Background

At no other time in its short history as an independent nation, has Papua New Guinea (PNG) inevitably forced to critically re-examine the role of it’s National Education System (NES) in the context of the overall national developmental objectives than it is today, mainly for two reasons. Firstly, it is not only a must but also necessary that it take stock of its educational achievement so far in terms of both the successes and the failures as the nation fast approaches the end of the current millennium. Secondly, based on the achievement so far, to make well-informed choices in planning and executing the overall direction that education would take as it prepares to meet the challenges ahead in the new millennium. Such an exercise is timely, particularly in the light of efforts being made at different levels of the education system in trying to create a balance between the so-called “Western” values of education and the existing “Traditional” values after more than two decades of nationhood.

For mathematics education, it is also not immune to such pressures, particularly in the light of two emerging problem areas. The first of these relates to the direction which mathematics education in PNG should take in the face of the increasing presence of computers and calculator-related technology in society. The second problem area relates to concerns, particularly among mathematics educators, for the children whose home and family cultures do not fully resemble that of the formal school system and the wider society.

Both of the above problems are related and thought-provoking, with the first one raising questions about educational values, about the importance of different kinds of knowledge attached by society, and the relationship individuals have with that knowledge. It is however the second problem area that is of particular concern to this discussion paper. This is because it raises questions about the role of mathematics education in society and how successful it has been in fulfilling its societal role in terms of the stated objectives, the underlying educational assumptions, and the type of classroom practices it envisages. Moreover, there are also a number of research findings (e.g. Bishop, 1991b; Masingila, 1993; Nunes, 1992; Roberts, 1976) which indicate that the influence of informal education is much greater than the formal education. Furthermore, the fact that majority of the students in PNG find formal mathematics to be both difficult and boring is also indicative of the problems associated with the current approach to mathematics teaching leading to an under-achievement in mathematics.

The above situation inevitably lead to concerns raised by individuals as well as the stake holders in education calling for a critical review of the societal role of mathematics education, a concern which is the main argument of this discussion paper. The need to re-examine the current overall objectives and the prescribed mathematics curriculum which legitimise the type of teaching practices adopted to both crucial and necessary given the fact that they are heavily based on the ‘Western values’ of education. This bias is reflected through the adoption of the type of teaching and learning theories and models of education. According to Masingila (1993), one of the main problems of the current teaching practices is that the acquisition of knowledge all too often grows out of a transmission paradigm of instruction which employs the dominant explain-practice method of teaching and is therefore largely devoid of ‘meaning’, in that, such knowledge lacks the appropriate contextual meaning, relevance and specific goal. Thus much of the past and present mathematical learning difficulties experienced by majority of the PNG students can be attributed to the teaching of mathematics biased towards the transmission paradigm of instruction. Because of its close link to the absolutist ideology of mathematical knowledge, this approach therefore views mathematics as being both culture-and value-free. Consequently it pays very little attention to the rich every day mathematical experiences that students bring to the formal classroom from diverse cultural settings.

Before going any further, it is both important and necessary to define the three key terms which will be fused extensively throughout this discussion paper, namely, Culture, Mathematics and Mathematics Education mainly because of their diverging meanings attached by different individuals depending on the context in which each term is used.

Culture

The notion of culture has been defined in many different ways based on the context in which it is being used. For the purposes of my discussion in this paper, the most appropriate meaning is the one provided by Bullivant (1981). He defined ‘culture’ as,

“the knowledge and conceptions, embodied in symbolic and non-symbolic communication modes, about the technology and skills, customary behaviours, values, beliefs, and attitudes, a society has evolved from its historical past, and progressively modifies and augments to give meaning to and cope with the present and anticipated future problems of its existence” (p. 19).

In other words, culture encompasses everything that is found within the physical and spiritual environment of a particular social group or society. Thus, if culture is to be located on the continuum of both space and time, it represents a collective way of coping with the problems of surviving in the spatial environment by its members (Bullivant, 1981). When culture is seen in this context as a form of problem solving and survival program that has been developed over time by members of any one society, it opens up the gate to other wider issues. One of these is the organisation and processes, which enable cultural knowledge to be transmitted from one generation to the next, a notion technically known as ‘enculturation’ (Bishop, 1991; Bullivant, 1981). It is this notion that forms the main argument in this discussion paper, particularly, the relative issue of ‘cultural context of learning and thinking’ and how such notion affects the enculturation process in PNG with specific reference to mathematical knowledge. Naturally, such discussion will involve the more familiar formal school system thereby leading one to raise questions relating to how effective and successful schooling has been responsible in achieving culture transmission.

Mathematics

The term mathematics as defined by Bullivant (1981) is a symbolic technology resulting from the demands of the environment as experienced by members of one specific cultural group or society through what Bishop (1991, p. 32-33) calls the ‘universal’ cultural activities of mathematics which he describes them 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.

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 quantities 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 copy able ‘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.

A careful analysis of the above cultural activities reveals that mathematics after all is a systematic symbolic representation of problem-solving methods specifically developed by any one society or cultural group over time as means to cope with problems of everyday existence of its members in direct relation to the environment. It is also important to note that the above universal cultural activities need not necessary be identical because of the staggering cultural diversity across different cultural groups.

Mathematics Education

Mathematics education on the other hand is the term which is so often characterised by what Bishop (1991, p. 89) defined as the “formal mathematical enculturation” into the culture of mathematics. In other words, it is the process of inducting children (learners) into the culture of symbolisations, conceptualisations and values of mathematics. Naturally such characterisation would clearly involve both the process and content of mathematics because of the culture’s frame of knowledge. Bishop (1991) therefore further argues that the formal mathematical enculturation cannot be just process-oriented, nor should it just attend to content knowledge, since education is more than just the mere transmission of knowledge. This simply means that the enculturation process must be designed and structured in such a way that it is responsible to both the child and culture, in that, it should respect the individuality and personality of the child at the same time the characteristics of the culture. In mathematics, the best description of the characteristics of an ideal enculturation process is provided by Bishop (1991b, p.3) stating that:

Educating people mathematically consists of much more than just teaching them some mathematics. It requires a fundamental awareness of the values, which underlie mathematics, and recognition of the complexity of educating children about those values. It is not enough merely to teach them mathematics; we need to also educate them about mathematics, to educate them through mathematics, and to educate them with mathematics.

The immediate problem of a mathematics educator is to make a well-informed choice concerning not only the type of teaching practices to be employed in dealing with the procedural knowledge of mathematics, but also be mindfully selective in deciding the type of content knowledge that will enhance meaningful development of conceptual knowledge by learners as a prerequisite to becoming an effective everyday problem solver.

The cultural context of mathematics learning and thinking

Based on the analysis of the definitions of culture, mathematics and mathematics education, it is obvious the three terms are highly inter-related. While culture may be seen as a way of life in its entirety for a particular cultural group or society, mathematics on the other hand, is a systematic problem solving methods purposely developed to solve the everyday problems of the existence of its members. Mathematics education can be seen as the organisation and processes, which enable the cultural knowledge, including mathematics, to be transmitted from one generation to the next.

Though not only limited to mathematics, the ‘cultural context of learning’ can be defined as the process of transmitting cultural knowledge which explicitly utilises the examples, illustrations, cultural values and the type of learning environment that are ‘familiar’ to the learners’ own social and cultural environment. This would essentially include all the informal learning that takes place outside of the formal classroom environment but within the boundaries of a particular cultural group’s everyday activities. The need to utilise the local (informal) knowledge of mathematics in PNG has the advantage in that it is not only familiar to the learners, but most importantly, it provides the necessary contextual meaning to many abstract mathematical ideas and concepts taught in the formal classroom. Utilising the indigenous knowledge within the formal classroom will not only enable the learners to construct meaningful mathematical relationships, but also provide an opportunity for interactions to occur between the learners themselves, as well as the teacher in explaining the many contrasting differences that exist across different cultural groups.

Discussion

Throughout history despite the great variety of their cultural forms, it is not uncommon to find communities of man kind have made it their utmost obligation to not only ensure that their cultural heritage is transmitted from one generation to the next, but have also made it their prime responsibility to establish conditions necessary for its growth and continuity (Kimball, 1974).

An analysis of the relationship between each of the three terms defined earlier further reveals that there are two distinct views that subsequently give rise to the type of approaches to be used in studying the cultural influences on mathematical knowledge which hage definite implications for mathematics education in PNG (Matang, 1996).

The first of these views contends that the definition of mathematical knowledge is somewhat implicit, in that, 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 (Stigler and Baranes in Matang, 1996). Thus cognitively speaking, the advocates of this view 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 other words, the mathematical skills that children learn in school are the results of the combination of previously acquired (or inherited) knowledge and skills, and new cultural input, rather than logically constructed on the basis of abstract cognitive structures. Accordingly, the notion of culture 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.

The second view contends that the analysis of cultural influences on mathematical knowledge can demonstrate both the differences and invariance in mathematical knowledge across cultures (Bishop, 1991a; Kimball, 1974; Nunes, 1992). 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 by way of making inferences using mental representations. Moreover, it argues that there is no need to manipulate such reality any further for the purpose of verifying this new knowledge. The most important consideration however is the ability to make mathematical inferences on the basis of these logical structures, rather than the content of knowledge the distinguishes mathematical knowledge.

Concerning the importance of the different kinds of knowledge attached by society, the above two views lend themselves, at least in mathematics, to two related but different types of knowledge, namely the procedural and conceptual. According to Kimball (1974), the former includes all informal and institutional arrangements provided by society for the purposes of instruction. It includes the organisation of the presentation of subject matter and the relationship between teacher and learner. In short, it deals with the question of how to do it. The conceptual conditions, though under the current arrangement is given very little prominence, are of far greater significance because it is these conditions that provide humanity with its sense of identity. It is also from these conditions that the perspective about the worldview is reflected. The rules that govern the organisation and evaluation of knowledge including the explanation of the processes of change are also to be found in them. In short, it is the conceptual conditions that make experience more meaningful.

Pedagogical Implications

The comparison of the pedagogical implications between the current approach to teaching mathematics in PNG and the one advocated in this paper can best be described by an analogy [illustration] involving two individuals; one a champion athlete and the other a novice, competing in, say, a 100 meters race. The race is conducted in such a way that the champion is placed some 50 meters up the lane while the novice has to start the race at the starting point. Based on normally expected human abilities and performance in the light of the background of the two competitors, it is obvious who the winner will be, let alone the question of whether the race can be considered as being fair.

Interpreting the above analogy in the context of education in general, or more specifically mathematics education, the chamption athlete is the child (or student) from a western cultural background while the novice is the child with PNG cultural background. The 100 meters race can be viewed as the education system in general encompassing everything in terms of the overall objectives, curriculum, and most importantly within the boundaries of the classroom, the type of learning environment promoted.

Although the first two aspects of any education system are important, the main focus of this paper however will be on the type of learning environment promoted by the current education system. In terms of the teaching and learning process, Kimball (1974) argues that “learning process must be viewed as an aspect of the cultural milieu.” This is because culture is transgenerational in that it provide clues to what and how the child learns at the same time we must be mindful in abstracting from culture what is of universal relevance in terms of its educational value.

Given the above background, the current teaching practices seem to be counterproductive in that they do not enable majority of the learners to be mathematically literate citizens. One reason is that it contradicts with the rationale behind the generally accepted educational theory of teaching from known to unknown, or in the case of mathematics, teaching from concrete to abstract. It is also in conflict with the logical explanations of many everyday activities or phenomena within cultural groups in PNG in that when an idea, or for that matter any phenomena, fails to exist in reality then no further discussion takes place on it (Matang, 1998). One classical example highlighting this situation is the universal absence of a word label or symbol representing the notion of zero in the traditional counting systems in PNG. It is interesting to note that the non-representation of zero by the traditional counting systems in PNG is to direct relevance to the teaching of the topic on number concept. This is because it provides the contextual meaning in explaining the abstract reasoning supporting the argument why zero is not a member of the subset of the set of real numbers called natural or counting numbers.

In fact the current approach to teaching of mathematics is largely seen to be directly opposite, in that, it promotes the teaching of mathematics based on abstract thinking that is independent of the real-life experiences of the students. One reason is that the current teaching approach is strongly rooted in the absolutist ideology of mathematical knowledge (Ernest, 1991) which views mathematics as both cultural-and value-free.

Conclusion

From the discussions, it would not be unfair to suggest that the current mathematics teaching practices in PNG have not been effective in helping to achieve its fundamental obligation of transmitting the necessary cultural knowledge of mathematics in making PNG a mathematically literate society. In the light of the ever-increasing computer-related technology, this situation is unacceptable. One of the major contributing factors is that the current teaching practices fail to acknowledge at the sme time utilise the local knowledge of mathematics that have the potential to provide the necessary contextual meaning to many of the abstract mathematical concepts taught in the formal school system.

In addressing the above problem, the paper therefore strongly suggests for a greater utilisation of the rich and diverse cultural settings of the learners as the basis of teaching mathematics simply because it is a product of culturally based human social activities. While the current teaching practices have their share of marits in emphasising procedural understanding necessary for skill development, it has however given very little attention in developing a conceptual knowledge base among its citizens. It is now 25 years since independence and it is timely for the nation to take stock of its educational achievement to enable it to set the next course of action for the next millennium. The paper therefore suggests for a shift in emphasis from what is now an emphasis on procedural knowledge to an emphasis on conceptual knowledge that utilises the contextual meaning from diverse contrasting situations found in the cultural settings of the learners.

The advantage of such an approach is that it will not only provide a variety of contrasting situations from which meaningful comparisons can be made in furthering the development of knowledge, but also promote self-esteem and intellectual property on the part of the learners thereby portraying mathematics as a meaningful and reflective subject.

References

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