 
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 sociocultural 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 outofschool
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 Sociocultural 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. 93120) refers to
these as “utilitarianism and mathematical anthropology”. Bishop
(1992a : p.2), however, identifies three foci of research on “ethnomathematics”
. They are :
 mathematical knowledge in traditional societies (anthropology),
 mathematical developments in non western cultures (history),
 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 everyday 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 outofschool
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 outofschool 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 nonformal 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.1517).
As a followup to a study which investigated school
failure, the study of young street vendors in the Northeast 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 reinvent 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.170171), the key question
is not whether mathematics is learnt out of school but how the nature of outofschool
mathematics is different from that of school mathematics. She warns about
the false dichotomies of “outofschool” 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 “outof
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 preformulated problems
and contains prerequisite 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 logicomathematical 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 sociocultural 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.516). Barton (1992b : p.4) suggests that “ethnomathematical concepts”
exist in sociocultural 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.180181; 1991 : pp.3031) 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 :

Counting  eg. number systems ,algebraic representation, probabilities

Locating  eg. orientation, coordinates, bearings, angles, loci

Measuring  eg. comparing, ordering, measurements, approximations

Designing  eg. projections of objects/shapes, geometric shapes, ratio

Playing  eg. puzzles, paradoxes, models, games, hypothetical reasoning

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.57; 1991b : pp. 369377; 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
semicycles). 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 socioculture
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 sociocultural 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 sociocultural
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.7374)
Vithal (1992
: pp.1213) 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 sociocultural contexts. These mathematical ideas, knowledge
and activities are acquired and practiced by various sociocultural 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 outofschool
or in sociocultural contexts. The term “cultural mathematics”
therefore will mean the mathematical ideas, knowledge and practices that can
be identified in sociocultural 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 “nonEuropean 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
prerequisites 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 sociocultural
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.

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.
 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)
 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).

Aristolean views about
the nature of mathematics regard mathematics as a personally constructed
or internal set of knowledge.
 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.
 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.

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 :
 mathematics learning
 mathematics teaching, and
 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.140141).
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 socalled 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 procultural mathematics
views. By the same token, it is assumed that the teachers with strong external
conceptions about the nature of mathematics will have proschool 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 outofschool mathematics that may be identified as the mathematical
ideas, knowledge or practices of a sociocultural 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.137138) 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 preservice 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 sociocultural 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 preservice 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 causeandeffect 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 sociocultural 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 nonavailability) 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.
Sociocultural constraints.
Sociocultural 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 socalled
“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 (Inservice) 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 outperform 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 extracurricular 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 nonmaterial
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.1831) :
 the desirable ends of the educational enterprise.
 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).
 children and childhood.
Conceptions of education are sometimes expressed in terms of how a child
develops and the educational implications of the development
 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.
 Progressivism  values education
as a means of meeting individual needs and aspirations.
 Instrumentalism  stresses the responsiveness of education to the requirements
of socioeconomic order.
 Reconstructionism  conceives of education as an important way of moving
society in desired directions.
 Classical humanism  education serves
the function of transmitting cultural heritage.
 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.194195). 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 reexamining 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.):
 All groups of students from the participating countries found the tests
difficult.
 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).
 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).
 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, 19801982) 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) :

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).

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 studentteacher 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.

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 pretest to posttest
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 multiplechoice items selected on the basis
of international curriculum grids. These items, intended to measure cognitive
behaviour at all levels, also included some openended 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 (canonicalstructuralist)
and the ethnomathematical approach (Bishop, 1992a). “His research involved
teachers in creating six ethnomathematical microcurriculum 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 languagerelated.”
(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. monolingualism 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 sociocultural 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 outofschool, 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 nonEnglish 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 inservice 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 outofschool 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 base10, 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 pretests and posttests and there were increased
scores between posttests 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 :
 Due recognition given (both by inclusion and by respect for its use)
to traditional counting and measurement systems.
 Development of a case for standard measures and ways of comparing in order to achieve
precision, allow comparison and broaden application.
 Study of traditional pastimes and practices, where appropriate, in a mathematical
way.
 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.
 Use of PNG currency, place names, food stuff in book work examples and
exercises.
 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’
outofschool 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 outofschool
knowledge in the school. Nunes (1993 : p.35) suggests that bringing outof
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 outofschool mathematics incorporate these into school
mathematics ? Why ? Will it not pose the same problems that we accuse school
mathematics of ? Why incorporate identified outofschool 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 outofschool 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 
Multicultural 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 
Antiracist 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, bicultural teaching 
Bicultural 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 outofschool
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 nonwestern 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
microcurriculum 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.

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?

What is the status of SM and CM ?

Are there any conflicts between teachers’ conceptions
of CM and school mathematics ?

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 ?

Are there any differences in the classroom practices
of teachers with differing conceptions about mathematics (particular those
with positive views about CM) ?

How do teacher conceptions of CM or SM affect students’
conceptions of mathematics ?
