CHAPTER SIX. ANALYSIS AND DISCUSSION
OF RESULTS.
Introduction.
This chapter provides an analysis of the descriptive
results of chapter five in the context of the following research questions
:
 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 (particularly those with positive
views about CM) ?
 How do teacher conceptions of CM or SM affect students”conceptions
of mathematics ?
The first research question was addressed in section
5.4 in chapter five. The focus of the analysis in this chapter however, is
on teacher beliefs about SM and CM and their influences on teacher
practice in the classroom. The discussions of the descriptive results in section
5.2 of chapter five showing the teachers” student teachers”and
students”responses to individual items gives some idea about what their
beliefs were about SM and CM. Student responses are further analysed in the
context of influences of teacher conceptions of SM and CM on student conceptions
(section 6.3). Discussions of related research questions addressed here reveal
the differences between teacher conceptions of SM and CM (discussed in section
6.1.1).
6.1 Teacher beliefs about SM & CM.
The factor analysis in chapter five (section
5.3.1) showed that teacher beliefs about SM and CM is a meaningful classification.
Certainly, the research literature (see chapter two, for example) alludes
to the existence of this “other mathematics”refering to it as
“mathematical ideas”or “mathematical practices”of
sociocultural groups and “out of school mathematics” The basic
research question investigates teacher perceptions about the existence of
this “other mathematics”which is distinct from “the mathematics”or
school mathematics as it is commonly known.
The questionnaire referred to the two kinds
of mathematics  “school mathematics”(SM) and “cultural
mathematics”(CM). Teacher responses to individual items will therefore
be discussed in terms of teacher beliefs about SM and CM and their beliefs
about the nature of mathematics. In the first part of section 6.1.1, teacher
conceptions about SM and CM are analysed in the context of mathematics learning
and teaching. Teacher conceptions about the nature of mathematics are also
discussed in the last part of this section.
6.1.1 A comparison of teacher beliefs about SM
and CM.
In this section, teacher beliefs are discussed according
to the three categories that were identified by the factor analysis in chapter
five. These are teacher beliefs about SM, CM and the nature of mathematics
(NM). The first part of this section (6.1.1), answers the research question
: what are the teacher beliefs about SM and CM? Table 6.1, constructed using
data from Table 5.3 and Table 5.4 in chapter five, shows teacher responses
to items about SM and CM under the first two categories of : Teaching beliefs
about mathematics teaching and teacher beliefs about mathematics learning.
Teacher beliefs about SM and CM are compared. The last part of this section,
discusses teacher beliefs about the nature of mathematics in the context of
teacher beliefs about SM and CM.
Table 6. 1 Responses to SM and CM statements according to
categories
SM 
SA/
A
% 
CM 
SA/A
% 
Teacher
beliefs about mathematics learning  Locus 
(21)
Mathematics is learnt in schools only 
4 
(9)
Students can learn mathematics out of school while participating in
ordinary everyday activities 
91 
(13) Mathematical
knowledge is found only in mathematics textbooks 
7 
(1) Mathematics
can also be found in traditional cultural activities 
100 
(5)
The only mathematics students learn are those taught to them by teachers
in schools 
26 
(17)
Mathematical knowledge can be gained (learnt) by taking part in traditional
cultural activities such as fishing, building traditional houses etc. 
91 


(10) Traditional practices such as counting,
measuring, drawing are also mathematical 
100 
(24)
Students come to school to learn “school mathematics” not
cultural mathematics 
27 
(28)
Learning how to count in your own mother tongue is as important as counting
in English 
91 
(29)
All of the students prior knowledge is learnt in schools only 
10 
(25)
Some of the students”prior knowledge is learnt out of school while
participating in traditional cultural activities 
93 
Teacher
beliefs about what should be taught 
(15)
In schools, teachers should teach only the mathematics that is prescribed
in the syllabus and textbooks 
16 
(27)
Some mathematics identified in cultural activities should be included
in the secondary mathematics curriculum. 
82 
(11)
Teachers should show how (school) mathematics is used in cultural contexts 
95 
(7)
Traditional mathematics found in ones own culture should not be taught
in schools 
20 
(26)
School mathematics should teach students about values in life. 
85 
(4)
Mathematics identified in traditional cultural activities should also
be taught in schools 
82 


(19)
When teaching mathematics teachers should take into account students
prior knowledge learnt out of school 
90 
Why
that mathematics should be taught 
(3)
School mathematics is useful in traditional societies 
82 
(36)
Mathematics identified in traditional cultural activities is useful
to a modern PNG society 
68 


(12)
Showing examples of mathematics in traditional culture will give students
a sense of cultural identity 
91 


(20)
Mathematical concepts found in traditional culture will be lost if they
are not taught in schools 
69 


(8)
Mathematics found in traditional cultural activities is not as important
as the “real mathematics”that is learnt in schools 
19 
Key : SA / A = Strongly Agree and Agree (combined)
percent of teacher responses to the item; SM = school mathematics; CM = cultural
mathematics.
It is clear from Table 6.1 that teachers are not only
aware of CM but have distinct views about SM and CM. There is strong support
for the fact that the above is a meaningful distinction and that teachers
manifest distinct perceptions about these two constructs. The distinction
is also supported by positive teacher responses to the locus scale which indicated
that the teachers in the study did believe that mathematics can also be learnt
outside of the established systems of learning. That teachers believe mathematics
is learnt “out of school” acknowledges the existence of the other
mathematics  one that is learnt out of school, even in traditional cultural
activities and which is distinct from school mathematics.
For example, twenty seven (27 %) percent of the teachers
agreed that students come to school to learn school mathematics, not cultural
mathematics and twenty six percent (26%) agreed that the only mathematics
students learn are those taught to them by teachers in schools. What this
means is that although almost all of the teachers acknowledged that mathematics
exists out there  out of the normal established systems of learning, about
a fifth of them still believed that the only mathematics students learn are
those taught to them by teachers and that students come to schools to learn
this “school mathematics” not the “other mathematics”
While this seems contradictory, it also indicates that ( at least in the
minds of a fifth of the teachers), there exists two kinds of mathematics 
one that is learnt in schools and one that is learnt “out of schools”
Certainly in the interviews, most of the respondents agreed that the mathematics
that is found outside school in traditional activities (CM) was too simple.
This is further supported by the teacher responses
to Section B of the questionnaire where teachers were asked to indicate whether
the person performing the task needed no, some or a lot of mathematics.
Table 6. 2 Responses to traditional and non traditional
activities.
Activity 
“No
maths”
% 
“Some
maths”
% 
“Lot
of maths”
% 
pilot flying an aeroplane 
1 
2 
97 
carpenter
building a house 
1 
29 
70 
estimating
the height of a tree 
3 
70 
27 
measuring
the height of a student 
4 
81 
15 
selling
(betel nut) buai 
8 
82 
10 
the
teacher counting the number of students in the classroom. 
10 
78 
12 
children
playing a traditional game 
42 
58 
0 
making
patterns on bamboo walls 
34 
56 
10 
woman
weaving a mat 
27 
64 
9 
painting
a haus tambaran 
25 
60 
15 
villagers
building a traditional house 
17 
64 
19 
villager
using the stars to navigate by canoe from one island to another 
17 
55 
28 
building
a canoe 
16 
70 
14 
the
warrior counting his arrows using own counting system 
13 
74 
13 
Note : The traditional activities are shaded and the non traditional
activities are unshaded.
The above results clearly show teacher perceptions
of CM. The overall response to the traditional activities is that most teachers
believe that one would need at least some mathematics (“some maths”and
“lot of maths”combined) to perform the task. The lowest percentage
of “some maths”plus “a lot of maths”combined is 58.
In fact the percentage range of at “least some maths”is 58 
87). It was interesting to note that twenty eight percent (28%) of the teachers
believed that the villager using the stars to navigate by canoe from one island
to another uses a lot of mathematics. This ranks as third in the “a
lot of maths”response.
However, there were still many teachers who thought
that one does not use any mathematics in performing traditional activities.
Note the differences in the response pattern to the traditional and the non
traditional activities in the “no maths”column. The percentage
of teachers who believed that one does not need any mathematics to perform
the activity was higher for the traditional activities than it was for the
non traditional activities. In other words, many of the teachers believed
that no mathematics is used in many traditional activities.
A specific example is the response to children playing
traditional games. Forty two percent of the teachers (42%) believed that no
mathematics is involved while fifty eight (58%) thought that some maths is
involved while none of the teachers believed it involved a lot of mathematics.
Yet research clearly shows that it involves a lot of mathematics (eg. Zaslavsky,
1973; Ascher, 1991; Carraher, 1991; Nunes, 1992, Gerdes, 1994a, 1994b). The
implication for teaching is that if teachers do not think there is any mathematics
involved in those activities, they may not use the content as a teaching strategy.
The interview data support this differentiation. For
example, one of the teachers interviewed agrees that there is 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 teacher’s perception, mathematics in culture was different, yet
to be discovered. Certainly almost all the teachers interviewed agreed with
the qualification that this mathematics is simple, basic arithmetic level
mathematics. Almost sixty percent of the respondents to the questionnaire
either strongly agreed or agreed that mathematics identified in traditional
culture is too simple (thirty percent disagreed, while ten percent were not
too sure).
Teacher responses to statements about the nature of
mathematics also indicated that teachers held distinct views about CM, not
just about SM.
Teacher beliefs about the Nature of mathematics.
In the descriptive results about teacher beliefs regarding
the nature of mathematics (section 5.4.3 in chapter five), it was revealed
that the majority of the PNG teachers in the sample displayed strong internal
views. These results differ from other studies which found that classroom
teachers held strong external views. For example, studies reported by Dossey,
1992; Thompson, 1992; Nickson, 1992; Ernest, 1992; Steinberg et al., 1985
and Lerman, 1983 (see chapter two and three) found that teachers viewed mathematics
as consisting of a unified body of knowledge, immutable truths bound together
by logic and unquestionable certainty, knowledge which is absolute, value
free and abstract (the absolutist view). These views correspond to
external Platonic views which regard mathematics as an externally (external
to the learner) existing body of knowledge and facts which are discovered,
and are available from syllabi and textbooks or curriculum material, or as
“mathematics of the curriculum”span> where maths is arithmetic,
algebra, geometry. However, the results from this study do not differ from
the study by Boeha (1990) which found that PNG students at senior high schools
had Aristloean (internal) views about physics.
In chapter two it was pointed out that views about
the nature of mathematics fall into variations of an internal and external
continuum. In the construction of the Questionnaire (see section 3.2.1, chapter
three), it was assumed that those strongly supporting SM would also have external
views and those supporting CM would have internal views. For example, in the
statement, “Mathematics is culture free” a strongly agree/agree
response would denote external views (proSM) while a strongly disagree/disagree
response would denote internal views (proCM). This assumed relationship was
illustrated in Figure 3.3 in chapter three.
Table 5.4 in chapter five shows that the nature of
mathematics category contained a greater variation in the teacher responses
than the other categories. It was decided to investigate whether the individual
teachers consistently held internal and external views about the nature of
mathematics or whether the beliefs are in fact mixed. Individual questionnaires
were examined to check the pattern of responses. That is, to see if those
who had internal views also supported SM and those with external views supported
CM. A perusal of the responses in the questionnaires, however, revealed that
although some teacher beliefs were identified as falling into the above categories,
there were those who supported SM but had both internal and external views
while others who supported CM also had internal and external views. In other
words, most teachers in the sample had mixed views about the nature of mathematics.
This finding is supported by data on individual teacher profiles (Section
6.2) which revealed that the teachers did have mixed conceptions about the
nature of mathematics. The fact that these teachers had mixed views about
the nature of mathematics also confirms the results of the findings of the
factor analysis reported in section 5.3.1 in chapter five, which showed that
teachers”views about SM and CM are not necessarily at either ends of
the continuum but were quite distinct.
It would seem from the above evidence that teacher
views fall into the categories illustrated below.
Figure 6. 1 SM and CM beliefs continuum.

External Views
SM 
Internal
Views 

External Views
CM 
Internal
Views 
In other words, the teachers have distinct views about
SM and CM. These views may be internally or externally oriented. It is possible
for a teacher to have the following combinations of teacher beliefs.
Table 6. 3 Combinations of teacher beliefs

Internal 
External 
SM 
SI 
SE 
CM 
CI 
CE 
Key : SI = Internal views about school mathematics,
SE = external views about school mathematics, CI = internal views about cultural
mathematics, CE = external views about cultural mathematics.
Teachers may also have mixed views about mathematics.
Note that “mixed views”constitute a number of possibilities. For
example, views about either SM or CM may have mixed internal and external
views. A teacher may have internal views about CM but external views about
SM (or vice versa). For examples of mixed views by PNG teachers, see section
6.2.1.
The finding that the PNG teachers in the sample had
mixed views about mathematics is collaborated by findings from the Sosniak
et al. (1991) study which found that teachers did not seem to hold theoretically
coherent points of views. The Sosniak et al. study analysed data collected
from SIMS (questionnaires given to teachers as part of the Second International
Mathematics Study, see section 2.3.3 in chapter two). The teachers were asked
to state how much emphasis they placed on a set of curricular objectives,
or to rate the importance they placed on a list of what the teachers thought
made their teaching more effective, to state the amount of time they spent
on student activities and to rate on a scale of 1 to 5 statements that emphasised
the dynamic process or the static nature of mathematics. It was hypothesised
that teachers with an orientation towards a particular point of view (eg.
progressive v/s traditional, studentcentred progressively oriented v/s traditional
curricular) would respond in certain way to the questionnaire items. The results
showed that only a few teachers exhibited a consistent point of view in their
responses. The study concluded that the teachers do not seem to hold theoretically
coherent points of views. In the PNG study, the teachers”unanimous responses
to two categories of teacher beliefs (mathematics teaching and mathematics
learning), seem to indicate that teachers have 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.
For this study, it is important to note that the profiled
teachers included representatives of teachers having mixed views about mathematics.
However, there were two teachers whose views about SM and CM were external
views and one other teacher whose views about CM were consistently identified
as internal views. Further discussions of the beliefs orientations of these
teachers is presented in section 6.1.
It can also be noted that the scale was characterised
by a higher percentage of “not sure”responses. This may indicate
that the teachers were genuinely not sure whether for example, mathematics
was culture free because they had not been confronted with the issues dealing
with mathematics and culture. More importantly, it may reflect the conflict
that existed between their perceptions of mathematics and the mathematics
they portrayed in class, usually one that is culture free. It also illustrates
how practice shapes their beliefs. In the lessons that were observed, most
teachers taught mathematics as if it were a body of knowledge whose truths
should not be questioned (which supports an external view of mathematics).
Of those observed, only one (out of five) used methods which for example,
portrayed mathematical solutions as negotiable and presented mathematics as
a debatable subject (see teacher profiles in chapter five).
The “not sure”responses also illustrate
the conflict between their perceptions of SM and CM. One perception is culture
free, while the other is embedded in cultural activities. These responses
also show that, contrary to the expectation of the researcher, teachers do
have distinct views about CM, not just about SM. The expectation of the researcher
was that teachers would express strong SMO views about the nature of mathematics.
For example, views about the nature of mathematics which are consistent with
other studies which found teachers had formal views and saw mathematics as
“mathematics of the curriculum” a “static discipline”
a “bag of tools, rules  instrumentalist views”(Lerman, 1983;
Steinberg et al., 1985; Thompson, 1984; Ernest, 1992 ).
6.2 Teacher beliefs and practices.
In the first part to this section, the differences
in the beliefs of the profiled teachers are identified. The revised categories
are also used to place teachers on an imaginary beliefs continuum to show
the differences in the beliefs of the profiled teachers (section 6.2.1). In
section 6.2.2, the discussion centres around the classroom practices of the
teachers who were observed. An analysis of the relationship that existed between
these teacher beliefs and their practices is provided in section 6.2.3.
6.2.1 Differences in the profiled teacher beliefs.
In this section the differences and similarities between
different teacher beliefs are discussed. The discussions focus on the profiled
teachers who were observed.
The five teachers whose profiles are presented here
are the same teachers who were observed. The five teachers to be observed
were selected at the early stages of the field work because their views were
identified as strongly supporting cultural mathematics or strongly supporting
school mathematics. This was based on the summation of their responses to
items on Section A of the questionnaire (see also selection of the teachers
to be observed, Section 4.3.1, chapter four). The theory was that the summation
of the responses would form the basis of a beliefs continuum. It was hoped
that the observed teacher beliefs could be placed on a continuum. The analysis
of item reliability (described in section 5.3 in chapter five) indicated
that there was no statistical basis for adding the scores in the original
scales. However, using the items in the revised categories (teacher beliefs
about SM and CM) which were identified by the Varimax factor analysis (see
section 5.3.1), the profiled teachers were placed on a beliefs continuum.
The justification for the use of these categories in the beliefs continuum
and the results are presented in the section on “Differences in teacher
beliefs” Responses to individual items based on the basic categories
of teacher beliefs about mathematics, were also used to identify the beliefs
orientation of the teachers (see justification for retaining basic categories
in Section 5.3.2).
Similarities in teacher beliefs.
It is important to make this point about similarities
in teacher beliefs before the discussions on differences in teacher beliefs.
In general, there were similarities in the teacher beliefs about school mathematics
(SM) and cultural mathematics (CM). There were unanimous or near unanimous
responses for most of the items in the “locus”category. For example,
100 % of the sample either strongly agreed or agreed to the statements, “Mathematics
can also be found in traditional cultural activities”and “Traditional
practices such as counting, measuring, drawing are also mathematical”
Other statements in the category yielded equally high percentages of similar
responses. This shows that the teachers overwhelmingly supported the idea
that mathematics exists in traditional cultural activities and that CM should
be taught in schools or included in the mathematics curriculum (see discussions
in Section 6.1). Therefore, in relation to PNG government education policies
which encourage a “community oriented and a culturally based”education
and curriculum (discussed in Chapter One), the expressed beliefs of the PNG
teachers in the sample support the government policies on the need for a culturally
based curriculum.
In summary, similarities (near unanimous responses
to items which indicate general agreement) in teacher beliefs had to do with
their beliefs about :
 Mathematics learning  Locus . Teachers generally agreed that mathematics
can be identified in traditional cultural activities, that traditional practices
such as counting and measuring etc. are also mathematical and that mathematics
can be learnt outside of the official systems of learning (out of schools).
 Mathematics teaching. Teachers generally agreed that CM should be taught
in schools (although there was disagreement about the level  primary or
secondary  at which CM could be taught) and that teachers should show applications
of mathematics including showing examples of mathematics in a cultural
context. On the first point, it is important to note that according to the
interviews, most teachers believed that although CM should be taught in
schools, it is too simple and is suitable to teach at the primary level
only.
However, differences in teacher beliefs do exist.
If the teachers are placed on a beliefs continuum, one could see the differences
in the beliefs of the profiled teachers. The “profiled”teachers
responses in Table 6.4 below further highlight these belief differences and
identify the teacher beliefs orientations. The profiled teachers are used
because their observation and interview data could be used to make comparisons
between beliefs and practice.
Differences in Teacher Beliefs.
Using the items under the categories identified by
the Varimax factor analysis (section 5.3.1 in chapter five), two beliefs scales
were developed (this involved the summation of the response scores to the
individual items in the categories) to form the continuum. The justification
for summing the scores is that these items were identified by the factor analysis
as loading onto the SM and CM factors. Two separate beliefs continua were
used on the basis that teacher beliefs about SM and CM are two separate factors
(point made in section 5.3.1 in chapter five). That is, teacher beliefs about
SM and CM are not necessarily at either ends of a continua. The profiled teachers
were then placed along these imaginary beliefs continua (on the basis of their
scores). Their relative positions on the continua were then noted. The results
are shown in Table 6.4.
Table 6. 4 Profiled teacher positions in SM and CM Beliefs
continuum
Profiled
teacher 
CM
score 
CM
position, out of 135 
SM
score (*) 
SM
position, out of 135 
Titus 
40 
3 
42 
125 
Petrus 
37 
30 
41 
121 
Maria 
36 
54 
20 
2 
Tina 
36 
55 
19 
1 
Markos 
26 
128 
26 
8 
Key : CM score = Total response score from
items in the CM category; SM score = Total response score from items in the
SM category; CM position = position of teacher in relation to the other 135
teachers in the CM beliefs continuum; SM position = position of teacher in
relation to the other 135 teachers in the SM beliefs continuum; (*) Note that
the scoring system was such that the higher scores indicated a CM oriented
response and lower scores indicated an SM oriented response.
It is of importance to note the relative positions
of the teachers in the CM and SM continuums. Titus is near the top end of
the CM continuum. Tina and Maria are at the top end of the SM continuum while
Titus is near the lower end of the SM continuum. In other words, on the SM
continuum, Titus is on one end of the continuum while Tina and Maria are on
the other end of the continuum. This would seem to indicate that Tina and
Maria are more “school mathematics”oriented while Titus is more
“cultural mathematics”oriented. Petrus is at the top end of the
CM continuum and while Markos is at the lower end of the CM continuum. Their
relative positions change in the SM continuum.
Were the right teachers observed ?
One point to note relates to the question about whether
the right teachers were observed. The selection of the teachers to be observed
was based on the original scales which were subsequently found to be unreliable
(factor analysis). The intention was to ensure that the selection included
teachers who were SM oriented, CM oriented and those who were in the “middle”
Based on the above analysis, it can be seen that the selected teachers do
indeed fulfil the criteria so it can concluded that the right teachers were
observed.
A closer look at the teachers”actual responses
verifies the teachers beliefs orientations.
Table 6.5 below only includes the items which showed
sufficient differences in the responses of all the five teachers who were
profiled (items with unanimous or similar responses were discussed in the
section on similarities in teacher beliefs). That is, the items missing from
Table 6.5 below showed no disagreement between these teachers (compare also
with Table 5.14 : Profiled Teacher responses to individual items in Section
5.7, Chapter five).
Table 6. 5 Differences in profiled teacher responses
No 
Questionnaire
Items 
Maria 
Tina 
Markos 
Petrus 
Titus 

Locus 





5 
The only mathematics
students learn are those taught to them by teachers in schools 
A 
A 
D 
D 
D 
13 
Mathematical knowledge
is found only in mathematics textbooks 
A 
A 
D 
SD 
SD 
24 
Students come to
school to learn “school mathematics” not cultural mathematics 
SA 
SA 
D 
A 
D 

Mathematics
teaching
(what maths should be taught) 





4 
Mathematics identified
in traditional cultural activities should also be taught in schools 
D 
D 
A 
SA 
SA 
15 
In schools, teachers
should teach only the mathematics that is prescribed in the syllabus
and textbooks 
SA 
SA 
SA 
D 
SD 
19 
When teaching mathematics
teachers should take into account students prior knowledge learnt out
of school 
D 
D 
A 
A 
SA 

Mathematics
teaching
(Why that maths should be taught.) 





3 
School mathematics
is useful in traditional societies 
SD 
SD 
SA 
A 
SA 
8 
Mathematics found
in traditional cultural activities is not as important as the “real
mathematics” that is learnt in schools 
A 
A 
D 
SD 
SD 
36 
Mathematics identified
in traditional cultural activities is useful to a modern PNG society 
NS 
NS 
A 
A 
A 

Structure 





2 
Mathematics consists
of a body of knowledge whose truths should be questioned 
SD 
NS 
A 
A 
SA 
6 
School mathematics
is made up of abstract concepts and ideas which are value free 
NS 
NS 
D 
A 
D 
14 
Mathematics is about
learning arithmetic, algebra and geometry. 
A 
SA 
A 
D 
D 
16 
Mathematics identified
in traditional culture is too simple (at the arithmetic level) 
SA 
SA 
SA 
A 
NS 
22 
Rules are the basic
building blocks of all mathematical knowledge 
SA 
SA 
SA 

D 
23 
Mathematics is about
knowing when to use rules and formulas to find answers to problem 
SA 
SA 
A 
D 
D 
32 
Mathematics is culture
free 
NS 
NS 
SD 
SA 
SD 
Key : SA = strongly agree; A = agree; SD =
strongly disagree; D = disagree ; NS = not sure.
The teacher responses to the “sufficiently different”items
on Table 6.6 above also indicate where there were differences. These teachers”beliefs
can be classified into three main groups  two opposing views and one group
with views from both groups. Tina and Maria have similar responses to most
of the items so they form one group. Titus”responses differ from Tina
and Maria for almost all of the items and it forms the other group. These
two groups represents those with opposing views. Markos and Petrus represent
the third group which exhibit mixed beliefs (mixed responses) to most of the
items (which may also be termed as “middle of the ground”views).
The teacher beliefs orientations can be identified
from their responses.
 Tina and Maria are identified as having strong positive views about school
mathematics or “school mathematics oriented”or SMO views. The
group will therefore be known as the “school mathematics oriented”or
SMO group.
 Titus is the only profiled teacher in this group. He is identified as
having strong positive views about cultural mathematics and so this group
will be called the “cultural mathematics oriented”or CMO group.
 Markos and Petrus have mixed SMO and CMO views. Their responses to some
items indicate SMO views while their responses to other items indicate CMO
views. This group will therefore be described as the “intermediate”or
INT group.
It is important to note the point that was made in
section 5.3.1 of chapter five, that strong teacher beliefs about SM do not
necessarily mean the teacher will have less strong beliefs about CM. It is
possible to have teachers who have strong beliefs about SM and CM. That seems
to be the case with the majority of the teachers in the sample. The third
group here will fall into that category. However, in the case of the first
two groups (SMO and CMO), teachers seem to have strong beliefs about one factor
and less strong beliefs about the other. A detailed discussion of the beliefs
orientations of these groups follows.
School Mathematics Oriented (SMO) views.
The beliefs orientations of this group is described
here. In the description of this group (and the other groups), Pompeu’s
(1992) categories are used to describe these beliefs orientation (for details
of Pompeu’s categories, see section 3.1.1, chapter three) so that a
comparison can be made between these PNG teacher beliefs and the Pompeu category
of beliefs. This also helps in the identification of the teacher beliefs orientations.
Table 6. 6 Teacher Beliefs Profile 1  SMO (Tina and Maria)
Questionnaire
Items 
Maria 
Tina 
comments 
Math
learning : Locus  where math learning takes place 



The only mathematics
students learn are those taught to them by teachers in schools 
A 
A 
Their agreements to these statements show
strong SMO views. 
Mathematical knowledge
is found only in mathematics textbooks 
A 
A 
eg. Math ”from text books and 
Students come to
school to learn “school mathematics” not cultural mathematics 
SA 
SA 
Schools for learning SM, not CM 
Mathematics
teaching
(what maths should be taught) 



Mathematics identified
in traditional cultural activities should also be taught in schools 
D 
D 
Again their responses here show SMO views 
In schools, teachers
should teach only the mathematics that is prescribed in the syllabus
and textbooks 
SA 
SA 
eg. math knowledge taught from standard
maths texts or 
When teaching mathematics
teachers should take into account students prior knowledge learnt out
of school 
D 
D 
maths knowledge does not rely on knowledge
students bring 
(Why
that maths should be taught.) 



School mathematics
is useful in traditional societies 
SD 
SD 
As opposed to maths as a practical subject.
SM not useful to traditional society 
Mathematics found
in traditional cultural activities is not as important as the “real
mathematics” that is learnt in schools 
A 
A 
CM is not important 
Mathematics identified
in traditional cultural activities is useful to a modern PNG society 
NS 
NS 
From Interview data  response could easily
have been A/SA. 
Nature 



Mathematics consists
of a body of knowledge whose truths should be questioned 
SD* 
NS* 
Minor difference, based on universal truths 
School mathematics
is made up of abstract concepts and ideas which are value free 
NS 
NS 
NS responses but from interviews response
could easily have been A/SA 
Mathematics is about
learning arithmetic, algebra and geometry. 
A* 
SA* 
Math made up of separate entities of arithmetic,
algebra, geometry
Instrumentalist view, maths of the curriculum 
Mathematics identified
in traditional culture is too simple (at the arithmetic level) 
SA 
SA 

Rules are the basic
building blocks of all mathematical knowledge 
SA 
SA 
Emphasise procedures, rules facts, methods.
Instrumentalist view, math as a bag of tools 
Mathematics is about
knowing when to use rules and formulas to find answers to problem 
SA 
SA 
Same as above. Their agreements to this
and the preceding three statements show their views to be strongly SMO 
Mathematics is culture
free 
NS 
NS 
NS responses but from interviews  A/SA 
Key :
Comments = researcher’s own comments; A = Agree; SA = Strongly agree;
D = Disagree; SD = strongly disagree; NS = not sure; * Indicates minor differences.
Tina and Maria are identified as having SMO views. They both expressed
strong positive views about SM. Their views included seeing mathematics learning
as from teachers only (one way subject, according to Pompeu’s category’s
 in brackets), from text books (reproductive subject). They gave this view
of mathematics in spite of agreeing that students can learn mathematics out
of school or that mathematics is found in traditional culture. They also believed
that mathematics knowledge does not rely on knowledge students bring from
outside of school (separated subject), SM is not useful to a traditional society
and that CM is not as important as SM. These perceptions will be referred
to as the “school mathematics oriented”or SMO views about mathematics
teaching and learning.
Their views about the nature of mathematics show that
although they had a “not sure”response to the statement, “mathematics
is culture free” their responses to the other statements and the interview
data indicate that they regard mathematics, especially SM, as culture free.
This view is consistent with their responses to other statements which show
that they viewed mathematics as based on universal truths which are absolute
and are independent of any kind of cultural or social factors (universal and
culture free subject). They also viewed mathematics as emphasising rules,
facts, procedures and methods  views which can also be described as “instrumentalist”span>
views where mathematics is seen as a bag of tools or as “mathematics
of the curriculum” where mathematics consists of separate entities of
arithmetic, algebra or geometry with no structural relationship (Ernest, 1992;
Dossey, 1992).
In studies reported by Dossey (1992), these conceptions
were also described by Steinberg (1985) and Thompson (1985) as dualistic views
(according to a modified version of Perry’s stages of intellectual development
where individuals pass through stages; from dualism to multiplistic perspectives
to relativistic perspectives. See also Section 3.2.1.1). It can be said that
Tina and Maria have dualistic views about the nature of mathematics. These
conceptions are referred to as SMO views about the nature of mathematics.
These dualistic views begin with teacher perceptions
of SM and CM. That they regarded SM and CM as separate entities is supported
by interview data. In response to the interview question, “Is it possible
to have a culturally oriented curriculum” Tina replied , ”i>...maths
in PNG (culture) deals with arithmetic ... counting . Maths we’re teaching
now is something new, nothing to do with cultural background. ... For that
reason students can’t even apply what we’re teaching them. If
there is any maths that has been found in PNG (culture), then it’s okay”
Their regard for SM and CM as separate entities extended to their perceptions
about the nature of mathematics where they see mathematics as consisting of
separate entities. In other words, CM is just another topic of mathematics
but is more equated to arithmetic.
The explanation for the high percentage of “not
sure”responses in the sample (25%) given in section 6.1.1 (Teacher beliefs
about the nature of mathematics) certainly holds true for the SMO teachers.
They were not sure whether mathematics is culture free but this also reflects
the conflict between their perceptions of SM and CM.
Their concept of mathematics as “school mathematics
oriented”is reinforced by their responses to Section B of the questionnaire
where they indicated that one would need some mathematics only for those activities
which are assumed to be done by those who have had some schooling. For example,
they believed that only the pilot flying an aeroplane needs a lot of mathematics.
Only some activities such as measuring the height of a student, selling betel
nut and estimating the height of a tree would require a small amount of mathematics.
For the cultural activities such as making patterns on bamboo walls, children
playing a traditional game, building a canoe and painting a Haus Tambaran
(spirit house), they indicated that one would not need any mathematics at
all.
In the interview, Tina gave the following reasons
for the above responses to Section B. ”.. Pilot  need a lot of maths,
can’t even get a grade 10 or grade 6 to fly an aeroplane; carpentry
 arithmetic; estimating height of tree ... using trig  lot of maths; making
patterns on bamboo walls  some skill, no schooling, no maths; children traditional
games  no maths; building a canoe  skill but not maths skill”
Maria gave the following explanation; ”...Pilot  lots of maths;
For no maths activities...  learn these activities in the village (village
activities) ... can be any person who does not know how to read or write ...”.
The views expressed here by Tina and Maria about mathematics
teaching/ learning and the nature of mathematics should be seen in the context
of the responses by others in the sample. Table 6.7 compares the responses
of this group’s views with responses of the overall sample. If statement
5 is used as an example; the SMO groups response to this statement was A or
SA (from Table 6.6), using this Table 6.7 we note that 26 % of the sample
had the same views (A/SA  SMO groups response is underlined).
We use Table 6.7 below to find the percentage of the
sample that share this view.
Table 6. 7 Sample Responses to selected items.
Mathematics
teaching /learning 
A/SA
( %) 
NS
(%) 
D/SD
(%) 
(5) The only mathematics
students learn are those taught to them by teachers in schools 
26 
1 
73* 
(13) Mathematical
knowledge is found only in mathematics textbooks 
7 
1 
92* 
(24) Students come
to school to learn “school mathematics” not cultural mathematics 
27 
3 
70* 
(4) Mathematics
identified in traditional cultural activities should also be taught
in schools 
81* 
4 
15 
(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 
(8) Mathematics
found in traditional cultural activities is not as important as the
“real mathematics” that is learnt in schools 
19 
2 
79* 
Nature
of mathematics 
A/SA
( %) 
NS
(%) 
D/SD
(%) 
(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* 
(32) Mathematics
is culture free 
39 
25 
36* 
Key : A/SA = agree and strongly agree; NS =
not sure; D/SD = disagree and strongly disagree; underlined numbers = percent
of same responses as SMO group; * = percent of same responses as CMO group.
As can be seen from Table 6.7 above, SMO views about
mathematics teaching/learning were shared by only a small percentage of the
sample. For example, for the statement, “The only mathematics students
learn are those taught to them by teachers in schools” 26 % agree or
strongly agree as did Tina and Maria. For other examples, see the above table
where the percentage of the sample with similar responses to the item is underlined
.
Some of their views about the nature of mathematics
were shared by small proportion of the sample while their other views were
shared by a larger proportion of the sample. Their disagreement to the statement
that mathematics consists of a body of knowledge whose truths should be questioned
was shared by a small percentage of the sample (19%). Their “not sure”responses
to the statements that SM is value free or that mathematics is culture free
were shared respectively by thirteen and twenty five percent of the sample
(comparatively, the lowest percentage for the statements). However, their
agreement to the statements, “Rules are the basic building blocks of
all mathematical knowledge”and “Mathematics is about knowing when
to use rules and formulas to find answers to problems”was shared by
a higher percentage of the sample (74 % and 71% respectively). This shows
that a large proportion of the sample share their SMO view, which sees mathematics
as emphasising facts, rules and formulas.
Cultural mathematics oriented (CMO) views.
The differences between the SMO views and the CMO
views are noted here. Pompeu’s categories will also be used to help
identify their beliefs orientation. Interview data are also used to clarify
the differences between the SMO and the CMO group.
Table 6. 8 Teacher Beliefs Profile 2  CMO (Titus).
Questionnaire
Items 
Titus 
comments 
Math
learning : Locus  where math is learnt. 


The only mathematics
students learn are those taught to them by teachers in schools 
D 
His responses to most of
the statements here are in direct contrast to the SMO groups responses.
His disagreement to these statements shows that he has CMO views. 
Mathematical knowledge
is found only in mathematics textbooks 
SD 

Students
come to school to learn “school mathematics” not cultural
mathematics 
D 

Mathematics
teaching
(what maths should be taught) 


Mathematics identified
in traditional cultural activities should also be taught in schools 
SA 
He supports the idea that
CM should be taught in schools and also that students out of school
knowledge should be taken into account. 
In schools, teachers
should teach only the mathematics that is prescribed in the syllabus
and textbooks 
SD 
Again these views in contrast
with SMO views. 
When
teaching mathematics teachers should take into account students prior
knowledge learnt out of school 
SA 

Mathematics
teaching
(Why that maths should be taught.) 


School mathematics
is useful in traditional societies 
SA 
These responses show that
his views about math teaching are consistent with CMO views where CM
is held in high regard. 
Mathematics found
in traditional cultural activities is not as important as the “real
mathematics” that is learnt in schools 
SD 

Mathematics identified
in traditional cultural activities is useful to a modern PNG society 
A 

Nature 


(2) Mathematics consists of a body of knowledge
whose truths should be questioned 
SA 
His views about the nature
of math are consistent (again views are in contrast to SMO group). 
(6) School mathematics
is made up of abstract concepts and ideas which are value free 
D 

(14) Mathematics
is about learning arithmetic, algebra and geometry. 
D 
Dynamic problem driven view
of maths as opposed to maths of the curriculum or SMO views. Math not
necessarily seen as separate entities of arithmetic, algebra, geometry 
(16) Mathematics
identified in traditional culture is too simple (at the arithmetic level) 
NS 

(22) Rules are the
basic building blocks of all mathematical knowledge 
D 
Math is much more than just
rules etc. His views here contrast with SMO views 
(23) Mathematics
is about knowing when to use rules and formulas to find answers to problem 
D 
Dynamic problem driven view
of maths as opposed to instrumentalist view 
(32)
Mathematics is culture free 
SD 
Strong CMO view. 
Key :
A/SA = agree and strongly agree; NS = not sure; D/SD = disagree and strongly
disagree.
In contrast to Tina and Maria, Titus is identified
as having strong CMO views. His views about mathematics teaching/ learning
include seeing mathematics as (Pompeu’s category’s in brackets)
: applicable and useful (practical subject), a subject which investigates
environmental situations (exploratory and explanatory), a subject which includes
knowledge which pupils bring from outside of school (complementary). On the
whole these views of mathematics can be described as dynamic problem driven
(Dossey, 1992; Thompson, 1992; Nickson, 1992; Ernest, 1992; Lerman, 1983).
These perceptions shall be referred to as CMO views about mathematics teaching/
learning.
Titus had consistent internal views about the nature
of mathematics. For example, he strongly agreed that mathematics consists
of a body of knowledge whose truths should be questioned and disagreed that
mathematics is culture free or that school mathematics consist of concepts
and ideas which are value free. He disagreed that “Mathematics is about
learning arithmetic, algebra and geometry” or that “Rules are
the basic building blocks of all mathematical knowledge” or that “Mathematics
is about knowing when to use rules and formulas to find answers to problem”
Titus”perception, although strongly acknowledging the existence of CM,
seemed to go beyond the level of perceiving mathematics as separate entities
 SM and CM. His responses about the nature of mathematics indicated that
he has clear perceptions about mathematics at the generalisation level. Titus
can be identified as having multiplistic views (Dosey, 1992). These views
are referred to as CMO views about the nature of mathematics.
That Titus”concept of mathematics is “cultural
mathematics oriented”is reinforced by his responses to Section B of
the questionnaire which indicated that he is aware of the mathematics that
can be found in “non school”or “cultural”activities
(see Table 5.15 in Section 5.7.1, Chapter Five for details). For example,
for all of the activities, he indicated that one would need at least some
mathematics. He also indicated that, together with the pilot flying an aeroplane,
one would need a lot of mathematics for the following activities; villagers
building a traditional house, villager using the stars to navigate by canoe
from one island to another, painting a haus tambaran (spirit house) and the
carpenter building a house.
Titus provided the following explanation in the interview
for his responses to Section B : For lots of maths responses, “…
thinking level high, highly skilled, person needs a lot of maths. Some maths
for all activities...”
Again Titus”responses about mathematics teaching/
learning and the nature of mathematics (Table 6.8) are compared in the context
of the responses by others in the sample (Table 6.7).
As can be seen from Table 6.8, Titus”responses
to items in the mathematics teaching and learning category, marked with an
asterisk (*), are also in direct contrast to those in the SMO group. His views
are also shared by a large proportion of the teachers in the sample. For example,
his disagreement with the first three statements in Table 6.8 is shared respectively
by 73, 92 and 70 percent of the teachers in the sample.
The pattern of response about the nature of mathematics
begins with the majority (68 %) of the sample sharing his view that mathematics
consists of a body of knowledge whose truths should be questioned and a higher
percentage (53 % as compared to 43 %) also disagreeing (as did Titus) that
mathematics is about learning arithmetic, algebra and geometry. Other statements
in this category had a relatively smaller percentage of the sample with the
same response. For example, only 19 % of the teacher sample also disagreed
or strongly disagreed that rules are the basic building blocks of all mathematical
knowledge and 26 % disagreed (as did Titus) that mathematics is about knowing
when to use rules and formulas to find answers to problem.
In general, the CMO views about mathematics learning
and teaching were shared by a large proportion of the sample. However, the
CMO views about the nature of mathematics were shared by a smaller percentage
of the sample, when compared with the SMO group.
Intermediate (INT) views.
Comparisons will be made with both the SMO and CMO
groups. Pompeu’s categories and the interview data are also used to
verify their beliefs orientation. This group is characterised by the mixed
SMO and CMO views.
Table 6. 9 Teacher Beliefs profile 3  Intermediate (Markos,
Petrus)
Questionnaire
Items 
Markos 
Petrus 
comments 
Math
learning : Locus  where math learning takes place 



The only mathematics
students learn are those taught to them by teachers in schools 
D 
D 
Basically, these views about math learning
are same as for CMO group 
Mathematical knowledge
is found only in mathematics textbooks 
D 
SD 

Students
come to school to learn “school mathematics” not cultural
mathematics 
D 
A 

Mathematics
teaching
(what maths should be taught) 



Mathematics identified
in traditional cultural activities should also be taught in schools 
A 
SA 
CM should be taught in schools 
In schools, teachers
should teach only the mathematics that is prescribed in the syllabus
and textbooks 
SA 
D 
Markos may believe that CM should be taught
as long as in syllabus 
When
teaching mathematics teachers should take into account students prior
knowledge learnt out of school 
A 
A 
Again these views basically CMO. 
Mathematics
teaching
(Why that maths should be taught.) 



School mathematics
is useful in traditional societies 
SA 
A 
Both express CMO views here 
Mathematics found
in traditional cultural activities is not as important as the “real
mathematics” that is learnt in schools 
D 
SD 

Mathematics identified
in traditional cultural activities is useful to a modern PNG society 
A 
A 

Nature 



(2) Mathematics
consists of a body of knowledge whose truths should be questioned 
A 
A 
Both have mixed views about the nature of
math  both express SMO and CMO views. For this statement both express
CMO views. 
(6) School mathematics
is made up of abstract concepts and ideas which are value free 
D 
A 
Markos contradicts above 
(14) Mathematics
is about learning arithmetic, algebra and geometry. 
A 
D 
Markos views  emphasise rules, facts and
methods, mathematics of the curriculum  SMO view. Petrus expresses
CMO view 
(16) Mathematics
identified in traditional culture is too simple (at the arithmetic level) 
SA 
A 

(22) Rules are the
basic building blocks of all mathematical knowledge 
SA 
 
Markos  Instrumentalist view, bag of tools
to use  SMO view 
(23) Mathematics
is about knowing when to use rules and formulas to find answers to problem 
A 
D 
Markos  same as above while Petrus  CMO
view 
(32) Mathematics
is culture free 
SD 
SA 
Markos expresses CMO view while Petrus expresses
SMO view 
Key :
A/SA = agree and strongly agree; NS = not sure; D/SD = disagree and strongly
disagree
Markos and Petrus are identified as having rather
mixed SMO and CMO views about both mathematics teaching/ learning and the
nature of mathematics.
Both Markos and Petrus disagreed that “The only
mathematics students learn are those taught to them by teachers in schools”and
“Mathematical knowledge is found only in mathematics textbooks”
The responses indicate that they viewed mathematics as investigating environmental
situations (exploratory and explanatory, according to Pompeu). They also agreed
that CM should be taught in schools (complementary subject) and that teachers
should take into account students knowledge learnt out of school (practical
and useful subject). These views can be identified as CMO views about mathematics
learning/ teaching .
However, both of them also manifested SMO views about
mathematics teaching / learning. For example, Petrus agreed that “students
come to school to learn school mathematics, not cultural mathematics”while
Markos strongly agreed that teachers should teach only the mathematics that
is prescribed in sylla bus and text books.
Further mixed views were manifested in their responses
to items about the nature of mathematics. Markos, for example, expressed strong
CMO views when he agrees that mathematics consists of a body of knowledge
whose truths should be questioned and disagrees that SM is made up of abstract
concepts and ideas which are value free. He also strongly disagrees that mathematics
is culture free (mathematics as socially / culturally based). However, he
displayed strong SMO views by agreeing or strongly agreeing to the statements;
“Mathematics is about learning arithmetic, algebra and geometry”
“Rules are the basic building blocks of all mathematical knowledge”and
“Mathematics is about knowing when to use rules and formulas to find
answers to problem”(views of mathematics as an informative subject).
These responses indicate that he saw mathematics as emphasising rules, facts
and methods  strong SMO views which can also be described as instrumentalist
views where mathematics is seeing as a bag of tools (Ernest, 1992) or as “mathematics
of the curriculum”where mathematics is arithmetic, algebra or geometry
(Thompson, 1992). He can be described as having dualistic views.
Petrus also expressed views which were contradictory.
He manifested strong CMO views when he agreed that mathematics consists of
a body of knowledge whose truths should be questioned, and disagreed that
mathematics is about learning arithmetic, algebra and geometry or that mathematics
is about knowing when to use rules and formulas to find answers to problems.
However, he expressed SMO views by strongly agreeing that mathematics is culture
free and that it is made up of concepts and ideas which are value free.
Although both are in the intermediate category, it
is to be noted, especially from the responses about the nature of mathematics,
that Petrus”perceptions were biased towards CMO views while Markos”conceptions
lean more towards SMO views. These inclinations were further confirmed by
their responses to section B of the questionnaire.
Markos indicated that one would need some or a lot
of mathematics only for those activities which are assumed to be done by those
who have had some schooling (for example, flying an aeroplane, building a
house, selling betel nut, estimating the height of a tree) while some activities
(such as counting) and in particular the “cultural activities”(such
as making patterns on bamboo walls, children playing a traditional game, woman
weaving a mat ) required no mathematics. The important point that revealed
his SMO inclinations is the fact that he thought some activities do not require
any mathematics at all. When asked about these responses in the interview,
Markos”reply was, ”.. Carpentry  lot of calculation; estimating
 need a lot of maths technical people need that. No maths for other activities
 don’t really need maths for counting, any old person can come and
count, does not need maths for that ...”In other words, because old
people in the village can count, there is no mathematics in counting. In the
interview, his response to the question, “Is it possible to have a culturally
appropriate curriculum” was, “I don’t think so because in
village situation no formal maths like what we have now... Own counting system
but that is all.”o:p>
Petrus on the other hand indicated that all the activities
would need at least some mathematics, with the pilot flying an aeroplane,
the carpenter building a house and the villager using the stars to navigate
from one island to another requiring a lot of mathematics. Again his CMO inclinations
were revealed by the fact that he thought there was some maths in all the
activities listed. Petrus”explanation for his responses to Section B
was : ”.. Pilot  needs lots of maths .. must be good in maths; building
bamboo wall  don’t need a lot of maths ... repetition; villager using
stars to navigate requires a lot of maths; knowledge of astrology ... stars
... position where he is going, ... distance, destination. All of activities
require some mathematics.”o:p>
Summary.
In summary, the teachers”beliefs about mathematics
seemed to fall into three basic categories  school mathematics oriented views,
cultural mathematics oriented views and views that are a mixture of these
two views (intermediate views).
In the context of this study, the “school mathematics
oriented”teachers saw mathematics teaching and learning as from teachers
or textbooks only and believed that mathematical knowledge does not rely on
knowledge brought from out of school. In other words, these teachers were
aware of the existence of CM but they still believed that CM cannot be taught
in schools. The SMO teacher views about the nature of mathematics were that
mathematics is culture free, and is based on universal truths which are absolute,
emphasising facts, rules and formulas.
The “cultural mathematics oriented”or
CMO teacher was not just aware of the existence of the “other mathematics”or
CM (almost all of the teachers in the sample agreed that mathematics exists
in traditional cultural activities, for example). What distinguished the CMO
teacher from the SMO teacher is that the CMO teacher did not see mathematics
teaching and learning as coming from teachers only but felt that it also depends
on knowledge the students bring from outside of school. Of particular interest
is the teachesrs”view about the nature of mathematics. The CMO teacher
view about the nature of mathematics was that mathematics is not culture free
and that it consists a body of knowledge whose truths should be questioned.
He did not see mathematics as about learning algebra or geometry etc. nor
did he see rules as the basic building blocks of all mathematical knowledge.
A distinction needs to be made here in the use of
terms “culturally aware”and “cultural mathematics oriented”
The culturally aware teacher acknowledges the existence of CM but the CMO
teacher also believed that CM can be taught in schools and his views about
the nature of mathematics were CM oriented (or “internally”oriented).
The teachers with the INT view were characterised
by the fact that they were not just aware of CM but they had mixed beliefs
about mathematics teaching and learning and about the nature of mathematics.
In general the majority of the PNG teachers in the
sample held CMO views about mathematics learning and teaching. While their
views about the nature of mathematics were mixed, they seemed to fall into
two categories. The strong SMO views had to do with agreeing (approximately
70 %) that mathematics is about rules and formulae. However, approximately
70 percent of the sample also expressed strong CMO views by agreeing that
mathematics consists of a body of knowledge whose truths should be questioned.
6.2.2 Differences in classroom practice.
In this section the differences in the observed teacher
classroom practices are reported using teacher profiles. The main question
that is addressed is, Were the differences in practice due to teacher beliefs
?
Analysis of observed mathematics lessons
In discussions on student teacher training, Bishop
and Goffree (1986) provide a conceptual structure for the typical “mathematics
lesson”which they describe in terms of the “lesson frame”
Teacher activities in the “typical”lesson include, ””instruction,
exposition, ‘chalk and talk” board work, question and answer,
”together with seat work, practice and the individual help of those
children who need help.”(Bishop & Goffree, 1986: p.311). They also
quote the 1979 Inspectors”report (H.M.I., U.K) which described mathematics
lessons as, “… predominantly teacher controlled : teacher explained,
illustrated, demonstrated, and perhaps gave some notes on the procedure and
examples. ”A common pattern, ”was to show a few examples on the
board at the start of the lesson and then set similar exercise for the pupils
to work on their own.”(Ibid. p.313).
Bishop & Goffree (1986) 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.” (Ibid. : p.315). This orientation
includes the following features :
 it puts the teacher in relation to the whole classroom group;
 it emphasises the dynamic and interactive nature of teaching.;
 it assumes the interpersonal nature of teaching
 it recognises the shared idea of knowing and knowledge, reflecting
the importance of both content and context;
 it takes into account the pupils”existing knowledge, abilities
and feelings, emphasising a developmental rather than a learning theoretical
approach (Ibid. : p.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 individuals”existing knowledge. The teacher role is
therefore to manage activities and provide opportunities for pupils to create
their own mathematical meanings.
They proposed three main components of the mathematics
classroom :
 activity  emphasis on learners involvement rather than on teachers presentation
of mathematical content;
 communication  underlies all teaching and is essential to shared meanings,
for example, in the teacher’s explanation and interpretation of mathematical
ideas;
 negotiation  of mathematical meaning where the emphasis is on goal directed
interaction of classroom teaching whereby teacher and learner seek to attain
respective goals.
The “social construction frame”provides
a useful way of analysing the mathematics lessons as it is sensitive to the
cultural aspect of mathematics teaching. The basic model is used in the analysis
here although some interpretations of the above categories will differ. The
analysis here considers how mathematics is portrayed in the mathematics
lesson at the secondary level in the following “teacher activities”:
 Classroom activities or mathematical exercises. This refers to
the activities that the teacher gets the students to participate in or the
mathematical exercises that the teacher gives to them. Does the teacher
provide opportunities for the pupils to “create or construct”their
own mathematical meanings (Bishop & Goffree, 1986) or for the students
to come up with their own solutions ? In the exercises, are the examples
CM oriented? This also includes what resources the teacher uses.
 Explanation or Exposition or Interpretation. The emphasis here
is on “communication”(equal to the Bishop & Goffree, 1986,
communication category) and will include :
( a ) explanation of mathematical concept, idea, topic or theory and the
examples used in the explanation;
( b ) interpretation of mathematical ideas, of representations, of symbols,
use of rules and formulas, and
( c ) the kind of language used. Does the teacher encourage communication
between pupils and teacher ?
What type of questions does he ask ? Are they just questions about stating
facts, rules and formulas ? Or are they questions of the type, “why
do you do that ?”Does the teacher probe the wrong answers and try
to understand why the students got something wrong ? (as opposed to saying
the answer is wrong because the rules say so). Does the teacher listen and
tries to make connection ? Does he/she explain terms and formulae, not just
state facts (explaining rather than telling)?
 Demonstration or Illustration and Review or Correction of exercises.
The emphasis here will be on the teacher portrayal of mathematics,
especially how the nature of mathematics is portrayed in the methods of
solution. Are they negotiable ? Negotiation here differs in meaning from
the way used by Bishop & Goffree. It is similar in meaning to Hoyles”(in
Harris & Evans, 1991) description of the formal and informal mathematics,
which formed the basis of the “Observation Schedule”that was
used as a guideline for the classroom observations conducted for this current
research (see chapter three). Negotiation therefore, refers to the methods
of solution and the approach adopted by the teacher in the classroom. For
example, are the illustration the teacher uses, the demonstration of the
methods of solutions and the correction of mathematical exercises all negotiable
? Does the teacher allow or use student suggested solutions ?
The three activities listed above are key teacher
activities in the classroom which constitute “teacher practice”
The mathematics lessons that were observed by the
writer at the PNG secondary schools are analysed using the above categories.
The use of these categories in the analysis will help identify the classroom
practices of the culturally aware teacher.
The aim of the analysis is to investigate the above teacher activities
at the classroom level to see if there are any differences in teacher practices
and eventually to investigate the relationship that exists between these teachers
beliefs and their practice. Of particular interest is to see if there were
any differences in the practices of the “cultural mathematics oriented”(CMO)
and the “school mathematics oriented”(SMO) teachers. In other
words, does being aware of this “other mathematics”(CM) make any
difference to their classroom practice or is there any difference between
the practices of the SMO and CMO teachers.
In the classroom observations, the researcher did not expect to see examples
of CM because of the constraints of the PNG secondary mathematics curriculum
which can be described as “canonical”and so the “intended”mathematics
content was SM (although the absence of CM from text books did not exclude
any teacher from using CM examples in their teaching).
The emphasis of the analysis is therefore on “how mathematics was
portrayed”in the lessons that were observed. This is based on the premise
that teacher beliefs about nature of mathematics are manifested in their portrayal
of mathematics (Copes, 1979; Kesler, 1985; Cooney, 1985 in Thompson, 1992;
see also “theoretical construct”in Section 3.2.1 in chapter three).
It is recognised that “how mathematics is portrayed”is highly
inferential but by using the above categories, inferences about the nature
of mathematics that is portrayed can be made. It is proposed to investigate
“how mathematics was portrayed”in the classroom by examining teachers”explanation
of the concept, in the methods of solution (are they negotiable), in the examples
that were used, in the use of the formulae and rules, the language used, the
resources used. Was it portrayed as a body of knowledge which is absolute,
value free and abstract  formal, external view of the nature of mathematics
? Or was mathematics portrayed as developing through a process of inquiry,
where questioning of knowledge and uncertainty is accepted as part of the
discipline.
Teacher Practice Profiles .
The data that were used to collate the following are from the Teacher
profiles in Section 5.8, Chapter five (For examples of the points that are
given for each of the teacher profiles, see the above section). The teacher
profiles are based on three key teacher activities or practices in the classroom
which were listed above.
Table 6. 10 Classroom Activities or Mathematical Exercises.

Classroom activities 
Examples, exercises used 
Resources used 

SMO 
Tina 
Teacher dominated lessons Impression of student
learning, active learners. Lot of communication between teacher and
students, mostly to teacher directed questions. 
Formal mathematics examples  from textbooks, work sheets.
No example of mathematics from traditional culture

Activities and exercise used from text book
: 5/5 work sheets photocopied from texts
Text book main source of knowledge 
Maria 
Teacher dominated lessons Impression of student
learning, passive learners. Not much communication between teacher and
students  communication one way. 
Formal mathematics examples  from textbooks, work sheets.
No example of mathematics from traditional culture 
Use of textbooks  sticks to textbook,
activities & exercises from text books
Activities and exercises used from text books : 6/6 (work sheet supplemented
2/6 ). 

CMO 
Titus 
Teacher gets students to suggest ways of finding a solution,
not call out answers only, gets students to actually participate in
finding solution. Solves problems together with students. Teacher
fields suggestions eg. Sum of numbers from 1 to 20.
Contextualises problem eg. Actual dramatisation of handshake
problem in 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 was actually used. 
Activities / exercises used from text book : 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. 

INT 
Petrus 
Typical math lessons. Topic from text book,
explain  example  exercise from text book, teacher corrects exercise,
gives extra work. 
Most exercises from text book. Teacher also
gives own exercises which are similar to text book. 
Activities / exercises used from text book : 6/6 lessons
Used own examples : 4/6
Textbook only resource used this week
Predominant use of textbook.
Most exercises from text book. Mathematics from textbook
only. 
Markos 
Mathematics lessons are all about going through the text
book.
Mathematics presented as a reproductive subject.
Mathematics is portrayed as a one way subject (mathematical
knowledge transmitted from the teacher to the pupils). 
Examples and exercises used mostly from text book.
Opportunity to explore math in traditional artefacts etc.
was passed, (9A  24/10/95). Topic on shapes  cylinders, cones, shapes
of houses  text has picture of traditional round houses. 
Activities / exercises used from text book : 8/8 lessons
Used own examples : Once
Predominant use of textbook  study of mathematics centers around textbook.
The mathematics lesson seems to be all about going through the textbook.
Text consulted constantly to check for answers. 
Table 6. 11 Explanation, Exposition and Interpretation.

Explanation /exposition 
Use of rules, formulas, (interpretation
of rules , symbols) 
Language used 

SMO 
Tina 
Explanations consist of statements of facts,
rules. Typical “mathematical”expla nations. 
Rules reliance, emphasis on rules and formulas 
Language precise, carefully differentiated
 mathematical language 
Maria 
Statements of facts, rules predominant in explanations.

Rules reliance.
Writes out rules and formulas, gets students to copy (area of rectangle
and square). There is really one way to solve  mathematics is about
rules, formulas. 
Typical mathematical language 

CMO 
Titus 
Encourages pupils to give their ideas (answer
questions) and builds on the answers, probes wrong answers. Uses own
words to explain questions and terms. Two way communication.
Asks quite often, why? How did you do it ? what is this pattern, where
did the ... come from? How did you get the number ...? 
Did
not seem to totally rely on rules. Gives the impression there are many
ways to solve a problem. 
Uses language which is familiar to students,
every day language. Uses metaphors, situations to explain problem. 

INT 
Petrus 
Formal explanations of rules, formulas, typical
explanations 
Relies on rules  resorts to rules for explanation. 
Uses mathematical language 
Markos 
Explanation of terms and concepts  very formal,
uses text book language, refers to text book for explanations. Lack
of probing of student answers. Explains concepts in terms of facts,
rules and formulas.
One way communication. Type of questioning mostly to do with statement
of facts. Eg. “what is ” etc. 
Definitely relies on rules and formulas, constantly asking
students about the formulas that are required for solution . Mathematics
portrayed as decontextualised, manipulation of numbers, consists of
undisputed facts. 
Predominant use of formal mathematical language, language
of the textbooks. 
Table 6. 12 Demonstration or Illustration and Review
or Correction of Exercises.

Method of solution 
Assessment procedures (formal / informal) 
SMO 
Tina 
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.
When giving solutions, students are asked, ”i>Why
is....?”/i>.
Mathematics is so well organised, defined, no room for mistakes,
all about facts.
Impression math does not rely on students out of  school
knowledge 
Informal assessment carried out regularly
throughout the lesson, to check if students understand  by asking questions
. Does ask lots of questions. 
Maria 
Asks for solutions to problem but as if there’s only
one way to solve it.
e.g. There is really one way of finding areas of rectangle
and squares  by using rules, formulae.
Was heard, There are other ways of writing ... algebraic
solution but did not explore, pursue this line.
Mathematics is precise with rules, solutions with algebraic
manipulations. 
Routinely checks student exercise books 
CMO 
Titus 
Methods of solution are negotiable, allows
students to use own methods of solution. Presents mathematics as a debatable
subject. Shows steps to solution by asking questions.
Approach suggests  math relies on students
outofschool knowledge 
Asks a lot of questions  how did you
get ..? why ? what is ... , explain.. How did you do it ... explain.
Directs questions at student, around room. 
INT 
Petrus 
Sticks to formal methods of solution 
Questions of the type, what do I do next, which
formula do I use etc. 
Markos 
Teacher does not seem to promote that other methods of solution
are possible. There is one correct way to find the solution., through
the use of formulas. Other possible methods of solution are not explored. 
Questions were mostly of the
type : which question in text would you like me to go through, Do you
understand. 
Summary
There were no differences in the classroom practices
of the SMO and the INT groups. Although there were differences amongst these
two groups, these were nominal differences. The practices of these two groups
are referred to here as the “school mathematics oriented”practices
where the lessons are typical mathematics lessons which fitted into the “lesson
frame”category (Bishop & Goffree, 1986).
Differences were observed between the classroom
practices of the CMO and the SMO / INT groups. There were differences in the
following areas :

Activities. The CMO teacher provided opportunities
for the pupils to come up with own solutions, contextualise (or dramatise)
mathematical problems. The CMO teacher had students actually participate
in finding solutions, and he did it together with students.

Explanations. The CMO teacher encouraged pupils to
give their own ideas and built on the answers, using his own words to
explain terms and questions. Use questions of the type, Why do you do
that ? How did you get that ?

Methods of solution . The CMO teacher presented methods
of solutions as negotiable, presenting mathematical facts, rules, formulae
through the process of inquiry. Portrayed mathematics as consisting of
facts or rules which are questionable; his methods /approach in classroom
also suggested that they are. Also showed steps to solutions by asking
questions.
In other words, although the basic lesson activities
and the contents were the same, the CMO teacher’s classroom approach
differed from that of the SMO / INT group. His approach suggests that mathematics
is a debatable subject where the knowledge also depends on what the students
bring from outofschool. The SMO teachers”classroom approach is typical
of the mathematics lesson in the “lesson frame”which portrayed
mathematics knowledge as consisting of unquestionable facts, rules and methods
of solution where the teachers and the text books are the predominant source
of information and students”outofscho ol knowledge is not taken into
account.
Why the difference ? Explanations for the difference
is provided in the next section.
Further to the discussions (in the summary to
Section 6.2.1) about the differences between the “culturally aware”and
the “cultural mathematics oriented”span> teacher, we can now
add that as far as practice is concerned, although the former may be aware
of CM, he/she is not inclined or oriented to do something about it. The CMO
teacher is distinguished from the “culturally aware”because the
CMO teacher beliefs orientation is manifested in his classroom practice.
6.2.3 Relationship between teacher beliefs and classroom practice
In the issue of beliefs and practices, there are two
positions that were considered. One is to assume that beliefs influence practice.
The research implications are : To identify differences in teacher beliefs
about SM and CM. If there are differences, to carry out observations to determine
if these differences are manifested in the teachers”classroom practices.
The other position is, not to assume that beliefs influence practice but rather
to identify teacher differences in beliefs (via questionnaire and interviews),
then to identify differences in teacher practices (through observations).
The researcher’s role is to investigate the relationship between the
beliefs and the practices. In this study, the role adopted is as described
in the second category. It was established that there were differences in
teacher beliefs, there were also differences in teacher practices. Thus, in
this section some important issues about the relationship between beliefs
and practice are addressed. For example, Is there any evidence of the influence
of teacher beliefs on their classroom practice ? How do the teacher beliefs,
expressed in their responses to the questionnaire and the interviews manifest
themselves in their classroom teaching ?
It was also established in the last section, that
there were differences in the classroom practices of the SMO / INT and the
CMO teachers. An important question that needs to be addressed in this section
is therefore, whether teacher beliefs is likely to have been a factor that
caused the difference. Other issues are: Were there any mismatches between
their beliefs and their practices ? What is the likely effect of practice
on teacher beliefs ?
The main points to be addressed will therefore be
:
 Is there any evidence of the influence of teacher beliefs on their classroom
practice ?
 Why is there a difference in the practices of the CMO and the SMO/ INT
teachers ? Was this due to teacher beliefs ?
 What is the explanation for the mismatch between some beliefs about mathematics
and practice ?
 Is there any evidence of the influence of views about the nature of mathematics
over other beliefs about mathematics.
 Influence of beliefs about mathematics on practice and the influence
of practice on these beliefs.
 Generally, what is the relationship between beliefs and practice ?
Beliefs”influence on practice
Was there any evidence in this study about the influence of teacher beliefs
on their classroom practice ? Why were there difference in the practices of
the CMO and the SMO/ INT teachers. Were they due to teacher beliefs ? These
important questions are addressed by linking beliefs that were manifested
in practice and analysing the beliefs that were not manifested in practice.
Beliefs manifested in practice.
Table 6.13, constructed using data from Tables 6.10,
6.11, 6.12 in Section 6.2.2, shows where the beliefs were observed to be manifested
in practice. The next section considers the dissonance between belief and
practice.
Table 6. 13 Beliefs manifested in practice
Beliefs 
Classroom
Practice 
SMO 
Tina, Maria :
Have instrumentalist views  View math as emphasising rules,
facts, procedures.
Mathematics based on universal truths which are independent
of any kind of cultural or social factors, separate entities of arithmetic,
algebra, or geometry SMO view  Nature of mathematics
View math learning as from teachers only,
From text books
Math knowledge does not rely on student’s
outofschool knowledge. Schools for SM, not CM 
Rules reliance, emphasis on rules and formulas, explanations
consist of facts, rules
Teacher is dominant in classroom, main source of math knowledge,
predominant use of text book, no negotiation which builds on students”out
of school knowledge 
CMO 
Titus
Views  math knowledge should be questioned, math as debatable;
Mathematics is not culture free, math consists of body of
knowledge whose truths should be questioned,
Disagreed that math is about learning arithmetic, algebra,
geometry or is about knowing when to use rules and formulas to find
answers to problems  CMO view  Nature of mathematics
Math as subject which includes students”out of school
knowledge. 
Classroom approach suggest that methods of solution are
negotiable, presents math as debatable subject, encourages students
to give their own ideas, contextualises problems, uses students”outof
school knowledge, hardly mentions text book 
INT 
Petrus
Agrees that math is culture free and
that it is made up concepts and ideas which are value free; SMO view
 Nature of mathematics (mixed)
Also agrees that students come to school
to learn SM, not CM 
Typical math lessons, topics, exercise from text books,
formal methods of solution, relies on rules and formulas, formal explanations,
math portrayed as a decontextualised subject. 
Markos
Instrumentalist views where math is rules, facts and procedures,
is all about learning algebra, geometry etc. SMO view Nature of
mathematics views (mixed)
Teachers should teach only the mathematics that is found
in syllabus 
Definitely relies on rules and formulas, constantly asking
students about formulas that are required for solution
Mathematics lessons are about going through the text books.
Text book predominant source of knowledge 
Table 6.13 shows where stated teacher beliefs were
manifested in observed practices. This match between beliefs and practice
shows that there is a relationship between beliefs and practice although how
they are related is not clear. It is also recognised that the existence of
this relationship does not necessarily denote a causeandeffect relationship.
The main point to be noted here is that teacher beliefs about mathematics
are “manifested”in their classroom practices, especially in the
portrayal of mathematics.
The diagram below summarises the main point that emerges
from the Table 6.13 when the individual teachers are considered. It shows
that the SMO and INT views result in the same practice (SMO).
Figure 6. 2 Beliefs about nature of mathematics and corresponding
practice.
Beliefs about the nature of mathematics influence practice.
An important point that comes out of the Table 6.13
is the role of the nature of mathematics in the way mathematics was portrayed
in the classroom. As can be noted from Table 6.13, the most notable manifestations
of their beliefs in classroom practices were the teachers”beliefs about
the nature of mathematics. The manifestation of the beliefs in practice was
clearly seen where teachers have definite SMO and CMO views about the nature
of mathematics. Teachers with INT views (mixed) used the same classroom practices
as those with SMO views but only where their views about the nature of mathematics
were the same as SMO views. In other words, for these teachers, it was the
beliefs about the nature of mathematics which are manifested in practice.
The evidence here suggests that teacher beliefs about the nature of mathematics
may have a greater influence on their practice.
The finding that teacher beliefs about the nature
of mathematics influence classroom practice is supported by literature reported
in chapter two. For example, Thompson (1992: p.127) cites Hersh 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. Mayers (1994)
suggests that teachers”beliefs about mathematics, rather than beliefs
about mathematics teaching, has a greater influence on their practice. Schoenfeld
(1992: p.341) states that what one thinks is the nature of mathematics will
shape his/her practice.
The discussion of cases below provides further insights
into how beliefs about mathematics teaching and learning, in particular definite
beliefs about the nature of mathematics, are manifested in practice.
Case 1. Teachers with different beliefs but same practice
The cases where teachers with different beliefs have
the same practice are considered first. In this study, examples of these cases
include the SMO teachers versus the INT teachers. The SMO and INT teachers,
although having different beliefs had the same practice. Even within the INT
group, the teachers, although having mixed beliefs, had different beliefs
orientations. One was more CM oriented while the other was more SM oriented.
Another example of the above case would be INT A (INT teacher from school
A) versus INT C (INT teacher from school C).
Why were there no differences in the practices of
the SMO and INT groups ? The similarities in practice is usually explained
in terms of the constraints on teacher practice. For example, teachers”choice
of practice is influenced by curricular constraints, school values, other
teachers, social pressures, social context etc. (see section on constraints
on practice in chapter two and also the next section in this chapter). A similar
example is reported in chapter two where Ernest (1988) noted the effect of
social context on instructional practice where teachers in the same school,
although having differing views, adopted similar classroom practice.
While the above explanation is possible, it is suggested
here that another reason for there being no difference in the SMO and INT
practices is because the INT group had no definite beliefs about the nature
of mathematics or mathematics teaching /learning (remember the INT had mixed
beliefs with some beliefs similar to the SMO group). In other words, their
views overlapped. Furthermore, this indicates that teacher beliefs about mathematics
teaching / learning are easily influenced by contextual factors. For example,
in the case where the INT beliefs were more CM oriented, because the prevalent
conditions favoured the SMO approaches and the contextual factors were more
SM oriented, their classroom practices were more SM oriented. The SMO and
INT groups had beliefs which also matched contextual factors, or the existing
curricular situations which were more SM oriented so it is not surprising
their practices were SM oriented. Their beliefs about math teaching/ learning
were not manifested in practice because these beliefs are easily overruled
by contextual factors. As will be seen from the mismatch Table 6.14, there
are situations where beliefs about math teaching / learning are not manifested
in practice
But what is the explanation for the situation where
there are differences in the beliefs and the practices of teachers,
especially if the teachers were from the same school where the context is
assumed to be similar, as in the case of the CMO teacher and one INT teacher
?
Case 2. Teachers with different beliefs and different practices.
Consider the case where teacher with different beliefs
have different practices. In this study, examples of cases include the CMO
versus the SMO teachers, and the CMO versus the INT teachers. In the SMO v/s
the CMO case, the teachers were from different schools (where the contextual
factors may be different) but the CMO versus the INT A teacher is interesting
because they were both from the same school (which is assumed to have been
a similar context).
Explanations in terms of contextual constraints on
practice are inadequate for the above situations. It is suggested here that
the difference in practice was because their beliefs about the nature of mathematics
were different. The CMO teacher had definite views about the nature of mathematics
while the INT teacher had mixed views about the nature of mathematics. In
the case of the SMO and the INT group, it has already been explained that
there were similarities in their beliefs.
Teacher beliefs about the nature of mathematics is
a likely factor that explains the differences in teacher practices. In
Section 6.2.2, it was noted that there were differences in the practices of
the CMO and the SMO or INT groups. Table 6.14 shows that differences in practice
are observed where there were also differences in their beliefs about the
nature of mathematics. For example, where there were differences in practice
between CMO and SMO/INT groups, there were also definite differences in their
beliefs about the nature of mathematics (although teachers who have distinctly
different views about the nature of mathematics would also have differing
views about math teaching / learning ). Beliefs about math teaching / learning
were not manifested in practice.
It is to be noted that cases having Teachers with
the same beliefs and same practice have been discussed above (eg. SMO
teacher A v/s SMO teacher B, also SMO v/s INT). No cases were observed in
this study for situations that had Teachers with same beliefs but different
practice.
Beliefs not manifested in practice.
Table 6.14 also constructed using data from Table
6.10, 6.11, 6.12 in Section 6.2.2, and reveals the beliefs that were not manifested
in practice.
Table 6. 14 Beliefs not manifested in practice
Beliefs 
Classroom
Practice 
SMO 
Tina, Maria
No observed mismatch between beliefs and practice 
 
CMO 
Titus
No observed mismatch except for the statements: CM should
also be taught in schools, disagrees that students come to school to
learn SM only  CMO view 
No actual example of CM used in lessons. Example only of
SM although math was portrayed as debatable subject, methods of solution
negotiable etc. 
INT 
Petrus
Disagrees that the only mathematics students learn are those
taught by teachers only and that Mathematical knowledge found only in
text books. CMO views
Agrees that CM should be taught in schools, teachers should
take into account knowledge learnt outofschool  CMO views
Agrees that Math knowledge should be questioned and disagrees
that mathematics is about learning arithmetic, algebra, geometry or
that math is about knowing when to use rules and formulas  CMO view 
Lessons are teacher and text book dominated,
Classroom approach where student outofschool knowledge is not too
important
Teaching approach does not take student
out  of  school knowledge into account.
Relies on rules and formulas, formal explanation of rules,
formulas, sticks to formal methods of solution, portrays math knowledge
as unquestionable. 
Markos
Disagrees that the only mathematics students learn are those
taught by teachers only and that Mathematical knowledge found only in
text books. CMO views
Agrees that CM should be taught in schools, teachers should
take into account knowledge learnt outofschool.  CMO views
Disagrees that Mathematics body of
knowledge, should be questioned or that math is culture free  CMO
view 
Classroom approach is teacher and text book dominated, where
student knowledge is not too important. Lesson are about going through
math text books
No example of CM,
Mathematics portrayed as culture free,
consisting of unquestionable facts, rules, methods of solution unquestionable 
There are two important points to note from Table
6. 14. Firstly, the SMO teacher had no belief statements that were not manifested
in practice. It is not surprising that there were no observed mismatches between
the beliefs and practices of the SMO group. They had school mathematics oriented
views and the contextual factors favoured the SMO approaches in the classroom.
In this case, the contextual factors cannot really be considered as “constraints”to
practice but as “facilitators”of practice. The CMO teacher had
only two beliefs statements which were not manifested in practice. In the
case of the CMO teacher, the contextual factors acted as constraints which
explains why the beliefs were not manifested in practice. For example, the
CMO teacher agreed that CM should be taught in schools but in practice taught
mostly SM content.
Secondly, the INT group (those with mixed views) had
a lot of beliefs statements that were not manifested in practice. This shows
that where there are mixed views about mathematics teaching and learning and
mixed views about the nature of mathematics, beliefs are not likely to be
manifested in practice. These beliefs that were not manifested in practice
are beliefs that were identified as CMO views. The INT group’s practice
was identified as SM oriented. It is likely that beliefs are manifested in
practice only where there is a match between the contextual factors and the
beliefs.
The dissonance between beliefs and practice may therefore
be explained in terms of constraints. Beliefs are not manifested in practice
when contextual factors are different to teacher beliefs; these factors impede
or constrain practice.
Constraints on practice
Some of the mismatches (from Table 6. 14) can be easily
explained in terms of the curricular constraints. For example, Titus, Petrus
and Markos all agreed that CM should be taught in schools but the prescribed
curriculum content is mostly SM so it is not surprising that there were no
examples of CM being taught in the lessons that were observed. As noted in
chapter two, practice is largely determined by curricular context, curricular
constraints, situations. Curricular constraints such as, pressures of external
examinations, pressure to “cover syllabus” ensures that the teachers
do not deviate from teaching the prescribed content. Certainly, in the case
of PNG, 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).
These constraints were described in chapter two as
the contextual factors or the context at which practice takes place and which
may act as constraints to practice. They include, for example, curricular
constraint  pressures of exams, curriculum (intended  content, implemented,
attained), headmaster, peers (fellow teachers); sociocultural context; internal
constraints  own beliefs, knowledge, experience. Thompson (1992 : p.138)
noted the social context as one source of influence at work ;
“… 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.”o:p>
Other situations where the teacher beliefs are not
manifested in practice are not as easy to explain. For example, Petrus and
Markos disagreed that the only mathematics students learn are those taught
to them by teachers and that mathematical knowledge is found only in text
books. Yet their classroom approach is teacher and text book dominated where
student knowledge is not taken into account. They also agreed that mathematical
knowledge should be questioned but in practice mathematics was portrayed as
consisting of unquestionable facts and methods of solution. Petrus disagreed
that mathematics is about learning arithmetic, algebra, geometry or that mathematics
is about knowing when to use rules and formulae and yet in practice he relied
on rules and formulas, made formal explanation of rules and stuck to formal
methods of solutions. Markos disagreed that mathematics is culture free but
the classroom approach portrayed mathematics as culture free, consisting of
unquestionable facts and rules.
There is evidence that there are some teacher beliefs
which were manifested in classroom practice. However, it was also noted that
there are other teacher beliefs which are not manifested in practice. These
can be explained in terms of constraints. Perhaps the two most important constraints
on practices in PNG are:
 Allowed practice  what the system allows one to do (curricular constraints
from the system, curricular constraints).
 Expected practice  pupil, parent, school, senior teacher expectations
of what one should do. For example, What to teach, what knowledge is.
These constraints to PNG teacher practice were explained
in section 2.2.6 in chapter two.
The results from this study confirms what has been
reported by other studies (eg. Howson & Wilson, 1986; Garden, 1987; Travers
& Westbury, 1989, Robitaille & Travers, 1992). 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 schools and claim
SM is used in a cultural context, in practice this does not happen. In Section
6.4.2 we provide further insight into teacher beliefs and practice in the
context of curriculum implementation. It is suggested in that section that
how the teacher implements the curriculum depends on how the teacher interprets
the intended curriculum but how the teacher interprets the curriculum may
depend on his/her beliefs about mathematics.
Three points that were made in the section on matching
beliefs and practice are reinforced here :
 Where the views matched the existing contextual factors, there were no
mismatches between beliefs and practice. Contextual factors either facilitated
or constrained practice.
 Where there was no definite view (or where there are mixed views), curricular
context determined practice.
 Beliefs about mathematics teaching / learning were not necessarily manifested
in practice.
Practice influencing beliefs
It is possible of course that practice influences
beliefs. There is some support for the contention that practice influences
beliefs. This is evident by the way mathematics was portrayed in the class.
The textbooks portray the nature of mathematics as an externally existing
body of knowledge, which shapes teacher views about the nature of mathematics.
Teaching this content helps to shape their beliefs about the content and nature
of mathematics. The fact that many PNG teachers have SM oriented conceptions
about the nature of mathematics only reinforces the idea that practice may
influences their beliefs.
However, there are still some unanswered questions.
If practice influences their beliefs, what is the explanation for teachers”strong
beliefs about CM ? Why did their practice (eg. of sticking to text books etc)
fail to influence their beliefs about “locus”or mathematics teaching
? A likely explanation is that their experiences or practices inside the classroom
influenced their beliefs about the nature of mathematics while their “outside”experiences
influenced their beliefs about “locus”and what should be taught
(intended curriculum). It is also true that when the teacher is teaching (practice),
feedback from students may result in the teacher changing practice and this
in turn may influence his/her beliefs.
The relationship between beliefs and practice
As noted in chapter two, it is recognised that the
relationship between beliefs and practice is rather complex. This complexity
of relationship is illustrated by the Grouws and Koehler model (1992, see
Figure 3.1 in chapter three). The factors that may influence teacher practice
in the classroom include teacher beliefs about teaching and mathematics, teacher
attitudes and teacher knowledge of student learning, pedagogy and content.
Pupil behaviour and characteristics also influence teacher classroom practice.
Constraints such as pressures of exams, curriculum (intended  content, implemented,
attained), headmaster, peers (fellow teachers) and sociocultural context;
internal constraints (own beliefs, knowledge, experience) also influence teacher
practice in the classroom.
The results from this study show that teacher beliefs
were manifested in classroom practice if the teachers had definite beliefs
about mathematics teaching and learning and in particular definite beliefs
about the nature of mathematics. The dissonance between beliefs and practice
may be explained in terms of constraints. Knowledge is also important to practice
as shown by the CMO teacher who was the only one who had taken a course on
ethnomathematics during teacher training. His classroom practice was different
from that of the other teachers who were basically SM oriented. Knowledge
of mathematics as an “internally”existing subject (as is usually
portrayed in schools) may lead to “school mathematics”oriented
practice.
Summary

What this study showed is that for the participant
teachers, beliefs about mathematics teaching/ learning are not crucial
to practice, whether they are put into practice is to a large extent determined
by contextual factors (constraints)  curricular constraints, context,
sociocultural context.

Beliefs about nature of mathematics are crucial to
practice  where one has definite beliefs. It does not matter what the
context or content is, mathematics will be portrayed according to the
teacher beliefs about the nature of mathematics. Unfortunately, most teachers
do not have definite views but have mixed views.

Where teacher beliefs are manifested in practice,
the contextual factors match the teacher beliefs. Contextual factors can
therefore act as either determinants or facilitators of practice.

Where teacher beliefs are not manifested in practice,
this may be explained in terms of constraints. Mismatch occurs when contextual
factors are different from teacher beliefs, these impede or act as constraints.
For example, the teacher may believe that CM should be taught in schools
but if the intended curriculum (content) is SM, this may act as a constraint
to implementing the teacher belief.

Teachers may have mixed views, not definite views.
A large number of teachers in the PNG sample fall into this category .
6.3 Relationship between teacher conceptions of SM and CM and student conceptions.
One of the research questions aimed to investigate
the influence of teacher beliefs on student beliefs about mathematics. Do
teacher beliefs have any influence on student conceptions of mathematics ?
How do teacher conceptions of cultural mathematics (CM) or school mathematics
(SM) affect student conceptions of mathematics ? These issues are addressed
in this section.
The analysis of variance (ANOVA) of student responses
to individual items in the student questionnaire (see section 5.4.3 in chapter
five) showed that there were significant differences (p £ 0.05) in many of
the items amongst the students for the following variables : schools, grades,
classes according to teachers (the results are included as Appendices 12 to
14). All the tests of significance for differences between students were based
on the students”responses to the individual items to section A of the
student questionnaire. Because one of the interests of this study is to investigate
teacher beliefs and their influence on student beliefs, it was decided to
further explore the “teacher”variable (ie. classes when grouped
according to teachers) as a factor in explaining these differences.
Further tests of significance (p £ 0.05), using the
ANOVA models below, showed that there were significant differences amongst
the grade 8 classes, if grouped according to their teachers. It also confirmed
the fact that there were significant differences (p £ 0.05) across grades,
even in the same schools (see also Appendix 18).
Student sample ANOVA models.
Note that the observations were carried out in three
schools and the student questionnaires were given only to the students taught
by the observed teachers; hence the use of three schools only in the student
ANOVA models. These schools are referred to as schools A, B and C. Although
the student sample ANOVA model shows differences across different grades (eg.
Grade 8 v/s Grade 9 in School A), the discussions center around differences
in the same grade (in this case, the Grade 8 classes). This is because the
interest is in the teacher factor as a likely source for the differences in
Grade 8.
Note that the tests of significant differences
for the following model were carried out according to student responses
to questionnaire items.
Figure 6. 3 ANOVA  School A.
Key : CMO = cultural mathematics oriented; INT
= Intermediate (teacher with mixed SMO and CMO views). The INT teacher
is also referred to as INT Teacher A which means the INT teacher from school
A; SMO = school mathematics oriented.
School A. (G.8: n =100, m=70, f =30; G9 : n =28, m
= 15, f = 13)
There were no significant gender differences. There
were significant differences (p £ 0.05) between Grade 8 classes when grouped
according to teachers (see ANOVA results in Appendix 15). There were also
significant differences between Grade 8 and Grade 9 (see Appendix 16) although
it is interesting that there were fewer items showing significant differences
between Grade 8 and Grade 9.
The important question is; Were the differences due
to the teacher factor ?
Figure 6. 4 ANOVA  School B.
Key : SMO = school mathematics oriented; Teachers B1, B2 = teachers
1 and 2 in school B
School B. (G8: n = 55, m = 13, f = 42)
There were no significant differences
when tested for teacher factor (see Appendix 17). There were also no significant
gender differences.
Figure 6. 5 ANOVA  School C
Key : INT = Intermediate (mixed SMO and CMO views); Teachers
C1 and C2 = teachers 1 and 2 in school C.
School C. (G8 : n = 50, m = 33, f = 17; G10 : n = 52, m = 25,
f = 27).
There were no gender differences. There were also
no significant differences (p £ 0.05) between two Grade 8 classes (see ANOVA
results in Appendix 19), although there were significant differences between
Grade 8 & Grade 10 (see Appendix 18).
Teacher influences on student beliefs.
Because the analysis of variance showed that there
were significant differences across the grades (eg. Grade 7 v/s Grade 8, Grade
8 v/s Grade 9), it was decided to use the Grade 8 classes only to investigate
the teacher factor as a likely source for the differences. Using the same
grades eliminates other factors which may be likely sources for the difference
(eg. school differences, grade differences).
The above student ANOVA models show that there is
some evidence of teacher influence on students”beliefs. Of particular
interest is the situation in School A (Figure 6.5), where no significant
difference (p £ 0.05) was observed between the two Grade 8 classes (n = 66)
taught by the CMO teacher but there were significant differences (p £ 0.05)
between these two classes and the other Grade 8 class (n = 34) taught by the
INT teacher. This is an important result considering that the other factors
(eg. school, different grades) that were likely explanations for the differences
have been eliminated (ie. same school, same grade). This shows the teacher
factor as a likely source for the significant differences between these grade
8 classes from the same school.
The significant difference could be attributed to
the fact that teachers who taught the students in School A had different beliefs
orientation (CMO v/s INT) and their classroom practices were different
(remember, INT teachers had SMO practices). It is assumed here that teacher
conceptions are communicated to students through classroom practices. That
the beliefs orientation may be a factor is further corroborated by the fact
that in School C, there were no differences in the grade 8 classes taught
by teachers with the same beliefs orientation (SMO). Of course, one can never
be certain that the differences observed were caused by the teacher variable
because the possible extraneous variables were not controlled.
Table 6.15 below shows the questionnaire items that
showed significant differences (p £ 0.05) between the grade 8 students if
grouped according to the CMO and INT teachers (ANOVA results are given in
Appendix 15).
Table 6. 15 Significant differences : CMO and INT Students
( School A).

SA/A
% 
NS
% 
D/SD
% 


CMO 
INT
A 
CMO 
INT
A 
CMO 
INT
A 
Sig.
level 
5.04 Mathematics
identified in traditional cultural activities should also be taught
in schools. (+) 
70 
62 
17 
15 
13 
23 
.05* 
7.08 Mathematics
found in traditional cultural activities is not as important as the
“real mathematics”that is learnt in schools (). 
38 
59 
13 
9 
49 
32 
.05* 
14.20 Mathematical
ideas found in traditional culture will be lost if they are not taught
in schools.(+) 
86 
44 
5 
9 
9 
47 
.001* 
30.23 Mathematics
is about knowing when to use rules and formulas to find answers to problems.() 
86 
100 
3 
0 
11 
0 
.024* 
16.24 Students
come to school to learn “school mathematics” not cultural
mathematics () 
48 
65 
5 
9 
47 
26 
.024* 
24.35 Mathematics
is about knowing when to count, measure or make patterns in your own
culture (+) 
77 
59 
11 
12 
12 
29 
.018* 
19.0 School mathematics
has no real use in a traditional society. () 
18 
35 
9 
15 
73 
50 
.037* 








Key : (+) = a positive symbol after each statement
denotes that the statement is a positive statement, relative to cultural mathematics
(it can also be considered a pro  CM statement).
() = a negative symbol after each statement denotes
that the statement is a negative statement, relative to cultural mathematics
(it can also be considered a pro  SM statement).
CMO = students taught by the CMO (cultural
mathematics oriented) teacher.
INT A = students taught by the INT (intermediate)
teacher in school A (the A refers to school A, the differentiation is necessary
because there are other INT teachers). Note also that in the above table,
CMO and INT do not describe student beliefs orientations. These are
labels that describe the beliefs orientations of the teachers who taught them.
SA/A = strongly agree/ agree; D/SD = disagree/
strongly disagree; NS = not sure
Also note that the ANOVA was based on the items scores
and not on the percentages listed on the tables. The response percentages
are included to give some indication of the pattern of response.
It is to be noted from Table 6.15 above that the overall
student response is proCM. The individual group response pattern for both
the CMO and the INT students shows that the proCM responses are higher than
the proSM responses. This is consistent with the student response data in
Chapter Five which showed that the overall student sample were CM oriented.
However, if a comparison is made of the CMO and the
INT students, it can be seen that students in the CMO group (those taught
by the CMO teacher), had a higher percentage of proCM responses. For example,
seventy percent of the students taught by the CMO teacher agreed or strongly
agreed that mathematics identified in traditional culture should be taught
in schools while only sixty two percent of the students taught by the INT
teacher agreed. For the same statement, only thirteen percent of the CMO group
disagreed or strongly disagreed while twenty three percent of the INT group
disagreed or strongly disagreed. For the negative statement, mathematics found
in traditional cultural activities is not as important as the “real
mathematics”that is learnt in schools, almost fifty percent of the CMO
group disagreed or strongly disagreed, while only thirty two percent of the
INT group disagreed or strongly disagreed. For the same statement, only thirty
eight percent of the CMO students agreed or strongly agreed while almost sixty
percent of the INT students agreed or strongly agreed.
So the items showing significant differences also
provides some evidence of the CMO teacher’s influence on the students”conceptions
about mathematics.
But were there any differences between the CMO and
the SMO students ?
The test of significance carried out between the CMO
students in School A and SMO students in School B showed significant differences
(p £ 0.05) between the CMO and the SMO groups (see ANOVA results in Appendix
20). The items showing significant differences are shown in Table 6.16. The
CMO students (n = 66) were taught by the CMO teacher at school A (all Grade
8 students) and the SMO students (n = 55) were taught by the SMO teachers
at school B.
A word of caution about the significant differences.
The students came from two different schools so the school factor as a possible
source of explanation for the significant difference cannot be discounted.
The schools were from different regions and one of the schools was an urban
school while the other was a rural school, all factors which showed significant
differences.
Ideally, it would have been better if the SMO and
CMO teachers had been from the same school, preferably teaching the same grade.
The school factor would then have been eliminated. This is an area which further
research should address.
Table 6. 16 Significant differences : SMO and CMO students
(Schools A & B).

SA/A
% 
NS
% 
D/SD
% 


SMO 
CMO 
SMO 
CMO 
SMO 
CMO 
Sig.
level 
3.01 Mathematics
can also be found in traditional cultural activities. (+) 
60 
92 
22 
5 
18 
7 
.001* 
5.04 Mathematics
identified in traditional cultural activities should also be taught
in schools. (+) 
47 
70 
33 
17 
20 
13 
.037* 
21.11 When teaching
mathematics, teachers should show examples of mathematics from traditional
culture. (+) 
36 
70 
29 
6 
35 
24 
.005* 
12.16 Mathematics
identified in traditional culture is too simple. () 
55 
79 
31 
13 
14 
8 
.033* 
14.20 Mathematical
ideas found in traditional culture will be lost if they are not taught
in schools. (+) 
55 
86 
18 
4 
27 
9 
.009* 
18.27 Some mathematics
identified in cultural activities should be included in the high school
mathematics text books. (+) 
55 
79 
16 
11 
29 
11 
.017* 
24.35 Mathematics
is about knowing when to count, measure or make patterns in your own
culture .(+) 
53 
77 
27 
11 
20 
12 
.009* 








Key : Same as for Table 6.15.
It can be seen from Tables 6.15 and 6.16 that, in
all the statements which showed significant differences between the CMO students
and the SMO and INT students, there is clearly a higher percentage of proCM
responses from the students taught by the CMO teacher. This provides some
evidence of the influence of the CMO teacher on the students”conceptions
about mathematics.
This could be attributed to the fact that the “cultural
mathematics oriented”(CMO) teacher had strong (definite) beliefs about
mathematics and was able to portray mathematics according to these beliefs.
It is possible that teacher conceptions of mathematics were communicated to
students through the way mathematics was portrayed in the classroom. In the
classroom practice, the CMO teacher portrayed mathematics as a debatable subject,
consisting of facts which were questionable, decontextualised mathematical
problems and negotiated methods of solution (see classroom practices, section
6.2.2 in this chapter). In this approach an “alternative”view
of mathematics was portrayed (alternative to the “normal” “school
mathematics oriented”or SMO view of mathematics). The fact that a higher
percentage of students (CMO) supported the “alternate”(proCM)
views, provides some evidence of the influence of teacher beliefs on students”beliefs.
The emphasis here is on the CMO teacher influences
on students”beliefs. This is because the CMO teacher practice was different
from the other teacher practices (SMO and INT) which portrayed mathematics
in the “normal”SM oriented way .
It is to be noted also that there is a higher percentage
of “not sure”responses by SMO students in Table 6.19. The “not
sure”responses could denote that this group of students are in a transitional
stage. They could not make up their minds one way or another about the concept
of “cultural mathematics” There is a conflict between their concepts
of mathematics and the mathematics they learn in schools. There is a possibility
that in the Papua New Guinea situation, the SMO practices which portray mathematics
in the “SMO”way actually result in conflicting conceptions about
mathematics in the students”minds. The CMO students”concepts of
“cultural mathematics”are reinforced by the influence of the CMO
teacher portrayal of mathematics in the classroom.
The above conclusions about the teachers”influence
on student conceptions of mathematics support the research findings that were
reported in chapter two which suggest that as far as mathematics is concerned,
it is the teacher’s conceptions about the nature of mathematical knowledge
that is transmitted to the students. Dossey (1992) for example, states that
the conception of mathematics held by teachers has a strong impact on the
way mathematics is approached in the classroom. Cooney (1985, 1987) and others
who support this view (Thompson, 1992; Schoenfeld, 1992; Nickson, 1992) assert
that the nature of mathematics which is portrayed through school mathematics,
gives a formal and external view of mathematics which is communicated to children
and affects their views of mathematics.
The above in no way assumes that the influence of
teacher beliefs on student beliefs is oneway. It is acknowledged that student
beliefs can also influence teacher beliefs (see section 6.2.3  constraints
on teacher practice and the relationship between beliefs and practice). Teacher
practice in the classroom can be constrained by the students”expectations
of the teacher, which in turn may influence teacher beliefs about mathematics.
For example, the teacher may include examples of cultural mathematics (CM)
in his or her teaching. However, if the students do not accept CM as “mathematics”
then the teacher may desist from using further examples of CM. This in turn
is bound to influence the teacher’s views about what constitutes mathematics.
That the student influence on teacher conceptions
is also an important factor is supported by studies reported by Thompson (1992)
in Chapter Two. The Grouws and Koehler (1992) research model shows where teacher
behaviour in the classroom is influenced by factors which include pupil behaviour
in the classroom, pupil characteristics which in turn may influence teacher
beliefs.
However, the same model (Grouws and Koehler, 1992)
shows that there is also a relationship between teacher beliefs and teacher
behaviour in the classroom which in turn affects student behaviour in the
classroom and student attitudes towards mathematics. What this study suggests
is that there is some evidence that the teacher conceptions about mathematics
are manifested in the way mathematics is portrayed in their classroom practice
which in turn influences students ”conceptions of mathematics.
The example that was observed in this study shows
some evidence of the “cultural mathematics oriented”(CMO) teacher’s
influences on students”beliefs. This is an area for further investigation.
There is a need to observe more cases where CMO teacher influences on students”conceptions
are noted.
6.4 A Synthesis of teacher beliefs and their
influence on the mathematics curriculum.
In this section, we put together the ideas emerging
from the findings dealing with teacher beliefs about CM reported earlier in
this chapter. Teacher beliefs about CM are considered in the context of the
thesis  The cultural dimension of the mathematics curriculum in PNG : Teacher
beliefs and practices. The cultural dimension of the curriculum was defined
in Chapter One as the “intent to include values, beliefs and knowledge
from traditional culture into the school curriculum” This thesis focuses
on teacher beliefs about the cultural dimension of mathematics. Of particular
interest is the role the CMO view played in the implementation of the mathematics
curriculum in the classroom. For that reason, the emphasis in this section
is on teacher beliefs about CM.
6.4.1 Teacher beliefs about CM and the school mathematics curriculum
In the preceding sections of this chapter it was established
that there is a SMO and a CMO view about mathematics teaching / learning and
about the nature of mathematics. While the SMO views represent “normal”views
about school mathematics, CMO views may be considered as “alternate”views
about mathematics. How these beliefs are manifested in teachers”classroom
prac tice was also discussed.
The questions that are of interest are; What was the
role that the CMO view played in the implementation of the mathematics curriculum
in the classroom ? What is the impact of the CMO view (or ethnomathematics
in general) on the mathematics curriculum ?
Discussions on cultural mathematics, and in general
ethnomathematics, nearly always end up with deliberations about the implications
for the intended curriculum. It is assumed that the usefulness of CM is in
its inclusion in the curriculum or if it is taught in schools. For examples,
see Gerdes (1988, 1994a, 1994b), Pompeu (1992) and Begg et al (1993), as reported
in Chapter Two. Barton and Fairhall (1995) present the issues dealing with
mathematics in Maori education in New Zealand. Vithal and Skovsmose (1997)
examined the issues on the impact ethnomathematics had on the school curriculum.
Lancy (1983) and Souviney (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” In this current study, PNG teacher responses to the questionnaire
where the majority agreed that CM should be included in the school curriculum
indicates that they are very much interested in how CM could be used in the
classroom.
While the implications of CM for the mathematics
curriculum is an important consideration (and proposals on how this can be
done are presented in chapter seven), it is by no means the only aspect of
cultural mathematics that needs to be carefully thought about. It is suggested
here that of equal importance is the fact that ideas inherent in “cultural
mathematics”(and in general, ethnomathematical ideas), or the views
and beliefs about cultural mathematics, are challenging views about mathematics
in general and in particular, views about the nature of mathematics. It is
important to have inquiring minds that are critical of mathematical truths,
however true the facts may be. Of particular interest is the notion that ideas
linked to the CMO views can influence teacher beliefs about the nature of
mathematics. How mathematics is portrayed in the classroom depends on the
teacher’s interpretations of the intended curriculum which depend on
the teachers”views about the nature of mathematics (see section 6.2.3).
The above point is illustrated by the fact that the
CMO teacher, despite the strong SM oriented curriculum, was still able to
portray mathematics as a subject whose methods of solutions are negotiable,
consisting of facts which are questionable and debatable (see section 6.2.2).
Proposals about how teacher views can be changed is made in section 7.3 in
the next chapter.
The educational implication involves a rethinking
of the way mathematics is taught, especially at the primary and lower secondary
levels of schooling. It has further implications for teacher education and
the curriculum. For example, questions about what mathematics should be taught,
are bound up in “background”and “foreground”arguments
(Vithal and Skovsmose, 1997) which encompass issues about knowledge the students
come to school (background knowledge) with and what knowledge is necessary
for future use (foreground knowledge  see also section 6.1.1.2 and section
7.1.4).
6.4.2 Teacher beliefs in the context of the “curriculum
framework”
In the synthesis of PNG teacher conceptions of CM
and SM and their role in the implementation of the mathematics curriculum
in the classroom, the intended, implemented and attained curriculum framework
is used (Robitaille & Dirks, 1982; Garden, 1987; Travers & Westbury,
1989; Pompeu, 1992; Robitaille & Travers, 1992; as noted in Chapter two).
This framework offers a broader interpretation of “curriculum”
According to the Travers and Westbury (1989) model reported in chapter two,
the intended, implemented and attained curriculum takes place in various contexts
and levels. What the intended, implemented and the attained curriculum means
was defined in Section 3.1.2 in Chapter three.
How do teacher beliefs relate to the curriculum framework
? This study investigated teacher beliefs about CM and SM (the intended curriculum)
and how these beliefs translate to practice (what actually happens at the
implemented curriculum) at the classroom level and the teacher influence at
the attained curriculum level.
The main ideas emerging from this study suggest that
as far as the teacher is concerned, there is an alternative way of looking
at the curriculum framework. It is suggested here that teacher beliefs about
mathematics have some influence on the way the mathematical ideas and symbolisms
in the “intended”curriculum are interpreted and then implemented.
There seems to be an intermediate level between the “intended”and
the “implemented”curriculum framework which may be called the
“Interpreted curriculum”and two sublevels to the implemented
curriculum which we term the “taught curriculum”and the “portrayed
curriculum” The following is therefore an alternate view of the curriculum
levels :

Intended curriculum as “framework”

Interpreted curriculum. How the teacher interprets
the intended curriculum. Interpretation of the mathematical ideas, of
representations, of symbols, use of rules and formulas.

Implemented curriculum.
The implemented curriculum (the content that is actually taught or implemented
in the classroom) will also include teacher portrayal of mathematics.
It is suggested here that within this level, there are two sublevels;
the “taught”curriculum and the “portrayed”curriculum.
Teacher interpretation of the curriculum is manifested in the implemented
stage by how one portrays mathematics.

Attained curriculum. What the students actually learn
(content). It also includes teacher communicated conceptions of mathematics
the students learn.
An elaboration of these levels is given below.
The intended curriculum.
The intended curriculum (eg. official syllabus) provides
the framework to the teachers for teaching mathematics. It is a framework
in the sense that although the intended curriculum includes the mathematical
content (or topics) for teaching, it is a guideline for teacher action in
the classroom. The teacher still needs to make decisions about what to actually
teach, when to teach, how to teach, what to exclude or include, the depth
of topic coverage etc. More importantly, the idea of the curriculum as a “framework”is
promoted because it allows the teacher to “interpret”the mathematical
concepts and ideas that these topics embody.
The official “intended”PNG curriculum
: As noted in chapter one , the content of the PNG mathematics curriculum
was basically SM. The view that the curriculum depicts of mathematics is SM
oriented or the “external”view of mathematics. The curriculum
system in PNG is centralised and the curriculum is determined centrally by
subject Syllabus Advisory Committees (for example, Mathematics Syllabus Advisory
Committee). Teacher representation on the mathematics committee is minimal.
Usually the framework for the curricular content is determined by the Mathematics
Curriculum Unit of the Curriculum Division of the National Department of Education.
The content of the current secondary mathematics curriculum is an adaptation
of the Harcourt Brace Jovanovich (HBJ) textbooks from Australia.
Teacher beliefs about the intended mathematics curriculum.
It is of interest that, as noted in section 6.1.1
(Teacher responses according to categories), eighty two percent of the teachers
in the sample believed that CM should be included in the mathematics curriculum.
In relation to teacher beliefs about mathematics learning  where mathematics
learning takes place, most teachers (90 %) believed that mathematics can also
be learnt in a sociocultural context. In terms of teacher beliefs about the
intended curriculum, the majority of teachers believed in the existence of
CM and so can be described as being “culturally aware”
The interpreted curriculum.
An important way to think about this idea is to recognise
that the teachers may view the intended curriculum quite differently. It is
referred to as the “interpreted curriculum”because of the teacher’s
interpretation of the mathematical concepts and ideas, of representations,
of symbols, the use of rules and formulae. The term “interpretation”is
used here in the same way that was used by Bishop & Goffree (1986; see
section 6.2.2). Each teacher may interpret these concepts, ideas, rules and
formulas etc. quite differently. In this regard, teacher beliefs are important
because the curriculum is interpreted according to teacher beliefs. It is
suggested here that one of the key factors that determine how the teacher
interprets the curriculum is teacher beliefs about the nature of mathematics.
If we consider this point in the context of the CMO,
SMO and INT views described in section 6.2.1 of this chapter, teachers having
these views may interpret the “official curriculum”(intended curriculum)
in different ways. It is suggested here that the teacher’s interpretation
includes the interpretation of the curriculum framework (what the teacher
sees the purposes and the intentions of the curriculum to be) and the interpretation
of the mathematical concepts, symbols etc.
The SMO teacher sees mathematics learning as predominantly
from teachers and text books, where mathematics does not rely on knowledge
students bring from outside of school. Mathematics is viewed as a subject
which emphasises rules, formulas, procedures and facts which are unquestionable.
The interpretation of the mathematical concepts and ideas, rules and formulas
embedded in the intended curriculum is that these are facts which are unquestionable,
based on universal truths which are absolute, independent of social and cultural
factors (see SMO views in section 6.2.1). These views are consistent with
what Ernest (1992) described as instrumentalist views where mathematics is
seeing as a bag of tools where mathematics consists of separate entities of
algebra, geometry or arithmetic with no structural relationship or where mathematics
is viewed as “mathematics of the curriculum” The intended curriculum
is therefore viewed as a guideline which must be adhered to, so that there
is not much deviation from this framework. The framework contains the mathematical
knowledge that should be learnt.
The CMO teacher interpretation of the content of the
framework is that mathematics consists of concepts and ideas, rules, formulae,
symbols or a body of knowledge whose truths should be questioned. Mathematics
is not just about learning rules, formulas, algebra or geometry. Mathematics
is seen as a subject which investigates environment situations and which includes
knowledge which the pupils bring from outside of school (see CMO views in
section 6.2.1). The CMO teacher uses the curriculum framework as a guideline,
but could diverge from it. The intended curriculum framework is presented
by the CMO teacher as offering opportunities, for example, to demonstrate
that mathematics consists of a body of knowledge whose truths should be questioned,
that the methods of solution are negotiable, that there are many ways to solve
a problem. The framework forms the basis to explore further what the students
know, seeking opportunities to use this knowledge and involve students in
solutions. Certainly the CMO teacher classroom approaches suggested they are
(see next section on “portrayed curriculum” implementation stage).
It is the assertion here that the CM teacher was able to interpret the curriculum
as described above because of his beliefs about culture and mathematics.
The INT teacher’s interpretation of the intended
curriculum framework is basically the same as that of the SMO teacher. Mathematics
is viewed as emphasising rules, procedures, formulae and is viewed as “mathematics
of the curriculum”where mathematics is arithmetic, algebra and geometry.
The difference between the SMO and INT teachers is that the INT teacher shares
some of the CMO teachers interpretation of the mathematical concepts (see
section 6.1 and 6.2.1 where as mixed conceptions are described). Although
the INT teacher agrees to the two statements; CM should be taught in schools
and teachers should teach only the mathematics that is prescribed in the curriculum.
This can be taken to mean that CM should be taught in schools as long as it
is included in the school curriculum. In other words, their view seems to
be; “It’s nice to be idealistic but the reality is, this is the
curriculum that we have” We won’t diverge much from the guideline.
Most of the PNG teachers in the sample would fall into this category. The
SMO and the CMO teachers represent teachers with extreme views.
Teacher interpretation of the intentions of the curriculum
seems to be an important stage in the teaching process. It influences the
actual implementation of the curriculum. In a study of the National Mathematics
Curriculum in England, Johnson and Millet (1996) noted that teachers”implementation
of the intentions of the curriculum (or as they called it, “Using and
Applying Mathematics” UAM) depended on the teachers”interpretations
of the UAM texts. Although the UAM represented changes to the traditional
pedagogy and content of the mathematics curriculum, the implementation was
hindered because of teacher interpretations of the intentions of the curriculum.
These interpretations depended on the beliefs the teachers had.
It is also important to recognise that the teachers
interpretation of the curriculum is further manifested in the implemented
stage by how he/she portrays mathematics (see notes on “Portrayed curriculum”in
the next section  The implemente d curriculum).
The implemented curriculum.
It is suggested here that there
are two sublevels to the implemented curriculum in the context of teacher
beliefs and practice (see Figure 6.7).
 The mathematics content that is actually taught (implemented) in the
classroom. This stage is referred to here as the ”taught curriculum”
 The mathematics that is portrayed. This we refer to as the ”portrayed
curriculum”
These two aspects of the implementation
level are discussed in detail below.
Taught curriculum.
This refers to the mathematical content the teacher
actually teaches in the classroom  which is the usual interpretation of the
“implemented”curriculum .
As noted by Ernest (1988) and Thompson (1992), how
the curriculum is implemented depends on the social context (see section 6.2.3).
For example, social factors such as the values, beliefs and expectations of
students, parents, senior teacher, peers etc. play a role in determining
the content for what the teacher actually teaches in the classroom. Some of
the other social factors listed here ensure that the teacher cannot deviate
too far from the intended curriculum. But of course, the intended curriculum
plays the biggest role in determining what the teacher actually teaches
in the classroom.
For example, in the case of PNG, there is pressure
from external examinations which are conducted at various “cut off”points;
grade six, grade eight, grade ten and grade twelve. These examination results
are used as criteria for selection into higher grades or to universities and
because there are limited places available at these cut off points (eg. only
40 % of grade six students go on to grade 7; NDOE, 1989), pressure from parents,
students, senior teachers and headmasters ensure that the teacher ‘sticks
to the syllabus”
In the classroom observations that were carried out
for this study, the content (topics taught) was mostly from the recommended
text books that were a part of the secondary mathematics curriculum in PNG.
Although the teachers were identified as having different beliefs, the content
taught was basically the prescribed content from the text books. Even the
teachers who used their own examples or work sheets used mathematics that
is basically similar to that found in the prescribed text books. Even the
CMO teacher used examples which were typically text book examples. Where there
were differences was in the way mathematics was portrayed.
The portrayed curriculum.
The idea of a “portrayed curriculum”is
based on the premise that when teaching mathematics, other ideas and skills
are also communicated. In the process of teaching the content, different conceptions
or views about the nature of mathematics are communicated. The “portrayed”curriculum
is different from the “taught”curriculum because two teachers
may teach exactly the same topic but may portray different conceptions about
the nature of mathematics. The reference here is not to the actual mathematics
content that the teacher teaches but what conceptions of mathematics the teacher
portrays when teaching the mathematics. For example, in the explanation, in
the interpretation of the rules, in the methods of solution, what view about
the nature of mathematics does the teacher portray ? Teacher conceptions about
mathematics are communicated by how mathematics is portrayed.
It is suggested here that one can deduce the conceptions
of mathematics that are communicated by observing how mathematics is portrayed
in the classroom. Inferences can be made about what mathematics is portrayed
by how the teacher actually presents these mathematical ideas, representations
and symbols. The portrayal of mathematics is a manifestation of the interpreted
curriculum.
As noted in section 6.2.2, “the portrayal”of
mathematics is highly inferential but these inferences are based on categories
of observed teacher behaviour in the classroom (Bishop & Goffree, 1986).
The observations of the teachers revealed how mathematics was actually portrayed
in the classroom by the PNG teachers (see Tables 6.11, 6.12 and 6.13 and the
summary in section 6.2.2). There were differences observed between the classroom
practices of the CMO and the SMO / INT teachers. The SMO and INT teachers”portrayal
of mathematics was similar to the “typical”mathematics classes
in the “lesson frame”(Bishop & Goffree, 1986). For example,
the SMO and INT teachers portrayed mathematical knowledge as consisting of
unquestionable facts and rules, presented formal explanation of rules, portrayed
methods of solution as not negotiable (stuck to formal methods of solution);
teachers and text books are the predominant source of information and the
students”outofschool knowledge was not taken into account. The impression
given is that mathematics is so well organised and defined that there is no
room for mistakes.
The CMO teacher’s portrayal of mathematics differed
from the SMO/ INT group. For example, in the activities, the CMO teacher provided
opportunities for the pupils to come up with their own solutions or participate
in finding solutions. In the explanations, the CMO teacher encouraged pupils
to give own ideas and built on the answers and asked questions of the type,
why do you do that ? The CMO teacher also presented the methods of solution
as negotiable, presenting mathematical facts, rules, formulas through the
process of inquiry. The Portrayed mathematics consisted of facts or rules
which were questionable, the methods / approaches in classroom suggested the
same. The CM teacher was able to portray mathematics as described above because
of his beliefs about culture and mathematics.
The idea of the teacher “portraying”a
particular view about the nature of mathematics in classroom practice was
noted in chapter two. For example, Cooney’s (1987) analysis of the nature
of mathematics portrayed in the classroom used Goffree’s (1985) work
and concluded that school mathematics is portrayed in a way which promotes
the formal and external view (what is referred to here as the SMO view) of
mathematics. These studies were also referred to by Dossey (1992) in his analysis
of the role and influence of the nature of mathematics. Brown (1985) noted
student reaction as a factor that influences teacher portrayal of the nature
of mathematics in class. The study by Goffree (1985) concludes that even text
books “portray”external views about the nature of mathematics.
It is suggested here that teachers portray mathematics in their classroom
practice. A SMO curriculum may encourage teachers to portray an external view
of mathematics but as seen in the case of the CMO teacher, he was still able
to portray the alternative view (internal or CM oriented) even though the
curriculum was SM oriented. That is why it is further suggested that it is
the teachers beliefs about the nature of mathematics that determines how mathematics
is portrayed in the classroom. See section 6.2.3 where teacher beliefs about
the nature of mathematics were identified as a likely factor that explains
the differences in teacher practices and portrayal of mathematics.
The attained curriculum.
What the students actually learn (attained curriculum)
also includes conceptions of mathematics portrayed by the teacher.
There is probably an interpretive stage in the process
of attainment where the students interpret content (what was taught) and may
rebuild the knowledge according to their own understanding. The student also
interprets the conceptions of mathematics that are portrayed by the teacher.
However, it is not within the scope of this thesis to discuss this issue.
This is an area for further research. For example, how do the students interpret
the content and the conceptions that are portrayed by the teacher ?
This study noted however, that there was evidence
of the influence of teacher conceptions of mathematics on student conceptions
of mathematics (see Section 6.3). The CMO teacher was able to portray an
alternative view of mathematics in his classroom practice. He was able to
communicate his conceptions of mathematics through the classroom practice.
Tests of significance between his students and the students taught by the
other teacher in the same school (same grade), showed significant differences
(p £ 0.05) between the students. The CMO teacher’s students gave a higher
percentage of proCM responses to the questionnaire items. This shows the
CMO teacher’s influence on the mathematics the students learnt (the
attained curriculum).
Summary.
Figure 6.6 below provides a summary of the main ideas
that were presented in this section.
Figure 6. 6 The curriculum framework
The unshaded area shows the curriculum framework as
it is commonly known. The shaded area shows an alternate way of looking at
the curriculum framework which is an extension of the existing framework.
The following points summarise the insights regarding
curriculum that have been gained from this study.

The intended curriculum acts a framework (or guideline)
to the teacher for the mathematical content that is to be taught.

The teachers”interpretation of the mathematical
concepts and ideas, representations and symbols, rules and formulas (that
is, the intentions of the curriculum) is referred to as the interpreted
curriculum.

There are two sublevels within the implemented level.
The taught curriculum which refers to the actual mathematical content
the teacher teaches and the portrayed curriculum which refers to the conceptions
of mathematics that the teacher portrays.
The teacher interpretation of the curriculum is manifested in the implementation
stage by how he portrays mathematics.

Teacher conceptions about mathematics is communicated
in the classroom by how mathematics is portrayed. How mathematics is portrayed
depends on the teacher’s conceptions about mathematics. This is
where the link is between CMO views and portrayal of mathematics. It is
asserted here that having beliefs about cultural mathematics influences
the way one “interprets”the curriculum and hence how mathematics
is portrayed in the classroom.

What the students actually learn (attained content)
also includes their interpretations of the mathematics portrayed by the
teacher.
6.5 Summary according to research questions.
In this section, a summary is presented of the findings
reported in this chapter. These findings are presented according to the research
questions.
 What beliefs and conceptions do teachers, student teachers and students
have about school mathematics (SM) and cultural mathematics (CM), the nature
of SM and CM and the perceived usefulness of SM and CM?
 What is the status of SM and CM ?
 Are there any conflicts between teacher conceptions of CM and school
mathematics ?
 What influence do these teacher conceptions and beliefs about CM (or
school mathematics, nature of mathematics and CM, 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 the classroom ?
 Are there any differences in the classroom practices of teachers with
differing conceptions about mathematics (particularly those with positive
views about CM) ?
 How do teacher conceptions of cultural mathematics (CM) or school mathematics
(SM) affect students”conceptions of mathematics ?
Research
question 1. What beliefs and conceptions do teachers, student teachers
and students have about SM and CM, nature of SM and CM and the perceived usefulness
of SM and CM?
Because this study concentrates on the cultural dimension
of teacher beliefs and practice, the emphasis here was on the teachers rather
than the student teachers or the students.
Teacher conceptions of cultural mathematics (CM) and school mathematics
(SM).
The first research question is a basic status question
which sought to establish what actual beliefs PNG teachers had about SM or
CM. Detailed answers to this question were provided in section 6.1.1 where
teacher responses to SM and CM are presented according to scales : Teacher
beliefs about mathematics learning  locus, mathematics teaching  content,
why that mathematics should be taught and teacher beliefs about the nature
of mathematics. Teacher responses to individual items are given in Appendix
8 and are discussed in section 5.2.1 and 5.4 in chapter five.
Teacher beliefs about where mathematics learning
takes place : The results show that the majority of the teachers believe
that mathematics can be learnt in a cultural context. For example, a hundred
percent agree that mathematics can be found in traditional cultural activities
or that mathematics can be learnt by taking part in traditional cultural activities
(91 %) or out of school activities (91 %). These results show that the majority
of the teachers are “culturally aware”
Teacher beliefs about what mathematics should be
taught and why that mathematics should be taught : Again the teachers”responses
indicate proCM orientation. For example, eighty one percent agreed that CM
should be taught in schools or that CM should be included in the secondary
mathematics curriculum while seventy six percent disagreed to the statement
that CM should not be taught in schools. It is also true that the majority
of the teachers believe that SM is useful in traditional PNG society (82 %)
or that CM is useful to a modern PNG society (68 %). Seventy nine percent
disagreed that CM is not as important as SM.
Teacher beliefs about the nature of mathematics
: Teachers had differing responses to statements in this category. The
responses here indicate that teachers do not necessarily hold theoretically
coherent views about the nature of mathematics. For example, sixty eight percent
agreed that mathematics consists of a body of knowledge whose truth should
be questioned but forty five percent agreed that school mathematics is made
up of abstract concepts and ideas which are value free (41 % disagreed). Responses
to the statement “mathematics is culture free”where thirty nine
percent agreed, thirty six percent disagreed and twenty five percent were
undecided, reveal further mixed views about the nature of mathematics.
Student teacher conceptions of cultural mathematics (CM) and school mathematics
(SM).
Student teacher responses to individual items are
given in Appendix 9 and a description is given in section 5.2.2 in chapter
five. A comparison of teachers”and student teachers”responses
was made in sections 5.4 and 5.5 in chapter five. The majority of the student
teachers believe that mathematics exists in traditional cultural activities
and that CM should be taught in schools or included in the secondary mathematics
curriculum but at the lower levels. There were unanimous or near unanimous
responses to a lot of the items to the “locus”statements.
Students conceptions of cultural mathematics (CM) and school mathematics
(SM).
Students”responses to individual items are given
in Appendix 10 and a description is given in section5.2.3. A comparison of
teachers and students teachers responses to was made in section 5.4 and 5.5
in chapter five. Student responses to the questionnaire items showed a lot
more variability. The majority of the students (like the teachers and the
student teachers) believe that mathematics can be found traditional cultural
activities. However, the students were divided in their responses to statements
about teaching CM in schools or the inclusion of CM in the secondary schools.
In comparison to teachers and the student teachers, the students views about
mathematics could be identified as more school mathematics oriented.
Research
question 2. What is the status of school mathematics (SM) and CM ? (according
to the teachers).
In section 6.1, it was seen that teachers have distinct
views about SM and CM. Teachers do believe that mathematics can be learnt
outside of the established systems of learning and support the idea that CM
should be taught in schools or included in the secondary mathematics curriculum
and generally support the notion of the existence of this “other”mathematics.
However, most of the teachers seem to think that CM is simple and should be
taught at the primary and lower levels of secondary schools.
Research
question 3. Are there any conflicts between teacher conceptions of CM
and SM ?
The fact that teachers have distinct views about SM
and CM seems to be the source of some conflict in their perceptions of mathematics.
For example, teachers have mixed internal and external views about the nature
of mathematics. The fact that a lot of teachers think of CM as simple, elementary
level mathematics means that they think of SM as the “premium”mathematics.
It was also noted that the nature of mathematics category was characterised
by a higher percentage of “not sure”responses. For example, 25
% of the teachers were not too sure that mathematics was culture free. This
reflected the conflict that exist between their perceptions of mathematics
and the mathematics they portray in class, usually one that is culture free.
Research
question 4. What influence do these teacher conceptions and beliefs about
CM (and SM) have on their classroom practice ?
The summary discussions here are limited to evidence
of the influence of teacher beliefs on their classroom practice. Differences
in the profiled teacher practices in the classroom are discussed in the context
of the next research question.
The overall teacher sample responses to CM and SM
statements in the mathematics teaching and learning categories (section 6.1.1)
were scrutinised to see if there was any evidence of these beliefs in their
classroom practice, particularly teacher beliefs about CM and the influence
of these beliefs on their classroom practice. It is interesting to note that
the teachers are “culturally aware”as indicated by teacher responses
to statements in the mathematics teaching and learning categories, responses
which were also described as pro CM responses (see section 6.1.1). For example,
eighty one percent (81%) agreed that CM should be taught in schools. However,
observations revealed that teachers did not use examples of CM in their lessons.
It is highly unlikely that the overall situation in PNG is different from
the observed lessons. The interview data where teachers were asked if they
used examples of CM in their teaching also confirmed the above.
This question was also addressed in section 6.2.3
(Beliefs influence on practice). Here the beliefs of the profiled teachers
(SMO, CMO, INT) were matched against their practices (Tables 6.14 & 6.15
in this chapter). It was seen here that the SMO and the INT teachers exhibited
the same practice (SM oriented) while the CMO teacher practice was different.
The conclusion here was that teacher beliefs about the nature of mathematics
plays a big role in the way mathematics is portrayed in the classroom. Teacher
beliefs about mathematics teaching and learning are not necessarily manifested
in practice.
However, what was also acknowledged was the complexity
of the relationship that exists between beliefs and practice. Practice can
also influence beliefs. There are also important constraints on teacher practice.
Associated
questions : In particular, to what extent do the teachers bring CM into
the classroom ?
In the lessons that were observed, no examples of
CM were used. The content was basically SM. One teacher did mention to the
researcher that he had used examples of CM in the introduction to one of his
lessons.
What do teachers view as constraints to bringing CM
into classroom ?
The teachers see the following as factors which may
hinder one from using examples of mathematics from culture in the classroom
: teachers being from different background to students; students being from
diverse backgrounds; lack of time as a lot of effort is needed to organise
or include in curriculum; lack of knowledge of local culture and more research
needed to identify and put ideas (mathematical concepts) together (see section
6.1.1.2).
Research
question 5. Are there any differences in the classroom practices of teachers
with differing conceptions about mathematics (particular those with positive
views about CM ) ?
Five teachers with three categories of distinct beliefs
were identified and observed  SMO, CMO and INT teachers (see section 6.2.2).
Two teachers were identified as having school mathematics oriented (SMO) beliefs,
one was identified as having “cultural mathematics oriented (CMO) beliefs
while two others had mixed or “intermediate”(INT) beliefs  see
section 6.2.1.
Observations of the teacher practices in the classroom
revealed that there were no differences in the practices of the SMO and INT
teachers (their practices were SM oriented). It is also likely that the SMO
practice is the predominant practice in PNG classrooms. The CMO teacher practice
was different  it was CM oriented. Despite the strong SM oriented curriculum
and the other constraints, the CMO teacher was able portray a CMO view of
mathematics. The differences in classroom practices could be attributed to
the differences in the teachers”beliefs orientations.
The conclusion was that it was important for the teacher
to have definite (strong) views about nature of mathematics  important to
the portrayal of mathematics in the classroom. Just being aware of CM would
not necessarily result in CMO practice. In order to portray an alternative
view (to the SM view of mathematics) of mathematics, one must have strong
CMO views about the nature of mathematics (see section 6.2.3)
Practice is very much influenced by constraints from
the system (what the system allows you to do) and constraints from
the students (what the students expect you to do). These factors act
as constraints on belief. The system that is in place is very much in favour
of SMO beliefs. What cannot be easily influenced by these constraints is the
way mathematics is portrayed in the classroom, if the teacher has definite
CM views about the nature of mathematics.
Research
question 6. How do teacher conceptions of cultural mathematics
(CM) or School Mathematics (SM) affect students”conceptions mathematics
?
As seen in section 6.3, tests of significance were
carried out between two groups of grade eight students in School A who were
grouped according to the teachers with different belief orientation (CMO and
INT). These tests showed significant difference (p £ 0.05) between these
two groups of students. The difference could be attributed to the fact that
the teachers had different beliefs orientation. That the beliefs orientation
was a likely source of difference was confirmed by tests of significance carried
out between two groups of grade eight students in School B who were taught
by teachers with the same beliefs orientation which showed that there were
no significant differences between these students.
When a comparison was made of the CMO students (students
taught by CMO teacher) and the INT students (students taught by the INT teacher)
in School A, it was seen that for all the items that showed significant difference,
the CMO group had a higher percentage of proCM responses. This provides some
evidence of the CMO teacher influence on the students”conceptions about
mathematics.
Teacher conceptions of mathematics were communicated
through the way mathematics was portrayed the classroom. The CMO teacher portrayed
mathematics as a debatable subject, consisting of facts which were questionable,
decontextualised mathematical problems and negotiated methods of solution
(see classroom practices, section 6.2.2 in this chapter). In this approach
an “alternative”view of mathematics was portrayed (alternative
to the “normal” “school mathematics oriented” Eor
SMO view of mathematics).
In summary, one of the important findings of this
study is that there is an intermediate level between the intended and the
implemented curriculum  the interpreted curriculum. How the teacher interprets
the intended curriculum determines how he/she “portrays”the mathematics
in the classroom. The teacher interpretation of the curriculum is shaped by
the teacher’s beliefs about mathematics. Ethnomathematical ideas (or
ideas about mathematics and culture) can help shape teacher beliefs about
the nature of mathematics to be more “cultural mathematics oriented”