EASTERN CAPE TEACHER S BELIEFS REGARDING THE NATURE OF MATHEMATICS

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1 Buffler, A. & Laugksch, R.C. (Eds.) (2004). Proceedings of the 12th Annual Conference of the Southern African Association for Research in Mathematics, Science and Technology Education. Durban: SAARMSTE. EASTERN CAPE TEACHER S BELIEFS REGARDING THE NATURE OF MATHEMATICS LYN WEBB & PAUL WEBB Department of Science, Mathematics and Technology Education, University of Port Elizabeth, South Africa cenmlw@upe.ac.za ; cenppw@upe.ac.za Various studies have shown that what teachers consider as optimal ways of teaching mathematics is influenced by their beliefs about the nature of mathematics and that it is advantageous to determine te achers conceptions of the nature of mathematics before developing curriculum interventions. In this study various methods were employed to stimulate teachers to both reflect on their beliefs and to make them explicit. A Likert -scale questionnaire was administered to 339 in-service teachers in urban and rural areas of the Eastern Cape. A sample of ninety -five of these teachers completed a questionnaire based on videotapes of lessons recorded during the TIMSS (1995) study that they had viewed. These teachers also ranked their own teaching on a continuum ranging from traditional to constructivist approaches and provided explanations for their ranking. A further sub -sample of thirty-six teachers participated in individual interviews, which explored their percep tions of the nature of mathematics and their own teaching practice. In order to investigate whether these beliefs are mirrored in practice, four teachers were videotaped in their classrooms. The data generated by these videos support the findings of simila r studies, i.e. that teachers beliefs of the nature of mathematics are often not reflected in their practice. INTRODUCTION Recent research by Chapman (2002), Hart (2002), Llinares (2002), Lloyd (2002), Philippou and Christou (2002) and Wilson and Cooney (2002) draw a strong link between teachers beliefs and practices and they suggest that changing teachers practices depends on changing their beliefs. In South Africa the Revised National Curriculum (2002) states that teachers are key contributors to the t ransformation of education. As such, it follows that any attempts to effect the proposed transformation requires that the beliefs of teachers be scrutinized (Taylor 2002). In the light of the above, this study investigates the beliefs about the nature of mathematics of teachers in the Eastern Cape, South Africa. It also investigates how these beliefs about the nature of mathematics are related to their beliefs concerning the teaching and learning of this subject and whether there is any correlation between these beliefs and their classroom practice. LITERATURE REVIEW Thompson (1992) maintains that teachers differ a great deal in their beliefs about both the nature of mathematics and how they view the teaching and learning of mathematics. Teachers conceptions of the nature of mathematics range from seeing mathematics as an absolute, fixed body of knowledge to viewing the discipline as a fallible and expanding human invention. Schoenfeld (1985) concludes that most students view mathematics as a body of knowledge to be memorised, despite the fact that teachers often emphasise the importance of understanding the subject. Schoenfeld (1985) also notes that learners experience neither understanding nor a perception of utility of the subject in practice. These findin gs are supported by Morars (2000) study with teachers in South Africa who, despite professing to believe in a constructivist paradigm, used a traditional approach where learners experienced mathematics as a subject to be memorised. Lerman (2002) cautions that a number of research studies start from the premise that espoused beliefs and teachers practice are consistent. However, this is frequently not the case and Wilson and Cooney (2002) highlight the methodological weakness in presuming that questionnair es and interviews will reveal beliefs, 1025

2 or systems of beliefs, that determine a teachers actions in the classroom and they note that there appears to be a shift away from purely pen-and-paper or quantitative interview techniques towards a wider acceptance of qualitative research designs. Nevertheless, the data generated in this study by the context specific traditional pencil and paper Likert-scale questionnaire (designed with the respondent s backgrounds and experience in mind) were valuable in determi ning leads to be followed and in identifying understandings that required further unpacking. These data were then complemented and supplemented by the more qualitative findings from the video clip discussions, teacher reflections, qualitative questionnaire and interview data. The combined data sets allowed for more reliable and valid inferences to be drawn. PHILOSOPHIES OF MATHEMATICS Ernest (1989, p. 250) maintains there are three philosophies of mathematics and defines them as follows: Instrumentalist mathematics is a set of unrelated but utilitarian rules and facts. It is an accumulation of facts, rules and skills to be used in the pursuance of some external end. Platonist mathematics is a static, but unified body of certain knowledge. Mathematic s is therefore discovered not created. Problem-solving view of mathematics dynamic, continually expanding field of human creation, a cultural product. Mathematics is therefore constantly being revised and constructed. Mathematics is a process of inqui ry and coming to know, not a finished product, for its results are open to revision. Ernest (1989) sees the above three viewpoints as a hierarchy with instrumentalism at the bottom and problem solving at the top, and links teachers views of the nature of mathematics with their models of teaching and learning. He maintains that teachers conceptions of the nature of mathematics forms their philosophy of teaching and learning mathematics, despite the fact that they may be unable to fully articulate their beliefs as they are often implicitly held (Ernest 1989). He notes that the teacher who views the nature of mathematics as a problem-solving activity (Ernests highest level) will act as a facilitator, will regard learning as an active construction of understa nding, and will possibly even see learning as an autonomous problem-posing and problem-solving activity (Ernest 1989), challenges inherent in the demands of the South African Revised Curriculum Statement (2002). METHODOLOGY Taking Lermans (2002) caveat about not believing that espoused beliefs and teachers practice are consistent into account, and Wilson and Cooneys (2002) warning against presuming that questionnaires and interviews will reveal beliefs, or systems of beliefs, that determine a teacher s actions in the classroom, the decision was taken to design this research using the following data collection methods: A Likert-scale questionnaire, consisting of multiple choice and open -ended questions (n = 339). Getting teachers to compare the teaching pr actices of an American lesson and a Japanese lesson shown in the TIMSS videos ( n = 95). Individual interviews ( n =36). Written paragraphs by teachers on what they believe constitute mathematics and why it is taught ( n = 32) Reflective essays by teachers on their own teaching practice and self -ratings of their own teaching on a scale from 1 (highly traditional) to 10 (highly constructivist) ( n = 95). Videotaping of teachers in the classrooms to assess whether there was a correlation between their espoused beliefs and their enacted practices ( n = 4). All of the participating teachers were University of Port Elizabeth students who were completing a BEd upgrade qualification in mathematics, science and technology education. The quantitative results from the Likert-scale questionnaire were analysed while the qualitative data, i.e., the essays, paragraphs, comparisons and transcripts of the interviews and classroom visits were 1026

3 interrogated in order to broaden and enrich the quantitative data and give a deeper unde rstanding of the problem under investigation. RESULTS The results of the Likert-scale questionnaire, TIMMS video clip responses, interviews, teachers rating of their own teaching practice and classroom observations are described below. Likert-scale questionnaire From the qualitative data, with a possible range from 1 to 4, the mean score for the teachers views of the nature of mathematics (n = 338) was 3, whereas the mean for learning ( n = 337) was 3.2 and the mean for teaching (n = 325) was 3.1. There was a positive correlation of 0.61 between views on the nature of mathematics and views on learning mathematics. The correlation between beliefs concerning the nature of mathematics and the teaching of mathematics was 0.59 and the correlation between the na ture of mathematics and teaching was These data suggest that teachers who view the nature of mathematics through constructivist eyes believe that learning takes place actively and teaching should be learner - centred, and that those who view the nature of mathematics as instrumentalists believe that learning takes place through mastering algorithms and skills through teacher dominated instruction. TIMSS video clips From the analysis of the quantitative video responses, it appears that the teachers recog nised that their own teaching practice was similar to that presented in the American model. They were able to reflect on the shortcomings evident in both models, but were not able to meaningfully discuss methods that would be more appropriate and, probably, more successful than the ones shown in the videos. To a large degree the teachers were initially convinced that the USA teacher was using more constructivist methods than the Japanese teacher as they commented that her students were seated in groups an d she moved among the groups giving individual attention, whereas the Japanese teacher stood in front of the straight rows of students for the majority of the lesson. They described constructivism through the physical trappings and had to be guided, through later discussion, to unpack the concept of problem solving and active learning - with the teacher as a facilitator and not an instructor of algorithms. On reflection, participating teachers were able to recognise that the American example relied on skill s whereas the Japanese lesson emphasised understanding a single problem in depth. Interviews Every teacher stated that their views about the nature of mathematics, teaching and learning had changed over the period of their BEd studies at UPE as they had be en exposed to new ideas and concepts. Previous to the commencement of their studies, they had neither been challenged to describe mathematics, nor to reflect on their current practices. Rating of their own teaching practice Ninety-five teachers rated their teaching on a scale from 1 (highly traditional) to 10 (highly constructivist). The mean score was 5,49 with the minimum score being 3 and the maximum score being 8. Classroom visits Four teachers were visited in their classrooms. All had been scored as having constructivist views of the nature of mathematics in terms of the Likert-scale questionnaire. Apart from the above reason, they were also chosen because they taught a range of grades and because their schools were situated in Port Elizabeth and were therefore easily accessible. 1027

4 Although the teachers had professed to have constructivist beliefs about the nature of mathematics, they did not demonstrate in their classroom practice that their role, and the intended outcome of instruction, was that of being a facilitator and learner-centeredness. On the contrary, all seemed to teach in instructor mode, where the emphasis was on skills mastery and only correct performance was acceptable, and there was little correlation between their espoused beliefs as reflected in the Likert-scale questionnaire and their classroom practices. DISCUSSION The teachers in the study expressed consistent constructivist views about the nature of mathematics, learning and teaching in a variety of ways, however, their practice di d not mirror their espoused beliefs. Stoker (2003) has commented on teachers who have learned the rhetoric but who have not internalised the reality of constructivist approaches to teaching and learning. The apparent discrepancy between the participating teachers espoused beliefs and their classroom practice may possibly be attributed to the fact that they were University of Port Elizabeth students who were completing an upgrade qualification in mathematics, science and technology and who, as such, had recently been exposed to the rhetoric of constructivism. This may have skewed their responses towards a constructivist approach without being sufficiently internalised to change their practice in the classroom. Participating teachers (n = 36) were asked why they felt this schism existed. Reasons expressed for not being able to implement better teaching and learning strategies included deficient initial teacher training, and that they had no recourse to appropriate resources and on -going professional development to make their methods more successful. As such, it appears that the teachers participating in this study are paying lip service to the rhetoric of constructivism, and that their espoused beliefs are not being translated into enacted beliefs. Ernest (1989) states that changes in the key beliefs of the nature of mathematics, and those on the nature of mathematics teaching and learning are connected to reflection and autonomy on the part of the mathematics teacher. Thus, interventions aimed at getting teachers to reflect on their beliefs and practices, and to become autonomous agents of change, is of vital importance and should have far reaching consequences in South African education where a new, constructivist orientated curriculum is in the process of being implemented. ACKNOWLEDGEMENTS Appreciative acknowledgement is made to the South AfricanNetherlands Research Programme on Alternatives in Development (SANPAD) for financial support for this study, which forms a sub -set of a larger SANPAD research initiative on Science and Mathematics Teachers Perceptions of the Nature of Science. REFERENCES Chapman, O. (2002). Belief structure and in-service high school mathematics teacher growth. In G. Leder, E. Pehkonen & G. Törner (Eds.), Beliefs: A hidden variable in mathematics education? Dortrecht: Kluwer Academic Publishers. Ernest, P. (1989). The impact of beliefs on the teaching of mathematics. In P. Ernest (Ed.), Mathematics teaching: The state of the art. London: Falmer Press. Ernest, P. (1991). The philosophy of mathematics education. London: Falmer Press. Hart, L. (2002). A four year follow-up study of teachers beliefs after participating in a teacher enhancement project. In G. Leder, E. Pehkonen & G. Törner (Eds.), Beliefs: A hidden variable in mathematics education? Dortrecht: Kluwer Academic Publishers. Lerman, S. (2002). Situating research on mathematics teachers beliefs and on change. In G. Leder, E. Pehkonen & G. Törner (Eds.), Beliefs: A hidden variable in mathematics education? Dortrecht: Kluwer Academic Publishers. Llinares, S. (2002). Participation and reification in learning to teach: the role of knowledge and beliefs. In G. Leder, E. Pehkonen & G. Törner (Eds.), Beliefs: A hidden variable in mathematics education? Dortrecht: Kluwer Academic Publishers. Lloyd, G. (2002). Mathematics teachers beliefs and experiences with innovative curriculum materials. The role of curriculum in teacher development. In G. Leder, E. Pehkonen & G. Törner (Eds.), Beliefs: A hidden variable in mathematics education? Dortrecht: Kluwer Academic Publishers. 1028

5 Morar, T. (2000). Evaluating the contribution of 2 key teachers to the systemic transformation of educational support and the professional development of their colleagues in the Eastern Cape Province. Paper presented a t the 8th Annual Conference of the Southern African Association for Research in Mathematics and Science Education, Port Elizabeth. Philippou, G. & Christou, C. (2002). A study of the mathematics teaching efficacy beliefs of primary teachers. In G. Leder, E. Pehkonen & G. Törner (Eds.), Beliefs: A hidden variable in mathematics education? Dortrecht: Kluwer Academic Publishers. Department of Education. (2002). Revised national curriculum statement Grades R-9 (Schools). Pretoria: Department of Education. Schoenveld, A.H. (1985). Students beliefs about mathematics and their effects on mathematical performance: a questionnaire analysis. Paper presented at the Annual Meeting of the American Educational Research Association, Chicago. Stoker. J. (2003). An investigation of mathematics teachers beliefs and practices following a professional development intervention based on constructivist principles. Unpublished PhD dissertation, Curtin University of Technology. Taylor, M. (2002). Implementing the standards: Keys to establishing positive professional inertia in pre-service mathematics teachers. School Science and Mathematics, 102(3), Thompson, A.G. (1992). Teachers beliefs and conceptions: A synthesis of the research. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning. New York: Macmillan Publishing Company. TIMSS. (1995). National Center for Education Statistics. Institute of Education Sciences. Washington DC: Department of Education. Wilson, M. & Cooney, T. (2002). Mathematics teacher change and development. The role of beliefs. In G. Leder, E. Pehkonen & G. Törner (Eds.), Beliefs: A hidden variable in mathematics education? Dortrecht: Kluwer Academic Publishers. 1029

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