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 :

  1. What beliefs and conceptions do teachers, student teachers and students have about school mathematics (SM) and cultural mathematics (CM), nature of SM and CM, perceived usefulness of SM and CM?
  2. What is the status of SM and CM ?
  3. Are there any conflicts between teachers”conceptions of CM and school mathematics ?
  4. What influence do these teacher conceptions and beliefs about CM (or school mathematics, nature of mathematics, perceived usefulness of mathematics and CM) have on their classroom practice ? In particular to what extent do the teachers bring CM into the classroom.
    What do teachers view as constraints to bringing CM into classroom ?
  5. Are  there any differences in the classroom practices of teachers with differing conceptions about mathematics (particularly those with positive views about CM) ?
  6. 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 socio-cultural 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 (pro-SM) while a strongly disagree/disagree response would denote internal views (pro-CM). 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, student-centred 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 :

  1. 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).
  2. 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.

  1. 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.
  2. 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.
  3. 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 :

  1. it puts the teacher in relation to the whole classroom group;
  2. it emphasises the dynamic and interactive nature of teaching.;
  3. it assumes the interpersonal nature of teaching
  4. it recognises the shared idea of knowing and knowledge, reflecting the importance of both content and context;
  5. 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 :

  1. activity - emphasis on learners involvement rather than on teachers presentation of mathematical content;
  2. communication - underlies all teaching and is essential to shared meanings, for example, in the teacher’s explanation and interpretation of mathematical ideas;
  3. 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”:

  1. 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.
  2. 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)?
  3. 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 out-of-school 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 : 

  1. 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.
  2. 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 ?
  3. 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 out-of-school. 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”out-of-scho 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 :

  1. Is there any evidence of the influence of teacher beliefs on their classroom practice ?
  2. Why is there a difference in the practices of the CMO and the SMO/ INT teachers ? Was this due to teacher beliefs ?
  3. What is the explanation for the mismatch between some beliefs about mathematics and practice ?
  4. Is there any evidence of the influence of views about the nature of mathematics over other beliefs about mathematics.
  5. Influence of beliefs about mathematics on practice and the influence of practice on these beliefs.
  6. 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 out-of-school 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”out-of -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 cause-and-effect 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 out-of-school - 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 out-of-school 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 out-of-school. - 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); socio-cultural  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:

  1. Allowed practice - what the system allows one to do (curricular constraints from the system, curricular constraints).
  2. 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 :

  1. Where the views matched the existing contextual factors, there were no mismatches between beliefs and practice. Contextual factors either facilitated or constrained practice.
  2. Where there was no definite view (or where there are mixed views), curricular context determined practice.
  3. 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 socio-cultural 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

  1. 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, socio-cultural context.
  2. 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.
  3. 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.
  4. 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.
  5. 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 pro-CM. The individual group response pattern for both the CMO and the INT students shows that the pro-CM responses are higher than the pro-SM 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 pro-CM 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 pro-CM 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”(pro-CM) 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 one-way. 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 sub-levels 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 :

  1. Intended curriculum as  “framework”
  2. Interpreted curriculum. How the teacher interprets the intended curriculum. Interpretation of the mathematical ideas, of representations, of symbols, use of rules and formulas.
  3.  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 sub-levels; the “taught”curriculum and the “portrayed”curriculum. Teacher interpretation of the curriculum is manifested in the implemented stage by how one portrays mathematics.
  4. 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 socio-cultural 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 sub-levels to the implemented curriculum in the context of teacher beliefs and practice (see Figure 6.7).

  1. The mathematics content that is actually taught (implemented) in the classroom. This stage is referred to here as the ”taught curriculum
  2.  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”out-of-school 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 pro-CM 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.

  1. The intended curriculum acts a framework (or guideline) to the teacher for the mathematical content that is to be taught.
  2. 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.
  3. There are two sub-levels 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.
  4. 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.
  5. 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.

  1. 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?
  2. What is the status of SM and CM ?
  3. Are there any conflicts between teacher conceptions of CM and school mathematics ?
  4. 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 ?
  5. Are  there any differences in the classroom practices of teachers with differing conceptions about mathematics (particularly those with positive views about CM) ?
  6. 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 pro-CM 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 pro-CM 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”

 

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