CHAPTER SEVEN. IMPLICATIONS AND RECOMMENDATIONS.

 

Introduction

“The cultural perspective requires us to culturalise the curriculum at each of the levels, and demonstrate that no aspect of the mathematics teaching can be culturally neutral. The cultural ‘messages’ in the educational enterprise are created and manifested by people. People create the national and local curriculum statements, people write the books and computer programs, people bring their cultural histories into the classroom, and people interpret and reconstruct the various messages.” 

(Bishop, 1992b : p. 185).

The Education policies of the PNG government manifest intentions which are certainly in line with the stated “philosophy of education”. These education policies advocate a transmission of knowledge, skills and beliefs from PNG culture to students (Matane, 1986; DOE, 1991a, 1991b,1992a), what we refer to in this study as the cultural dimension of the education policies. Despite the good intentions of the government which are manifested in these policies, the mathematical knowledge (and the associated beliefs about mathematics) and what the students learn is not “culturally oriented”. This study investigated the beliefs and practices of the major players in the implementation of these government policies - the teachers. The intended, implemented and the attained curriculum do not seem to reflect these policies. This is the “problematique” that was described in chapter one.

The problem.

The “problematique” described in chapter one and restated at the end of chapter two noted the dissonance in the “cultural dimension” (defined in chapter one) of the education policies and the  mathematics curriculum. The problem was that the government had educational policies which encouraged culturally based curricula : 

• but which did not seem to extend to the mathematics curriculum. An examination of the official PNG (intended) mathematics curriculum showed that the curriculum was not “culturally oriented”. The curriculum was no different to those found in other countries - assumed to be culture free and “canonical” (Howson & Wilson. 1986).
• and that in practice (the implemented curriculum), the mathematics that was taught was not “culturally oriented” and did not have examples of knowledge and skills from traditional PNG culture. Ninety percent of the lessons that were observed were typically “school mathematics oriented”. For example, in the activities, the explanations, the resources, the methods of solution, mathematics was portrayed as consisting of facts, rules and formulas which are not negotiable - factors which encourage a SMO view of mathematics. 
• and that the actual mathematics that was learnt (attained curriculum) was not culturally oriented. The content of the mathematics in the lessons that were observed were typically school mathematics. As a result a lot of the students develop school mathematics oriented views.

In this chapter, the findings from this study are used to address the issues that were raised in the discussions of the “problematique”. In section 7.1 we note the implications of the findings for the education policies. The findings from the study of the cultural dimension of the school mathematics curriculum and the implications for the intended curriculum, teaching, teaching materials and for teacher education are presented in section 7.2. The findings of teacher beliefs and practice and the implications for pre-service and in-service teacher education are presented in section 7.3. In section 7.4, we discuss the limitations of the study and implications for further research.

7.1. The Education policy

7.1.1 The Cultural dimension of the educational policies.

The issues dealing with the cultural dimension of the education policies that were noted above are political as well as educational. The political issues have two seemingly conflicting thoughts. The first has to do with the preservation of cultural heritage, knowledge, skills, values and beliefs while the second has to do with the acquisition of new knowledge, skills, values and beliefs which are needed for advancement in a technological world. The two are conflicting thoughts because in the PNG situation, knowledge and skills needed in a traditional society are different to knowledge and skills which are needed in a technological society. This poses a dilemma because while there is a need to preserve cultural knowledge and skills, on the other hand, there is a need to acquire new knowledge and skills if one is to advance in a technological world.

Statements by political leaders and prominent Papua New Guineans illustrate that they are very much aware of the existence of the dilemma. The Vice-Minister for Education, Science and Culture, Dr. John Waiko had this to say at the opening of the “Waigani Seminar” on “Information and the Nation” at the University of Papua New Guinea (Waiko, 1997b) :

“Indigenous knowledge is a living treasure and Papua New Guineans must learn to accept, preserve and promote the traditional knowledge which has been inherited from our ancestors …PNG …and its … diversity is a great potential source of scientific research which would benefit from a high technology awareness, while at the same time save traditional knowledge from extinction … PNG must protect, promote and harness her living treasures inherent in the indigenous cultural diversity ... national reforms currently in  place are aimed at integrating traditional knowledge into the national educational curriculum”.

Mr. Bernard Narakobi, a philosopher, lawyer, and the Leader of the Opposition in the National Parliament (PNG) had this to say at the “Waigani Seminar” (Narakobi, 1997):

“(Narakobi) … urged institutions to develop credible links between knowledge, technology and Melanesian history….. teach their subjects with a historical perspective and not jump straight to modern technology… PNG cannot make intelligent decisions unless they know the components, uses, and the range of different information technology, …PNG was on the receiving end of technology and information and challenged the authorities and policy makers to  develop the right concepts.”

For mathematics this dilemma is more pronounced because mathematics is seen as a subject for “advancement” in a technological society. The researcher was reminded of this dilemma by a senior education official during “informal” talks about this research project. The sentiment expressed was that although one could find examples of mathematics from traditional culture to teach in schools, the education system should also equip the students with the necessary mathematical skills for further education and for advancement in a technological society.

This dilemma also raises some educational issues. These issues are really about what Vithal and Skovsmose (1997) call student “background” and “foreground” arguments. For example, questions about what mathematics should be taught encompass issues about knowledge the students come to school with (background knowledge) versus what mathematical knowledge is necessary for future use - foreground knowledge. These issues certainly have educational implications for teacher education and the curriculum and will be considered in the later sections of this chapter.

The conflict that this dilemma creates for the mathematics teacher is this. On the one hand we have policies that encourage cultural knowledge which implies cultural mathematics (CM). The message from the political leaders and the public (see chapter one) is to promote cultural knowledge and skills. This study shows that the teachers are also in favour of some cultural influences on the mathematics curriculum. But on the other hand, they are expected to equip the students with the mathematical skills that are necessary in a modern technological society. The curriculum they have to work with is definitely SM oriented.

What happens in situations where there is this political and educational dilemma ? It is not only the question about what mathematics should be taught but other questions become important. For example, Whose (and which) values and beliefs about mathematics should be conveyed to students? Whose (and which) values and beliefs about mathematics are actually transmitted to or learnt by the students ?

Teacher beliefs in relation to government policies

This research investigated what beliefs the teachers have about mathematics, what their classroom practices are and what beliefs are actually transmitted to students at the classroom level. Teacher beliefs were investigated because the teachers are the major players at the implemented curriculum level and this study looked at the cultural dimension of the mathematics curriculum in the context of teacher beliefs and practices.

The findings of the research (reported in the preceding chapters) show that the teachers’ expressed beliefs are clearly in line with government policies. Teacher responses to the questionnaire indicated that the teachers were “culturally aware”. For example, ninety percent of the teachers in the sample agree that mathematics can be found in traditional cultural activities and eighty two percent believed that CM should be included in the mathematics curriculum. However, it is important to note that the beliefs the teachers had were not strong enough for them to portray mathematics as “culturally oriented” (as the policy intended). The teachers were “culturally aware” but were not “cultural mathematics oriented” (for example, they did not have definite CMO beliefs about the nature of mathematics). The constraints (for example, the SM oriented curriculum) were such that, for most of the teachers, these beliefs were not implemented or manifested in their classroom practices (see section 6.2.3).

The findings of this research confirm what was reported in chapter two. Research on educational policies and practice (Curriculum policy and practice) suggests that external policies do not have much impact on teacher practice; teachers interpret policies according to their own beliefs. In other words, even if the policy encouraged a culturally based curriculum, the official curriculum was not culturally based. Even if the official curriculum was culturally based, or the values and beliefs about mathematics that we wished to transmit to students were culturally based, there would be no guarantee that the cultural dimension would be implemented because it is the teachers beliefs and values that really matter since these are transmitted to students.

Implications for educational policies

The government should have clear policies about the cultural dimension of the mathematics curriculum. It should have policies that encourage the cultural dimension of the mathematics curriculum at the primary level. In the current structure of the education system, this would be from pre-school through to grade 8. This would also be in line with the beliefs of the majority of the teachers who clearly believed that CM should be taught in schools but at the primary and lower secondary levels only. That is not to say that teachers at the upper secondary levels should not be aware of the “cultural aspect” of mathematics learning. Opportunities do exist at the secondary schools where CM oriented ideas can shape teacher practice in the classroom so that an alternative view of mathematics can be portrayed. 

However, having policies alone will not necessarily have an impact on classroom practice. As noted at the end of section 2.3.2, there may be little or no impact at all of policy on the classroom practice. The educational policies in PNG were more to do with institutional policies or policies that dealt with organisation rather than curriculum or practice. The policy resulted in a change to the educational system (the education system was restructured). However, there is no evidence that the policies resulted in changes to the curriculum or teacher practices in the classroom. There needs to be more than just a change in education policies. The curriculum needs to change as well as the incorporation of ethnomathematical ideas into teacher education programs. These ideas are further explored in section 7.2.

7.2 The Cultural dimension of the school mathematics curriculum.

In chapter one, it was noted that there was a mismatch between government intentions and the intended mathematics curriculum, the implemented mathematics curriculum and the attained mathematics curriculum. Section 7.2 is divided into two subsections. In the first subsection we make recommendations for the intended mathematics curriculum in PNG. In the second subsection we make proposals for PNG teacher action in the classroom (implemented curriculum). 

However, it should not be assumed that changing to a CM oriented curriculum only involves including some examples of CM in the curriculum. There are also important questions, first noted at the end of chapter two, that need to be addressed. For example, considerations need to be given to the following : Is it possible to have a culturally oriented curriculum in the PNG context where the diversity is greater even in localities ? (especially if teachers are required to use local examples of traditional mathematics). Should we after identifying CM or out-of-school mathematics incorporate into school mathematics ? Why ?

It should  also be noted that an example of a culturally oriented mathematics project had already been trialed in PNG. The Indigenous Mathematics Project (IMP) whose intention was “utilising complementary aspects of indigenous and western mathematics as basis for developing culturally relevant student materials, instructional aids, and teachers guides which reflected the practical constraints of the community school environment.” (Souviney, 1981). The fact that by 1985 new adapted Australian text books were proposed for the primary schools (Clarkson & Kaleva, 1993) suggests that although the trial IMP project showed successful results (Souviney, 1983), the lessons or ideas and the wealth of information gathered from that project were never fully incorporated into the primary mathematics curriculum. In the absence of a follow-up evaluation of the project (according to the knowledge of the writer who also did a literature search which failed to find any reports of an evaluation of the project), we may never know why the good intentions were never incorporated into the PNG mathematics curriculum.

It is interesting to note that in the only official evaluation of the PNG High school mathematics curriculum, Hayter, (1982, see section 2.3.6 in chapter two), made recommendations which included the following :

1. Due recognition given (both by inclusion and by respect for its use) to traditional counting and measurement systems.
2. Study of traditional pastimes and practices, where appropriate, in a mathematical way.
3. Applications of skills and techniques developed in mathematics to local situations; to aspects of national life which will impinge at the village level, or a likely to affect high school graduates (eg. postal charges, timetables); and to problems which arise at school and at village level.
4. Use of PNG currency, place names, food stuff in book work examples and exercises.

In Chapter two (see Table 2.4), Bishop (1992a) provides a framework of cultural conflict and the responses to this conflict in a mathematics education context. In discussing educational implications of ethnomathematical research, Gerdes (1996: pp.928-931) identified the following trends in the applications of these findings in educational settings. The trends (although interrelated are grouped into three categories by this writer) and are listed below.

1.   Curriculum applications.

1. Incorporation into the curriculum elements belonging to the socio-cultural environment of the pupils and teachers, as staring points for mathematical activities in the classroom.
2. Incorporation into the curriculum material from several cultures, thereby valuing the cultural background of all pupils and enhancing the self confidence of all. 
3. Use of ideas embedded in the activities of certain cultural or groups within a society to develop a mathematical curriculum for and with/ by this group.
4. Introduction in text books of cultural elements that facilitate learning by being recognised and appreciated by the pupils as belonging to their culture.

2.   Resource use.

1. Elaboration of materials on the mathematical heritage of the forefathers/mothers of the pupils, and the introduction of these in teacher education programs and/or school curricula
2. Elaboration of materials that explore the possibilities for mathematical activities starting with artistically appealing designs belonging to the culture …  of the students or of their forefathers/ mothers.

3.   Teacher education.

1. Alerting future mathematics teachers and teacher educators to the existence of mathematical ideas - understood by people with little or no education - which are similar to or different from, those in standard textbooks; learning to respect and learn from other human beings in other socio-cultural groups.
2. Preparation of future mathematics teachers who will investigate mathematical ideas and practices of their own cultural, ethnic, linguistic communities and who will look for ways to incorporate their findings to their own teaching.

The writer would like to suggest that Hayter’s recommendations (1982), Bishop’s (1992a) possibilities of a cultural oriented curriculum and Gerdes’ (1996) trends for educational application for ethnomathematical research raise important points which are relevant to mathematics education in PNG. These will be taken as guiding principles when making recommendations for the mathematics curriculum and teacher education in Papua New Guinea.

7.2.1 Implications for the intended curriculum : The CMO curriculum.

As noted in section 6.4.1 there was a need to change the SM oriented curriculum to a CMO curriculum. We make proposals on how this could be achieved.

The curriculum that is proposed for the lower secondary and the primary schools in PNG should be one of “assimilation”, according to Bishop’s approaches. This means that the assumptions are that the learners culture should be useful as examples, the curriculum will include some learners cultural contexts and that the teaching will require some modification for teaching some learners, the language used will be English, the official language of instruction although remediation will be a necessary part because English is a second language for most learners. This approach was chosen because the diversity is such that localities differ markedly. There is also the dilemma that was noted in the beginning of this chapter where “school mathematics” is seen as a subject for advancement in a technological society. The pressure to study “school mathematics” cannot be ignored because in the case of PNG this so-called “advancement”, has to be borne by Papua New Guineans themselves. The situation is somewhat different from other countries where some “minority” groups exist within other dominant cultures and so the “advancement” becomes the responsibility of the dominant socio-cultural group.

This curriculum will incorporate examples of mathematics from the cultural environment of the students or the teachers, curriculum materials from PNG cultures or from other cultures from around the world, use mathematical ideas found in other socio-cultural groups from around PNG, include in text books examples of cultural elements, traditional counting systems and measurement etc. from PNG.

7.2.2 Teacher implementation of the mathematics curriculum.

In classroom teaching, what can the teachers do ?

Implications for teaching

Bishop and Goffree (1986) offer “the social construction frame” as an alternative conceptualisation of the mathematics lesson. As noted in section 6.2.2 of chapter six, 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. Teacher role is therefore to manage activities and provide opportunities for pupils to create their own mathematical meanings. The “social construction frame” provides us with an alternative way of viewing the mathematics lesson (eg. three main components of the mathematics lesson are activity, communication and negotiation) and is sensitive to the cultural aspect of mathematics teaching, especially how mathematics is portrayed in the mathematics lesson at the schools during “teacher activities”. This means that the teacher will take on the role of the manger of activities, will be involved in negotiating knowledge, portray methods of solutions as negotiable, explanations to include student suggestions and use of student out of school knowledge. This last point will require the teacher to find out the students out of school knowledge.

Nunes  (1992 : p.557) makes suggestions on how the teacher could use out-of school knowledge in the school. Nunes (1993: p.35) suggests that bringing out of school mathematics into the classroom means giving students problems which they can “mathematicise” in their own ways and in so doing come up with results (methods, generalisations, rules etc.) which approach those already discovered by others. It is not simply taking into classroom an everyday problem and using algorithms learnt to solve problems. Nunes suggests that the teacher should start by determining which concepts he or she wants the students to learn and then identifying the everyday mathematics which has that concept. The teacher must also consider whether the students have used the concept in everyday life. Take everyday problems into classroom but remember no ready made methods or solutions should be expected of them. Nunes gives examples of approaches to teaching specific aspects of mathematics with support of everyday problems. Real life mathematics should be used in open ended way.

This approach can be applied by the teacher to identify the cultural knowledge students bring with them to school. Questionnaires could be used to help identify students’ local knowledge of CM. Included in the questionnaire would be items about the mathematical activities in the students’ culture, counting systems in the different languages, examples of designing, explaining, measuring and locating etc in the students cultures.

 Implications for teaching materials (resources).

PNG cultures have various designs (eg. designs on bamboo walls, basket weaving, artefacts, carvings) that can be used as starting points to explore the possibility of mathematical activities. Students can be asked to bring these objects to class for discussions (eg. symmetry, rotation). Other materials which may have mathematical significance (or mathematical attributes) from the students culture can be identified and used as a part of mathematics teaching.

 Implications for Teacher education

What to do during teacher training to prepare teachers for this possibility ?

Introduction of ethnomathematics courses during teacher education programs where their beliefs about mathematics would be challenged. Mathematics teachers and teacher educators to be made aware of the existence of mathematical ideas which are different from those which are portrayed by standard text books. Teacher education programs should also prepare teachers to identify the mathematics which exist in their own cultures or in other cultures like their students (eg. in cultural activities, in their languages) and look for possibilities to include these in their teaching.

7.3 Teacher beliefs and practice.

In section 6.4 (chapter six) it was noted how important it was to have CM views about the nature of mathematics in portraying an alternative view of mathematics (to the SM view) in the classroom.

What then is the role of CMO views in influencing teacher beliefs about the nature of mathematics ? As noted in section 6.4, there is link between the CMO views and the views about the nature of mathematics. This then provides a role for the CMO views in influencing teacher views about the nature of mathematics.

7.3.1 Teacher beliefs and the school mathematics curriculum

The answers to the following questions of interest were noted in section 6.4.1; What was the role that the CMO view played in the implementation of the mathematics curriculum in the classroom ? Did teacher beliefs about CM have any impact at all on the school mathematics curriculum ? How ?

 Teacher views about CM and the intended mathematics curriculum. 

Teacher beliefs about the intended curriculum: As noted in section 6.1.1 (Teacher responses according to scales), in terms of teacher beliefs about the intended curriculum, the majority of teachers believe in the existence of CM and so can be described as being “culturally aware”.

There is obviously a mismatch between teacher beliefs about the intended curriculum and the actual intended curriculum (this is the “problem” that was noted in chapter one and restated in chapter two). The majority of the teachers agree that CM should be included in the curriculum or taught in schools. The intended curriculum should therefore be changed. It is important to change the curriculum because if we are interested in portraying an alternative view (to the SM view) of mathematics (Table 2.2, chapter two),  then the intended curriculum should change to reflect this view.

7.3.2 Teacher beliefs about CM and the nature of mathematics.

 As seen in section 6.2.2, the majority of the PNG teachers teach mathematics in such a way that it portrays a “SMO” view of mathematics. As noted by numerous writers including Bishop (1993) and D’Ambrosio (1991a) in Section 2.1 of chapter two, for far too long mathematics was taught as a culture free subject, made up of universal truths which are absolute and unquestionable where the emphasis is on rules, facts, procedures and methods. It is true that the majority of mathematics teaching may still portray mathematics as described here. The PNG classroom observations in this research also confirm the above. However, there is now overwhelming evidence that suggests mathematics is not culture free. Mathematics does not have to be taught as if it is culture free.

In this research it was observed that, despite constraints to practice, in spite of contextual factors and the content being SM, the teacher with CMO views about the nature of mathematics was able to portray the methods of solution in mathematics as negotiable, mathematics as consisting of facts, rules which are debatable, to be questioned and decontextualised the problems (see section 6.2.2). This suggests that teacher beliefs about the nature of mathematics is an important factor in determining practice in the classroom. It seems from the above example that beliefs about nature of mathematics are crucial to practice - where one has definite beliefs. It doesn’t  matter what the context or content - mathematics will be portrayed according to how the teacher interprets the curriculum which may depend on the teacher beliefs about the nature of mathematics.

If it is important to have CMO views about the nature of mathematics, as suggested above, how then do we change teacher views to CMO views about the nature of mathematics.  This question is addressed in the next section.

The impact of the CMO view on the implemented mathematics curriculum.

As seen in section 6.4, the CMO teacher viewed mathematics as consisting of a body of knowledge whose truths should be questioned and that mathematics was not culture free. The teacher also had definite (not mixed) CMO views about the nature of mathematics.

Despite the strong SM view which is portrayed in the official intended curriculum, differences do exist at the implemented level.

Although the content was basically SM, there was some impact of the CMO views on how mathematics was portrayed or presented in classroom. The CMO teacher portrayed mathematics as a debatable subject, whose methods of solution were negotiable, contextualises problems and uses students ideas for solutions.

It is very important therefore to understand why this teacher was able to do this. If we are interested in changing the situation we need to be clear about the possibilities for change. We have clearly  seen in the previous section that one way would be to change the intended curriculum.

Is that enough ?

The data from the five teachers suggests that it is not. Two of the teachers had mixed views and if the official curriculum had been of a CM nature, it is likely that they might have  had CM views. But would this necessarily have meant a change in their practice ? The evidence suggests not. Another way of looking at  this point is that, even if the mathematics curriculum was CM oriented, teachers having definite SMO or mixed views about the nature of mathematics would still teach the content  in a SMO way.

There is clearly a need to change teacher beliefs. This study shows that the majority of the teachers are culturally aware but this not manifested in practice.  These teachers have mixed views about the nature of mathematics. Thus the important factor seems to be the strength of the teachers view. By that we mean, for teacher belief to progress beyond the level of just being “culturally aware”. Although ninety (90 %) percent of the teacher sample are “culturally aware”, the classroom practice was predominantly SM oriented. The CMO teacher’s practice shows that it doesn’t matter what the content is  (in this case, SM); the approach will depend on whether the teacher had definite views about the nature of mathematics. If teachers were to portray an alternative view of mathematics (eg. as a subject whose truths are debatable or the methods of solution as negotiable), they must have definite (strong) CMO views about the nature of mathematics.

How then do we change teacher beliefs about the nature of mathematics ?

If we take the case of the CMO teacher , where did the teacher’s CMO ideas come from ? The teacher’s training background may have shaped the teachers beliefs about mathematics. The teacher was a recent graduate from the university and this was the first year of teaching. The teacher had enrolled in a course which introduced the ethnomathematical ideas (confirmed in interviews). Of the observed teachers, this teacher was the only one who had actually heard of the term “ethnomathematics” or participated in ethnomathematical assignments during pre-service training. This would have obviously helped in shaping the teachers beliefs about the nature of mathematics.

Kagan (1990) noted the need for “teacher education programs” (T.E.P), both pre-service and in-service, to challenge existing beliefs. About pedagogical beliefs, Kagan states that pre-service teachers go through T.E.P and come out with their misconceptions reinforced ; they will not change unless challenged by self reflective thought programs in university TEP that emphasise the questioning and restructuring of pre-existing beliefs, one must redefine the purposes of education- students natural knowledge structures and reasoning processes.

One of the important findings of this study is the role that ethnomathematics or ideas about culture and mathematics can play in changing conceptions about the nature of mathematics. Whether they meet ETM through pre-service teacher education programs or at in-service programs, teachers need to be shown examples of CM that exist at various socio-cultural contexts. The points that were made in section 7.2.2 - implications for teaching and for teacher education - also apply here. As noted in section 6.4.2 in chapter six, 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. As seen with the example of the CMO teacher, ethnomathematical ideas (or ideas about mathematics and culture) can help shape teacher beliefs about the nature of mathematics to be more “cultural mathematics oriented”.

One advantage of observing the teacher in the first year of teaching was that the teacher’s beliefs were still “intact”. It was quite possible that had he been observed a few years after the initial teacher training, the strong SMO nature of the mathematics curriculum and the predominant nature of the SMO practice in the schools would have influenced his own practice and views about the nature of mathematics.

That is why recommendations were made for “cultural mathematics” oriented curriculum. However, it is not enough to only change the school mathematics curriculum into a  CM oriented curriculum. Nor is it enough to only change teachers’ beliefs (about the nature of mathematics). A change in the content must also come out about with a change in teachers beliefs about the nature of mathematics. Existing teacher beliefs about mathematics must be challenged.

The idea that change in the curriculum should also be accompanied by a change in the teachers beliefs is supported by literature on management of change. Fullan (1986) for example, states that successful innovation  involves :

1. Learning material.
2. Practices or behaviour
3. Beliefs and understanding. 

He suggests that it is easy to equate innovative change as the first of the three but the more problematic are points 2 and 3 because these  involve skills and practices and the way teachers think.

As reported in chapter two, Pompeu’s research (1992) which involved teachers participating in micro-ethnomathematical activities showed that teachers who participated in the projects made significant changes in their views about mathematics teaching. This shows that it is possible to change teacher views about mathematics but if lasting change is valued then changing the SMO nature of the mathematics curriculum is an even greater priority.

7.4 Research implications

7.4.1 Limitations of study

Use of Questionnaires

There are disadvantages to using questionnaire in that it limits the responses, participants only in the way the questionnaire items are structured. Limitations of having Likert scales only in questionnaire. Should also use one other section eg. Do you use examples of CM in your teaching ? If yes, give examples. If no, why not. Although used in the interviews, the limitation was that the sample was small. If used in questionnaire, one could get responses from a bigger sample.

Also one is left wondering whether those who returned the questionnaire formed a representative sample. It is quite possible that those who returned the questionnaire were Pro-CM or anti-CM; those who didn’t care one way or another may not have bothered to return the questionnaire.

Scales

Because the factor analysis and the item reliability tests found the scales to be unreliable (see section 5.3 in chapter five), this resulted in a reclassification of the scales into categories. Also, the teachers could not be placed on a beliefs continuum (it would have required summing the teacher responses according to the scales). Regarding the assumption that each construct (or scale) will have alternate views, (ie. if one has strong SM views, then they will have weak CM views), there was no support for this assumption from the item and factor analysis.

Questionnaire items

Because the researcher wanted to keep the Questionnaire as short as possible, it limited the number of items that could have been included. The questionnaire did not include items on teacher beliefs about the implemented curriculum.

Interviews

The researcher could have interviewed a lot more teachers. Could have asked more questions relating to their questionnaire responses.

 Observations

Selection of teachers to be observed

Selection of teachers to be observed was to be based solely on addition of scores - beliefs continuum. However, other considerations though came into play eg. geography, location of school - financial consideration, regions, gender, urban,  rural.

Actual observations

In the observations, only one CMO teacher was observed. The researcher would have liked to observed more CMO teachers. There were no comparison of the CMO and SMO teachers in the same school, teaching the same grade (eg. CMO teacher to teach one Grade 8 class while the SMO teacher taught another Grade 8 class). For comparative purposes, it would have been desirable of the observations were carried out when teachers were teaching the same topic (especially word problems).

The researcher would have liked to use audio-visual equipment (videos) to get more details during the classroom observations but considered the novelty of the equipment as more of a distraction to “normal” classroom processes.

 Pilot study

Ideally the item analysis should have been done during the pilot study to make the questionnaire more reliable. Because where the research was conducted was so far from the home university, the researcher was not able to consult with “experts”. The researcher would have liked to use one other school.

7.4.2 Implications for further Research.

An area for further research would be to confirm the relationship that exists between teacher SMO and CMO views and to confirm the relationship between these teacher beliefs about mathematics and classroom practice. There is a need to identify more teachers who have definite SMO and CMO views  that could be observed to confirm the above relationship.

Another area for further investigation is where the teacher influence (both SMO and CMO) on students conception were noted. Especially where the CMO teacher influences on student conceptions of mathematics were noted. Ideally, it would have been better if the SMO and CMO teachers were from the same school, preferably teaching the same grade. The school factor is then eliminated. This is an area which further research should address.

Finally, the concept of the ‘portrayed curriculum’ needs further development. It clearly relates to the teaching of beliefs and values, and could be an important determinant of further changes in culturally aware teaching.

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