SECONDARY
TEACHER BELIEFS AND PRACTICES ABOUT MATHEMATICS IN THE PAPUA NEW GUINEA CONTEXT
Wilfred T. Kaleva, PhD
The University of Goroka
INTRODUCTION
For decades school mathematics was taught as a ‘culture free’ subject.
It cannot be denied that much of the teaching that goes on in the classrooms
still portrays mathematics as culture free. However, the realisation that mathematics
exists in cultural activities and that mathematics can be learnt outside of
the official school situations has led to the emergence of ‘ethnomathematics’.
DEFINITION OF ETHNOMATHEMATICS
There are a variety of definitions of ethnomathematics
(for examples, see D’ Ambrosio, 1990:22; Howson & Wilson, 1986;
Presmeg, 1996; Frankenstein, 1990; Borba, 1990; Pompeu, 1992). This paper
adopts Barton’s (1996) definition which states that “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” (p. 196).
This writer also supports the view that ‘ethnomathematics’
is not the mathematics of particular groups of people but is an academic
field of study of mathematical ideas (knowledge), practices, activities
which can be identified in sociocultural contexts. It is not a mathematical
field of study as such, but is more like anthropology or history.
These mathematical ideas, knowledge and activities
are acquired and practised 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 paper 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 in outofschool or in sociocultural contexts. The term ‘cultural
mathematics’ (or CM) will therefore 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’ (or SM).
Examples of Mathematics from Other Cultures
In this paper actual examples from different cultures
will not be given. However, references are given of examples of mathematics
from some cultures. For example,
Gerdes (1988 & 1990) 
 
Mozambique weaving to illustrate mathematics
and geometric thinking and mathematics in traditional sand drawings;

Keitel, Damerow & Bishop (1989) 
 
Mathematics from Africa and other parts of the
world; 
Finau & Stilman (1995) 
 
Geometric skills in tapa designs in Tonga; 
McMurchyPilkington (1995) 
 
Mathematical activities of Maori women in the Marae
kitchen; 
Ascher (1991) 
 
Bushong sand figures and Malekula and Vanuatu sand
tracing to illustrate graph theory, Inca strip patterns and Maori rafter
patterns to illustrate geometry, Iroquoi games to illustrate chance: 
Barton & Fairhall (1995) 
 
Mathematics from Maori designs, patterns and carvings; 
Lean (1994) 
 
Counting systems of PNG and the Oceania region;
and 
Kaleva (1995) 
 
Bamboo wall patterns from PNG for number patterns.

Educational Challenges for PNG
One of the challenges of education in PNG and
elsewhere is how school mathematics can take the learners’ outofschool
knowledge into account or how the curriculum can incorporate this ‘cultural
mathematics.’ Official education policies of the Papua New Guinea (PNG)
Government encourage a ‘community oriented and a culturally based’
education and curriculum. These policies support a ‘culturally oriented’
curriculum. A perusal of mathematics curriculum documents, however, shows
a curriculum which is similar to those found in other countries  assumed
to be culture free and ‘canonical’ (Howson & Wilson, 1986).
The mathematics taught in PNG schools and the classroom practice clearly shows
that the intentions of the government are not being implemented.
THE STUDY
At the forefront of any implementation process
are the teachers, which is why the research by Kaleva (1998) examined teacher
beliefs about mathematics and culture and the observed teacher classroom practices.
Sample and Participants
The sample size (n = 135) was made up of lower and upper
secondary mathematics teachers in PNG High schools and included 34 (25 %)
females. The respondents from 50 schools included 7 (5 %) Head Teachers and
12 (9 %) Subject Heads. The schools were from 16 of the 21 provinces and from
the four main regions. Fortysix percent (46 %) of the teachers had less than
three years teaching experience, 43 % had four to ten years while the rest
(14 %) had more than ten years teaching experience. Most of the teachers (67
%) had diplomas in Secondary Teaching while 25 % had first degrees. Twelve
teachers (including 4 females) were interviewed and 5 teachers (including
2 females) were observed in class.
Data Collection
Three main methods were used to collect data:
questionnaire, interview and observation. The questionnaire was divided into
two main sections. Section A was Likerttype, with statements about “School
mathematics” (SM) and “Cultural mathematics” (CM). These
statements were grouped according to four teacher belief categories (or scales)
based on a theoretical construct. The four scales were teacher beliefs about:
where learning takes place (Locus); mathematics teaching; the nature of
mathematics, and the status of SM and CM. Section B contained a list of
activities and the respondents indicated how much mathematical knowledge was
needed for the activity.
Teacher questionnaires were mailed to 112 PNG
high schools in early 1995. After the initial analysis of the returned questionnaires,
five teachers from three schools were selected for observations, based primarily
on the responses which determined the position of the teacher on an imaginary
‘beliefs’ continuum. Other considerations in the selection of
the teachers were where the teacher was teaching (urban or rural school, region)
and the gender of the teachers.
The observations of the lessons were conducted
in Term 3 of 1995 and an observation guide was used. All the lessons were
audio taped and each teacher was observed for a week. An interview schedule
with questions about their responses to the questionnaire and their use of
CM was used in the interviews.
QUESTIONNAIRE RESULTS
Table 1 below shows teacher responses to statements
in section A of the questionnaire as grouped according to the three beliefs
categories (see notes on Statistical test results which explains why
the three beliefs categories were used).
Table 1.
Teacher Responses to Selected Items in Section A of Questionnaire.


Item 
Mathematics
Teaching 
A/SA 
NS 
D/SD 
(%) 
NS 
(%) 
(4)

Mathematics identified in traditional cultural activities should also be taught
in schools 
81 
4 
15 
(7)

Traditional mathematics found in ones’ own culture should not be taught
in schools 
20 
4 
76 
(11) 
Teachers should show how mathematics is used in cultural contexts 
94 
2 
4 
(15) 
In schools, teachers should teach only the mathematics that is prescribed
in the syllabus and textbooks 
16 
1 
83 
(19) 
When teaching mathematics teachers should take into account students’
prior knowledge learnt out of school 
89 
1 
10 
(26) 
School mathematics should teach students about values in life. 
85 
10 
5 
(27) 
Some mathematics identified in cultural activities should be included in the
secondary mathematics curriculum. 
82 
7 
11 
Mathematics learning: where mathematics
learning takes place 

(1) 
Mathematics can also be found in traditional cultural activities 
100 
0 
0 
(5) 
The only mathematics students learn are those taught to them by teachers in
schools 
26 
1 
73 
(10) 
Traditional practices such as counting, measuring, drawing are also mathematical 
100 
0 
0 
(24) 
Students come to school to learn “school mathematics”, not cultural
mathematics 
27 
3 
70 
Nature of mathematics 

(2) 
Mathematics consists of a body of knowledge whose truths should be questioned 
68 
13 
19 
(6) 
School mathematics is made up of abstract concepts and ideas which are value
free 
45 
13 
41 
(14) 
Mathematics is about learning arithmetic, algebra and geometry. 
43 
4 
53 
(16) 
Mathematics identified in traditional culture is too simple (at the arithmetic
level) 
58 
10 
32 
(22) 
Rules are the basic building blocks of all mathematical knowledge 
74 
7 
19 
(23) 
Mathematics is about knowing when to use rules and formulas to find answers
to problem 
71 
3 
26 
(30) 
Mathematical knowledge consists of facts, theories and formulae which are
unquestionably true 
59 
10 
30 
(32)

Mathematics is culture free 
39 
25 
36 
Table 2 shows teacher responses to section B of
the questionnaire where respondents were asked to indicate how much mathematical
knowledge was needed for the activity.
Table 2.
Teacher Responses to Section B of the Questionnaire.
Activity 
% of “No math” response 
% of “Some math” responses 
% of “Lot of math” responses 

1 
2 
97 

1 
29 
70 

3 
70 
27 

8 
82 
10 

42 
58 
0 

34 
56 
10 

27 
64 
9 

17 
64 
19 

17 
55 
28 
STATISTICAL TEST RESULTS
A number of statistical tests were carried out
on teacher responses to the questionnaire items. Oneway Analysis of Variance
(ANOVA) tests of significance for various independent variables showed
that there were no significant differences( p <
0.05) for gender, teaching experience, teacher level, specialist areas, qualification,
schools, school location (urban/ rural or region) and school type. This indicated
that the teacher sample was fairly homogeneous. This was not surprising because
the response frequencies showed that there were no variances for many of the
items (i.e., high percentage of unanimous or near unanimous responses).
Item reliability tests were carried out
for the overall items and the four beliefs categories (scales which were based
on the theoretical construct of teacher beliefs about mathematics learning,
mathematics teaching and nature of mathematics). The Cronbach’s Alpha
(which measures the internal consistency of a test) for all the items in Section
A was 0.75. This is a measure of a single construct. For example, if the items
were a measure of teacher beliefs about cultural mathematics, the Cronbach’s
Alpha measurement would be 0.75. Although this was acceptable, the
Item / Total correlation (‘Pearson’s correlation coefficient’
between each item and the other items) showed that most of the items had correlations
of less than 0.4 which indicated a weak relationship between each item and
the other items, so there was no strong basis for considering the overall
items as the measure of a single construct: e.g., a measure of teacher beliefs
about cultural mathematics. The Cronbach’s Alpha for scales 1 4 were
0.49, 0.24, 0.38, 0.64 respectively. These results show that there was no
basis for the use of the scales. Factor analysis was therefore
used to group the items into the three categories (rather than scales) shown
in Table 1 (see also the section on teacher conceptions about SM and CM).
CLASSROOM OBSERVATION RESULTS
Table 3 gives a summary of the results of the classroom observations.
Table 3
Summary of Classroom Observation
Teacher 
Classroom Activities 
Examples/
Exercises used 
Resource Use 
Explanation/
Exposition 
Methods of Solution 
Portrayal of Maths 
Teacher A 
Typical classroom activities – Teacher gives examples, explains, exercise,
students do exercise. 
Formal mathematics examples – from textbooks, worksheets. No example
of mathematics from traditional culture. Activities and exercises used
from text book:
5/5 worksheets photocopied from texts 
Text books and work sheets 
Explanation of formulas, procedures with questions. When giving solutions,
students are asked, “Why is ? 
Methods of solution not necessarily negotiable but explored ways of getting
an answer. Impression seems to be, other methods of solution not possible,
they are well defined. 
Mathematics is so well organized, defined, no room for mistakes, all about
facts. 
Teacher B 
Impression of student learning passive learners. Not much communication between
teacher and students – interaction mostly one way. 
Activities & exercises used from textbooks: 6/6 (work sheet supplemented
2/6). 
Use of textbooks, sticks to textbooks, worksheets.
Activities and exercises used from textbooks. 
Asks for solutions to problem but as if there’s only one way to solve
it. There are other ways of writing … algebraic solution but did
not explore, pursue this line. 
There is really one way of finding areas of rectangle and squares –
by using rules, formulas. 
Mathematics is precise with rules, solutions with algebraic manipulations. 
Teacher C 
Teacher gets students to suggest ways of findng a solution, not call out answers.
Teacher fields suggestions eg. Sum of nos. from 1 to 20
Actual dramatisation of handshake problem in class. Contextualising problem
Handshake problem, 9 people, how many handshakes – relates handshake
problem to class. 
The kinds of examples used were typically textbook exercises but his approach
to teaching was to use appropriate procedures to solve problems eg.
Negotiate methods of solution. No examples of mathematics from culture
were actually used. 
Activities/
exercises used from textbook:
1/5 lessons. Used own examples: 4/5.
Mentioned textbook at end of one lesson. Did not mention textbook often, although
some problems were obviously from textbook. No other resources were
used. 
Shows steps to solution by asking questions, Does it together with students 
Methods of solution are negotiable, allows students to use own methods of
solution, gets students to actually participate in finding solution. 
Presents mathematics as a debatable subject. 
Teacher D 
Typical maths lessons: Example, explain, student exercise. 
Activities/exercises used from textbook: 6/6 lessons. Used own examples: 4/6.
Most exercises from textbook. 
Textbook only resource used this week. Mathematics from textbook only. 
Typical maths lessons, relies on rules and formulas. 
Formal methods of solution, formal explanations. 
Mathematics is all about rules and formulas. Maths as decontextualised subject. 
Teacher E 
Typical maths lessons: Open text book, do exercise on page …
Mathematics lessons are all about going through the text book. 
Activities/ exercises used from textbook: 8/8 lessons. Used own examples:
Once 
Uses mostly text books. Predominant use of textbook – study of mathematics
centres around textbook. Text consulted constantly to check for answers. 
Mathematics presented as a reproductive subject. Mathematics lessons are about
going through the textbook. 
Teacher does not seem to promote other possible methods of solution. There
is one correct way to find the solution, through the use of formulas. 
Mathematics is portrayed as a one way subject (mathematical knowledge transmitted
from the teacher to the pupils). 
DISCUSSION OF FINDINGS
The findings may be discussed under several sub
headings, including teacher beliefs about mathematics teaching and learning,
the nature of mathematics, teachers’ conceptions of school and cultural
mathematics, and appropriate classroom practice.
Teacher Beliefs about Mathematics Teaching
Eightytwo percent of the teachers agreed or strongly
agreed that mathematics identified in traditional activities should be taught
in schools and that some of this should be included in the secondary mathematics
curriculum (also 82 %). The result is confirmed by the seventysix percent
who disagreed with the converse statement, that traditional mathematics found
in one’s own culture should not be taught in schools. When asked if
they thought some examples of school mathematics could be found in traditional
activities, an affirmative answer was given citing geometry, patterns, measurement,
area, volume and counting systems as examples.
It is interesting to note that although the majority
of the teachers agreed that mathematics from culture (CM) should be taught
in schools, almost all the teachers interviewed agreed that this type of mathematics
was simple and was at the basic elementary level (remember that this sample
is made up of secondary teachers). Almost sixty percent of the respondents
to the questionnaire either strongly agreed or agreed that mathematics identified
in traditional culture was too simple (thirty percent disagreed, while ten
percent were unsure). It was clear from the interview data that although most
of the teachers agreed that mathematics from culture should be taught in schools,
they also believed that cultural mathematics was basic mathematics which was
best taught at the primary level. Many believed that if CM were to be included
in the secondary school curriculum, then the appropriate level would be the
lower levels (e.g., Grades 7 and 8 or what is now the upper primary).
Those who disagreed that CM should be taught in schools
(15 %) did so because they did not believe any mathematics had been found
in PNG culture that was worth teaching in high schools. For example, when
asked if mathematics from traditional culture should be included in the secondary
curriculum, the response from one of the interviewees was: “If there
is any maths that has been found in PNG (culture), then it’s okay”.
Even though eightythree percent disagreed that teachers should teach only
the mathematics that was prescribed in the syllabus and textbooks and ninety
four percent agreed that teachers should show examples of how mathematics
was used in a cultural context, the observations suggest that in reality the
prescribed mathematics is the only mathematics that most of them teach.
Teacher Beliefs about Mathematics Learning
Statements about mathematics learning sought to
establish whether PNG teachers believed that mathematics could be learnt outside
of the official systems of mathematics learning, particularly in a traditional
cultural context.
Ninetyone percent of the teachers believed mathematics
could be learnt by participating in traditional cultural activities and another
ninetyone percent agreed that mathematics could be learnt ‘outof–school’,
with the majority (94%) disagreeing that mathematics is learnt in schools
only or that it is found in mathematics textbook only (92 %).
The result that almost all the teachers believed
mathematics could be learnt in a cultural context, for example, by taking
part in traditional cultural activities, such as fishing or building traditional
houses, was surprising. It was surprising because mathematics is associated
with schools and school mathematics exercises almost always give examples
of applications of mathematics in activities associated with ‘western’
culture or a modernised society with hardly any examples of mathematics associated
with traditional cultural activities.
The most likely explanation for these results
is related to the home backgrounds of the teachers in the sample. Because
ninety percent of the PNG population live in rural areas where traditional
practices and activities are the norm, most of the teachers would have experienced
this lifestyle. For most, their first experiences of mathematics would have
probably been with activities which involved counting in their own languages,
measuring the length of a pandanus floor for the house, using a rope or cutting
out a round cricket sized ball from the cylindrically shaped soft tissue that
forms the inside of a fern.
Research elsewhere on children’s mathematical
knowledge shows that mathematics is acquired outside of the structured systems
of mathematics learning, for example, in every day activities out of school,
at work, in the street or informally (Nunes, 1992). These results show that
PNG teachers are culturally aware.
Teacher Beliefs about the Nature of Mathematics
Previous studies of teacher conceptions about
mathematics noted that views about the nature of mathematics fall into variations
of an internal and external continuum (Dossey, 1992). External views regard
mathematics as an externally existing body of knowledge, facts, principles
and skills available in syllabi or curriculum material while internal views
regard mathematics as a personally constructed or internal set of knowledge,
where mathematics is a process or a creation of the mind. There is a third
perspective which states that mathematical knowledge (facts, concepts and
skills) results from social interaction that relies heavily on context (Dossey,
1992; Bishop, 1985 & 1988).
PNG teachers had differing views to the statements
in this category. Responses to the statement: “Mathematics consists
of a body of knowledge whose truths should be questioned” showed that
68 % of the teachers agreed or strongly agreed while only 19% disagreed or
strongly disagreed (13% not sure), seeming to indicate that a lot of the teachers
had strong internally oriented views. The picture becomes more revealing when
one considers the responses to the other items in this category. For example,
in response to the statement: “School mathematics is made up of abstract
concepts and ideas which are value free”, 45% agreed or strongly agreed,
41% disagreed while 13% were not sure. In response to the statement: ‘Mathematics
is culture free’ 39 % agreed, 36 % disagreed while 25 % had a ‘not
sure’ response. The responses indicate that the number of teachers who
hold internally oriented and externally oriented views is about equal (40
%).
The higher percentage of ‘not sure’
responses (25%) may be noted. This may indicate that the teachers are genuinely
not sure whether, for example, mathematics is culture free because they have
not been confronted with the issues dealing with mathematics and culture.
More importantly, it may reflect the conflict that exists between their perceptions
of mathematics and the mathematics they portray in class  usually one that
is culture free.
The above results are also surprising as they
show that a lot more PNG teachers than expected have strong internally oriented
views about the nature of mathematics. Based on the writer’s experience
as a mathematics student and then as a mathematics educator, mathematics in
PNG (as elsewhere) is taught as an abstract and context free subject which
does not seem to have any connection whatsoever with traditional culture (Kaleva,
1992). The classroom observations also confirm this.
Teacher Conceptions of School Mathematics (SM) and Cultural Mathematics
(CM)
The teacher responses to the questionnaire show
that the teachers manifest distinct perceptions about school mathematics (SM)
and cultural mathematics (CM). It is important to make this distinction because
teachers’ expressed beliefs distinguish between SM and CM. Three distinct
teacher conceptions (including teacher conceptions about SM and CM) were identified
from the factor analysis (of the items from section A of the questionnaire).
They were teacher conceptions about: teaching and learning outofschool mathematics
(e.g., as in traditional culture or CM); teaching and learning in school mathematics
(SM); and mathematical concepts and knowledge, which refer to teacher conceptions
about the nature of mathematics (Kaleva, 1998).
An important point to note about the factors identified
above is that they are separate factors. It does not mean that the teachers
believe either one factor or the other. It cannot be assumed that if a teacher
has strong beliefs about teaching and learning CM, then this teacher will
have less strong beliefs about the teaching and learning of SM. It is quite
possible that the teacher who has strong beliefs about CM can also have strong
beliefs about SM. What this means is that the teachers in PNG will have distinct
beliefs about SM and CM, and at times these views may seem contradictory.
At the beginning of the study it was assumed that views supporting cultural
mathematics (CM) would be more internally oriented. It can be seen that this
is not so as the teachers’ views about SM and CM can be both internally
and externally oriented. This will also explain why teachers were not sure
in regard to the statements in the nature of mathematics category: ‘mathematics
is culture free’ and ‘school mathematics is made up of abstract
concepts and ideas which are value free’.
The interview data further support this distinction.
For example, one of the teachers interviewed agreed that there were mathematics
in culture. However, when asked if it should be taught in schools the teacher
replied: “Yes, if it can be found.” This indicated that according
to the teachers’ perception, mathematics in culture was different, yet
to be discovered. The idea that mathematics is an externally existing body
of knowledge, waiting to be discovered is consistent with ‘Platonic’
(external) views which teachers have about mathematics. The other teacher
conception of mathematics is the ‘Aristolean’ (internal) view
which regards mathematical knowledge as a personally constructed internal
set of knowledge (Dossey, 1992).
Teacher responses to section B of the questionnaire
(see Table 2), where teachers were asked to indicate how much mathematical
knowledge was needed for a list of activities, also show teachers’ distinct
perceptions of SM and CM. Many teachers still believe that one does not use
any mathematics in performing traditional activities. The percentage of teachers
who believe that one does not need any mathematics to perform the activity
is higher for the traditional activities than for the nontraditional activities.
A specific example is the response to children
playing traditional games. Fortytwo percent believed that no mathematics
was involved while fiftyeight thought that some mathematics was involved
while none of the teachers believed it involved a lot of mathematics. Yet
research clearly shows that it involves a lot of mathematics (Nunes, 1992).
Implications for teaching are that if teachers do not think there is any mathematics
involved in those activities, they will not use it as a teaching strategy.
The teacher responses to the questionnaire clearly
show that the teacher beliefs are in line with government policies. The teachers
are culturally aware and believe that CM should be taught in schools, and
that examples of CM should also be included in the curriculum. The important
question is: What happens in practice? Do the teachers use examples of CM
in their teaching?
Classroom Practice
The ‘social construction’ framework
by Bishop and Goffree (1986) was used to analyse the mathematics lessons that
were observed. These researchers offer “the social construction frame”
as an alternative conceptualisation of the mathematics lesson. This view recognises
the social aspect of classroom interactions and “ views mathematics
classroom teaching as controlling the organisation and dynamics of
the classroom for the purpose of sharing and developing mathematical
meaning” (Bishop & Goffree, 1986:315). An important aspect
of this view of classroom teaching is the concept that any new mathematical
idea only has meaning if it can make connections with an individual’s
existing knowledge. Bishop and Goffree proposed three main components of the
mathematics classroom as activity, communication and negotiation. The
use of this framework was important for this study because it was sensitive
to the cultural aspects of mathematics teaching. In particular, it helped
to identify “teachers’ portrayal of mathematics” and also
allowed for an analysis of the methods of solutions used in the classroom
(see Table 3).
The emphasis of the analysis on “how mathematics
was portrayed” in the lessons that were observed was based on the premise
that teacher beliefs about the nature of mathematics are manifested in their
portrayal of mathematics (Thompson, 1992). The researcher recognised that
“how mathematics is portrayed” is highly inferential, but by using
other categories, questions related to methods, examples, formulas and rules,
language and resources, inferences about how mathematics is portrayed could
be made.
The beliefs of the observed teachers were identified
before the observations. Teachers A and B were identified as having strong
school mathematics oriented views. Teacher C was identified as having strong
cultural mathematics oriented views. Teachers D and E had mixed SM and CM
oriented views.
In most (80%) of the lessons of Grades 7 and 8
classes that were observed the teachers did not use examples from traditional
cultural activities. Of the teachers who were interviewed and observed, only
one teacher indicated that he had used examples of CM to introduce a lesson.
The content of the mathematics taught by all the teachers consisted typically
of text book exercises but there were differences in their teaching approaches,
particularly in the methods of solution and the portrayal of mathematics.
As can be seen from Table 3, the methods of solution
used and how mathematics was portrayed by teachers A, B, D and E are typical
of most mathematics lessons. The methods of solution are not negotiable, formal
methods of solutions are used with formal explanations and emphasis on use
of formulas for solutions. Mathematics is portrayed as well organised, precise
with rules and formulas, and as a one way subject with knowledge transmitted
from teacher to pupils. There were no differences in the way mathematics was
portrayed in class between these four teachers. They employed school mathematics
(SM) methods. There were differences between these teachers and Teacher C.
Teacher C used methods of solution which were negotiable, allowed students
to use their own methods of solution and portrayed mathematics as consisting
of facts which were debatable.
Although the teachers unanimously agreed that
CM should be taught in schools, in practice that does not happen. This is
understandable considering that there are curricular constraints. Because
the mathematics curriculum is nationally prescribed and is centrally imposed,
the teachers teach according to the syllabus. The pressure of examinations
forces teachers to ‘cover the syllabus’.
What this suggests is that teacher beliefs about
mathematics teaching and learning are probably not as important to classroom
practice as teacher beliefs about the nature of mathematics. The educational
system under which the teacher has to operate can act as constraints to practice.
However, teacher beliefs about the nature of mathematics can have an influence
over the way mathematics is portrayed. As can be seen from the observations,
teachers with SM oriented views or mixed views portrayed mathematics as consisting
of facts, rules and formulas where the methods of solution were not negotiable.
Teachers with the CM oriented view negotiated mathematical solutions with
the students and portrayed mathematics as a debatable subject.
CONCLUSION
This study confirms what has been reported by
other studies (Howson & Wilson, 1986; Travers & Westbury, 1989). There
is a mismatch between teachers’ stated beliefs and their practice. For
example, although the teachers’ stated beliefs are that they should
teach CM in school and show how SM is used in a cultural context, in practice
it doesn’t happen.
The findings also support those of Sosniak et
al. (1991) who found that teachers did not seem to hold theoretically coherent
points of view. For example, teachers’ unanimous responses to two categories
of teacher beliefs (mathematics teaching and mathematics learning), seemed
to indicate that teachers had internally oriented beliefs. However, their
responses to the nature of mathematics category indicated that they had differing
views  some internally oriented while others had externally oriented views.
Teacher beliefs are not the single most important
factor that influences practice. Practice is largely determined by curricular
context, curricular constraints and situations. Curricular constraints, such
as, pressures of external examinations, or to ‘cover the syllabus’,
ensure that the teachers do not deviate from teaching the prescribed content.
The relationship between beliefs and practice
is not linear but circular. Practice changes beliefs and beliefs develop,
change and evolve. Curriculum context is necessary before beliefs can be put
into practice. For example, the availability of material on CM will determine
if teachers can include CM in their classroom practice.
Teachers in PNG manifest beliefs which indicate
that CM exists and that CM should be taught in schools. However, the curricular
constraints are such that it does not happen in practice. One of the curricular
constraints is the lack of available CM materials. If CM in the PNG context
can be identified and included in the curriculum, it ‘legitimises’
the mathematics and also provides teachers with the opportunity to put into
practice their beliefs. I repeat the statement made at the introduction that
one of the educational challenges for PNG in the 21^{th} century is
how school mathematics can take the learner’s outofschool knowledge
into account or how the curriculum can incorporate this ‘cultural mathematics’.
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