8.2. Lecturers' Beliefs About Mathematics
8.2.2. In What Ways Did These Lecturers Try To Present A Broader View Of
One lecturer, K said that he presented a broader view of mathematics through lectures and demonstrations where he and the pre-service teachers role-played as class teacher and students respectively. He expected the pre-service teachers to learn mathematics through these
recreational activities. He explained that "If they enjoyed doing it hopefiilly they would transfer
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it to their students." However he preferred the pre-service teachers "to read up to discover for themselves this kind of mathematics."
0 who had taught physics "98 % of the time" before joining the college as a mathematics lecturer found it difficult to incorporate activity based teaching into her mathematics lectures.
Compared to Physics which was "very experiment oriented", she found mathematics "not so activities based". O's account suggested that she equated the structured step-by-step physical activities conducted by students in Physics experiments with the activities engaged by students in mathematics. O's structured view of mathematical activity was emphasised in her response to the intervention activities (see Section 11.5.). I know that students in Physics classes were given step-by-step instructions on how to proceed with experiments presented in their
workbooks. These activities are in sharp contrast with the mathematical activities suggested in paragraph 243 of the Cockcroft Report (DES, 1982) and also in the mathematical literature (for example see ATM, 1991; Burton, 1994, 1984, 1980; Countryman, 1992; Brown and Walter, 1990; Borasi, 1987; Haylock and Cockburn, 1989; Van den Brink, 1987; Waywood, 1988).
0 explained that the pre-service teachers were engaged in activities where they were more involved in searching the library for known recorded mathematical information than activities such as investigative mathematics. She "usually let the pre-service teachers get the information from the library and then they come back and present to the class". O's response to my question
"Do the pre-service teachers work in groups and discover something mathematical from the activity?" was "No, those activities, no." W explained that pre-service teachers were often
"given topics such as history of time, history of money, history of mathematics where they have to work in groups to find information pertinent to their topic. They then shared their knowledge with their peers, in their groups and in class discussions." All four lecturers admitted that the pre-service teachers had not been exposed to mathematical ways of working, such as investigations and problem solving. The evidence of their teaching suggests that their
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own understanding of and experiences with mathematics had influenced the kinds of mathematics in which they engaged the pre-service teachers.
The lecturers' accounts suggest that they were still 'demonstrating', 'imparting' and 'transferring' knowledge to their students in the hope that these pre-service teachers would pick up the information that was being "broadcast" (Cooney, 1988). This suggests that the lecturers "read up" and widened their mathematical knowledge, but because they themselves did not construct the learning of what it meant to have a broader view of mathematics, they could only expose their pre-service teachers to a very limited form of mathematical activity. It further suggests that these lecturers equated mathematical activity with physically collecting knowledge.
As the English lecturers had no objection to my sitting in their classes, I was able to observe them teaching. In one particular lesson, J was interested in how the pre-service teachers would introduce division to their students. The pre-service teachers were seated and working in
groups. J gave them time to think about that situation and then he elicited responses from them.
Another lecturer A used geography and maps to introduce the idea of scales to the pre-service teachers. The classroom behaviour of these lecturers suggested that they were trying to engage their pre-service teachers in mathematics where they as learners were developing a sense of mathematics, sharing and constructing the knowledge for themselves. The English pre-service teachers were suggesting different ways to approach the teaching of division in a primary class.
My view was that these English pre-service teachers were made aware that there was more than one way to understand and teach division. More importantly the responses were originating from the English pre-service teachers and not from J, the lecturer.
8.2.3. Findings Based On The Lecturers' Accounts And The Input By The Head Of Mathematics Department
The lecturers' accounts suggest that, though they wanted the pre-service teachers to leave college with a broader view of mathematics, they themselves had difficulty explaining their own personal construct of a 'broader view of mathematics'. The lecturers' inability to articulate
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their views of mathematics was perhaps similar to that expressed by Davis and Hersh (1981) in their book "The Mathematical Experience":
I had been teaching mathematics for some 15 years, at all levels and in many different topics, but in all my other courses the job was not to talk about mathematics, it was to do it. Here my purpose was not to do it, but to talk about it. It was different but frightening. (p.3)
Davis and Hersh identified a possible reason why university lecturers were inarticulate about their views of mathematics. They were constantly engaged in the doing of mathematics and when they were asked for their views about mathematics they could not articulate them.
Similarly the Malaysian lecturers taught mathematics as a formal school discipline first to their students and then to the pre-service teachers. However, the two English lecturers were more articulate in their views because they had the opportunity to engage in continuous and
challenging discussion with mathematics educators who were equally committed to challenging the kind of mathematics and how it was being taught in schools and in teacher education.
The Malaysian lecturers and the pre-service teachers owed their "cognitive loyalty" to the tradition and culture of learning that they had experienced in school (Buchmann, 1987, Ernest,
1989) and which had led to their success so far. The lecturers had not been exposed to, nor were they actively engaged in, discovery of a broader view of mathematics. Perhaps the narrow KBSR mathematics syllabus did not help to challenge and perturb the lecturers' views of mathematics. The lecturers remained loyal to their own personal epistemology. Since they extended their view of mathematics through reading, they believed their epistemology would transfer to the pre-service teachers if they told the pre-service teachers to do likewise, despite the fact that they knew the pre-service teachers were limited by their weak subject matter- knowledge, poor command of English and limited resources.
Not only was there a lack of up-to-date resources in terms of international and local periodicals, there was also a distinct lack of opportunities for these lecturers to engage actively in
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discursive dialogues with others in the field. In my view continuous and frequent opportunities to listen, to engage and to be confirmed and challenged by others of similar and dissimilar views are important factors that encourage teachers and learners of mathematics to reflect and question their views of mathematics. Waywood (1988) put it succinctly "... it is through expressing that students locate themselves in our mathematical culture" (p. 52). The English lecturers had such opportunities where interested mathematics educators took time and
convened to learn from each other. There was a community of researchers who were willing to share their findings with their colleagues. There was a climate of support from interested colleagues. The study by Wilcox et al. (1991) has provided evidence that a supportive learning community helped learners, in their case pre-service teachers, acquire a more positive stance towards mathematics, teaching and learning. Wilcox et al.'s findings, my own interactions with the English lecturers and their support group, together with my observation of the lack in the Malaysian lecturers community all suggest that lecturers can only benefit from such
experiences, and be disadvantaged from their absence.
A climate of support for research was not so apparent in the Malaysian TTC. The responses of the lecturers suggested a climate of suspicion on the part of the Malay lecturers and authorities, resentment and fear on the part of the non-Malays. HD confided that he had conducted a substantial amount of research about the mathematical experiences of children of Indian rubber estate workers. He felt he could not publish these research findings as they were racially sensitive. In present day Malaysia there are legal and constitutional sanctions against raising and discussing racial issues (Wang, 1983). The lecturers' motives and the results of on-site research conducted by them were often questioned. The results often showed that Malay pre- service teachers' mathematics performance was below other racial groups and the lecturers who conducted the research were non-Malays. Lecturers sensed that the Malay lecturers were of the opinion that the lecturers were using mathematics grades to show that the Malays were not good in mathematics as compared to other races in the country.
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When the lecturers aired their views they often lowered their voices when they felt that they were discussing sensitive issues. These actions suggested lack of support from authorities and a climate of fear and suspicion that permeated the college (as Wang, 1983 had shown among university lecturers). Consequently the lecturers were inhibited from entering into discussions on issues that were pertinent and important to improving the kinds of mathematics taught and how it should be presented to the pre-service teachers.
Under such circumstances the lecturers saw no incentive to take the initiative to explore the mathematics syllabus they were teaching. It was better and safer to teach what was defined by
"experts" (Burton, 1992) allowing them "to dictate the syllabus than in taking on the responsibility themselves" (Burton, 1992a, p. 379).
American civil servants who were authorised to select textbooks to be used in schools had a significant role in determining what kinds of mathematics books were presented to the schools and public at large (National Research Council, 1989). In Malaysia a similar body selects and approves textbooks to be used in schools. This body has significant authority to select books as Malaysia practises a textbook loan scheme. Under this scheme economically disadvantaged students are loaned textbooks for the academic year. Thus authors of books, in order to secure approval for their textbooks and huge government contracts, stuck closely to the KBSR mathematics syllabus and the examination requirements, largely made up of drill questions, aimed to increase the speed and proficiency of the students. Davis and Hersh (1981) found similar trends in the American mathematical textbooks. They said
The presentation in textbooks is often 'backward'. The discovery process is eliminated from the description and is not documented.
After the theorem and its proof have been worked out, by whatever path and by whatever means, the whole verbal and symbolic
presentation is rearranged, polished and reorganised according to the canons of the logico-deductive method. The aesthetics of the craft demands this. Historical precedents - the Greek tradition - demands it.
It is also true that the economics of the book business demands
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maximum information in minimum space. Mathematics tends to achieve this with a vengeance. Brevity is the soul of mathematical brilliance or wit. Fuller explanations are regarded as tedious. (p. 282)
Because lecturers, at both personal and departmental level, were more concerned about pre- service teachers' weak subject matter knowledge they concentrated on improving such knowledge but their approach was didactic. The accounts of the lecturers suggest that the pre- service teachers left college with textbook and syllabus based mathematics. Lecturers offered the pre-service teachers mathematical activities that involved them in collecting known information. The lecturers' epistemology was based on their past teaching and learning
experiences, their own understanding of mathematics as well as from their own reading. These lecturers said that they were against mathematics being taught for the sole purpose of
assessment.
The lecturers also had high expectations of what pre-service teachers should do to improve their subject-matter knowledge as well as to extend their view of mathematics. They expected pre-service teachers to read up to improve their mathematics, and to discuss among their friends. At the same time they had a very low estimation of the pre-service teachers' ability to understand what they had read. Furthermore, they stated that the available resources were either not effective, difficult for the pre-service teachers to access, to read in English and to
understand. It seemed that the lecturers' expectations of the pre-service teachers' activities were in conflict with their assessment of the pre-service teachers' abilities.
The English lecturers, however, were actively engaged in trying to extend the English pre- service teachers' mathematical knowledge by engaging them in problem-solving activities and investigation work. J acknowledged that changes took time as he was aware that their pre- service teachers entered college "so much indoctrinated by school mathematics, they have a very narrow view". Nevertheless J pointed out that by the end of their third year eN en those pre- service teachers who were non-enthusiastic about the mathematics they were learning showed an enthusiasm and confidence that they lacked when they first entered college. The English
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pre-service teachers that I interviewed said that they were enjoying the mathematics they were engaged in during their college mathematics courses.
I found that there was no conflict between the Malaysian lecturers enacted and 'needs beliefs'.
They all had 'aspired beliefs' about mathematics which most of them could neither explain nor implement. The English lecturers tried to connect their enacted beliefs with their espoused beliefs.