Mathematics Education Research Journal 2005, Vol. 17, No. 2, 69–90 A Typology of Mathematics Teachers’ Beliefs about Teaching and Learning Mathematics and Instructional Practices Anastasios (Tasos) Barkatsas John Malone National University of Athens, Curtin University of Technology Greece and St Joseph’s College, Australia The primary purposes of the study reported here were to investigate Greek mathematics teachers’ beliefs regarding the nature of mathematics and mathematics teaching and learning, and to explore the various links between these beliefs and instructional practice. Two orientations that are characteristic of secondary mathematics teachers’ beliefs were identified: Acontemporary – constructivist orientation, and a traditional – transmission – information processing orientation. Acase study of a veteran teacher demonstrated that classrooms can be complex sites of political, historical, social and cultural influences, and that the teacher’s beliefs about mathematics learning and teaching were less traditional than her actual teaching practice. Introduction Attitudes and beliefs about teaching and learning have attracted considerable interest in relatively recent research studies (Leder, 1993; Leder, Pehkonen, & Törner, 2002; McLeod, 1989, 1992). Pajares (1992) argued that knowledge and beliefs are inextricably interwoven and that beliefs strongly affect an individual’s behaviour. Hollingworth (1989) reported that the way teachers implement new methods or programs in their classrooms relates to whether teachers’ beliefs are congruent with the proposed new methods or programs. Various other researchers have suggested a number of definitions or descriptions of the term beliefs over the past two decades. The term has been a particularly difficult one to define in the educational and psychological literature. McLeod’s description (1992) was considered adequate for this research study: Beliefs are largely cognitive in nature, and are developed over a relatively long period of time. Emotions, on the other hand, may involve little cognitive appraisal and may appear and disappear rather quickly … Therefore we can think of beliefs, attitudes and emotions as representing increasing levels of affective involvement, decreasing levels of cognitive involvement, increasing levels intensity of response, and decreasing levels of response stability. (p. 579) Other studies have investigated various aspects of teachers’ beliefs – for instance how teachers’ beliefs impact on teachers’ daily practices (Buzeika, 70 Barkatsas & Malone 1996); how to encourage teachers to reflect on their teaching practices (Malone, 1995); alternative models for mathematics teaching (Malone, 1995; Malone, Thornton, Langrall & Jones, 1997); espoused primary and secondary teachers’ beliefs about mathematics and the learning and teaching of mathematics (Howard, Perry & Lindsay, 1997; Perry, Howard, & Conroy, 1996; Perry, Howard, & Tracey, 1999). In the next section, important aspects of the research literature on beliefs and conceptions of mathematics teachers are examined. Mathematics teachers’ beliefs about teaching and learning One obstacle to progress in this domain has been the non-alignment of terminology used by mathematics education researchers. Schoenfeld (1992), in an attempt to develop a theory of mathematical thinking and problem solving, listed beliefs and affects as one of five aspects of cognition. Leder (1993) provided a definition of “affect” as a term used to: “Denote a wide range of concepts and phenomena including feelings, emotions, moods, motivation and certain drives and instincts” (Leder, 1993, p. I–46). Despite a lack of consensus among mathematics education researchers on the use of terminology, the affective domain is generally regarded as referring to constructs that, according to McLeod (1992), go beyond the cognitive domain, and that beliefs, attitudes and emotions can be considered as subsets of affect. Pajares (1992) cited a number of constructs which can be considered as subsets of the broadly defined “educational beliefs” term, the most commonly used being: teacher efficacy, epistemological beliefs, attributions, anxiety, self- concept, self-esteem, self-efficacy, and specific subject-matter beliefs. The clarification needed here is, according to Thompson (1992), a careful consideration of the concept of beliefs from both a philosophical and a psychological perspective. Green (1971) proposed a multidimensional perspective on the structure of beliefs – one that incorporated both philosophical and psychological constructs. He claimed that there are three dimensions of belief structures: the quasi-logical relation between beliefs, the central-peripheral dimension, and the premise that beliefs are held in clusters. Rokeach (1968, cited in Pajares, 1992) pioneered the introduction of the notion of belief systemsand his analysis included three assumptions: Beliefs differ in intensity and power; beliefs vary along a central-peripheral dimension; and, the more central a belief, the more it will resist change… Rokeach defined centrality in terms of connectedness: the more a given belief is functionally connected or in communication with other beliefs, the more central the belief… He proposed four assumptions for connectedness that form a set of priorities for the perceived importance of the self. Beliefs on an individual’s identity or self are more connected, as are beliefs one shares with others. Derived beliefs are learned from others; underived beliefs are learned by direct encounter with the belief object. Underived beliefs have more functional connections, partly because the: “I saw with ATypology of Mathematics Teachers’Beliefs about Teaching and Learning Mathematics 71 my own eyes” phenomenon is existential and connected to one’s sense of self. (p. 320) Pajares (1992) agreed with this connectedness notion, arguing that beliefs are prioritised according to their connections to other cognitive and affective structures. Mathematics teachers’ beliefs have an impact on their classroom practice, on the ways they perceive teaching, learning, and assessment, and on the ways they perceive students’ potential, abilities, dispositions, and capabilities. Van Zoest, Jones, and Thornton (1994) compared the beliefs about mathematics and mathematics teaching of pre-service primary mathematics teachers involved in a mentorship program to the beliefs of peer pre-service teachers not involved in such a mentorship program. The mentorship intervention was grounded on a socio-constructivist approach to mathematics instruction, and the treatment group teachers participated in small-group teaching activities. Van Zoest et al. (1994) used a variation of Kuhs and Ball’s (1986) model for their study. They found that the intervention group acted as if, initially, its views had been influenced by the socio-constructivist approach to mathematics instruction promoted by the mentorship program. Later on in the program however, their actions indicated that they favoured a more traditional set of beliefs about mathematics teaching. Perry and his colleagues (Howard et al., 1997; Perry et al., 1996; Perry et al., 1999) contributed to research on teachers’ beliefs by providing a model that aimed to describe the espoused beliefs of both primary and secondary mathematics teachers, as a result of a series of studies they conducted in Australia. They identified two factors, the transmission view and the child- centredness view, which they considered sufficient for a full description of the espoused beliefs of mathematics teachers. Perry et al. (1999) stressed that they considered the two factors making up their model as being distinct factors and not as the two ends of a continuum of one belief factor. Similarly, Fennema, Carpenter, Franke, Jacobs, & Empson (1996) conducted a study of mathematics teachers’ beliefs and instructional practices as they learned about students’ thinking and decided how to utilise that knowledge to enable them to finalise instructional decisions. They categorised teachers’ beliefs into four levels: • Level A: Teachers in this category believed that students learn best by being told how to do mathematics. • Level B: Teachers in this category were beginning to question the idea that children needed to be shown how to do mathematics, butthey have conflicting beliefs. 72 Barkatsas & Malone • Level C: Teachers thought that children would learn mathematics asthey solved many problems and discussed their solutions. • Level D: The beliefs of teachers in this category were characterised by the acceptance of the idea that children can solve problems without direct instruction and that the mathematics curriculum should be based on children’s abilities. The researchers reached the conclusion that with regard to the relationship between levels of instruction and beliefs; “there was no consistency in whether a change in beliefs preceded a change in instruction or vice versa” (Fennema et al., 1996, p. 423). This scheme has been used in the analysis of the case study in this project. Based on their research, Cooney and his collaborators (Cooney, Shealy, & Arvold, 1998) developed a characterisation of secondary mathematics teachers’ belief structures that has also been used in the analysis of the case study. These authors acknowledged the contributions earlier researchers have made on shaping the notion of reflection, and proposed the following partial scheme for the characterisation of teachers’ knowledge belief structures: • Isolationist:Teachers in this category tend to have belief structures insuch a way that beliefs remain separated or clustered away from others. Accommodation is not a theme that characterises an isolationist. • Naïve idealist:Teachers in this category tend to be received knowers in that, unlike the isolationists, they absorb what others believe to bethe case but often without analysis of what he/she believes. • Naïve connectionist:This position emphasises reflection and attention to the beliefs of others as compared to one’s own. The naïve connectionist fails, however, to resolve conflict or differences inbeliefs. • Reflective connectionist:This position emphasises reflection and attention to the beliefs of others as compared to one’s own. Thereflective connectionists, however, resolve conflict through reflective thinking. Cooney (1999) stated that the “inculcation of doubt and the posing of perplexing situations” (p. 173) would seem to be central to the shift from being a naïve idealist (or even an isolationist) to being a reflective connectionist. Inciting doubt and making the previously unproblematic problematic can have significant impact on a person’s world and could lead to varied and perhaps unsettling responses. ATypology of Mathematics Teachers’Beliefs about Teaching and Learning Mathematics 73 Relationship between teachers’ beliefs about mathematics and mathematics teaching and learning, and their instructional practice Research findings (Cooney, 1999; Ernest, 1989, 1989b; Nespor, 1987; Pajares, 1992; Thompson, 1992) suggest that teachers’ beliefs about the nature of mathematics and mathematics teaching and learning, have an impact on teaching practice. Some researchers have stressed the importance of teachers’ beliefs on the success or otherwise of contemporary mathematics curricular reforms (Askew, Brown, Rhodes, Johnson, & Wiliam, 1997; Pajares, 1992; Thompson, 1992; Van Zoest et al., 1994). Pre-service high school mathematics teachers experienced difficulties in the implementation of their espoused beliefs in practice, according to Parmelee (1992, cited in Van Zoest et al., 1994). Nisbet and Warren’s (2000) review of the relevant research investigating potential interrelationships between mathematics teachers’ espoused beliefs and their classroom practice offers some insights into the current state of affairs. For example: • the relationship between teachers’ beliefs and classroom practice isdynamic with each influencing the other; • teachers’ practices are shaped by their beliefs about the nature ofmathematics teaching and learning; • changes in teachers’ beliefs about teaching and learning are derived largely from classroom practice; and • changes in teachers’ beliefs about teaching and learning are influenced by the production of valued outcomes (i.e., student learning) resulting from classroom experimentation. (Adapted fromNisbet & Warren, 2000) Regarding the causes of disparities between teachers’ espoused (stated) andenacted beliefs (classroom actions), three possible explanations have been offered by Ernest (1989), namely: • the depth of the teacher’s espoused beliefs and the extent to which they are integrated with his/her other knowledge and beliefs, especially pedagogical knowledge; • the level of the teacher’s consciousness of his/her beliefs and the extent to which the teacher reflects on his or her teaching practice; and • the influence of the social context and its effect on teachers’ actions and behaviours. Raymond (1997) proposed a model of relationships between teachers’ mathematics beliefs and their teaching practice which was used in this study. The model is presented in Figure 1 and provides a useful starting point and a way of structuring discussion. 74 Barkatsas & Malone Past school Teacher education Social teaching Teacher’s life experiences program norms outside school Mathematics Mathematics beliefs teaching practices Immediate classroom situation Personality traits ofthe teacher Early family Students’ lives experiences outside school Indicates strong influence Indicates moderate influence Indicates slight influence Mathematics beliefs:About the Students’lives:Home environment, natureof mathematics and parents’ beliefs (about children, school, mathematics pedagogy and mathematics) Mathematics teaching practices: Teacher education program: Mathematical tasks, discourse, Mathematics content courses, field environment, and evaluation experiences, student teaching Immediate classroom situation: Past school experiences:Successes Students’ (abilities, attitudes, and inmathematics as a student, past behaviour) time constraints, the teachers mathematics topic at hand Early family experiences:Parents’ Social teaching norms:School view of mathematics, parents’ philosophy, administrators, educational background, interaction standardised tests, curriculum, with parents (particularly regarding textbook, other teachers, resources mathematics) Teacher’s life:Day-to-day occurrences, Personality traits:Confidence, other sources of stress creativity, humour, openness to change Figure 1. Raymond’s (1997) model of the relationships between teachers’ mathematics beliefs and their teaching practice. ATypology of Mathematics Teachers’Beliefs about Teaching and Learning Mathematics 75 The model suggests complex relationships between mathematics teachers’ beliefs and their practice, and it contributes to a better understanding of some factors that are considered to be important when teachers attempted to explain the existence of inconsistencies between them. Raymond (1997) suggested that the main causes of the inconsistencies reported in her study were not so much teachers’ beliefs, but school and classroom practices. The cumulative effect of these other influences (i.e., past school experiences, teacher education program, personality traits of the teacher, social teaching norms, etc.) was the main cause of inconsistencies between beliefs and practice. The manifestation of teachers’ beliefs in mathematics classrooms would be likely to be influenced by: (a) the teacher’s prior experiences of learning mathematics, classroom teaching, peer interactions, awareness of research results in mathematics education, using mathematics in other life experiences, teacher education programs; and (b) by various socio-historical, socio-economic, socio-political, and socio- cultural factors. Purpose of the study and research questions The central concern of this study was to contribute to an understanding of secondary mathematics teachers’ beliefs about mathematics, mathematics teaching, mathematics learning and their teaching practice. To explore these complex affective factors, this study utilised both qualitative and quantitative research methods. The research questions were as follows: (1) What are the beliefs of Greek secondary mathematics teachers with regard to mathematics as a discipline, and to the learning and teaching of mathematics? Specifically, does there exist a typology of mathematics teachers’ beliefs that correspond to the categorisations postulated in the research literature? (2) Are there inconsistencies between the teaching practice of Greek secondary mathematics teachers and their beliefs about mathematics and mathematics learning and teaching? An extensive search of the literature revealed that no studies investigating the relationship between teachers’ beliefs about mathematics and mathematics teaching and learning, or their instructional practices, had been conducted in Greece. Methodology This section describes the methods used to investigate the research questions. The study was conducted in two parts. The first part was a qualitative survey and the second part a case study of an experienced teacher. Part 1: Survey Asurvey was administered to explore the areas of interest and to produce data to examine the research questions. The survey was sent to a random selection of six hundred grade 7–12 mathematics teachers. The return rate 76 Barkatsas & Malone was 78%, (n = 465) and the resulting sample comprised 465 (276 males, 145 females, 44 no gender specified) participants, which included 431 secondary mathematics teachers (244 males, 143 females), 24 secondary mathematics teachers holding a principal’s position (22 male, 2 female), and 10 regional mathematics consultants (10 males), in 45 State High Schools throughout Greece. The returned surveys reflected a reasonably well- balanced distribution of grade-level experience. The 34-item questionnaire, designed by the first author, was administered during 1999–2000 in State High Schools in Greece, and covered the following participant demographics: gender, age, professional development background, length of teaching experience, position held, postgraduate studies background, and beliefs about mathematics, mathematics learning, and mathematics teaching. A Likert-type scoring format was used, and teachers were asked to indicate the extent to which they agreed (or disagreed) with each statement presented. A five point scoring system was used – strongly disagree (SD) to strongly agree (SA). A score of 1 was assigned to the SA response and a score of 5 to the SD response. The survey contained three subsets of items: Beliefs about Mathematics (BM: 6 items), Beliefs about Mathematics Learning (BML: 7 items) and Beliefs about Mathematics Teaching (BMT: 21 items). Aspace was also provided for teachers to comment on any aspect (of each part) of the instrument and its items. Part 2: Case Study In the second part of the study, a case study focused on the work of a veteran secondary mathematics teacher in Greece named Ann (pseudonym). Data were collected through lesson observations and pre- and post- lesson interviews. Excerpts from the case study are reported in a later section. The analyses drew on her own words as well as on detailed analyses of her lessons and the pre- and post- lesson interviews. Transcripts of audiotapes and videotapes were used to ensure that the analysis was based as closely as possible on the reality of the classrooms observed. Professionals transcribed the dialogue as completely and accurately as the equipment would permit, and the final transcript was double-checked by the first author. The transcription process was completed when a professional translator, an Englishman who has lived and worked in Athens as a translator for thirty years, translated all Greek transcripts into English. These were in turn double-checked by the first author using the back-translation technique and by paying special attention to the accuracy of mathematical nomenclature. Verification procedures When the analysis of the videotaped sessions was completed, Ann was given a copy and she was asked to comment on its veracity and accuracy. She expressed no disagreement or concern with the analysis. Two further checks ATypology of Mathematics Teachers’Beliefs about Teaching and Learning Mathematics 77 of the transcripts of the videotaped sessions were administered. Randomly selected sections were analysed by another experienced coder (a Senior Lecturer in mathematics education at the University of Athens) for inter- coder reliability. The entire analysis was then re-examined “blind” by the first author to provide an intra-coder reliability check. Data analysis Questionnaire responses regarding beliefs about mathematics, mathematics teaching and mathematics learning were analysed using SPSSwin. Principal Components Analysis, Cluster Analysis, Multiple Discriminant Analysis, and Trends Analysis were used in the study. In what follows, only the Principal Components Analysis results are presented. Detailed analyses of the extensive statistical procedures used in the study can be found in Barkatsas (2003). Principal Components Analysis Aconfirmatory Principal Components Analysis (PCA) was used in order to interrogate the 34 questionnaire items for a typology of teachers’ espoused beliefs. Given the exploratory nature of the study and guided by the Analysis of Variance (ANOVA), the scree plot and the interpretability of the factors, a five components orthogonal solution (Table 1) was accepted after the extraction of principal components and a Varimax rotation. The solution accounted for 39% of the variance, and 22 of the 34 items were used to delineate the components. For this study, the Kaiser-Meyer-Olkin Measure of Sampling Adequacy was 0.679, and Bartlett’s Test of Sphericity was significant (p < .01), so factorability of the correlation matrix was assumed. The analysis yielded 13 components with eigenvalues greater than 1. Given the exploratory nature of the study and guided by the interpretability of the components, as well as the scree plot, a five-component orthogonal solution was accepted after the extraction of principal components and a Varimax rotation. The PCA depicted in Table 1 followed the elimination of psychometrically “poor” items. Variables loading on more than one component were eliminated, Cronbach’s alpha coefficient of internal consistency was used to ensure that the items comprising the resulting components produced a reliable scale, and a normality test was carried out to assess normality among single variables by examining their skewness and kurtosis. If a variable has substantial skewness or kurtosis, it needs to be transformed (Tabachnick & Fidell, 1996). In this study, no variable had any substantial skewness or kurtosis, so no transformation of variables was necessary. The rotation method was a Varimax with Kaiser Normalisation. The rotation converged in seven iterations. 78 Barkatsas & Malone Table 1 Components Related to Views about Mathematics, Mathematics Teaching and Mathematics Learning. Item Item description Loading Factor 1:Asocio-constructivist orientation to mathematics, mathematics teachingand mathematics learning 24 It is important for students to be provided with opportunities .594 to reflect on and evaluate their own mathematical understanding 14 Ignoring the mathematical ideas generated by the students can .594 seriously limit their learning 32 The education system should be preparing critically thinking .583 citizens who are able to utilise their mathematical skills 17 Mathematics teachers should be fascinated with how students .482 think and intrigued by alternative ideas 16 Teachers should encourage their students to strive for elegant .385 solutions when they solve problems 1 Justifying the mathematical statements that a person makes 306 is an important part of mathematics Factor 2:Adynamic problem-driven orientation to mathematics, mathematics teaching and mathematics learning 13 Mathematics learning is enhanced by challenging activities within .650 a supportive environment 30 The comprehension of mathematical concepts by students should .608 correspond to their cognitive development and it should be a decisive factor in the content sequence to be taught 34 Teachers should respect the mathematical knowledge of their .558 students, which is consisting of a nexus of experiences, beliefs, attitudes, representations, concepts, strategies, connections, values, judgements and emotions 6 Mathematics is a beautiful, creative and useful human endeavour .463 that is both a way of knowing and a way of thinking 19 Teachers always need to hear students’ mathematical explanations .447 before correcting their errors 4 Mathematics is the dynamic searching for models and problems .358 and their results are open to review Factor 3:Astatic-transmission orientation to mathematics, mathematics teaching and mathematics learning 17 The most effective way to learn mathematics is by listening .718 carefully to the teacher explaining a mathematics lesson