of one's epistemological commitment in relation to mathematical knowledge, and not merely one of expediency in response to societal pressures or of pedagogical convenience (Lerman, 1983, p. 59)
Research shows that while some teachers were consistent in their professed conceptions of mathematics and in their practice, others professed certain conceptions of mathematics but their instructional practice demonstrated otherwise. As an example of the former case, Thompson (1984) found a high degree of consistency among the teachers she observed. She acknowledged that
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Although the complexity of the relationship between conceptions and practice defies the simplicity of cause and effect, much of the contrast in the teachers' instructional emphases may be explained by
differences in their prevailing views of mathematics (p.119).
She went on to provide evidence to support her view. Lynn, who was characterised as having an instrumentalist view of mathematics, was prescriptive in her teaching, emphasising
demonstrations of rules and procedures. Thompson said that Lynn's instructional practice was consistent with her remark that
Mathematics is cut and dried. This is the answer. Follow this procedure and this is the answer. (p. 116)
Jeanne who saw mathematics as a coherent subject of logically interrelated topics, taught accordingly, emphasising mathematical meaning of concepts and the logic of mathematical procedures. Kay, who had a problem-solving view, focused more on activities that engaged her_ _ students in the generative processes of mathematics.
Cooney (1985) examined the beliefs about problem-solving of Fred, a secondary mathematics teacher, while he was finishing his pre-service training and during his first three months of teaching. Though he had professed his belief that teachers should motivate students and focus on problem-solving, his instructional practice did not reflect his professed conceptions of mathematics. Fred believed in the efficacy of problem-solving as he was influenced by his mathematics and mathematics method courses during his preparatory courses. To Fred, "The principal activity of mathematics is solving problems ... A central point of teachin g problem solving is teaching heuristics" (Cooney, 1985, p. 328). Though he espoused these beliefs about mathematics he also confounded his beliefs with his reason for usinv, problem-solving activities in his class. He justified his use of the problems he chose on two separate occasions.
My intention was not so much a problem-solving situation but as sort of an interest creator. ....[In another case] I intentionally used that [a particular number-theoretic problem] during the first period class just
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as something fun for us to look at before we got down to brass tacks.
With the second period, if I recall correctly, it was a sort of a stop gap - if we have some time left, let's do something different. Both were motivational devices but maybe more so for the first period class; the second period was more of a time filler. (p. 331, original emphasis)
To Fred, a "teacher's chief responsibility is to motivate students" and he felt that it could "best be done by using recreational problems" (Cooney, 1985; p.329). Fred's use of problems as interest creators was consistent with his belief about recreational problems; it seemed
inconsistent with his view that "solving problems is the essence of mathematics". (p. 328) After introducing the problem he heavily emphasised the algorithms to solve it. Fred was described as practising textbook based teaching, and heuristics were not central to Fred's teaching style though he criticised teachers and texts that required too much drill and practice. Underhill (1988) maintained that Fred had two views: "a highly authoritarian approach characterised by rigid control and boredom, and a problem solving approach (which he tried to implement) characterised by motivation, fun, and casualness" (p. 50).
Strong relationships between the subject-matter knowledge base of novice teachers and expert teachers and their instructional practice were documented in studies by Leinhardt et al. (1991) and Lampert (cited in Ball, 1991). Leinhardt et al (ibid.) illustrated the role of subject-matter knowledge by comparing novice and expert teachers concerning
their general agendas describing the major purposes and strategies for the lesson as a whole;
curriculum scripts that represent their goals and actions for teaching particular topics within the lesson;
• their explanations of particular mathematical content;
• their representations of particular concepts, meanings, or procedures.
In each of these areas, the expert teachers' knowledge tended to be better differentiated and better organised, as well as more accessible for use when needed and more adaptable to the
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particular circumstances that arose as the lesson developed. These findings suggested that if I looked at pre-service teachers teaching mathematics in school, their less organised subject- matter knowledge would lead them to conduct mathematics activities that were less coherent and less organised.
I feel that the Malaysian belief in the simplicity of teaching primary mathematics is related to the equation of subject-matter knowledge with the four basic rules of arithmetic, formulae and algorithmic procedures. What is so difficult about teaching children mathematics when it is a simple case of teaching them how to add, subtract, multiply and divide? Malaysians, myself included, learn algorithmic mathematics without understanding the whys of the algorithm.
Ball's study (1988, 1990a, 1990b) demonstrated that in-service and pre-service teachers alike assumed the ability to apply a certain algorithm was evidence of their understanding of the mathematics involved. Consequently, they applied the same assumptions to their students, taking their students' ability to demonstrate the taught algorithm as an indication of their understanding. Moreover teaching students algorithmic mathematics made assessment easier, for teachers took the facility to demonstrate application of the algorithm in their work to be understanding. Ball (1991) provided very clear reasons why this assumption was false. She said that
many people ... had taken subject-matter knowledge for granted in teaching mathematics. Our reconsideration of the role of subject matter calls this assumption sharply into question. First, learning to do mathematics in school, given the ways in which it is taught, may not equip even the successful student with adequate or appropriate knowledge of or about mathematics. Second, knowing mathematics for oneself may not be the same as knowing it in order to teach it.
While tacit knowledge may serve one well personally, explicit understanding is necessary for teaching. Finally, subject-matter knowledge does not exist separately in teaching, but shapes and is shaped by other kinds of knowledge and beliefs (p.40).
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Ball warned all those who could do mathematics not to misconstrue and equate their knowledge of how to do mathematics with their ability to teach the subject. Her findings suggest that classical ways of teaching and learning school mathematics do not prepare individuals well enough to teach mathematics. Even successful mathematics students were not necessarily capable of teaching mathematics.
Ball (1991) used Lampert, an elementary teacher who was also a researcher and a university professor, as an example to demonstrate the effects of rich disciplinary knowledge of mathematics on instructional practice. According to Ball, Lampert's rich mathematical knowledge helped her in
Making the judgements about which student suggestions to pursue, developing the tasks that encourage certain kinds of exploration, and conducting fruitful class discussions - all these tasks depend heavily on the teacher's subject matter knowledge. (p. 39)
I want to add that Lampert's rich subject-matter knowledge gave her the confidence to work with her students as documented in the study. Her confidence gave her the power to decide which suggestions to pursue, which to ignore. Her power gave her insight as to where their suggestions would lead. If she followed a particular suggestion would it lead to a dead end? If it did, her rich knowledge enabled her to follow up other suggestions that would lead to other paths and other possibilities. Her rich subject-matter knowledge allowed her to take on the role of facilitator rather than the traditional role of a didact and gave her the confidence to work with groups without the worry that she would not be able to answer questions posed by
different groups. Thus I propose that her rich knowledge shaped and interacted with her beliefs of teaching and learning mathematics.
I feel that teachers' insecurity about their weak subject-matter knowledge coupled with their 'simplicity belief encourage them to be more authoritarian. They are more likely to enforce individual work or work that is very structured on their students so that they know that every
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student will pursue the work along certain paths. Cooney (1988) found that this was the case with secondary pre-service teachers in his study. These secondary pre-service teachers admitted that they became more authoritarian because they wanted to maintain order in the class, not only so that lessons could be conducted, but also because they were afraid that "someone will ask a question that I don't know the answer to" (Cooney, 1988, p. 357). Ball's (1988) work with pre-service secondary teachers showed that they themselves recognised that broader and deeper understanding of mathematics is necessary for conceptual teaching.
During my teacher education courses my mathematics method lecturers told us that we should introduce recreational elements into our lessons to make our lessons interesting. A token example was given and we were left to our own devices. If my experience, which I admit is limited, were typical, I would conjecture that Malaysian mathematics lecturers in the teaching colleges were quite likely to encourage their pre-service teachers to make their lessons 'interesting' by introducing recreational elements into their lessons with the customary token examples. Recreational activities are perceived as distinct from mathematical activities. I am not criticising the lecturers; they could only do as much given their own limited experiences of mathematics and methods of making mathematics interesting. I would expect to hear from the lecturers and the pre-service teachers alike that they would want to make their mathematics lessons interesting, but their actual practice might not reflect these espoused views.
To summarise, research shows that teachers' beliefs about mathematics affect their practice.
The traditional style of teaching mathematics, focusing on rules and algorithms, often meant that individuals could apply the correct rules and algorithms but might lack the ability to teach them. Rich subject-matter knowledge of mathematics enabled expert teachers to adopt a more facilitating role than was possible for those with a rule-based knowledge of mathematics.
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4.7. Relationship Between Espoused Beliefs About Teaching And Learning And Actual Instructional Practice
Why do teachers experience difficulties putting into practice their professed beliefs about teaching and learning? Thompson (1992) said that
What a teacher considers to be desirable goals of the mathematics program, his or her own role in teaching, the student's role,
appropriate classroom activities, desirable instructional approaches and emphases, legitimate mathematical procedures, and acceptable outcomes of instruction are all part of the teacher's conceptions of mathematics teaching. (p. 135)
Clark (1988) emphasised this point when he highlighted the fact that
Research on teacher thinking has documented the fact that teachers develop and hold implicit theories about their students (Bussis, Chittenden, & Amarel, 1976), about the subject matter that they teach (Kuhs, 1980, Elbaz, 1981, Ball, 1988) and about their roles and responsibilities and how they should act (Ignatovich, Cusick & Ray,
1979, Olson, 1981). These implicit theories are not neat and complete reproductions of the educational psychology found in textbooks or lecture notes. Rather, teachers' implicit theories tend to be eclectic aggregations of the cause-effect propositions from many sources, rules of thumb, generalizations drawn from personal experience, beliefs, values, biases, and prejudices (p.6).
The findings of the various studies that have looked at the relationship between teachers' professed beliefs about teaching and their instructional practice were not consistent across studies or across teachers. Grant (1984) found that the beliefs of the three teachers that he studied were congruent with their teaching practice. Studies by Cooney (1985) and Thompson (1982, 1984) showed that there were sharp contrasts between some of the teachers' professed beliefs and their instructional practice.
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Could it be possible that teachers express certain beliefs about mathematics teaching that they themselves wish to put into practice but can not do so as their colleagues, students and school administrators do not share such beliefs? Receiving neither support nor understanding, teachers, particularly novices, possibly find it is easier to follow the accepted institutional practice than to go on alone. Thompson (1992) wrote that
The inconsistencies reported in these studies indicate that teachers' conceptions of teaching and learning mathematics are not related in a simple cause-and-effect way to their instructional practices. Instead, they suggest a complex relationship, with many sources of influence at work; one such source is the social context in which mathematics teaching takes place, with all the constraints it imposes and the opportunities it offers. (p.138)
Sosniak, Ethington and Varelas (1991) who used the data from the Second International Mathematics Study to explore teachers' beliefs found teachers' instructional practice inconsistent with their professed beliefs. They found that
... eighth-grade mathematics teachers in the US apparently teach their subject matter without a theoretically coherent point of view. They hold positions about the aims of instruction in mathematics, the role of the teacher, the nature of learning, and the nature of the subject matter itself which would seem to be logically incompatible. (p.127, original emphasis)
Glidden (1991) was concerned that Sosniak et al's. findings might have been influenced by the instruments used, a point accepted by Sosniak et al. themselves. However Sosniak et al. said that the interconnectedness between teachers' professed beliefs and their instructional practice also depended on how closely related the issues examined were to actual classroom teaching.
They said
We have a set of findings regarding teachers' curricular orientations which shift systematically from "progressive" to "traditional" as the teachers move from considering the issue most distant from schooling
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and classroom instruction to the issue most central to schooling and classroom instruction (p.129).
This could be related to teachers' awareness of the actual realities of classroom teaching acting as a restraint on their espoused views of teaching and learning in the classroom. In my view, teachers are more likely to suppress espoused beliefs if these are seen to be over-ruled by the immediate needs of the students, a case of 'needs belief being incompatible with espoused beliefs.
Lerman (1986) and Ernest (1989) acknowledged that the social context had powerful effects on teachers' instructional practice. Ernest (1989) said
These sources lead the teacher to internalise a powerful set of constraints affecting the enactment of the models of teaching and learning mathematics. The socialization effect of the context is so powerful that despite having differing beliefs about mathematics and its teaching, teachers in the same school are often observed to adopt similar practices. (p. 252-253)
Brown (1985, see also Cooney ,1985) showed how events in the social context influenced Fred, a novice, in his role. Cooney (1985) described the tensions experienced by Fred. Though Fred tried to use problem-solving activities as a motivational rather than a mathematical objective, he had to change his instructional practice in his classes when the general stream mathematics pupils saw these activities as a waste of time. The general mathematics pupils argued that if Fred really cared for the class, he would not engage the class in recreational mathematics.
Rather he would be teaching them mathematics which, according to them, involved the four basic rules. Moreover they should be practising for their examinations and not wasting time.
Therefore Fred had to compromise his initial espoused belief of using problem-solving activities as motivational objectives, reverting to a more authoritarian instructional style to enable him to cover the syllabus and maintain class control.
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Similarly, Malaysian mathematics teachers were highly influenced by the demands of the administrators, parents and students. Teachers had to project yearly performance of the students they were teaching, forecasting the percentage of passes they expected to achieve. Students similarly demanded that their teachers help them not only to pass their mathematics papers, but also to achieve as good a grade as possible. Parents added to the pressure by emphasising to the heads of schools that they expected the best results from the teachers. The linking of students' performance with remuneration and promotion prospects has become a crucial and motivating force in the effort to achieve better examination results. Though many of my colleagues, especially those teaching the weaker students, had expressed their wish to take more time over certain mathematics topics so that their students could explore and develop a better
'understanding', these teachers were also driven by the need to complete the syllabus, partly because of performance-related pay. Thus, though there were teachers who believed that students should understand the mathematics they learned and how it was applied in real-life situations, they had to subordinate that belief and cater to the more observable and immediate demand of the administrators, parents and pupils. Lerman (1986), Ernest (1989) Hoyles (1991) and Thompson (1992) have reported similar findings in their reviews.
Thompson (1992) said
the current reform movement in mathematics education, and the publication of documents such as NCTM's Curriculum and
Evaluation Standards for School Mathematics, may have an influence on teachers' verbal expressions regarding their views of mathematics teaching and learning. (p. 138)
Practising and pre-service teachers might profess rather idealistic beliefs about mathematics teaching. Their professed beliefs about mathematics teaching could have been affected b:s what they have read in the literature, or what they were told by their methods lecturers during their university or teacher training days. Thus Shaw (1989) showed that teachers' \ erball„s proffered beliefs were consistent w ith "hat they belies ed about teaching in the abstract, but in reahtn
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they knew that they could not put them into practice. Thompson (1992) said that
inconsistencies of such a nature were less likely to occur among more experienced teachers because they would have had the opportunities to put their beliefs to the test and would be more aware of the constraints existing in the classrooms that mitigate against the implementation of their professed beliefs. This suggests that it is methodologically unsound to take what is verbally proffered by teachers and pre-service teachers as both their espoused and enacted beliefs. Scheffler (1965) also warned against taking verbal evidence of beliefs. He wrote
it seems particularly important to avoid mistaking verbal dispositions for belief. To this end, it is crucial that we recognize not only the ramifications of belief in conduct but also the influence of motivation and social climate on verbal expression. (p.90)
The empirical literature demonstrates that students' beliefs could very easily and subtly subvert the instructional practice of teachers (Brown and Cooney, 1985). This subversion was also evident among pre-service teachers. For example, Foss' study (1993) showed that the
"prevailing conceptions held by preservice teachers not only remained constant, but confounded the pursuit of learning to teach mathematics .... The methods instructor ... was transformed from hopeful constructivist to desperate lecturer. Faced with the preservice teachers' dissonance, interactive lessons designed to lead students from concrete to abstract thinking were replaced with formal lectures".
Studies that looked at countries where reforms in teaching of mathematics had been introduced showed that teachers and pre-service teachers expressed beliefs that were congruent with the official documents. There could be a number of reasons to explain this. It could be possible that in-service and pre-service teachers thought it was the appropriate thing to do because they believed that the authorities 'knew better'. They could be so disillusioned with the present style of teaching that they were willing to believe anything new and innovative, regardless of whether they themselves practised it. Pre-service teachers, unlike their more experienced counterparts who had knowledge of how physical and social constraints affected
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