Chapter 3

The Current Practices of Mathematics Education

in PNG

Papua New Guinea is a strange, fascinating country which is at present going through an amazing period of change. All countries experience change, but it is possible that few have ever experienced change so rapid as that in Papua New Guinea (PNG). “From stone-age to twentieth century in one lifetime” is no overstatement. Apart from those living in the few small towns the majority of the population have little contact with the technological society and culture which we know so well. [Bishop, 1979, p. 135]

3.1 Background

The above introductory remarks made by Bishop (1979), from an outsider’s point of view, emphasise the rate of change that has taken place in the last one hundred and twenty five years of PNG’s development since the first permanent European settlement in the 1870s. In particular the role of education, which is often the forerunner of, and to some extent seen to be synonymous with any process of change, both within the developed and developing world, is probably more apparent in PNG than in similar situations elsewhere. Furthermore, it is also possible that few countries (if any) have experienced cultural diversity as rich as that found in PNG, particularly in a country comprising well over 700 different subcultures, each having its own distinct language, values, beliefs, customs, aims and traditions (Lancy, 1983). Thus, while in educational terms such diversity on the one hand is seen to be an advantage, on the other hand it has provided the most difficult task that one could imagine for the providers of educational services. Mathematics education for its part is no exception in this regard, particularly in the context of the various conflicting cultural values created by the introduced ‘Western’ and the existing indigenous ‘Traditional’ societal aims and beliefs systems.

Politically, such cultural diversity is a catalyst for fragmentation, a view strongly held by many people within Australian Government circles during the pre-independence period (Sullivan, 1981). However, despite all these diverse cultural differences, including the contrasting world opinion, the country had successfully survived its first two decades of nationhood without fragmentation and bloodshed. While such an achievement is a credit to all the leaders in the country, both at the political and bureaucratical level, there is however one thing that stands out at the heart of national unity in PNG, notably, education.

3.2 The Underlying Assumptions of Education in PNG

3.2.1 The instrumental view of education

In the light of the above background, it is not surprising to find that education in PNG has been firmly directed towards achieving the national developmental objectives and principles, from the early colonial days up until the present time. Thus, the emphasis placed in education by the political leaders in PNG was not only considered crucial for promoting nationalistic aims, but also seen as a means for achieving and cementing national unity. As in many other countries, the education system in PNG stressed education’s instrumental value in fostering the social goals of living and working together for the common good (Howson, Keitel & Kilpatrick, 1981). This approach to education is underpinned by an instrumental ideology which views all knowledge as neutral and value-free (Julie, 1993). Such emphasis on education by both the previous and current executive governments in PNG has been particularly apparent during the period immediately before independence in 1975, as well as in the last two decades of nationhood, subsequently introducing an element of ‘elitism’ into the school system (Ernest, 1991; Sullivan, 1981).

3.2.2 The Old Humanists educational ideology

If the above emphasis is taken as an indication of the underlying assumption of education in PNG, then it is somewhat seen to be characteristic of the Old Humanist Educational Ideology which furthermore has its roots in the overall Purist Ideologies group (Ernest, 1991). This group, as the name suggests, considers pure knowledge to be worthwhile in its own right. Subsequently, in terms of educational aims, the most central educational goal of the Old Humanists’ position is the transmission of pure knowledge and high culture and its associated values. Accordingly, in terms of educational process, it aims to produce the “liberally educated person, with an appreciation of culture for its own sake, and the discriminatory powers and tastes that accompany it” (Ernest, 1991, p. 170). This logically means that eventually only a minority will achieve this, leading subsequently to educational aims being characterised as being elitist in nature. It has to be however pointed out that, although the educational emphasis in PNG to a greater degree reflects many of the characteristics of the Old Humanists ideology, to some extent there also exist an overlap of these characteristics that seem consistent with the ‘Technological Pragmatist’ and ‘Progressive Educator’ ideologies. This is particularly evident with regard to views of mathematics, theories of learning, theories of teaching mathematics and theories of assessment in mathematics held by the advocates of these two educational ideologies (Ernest, 1991).

3.3 The Underlying Assumptions of Mathematics Education in PNG

3.3.1 Education of a mathematical elite

Mathematics, which is too often viewed as part and parcel of the prescribed curriculum following the Western-style education system, is no exception with regard to the overall underlying assumptions of education in PNG. This is particularly apparent in the context of the underlying educational assumptions of the Old Humanists pursued by the respective PNG governments in the past, notably, towards producing an educated elite to cater for the various skilled occupations both within the public service and in the private sector (Sullivan, 1981). This subsequently led to a considerable emphasis being placed on the selection and education of a mathematical elite (Howson, Keitel & Kilpatrick, 1981) who would eventually participate actively in PNG’s industrial development as well as in the Government bureaucracy. Thus, it was implicitly assumed that the achievement of many of the country’s developmental objectives stated in the Eight Point Plan would not have been possible without the indigenous public servants and bureaucrats having some understanding of mathematical knowledge. Therefore, like education in general, mathematics education in PNG was also seen to be useful in achieving the country’s developmental objectives by many of the leaders, mainly because of its instrumental value in transmitting mathematical knowledge (Sullivan, 1981).

When seen in the above context, it is therefore not uncommon to find that many of the mathematics educators and classroom teachers in particular see school mathematical knowledge, like the discipline itself, to be a pure, hierarchically structured self-subsistent body of objective knowledge (Ernest, 1991). Hence, higher up the hierarchy, mathematics too often becomes increasingly pure, rigorous and abstract, characteristics of an absolutist view of mathematical knowledge. Students on one hand are encouraged to climb up this hierarchy as far as possible, but in practice, achievement of this depends very much on their ‘mathematical ability’ (Ernest, 1991). Such practices are therefore seen to be consistent with the Mathematical Aims and Ideology of the Old Humanists, who contend that:

The goal is to transmit pure mathematics per se, with an emphasis on the structure, conceptual level and rigour of the subject. The aim is to teach mathematics for its intrinsic value, as a central part of the human heritage, culture and intellectual achievement. This entails getting students to appreciate and value the beauty and aesthetic dimension of pure mathematics, through immersion in its study. (Ernest, 1991, p. 176)

A very important branch of this aim is the education of future mathematicians, one of the key features of the elitist mathematics education, with teaching methods dominated by an emphasis on subject knowledge and technical performance (Bishop, 1991b; Ernest, 1991).

3.3.2 The instrumental approach to mathematics education

Before going any further, it is probably worthwhile to point out for the purpose of clarity that the definition of the term instrumental used in this paper, particularly in the context of mathematics education, is the one provided by Skemp (1978, p. 9) which he described as “rules without reasons”. Thus, according to Skemp, the instrumental approach to mathematics education, in direct contrast to relational mathematics education, therefore places strong emphasis on the learning of rules rather than on understanding the underlying conceptual knowledge associated with such rules. This subsequently means that for the ‘instrumental’ mathematics teacher, teaching is based strongly on the ‘established’ rules and procedures rather than on explaining the underlying ideas and concepts associated with these rules. For students of the instrumental approach, all that is required is some kind of rule for getting the right answer. As soon as this is reached, they adhere strongly to it and ignore the rest.

The implication of such an approach to mathematics education as pointed out by Skemp (1978) is that it generates two kinds of mathematical mis-matches. The first one involves learners whose goal is to understand instrumentally, but are taught by a teacher who wants them to understand relationally. The second mis-match is the vice versa of the first one, whereby the goal of the learners is to understand relationally but they are taught by a teacher whose expectation of the learners is based on instrumental understanding. While the first situation will cause fewer problems ‘short-term’ to the students, though it will be frustrating to the teacher, it is however the second one that can be of a more damaging one in the long term (Skemp, 1978). This is because, for a start, it is very difficult, if not impossible, to understand relationally on the basis of instrumental mathematics teaching. This approach is certainly true in the context of mathematics education in PNG, particularly if this author’s own student experiences are taken as any indication. Furthermore, this approach to mathematics education is also one of the key factors that has contributed to the mathematics learning difficulties experienced by many of the Papua New Guinean students. Hence, this notion has subsequently prompted the investigation of the notion of ‘ethnomathematics’ in this literature survey.

3.4 Mathematics Curriculum Development in PNG

3.4.1 The underlying assumption of curriculum development

In an attempt to describe the types of mathematics curriculum development models that are in use today with specific reference to PNG, one has to realise that, irrespective of the diversity of forms of mathematics classrooms around the world, they all have one thing in common; they operate in society and are therefore part and parcel of that society’s educational system. It is because of these rigid societal constraints that mathematics teaching, like any other academic teaching subject, is most often guided by some form of curriculum which incorporates the respective society’s beliefs, values, aims and traditions. Subsequently, there are many factors that operate in a society which can either enhance or constrain any curriculum development activities within the respective educational systems.

In the light of the above background, the overall main emphasis of past and present mathematics curriculum development activities in PNG has therefore reflected the country’s nationalistic aims, subsequently adopting the instrumental approach to mathematics education as described in section 3.3.2. Moreover, because of the ‘purist’ characteristics of the Old Humanists ideology this leads to a restricted view of mathematics and of the uses of the resources appropriate for school mathematics (Ernest, 1991). Thus it is not uncommon for teachers in PNG to resort to heavily relying on textbooks and traditional aids in teaching pure mathematical constructions. This approach has subsequently led to the mathematics curriculum being strongly directed towards the performance of techniques with greater emphasis being placed on the content knowledge of the subject (Bishop, 1991b; Clarkson & Kaleva, 1993).

3.4.2 The curriculum development model in PNG

The ground work for the establishment of the PNG Curriculum Unit, in terms of organisational and institutional structure within the National Department of Education in PNG, significantly took place during the period immediately before independence in 1975, thus paving the way for many of the current curriculum development activities (Clarkson & Kaleva, 1993). Although PNG has a decentralised system of government with many of its functions carried out by a provincial government system, the national government still retains control over certain functions which include, among others, control over the curriculum through the Curriculum Unit. As a result, the dissemination of curriculum materials including that of mathematics have somewhat followed the centre periphery model (Howson, Keitel & Kilpatrick, 1981). In this model, the government not only supports the development of a curriculum but also mandates its use in schools. Thus, a small number of competent specialists combine their expertise to develop new course materials at the centre, which are then made generally available to teachers in the school system. Moreover, of all areas of the curriculum development activities in PNG, mathematics has been also influenced most heavily by the Research, Development and Diffusion (RDD) model, because of the adoption of the curriculum dissemination model noted above (Howson, Keitel & Kilpatrick, 1981; Lancy, 1983).

3.4.3 Organisation and structure of curriculum development

Nearly all the curriculum development activities in PNG are subject centred. Each subject area has a Syllabus Advisory Committee (SAC) whose functions include reviewing and recommending changes to the syllabi as well as helping in the development of other teaching resources. However in the last decade or so, textbook development activities have somewhat become a priority area in curriculum development in PNG, hence the development of other teaching and educational resources has virtually become non-existent (Clarkson & Kaleva, 1993). As for other prescribed teaching subjects, the Secondary Mathematics Syllabus Advisory Committee is made up of representatives from the Mathematics Department of the Curriculum Division, UPNG, a couple of teachers from the secondary schools and a textbook writer. This committee formed the structural framework for the channelling of all curriculum development concerns, suggestions and activities. In practice however, experience has shown that such arrangements allow for the inclusion of little input from the ordinary classroom teachers. Thus, this has been considered to be one of the key factors responsible for the failures of many of the past curriculum implementation programs.

3.4.4 Textbook development

Prior to 1957, the mathematics syllabi and textbooks used in secondary schools in PNG were mainly Australian. Although the efforts to modify the syllabus began in the later part of that year, it was not until 1975 that the development of PNG secondary textbooks commenced with the recruitment of a UNESCO curriculum specialist, namely Murray Britt (Clarkson & Kaleva, 1993). His first task not only involved the writing of syllabi for grades 7 to 10 but also writing PNG text material as a result of the interest shown by the teachers for PNG’s own mathematics textbook. The production of the first two series of booklets somewhat formed the basis for the publication of the grade 7 and 8 textbook series Mathematics Our Way (MOW) from 1980 to 1982. Britt was also responsible for drafting the grades 9 and 10 textbooks. However, their actual trial, modification and subsequent publication became the responsibility of the SAC and the Curriculum Unit after his departure in 1981(Clarkson & Kaleva, 1993).

The key feature of the MOW textbooks was that they involved an activity based approach to mathematics. Thus, they included the use of pictures, symbols and practical activities. While such an approach was becoming common elsewhere at the time, these textbooks were however rejected by a number of teachers on the grounds that they used unfamiliar and untried methodologies, activities that were time consuming, appeared too simple and did not have enough exercises. According to Clarkson and Kaleva (1993), these reasons seem to be superficial, claiming that factors which actually hampered their acceptance included the slowness of production, the conservativeness of the teaching force and more importantly, the lack of inservice training in the use of the textbooks.

With Bryn Roberts replacing Britt in 1983, a new phase of textbook development took place with the writing of the Grades 4 to 6 texts being his first task. It was also during this time that the secondary school mathematics syllabus was reviewed from 1985 to 1986. This subsequently paved the way for the adaptation and eventual publication of the currently used Secondary Schools Mathematics textbook series for grades 7 to 10. These textbooks are based on the Australian series published by the Shakespeare Headpress, namely, the Modern Mathematics series. By 1991, the grade 7 and 8 textbooks were being used in schools, with the grade 9 and 10 textbooks following in 1992 and 1993 respectively. The emphasis of this set of textbooks is on problem solving, practical activities and investigative methods, that is, an activity oriented approach. To some extent, this approach is based on the assumption that equipping students with problem solving skills will in some way contribute to the notion of integral human development, the underlying emphasis of the current philosophy of education for PNG noted in chapter 2 (Clarkson & Kaleva, 1993).

3.4.5 The current ‘technique-oriented’ (TO) curriculum

As noted in sections 3.2 and 3.3, the first major area of concern is the existing PNG mathematics curriculum which is strongly directed towards the performance of techniques, owing much of these to its underlying educational assumption of the instrumental approach based on the Old Humanist ideology. This subsequently leads to arithmetical computation being firmly entrenched as the basis of much of the mathematics curriculum, “with the ‘four rules’ gradually being developed to handle more and more complicated ‘numbers’ - natural, integer, fractions, decimals, complex and, later, matrices and vectors” (Bishop, 1991b, p. 7). As a result, algebraic work is seen as developing the skills of solving more and more complicated equations and of rearranging complicated expressions so that they can be solved. Geometry for that matter is also developed as an area to which one is able to apply arithmetical and algebraic techniques, be it in trigonometry or co-ordinate geometry. For those who have succeeded at, or survived these tests, there is further skill development that involves calculus with its vast number of integrals and differential equations waiting to be recognised, classified, and of course solved (Bishop, 1991b).

The above description concerning the current situation in mathematics teaching is not uncommon in many of the mathematics classrooms that follow the ‘Western-style’ education system having inherited the type of curriculum that is characterised by what Bishop (1991b, p. 7) calls the “technique curriculum”. This curriculum, which subsequently portrays mathematics as a ‘doing’ subject therefore consists of procedures, methods, skills, rules and algorithms resulting in a situation well-expressed by Bishop (1991b, p. 7) who further notes that:

Mathematics is therefore not portrayed as a reflective subject. It is not a way of knowing. ... within this curriculum, ... [there] is a limited and constrained type of thinking, related to adopting the appropriate procedure, using the correct method of solution, following the rules and obtaining the correct answer. It is therefore a curriculum in which ‘practice makes perfect’ with examples to be emulated, and exercises to be carried out.

The most notable feature of this curriculum is that it is a ‘users’ curriculum and therefore it aims at developing a comprehensive as well as wide-ranging ‘tool-kit’ for the user. While on the one hand, the aim is for the learner to be able to use these techniques both inside and outside mathematics, on the other hand, ‘development’ in terms of this curriculum is somewhat seen as the mastering of ever more complex and wide-ranging set of techniques. This therefore logically leads to the notion of ‘mastery’ which is somewhat becoming synonymous with the evaluation process in the technique-oriented curriculum (Abraham & Bibby, 1988; Bishop, 1991b).

3.5 Current Mathematics Classroom Practices in PNG

3.5.1 Theoretical assumptions of classroom practices

As described in the earlier sections of this chapter, many of the mathematics classroom practices adopted by mathematics educators in PNG, particularly the ordinary classroom teachers, are heavily influenced by the overall emphasis of the instrumental approach to education advocated by many of the country’s leaders. In addition, these classroom practices are also influenced by the underlying theoretical assumptions of the Old Humanists educational ideology. Accordingly, mathematics is therefore viewed as a body of pure objective knowledge, based on reason and logic. Subsequently, it is seen as a logically structured body of knowledge, leading to the view that it is hierarchical in nature, involving “a system of rigour, purity and beauty, and hence neutral and value-free” (Ernest, 1991, p. 169). When seen in this context, it is evident that many of the mathematics teaching practices in PNG are highly structured, particularly if the author’s experiences are taken as any indication. Thus, it is not uncommon to find many of the mathematics lessons employing the expository methods of teaching characteristic of the instrumental approach to Mathematics education noted in section 3.3.2.

3.5.2 Expository teaching method

As one would expect, the predominant mathematics classroom practice in terms of teaching methods used in many of the mathematics lessons in PNG have too often followed the most common sequence of lecture-example-exercises pattern (Clarkson & Kaleva, 1993). Such a method, though not only limited to PNG, is however viewed by many educators to be one of the common characteristics of the expository teaching method employed by many of the schooling systems around the world. Such a teaching method is particularly more apparent within the school systems which strongly emphasise skill acquisition rather than conceptual understanding of mathematical knowledge as the main focus of mathematics education. Furthermore, it is also consistent from the perspective of the Old Humanists theory of teaching mathematics in that the teacher’s role is viewed as that of lecturer and explainer, communicating the structure of mathematics meaningfully to his/her students (Ernest, 1991). In other words, the assumption according to this view is that the teacher is somewhat implicitly expected to inspire students through an exciting delivery which should enrich the mathematics course through additional problems and activities similar to a structured textbook approach. Such an approach is particularly evident in mathematics teaching in PNG where many of the mathematics classroom activities are seen to be not only textbook based but are done at the expense of various other productive teaching approaches (Clarkson & Kaleva, 1993).

3.5.3 Impersonal Learning

As noted in section 3.3, because of the instrumental approach to education based on the Old Humanist ideology, many of the learning experiences provided in the mathematics classrooms in PNG are most often characterised by what Bishop (1991b) correctly described as “impersonal learning” (p. 9). The key feature of this type of learning is that the task of the learner is conceived of as being independent of the person of the learner. In other words, the main emphasis of ‘impersonal learning’ is on the learner learning mathematics and not on the learner striving to find some personal meanings from mathematical education. This is not to say that teachers are entirely responsible for such learning, because the whole system of mathematics education starting from syllabuses, examinations, textbooks, teacher training and even research are firmly entrenched in this view of learning, eventually resulting in classroom practices being dominated by the strong emphasis on subject knowledge and technique performance (Ascher & D’Ambrosio, 1994; Bishop, 1991b, Ernest, 1991).

It has to be also noted that there is a strong connection between the impersonal learning and the technique-oriented curriculum described in section 3.4.5 in that, because the emphasis is on getting only the right answers, the technique curriculum does not provide the necessary opportunity for students’ personal interpretation and invention (Ascher & D’Ambrosio, 1994; Bishop, 1991b). This subsequently leads to many mathematics classroom practices in PNG being characterised by the learning of rules, the acceptance of established procedures, and eventually skills practised through a vast number of exercises, starting from the simple to the most difficult ones. This logically means that there is very little consideration (if any) given to the learner’s personality and concerns, because of the pre-conceived assumption that mathematical ideas and knowledge are still the same the world over, irrespective of the learner’s personal characteristics, experiences and the cultural background. As Bishop (1991b, p. 9) further notes:

It doesn’t matter what sort of person the learner is, the mathematical result is the same. It doesn’t matter ultimately if you are a visualiser or someone who prefers analysing the logic of the situation, because will still equal . It doesn’t matter what the learner brings to the situation, as long as they take away the same thing.

The above situation implies that there is no need for discussion, no need for ‘views’ and ‘opinions’ to be expressed and therefore no real need to provide opportunities for personal conversations to take place between the teacher and learners. Any questions asked demand a specific pre-determined answer (usually known by the teacher), problems in the textbook demand certain kinds of solution (too often shown in the text). Any slight deviation from these procedures are most often regarded as unacceptable for the simple reason that they do not conform to the established ‘rules’ of the ‘game’. Under this assumption, it is expected that anyone who has been taught ‘properly’ would surely be expected to know the kind of answer that is acceptable.

Interestingly, there is also another aspect of impersonal learning which seem to be appealing to many learners, be they children or adults. This relates to the notion of security associated with right answers and correct procedures, which is too often seen as the strength of mathematics itself. In this context, mathematical truth is perceived as being both geographically and personally independent, and as such, it can be verified (in theory) by anyone (Bishop, 1991b).

3.5.4 Personal mathematical meanings

On the other hand this author is in total agreement with Bishop (1991b) and Ascher and D’Ambrosio (1994), who argued that learning these mathematical truths does not in itself constitute an adequate mathematics education. In other words, there is no reason to assume that mathematics education should be the same everywhere and for everyone, just because mathematical truths seem to hold everywhere. Even if they are, that does not mean that the individuality of the learner, nor the social and cultural context of education should be ignored by mathematics education. In this respect, mathematical education needs to do more than merely inform learners of these truths.

Furthermore, while this presentation acknowledges that there is an ‘agreed’ aspect of mathematical learning in terms of shared meanings of mathematical truth, on the other hand however, there is an equally important personal side associated with those meanings as well, which are nearly always ignored by the current emphasis of the learning process. As Bishop (1991b, p. 10) correctly described:

Meaning is about connections we make between ideas, and only some of those connections will be agreed, shared, ‘official’, mathematical connections and meanings. Others will be personal connections, of imagery and metaphor, of examples from home or of other experiences, of significant events from learning other subjects, or associations with other people. We all construct personal meanings for ourselves, which give significance to our lives.

It is therefore evident from the above discussions that impersonal mathematics learning totally ignores these connections and the personal meanings, and by doing so “depersonalises the learning process. ‘No personal meanings’ means that no ‘persons’ are in this mathematics classrooms; what you have is a teacher of mathematics and several learners of mathematics” (Bishop, 1991b, p.10). Subsequently, the task of the teacher is seen as that of communicating ‘the mathematics’ as effectively and as efficiently as possible, so that the learners can learn ‘the mathematics’. Thus, ‘the mathematics’ eventually becomes an impersonal object to be transmitted in a one-way communication (Bishop, 1991b). Furthermore, as a consequence of the impersonal learning, teacher’s personal views and meanings are also seen to be irrelevant because of the pre-conceived assumption that they will only become barriers to what is already considered a universal mathematical knowledge independent of human social activity (Abraham & Bibby, 1988; Ernest, 1991). As a result, they are rarely allowed to express personal feelings, personal intuitions, personal meanings and personal interpretations (Bishop, 1991b).

3.5.5 Text-based Teaching

The text-based teaching is also one of the major area of concern within the teaching-learning process and logically follows on from the previous section, that is, from impersonal learning to impersonal teaching characterised by what Bishop (1991b, p. 10) called “text teaching”. As in many other countries, mathematics classrooms in PNG also bear witness to this type of teaching whereby teacher teaching is viewed to be synonymous with textbook teaching such that it is usually rare to find any mathematics teacher teaching without the textbook. As noted in section 3.4.4, although on the one hand, the emphasis of the current secondary school textbooks in PNG is on problem solving, practical activities and investigative methods, on the other hand however, just how and to what extent this activity approach to mathematics teaching is implemented in the actual classroom situation is anybody’s good guess (Clarkson & Kaleva, 1993). Moreover, the debate on who should control the textbook is also not an unfamiliar one. In practice, this however means that the actual teaching-learning process in the classroom is very much limited and inflexible. In other words, if the teachers are to be held accountable for what goes on in the classroom, then they must not at the same time be controlled by the textbook. On the other hand, they must be helped and supported with materials and activities over which they have control, in order that they can help their students (learners) to be successful, since there is nobody else beside the teacher who is in the better position to judge both the failures and successes of the learners within the classroom context (Bishop, 1991b).

3.5.6 Assessment practices in mathematics education

As noted in chapter 2, since many of the educational practices in PNG can be traced back to the Australian system, many of its assessment practices are therefore based on the use of formal assessment procedures entrenched in written examinations (Clarkson & Kaleva, 1991). Furthermore, in keeping with the Old Humanist ideology, it is also not uncommon to find in PNG two predominant formal assessment practices being used extensively across the whole curriculum namely, formative and summative (Ernest, 1991). Formative assessment, while it may vary from one school to another and involve a range of methods during the teaching of specific units or topics, is however used mainly by classroom teachers. Summative assessment on the other hand usually involves external examinations set by the central agency (Ernest, 1991). In the case of PNG, the external examinations in the core subjects (i.e. English, Mathematics, Science, Social Science) are administered by the Measurement Services Unit (MSU) of the National Department of Education (NDOE) which is also responsible for both the writing and trialing of examination items and the certification of students at the completion of each level of education. This occurs at the end of grade 6 in community (primary) schools, grade 10 in provincial high schools and grade 12 in national high schools. Prior to 1977, the then two national high schools ran their own examinations. In 1978 however, the first common mathematics examinations for year 12 was produced and administered by the Mathematics Education Centre at the PNG University of Technology. This task was then taken over by the Mathematics Department of the University of Technology in 1985 with the result that, in recent years a pattern of setting these examinations is shared between the two universities, namely UPNG and PNGUT (Clarkson & Kaleva, 1993).

It has to be pointed out that assessment in mathematics in particular, because of the Old Humanists ideology, is determined by a hierarchical view of the subject matter and students’ mathematical ‘ability’ (Ernest, 1991). Thus, while many of these examinations are found to be very difficult by the majority of the learners, they however often fulfil the objective of being very selective in identifying those learners who wish to further pursue pure, abstract and rigorous mathematics. This is true in the case of PNG where only a small number of students continue to take further mathematics at the University level, particularly if the author’s experiences and observations are any indication.

3.6 The educational implications of current practices

3.6.1 A critique of mathematics education practices

The current practices of mathematics education in PNG have far reaching consequences of negative educational outcomes affecting all facets of the teaching-learning process, starting from curriculum development to teacher education activities. In addition, they are also seen to be deep-rooted in the minds of many of the mathematics educators and teachers in PNG who still want to maintain the traditional absolutist view of mathematics. To some extent this is understandable, particularly when these practices have not only become the ‘norm’ during the teaching of many mathematics lessons in PNG, but are also seen to be successful in achieving the objectives of these highly structured lessons.

Contrary to the above conservative reasons and sentiments, there is ample research evidence to date which not only shows that much mathematical misunderstanding has occurred as a result of the kind of teaching practices that have been adopted, but also show how limited that understanding is, even in situations where it is considered ‘correct’ (Bishop, 1991b). Such situations as these unfortunately result in individuals “rejecting mathematics, fearing it, disliking it and, if they continue to study it (which they don’t), resorting to rote and instrumental methods to cope with the examination-oriented demands” (Bishop, 1991b, p. 2). This is particularly true in the case of many young people in PNG who do not succeed in mathematics, thereby seeing it as being difficult, mysterious, meaningless and boring (Bishop, 1991b; Flaherty, 1990). This claim is also supported by evidence from other research studies (e.g. Adetula, 1990; Clarkson, 1992; Dave, 1983; Lean, Clements & del Campo, 1990; Souviney, 1983) citing language, culture and learning modality as the three key areas contributing to greater mathematics learning difficulties experienced by students from diverse cultural backgrounds. Such experiences are particularly apparent in educational systems where mathematics teaching is done through a second language other than the learners’ own native language.

Furthermore, the above situation is not only created by the types of teaching approaches adopted, but also as a result of the overall underlying assumptions of education based on the Old Humanists ideology. In this regard, it is therefore open to criticisms in three areas. According to Ernest (1991), the first criticism relates to the purist-absolutist view of mathematics which denies the connection between pure mathematics and its applications, thereby arguing that: “To view mathematics as a pure entity, divorced from the base shadow of its applications is to subscribe to a dangerous, unsustainable myth” (p. 179). The second criticism relates to the hierarchical academic structure of the subject and its associated elitist position which denies that mathematics has any involvement in, or responsibility for, broader social issues. In this respect, it is therefore further argued by Ernest (1991, p. 179) as being “morally irresponsible and incorrect”. The third criticism of the Old Humanists ideology is directed at the strongly held assumption of a fixed view of human ability, related to a stratified and elitist view of society and human nature at large. Consequently, this assumption is therefore seen to be in direct conflict with the now widely accepted view that, apart from whatever part inheritance plays in human and mathematical ability, environmental influences have a major impact on its realisation (Bishop, 1985; Ernest, 1991; Stigler & Baranes, 1988).

3.6.2 Negative educational outcomes

The weaknesses of current practices of mathematics education in PNG, and its underlying Old Humanists educational ideology, have grave consequences for education in three key areas (Ernest, 1991). Firstly as noted in section 3.4.5 there are problems stemming from the ‘top-down’ view of the current ‘technique-oriented’ mathematics curriculum, which sees the primary function of teaching mathematics as that of preparing students for higher level educational institutions, notably universities. One of the unacceptable implications of such a curriculum is that too often it is inappropriate and designed without either the needs or interests of majority of the students in mind, particularly their social and cultural environment.

A second consequence of this perspective is that, since mathematics is presented to learners as an objective, external, cold, hard and remote knowledge, it leads to a powerfully negative effect on attitudes and affective responses to mathematics by the learners. In particular, the view of mathematics as being separate from the societal concerns is considered to be largely responsible for females’ negative attitudes to mathematics, thereby leading to their subsequent underparticipation (Ernest, 1991; Flaherty, 1990).

Thirdly, the assumption that mathematical ability is determined by inheritance is quite often very damaging for those not labelled as mathematically gifted. In other words, this assumption can be taken to mean that mathematics has no concern for human values and feelings, thereby leading to what can best be described as a ‘self-fulfilling’ prophecy (Bishop, 1991b; Ernest, 1991). At best, it leads to a reduced level of attainment by those learners labelled as low-ability, thus damaging their mathematical achievement.

3.7 Conclusion

In summary, if mathematics education in PNG is about helping young Papua New Guineans to relate better to their environment, then it is clearly failing in this task. This situation is a direct result of the type of teaching practices adopted by mathematics educators in PNG which are based on the Old Humanist educational ideology. This ideology, because of its link with the overall Purist Ideologies group, views mathematics as being a pure, rigorous, abstract and hierarchically structured objective knowledge. In terms of the teaching-learning process, this view logically leads to emphasis being placed on the content knowledge of the subject. Thus, it is not uncommon to find many mathematics classrooms in PNG employing expository teaching methods that involve performance of procedures, methods, skills, rules and algorithms. On the one hand, such teaching methods are seen to be somewhat consistent with the underlying assumptions of PNG’s national developmental objectives. On the other hand however, it portrays mathematics as a non-reflective subject with many learners viewing it as being difficult, meaningless and boring. This most unacceptable situation is a direct result of the current approaches to mathematics education which also view mathematics as being independent of the learners’ social and cultural environment because of its strong links with the absolutist view of mathematical knowledge. Thus, if the above situation is of any significance, it suggests that mathematics education in PNG is not fulfilling its overall societal obligations in producing a mathematically literate society.

Furthermore, in the view of recent research studies of mathematics learning difficulties by students from diverse cultural backgrounds, citing language, culture and learning modality as the three key areas is to be taken as any indication, it strongly suggests that the current practices are also not relevant within the context of PNG situation. In this respect they are therefore seen to be directly contradicting the overall underlying principle of the current philosophy of education for PNG noted in chapter 2 namely, Integral Human Development. In particular, the fact that the current philosophy of education strongly emphasises all aspects of the person as an important and integral part of any educational process strongly suggests that the current practices of mathematics education in PNG are not only irrelevant, but are also detrimental in achieving the above fundamental educational assumption. In other words, because the current approach to mathematics education in PNG ignores the social and cultural aspects of the learners as stipulated in the current philosophy of education, the current mathematics education practices starting from curriculum to teacher education via the teaching-learning process need to be critically re-examined.

It is both in the light of the above background and in contrast to the underlying educational assumptions of the current practices that the next three chapters are devoted to considering an alternative approach to Mathematics education in PNG, notably, the socio-cultural perspective of mathematics education, developing from the fact that mathematics is a cultural phenomenon (Bishop, 1991b) through the notion of ethnomathematics (D’Ambrosio, 1990a). The main tenet of this approach is the argument that since mathematics is a socially constructed knowledge resulting from human activities, it is an educational injustice not to significantly represent that knowledge in the formal educational process in PNG. In this context, the proposed ‘Ethnomathematical Approach’ to mathematics education in PNG can be considered relevant in that it envisages the adoption of the underlying educational assumptions of the current Philosophy of Education for PNG as its theoretical base.