CHAPTER TWO : LITERATURE REVIEW.

 

Introduction.

In chapter one, it was noted in the “problematique” that there was a mismatch between PNG government policies which encouraged “a community based and culturally oriented curriculum”, and the existing mathematics curriculum. It would seem that the official mathematics curriculum (content), what the teachers teach in the classrooms and what the students learn, do not fit into the category of a community based or a culturally oriented curriculum. It is of interest to investigate why there is this mismatch but especially to focus on one of the principal players in the education process - the teacher. It is for these reasons that the literature focuses on three main areas of research : mathematics in a socio-cultural context, teacher beliefs and practice, educational policy and curriculum practice.

This chapter reviews the three areas of research which are important to this study. In section 2.1, the literature review focuses on research into mathematics in a social and cultural context, in and out-of-school mathematics and the ideas associated with “ethnomathematics”. Section 2.2 examines studies on teacher beliefs about mathematics and how these beliefs relate to instructional practice including studies which investigated teacher practices in Papua New Guinea. In the third section  (section 2.3), a review is made of research literature on educational policy and curriculum practice. Particular emphasis is given to reports on the International studies (IEA studies) which focus on factors influencing student mathematical achievement. These studies used the “intended, implemented and attained” curriculum framework to investigate these factors. A report on PNG studies on the mathematics curriculum is also given in this section. In the last two sections of the chapter (sections 2.4 and 2.5), the problem is restated and the research questions are proposed.

2.1 Mathematics in Socio-cultural contexts.

Mathematics was for a long time regarded as “neutral - culture free, societally free and value free” (Bishop, 1993 : p.11; D’ Ambrosio, 1991a: p.15). It was always taught in schools as a  “culture free” subject which involved learning supposedly universally accepted facts and concepts. This view of mathematics was so predominant that for many this was the only conception about the nature of mathematics (ie. it was thought to consist of a body of knowledge of facts, algorithms, axioms and theorems) and this perception was reinforced by the learners experience of the way the subject was taught in schools. Dossey  (1992) quotes Cooney (1987) and Brown, Cooney & Jones (1990) who suggest that the “teachers’ view of mathematics is transmitted to the students and helps shape their views about the nature of mathematics” (p.39). Even though the universality of mathematical truths is not questioned, it is only in this decade that the view of mathematics as culture free has been challenged (Bishop, 1988a : p.180).

In the introduction to the UNESCO publication, “Significant influences on children’s learning of mathematics” Bishop, Hart, Lerman & Nunes (1993 : p.1) makes the bold statement that “there is no sense in regarding mathematics learning as abstract and culture free ... the learner cannot be abstract and context free (ie. free of societal influence)”. Recent studies (eg. Nunes, 1992; Carraher, 1991; Harris, 1991, Gerdes, 1994a) which looked at mathematics in a variety of contexts seem to confirm the above.  It is worth noting that the contextualisation of mathematics has been described in two ways (although they are not seen as distinct at times); use of mathematics in context (eg. mathematics in work places) and the identification of mathematics in context (eg. mathematical activities in culture). Dowling (1991 : pp. 93-120) refers to these as “utilitarianism and mathematical anthropology”Bishop (1992a : p.2), however, identifies three foci of research on “ethnomathematics” . They are :

  1. mathematical knowledge in traditional societies (anthropology),
  2. mathematical developments in non western cultures (history),
  3. mathematical knowledge of different groups in society (social psychology).

Section 2.1 reports on research which focus on aspects of mathematics under these three categories.

2.1.1 Social context.

This section begins by considering the mathematical knowledge of different groups in societies or mathematics in a social context. The learning of mathematics has always been associated with the schooling process. That is, it was thought that mathematical concepts and skills were acquired only if one went to school. However, analysis of children’s mathematical knowledge has led researchers to conclude that mathematical knowledge is also acquired outside the structured systems of mathematics learning such as schools (Nunes, 1992). Mathematics in a social context refers to the use of mathematical skills outside of schools and the acquisition of mathematical skills other than from schools. For example, in the context of every-day activities outside of school (Carraher, 1991), at work (Harris,  1991), in the street (eg. street mathematics - informally learned mathematics; Nunes, 1992), folk mathematics (Maier, 1991).

Mathematics in the work place.

Dowling (1991) refers to the use of mathematics in the context of  the workplace and the notion of mathematics being a set of tools as “utilitarianism”. He cites research by Sewell (1981), University of Bath (1981) which investigated how mathematics is used in everyday practices and working life (the utilitarian activities). A part of the Bath study listed the mathematics used in various occupations. According to Dowling, the “studies only demonstrated the discontinuity in the practices they observed”. The research showed that everyday and work practitioners did not recognise they were doing mathematics. He argues that it is no longer valid to regard mathematics as a set of tools (utilitarian) if the working practitioners do not see it as such. Bishop (1993) reinforces this idea by stating that the utility argument can no longer be valid because computers and calculators have taken over previously needed skills. On another aspect, “mathematical anthropology”, or the identification of activity as mathematical in diverse practices, Dowling asserts the researchers tend to define cultural activities in their own terms (based on their own mathematical experiences). For example, in the case of Gerdes (eg. 1988), he defines Mozambique basket weaving in terms of school mathematics.

2.1.2 In and out -of- school mathematics.

There is evidence of the existence of out-of-school mathematics. That is, mathematics that is different to school mathematics and which is not necessarily learnt in school (Carraher, 1991; Nunes 1992). According to Carraher, that mathematics exists out-of-school is shown by the fact that children develop understanding of numbers before they come to school (She quotes Piaget, 1952; Gelman & Gallaistic, 1978; Hughes, 1986) and unschooled adults perform calculations at work (cites Scribner, 1984; Carraher, Schlieman & Carraher, 1988). Bishop (1993) adapts Coombs’ (1985) definitions of non-formal mathematics education (NFME) as “an organised, systematic, mathematics education activity carried on outside the framework of the formal system” (Bishop, 1993 : p.15). NFME activities include adult numeracy, special courses for gifted children, television programs and vocational training courses. Bishop contrasts this with formal mathematics education (FME) and informal mathematics education (IFME) which is mathematical knowledge that is gained informally from, for example, adults in traditional societies or from the media in industrialised societies (Bishop, 1993 : pp.15-17).

As a follow-up to a study which investigated school failure, the study of young street vendors in the North-east of Brazil was instigated to find out something about their knowledge of “street mathematics” - street algorithms as compared to normal school computations. They found that there were differences in the success rates across the two settings. The vendors were more successful in correctly solving street setting and verbal problems but were not so successful in solving straight out computation problems. The procedures for solution were also different from those taught at school. Further studies showed the same results. Carraher (1991) suggests that this shows that important mathematical concepts seem to develop outside of school without specific instructions : “The concepts and procedures would appear to arise through an individual’s social interactions in everyday activities such as commerce and production of goods” (p.183). Based on research with Brazilian vendors/ American adults, Lave  (1988) concluded that  “mathematics used outside is a process of modeling rather a mere process of manipulation of numbers” (p.30).

Studies also reveal the existence of mathematics in everyday activities. Nunes (1992 : p.570) gives examples of studies which show that everyday activities such as building houses, exchanging money, weighing products, and calculating proportions for a recipe involve numbers, calculations and precise geometrical patterns. Nunes cites examples of research into three mathematical concepts (chosen because they are similar to those encountered by pupils at the elementary level) - numeration and measurement systems, problem solving and computation, inversion and modeling - to illustrate mathematical differences and invariance embedded in everyday situations.  For example, Nunes quoted studies (by Saxe, 1982; Carraher and Schleiman, 1990; Carraher, 1985,1989) on counting and measuring in everyday activities, which :

“…indicate that subjects can re-invent the concept of grouping into units for counting in ways appropriate to their activity. Further, this ability is independent of the base structure in the particular numeration system. Everyday practices also create opportunities for individuals to make inferences typical of classroom mathematical activities.” (Nunes, 1992 : p.561).

Although some of these concepts were acquired without schooling, schooling did accelerate the learning of these concepts (in particular, inverse proportion word problems).

Differences between in and out- of- school mathematics.

According to Carraher (1991: pp.170-171), the key question is not whether mathematics is learnt out of school but how the nature of out-of-school mathematics is different from that of school mathematics. She warns about the false dichotomies of “out-of-school” and “in school” mathematics because in reality there is no clear distinction and there are areas of overlap.

The applications of mathematics outside of the school differ from applications shown in school mathematics. Several writers highlight these differences between “school mathematics” and “out-of school mathematics”. For example, references are made to academic mathematics, formal mathematics, abstract mathematics versus ethnomathematics (see Taylor in Julie, Angelis & Davis, 1993; Ernest 1991). Hoyles (in Harris & Evans, 1991 : p.129) provides a description of  “informal” mathematics and school mathematics (see Table 2.1 below).

Table 2.1 Informal v/s school mathematics (Hoyles in Harris & Evans, 1991 : p.129).

 Informal mathematics.

School mathematics.

embedded in task

decontextualised

motivation is functional

motivation is intrinsic

objects of activity are concrete

objects of activity are abstract

processes are not explicit

processes are named and are the object of study

data is ill defined and  noisy

data is well defined and is presented tidily

tasks are particularistic

tasks are aimed at generalisation

accuracy is defined by situation

accuracy is assumed or given

numbers are messy

numbers arranged to work out well

work is collaborative, social

work is individualistic

language is imprecise and responsive to setting

language is precise and carefully differentiated

correctness is negotiable

answers are right or wrong

According to Maier (1991), mathematics is learnt out of school in different forms. He uses the term “folk mathematics” when making reference to the way “ordinary folks” handle mathematics related problems arising in everyday life and explains how and why it is done differently. Maier suggests that some differences are that school mathematics consists of exercises unrelated to anything outside of school; what is learnt inside does not apply outside; school mathematics uses pencil and paper while folk mathematics uses mental computation and estimations or algorithms that lend themselves to mental use and school mathematics has pre-formulated problems and contains pre-requisite data. The problems themselves differ in nature. Reference by Nunes (in Bishop, 1993) has already been made to the purported  differences in the social organisation of street and school mathematics and the similarities in their logico-mathematical structures. Nunes points out that the “important difference is that mathematics used outside school is a tool in the service of some broader goal ... and that the situation in which mathematics is used outside of school gives it a meaning , making mathematics ... a process of modeling rather than a mere process of manipulation of numbers” (p.30). There are also differences in methods of solution (eg. as in proportion - ratio problems).

2.1.3 Cultural context.

In this section a review is made of the literature about mathematical knowledge found in traditional societies (anthropological studies). The applications of mathematics in a variety of contexts is not a new concept. In fact, from the late 1970’s the emphasis of mathematics curricula was to show how mathematics was used in a variety of contexts in working and everyday life (Dowling, 1991 : p.93). What is new is the concept that mathematics can be identified in cultural activities in traditional societies.

Mathematical Practices and ideas.

“Mathematical practices” such as  counting, ordering, measuring, inferring etc. have been identified (by anthropologists) in socio-cultural groups (Howson & Wilson, 1986; Gilmer, 1988 in Vithal, 1992; Ascher  1991; D’Ambrosio, 1985). D’Ambrosio (1991a: p.20) states that “practices such as ciphering and counting, measuring, classifying, ordering, inferring, modeling ... constitute ethnomathematics”.

Gerdes(1988) uses Mozambique weaving to illustrate what he calls “frozen” mathematics and geometric thinking. He argues that the weavers already engage in complex mathematical thinking although they may not know they are doing mathematics (p.140). Incorporation of mathematical traditions into curriculum will help to get rid of psychological blockade... uncover hidden mathematics in geometric forms and patterns of traditional object like baskets, mats, pots, houses, fish traps by asking why they possess these forms (p.141). Gerdes (1990) also gives  examples of mathematics in traditional designs/drawings (“sona”) of the Tchokwe. Gerdes believes  “identifying” mathematical knowledge frozen in African production activities will lead to cultural confidence. Further examples of mathematical ideas existing in indigenous African cultures are shown by Zaslavsky  (1973).

Ascher(1991) suggests that mathematical ideas are embedded in cultural activities and that the emphasis of fundamental activities may differ between cultures. Ascher gives examples of  “mathematical ideas” which involve number, logic, spatial configuration, and suggests any combination of these can form structures or systems (p.185). Ascher gives examples from different cultures which include Bushong sand figures and Malekula sand tracing which illustrate graph theory ; Inca strip patterns and Maori rafter patterns - geometry ; Iroquoi games - chance; Maori games - strategy. The appendices to the conference papers edited by Barton and Fairhall (1995) give examples of mathematics in Maori culture (eg. Maori designs, patterns and carvings). Mathematics from many cultures around the world is also the subject of a book by Irons, Burnett, & Foon (1994).

These mathematical ideas are sometimes referred to in literature as mathematical traditions, indigenous mathematics or traditional mathematical thought (see Gerdes, 1988; Mtetwa & Jaji in Barton  1992b : pp.5-16). Barton (1992b : p.4) suggests that “ethnomathematical concepts” exist in socio-cultural groups but that conceptions can be overridden by new conceptions. He makes a distinction between ethnomathematical conceptions and mathematical practices :

“... mathematical conceptions have very broad areas of applicability, are very generalised and can therefore explain a wide range of activities ... Mathematical practices embody implicitly the mathematical conceptions. It takes the interplay of many practices to carry the full impact of the conceptions behind them. This explains why mathematics of minor cultures become subsumed or colonised. The mathematical practice with the wider range of applicability will accommodate different practices more readily than the minority will accommodate its opposing practices.” (p.5)

Borba (1990 : p.40) asserts that every culture has mathematics. This assertion is supported by Bishop (1988a : pp.180-181; 1991 : pp.30-31) who states that mathematical ideas are found in all cultures (also supported by Ascher, 1991). Bishop (1988a : p.182) lists six fundamental mathematical activities which he suggests are common to all cultural groups and which contributed to the following ideas. They are :

  1. Counting - eg. number systems ,algebraic representation, probabilities
  2. Locating - eg. orientation, coordinates, bearings, angles, loci
  3. Measuring - eg. comparing, ordering, measurements, approximations
  4. Designing - eg. projections of objects/shapes, geometric shapes, ratio
  5. Playing - eg. puzzles, paradoxes, models, games, hypothetical reasoning
  6. Explaining - eg. classification, conventions, generalisations, symbolic explanations

It is asserted that from these basic notions the “western mathematical knowledge” can be derived.

Several writers make reference to the fact that mathematics has a cultural history (eg. Bishop, 1988a : p.180; D’Ambrosio, 1991a : pp.5-7; 1991b : pp. 369-377; Barton, 1992a : p.3). Ernest (1992 : p.100) refers to it as the social origins of mathematics. D’Ambrosio (1991b) specifically calls this “western history”, as opposed to the history of other cultures. D’Ambrosio (1991a) states that establishing a bridge between anthropologists and historians of culture and mathematics is a step towards recognising that different modes of thought can lead to different forms of mathematics. The cultural history of mathematics shows how early philosophies of what “ideal” education was and who should have access to this education influenced what mathematics should be taught. According to D’Ambrosio there were two kinds of mathematics - practical mathematics for the working masses and scholarly mathematics for a selected few (D’Ambrosio 1991b). Over time these two “branches” of mathematics evolved and developed to become “scholarly practical” which is known today as “academic mathematics” (mathematics taught and learnt in schools). He contrasts this with ethnomathematics as the mathematics that is practiced among identifiable cultural groups.

A look at the history of mathematics shows that a lot of ideas and concepts we know today arose out of the need to solve practical problems associated with cultural activities. For example, the Egyptians of the river Nile used knots on ropes to divide or survey land - the same principle which is known as the Pythagorean theorem on the 3,4,5 right angled triangle (Eves, 1990). So called “western” mathematics was developed from real world problems and has its origins in ancient Egypt, Greece and India etc. Mathematics at its beginning was not something which was organised, logical and well set out as in textbooks. Bidwell (1990) point out that the early calculus ideas of Leibniz and Newton were nothing like the refined calculus that is studied today. What we meet in schools today is mathematics that is organised, refined or “codified, institutionalised mathematics”.

PNG studies on mathematics in a cultural context.

PNG studies on mathematics in a cultural context focus on “school mathematics” and the effect of the cultural factors on learning school mathematics. Examples of these studies are discussed in section 2.3.4 - PNG studies on the curriculum. In this section, we report on studies which focused on identifying the “cultural mathematics - CM” or the acquisition of CM in a traditional cultural setting.

Lean’s (1986, 1994) extensive research into the counting systems in PNG (about 65 % of the 800 or so language groups were analysed) revealed the diversity of the counting systems. For example, cycles (or bases) range from two to sixty eight (as in body part counting systems, although there were identifiable semi-cycles). He was also able to study the indigenous counting systems of other Polynesian and Melanesian countries in the Oceania region. He documented and studied the relationship between 2000 or so different languages. Based on his findings, Lean was able to conclude that every distinct language had an unique counting system. He placed great emphasis on the fact that counting systems were a part of language and that language was a part of culture ( Ellerton & Clarkson, 1996).

Saxe (1982) reported the development of counting among the Oksapmin children of Papua New Guinea. He noted the use of the body part counting system and the conservation of number. He subsequently found that the introduction of money had an effect on the counting systems and that because of the use of money, these people had developed new ways of doing mathematics (counting). Lancy (1983) and Souviney (1981, 1983) report on the Indigenous Mathematics Project (IMP) in PNG which attempted to examine the cultural background of children, patterns of cognitive development and acquisition of school mathematics. Specific aspects of culture such as indigenous mathematics (number systems), classification, games etc. were identified. The intention was then to develop appropriate curricula to suit student needs. The development of a trial curriculum at the primary level (a result of the IMP studies)  is described in section 2.3.4 - PNG studies on the curriculum.

2.1.4 Ethnomathematics

The recognition that mathematics is learnt (and can be identified) in social and cultural contexts has led to the emergence of the area in mathematics education called “ethnomathematics”. The definitions suggest that “ethnomathematics” (ETM) is viewed by some as mathematics or mathematical knowledge or mathematical activity while others see it as a field of study. The following examples of ETM definitions illustrate this point.

Definitions.

According to D’ Ambrosio (1985), ethnomathematics is an activity which is practised in socio cultural groups:

“... the mathematics which is practiced among identifiable cultural groups, such as national - tribe societies, labor groups, children of certain age brackets, professional classes and so on ... may include mathematics as practiced by engineers ... which does not respond to concept of rigour and formalism ... Builders, well diggers, shack raisers in the slums also use examples of ethnomathematics”. D’Ambrosio(1985 : p.45; also in Harris 1991 : p.15)

 

D’ Ambrosio (1990 : p.22) later defines ethnomathematics in the following way :

Resorting to etymology, the term ethnomathematics is introduced as the art or technique (tics) of explaining, understanding, coping with (mathema) the socio-culture and natural (ethno) environment.” (also in Harris,1991 : p.23)

 

The Howson and Wilson (1986) definition suggests ETM is viewed as a mathematical activity :

“.... in any socio-cultural group there exist systematic practices of classifying, ordering, quantifying, measuring, counting, comparing, dealing with spatial orientation, perceiving time and planning activities, logical reasoning, inferring etc.” (Howson  & Wilson, 1986)

 

Borba’s (1990) and Pompeu’s (1992) definitions suggest a view of ETM as mathematical knowledge.

“Mathematical knowledge expressed in language code of  a given socio-cultural group is called ‘ethnomathematics’.”  Borba (1990 : p.40)

 

“ Ethnomathematics refers to any form of cultural knowledge or social activity characteristic of a cultural/social group which can be recognised by other groups like ‘western’ anthropologists, but not necessarily by the original group, as mathematical knowledge or as mathematical activity.” Pompeu (1992 : pp.73-74)

 

Vithal (1992 : pp.12-13) quotes examples of definitions which seem to portray ETM as mathematical knowledge found in the environment as well as an activity :

“At one level, it (ethnomathematics) might be called ‘math in the environment’ or ‘math in the community’. At another level, it is the way specific cultural groups go about the tasks of ciphering and counting, measuring, classifying, ordering, inferring and modeling.” (Gilmer in Vithal , 1992).

 

Ascher’s definition suggests a view of ETM as the study of mathematical ideas of various peoples :

“The study of mathematical ideas of traditional peoples is part of a new endeavour called ethnomathematics ... [These] mathematical ideas ... [are] embedded in some traditional cultures.” Ascher  (1991 : p.1 [Bracket added]).

 

Frankenstein’s (1990) definition of ETM as an emerging discipline also promotes the idea of a field of study :

“ This emerging discipline (ethnomathematics) analyses how people in our daily activities , think mathematically, showing that there are logical structures in a variety of mathematical practices in addition to those  in `academic’ mathematics. (Frankenstein, 1990 : p.107)

 

Presmeg (1996) provides an informal definition of ethnomathematics as :

“… the mathematics of cultural practice, a definition which includes ideas based on activities from all cultures, including ones own (and not specifically limited to those of ‘traditional people’)” (Presmeg, 1996).

 

Presmeg contrasts ethnomathematics with the “academic mathematics” that is traditionally taught in schools.

 

Barton (1996) clearly views ethnomathematics as a field of study and not as a kind of mathematics. He provides the following definition :

“Ethnomathematics is the field of study which examines the way people from other cultures understand, articulate and use concepts and practices which are from their culture and which the researcher describes as mathematical” (Barton, 1996 : p.196).

According to Barton, ETM is an academic field of study, although it is not a mathematical field of study but is more like anthropology  or history. This field of study is not restricted to university level research such as the identification of the mathematical ideas or practices of particular groups of people but may also include the work by children exploring the mathematical ideas of other groups (Barton, 1996 : p.197). He states that ethnomathematics is not the mathematics of a particular group of people but the study which examines mathematical ideas in their cultural context.

The present study adopts Barton’s view that “ethnomathematics” is not the mathematics of particular groups of people but is the study of mathematical ideas (knowledge), practices, activities which can be identified in socio-cultural contexts. These mathematical ideas, knowledge and activities are acquired and practiced by various socio-cultural groups (in all cultures). It is not the writer’s intention to give a detailed definition of ethnomathematics. However, in the context of this thesis the term “cultural mathematics” (rather than ethnomathematics) will be used to differentiate between the mathematics that is learnt in schools and the “other” mathematics that may be identified out-of-school or in socio-cultural contexts. The term “cultural mathematics” therefore will mean the mathematical ideas, knowledge and practices that can be identified in socio-cultural contexts through the study of ethnomathematics. Mathematics that is learnt in schools will be referred to as “school mathematics”.

Critique of ethnomathematics.

Munster, Nielsen, Nielsen & Simoni (1994) in their critique of ethnomathematics make the point that in trying to identify the mathematics in cultural activities of artefacts, it is possible to read too much into what mathematics is really there  (“knossos syndrome”). Another criticism is the use of “western” mathematics to identify mathematics in traditional activities which is supposed to remove cultural blockage. Some writers, in attempting to show the history of mathematics in various cultures, do not really show how mathematics is different from the European mathematics. For example, in referring to Indian mathematics, Joseph (cited by Munster et al., 1994) measures Indian work on a European mathematics scale. Doing this only succeeds in showing the “non-European roots of European mathematics”. No reference was made about the different basis upon which mathematics was developed. For example, the philosophical basis by the Greeks, the empirical basis by the Indian and the Chinese. Another point is the danger with a culturally oriented curriculum. It can be used to perpetuate existing systems, as was the case in South Africa where an inferior curriculum was written for the blacks in the guise of a culturally oriented curriculum which was supposed to be suitable for them (Munster et al. 1994 : p.100). Munster et al. proposed a new ethnomathematical approach which is similar to “everyday” applications. Rather than the historical, frozen mathematical approach, one should use knowledge that children possess.

ETM may have as one of its strongest point  the fact that one should build on the knowledge that children possess. Chevallard (1990) states that  cultural issues do not just concern the mathematics found in cultures that are disadvantaged, (for example, former colonies, third world countries only) but should include all societies. He suggests that ethnomathematics has no strong theoretical basis which can be applied to all societies and that it is not scientifically legitimate. His argument is as follows : Mathematics as an activity if not as a  body of  knowledge is not culture free; learners’ cultural equipment may be at variance with cultural pre-requisites of mathematics learning; because of the above, learning difficulties follow. Chevallard suggests that the point about learning difficulties should be ignored because anybody introduced to a new body of knowledge will experience difficulties. The point that Chevallard misses is the acknowledgment that children from these different backgrounds (traditional) possess knowledge that one can build on.

Chevallard admits that all cultures develop what he calls “protomathematics” but adds that few developed into fully fledged sciences. He questions some historical epistemologies of mathematics and argues that mathematics went through a turbulent history to get to where it is today . There was no easy way from Babylonian/Egyptian protomathematics to Greek mathematics to today’s mathematics and acknowledges the contribution of societies of all cultures to the make up of today’s mathematics.

On Bishop’s six fundamental activities (common to all cultures) from which western mathematics was derived, Chevallard states they may provide the learner with cultural confidence and motivation but are of little help in solving the main problems that mathematics education face. This writer disagrees with Chevallard’s contention because the biggest problem facing mathematics educators in developing countries are the problems associated with teaching or learning mathematics in cultural contexts and to make learners more confident in learning mathematics or to see the relevance of learning mathematics. The fundamental activities also help in redefining ideas about what mathematics is. Thinking of mathematics in terms of the six fundamental activities gives an alternate view of what constitutes mathematics. These activities also form the basis or starting points for many classroom activities and can also be used to identify mathematics in socio-cultural contexts.

2.2 Teacher beliefs and practice.

Introduction

At the end of section 1.1 in chapter one where the “problematique” was summarised, it was noted that “mismatch” occurs at three levels : between Policy and “intended” curriculum, between policy and the implemented curriculum (taught curriculum), between policy and the attained curriculum (learnt curriculum). This section reviews literature on teacher beliefs and practice. The focus is on teachers because they are the link between policy and practice. Teachers “operate” at the classroom level and are therefore major players in the implementation of policies. An important consideration should therefore be, what do the teachers think about these policies ? Do they think it is possible to have a “culturally oriented” mathematics curriculum ? It is therefore important to consider teacher beliefs about the cultural dimension of the policies and the curriculum and how these beliefs relate to their classroom practices. It is for these reasons that the literature review focuses on research into teacher beliefs about mathematics and  how these beliefs relate to their practices.

The literature review begins by examining teacher conceptions about mathematics (section 2.2.1 and section 2.2.2). Section 2.2.3 looks at teacher beliefs and how these beliefs relate to their practice. Constraints on practice is considered in section 2.2.5 while studies on teacher practices in PNG is reviewed in section 2.2.6.

2.2.1 Conceptions about the Nature of Mathematics.

According to Dossey (1992), research shows that differing conceptions of mathematics (or the nature of mathematics) influence the way in which both teachers and mathematicians approach the teaching and the development of mathematics (quoted studies include Brown, 1985; Cooney, 1985; Good, Grouws and Ebmeier, 1983; Owens, 1987; Thompson, 1984).

Platonic and Aristolean conceptions of mathematics.

There is generally a lack of consensus about what the conceptions are about the nature of mathematics (Dossey, 1992 : p.42). This does not mean that the nature of mathematics has not been debated. On the contrary, discussions about the nature of mathematics date back to the fourth century B.C (p.39). Plato and his student, Aristotle offered varying views which form the basis of two major contrasting themes about the nature of mathematics. Plato’s view was that objects of mathematics existed on their own, outside the mind in the external world (external view). Aristotle’s view was that mathematical knowledge was obtained from experienced reality by experimentation, observation and abstraction (internal view). Construction of mathematical ideas came about as a result of experience with objects (p.40). Mathematicians do not necessarily think about the nature of mathematics in their work although they seem to hold “Platonic” conceptions of mathematics. When challenged, however, they express “Aristolean” views ( Dossey, 1992 : p.41).

Historians identify views about the nature of mathematics which tended to fall along a continuum between these two contrasting themes. For example, those identified as having more Platonic views were works by Gottlob Frege, (1884) on logicism (contents of mathematics were the elements of the body of classical mathematics, its definition, and its theorem -  “principia mathematica” written by Whitehead and Russel, 1910 -1913, who held similar views). Those identified as more Aristolean were works by Brouwer - intuitionism (mathematics content were the theories, that had been constructed from first principle via valid patterns of reasoning) and Hielbert - formalism (content made up of axiomatic structures developed to rid classical mathematics of its short comings). All three tended to view elements of mathematics as finished “products”. For more details see Dossey (1992 : pp.40 - 41).

Modern conceptions of mathematics.

Sowder (in Dossey 1992) identifies five modern conceptions of mathematics in mathematics education literature which are variations of the Platonic and Aristolean views and which fall along an external / internal continuum.

  1. Platonic (external) views regard mathematics as an externally existing, established body of knowledge, facts, principles and skills available in syllabi and curriculum material. There are two variations to this view.
    1. Those who view mathematics as a static discipline. The focus is on teaching methods (effective teaching) and the role of the teacher to convey knowledge to students. Cited research into teaching of concepts by Cooney, 1980; Cooney & Bradford,1976 and  Sowder, 1980, illustrate the focus. Research on effective teaching which used the classroom as their source of data include Brophy, 1986; Fisher & Berliner, 1985; Good, Grouws & Ebmeier, 1983; Slavin & Karweit, 1984,1985 (all cited by Dossey, 1992)
    2. The second variation to the external view takes a more dynamic view of mathematics. The focus is on adjusting the curriculum, especially through use of modern technology. The aim is to improve student understanding to achieve growth of individual knowledge of an existing portion of mathematics. Examples of studies which emphasise the above include Thorpe, 1989; Kaput, 1989; Wearne & Hiebert, 1988 (cited in Dossey, 1992).

     

  2. Aristolean views about the nature of mathematics regard mathematics as a personally constructed or internal set of knowledge.
    1. In the first variation to this view, mathematics is a process, knowing mathematics is the same as “doing” mathematics eg. experimenting, abstracting, generalising, specialising constitute mathematics (Von Glaserfeld, 1987). Knowledge and competence are products of individual’s conceptual organisation of individuals experience. The teacher role is not to dispense knowledge but to help guide the student in conceptual organisation (p.43). Cited studies include Steffe, 1988; Romberg, 1988; Polya, 1965, NCTM Agenda, 1985.
    2. The second Aristolean view describes mathematical activities in terms of psychological models, cognitive proceedures and schemata, cognitive science. Cognitive modelling approach which is a model for viewing structure of  mathematics learning. Dossey (1992) describes it as “identification of representatives for mathematical knowledge, of operations individuals perform on that knowledge and of the manner in which the human mind stores, transforms and amalgamates that knowledge” (p.45). The research cited by Dossey (1992) which supports this view include Bransford et al., 1988; Campione, Brown & Connell, 1988; Carpenter, 1988; Chaiklin, 1989; Hiebert, 1986; Larkin, 1989; Marshall, 1988; Nesher, 1988; Ohlsson, 1988; Resnick, 1987.

     

  3. The third perspective states that mathematical knowledge results from social interactions. Relevant facts, concepts, principles and skills are acquired as a result of social interactions that rely heavily on context. Cited works supporting this view include Bauersfeld, 1980; Bishop, 1985a, 1988b; Kieren, 1988; Lave, Smith & Butler, 1988; Schoenfeld, 1988, 1989 (cited by Dossey, 1992). Schoenfeld (1988) states that the nature of mathematics perceived by the student is a result of an intricate interaction of cognitive and social factors existing in the context of schooling. Resnick (cited by Schoenfeld, 1992 : p.340) sees mathematics learning as  notion of socialisation (enculturation - entering and picking up values of a community or culture), highlights importance of perspective and point of view as core aspects of knowledge. Mathematics education is seen as a socialisation process (rather than as instructional process).

Even within the last group there are differing views. Nunes (1992) notes two views on cultural influences on mathematical knowledge. According to Stigler and Baranes (1988, in Nunes, 1992):

“Mathematics is not a universal, formal domain of knowledge ... but rather an assemblage of culturally constructed symbolic representations and procedures for manipulation these representations .... Children ... develop ... representations and procedures into their cognitive systems, a process that occurs in the context of  socially constructed activities. Mathematical skills that the child learns in schools are not logically constructed on the basis of abstract cognitive structures but rather forged out of a combination of previously acquired (or inherited) knowledge and skills and new cultural input”.  (p.558).

According to Hersh (in Dossey, 1992) mathematicians proceed by intuition, exploring concepts and their interactions. The question of what is mathematics could be answered if mathematics is accepted as a human activity, not governed by one school of thought (eg logicist, formalist, constructionist). Hammer (1978) states that mathematics arose out of the need of organised society. “Attempts to suggest that mathematics is part of a safe, secure, logical structure existing independently of human experiences are erroneous” (p.250). Mathematics and logic cannot be divorced from the activities of people in society. Hammer cites Court (1961) and Polya (1952, 1954) as those who would agree with him and quotes Mannoury (1947) who “labels as pure superstition the notions that mathematics is absolute, perfectly exact, general and autonomous or in short being true or eternal” (p.255).

2.2.2 Teacher beliefs and conceptions about mathematics

Teacher conceptions and beliefs are discussed here. It is argued in the next section that teacher conceptions about the nature of mathematics are important because they relate directly to instructional practice in the classroom (although the relationship is somewhat complex).

Studies on teacher beliefs about mathematics generally agree that teacher beliefs about mathematics consist of their conceptions (in this chapter, no distinction is made between teacher beliefs and conceptions) about :

  1. mathematics learning
  2. mathematics teaching, and
  3. the nature of mathematics

See for example, studies reported by  Thompson (1992); Mayers, (1994); Van Zoest, Jones & Thornton (1994); Southwell, (1995); Buzeika, (1996); Perry, Howard and Conroy (1996). In this section, the discussions center around teacher conceptions about the nature of mathematics because, as will be seen, teacher conceptions about learning and teaching are enmeshed in teacher conceptions about the nature of mathematics. This theme is extended in chapter three where the above categories are used in the formulation of the “theoretical construct of teacher beliefs about mathematics” (see section 3.2.1 in chapter three - Theoretical construct of the questionnaire items).

Teacher conceptions about the nature of mathematics.

According to Thompson (1992 : p.132) “teachers’ conception of the nature of mathematics may be viewed as the teachers’ conscious or subconscious beliefs, concepts, meanings, rules, mental images, and preference concerning the discipline of mathematics. Those beliefs, concepts views and preferences constitute the rudiments of a philosophy of mathematics, although for some teachers it may not be developed into a coherent philosophy.” (cites Ernest, 1988; Jones, Henderson and Cooney, 1986). Thompson further suggests that the conceptions include mental structures, encompassing beliefs, teacher knowledge that influence experience - meaning, concepts, proposition rules, mental images (pp.140-141).

Although attempts have been made to categorise teacher conceptions of mathematics (eg. Ernest, 1988; Thompson, 1984; Lerman, 1983; Copes,1979; Skemp, 1978; cited by Thompson,1992: p.132) it would seem that the teachers’ conceptions of mathematics fall into two main categories (which admittedly will have variations). Dossey (1992 : p.43) suggests that these conceptions are variations of the Platonic and Aristolean views which he refers to as “External” and “Internal” conceptions of mathematics.

Table 2.2 below gives examples of differing conceptions of the nature of mathematics which fall into these two categories (including teacher conceptions).

Examples of conceptions about the nature of mathematics (including teacher conceptions). This Table was compiled from information provided by Dossey, (1992). It provides a summary of the so-called internal and external views.

Table 2.2 External and Internal views about the nature of mathematics

External view.

Internal view.

Platonic (in Dossey, 1992) - external existence of a body of knowledge to be transmitted to the learner

Aristolean (in Dossey, 1992) - mathematics as a game where symbols are manipulated according to societally accepted rules, knowledge gained from experienced reality with objects (or object created through sense perceptions), observations and abstraction.

Nickson (1992: p.103) - Formalistic tradition, foundations of mathematics knowledge lie outside human action, consisting of irrefutable truths and unquestionable certainty, mathematics waiting to be discovered

Nickson (1992) - “Growth and change” view of mathematics. Cites Lakatos (1976) and Popper (1972) who state that how knowledge comes into being (challenged, superseded, changed) is not only a social phenomenon but a cultural one .

Dossey (1992) - Modern conceptions of mathematics (especially mathematics education research), more Platonic - externally existing body of knowledge, facts available in syllabi and curriculum material: (i) Mathematics as a static discipline - focus on teaching methods to convey knowledge to students (ii) Dynamic view of mathematics, focus on curriculum, growth of individual knowledge to improve student understanding

Dossey (1992) - Modern conceptions of mathematics (internal) - more Aristolean.  Mathematics as personally constructed, internal set of knowledge. (i) mathematics is a process, knowing mathematics is the same as doing mathematics; experimenting, generalising, abstraction constitutes mathematics,  (ii) mathematical activities in terms of psychological models, use of cognitive modeling to view structure of mathematics learning (iii) mathematical knowledge results from social interactions, acquisition of concepts, principles, through social interactions that rely heavily on context

Ernest (in Thompson, 1992) - Two teacher conceptions of mathematics (related to philosophy of mathematics) (i) mathematics as static but unified body of knowledge, truths bound together by logic and meaning, discovered but not created (ii) mathematics as a bag of tools, set of facts, rules, to be used by trained artisan (instrumentalist view).

Ernest (in Thompson, 1992) - Dynamic problem driven view of mathematics as a continually expanding field of human creation, mathematics as a process of inquiry, knowledge is added (problem solving view)

Lerman 1983 (in Thompson, 1992) - Absolutist view - all mathematics is based on universal, absolute foundations, paradigm of knowledge certain, absolute, value free, abstract

Lerman, 1983 (in Thompson, 1992) - Fallabilist view - mathematics develops through conjectures, proofs, and refutations and uncertainty is accepted as inherent in the discipline.

Where do teacher conceptions about the nature of cultural mathematics (CM) fit into the above categories ?

It is assumed here that the teachers with strong internal conceptions about the nature of mathematics will also have pro-cultural mathematics views. By the same token, it is assumed that the teachers with strong external conceptions about the nature of mathematics will have pro-school mathematics oriented views. The distinction between “school mathematics” and “cultural mathematics” was made at the end of section on ethnomathematics (section 2.1.4). “Cultural mathematics” or CM in this thesis will mean the out-of-school mathematics that may be identified as the mathematical ideas, knowledge or practices of a socio-cultural group.

 This assumption is made because the external views about mathematics are views which are normally associated with “school mathematics”. For example, school mathematics is usually thought of as culture and value free, where mathematical knowledge consists of irrefutable truths, facts, rules which are available in syllabi and curriculum material. On the other hand, the internal views about mathematics support the conceptions about CM. For example, “cultural mathematics” knowledge results from social interactions that heavily rely on context; how knowledge comes into being is not only a social phenomenon but a cultural one; mathematics as a personally constructed internal set of knowledge where uncertainty is accepted.

This assumption forms the basis of the “theoretical construct” for the scales in the questionnaire (section 3.2.1 in chapter three). This assumption will be tested only after the questionnaire is given and item reliability tests and factor analysis is carried out (ie. to see how reliable these constructs are. See section 5.3 in chapter five).

PNG studies on conceptions about mathematics.

Studies into teacher conceptions about the nature of mathematics in the PNG context are limited. A perusal of the PNG Journal of Education (1967 to 1995), the PNG Bibliography of Education : 1981 - 1982 (Crossley, 1985); PNG Bibliography of Education : 1986 - 1990 (Eyford, 1992) and the Research on Mathematics Education and Mathematics (in PNG) : 1982 to 1984 (Clarkson, 1982, 1984) failed to reveal any studies on teacher beliefs about mathematics. Many of the reported PNG studies in mathematics concentrate on students. But again these studies were not necessarily about conceptions or beliefs about mathematics. The closest studies to beliefs or conceptions were attitudinal studies (see for example, Clarkson and Leder, 1984  and Wilkins, 1985 in section 2.3.5). Roberts (1989) conducted a study on the attitude of tertiary students to mathematics. His comparative study of PNG and Australian students showed that PNG students enjoyed mathematics more than the Australian students but that the PNG students were less satisfied with the high school preparation than the Australian students (for details of other PNG studies see section 2.3.5).

Student or student teacher conceptions were noted in research on science. Boeha (1990) for example, conducted a study into conceptions (and misconceptions) of physics knowledge that Grade 12  students in Papua New Guinea had. He noted that these students seem to hold Aristolean views about physics. Boeha (1991) was also able to observe some changes in the students’ beliefs (or misconceptions) about  momentum when he used a teaching /learning strategy to teach the concept of momentum. Vlaardingerbroek’s (1991) study reported on the views about “ethnoscience” held by student teachers at the Goroka campus of the University of Papua New Guinea. One of his findings was that the teacher trainees generally laughed off the charge of science education as cultural imperialism (see also Young, 1977; Young & Bartos, 1977). He concluded that the teacher trainees appreciate western science because of its usefulness; some even mentioned that people’s lifestyles improved because of  technology. The teacher trainees agreed that “science education may weaken traditional culture but regarded this as inevitable, not necessarily an undesirable price to pay for progress …” (p.33).

2.2.3 Teacher beliefs and practices

In this section we report on the studies that examine the relationship between teacher beliefs and practices. While some suggest there is no evidence of a cause and effect relationship between teacher beliefs and practices, other studies noted here, although acknowledging the complexities, report that there is some evidence of a relationship between teacher beliefs and their instructional practices.

The relationship between belief and practice.

Although it is said that there is congruency between belief and practice (eg. Bauch in Mayer 1985), it is not necessary that beliefs are a vision for or a guide to practice (p.8). Philosophy or educational beliefs do not necessarily shape their practice; beliefs may be a result of or a justification for practice eg. environmental factors could force the teacher to shape practice in a certain way. Curriculum could stipulate that something be taught in a certain way. According to review of research (by Mayer, 1985), there is no conclusive evidence that there is indeed a link or a relationship between belief and practice. If there is any link, it is a rather complex one.

However, Cooney (1985) states that the conceptions of mathematics held by teachers has a strong impact on the way mathematics is approached in the classroom. For example, it is suggested that the nature of mathematics is portrayed in class through school mathematics which gives a formal and external view of mathematics; through textbooks which portray the nature of mathematics as well as through how the teachers actually employ the textbooks in class. Teachers’ conceptions of mathematics and the way it is characterised in classroom communicate to children a certain view of mathematics (p.43). Cooney (1987) further suggests that innovative changes may not take place because of teacher beliefs about the nature of mathematics. Dossey (1992) cites Hersh (p.42) who suggests that how one teaches in the classroom is controlled by beliefs about the nature of mathematics, not by beliefs about the best way to teach; therefore to change situations, one must find alternative ways of conceptualising the nature of mathematics (also cited by Thompson, 1992 : p.127).

According to Heaton, Prawat, Putman & Remillard (1992), there is a growing body of evidence which has established that there is indeed an important relationship that exists between the knowledge and beliefs of the teacher and classroom practice. The study, which was a case study of four fifth grade teachers, looked at teacher knowledge and beliefs about mathematics and how it is best taught and learned. They studied the interrelationships between these beliefs and teacher practices in the classroom. Their study showed, for example, that a belief that mathematics should be enjoyable and engaging led teachers to work hard to motivate their students (p.225).

Hoffman (1989) cited by Schoenfeld (1992: p.341) discusses the importance of epistemological issues, and whether one is explicit about their epistemological stance. He states that what one thinks mathematics is, will shape the kind of mathematical environment one creates and thus the kinds of mathematical understandings that one’s students will develop. Schoenfeld  (1992) states that what one thinks is the nature of mathematics will shape his/her practice.

While some teachers seem to possess complicated systems of beliefs (philosophies), others appear not to have any. This view is supported by Thompson (1992: pp.137-138) and other cited studies. They also suggest that there are many sources of influence, one of which is the social context - which includes values beliefs, expectations of students, parents, fellow teachers, administrators, curriculum, assessment  procedures, values and philosophies of education systems at large. These factors can act to shape practice.

Nickson (1992)  describes the progressive steps which have been taken in the formalistic tradition. In reference to the views held by students and teachers, Nickson has this to say :

 “ The differing views held by teachers and students in relation to the nature of mathematical knowledge are an important component in the culture of mathematics classroom, since they are linked with the way mathematics is taught and received. One perspective may result in a classroom context which could be described as ‘asocial’ insofar as it emphasises the abstractness of mathematics to be done individually and more or less in silence by the pupils in the classroom. Another emphasises the social aspect of the foundations” (Nickson. 1992 : : p.105)

Studies on mathematics teacher beliefs and practices

Thompson (1984) studied three junior high school mathematics teachers. The aim was to investigate the relationship of teacher conceptions of mathematics to instructional practice. The results showed evidence of teacher differences in their beliefs, views and preferences they had regarding mathematics and its teaching. Thompson attributes the differences in the instructional emphasis on the differences in the prevailing view of mathematics.

Buzeika (1996) explored the relationship between teacher beliefs and practices as perceived by primary teachers implementing a new curriculum document. This study showed that their beliefs changed as result of their practice.

The following studies used Perry’s (1970) levels of intellectual and ethical development. Cooney (1985) studied teachers’ conceptions of the nature of mathematics using modified version of Perry’s levels of intellectual and ethical development (Stages of intellectual development - measures view points about their conceptions of knowledge which prevailed during the historical development of mathematics : absolutism, dualism, multiplism, relativism, dynamism). Cooney found teacher conceptions to be dualistic, multiplistic, relativist (see also Thompson, 1992 and Dossey, 1992).

Cooney, (1987) concluded (after analysing Goffree, 1985 and Perry’s work) that how mathematics is portrayed in class usually promoted the formal (external view) about the nature of mathematics. Owens (1987) and Bush (1982) found pre-service teachers’ dualistic or multiplistic views were strengthened by experience of the mathematics content courses at upper secondary or university level where the  teaching strongly supports the formalistic view of mathematics as an externally developed axiom. Teachers’ conceptions of teaching are likely to reflect their views about students’ mathematical knowledge and how students learn. A strong relationship is observed between teachers conceptions of teaching and their conceptions of students mathematical knowledge (Cobb, Wood, & Yackel, 1992; Carpenter, Fennema, Peterson & Carey, 1988)

Differences in teacher views of mathematics teaching appear to be related to differences in conceptions of mathematics (Thompson, 1984). Thompson (1992) reported on the studies by Copes (1979) which indicated that teaching styles communicate different conceptions of mathematics. Teachers’ view of the nature of mathematics is manifested in their beliefs about the way it should be taught. Thompson (1992) also reported the studies by Kesler (1985) which found that teachers viewed their role as dualistic, for example, as providing either right or wrong answers (according to Perry’s intellectual development scale). The teachers’ concern that students would perform their tasks was reduced to knowing how, rather than knowing why.

Brown (1985) identified student reaction as a strong factor in influencing teachers’ portrayal of mathematics in class eg. teacher initiates a problem solving method in class but reverts back to the usual (more expository method) when students respond negatively.

Whitman and Lai (1990) investigated similarities and differences in teacher beliefs about effective teaching of mathematics in Japan and Hawaii. This study highlighted the role of the socio-cultural milieu in the teaching and learning of mathematics (p.71). Their findings confirm studies by Stigler et al. (cited by Whitman & Lai, 1990) which showed that cultural values for mathematics held by teachers and hard work were important variables in pupils’ mathematics development.

Nebres (1988) stated that there was a need for further research to understand the cultural values that support learning of mathematics. Nebres suggested that such a study was important for the education of mathematics teachers regarding their beliefs about effective teaching. The findings showed that there were some similarities in beliefs about what constitutes effective teaching, but that the differences in teacher beliefs about effective teaching were greater.

Brousseau, Book & Byers (1988) investigated teachers’ beliefs and the cultures of teaching. The research made a comparison of pre-service and experienced teachers’ beliefs. The study found “experience of work” as  the variable that may impact on teacher beliefs about “teaching culture” (others did not have much impact).

While some of the studies quoted above suggest that there is no evidence of a cause and effect relationship between beliefs and practices, the writer agrees with the findings from other studies which, while acknowledging the complexities, report that there is some evidence of a relationship between teacher beliefs and their practices. The methodologies that were used in some of the above studies will be reported in section 3.1.1 in chapter three.

2.2.5 Constraints on practices.

Classroom practice is influenced by many factors and research should consider these other factors. The Grouws and Koehler Research model, (1992 : p.118) in section 3.1.2 in chapter three, sums up the factors influencing teacher beliefs and practices (teacher behaviour) in the classroom (classroom processes). For example, one cannot attribute teacher beliefs as the only factor to explain teacher behaviour in the classroom - other factors influence practice. Teacher beliefs, knowledge and experience may be considered as “internal” factors. Other factors such as pupil characteristics, curriculum, assessment, peer pressure are external to the teacher. These external factors may even be considered as constraints to teacher beliefs (internal factors), shaping teacher behaviour in the classroom.

In the previous section it was noted that the relationship between beliefs and practice is rather complex. Thompson (1992 : p.138) sums up the situation by noting the following :

“ … teacher conceptions of teaching and learning are not related in a simple cause-and-effect way to their instructional practice … complex relationship … many sources of influence at work … one … social context in which mathematics teaching takes place with all the constraints it imposes and the opportunities it offers. Embedded in this context are the values, beliefs, and expectations of students, parents, fellow teachers, administrators, the adopted curriculum, assessment practices, and the values and philosophical leanings of the educational system at large.”

This section discusses the “constraints” that hinder a teacher from putting his/her beliefs into practice. An attempt has been made to put these constraints into categories - curricular constraints (eg. curriculum system, examinations, peers) and socio-cultural constraints (what you are allowed to do, what you cannot do, cultural constraints or the value systems from society) .

Curricular constraints.

Curricular constraints refer to the educational system under which the teacher must practice. It includes the curriculum system, examination systems, pressure from fellow teachers, senior teachers, principals or school inspectors. It may include constraints imposed by the school organisations or the availability (or the non-availability) of materials such as text books, computers and calculators (Thompson, 1992). These are what the system allows or expects you to do.

An important constraint is student behaviour or the expectations of the students. If the students approve the teacher action in the classroom, the teacher is likely to repeat the action but if the students disapprove or fail to master the concepts, the teacher is likely to change his/her practice. For example, the teacher may believe that using the ‘investigative method” is the best way to teach a topic. If the students’ reaction suggests that they are not comfortable with this method then the teacher may not persist with this approach. Ernest (1988) reported the effect of social context on instructional practice where teachers in the same school, although having differing views adopted similar classroom practices.

Socio-cultural constraints.

Socio-cultural constraints refer to the pressures from the society at large which dictate what one is allowed to do or what one cannot do. Teacher practice in the classroom may be dictated by the expectations from groups of people in the society or by certain values inherent in the society. These values or beliefs dictate teacher practice in the classroom.

2.2.6 Teacher practices in PNG.

Avalos (1991) noted that there have been a number of studies in Papua New Guinea which looked at “teaching styles”, first defined by Beeby in 1966. Most of these studies indicate that the so-called “formalistic” style of teaching has not changed over time. Guthrie (1980), for example, adapted the Beeby teaching styles continuum and suggested five categories within the continuum. The teaching styles categories were authoritarian, formalistic, variation, liberal and democratic. He identified the PNG teaching styles as tending towards the  formalistic end of the continuum.

Other studies (e.g. Dunkin, 1977; Cheetham, 1979, Wilson, 1979, Otto, 1989) cited by Avalos (1991) of teaching in PNG found that lessons were based on questions and answers that were structured entirely by the teachers. Very rarely did the students spontaneously ask questions.  The Avalos (1991) study which observed primary teacher trainees during their teaching practice found that the teaching in almost all the lessons observed involved three main activities : teacher questioning, questioning by pupils to other pupils and seat work. Teacher questions were mostly factual, and left little scope for students to ask questions, nor did the student teachers seem interested in getting students to ask questions. There were few teaching materials which were used as teaching aids (other than the chalk and board).

Most of these studies were of primary teachers. However, Kaleva (1991) noted that the predominant teaching method in the secondary mathematics classroom is typically the expository method which follows the pattern : lecture - example - exercise. Kaleva (1991) suggests that the secondary mathematics teacher is at the formalistic end of Guthrie’s teaching styles continuum. The predominance of these teaching styles is illustrated in the following conversation the writer had with a student teacher. After observing some actual classroom teaching (while on teaching practice at a secondary school), the student teacher who had been encouraged to use other methods of teaching (eg. investigative, discussion etc), remarked; “Why weren’t the teachers taught about these other methods ?” (Clarkson & Kaleva, 1993). The student teacher’s observations of teacher practice in the classroom left him with the impression that the prevailing classroom practice was the expository method where the interaction between teacher and students is one way - predominantly from teacher to students.  In a study which was conducted as a part of evaluating the B. Ed (In-service) Science Education program at the University of Papua New Guinea, Haihue (1991) observed science teaching in selected provincial high schools. He was able to conclude that the most common style of teaching was the “recall / explanatory” type. There were few sequences involved, few higher order statements and questions requiring students to infer and question.

Constraints on teacher practices in PNG

As noted in the section on constraints to practice (2.2.5) above, practice is largely determined by curricular context, curricular constraints and situations. Curricular constraints such as the pressures of external examinations, pressure to “cover syllabus”, ensure that the teachers do not deviate from teaching the prescribed content. In a centralised curriculum system as in PNG, where the school curriculum is imposed on the teachers and the schools, the pressure to cover the syllabus is greater because the examination results are used as criteria for selecting students at various transitional stages (eg. from grade 6 to grade 7, from grade 8 to grade 9, from grade 10 to grade 11) and there are limited places available for students in the schools (eg. the transition rate from grade 6 to grade 7 in 1990 was approximately 40 %, see section 1.2 in chapter one).  Certainly the pressure and the expectations from the school (from fellow teachers, headmaster) and the parents for the school to out-perform other schools is very high. Guy (1994) identified the existing “structures” which define teacher practice as being, heavy teaching loads, a centralised and uniform curriculum system, large class sizes and extra-curricular duties.

The Secondary Schools Community Extension Project (SSCEP- 1978 to1982) in PNG is an example of a project which was initiated with good educational intentions - to train students so they can fit back into the community.  However, because of pressure from the community, this innovative scheme was discontinued. One of the aims of the project was to get students to be positively disposed to return to the villages upon completing the program (Vulliamy, 1981). However, the parents were not keen on seeing their children go to school just to learn skills to use back in their villages. Parents wanted their students to do well in the “academic” subjects that would enable the students to get paid jobs (this view could well be significant in the present study). There were also constraints from the existing high school system (Vulliamy, 1983). For example, because the core subjects were assessed nationally,  the students’ perception of the core subject teaching was in line with the existing syllabus ie. they  compared themselves with the students from the existing schools.

The Avalos (1991) study which focused on primary teacher trainees found that even if the student teachers wanted to alter their teaching styles, they found it difficult to do so because of the “constraints produced by the nature of school curriculum and its subject distribution, as well as their often limited knowledge of the content they have to teach” (p.180). The constraints were such that the teachers felt that they did not have the power to change what was presented as good practice in the guides (Avalos, 1989).

The cultural values that are within a society may also act as constraints to teacher beliefs and practice in the PNG situation. For example, the teacher who believes in “group discussions” may find it hard getting mixed groups of boys and girls because traditional “taboos” forbid girls from talking to boys or vice versa (a lot of these taboos are no longer practised, especially in the urban areas). The above would also explain why the male teacher may spend more time helping the boys rather than the girls; spending a lot of time helping the girls would be frowned upon.

It is important to consider the implications of these constraints to teacher practices in the classroom. Firstly there are constraints to do with the existing structures - what the system allows the teacher to do. Then there are the value constraints - teacher values, student values and parent values. Even if the teachers’ beliefs and values may be in line with government intentions (eg. to have a culturally oriented mathematics curriculum), the parents and student values may influence teacher implementation of the curriculum (practice). The research should not only investigate teacher beliefs are about the intended and the implemented curriculum, but also what the teachers see as constraints to implementing these beliefs. It should also consider the implications of the cultural values as constraints on teacher practice. In the wider context of the study, an important question to consider would be; whose values and beliefs are actually transmitted to students in the implementation of the mathematics curriculum ?

2.3 Educational policy and Curriculum practice.

Introduction.

The major problem which was identified in chapter one was the “mismatch” in the translation of the Philosophy of Education, or the expectations of the public into the curriculum. The curriculum is viewed as the vehicle through which parts of culture which include values and beliefs are transmitted (Taylor & Richards, 1985 : p.35), but an examination of the PNG curriculum suggests that the  curriculum does not comply with policy statements in official government documents. This section begins by looking at the relationship that exists between conceptions and ideologies of education and the curriculum (section 2.3.1). An important point that is noted has to do with teachers who also have conceptions of education and ideologies. It is suggested here that it does not matter what conceptions and ideologies are encompassed in the curriculum; it is probably the teachers’ ideologies that get transmitted to the students. The theories affecting the relationship between educational policies and curriculum practice are examined (section 2.3.2).

The second part to this section (section 2.3.3 reviews studies on the curriculum, in particular the IEA studies. These studies, which used the intended, implemented and attained curriculum framework, considered the numerous variables which affect student achievement. Section 2.3.4 considers the PNG studies on the curriculum. It also introduces the reader to the kinds of research that have been conducted in PNG in relation to the mathematics curriculum. In the last section (section 2.3.5), world wide examples of culturally oriented curriculum are given.

2.3.1 Conceptions, Ideologies and the curriculum>

The PNG Philosophy of Education is primarily derived from the National Constitution. But to formulate this philosophy, the views of various interest groups from a cross section of the community were sought. These views were then incorporated into what was to be known as the “Matane Report” (Matane, 1986). The views expressed by the people about their expectations of education represented their beliefs and conceptions about what the education system should be and the kind of education they would like their children to receive. These conceptions may rest on “different views regarding the desirable ends of the educational enterprise and different beliefs, often unacknowledged and implicit, about the nature of knowledge, children, teaching and learning.” (Taylor & Richards, 1985 : p.31). We look at conceptions and ideologies of education and how they relate to the curriculum.

Conceptions of Education

According to Taylor & Richards (1985: p.18) :

Curricula are means by which the young are systematically introduced to the material and the non-material world they inhabit. Curricula embody perspectives from human culture considered important enough to merit systematic transmission ... curriculum are found in specified institutional settings such as schools, colleges, universities .... they embody beliefs about education... embedded in them are conceptions of education, of what the enterprise is about and how it ought to be conducted .... these embedded conceptions inform the nature of the contexts (schools, workplaces, homes) created wholly for the transmission of education, give form to curriculum, result in different curricula emphases and lead to different practices”

They outline four basic conceptions of education (although many other conceptions and variations to these four exist). They are conceptions about (Ibid. pp.18-31) :

  1. the desirable ends of the educational enterprise.
  2. the nature of knowledge. Conceptions of education involve important ideas about the nature of knowledge (often not stated but implicit). For example, the internal and external conceptions about knowledge (see section 2.2.1 in chapter two).
  3. children and childhood. Conceptions of education are sometimes expressed in terms of  how a child develops and the educational implications of the development
  4. teaching. Conceptions of education are expressed in terms of  what good teaching is.

It is worth noting that these conceptions are held by societies and individuals. More importantly the teachers have conceptions of the desirable ends of education, nature of knowledge, children and childhood and what constitutes good teaching. Other key players in the educational enterprise who have definite conceptions are the policy makers and the curriculum developers. The development of curriculum and implementation therefore poses potential sources of mismatch.

Ideologies of education

The conceptions of education looked at in the previous section illustrate the differing views about the desirable ends of the educational enterprise, and beliefs about the nature of knowledge, children , teaching and learning. These conceptions (although varying, especially with individuals) make up the “educational ideologies ... systems of beliefs and values about the educational enterprise” (Ibid, p.32).

According to Scrimshaw, 1983 (cited by Taylor & Richards, 1985) there are five main ideologies of education.

  1. Progressivism - values education as a means of meeting individual needs and aspirations.
  2. Instrumentalism - stresses the responsiveness of education to the requirements of socio-economic order.
  3. Reconstructionism - conceives of education as an important way of moving society in desired directions.
  4. Classical humanism - education serves the function of transmitting cultural heritage.
  5. Liberal humanism - acknowledges importance of intellectual disciplines for all pupils, seeks to create common culture.

Curricula then are seen not only as transmitting part of cultural stock but as means of controlling people by exposing them to particular values and beliefs or conceptions of education. They are more than just bodies of knowledge but have cultural and political significance. Proposals about what should be taught are put forward by groups or individuals with similar conceptions and ideologies about education.

In the PNG context, concerns about the perceived failure of the education system to equip students with skills to fit back into the communities and the desire to include in the school curriculum some culturally oriented knowledge and skills indicates that the education system and the schools are regarded as institutions of social reproduction of the society’s cultures and values.

But the question one has to ask is, whose beliefs and values and conceptions of education do we wish to transmit to students ? Or indeed whose beliefs and values are actually transmitted to the students ? Is it possible that it does not matter what the curriculum is, what ideologies we wish to transmit through the curriculum, that the teachers’ beliefs and conceptions are the most important as these are transmitted to students directly ?

Research on educational policies and practice  suggest that policies have little impact on classroom practice, teachers will interpret policies and implement these according to their beliefs (see next section 2.3.2). The IEA studies (section 2.3.3) also found that there is mismatch between the “intended” and the “implemented curriculum” and between the “implemented” and the “attained curriculum”.

2.3.2 Educational policy, Curriculum and practice.

The theoretical aspects of the problem identified in chapter one (the mismatch in translating the philosophy of education to educational policy) are discussed here, in particular the mismatch between policy and practice (what is to be taught and what is actually taught). Various committees were formed by the PNG National Department of Education (NDOE) to review the different sections of the education system and to formulate policies in accordance with the philosophy of education with a view to restructuring the education system. The recommendations and policies were put together into a document titled the “Education Sector Review” (DOE, 1991). The policies were supposed to serve as directives for educational reforms including reforms in curriculum development. However there is a mismatch between the policy intentions and the intended curriculum.

Elmore and  Sykes (1992) use the term “public policy” and state that in the study of these policies one can identify policies “which have some bearing on the curriculum, either because they treat curriculum in some way or because they contribute directly to our understanding of policy”. (p.185). They define curriculum policy as the “formal body of law and regulation that pertains to what should be taught in schools .” (p.186). In this context, the writer’s use of the term “educational policy” will mean policies that relate to the formal educational processes and particularly in this context will include curriculum policies and may be equated with the term “public policy” (the inverse is also true, that is, public policy can be equated here with educational policy).

We begin by examining the relationship of policy to curriculum and the relationship of policy to practice. According to Cuban ( 1992 : p.221) :

“ Policy decisions produce curriculum. Formally adopted at the federal, state and district levels in the decentralised system of United States, educational governance, curricular policies are shaped by many forces ... These policies are expected to be put into practice in the schools and classrooms ... Principals and teachers reshape the adopted policies as they implement them in their sites”.

In contrast, in the centralised curriculum systems (such as in PNG), policy decisions are made at the national level and they are imposed on the other levels which are obliged to implement them. Curriculum determination is a function of the National government. It does not leave those at the lower levels (eg. state or provinces, schools) with much room for decision making that influence policies. A major feature at the lower levels is the implementation of these policies. This also serves as potential source of mismatch. Those who are implementing policies do not have any input into their formulation. Translating policy into practice will therefore always pose some problems.

Relationship between curriculum policy and practice.

The relationship between curriculum policy and practice discussed in greater detail because of its relative importance to understanding how policy is put into practice. What then is the relationship between curriculum policy and practice ? Elmore and Sykes (1992) suggest that the policy - practice connection, though complex, can be regarded in its simplest form as a system where the curriculum which originates from an authority is influenced by the institutions (such as schools) and established teaching practices and results in some learning. State and national governments may formulate policies which change teaching practices. The developed policy is imposed on the system by one or more forms of government actions. It is then left to be implemented by the “practitioners through their immediate working environments and has some impact on teaching practice, which in turn has some impact on student learning “ ( Ibid. : p.192). This is a commonly held perception of how policy is put into practice in a system or organisation.

Elmore & Sykes (1992) in their review of literature on research on curriculum policy state that policy analysts differ in their views about the influence of educational policy on classroom practice. They summarise the various views in the following way:

“Some argue that policy has a major impact on classroom practice, while others see only modest effects. Still others argue that policy impact varies across classrooms and schools. Analysts also differ on the nature of the nature of the changes in practice that results from policy. Some argue that educational policy has a negative impact on teaching. Others believe that policy can have a positive impact on teachers content decisions. Finally analysts offer contrasting explanations for the relationship they describe between educational policy and classroom practice. Some attribute the minimal influence of educational policy to the inherent conditions of the practitioner’s work, others to the nature of the educational organisation. Still others suggest  that the explanation for minimal impact lies in the nature of policies rather than in the nature of organisation or work” (pp.192 - 193 ).

Institutionalised structures and loose coupling. Elmore & Sykes (1992) discusses the work of Meyer and Rowan (1977, 1978) who make references to the “institutionalised structures and loose coupling which they argue has little impact on practice because of the weak core technology of teaching and lack of agreement about which instructional methods are most effective” (p.194). In this view, external policies focused on institutional structures that surround schools have minimal impact on teaching; policy is viewed as directed at sustaining the confidence of society in schooling as an important social institution. “Policy does not dramatically influence curriculum but functions instead to bolster public satisfaction with school systems” (p.194). Policy does not have much impact on practice because it is not directed on practice and because institutional structures buffer teaching practice from external policies

Multiple influences, weakly coordinated. The content determinant research (Porter et al., 1986; Schwille et al., 1983 in Elmore & Sykes, 1992) argues that “external policies exert limited influence on curriculum. Teachers’ content decisions are affected by informal factors such as student, parental pressures and teachers’ prior convictions and beliefs about what should be taught to whom “ (pp.194-195). External policies do not have much impact on teacher practices and anyway teachers interpret policies in their own ways according to their own beliefs about curriculum content and curriculum policies. It is suggested that the impact of local context (eg. teacher beliefs) can be minimised through use of multiple policies that are more “precise” and “prescriptive”, supported by authority recognised as legitimate by the teachers. Low impact of policies is regarded as being due to poorly structured policies.

Street level bureaucracies.This refers to teachers who are seen as street level bureaucrats. This view argues that external policies have only modest impact because conditions of work influences are more powerful than policy influence and result in uncertainty of outcome and the uncertain way one goes about fulfilling a goal (cites Lipsky,1980). This view (supported by Weatherly & Lipsky, 1977) suggests that policies have limited influence on what gets taught to whom. Teaching involves a degree of uncertainty about what should be taught and external policies provide little or no guidance to practitioners. The overworked teacher is usually forced to modify policies to fit the time and energy available (Elmore & Sykes,1992: p.194).

Remillard, (1992) reports on a case study of a primary teacher’s interpretation of policy (“A framework for teaching mathematics for understanding”). He reinforces the above contention by noting that the teacher’s interpretation of policy was influenced by his beliefs about mathematics teaching and learning. Other studies in teacher beliefs and practices in mathematics show that conceptions about the nature of mathematics will influence the way mathematics will be taught (Hersh in Dossey, 1992). However conflicts may result if there is a mismatch between policy intentions and policy action.

The above emphasises the important role the teacher plays in translating policy into practice. In relation to the mismatch (identified in chapter one) between educational policy and the curriculum, the contention here is that there is little impact of policy on the mathematics curriculum in PNG. It would be interesting to see how the policies which resulted in a change in the educational system affected teaching practice. The contention of this thesis is that there may be little or no impact at all of policy on the classroom practice. The educational policies in PNG were more to do with institutional policies or policies that dealt with organisation rather than curriculum or practice. The policy resulted in a change to the educational system.

The important questions for this study are : Did the policies result in the changes to the intended, implemented and attained curriculum ? What are the actual teacher practices in the classroom ? It is the intention of this study to investigate what the teacher beliefs are and what their classroom practices are.

Conceptions of teaching embedded in existing curriculum policy and the design of policies.

Research in policy and curriculum suggests that policies embody simplified notions of teaching. Policy analysts see teachers as “professionals”, “brokers”, “street level bureaucrats”. Policy makers use external influences such as rules, model curriculum, tests etc. to prescribe what should be taught, in what order and sequence. “A dominant view of teachers in curriculum policies is that teachers act as conduits for the delivery of socially approved knowledge” (Elmore & Sykes, 1992 : p.209). Future research could considerably expand our understanding of the relationship between curriculum policy and teaching practice by examining the implied or expressed models of teaching embedded in existing policies and by re-examining the relationship between inquiry versus policy based conceptions of teaching. For example, teaching for understanding is encouraged but nothing is known about what policies encourage this type of teaching. New policies may not rely on the mandate and inducement methods but on methods which influence curriculum by involving schools in the development of new models and on change incentives for schools to adopt new policies.

In conclusion, research on the relationship of policy to practice suggests that policy has little or no impact on practice because the teachers interpret policies according to their own beliefs about the intended curriculum (content), how it should be taught and that teachers have their own conceptions about the desirable ends of education and conceptions about the nature of knowledge.

2.3.3 Studies on the mathematics curriculum

In the first half of this section we will look at the International Association for the Evaluation of Educational Achievement (IEA) studies which investigated factors affecting students achievement. The IEA studies on mathematics (and other subjects) used the Intended, Implemented and the Attained curriculum framework which encompasses the teaching and learning of mathematics (this framework was formally adopted in the second stages of the studies, Travers & Westbury, 1989). In this section, reference to studies on mathematics curriculum will also include studies on the teaching and learning of mathematics. The IEA studies are of interest because the surveys were conducted in a number of countries. In the second half of this section, we look at examples of ethnomathematics based curricula.

IEA studies

The IEA, a consortium of about fifty research centers, was established in 1960 (Robitaille, 1993). The goals of IEA were to conduct cooperative international research studies in education and to create a pool of research expertise world wide. It sought to fulfil these goals by looking at student achievement against a wide background of school, home, student, societal factors. The ultimate aim was to inform educational policy and decision makers about alternatives in organisation and practice or those factors which are most likely to raise levels of achievement (Robitaille and Garden, 1989; Wilson, 1986).

One of the first international studies carried out by IEA was the survey on mathematics achievement. Other subject based surveys were also conducted (eg. science, the classroom environment study, the writing composition study, computer applications, reading literacy etc. - Robitaille and Travers, 1992). In this section a brief description is given of the three mathematics studies; FIMS, SIMS and TIMSS. Papua New Guinea was not involved in any of the mathematics studies but did participate in the Second International Science Study (SISS). We also describe PNG participation in SISS to illustrate how the IEA studies were conducted at the national (country) levels.

FIMS.

The IEA first carried out a survey of mathematics achievement - First International Mathematics Study (FIMS) in the early 60s in 12 countries. Mathematics was chosen as a subject that could be used to make comparisons amongst countries. “The primary aim was to examine differences among schools systems and how these differences related to achievement, interest, and attitude of students” (Garden, 1987). It was intended to measure mathematical achievement in students in member IEA countries (Cross sectional study). Internationally developed achievement tests were given to two student populations (those in their first year of secondary school and those in their last year of secondary school). The tests were content (topics) based and were pitched at five cognitive levels (ie. Knowledge and information - recall of definitions; techniques and skills -  solution;  translation of data into symbols or schema;  comprehension and inventiveness).  In addition, descriptive and attitude tests were given to students, teachers, school principals and an expert on the education system (Robitaille and Travers, 1992).

Some of the major findings of FIMS were that (Ibid.):

  1. All groups of students from the participating countries found the tests difficult.
  2. Males outperformed females in both age groups (although recent studies have shown that achievement differences between males and females have narrowed significantly in those countries over the last twenty years).
  3. For the 13 years of age group, parents’ level of education was found to be positively correlated with students’ achievement. With the older students there was much less variability (parents of students at this level were much more homogeneous with regard to educational level).
  4. Data on student attitudes indicated that the thirteen year old students in all participating countries had a more positive view about mathematics as a process than the senior students.

A problem with the FIMS results is that although one can talk about the students’ achievement after a given number of years, one cannot compare rates of growth or link teachers’ practice to the achievements of their students (Ibid. : p.693).

SIMS.

The Second International Mathematics Study’s (SIMS, 1980-1982) aim was to produce an “international portrait of mathematics education with a particular focus on the mathematics classroom” (Garden,1987). SIMS was a comprehensive survey of teaching and learning mathematics in twenty countries. It was to investigate the mathematics at three curriculum levels; Intended, Implemented and Attained (see Travers & Westbury model; 1989, below). The study was basically longitudinal (data collected over the school year) while the reduced version of the study was cross sectional which meant that data was collected at the end of the year. Students, teachers and administrators from a number of countries participated.

Table 2.3 Travers and Westbury 1989 model of the curriculum  framework for SIMS

Curricular antecedents

Curricular contexts

Curricular content

Level

System features and conditions

Institutional settings

Intended

System

Community, school and teacher characteristics

School and Classroom conditions and processes

Implemented

School or classroom

Student background characteristics

Student behaviours

Attained

Student

The Intended curriculum was defined at the national or system level, as the content as prescribed in curriculum guides and text books that are approved for teacher use. The Implemented curriculum was the content that was actually taught by the teachers in the classroom.  The Attained curriculum was what the students actually learnt as manifested by their achievements and attitudes.

Two population groups of students participated in SIMS. Population A consisted of students in the grade where the majority of the students were 13 years and 11 months by the middle of the school year. Population B consisted of students who were in the grade which was the last year of the secondary education system and who were studying mathematics as a substantial part of their academic program. Again some difficulties were encountered internationally in placing students into the two populations. A content (topic) by cognitive behaviour grid for the tests was developed (the four categories of cognitive behaviour were computation, comprehension, application and analysis).  Other instruments included questionnaires for teachers, schools and about the education system.

Some of the major findings of the SIMS according to the major levels of the study are (Robitaille and Travers, 1992) :

  1. At the country level, there were wide differences (even within developed countries) in the level of opportunity provided for students to complete education up to grade 12 or equivalent. There were also large between country differences at the senior high school level with respect to the proportion of the age group who were studying mathematics necessary for post secondary studies. There were similarities in so far as the topics studied were concerned at the population A level (although they were not necessarily studied in the same order).
  2. At the school or teacher levels, the findings were that the majority of the teachers at both the Population A and Population B levels were experienced and well qualified to teach mathematics. Teacher responses to questionnaire items concerning their teaching practice indicated that teaching mathematics was largely  “chalk and talk’, with teachers using whole class instructional techniques, relying heavily on prescribed text books. There were differences between countries in the amount of time the teachers devoted to review (especially at grade 8 level). For example, teachers in North America indicated that they spent a lot of time to review while teachers in France and Japan indicated that topics were not reviewed because it was presumed to have been taught in a previous grade. Other differences included class sizes and student-teacher ratios. Another interesting finding was the result that grouping students according to ability  did not necessarily enhance student achievement. Data from achievement tests showed that students from countries with mixed ability groups (which did not group students according to ability) performed very well.  That is, there is no evidence from SIMS that grouping students according to ability enhances performance.
  3. Findings at the student level indicated that most students in Population A and Population B believed that mathematics is important and they indicated that they wanted to do well in mathematics and that a good knowledge of mathematics was important to their careers. They also indicated that their parents shared these opinions and encouraged them to do well in mathematics. Students’ opinion of mathematics were not necessarily negative or overly enthusiastic. Population A students found the achievement tests to be fairly difficult. The longitudinal versions of SIMS indicated that growth in student achievement from pre-test to post-test was modest. Gender differences in achievement at the Population A level indicated that girls tended to outperform boys in computational skills and in Algebra while boys outperformed girls in geometry and measurement. It was thought that these differences could be attributed to the differences in boys’ and girls’ spatial ability.

It is to be noted that comparisons of performance levels between countries was extremely difficult, particularly at the Population B level where in addition to differences in retention rates, significant differences exist in  the curriculum. It also took a long time for the SIMS results to be published (approximately 12 years after the first date of planning).

Second International Science Study in PNG (SISS).

Papua New Guinea did not participate in the IEA studies on mathematics but did participate in IEA studies on science. The SISS project in PNG (Wilson, 1986) is described as an example of a large scale curriculum research project which was carried out in PNG. It also gives an example of what happened in the IEA studies at the national (or country) level.

Following the First International Science Study (FISS), SISS began with initial discussions in 1981. SISS used the survey research methodology. It used achievement tests (administered internationally), questionnaires, pencil and paper instruments to investigate factors which may affect achievement. The ultimate aim was to provide information that would assist education policy makers to manipulate factors most likely to raise levels of achievement.

The difficulties that were encountered in participating in an international study had to do with the suitability of international instruments for PNG situations, resources required and the level of participation of third world countries. The intended curriculum analysis was straightforward because PNG had a nationally prescribed curriculum. One other problem had to with the definition of students’ population levels (ages and level of grades). PNG had higher age groups so it became difficult to make comparisons across age groups levels. Which topics to include in international tests were determined by considering these groups.

The PNG curriculum case study which was prepared by the National Research Coordinator (who was also a University representative) included information such as the structure of the education system, school age levels, administration, teacher education, curriculum content, detailed description of nature and development of science curriculum at the primary and secondary schools. The international instruments included student, teacher and school questionnaires, attitude instruments for the students. The instruments were modified to suit the PNG situation (eg. language used so that the instruments were culturally and environmentally  appropriate). The instruments were then trialed at the three population levels. At the international meeting to consider the trial results, modifications suggested by the PNG trials were included in the change (eg. items that were considered inappropriate, wording of items, needs of individual countries to be catered for in spite of international study. The PNG versions of the tests were developed and sent to the schools.

The writer was not able to find any reporting of the results of the PNG study. Overall SISS results (international study results) can be found in studies reported by IEA (1988). The absence of any reports on the SISS in PNG could be attributed to the fact that the author of the above report (Wilson), who was also the university representative on the coordinating committee, left the country before the PNG results could be analysed.

TIMSS.

The Third International Mathematics and Science Study (TIMSS), also sponsored by the IEA was intended to be the main international assessment activity of the 1990’s (Robitaille & Donn, 1992). The intention was not only to measure students’ achievement in mathematics and science but also to investigate differences in curriculum and instruction and include alternative assessment options. TIMSS was designed for international comparisons to be done in a valid and reliable way, based on experience gained from previous IEA studies.

According to Robitaille and Donn (1992), TIMSS was to focus on the teaching and learning of mathematics and science at three levels of the school system : the grade at which most students attain the age of nine (not included in previous studies), the grade at which most students attain the age of thirteen, and the population of students completing the last year of secondary education. Participating countries carried out basic studies for the three populations but had options of investigations to choose from. For example, one option concentrated on the investigation of scientific and mathematical attainment of students completing secondary schools with specialisations in mathematics or science. Another option concentrated on students’ problem solving strategies. A third option was to investigate the linkage between teacher practices and students’ achievement and attitudes.

Data on student achievement in mathematics and science were collected through the use of multiple-choice items selected on the basis of international curriculum grids. These items, intended to measure cognitive behaviour at all levels, also included some open-ended questions as well as some performance tasks. Other questionnaires sought information about student and teacher background, school and classroom variables (including school climate, opportunity to learn, and time on task). For detailed description of TIMSS, see also Robitaille (1993).

The results of TIMSS were not published until the end of 1996 (Bodin & Capponi, 1996 : p.568). Some preliminary results are presented here. International comparisons of results showed that students in some Pacific Rim countries such as Singapore, Korea, Japan and Hong Kong did very well in mathematics (although the performance of Thailand was poorer). For example,  Scotland’s performance relative to the Pacific Rim countries was poor, especially in mathematics although Scottish pupils’ performance was better on certain aspects of mathematics and science (Scottish Office of Education and Industry Department, 1996). United States eight graders finished slightly below average in mathematics and slightly above average in science (Bracey, 1997a). Singapore students scored highest on the TIMSS (Bracey, 1997b)  In almost all the countries boys did better than girls in science and the difference was significant. The Australian results on TIMSS were reported by Lokan (1997).

Studies on curriculum - Ethnomathematics based

The studies that were noted in the previous sections were studies that focused on “school mathematics” based curriculum. In this section, examples are given of studies that were ethnomathematics (cultural mathematics) based.

Pompeu’s PhD thesis research looked at how cultural mathematics could be brought into the classroom in schools in Brazil (Pompeu, 1992). He was concerned with how to incorporate a cultural element into the mathematics curriculum. He made comparisons with the standard Brazilian curriculum approach (canonical-structuralist) and the ethnomathematical approach (Bishop, 1992a). “His research involved teachers in creating  six ethnomathematical micro-curriculum projects, and then using them in their classrooms. He has analysed the views of teachers, pupils and parents, regarding this approach using questionnaires, interviews and observations” (Bishop, 1992a). Each of these activities were based on the pupils knowledge and experience based on Bishop’s six fundamental activities. Results showed that the teachers who participated in the projects made significant changes in their views about mathematics teaching and the pupils became active learners of mathematics.

Pompeu used the intended, implemented and attained curriculum framework (Travers & Westbury, 1989 - see Table 2.3) to analyse the Brazilian curriculum (See also Pompeu’s Model - Figure 2.1)

Pompeu identified the Brazilian mathematics curriculum (intended, implemented and attained) as fitting into the “canonical structural” category. He compared this category  with the “ethnomathematical” approach where culturally oriented projects were initiated at the intended curriculum level and  teachers  taught (implemented) these projects. Results showed that the teachers who participated in the projects made significant changes (towards the  ethnomathematical approach) in their views about mathematics teaching (see chapter three, Section 3.1.1). Pompeu’s model is used here because the “canonical structural” and the ethnomathematical approach espouse concepts this study investigates - mathematics curriculum, cultural mathematics or ethnomathematical ideas, teacher beliefs about mathematics and culture and teacher practices.

Figure 2.1 approaches to the curriculum (Pompeu, 1992).

The canonical structural approach

The ethnomathematical approach

A GENERAL PERSPECTIVE
Mathematics should be seen as :

a) a theoretical  subject (it concerns abstractions and generalisations);

b) a practical subject (it is applicable and useful);

c) a logical subject (it develops internally consistent structures);

d) an exploratory and expalanatory subject ( it investigates environmental situations);

e)a universal subject (it is based on universal truths)

f) a particular subject (it is based on truths derived by a person or group of persons).

AT THE INTENDED CURRICULUM LEVEL
The mathematics curriculum should :

a) be culture free (its truths are absolute, and independent of any kind of cultural or social factors)

b) be socially/ culturally based ( its truths are relative, and dependent on social and cultural factors);

c) be informative ( it emphasises procedures, methods, skills, rules, facts, algorithms and results)

d) be formative (it emphasises analysis, synthesis, thinking, a critical stance, understanding and usefulness);

e) be conservative (it promotes control over the environment and the stability of the society)

f) be progressive (it promotes the growth about the environment and progress/change of the society).

AT THE IMPLEMENTED CURRICULUM LEVEL
Teachers should teach mathematics as :

a) a one way subject (mathematical knowledge is transmitted from the teacher to the pupils);

b) a debatable subject ( mathematical knowledge is discussed among pupils and teachers);

c) a separated subject (mathematical lessons do not rely on knowledge which students bring from outside of  school);

d) a complementary subject ( mathematics lessons are based on knowledge which pupils bring from outside school);

e) a reproductive subject (mathematical knowledge is taught from standard mathematics textbook):

f) a productive subject (mathematical knowledge is developed from the pupils’ own situations)

AT THE ATTAINED CURRICULUM LEVEL
Pupils should be able to :

a) find correct answers to problems (it is the pupils’ final answers to problems which are important);

b) analyse problems (it is the pupils understanding of the structure of a problem which is important);

c) Use the formal mathematical method to solve problems (these methods are the ones that will produce the right solutions);

d) Use appropriate procedures to solve problems (it is the pupils’ ability to determine the appropriate solution procedure which is important);

e) reason mathematically about problems (it is how to solve problems mathematically that is important for pupils to know);

f) Make mathematical criticisms about problems (it is why to solve problems that is important for pupils to know).

2.3.4 PNG Studies on Mathematics curriculum.

“Mathematics curriculum” is interpreted here to mean the intended, implemented and the attained curriculum. The PNG studies on the mathematics curriculum will therefore refer to the studies conducted in PNG on the mathematics content (intended curriculum), the teaching of mathematics (implemented curriculum) and the learning of mathematics (attained curriculum).

A perusal of literature indicates that there is an abundance of PNG studies which focus on factors influencing student achievement or learning of mathematics (at the attained curriculum level). These PNG studies differ from the IEA studies in that, although the aim of the IEA studies was also to investigate factors affecting student achievement, apart from the fact that they were large scale studies, the IEA studies also investigated the variables at the intended and the implemented curriculum levels. These included, for example, factors such as teacher variables and curriculum content variables which could also affect student achievement. The PNG studies appear to focus on the student variables.

Low attainment in mathematics has always been a concern to mathematics educators at all levels of education in PNG. The concern was heightened when some tests showed that PNG students performed less well in some tests than students in other countries. For example, Lean and Clements (1981) in summarising developmental studies cite studies by Shea (1978) which reported that generally speaking on Piagetian conservation tests, PNG students performed less well than selected western European groups. There was a three year delay which lengthened to six years with the more complex conservation tasks (p.2). They also cite Jones (in Lean & Clements, 1981) who investigated Engineering students at the University of Technology (Lae) and concluded that they performed roughly at the same level as ninth grade pupils at English Grammar schools. Although Lean & Clements concluded that there are no cultural differences in basic cognitive processes (as did Lancy, quoted by Lean & Clements,  who argued that there is no evidence to support the view that any cultural group lacks the basic processes of abstraction), the concerns about the low level of performance of PNG students in mathematics at all levels of education is very much a current issue as it was more than fifteen years ago.

In an attempt to provide explanations for low achievement in mathematics, researchers investigated several factors. For example, influencing factors investigated include : language (eg. Clarkson, 1984; 1987; 1992; Clarkson & Galbraith, 1992), language and /or  cognitive ability or styles of learning (eg. Clements & Lean, 1981; Lancy, 1981; 1983; Souviney, 1983; Saxe, 1991), spatial ability (eg. Bishop, 1979; Lean & Clements, 1981), Piagetian tasks - formal operational thought (Wilson & Wilson, 1981), logical reasoning ability (Wilson & Wilson, 1984; Wilson, 1988), influence of prior knowledge (eg. Saxe, 1985), student attitudes (Wilkins, 1985, Roberts, 1989) and attributions of success or failure (eg. Clarkson & Leder, 1984).

As can be seen from the above, the focus of much of the research in PNG was on language and cognitive ability styles of learning mathematics. Clarkson (1987) noted that it is not surprising that the effect of language in learning mathematics has been the focus of many studies in PNG. Over sixteen percent of the world’s languages are spoken there (according to the PNG Minister for Education, Science and Culture, there are over 800 languages in PNG - Waiko, 1997a). In PNG, school mathematics is usually taught and learnt in English, which for most students is a second (or even third) language. A language proficiency in and an understanding of the language of instruction were considered important factors in achievement in mathematics.

Clarkson’s (1987) review of literature on language and mathematics from Papua New Guinea notes the findings from some PNG studies on various aspects of how language affects mathematics learning. Studies cited showed the language factor as a possible cause of problems in mathematics learning include (see Souviney, 1983; Clarkson, 1983; 1984; Lean and Clements, 1981; and  Jones, 1982; also cited by Clarkson, 1987). The learning difficulties were usually associated with what Clarkson termed as “readability” problems (eg. the ability to read, comprehend and compute mathematical word problems). In addition to the above studies, Clarkson (1991) also makes references to studies by Suffolk (1986) and Sullivan (1983) which identified language as a factor which impinged on student achievement in mathematics. In a study of year 6 PNG students, Clarkson concluded that one third or more of the errors in students’ processing of word problems “could be classed as reading or comprehension errors and could therefore be language-related.” (Clarkson, 1991 : p.32). Based on his studies (eg. Clarkson, 1983, 1984, 1989, 1991) with PNG students, Clarkson was able to conclude that there was a connection between language competency and mathematics achievement (language competency is used here to mean that a certain level of proficiency is displayed by the student, for example, in reading, comprehending, understanding and speaking the language of instruction). A similar conclusion was reached by Saxe (1988) who suggests that there is a link between the students’ language background (eg. mono-lingualism and bilingualism) and their mathematical achievement .

Other PNG studies identified language competency plus one other factor as likely influences on mathematics achievement. In the examples noted earlier, language and cognitive ability were identified as likely factors by Souviney (1983), Lancy  (1981,1983) and Saxe (1991). Clements and Lean (1981) suggest that differences in performance level (between national and expatriate students attending primary schools in PNG) is a function of language and home background factors. Research on the cognitive ability of PNG students in learning mathematics was certainly an important part of the Indigenous Mathematics Project (IMP) which is described later in this section. The writer suggests that one aspect of language which the Lean (1986, 1994) study emphasised but which requires further investigation is the documentation of “mathematical vocabularies” in the different languages (not just the counting system) found in PNG. An understanding of the meanings and the derivation of the vocabulary could provide an understanding of the conceptualisation of mathematical knowledge by different socio-cultural groups in PNG.

It is not surprising that language ability and cognitive ability were the focus of many of the investigations on mathematics achievement. In studies world wide, especially with language minority students, explanations for low achievements in mathematics usually included these two factors as possible variables to consider (Cocking & Mestre, 1988 : p.21). Cocking & Mestre (1988) note that apart from the language and cognitive ability patterns, low socioeconomic status and culture (values, parental assistance and motivation) might contribute to achievement problems with language minority students. They propose a research model which incorporates all these factors in the context of three major categories of influences upon school learning : Entry characteristics of learner; Opportunities provided to the learner; and Motivation to learn (Cocking & Mestre, 1988 : p.20). For example, poverty, language skills and cognitive abilities are entry characteristics of learner. Cultural environment (background) factors fit into the “opportunity to learn” category, for example, the home culture level of support (eg. parental assistance) or institutional level of support (eg. ghetto schools may have different student teacher ratio, teachers with less training or teachers who are unable to speak the student’s home language). Cultural values or parental values and expectations fit into the motivation category. Other studies quoted by Segada (1992) suggest that there is a relationship between degree of proficiency and mathematics achievement in that language. Segada also quotes other studies which suggest a relationship between bilingualism and mathematics achievement, the degree of bilingualism and the learning of mathematics.

In this thesis the interest is in the influence of cultural factors on the learning of mathematics. In writing about issues dealing with linguistic and cultural influences on mathematics learning, Saxe (1988) argued that culture constitutes a complex of intertwined factors, one of which is the language background of children. In his study of children in the Oksapmin area (West Sepik Province) of PNG, Saxe (1988) observed childrens’ use of the body parts system (a way of counting in the local area where different parts of the body represented the numbers) to do sums or arithmetic learnt in schools. Children used and adapted knowledge which are part of their home culture to that culture presented in school. He suggests that “studying cultural supports for mathematics development and how children utilise different backgrounds in coping with school mathematics curriculum, can offer insights about the sources of language minority children’s successes and failures in the mathematics classroom.” (Saxe, 1988 : p.61).

In the PNG studies, for example, the influence of language on mathematical achievement, cognitive ability studies, studies on spatial ability and logical reasoning etc. are viewed by this writer as illustrations of cultural influences on the learning of mathematics. This would fit into the “entry characteristics of the learner” category (Cocking & Mestre, 1988). The writer suggests that the mathematical knowledge the learner has acquired out-of-school, in a traditional cultural context makes up the entry characteristics of the learner.  This mathematical knowledge may consist of the child’s ability to count, design, locate, measure, design, play and explain (Bishop’s, 1988a,  fundamental activities - see section 2.1.3). This knowledge is gained in a cultural context.

Language becomes an important cultural factor in learning mathematics, not just because of the inadequate language skills in the language of instruction that the learner may posses, but also because the learner possesses knowledge learnt in a different language, in a different cultural context with possibly different interpretations of the mathematical concepts. Clarkson (1991) makes the important point that “cultural differences can affect mathematics learning in number of ways : clearly through language impinging on the content of the curriculum directly, but also through ways of behaving and knowing that are also embedded in the language. These factors need to be considered if the non-English speaking students are going to have every opportunity to reach their full potential.” (p.49). As noted in the earlier section on “PNG studies of mathematics in a cultural context”, Lean (1994), who documented approximately two thousand counting systems in the South Pacific), placed a lot of emphasis on the fact that counting systems were a part of language and that language was embedded in culture. Ellerton and Clarkson (1996 : p.1017) quote Mousely, Clements and Ellerton who claim that “one of the most fundamental aspects of all cultures is language …. centrality of language factors in all aspects of mathematics teaching and learning”.

Other studies

The studies on students’ learning of mathematics mentioned above, however, only refers to one area of the curriculum - the attained curriculum level (if categorised according to the Travers & Westbury Model, 1989). PNG studies on mathematics teaching (eg. effective teaching, mathematics teacher styles or practices) and the content (intended curriculum) are limited, especially studies aimed at the secondary level. The Indigenous Mathematics Project (IMP) however, is an example of a  project which included all three levels of the curriculum although the emphasis was very much on the cognitive development of children (this project is described below). The evaluation of the high school mathematics curriculum by Hayter (1982) also had the potential to include all three levels of curriculum (we also give a brief description of this study).

The Indigenous Mathematics Project (IMP) : 1977 to 1981.

Souviney, (1981) describes the Indigenous Mathematics Project in the following way :

“The project was established in 1977 by the PNG government to investigate various aspects of traditional mathematics development. During the first phase (1977 - 1979) of the five year program, basic research in cross cultural cognitive development was carried out and indigenous counting, classification, and measurement systems used throughout the country were documented. During the second phase the project (jointly funded by the Department of Education and UNESCO), pilot instructional materials were developed and trialed. The intent was to assess the feasibility of utilising complementary aspects of indigenous and western mathematics as basis for developing culturally relevant student materials, instructional aids, and teachers guides which reflected the practical constraints of the community school environment. The results of the IMP work are intended to inform future curriculum development in an effort to provide more appropriate materials and learning aid for community schools throughout PNG”

(Souviney, 1981: p.1,2; see also special issue of PNG Journal of Education, 1978, volume 14).

As a part of the IMP, a trial project was carried out in PNG to develop appropriate material for teaching mathematics at the primary school level. Case studies were conducted at five schools at various locations around PNG (Souviney, 1983). Instructional materials were developed which were then trialed by teachers who had received in-service training. The six weeks implementation of these materials at the schools were observed and recorded by a team of researchers.

The rationale behind using locally available instructional materials aids in teaching mathematics was stated quite clearly by Souviney (1983). “… the primary task of the teacher becomes that of helping children adapt traditional numbering and measurement knowledge into the mathematical context encountered in school. Since  direct translation is rarely possible, it is frequently necessary to extend and adapt cultural knowledge to help children define concepts that are not expressible in local terms.” (p.184). The study also recognised that children attending will have prior knowledge learnt out-of-school so that one of the aims was to utilise “locally derived knowledge” (p.184). “When understood by the teacher, such invented procedures rooted in the child’s experience can provide a useful starting point for the development of symbolic algorithms.” (p.185). In the study, the students were presented with tasks which required manipulation of concrete models, graphics and symbols.

The classes at the five selected schools were observed in operation by a research associate who recorded the implementation efforts of the teacher and the responses of the students. The teachers were interviewed before and after each lesson (the lessons were also discussed). Several measures of achievement and cognition were also administered. Each school was selected because of its uniqueness and reflected a wide range of environmental variables which promoted student achievement (for example, language and cultural factors, staff experience and motivation, availability of instructional materials, accessibility and frequency of inspections etc. The student population was made up of students from Manus, Western Highland, West Sepik and, Southern Highlands provinces and the National Capital District.

The IMP staff developed 30 minute lessons for grades two, four and six. Each school scheduled five periods of IMP mathematics per week. To control for variance due to teacher factor, one teacher taught the three grades in each school. The lessons were adapted from the “Mathematics for Community schools” teacher’s guide. The emphasis on the lessons was on base-10, place value and number operations, activities which provided transition between concrete and symbolic representations of whole number operation algorithms, and classification activities which utilised locally produced attribute materials (Souviney, 1983 : p.197).

The achievement results showed significant differences between schools although the patterns of significance were not consistent. For example, the highlands  schools generally showed lower mathematical achievements than the coastal schools although the Southern Highland school consistently outperformed the other schools. The NCD school scored highest in English but showed lower placement in mathematics. Mean mathematics scores increased for all the students between the pre-tests and post-tests and there were increased scores between post-tests and retention tests. The students were also far more successful in solving measurement, number operations and mathematical language than those involving problem solving (p.203). Some conclusions from the study were that : although there were significant differences in overall levels of mathematics and language achievement among the five schools, all schools showed consistent gains over the six week period; mathematics and language achievement varied significantly, grade levels and sites and measures of English reading ability and cognitive development were highly correlated, especially in grade 6 (Ibid : p.209). Souviney (1983) concluded that “examples of counting and measurement systems indigenous to PNG should be incorporated into the development of appropriate concepts in number, operations, measurement and geometry” (p.210). He advocated a three step instructional procedure - concrete model, pictorial, symbolic manipulation - to introduce number and operation concepts and the mathematical content to reflect the cultural heritage of the country.

Evaluation of the High School mathematics curriculum.

Hayter (1982) conducted an evaluation of the high school mathematics curriculum in PNG. It is interesting to note that in the only official evaluation of the PNG High school mathematics curriculum, the following recommendations were made :

  1. Due recognition given (both by inclusion and by respect for its use) to traditional counting and measurement systems.
  2. Development of a case for standard measures and ways of comparing in order to achieve precision, allow comparison and broaden application.
  3. Study of traditional pastimes and practices, where appropriate, in a mathematical way.
  4. Applications of skills and techniques developed in mathematics to local situations; to aspects of national life which will impinge at the village level, or are likely to affect high school graduates (eg. postal charges, timetables); and to problems which arise at school and at village level.
  5. Use of PNG currency, place names, food stuff in book work examples and exercises.
  6. Showing mathematics to be an international language in which members of a developing country must achieve competence.

2.3.5 Culturally oriented curriculum.

This section begins by looking at the main themes that seem to emerge from the literature on the educational implications of ethnomathematics. In the later half of this section, the curriculum implications are looked at, in particular, illustrations of culturally oriented curricula from around the world are given.

Educational Implication.

“ One of the great educational challenges of the present time concerns how school mathematics teaching should take learners’ out-of-school knowledge into account.” ( Bishop et al., 1993: p.1).

The educational implications of the ethnomathematical ideas are considered in this section.

Bishop (1988a: p.187) refers to the mathematics education as the induction of the young into a part of their culture through the “process of cultural interaction” (Bishop, 1992a : p.4). He uses the terms enculturation (ie. induction of child into their home culture) and acculturation (induction of person into a culture which is different from their home culture). He queries whether a child’s induction to “western” mathematics, is a part of the home or local culture. The answer will differ depending on the home background. In some countries, “western” mathematics (school mathematics) and even the notion of schooling is regarded as alien. Bishop cautions against “intentional” acculturation which does not try to preserve the child’s home culture (p.188). That caution extends (this writer adds), to mathematics teaching which assumes that as far as mathematical knowledge is concerned, the child’s mind is “tabula rasa” and does not take into account the learners’ out of school knowledge. Bishop (1988a,1993) also suggests that the most important contributor to new information is the extent of the previously learnt knowledge but the most important prior knowledge may be what is learnt outside the school context and will be embedded in totally different social structures. This idea is supported by literature on research in ethnomathematics (see section 2.1.4). The prior knowledge may consist of  cultural mathematical knowledge the child brings to school (the nature of CM and mathematics is discussed in section 2.2.1).

The notion that learning new knowledge must build on prior knowledge the child possesses or brings to school is supported by proponents of constructivism. Under this idea learning is viewed as a personal construction where the  learner “constructs” his or her own understanding (Gunstone, 1993). Clement (1990) states that in the constructivist view, knowledge cannot be transmitted directly, it must be constructed by the student from elements of prior knowledge (ie. prior knowledge activated, combined, criticised and modified to form new knowledge). Resnick (1988 cited by Schoenfeld ,1992 : p.340) supports this idea but adds that it is also a social activity where, it is argued, mathematical knowledge is gained through “ meaning construction” as a result of the socialisation process (rather than through the traditional “instruction process” which is the teaching of bodies of knowledge). In other words, how one makes sense of the knowledge he or she gains will depend on the social context or culture.

Nunes (1992 : p.557) makes suggestions on how the teacher could use out-of-school knowledge in the school. Nunes (1993 : p.35) suggests that bringing out-of- school mathematics into the classroom means giving students problems which they can mathematise in their own ways and in so doing come up with results (methods, generalisations, rules etc.) which approach those already discovered by others. It is not simply taking into the classroom an everyday problem and using algorithms learnt to solve problems. The teacher should start by determining which concepts he or she wants the students to learn and then identify the everyday mathematics which use that concept. The teacher must also consider whether the students have used the concept in everyday life. Nunes gives examples of approaches to teaching specific aspects of mathematics with support of everyday problems. Real life mathematics should be used in open ended way.

Another issue which Bishop identifies deals with the mathematics curriculum in schools, particularly in societies with various ethnic groups (specifically minority groups). “To what extent should mathematical ideas from other cultures be used ?” How should the curriculum be structured for this to happen ? (Bishop,1988a : p.188). This question is also relevant to countries like Papua New Guinea where the majority of the children come from traditional backgrounds whose mathematical practices and ideas may be different to the “western” mathematics which is learnt in school and where there are also differences within PNG cultures. Bishop’s suggestion is to use mathematical ideas from the child’s home culture. His six fundamental activities (1988a : p.182) could serve as a useful framework for a curriculum which could be structured around these activities.

It is also suggested that in some cultures there is a psychological and cultural blockage when learning mathematics which is often viewed as a “western” subject (the implication being that it has nothing to do with their culture).  According to Gerdes (1988 : p.140), identification of mathematics in cultural activities and artefacts will help get rid of psychological and cultural blockage and give cultural confidence. The writer’s own comment is that in some cultures it is probably important to break down the barriers as a starting point by retrieving “frozen” mathematics but it is not good enough to stop there, otherwise it becomes a sentimental reason and being sentimental does not solve mathematics education problems. Gerdes (1988) provides examples from Mozambique where identified mathematics from traditional cultural activities formed the basis for a new mathematics curriculum.

It is also believed that showing examples of mathematics found in traditional culture would help eliminate the notion that mathematics is a “western” subject. Perhaps the point that is of importance is that made by D’Ambrosio (1990: p.23) who states that negative self esteem is particularly strong amongst minority groups. Ethnomathematics avoids the problem of negative self esteem. Barton (1991) states that it is good for those whose mathematical practices have been overtaken, subsumed by superior mathematics practice. Bishop (1993 : p.7) alludes to western mathematics as the weapon of western cultural imperialism and “gradual acculturation by dominant cultures and assimilation of new ideas believed to be more important than traditional ones.” It is true that mathematics that can be identified in a culture could give a sense of cultural identity, ie. to study mathematics because it is the mathematics that is practised in one’s culture. Mathematics which is practised by cultural groups is identifiable. D’Ambrosio (1985) refers to this mathematics as a different kind of mathematics; it differs from school mathematics in its historical origins and patterns of reasoning. But an important educational implication for ethnomathematics still remains identification of mathematics in culture which could be taught and building on the prior “cultural mathematical” knowledge the child brings to school.

There are of course questions that remain. For example, how do you identify the mathematics in different societies, in traditional cultural activities ? How do you identify the mathematical ideas that the child comes to school with ? Who identifies them, teachers or curriculum developers ? What about in societies which are very diverse, is it feasible to expect teachers (or curriculum developers) to identify mathematics from the child’s home culture ? Answers to these questions have educational and curricular implications. An easy answer to the last question would be to say that they are so diverse, there are no common grounds, so we should teach them some new knowledge. This would have the same implications as saying, as far as mathematics is concerned, treat the child as having no mathematical knowledge, the only mathematical knowledge the child has is that learnt in school. The alternative is to acknowledge that diversity exists in societies and that the individuals that come to school will have knowledge of a variety of mathematical ideas and practices learnt out of school. New knowledge should build on this existing knowledge.

Curriculum implications - Culturally oriented curriculum.

Discussions on cultural mathematics or ethnomathematics are not complete unless the issues that deal with the curriculum implications are addressed. There are some important questions that come to mind when the implications for the mathematics curriculum are considered. For example :

Should we after identifying cultural mathematics or out-of-school mathematics incorporate these into school mathematics ? Why ? Will it not pose the same problems that we accuse school mathematics of ? Why incorporate identified out-of-school mathematics into school mathematics curriculum ? Will the mathematics be of a sufficiently high level to incorporate into secondary school mathematics ? What is hoped will happen if out-of-school mathematics is included in the curriculum ? How could this be achieved ?

Bishop (1992a) offers the following framework of cultural conflict and the responses to this conflict in a mathematics education context. 

Table 2.4 Bishop’s Approaches to culture conflict.

Responses to cultural conflict

Assumptions

Curriculum

Teaching

Language

Examples

Culture free traditional view

No culture conflict

Traditional canonical

No particular modification

official

In many schools

Assimilation

Learners’ culture should be useful as examples

Some learners’ cultural contexts included

Some modification of teaching for some learners

Official, plus relevant contrasts and remediation for second language learners

Multi-cultural approach Zaslavsky Girl friendly mathematics

Accommodation

Learners’ culture should influence education

Curriculum restructured due to learners’ culture

Teaching system modified as preferred by learners

Learners’ home language accepted in class, plus official language support

Anti-racist Critical mathematics education Ethnomathematics projects

Amalgamation

Culture’s adults should share significantly in educational decision and provision

Curriculum jointly organised by teachers and community

Shared or team teaching

Bilingual, bi-cultural teaching

Bi-cultural- bilingual Maori Aboriginal Family mathematics

Appropriation

Culture’s community take over educational provision

Curriculum organised wholly by community

Teaching entirely by community’s adults

Teaching entirely by community’s preferred language

Radical ethnomathematics Gerdes, Pixten

From Bishop’s analysis it can be seen that there are several possibilities of having “culturally oriented” curricula.  The “culture free”, “canonical” curriculum is what was referred to in this study as the normal “school mathematics” oriented curriculum. Possible “cultural mathematics” oriented curriculum which take into account student knowledge learnt out-of-school include curriculum which takes into account some learners cultural context (assimilation), curriculum restructured due to learners culture (accommodation), curriculum jointly organised by teachers and the community (amalgamation) and curriculum organised wholly by the community (appropriation).

A question for the PNG context : Is it possible to have a culturally oriented curriculum in a PNG context where the diversity is vast even in localities ? (especially if teachers are required to use local examples of traditional mathematics). Which of the above “models” suit the PNG situation ? See section 7.2.1 in chapter seven for the recommendations that were made for the mathematics curriculum in PNG.

Examples of culturally oriented mathematics curricula

Examples of culturally oriented mathematics curricula from around the world illustrate that it is possible to construct and implement alternatives to the “canonical universal curriculum” (the term used by Bishop, 1993 : p.7; see also Howson  & Wilson, 1986).

Mozambique incorporates mathematical ideas from rural and non-western cultures into their mathematics curriculum.  For example, as reported in section 2.1.3 in Chapter Two, Gerdes (1988) used basket weaving patterns from Mozambique to illustrate “geometrical thinking”. He argued that incorporation of examples of mathematics from traditional culture into the curriculum helps get rid of “psychological blockage”. In the two volumes of “Sona Geometry”, Gerdes (1994a) shows examples of mathematical knowledge from the traditional sand drawings (“sona”) of the Tchokwe and suggests possible uses of “Sona” in primary and secondary mathematics. Gerdes (1994b) presents a compilation of articles by several writers which show other examples of the use of cultural mathematics (ethnomathematics) and ethnoscience in the classrooms in Mozambique.

In Brazil, Pompeu (1992) explored the possibility of bringing ethnomathematics into the classroom by involving teachers who created micro-curriculum projects and used them in their classrooms in Brazil. In New Zealand the indigenous issues confronting mathematics and science education have been addressed (Begg, Bell, Biddulph, Carr, Carr, Chesney, Loveridge, Mckinley & Waiti, 1993). Of particular interest was the development of curricula based on the Maori culture (language and knowledge). Barton and Fairhall (1995) present the issues dealing with mathematics in Maori education in New Zealand.

The New Zealand studies have reported some success in the teaching of mathematics using indigenous languages. Begg (1993), for example, noted the growth of bilingual teaching using the Maori language (the main language of instruction is English) in schools in New Zealand. While the teaching of mathematics in the Maori language has its limitation (eg. lack of standardised mathematics vocabulary in Maori, lack of resources for mathematics written in the Maori language, lack of teachers who are qualified to teach mathematics who are fluent in the Maori language), there were some positive effects from these bilingual programs in relation to social outcomes and attitudes to Maori language and culture while the effects on mathematics learning and attitudes were still positive but less impressive.

Barton and Fairhall (1995) give a description of the development of a formal Maori mathematics curriculum. The mathematics curriculum was to be more than just a translation of the English curriculum into a Maori language curriculum. Crucial to the development of the Maori mathematics curriculum was the development of the Maori mathematics vocabulary. The vocabulary issues were resolved with much community consultation. The development of the Maori vocabulary was such that the mathematics curriculum was developed to senior high level and this made it possible to do mathematics up to university level in the Maori language (Barton & Fairhall, 1995). 

Presmeg (1996) reported on an ethnomathematics research project which was conducted in a high school in the United States of America. “The purpose of the project was to work with students and teachers to develop viable ways of using the diverse cultural and ethnic background of students as a resource for the learning of mathematics” (Presmeg, 1996 : p.2). This project was initiated within the context of the National Teachers of Mathematics (NCTM, in the US) vision for the future of mathematics education which advocated connections between the mathematical content and the home background of learners. The project was conducted in a high school which had students from multicultural backgrounds. A “bank” of authentic student activities  (what the students actually did outside of class) was developed. This was collated by interviewing a number of students. The idea was to get students to see the links between mathematics and the everyday activities the students participate in. The cultural backgrounds of the students were also investigated.

The researcher was able to teach a mathematics lesson using student group activities to illustrate the link between mathematics and music (spatial and algebraic patterns of rhythm and pitch). The students were then interviewed about their views about the nature of mathematics.  It was concluded from this project that traditional mathematics teaching does not facilitate a view of the nature of mathematics which encourages students to see potential for mathematics outside of classroom and that an “introduction of ethnic and home activities into mathematics classrooms  needs to be accompanied by a recognition of the value of such activities, and such recognition may involve a change of belief about the nature of mathematics, on the part of students and teachers” (Presmeg, 1996 : p.4). However, on another aspect of the project, the students were able to describe the mathematical aspects of the cultural and home activities and identify the potential for mathematics in other activities.

Irons, Burnett and Foon (1994) in their book, “Mathematics from many cultures” give illustrations of mathematics from cultural activities from around the world and their possible use in the classroom. They give examples of topics of “school mathematics” which can be taught in conjunction with  these mathematical activities from diverse cultures. The book is accompanied by illustrative charts.

Example from PNG. Souviney (1981, 1983) and Lancy (1981, 1983), reported on the Indigenous Mathematics project (IMP: 1977 to 1981) in Papua New Guinea (PNG) which attempted to identify “indigenous” mathematics with the intention to “develop appropriate curriculum” (See description of the trial project in section 2.3.4 of this chapter). The primary school mathematics curriculum uses some examples of counting systems from some PNG cultures to introduce number concepts.

2.4 Problem restated.

Studies of “ ethnomathematics” or “cultural mathematics” (CM) over the last decade have created an awareness of and provided evidence of the “other mathematics” which is “out of school” and is distinct from school mathematics. However, the impact of CM  on the curriculum is not at all clear. Issues that are addressed include these questions : Should this “other mathematics” be taught in schools ? Should we include it in the school curriculum ? If so, how should it be used ? These questions and issues can be seen as a part of the overall debate about what knowledge to include  or exclude from the curriculum. Inherent in these content issues are questions about what values, beliefs and ideologies to transmit via the curriculum.

Taylor and Richards (1985) describe curricula (Section 2.2.1) as a medium for transmitting part of cultural stock and a means of controlling people by exposing them to particular values and beliefs or conceptions of education. The curricula then are more than just bodies of knowledge but have cultural and political significance. Proposals about what should be taught are put forward by groups or individuals with similar conceptions and ideologies about education. In the PNG context, concerns about the perceived failure of the education system to equip students with skills to fit back into the communities and the desire to include in the schools curriculum some knowledge and skills that are associated with some aspect of life in the communities (culturally oriented knowledge and skills), indicate that it is the wish of parents and the community at large, that education and schools be regarded as institutions of social reproduction of the society’s cultures and values (not just mere transmitters of the societies knowledge through the curricula). It would seem the PNG community at large views the school system and the curriculum in particular as media for transmitting the cultural values, beliefs and what they consider as “relevant knowledge”. This is what was referred to in chapter one as the “cultural dimension” of the curriculum. That is, the inclusion of knowledge and skills, beliefs and  values from traditional cultural activities in the school curriculum. However, the problem is, this “cultural bias” does not seem to extend to the mathematics curriculum.

As seen in the earlier sections of this chapter, conceptions (or ideologies) of  education depend on perceptions about desirable ends of the educational enterprise, nature of knowledge, children, teaching and learning which are often manifested in the educational policies. They also reflect the ideologies of the key players or pressure groups in the educational enterprise. These ideologies and conceptions are supposed to be encompassed in the curriculum. The PNG Philosophy of Education and the educational policies encourage “culturally  based” curricula but the policies do not seem to extend to the mathematics curriculum. The policies for the mathematics curriculum are either non existent or specifically encourage a culturally based curriculum only at the elementary level. Policy statements about the “cultural dimension” of the secondary mathematics curriculum are vague. This is in contrast to the other strands in the curriculum which are encouraged to be culturally based from the primary through to the secondary levels. There are also mixed reactions to the idea. While education officials are quite happy to talk about culturally based curricula, their views that mathematics is needed for advancement in a technological world and therefore should not be culturally based, seem to contradict these government policies. Other key players at various levels of the curriculum implementation process have differing views. This also illustrates the “problematique” which is the dynamic for this study.

It is accepted that teachers have their own conceptions about the curriculum content, how it should be taught and the desirable ends of education. This gives rise to two lines of thoughts. The first, as seen in section 2.2.1 of this chapter, is that it does not matter what ideologies and conceptions encompass the curriculum, it is the teachers’ conceptions that get transmitted to the students. Teachers’ interpretation of policy is influenced by beliefs about mathematics teaching and learning ( Remillard, 1992). Research on educational policies and practice (Curriculum policy and practice) suggest that external policies have little impact on classroom practice; teachers will interpret policies and implement according to their own beliefs.

The second thought acknowledges that it is the teachers’ conceptions about mathematics that gets transmitted to students but also recognises the complex relationship that exists between these beliefs and the context into which these beliefs are put into practice. These teacher beliefs are also influenced by “ … the values, beliefs, and expectations of students, parents, fellow teachers, administrators, the adopted curriculum, assessment practices, and the values and philosophical leanings of the educational system at large…. ” (Thompson, 1992 : p.138; emphasis added). In other words, these factors including the  values and philosophical leanings of the educational system are also important because they affect teacher beliefs and practices.

In both cases, it highlights teachers’ beliefs about mathematics as an important factor in not just what knowledge and skills but also what values and beliefs are actually transmitted to students.

The question is, what beliefs and values about mathematics are conveyed at the three levels or modes of transmission - via policy by administrators and policy makers, via intended curriculum by curriculum developers and writers, via practice (implemented curriculum) by teachers - to the attained level (attained curriculum - what belief is actually transmitted to the student).

Figure 2.2 below summarises the key issues at stake here

Figure 2.2 The problematique

Levels

   

Cultural dimension of knowledge
(content)

Which ?? to be transmitted to students

Whose values/ beliefs.

General curriculum

Mathematics curriculum

Concerns, expectations

value / beliefs

Parents / community

 

 

knowledge
--------------------
culturally based not culturally based ??

Policy

value / beliefs

Admin./ policy makers

   

knowledge

--------------------

culturally based

not culturally based

Intended Curriculum

value /beliefs

Curriculum developers / writers

   

knowledge

--------------------

culturally based

not culturally based

Implemented curriculum -Practice

value / beliefs

Teachers

 

 

knowledge
--------------------
?? ??

Attained curriculum

values / beliefs

Students

   

knowledge

--------------------

??

??


At the expectation / concerns level, we have the parents and the community at large with their expectations of the values and beliefs which they want transmitted to the students. Their expectations of the knowledge is that the curriculum be  culturally based.

At the policy level, we have the administrators and the policy makers with their expectations of the values and beliefs which they want transmitted to the students. Their expectation of the knowledge is that the general curricula be culturally based. This expectation extends to the mathematics curriculum but only at the elementary levels (not secondary).

If the problematique is viewed according to the Travers and Westbury (1989) curriculum framework, at the intended curriculum level, we have the mathematics curriculum which is not culturally based although the official policies encourage a culturally oriented curriculum. At this level also we have the curriculum developers and the writers with their expectations of the values and beliefs which they want transmitted to the students via the official curriculum.

At the implemented curriculum level, we have the teachers with their expectations of the values and beliefs which they want transmitted to the students. Their expectations and beliefs of the knowledge that is to be transmitted to the students, particularly their beliefs about the mathematics curriculum, are not known. One of the aims of the research is to investigate teachers’ beliefs about the mathematical knowledge which they want to transmit to the students. In fact this level will be the focus of the study.

At the attained curriculum level, we have the students with their values and beliefs about mathematics and the mathematics curriculum. One of the aims of this research is to investigate what these beliefs are and what the teacher’s role is in influencing these beliefs.

As can be seen from the above, there is an expectation from people at the various levels that cultural values and beliefs be transmitted via the curriculum although it does not seem to extend to the mathematics curriculum. The “cultural dimension” is deemed important as noted at the concerns level, at the philosophy and policy levels. The question one needs to ask is, What are the teachers’ (those at the implemented curriculum level) perceptions about the cultural dimension of the mathematics curriculum? The study should investigate teacher perceptions about mathematics and the mathematics  curriculum.

The overall questions should be : Whose beliefs and values and conceptions of education do we wish to transmit to students ? Or indeed whose beliefs and values are actually transmitted to the students ? If, as noted in Section 2.2.1 of this chapter, it does not matter what the curriculum is, what ideologies we wish to transmit through the curriculum, the teachers beliefs and conceptions are the most important as these get transmitted to students, then the question becomes, What beliefs about mathematics are actually transmitted to students ? In other words, What beliefs about mathematics are conveyed at the classroom level ?

It is clearly important to investigate teachers beliefs about mathematics and the mathematics curriculum (ie. at the implementation level). In general, the problem can be viewed as a mismatch between policy and practice (see Figure 1.2 in chapter one). The investigation will therefore concentrate on teachers who are the key  players in the practice or implemented curriculum level. It will concentrate on investigating teacher beliefs about mathematics and culture and what happens in practice.

Figure 2.3 below gives an illustration of the between teacher beliefs and practice

Figure 2.3 Proposed research - teacher beliefs and practice.

Teacher beliefs

 

Teacher practice

- mathematics curriculum
- intended, implemented, attained
- mathematics in general
(nature of mathematics)

 

What actually happens in classroom.

2.5 Research Questions.

The following were the research questions for this study.

  1. What beliefs and conceptions do teachers, student teachers and students have about school mathematics (SM) and cultural mathematics (CM), nature of SM and CM, perceived usefulness of SM and CM?
  2. What is the status of SM and CM ?
  3. Are there any conflicts between teachers’ conceptions of CM and school mathematics ?
  4. What influence do these teacher conceptions and beliefs about CM (or school mathematics, nature of mathematics, perceived usefulness of mathematics and CM) have on their classroom practice ? In particular to what extent do the teachers bring CM into the classroom.
    What do teachers view as constraints to bringing CM into classroom ?
  5. Are  there any differences in the classroom practices of teachers with differing conceptions about mathematics (particular those with positive views about CM) ?
  6. How do teacher conceptions of CM or SM affect students’ conceptions of mathematics ?

 

Preceding chapter



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