`to provide a framework around which systems and schools may build their mathematics curriculum, and it identifies important components of mathematics education for the majority of students. It is descriptive rather than prescriptive, does not provide a syllabus or curriculum and, indeed, possesses a structure which makes it inappropriate for direct use in that way.' (p. 1).Traditionally, the mathematics curriculum taught in schools in the different Australian States and Territories has been decided upon and produced in either one of two places: the elementary and middle-school curriculum by State Education Departments, and the senior high school curriculum by State Public Examination Boards acting on the advice of Mathematics Syllabus Committees whose members include both high school and tertiary mathematics teachers (Malone, 1992). After a complete overhaul of the mathematics curriculum in both the elementary school and the junior high school, where a strong emphasis was placed on the introduction of recently promoted teaching methodologies, the State of Western Australia instituted dramatic changes to its mathematics syllabi at the senior high school level commencing in 1988. These changes were motivated by the number of students staying onto school beyond the compulsory attendance age of 15 years and the consequent variance in ability levels; the need to revise and replace out-dated content; the need to review traditional views on pedagogy, and the need to attract more students (particularly females) to study mathematics. Eight years later, the new mathematics curriculum has been implemented. This paper will focus on just two aspects of the change - the first, an evaluation of pedagogical changes encouraged within the new curriculum: what this change involved, how it was `sold' to teachers, the degree to which it was taken up, and the success or otherwise of the innovation, while the second assesses the impact of the changes upon students.
The revised Western Australian mathematics curriculum aimed to `present mathematics as an organized body of useful knowledge and provide students with the skills and confidence necessary to apply this knowledge in practical situations' (Secondary Education Authority, 1989, p.1).Some of the stated objectives of the courses are that
`students will communicate mathematical ideas and results, in both oral and written forms; compare outcomes with expectations and verify the suitability and reasonableness of a result'.The guidelines go on to say that
`it is highly desirable that students interact in a constructive and cooperative manner with peers and teachers and respond constructively to advice' (p.3).This was to be achieved not only by the change in curriculum content, but also by a change in the approach to teaching and learning so that both high achieving and low achieving students would benefit.
Now for any innovation to be successful, the changes must be carefully prepared, more so if the changes are at the State level, and even more so if it involves a reconceptualisation of traditional approaches to teaching. For that reason teachers must understand why pedagogical changes are necessary and be convinced of the viability of these changes so that they can commit themselves in how to implement them (Tobin and Imwold, 1992). Giacquinta's (1973) description of the three stages in introducing a new program into schools (initiation, implementation and incorporation) is still relevant today, particularly his statement that successful initiation does not necessarily mean the successful implementation of the innovation. There may be many discussions and great enthusiasm about an innovation, but if teachers' and students' behavior do not change and if teachers do not accept the pedagogical philosophy underpinning this innovation then the implementation has no effect. Rejection of a new scheme is not only based on the personal characteristics of teachers, but is often influenced by organizational features, politics of change and the type of change (Waugh & Punch, 1987).
During the initiation stage and at the beginning of the implementation stage, the Ministry of Education of Western Australia introduced a Mathematics Syllabus Implementation Project (MSIP) to support teachers during workshops to familiarize themselves with new mathematics topics and concepts. While the new content aspect of the revised mathematics curriculum in Western Australia was well handled with a variety of publications, workshops and professional development activities for teachers so that they knew what was new, the same cannot be said about the pedagogical aspects. Teaching interpretations which were in line with constructivist ideas (von Glasersfeld, 1990) - for example: group work, problem solving, reflection, student/teacher consensus, student/student and student/teacher discussion - were encouraged, but no professional development was devoted to this type of pedagogical reform.
A qualitative approach (Erickson, 1986) was the principal methodology adopted for this study. Data sources for the interpretative aspects included the following: interviews with teachers, teacher educators, Ministry of Education officials and students; observations and personal discussions with a large number of students who started tertiary studies at Curtin University during the last few years. Assertions were developed by the researchers from the field notes recorded during observations, and confirming/disconfirming evidence was sought in subsequent observations and in interviews and discussions with teachers, students and school administrators.
A quantitative element was introduced through the use of a questionnaire developed for use with a more broadly based sample of teachers. Approximately 65% of the senior high school mathematics teachers in both government and non-government schools in Western Australia who initially indicated their willingness to participate in the survey completed the questionnaire (n = 400). The information derived from this source was considered in conjunction with the other qualitative data collected.
Concurrent with this teacher survey, approximately 900 Year 11 and 650 Year 12 students - the Years 11 and 12 are the final two years of Western Australian high schools - completed a questionnaire in which their opinion was asked about reasons for selecting mathematics in general and specific mathematics subjects in particular; factors which they associated with their success in mathematics, and people who they consider most influenced their choice of mathematics.
At the conclusion of each MSIP workshop, participating teachers were requested to complete a form in which they could express their thoughts on the session and the value of the material discussed. Some frank feelings were very enlightening.
A study of the results of a diagnostic test on the knowledge of mathematics by newly enrolled engineering and computer science students at Curtin University provided insight in the weaknesses and strengths in pre-calculus and calculus concepts.
A study was made of a similar implementation of new mathematics courses in Dutch high schools. These courses are based on the realistic approach to mathematics education (De Lange, 1987) which has a strong affinity with constructivist ideas. Educators who are proponents of this realistic approach were interviewed while approximately 25 classroom lessons were attended with the aim of determining how far this new approach to education has been adopted. Discussions with the teachers involved provided good indicators of their personal views, and this cross-cultural aspect of the study constituted a useful pilot for the teacher interviews in Western Australia.
The task of the MSIP was to organize workshops, where teachers were to be introduced to new concepts which were added to the syllabus, invite contract writers to prepare Teachers' Notes on these concepts and arrange the logistics of the introduction of the new courses based on a budget which was limited to $2 million over a period of two years. It ought to be remembered that the area of the State of Western Australia is comparable with the area of Western Europe from Scandinavia to and including the Iberian Peninsula with Germany and France as eastern borders. The logistics involved teacher relief and reimbursement of travel costs - airfares included. The project became a well-organized professional working relationship between mathematics staff from Government schools and Independent schools and mathematics educators from the local tertiary institutions.
The aim of the workshops was to provide teachers with both background knowledge and teaching strategies so that they will be able to implement the new courses. After some initial teething problems, the decision to have workshops organized by a tertiary lecturer and a practising high school teacher was widely applauded. The general response from teachers - it seemed that nearly all upper secondary mathematics teachers attended one or more workshops - was very favorable, although insufficient attention had been paid to alternative teaching strategies.
(a) What were the principal changes in pedagogy inherent in the new approach?
A careful survey of materials prepared for teachers and interviews with the teachers themselves indicated that ideals such as student autonomy, student-centered rather than teacher-centered activity, reflection, classroom consensus on mathematical issues through negotiation, and the withholding of direct answers to student questions by the teacher were encouraged although the term constructivism was never used in these materials. In our interviews and in the questionnaire teachers were requested to indicate their intended support or opposition to these ideals. Their responses showed a strong commitment in promoting teaching strategies where students are given time to reflect upon what has been taught, are encouraged to demonstrate their methods and solutions to the rest of the class, and are encouraged to comment on other students' methods and results. The teachers also indicated that they were more interested in the students' method of solution than in their final answers. There was divided support for promoting small-group work in the classroom.
(b) To what extent were the methodological changes adopted?
In this study it was clear that teacher cooperation was strongest when implementing syllabus content changes, undoubtedly because in-service courses and workshops which were available enabled them to seek assistance regarding any problems they faced. It became apparent, however, that in general these workshops dealt with new contents for various courses while insufficient attention was paid to new teaching strategies. Many teachers indicated that they did attempt the suggested reforms but would switch back to traditional methods frequently in the interest of class discipline and as a result of pressure from colleagues, parents and administrators. Change has proved to be difficult to implement because teachers are inclined to maintain well-tried strategies which they have developed. Obviously, changing existing strategies cannot be achieved after a few in-service courses but require a personal commitment of teachers over a long period of time.
(c) What were teachers' attitudes and opinions towards the changes?
Many teachers rejected the majority of the constructivist thrusts, again in the interest of discipline and their responsibility to ensure that students completed the syllabus and were successful at examinations. This resistance was evident among both young and older teachers.
In general, teachers agreed that the new courses were beneficial for the students because of a greater choice of suitable subjects, and they considered that the standards of mathematics in comparison with the previous courses had not been lowered. Opinions were divided with respect to personal benefits because of the extra work involved to familiarize themselves with the new courses. The same applied to an improved effectiveness of teaching after introducing new strategies. A strong wish was expressed in favor of future stability in the curriculum: using the new courses for many years to come.
(d) What were the teachers' actions regarding the changes?
The researchers are currently working in a one-to-one relationship with a sample of teachers and have achieved some initial significant results with a number of them. A majority of the teachers in the sample are becoming convinced that many elements of the new approach have the potential to improve students' understanding of mathematics and are striving to adapt their existing approaches to teaching to conform with the new approach. Overall, our results support those of Waugh (1993) in that they emphasize the importance of teachers' attitudes to a change in curriculum if such change is to be successfully implemented.
Other relevant findings of the study established that system-wide change is better received by teachers if:
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