else, are to relate man to his environment on the one hand, and to relate man to man, on the other. Thus, ethnomathematics is also a notion that is ideological, sociological and sentimental in nature, in that it is used by members of specific cultural group to represent and to explain cultural phenomena that could not be adequately defined in any practical terms.
In this respect, White (cited in Bishop, 1991a) subsequently went further to divide the components of culture into four categories, namely:
· ideological: composed of beliefs, dependent on symbols, philosophies;
· sociological: the customs, institutions, rules and patterns of interpersonal behaviour;
· sentimental: attitudes, feelings concerning people, behaviour;
· technological: manufacture and use of tools and implements.
(cited in Bishop, 1991a, p. 32)
Moreover, it has to be pointed out that while the above four components are inter-related, White (1959) strongly argues that the technological factor is the most basic one with all other factors dependent upon it. In other words, for White the technological factor determines in a general way at least the form and content of the social, philosophic and sentimental factors.
The significance of written language in terms of mathematical symbolism as one of its particular conceptual ‘tools’ is also highlighted by Bishop (1991a). Based on White’s schema, he argued that mathematics as an example of both a cultural phenomenon and a symbolic technology, offers an opportunity to explore ideology, sentiment and sociology, and therefore simultaneously attend to the notion of ‘values’ as well. In this respect, Ascher and D’Ambrosio (1994) contend that in the education of children, values are being taught through an emphasis on contextless numbers whereby everyday examples are phrased in terms of numerical equivalents. It is therefore not uncommon to find that for those learners who are alienated by mathematics, there is a feeling that mathematics is both emotionless and lacks the necessary human feelings that initially created it. According to Ascher and D’Ambrosio (1994), this therefore means that the most “important point that ethnomathematics adds, however, is that, nonetheless, expressions of mathematical ideas do have content and cultural context” (p. 39), a notion that has been a large source of misunderstanding in the past.
The above argument is strongly supported by the evidence from Buxton’s research (cited in Bishop, 1991c) concerning adults’ fear of mathematics, which suggested that mathematics is largely seen as:
1. Fixed, immutable, external, intractable and uncreative.
2. Abstract and unrelated to reality.
3. A mystique accessible to few.
4. A collection of rules and facts to be remembered.
5. An affront to common sense in some of the things it asserts.
6. A time test.
7. An area in which judgements not only on one’s intellect but on one’s personal worth will be made.
8. Concerned largely with computation.
(Cited in Bishop, 1991c, pp. 195-196)
If the above evidence is of any significance, this paper therefore strongly shares the view that the practices of mathematics and mathematicians are not fundamentally different from human thought as embedded in other domains (Stigler & Baranes, 1988). Subsequently, it therefore views the practice of mathematics as not the discovery of truths existing outside the realm of human activity. Rather it considers mathematics to be, as Stigler and Baranes (1988) put it: “domain-specific, context-bound, and procedurally rooted as are other forms of knowledge” (p. 258). Taken in this context, it therefore views mathematics learning to be not uniquely immune to the influences of culture, but rather as culturally dependent as learning in other domains.
4.3 Views of Culture and Mathematics
The analysis of the discussion so far in this chapter generates two distinct approaches to the study of cultural influences on mathematical knowledge. According to Nunes (1992), in the first approach which is consistent with the view suggested by Stigler and Baranes (1988), the definition of mathematical knowledge is somewhat implicit, in that, it appears to be based on the content of knowledge. Thus, according to Stigler and Baranes (1988), “mathematics is not a universal, formal domain of knowledge waiting to be discovered, but rather an assemblage of culturally constructed symbolic representations and procedures for manipulating these representations” (p. 258). They therefore argue that the incorporation of representations and procedures by children into their cognitive systems is a process that occurs in the context of socially constructed activities. In this context, they further argue that mathematical skills that children learn in school are the results of a combination of previously acquired (or inherited) knowledge and skills, and new cultural input, rather than logically constructed on the basis of abstract cognitive structures. According to this view, culture therefore functions not as an independent variable that can promote or retard the development of mathematical abilities, but is considered as an integral part of the mathematical knowledge itself (Stigler & Baranes, 1988). In this approach, the variations between cultures are therefore treated as a reflection of the differences in language and numeration systems (Nunes, 1992).
The second perspective, as noted by Nunes (1992) and advocated by Ascher and D’Ambrosio (1994), Bishop (1991a, 1991b) and D’Ambrosio (1990a, 1991), contends that the analysis of cultural influences on mathematical knowledge can demonstrate both the differences and invariance in mathematical knowledge across cultures. According to this view, ‘mathematizing’ reality is therefore seen as representing reality in such a way that more knowledge about the represented reality can be generated via inferences using mental representations as contended by Bishop (1991a) in his analyses of the six fundamental activities described in the previous section. Moreover, there is also no need to manipulate such reality any further for the purpose of verifying this new knowledge. In using this view, Nunes (1992) further argues that the invariant logical structures are therefore part and parcel of mathematical knowledge, regardless of the location where this knowledge is developed, be it in school or out of school. However the most important consideration is the ability to make mathematical inferences on the basis of these logical structures, rather than the content of knowledge, that distinguishes mathematical knowledge. As Nunes (1992, p. 558) clearly put it:
Merely reciting count words ... would not be considered mathematical knowledge in this perspective. Despite the fact that numbers are the content of knowledge ... no inferences result from mere recitation, thus no mathematical knowledge is involved.
If count words are used to represent sets and make inferences about relationships between sets, the count words bear on mathematical knowledge. Taken in this context, it therefore means that different cultures have found distinct solutions to the organisation of their count words. In educational terms, these differences, like those of other cultural knowledge have important effects on how quickly children learn the count words within the context of formal classroom situations (Nunes, 1992).
4.4 Mathematical Enculturation
Mathematics education, like those of other ‘academic’ educational disciplines, is so often characterised by what Bishop (1991b, p. 89) described as the “formal mathematical enculturation” into the so-called Mathematical culture. Because of this, it has as its goal, the induction of children (learners) into the symbolisations, conceptualisations and values of the Mathematical culture. Such characterisation clearly indicates the involvement of both the process and content of mathematics. Because of the culture’s frame of knowledge, the formal mathematical enculturation cannot be just process-oriented, but nor should it just attend to content knowledge, since education is more than mere transmission (Bishop, 1991b). Moreover, this equally means that the ‘enculturation process’ has the responsibility to both the child and culture, respecting the individuality and personality of the children at the same time the characteristics of the culture. As Bishop (1991b) explains further:
To ignore the [first] would lead to indoctrination, while to ignore the [second] would lead to anarchy. Mathematical enculturation needs to be conceptualised as a social interactive process carried out within a certain knowledge frame but with the goal of recreating and redefining that frame (p. 89).
It is with such a background in mind that mathematics, like any other academic teaching subject, must be guided by some form of curriculum. In short, it is the ‘objectified’ representation of mathematical culture for the purposes of the educational process. This not only helps in thinking about the kinds of activities to be used for enculturing the learners into the mathematics culture, but also provides a suitable structure for organising those activities into one coherent whole (Bishop, 1991b). For the purpose of this literature survey in particular, this logically means that the ‘culturally-oriented’ mathematics curriculum must therefore seek a way to define the knowledge frame which will not only allow personality to flourish, but most importantly, create a suitable learning environment necessary for effective social interaction to take place within the context of learners’ socio-cultural environment.
In view of the above suggestion, it is argued by Lester (1989) that the knowledge gained in out-of-school situations, is often characterised by five key features developing out of everyday activities which, as he puts it:
1. occur in a familiar situation;
2. are dilemma driven;
3. are goal directed;
4. use the learner’s own natural language; and
5. often occur in an apprenticeship situation allowing for observation of the skill and thinking involved in expert performance. (cited in Masingila, 1993, p.18)
On the other hand, knowledge acquired in formal classroom situation all too often grows out of a transmission paradigm of instruction and is therefore viewed to be lacking in meaning, that is to say, lack of context, relevance and specific goal (Ascher & D’Ambrosio, 1994; Boaler, 1993a; Masingila, 1993). Accordingly, it is argued by Resnick (cited in Masingila, 1993) that schools place too much emphasis on the transmission of syntax (procedures) rather than on the teaching of semantics (meaning) thereby discouraging children from bringing their own intuitions to at least have some bearing on the school learning tasks.
Students need in-school mathematical experiences to build on and formalise their mathematical knowledge gained in out-of-school situations. The mathematical experience in school in terms of the guidance and structure provided by teachers can be of significant benefit to students in that it can help them to make connections among different mathematical ideas. In this context, some mathematics educators (e.g. Bishop, 1991b; Julie, 1993; Masingila, 1993; Zaslavsky, 1994) have suggested that mathematics teachers should establish master-apprenticeship relationships with their students in order to help initiate them into the mathematics community. Working with others toward common goals, being actively involved in doing mathematics, and discussing and refining mathematical ideas are all characteristics of being part of the mathematics community. At present, however, according to Masingila (1993) many students become isolated from the mathematics community rather than becoming a part of it. A key reason for this isolation is the wide gap that exists between mathematical practices in school and in out-of-school situations which also includes ethnomathematics defined in chapter 4 of this presentation representing the ‘cultural practices of mathematics’.
Thus, in terms of the objective of this study, the implication is that this wide gap is also seen to be mainly responsible for greater mathematical learning difficulties experienced by many of the Papua New Guinean students. As noted in chapter 3, this is because of the current approaches to mathematics education in PNG which do not take into account the learner’s cultural and social environment in which many of these ethnomathematical practices are embedded. For one thing, these learners should not be characterised as empty mugs, but rather, as equal partners who, in their own small ways, contribute to shaping the educational process, thus leading to meaningful and realistic educational experiences.
With such a view in mind, the prime objective of this study is to close this gap between doing mathematics in school and doing mathematics in out-of-school situations. The implication of this objective is that the alternative mathematics education curriculum should, as one of its main objectives, incorporate the social history of mathematics not only to reflect on the historical developments of mathematics, but to use it as the basis to generate further mathematical inferences and knowledge in benefiting everyone in society. Moreover, such an approach will not only become a rich educational experience for the learners, but also preserve the rich cultural heritage of mathematics to benefit future generations.
The definition of ‘ethnomathematics’ as generated by the discussion in this chapter can be seen as the mathematics that is practised by members of different cultural groups embedded in a very broad range of human social activities which include their jargons, codes, symbols, myths and even specific ways of reasoning and inferring. Moreover, the analysis of these cultural practices also reveals that ethnomathematics can also be characterised as both a content and process of mathematics. The ‘process’ of mathematics notably involves the six universal mathematical activities namely, counting, locating, measuring, designing, playing and explaining. These fundamental activities are considered to be not only necessary, but are also sufficient for the generation and development of mathematical inferences and knowledge. Thus, throughout the history of a specific cultural group these fundamental activities have both individually and in interaction been instrumental in developing the complex symbolisations and conceptualisations of mathematics which also include the so-called ‘Western’ mathematics taught in schools.
Because of its significant educational implication, the ‘process’ aspect of ethnomathematics is therefore consistent with both the objectives of this study and the underlying educational assumption of the current philosophy of education for PNG which both aim to generate, transmit, institutionalise and diffuse mathematical knowledge to learners with a strong emphasis on their socio-cultural environment. Consequently, such an emphasis is therefore in direct conflict with the educational assumptions of the current approach to mathematics education in PNG, because of its view of mathematics as being independent of the learners’ socio-cultural environment. This approach to mathematics education is seen to be largely responsible for the dislike of mathematics by many learners in PNG, a major concern of many of the recent research studies. In this respect, ethnomathematics therefore aims to draw from the cultural experiences and practices of the individual learners, the communities, and the society at large in using them as vehicles to not only make mathematics learning more meaningful, but more importantly, to provide the learners with the insights of mathematical knowledge as embedded in their social and cultural environment. In doing so, it not only allows for an easier flow of mathematical ideas in reducing the effects of many of the social and cultural factors, but also creates a more meaningful learning environment that is conducive to any human social process of which mathematics is its highest symbolic technological product.
It is with such views in mind that the next chapter will further consider some specific ethnomathematical practices within the cultural context of PNG which have the potential to generate mathematical ideas or inferences that could be incorporated into the formal mathematics classrooms in PNG. In particular, the aim is to use them as the basis for making the process of teaching and learning mathematics noted in the previous section as ‘mathematical enculturation’ more meaningful to the learners. Most importantly though is for the learners to ‘relationally’ understand mathematical knowledge such that it can be applied in any unfamiliar everyday context of the learners’ own environment as determined by the overall societal needs and problems.