The Cycle of Teachers’ Mathematical Beliefs

This article examines the nature and role of teachers’ mathematical beliefs in instruction. It is argued that teachers’ mathematical beliefs can be categorised in multiple dimensions. These beliefs are said to originate from previous traditional learning experiences mainly during schooling. Once acquired, teachers’ beliefs are eventually reproduced in classroom instruction. It is also argued that, due to their conservative nature, educational environments foster and reinforce the development of traditional instructional beliefs. Although there is evidence that teachers’ beliefs influence their instructional behaviour, the nature of the relationship is complex and mediated by external factors.

Educationalists have attempted to systematize a framework for teachers’ mathematical belief systems into smaller sub–systems. Most authors agree with a system mainly consisting of beliefs about 

(a) what mathematics is, 

(b) how mathematics teaching and learning actually occurs, and

 (c) how mathematics teaching and learning should occur ideally (Ernest, 1989a, 1989b; Thompson, 1991). 

Certainly, the range of teachers’ mathematical beliefs is vast since such a list would include all teachers’ thoughts on personal efficacy, computers, calculators, assessment, group work, perceptions of school culture, particular instructional strategies, textbooks, students’ characteristics, and attributional theory, among others. The concept of progressive instruction is associated with a socio-constructivist view of teaching and learning mathematics. Socio-constructivism, which for the sake of brevity will be called just constructivism, gives recognition and value to new instructional strategies in which students are able to learn mathematics by personally and socially constructing mathematical knowledge. Constructivist strategies advocate instruction that emphasizes problem-solving and generative learning, as well as reflective processes and exploratory learning. These strategies also recommend group learning, plenty of discussion, informal and lateral thinking, and situated learning (Handal, 2002; Murphy, 1997). In turn, traditional instruction is associated with a behaviourist perspective on education. Behaviourist practices are said to emphasise transmission of knowledge and stress the pedagogical value of formulas, procedures and drill, and products rather than processes. Behaviorism also puts great value on isolated and independent learning, as well as conformity to established one-way methods and a predilection for pure and abstract mathematics (McGinnis, Shama, Graeber, & Watanabe, 1997; Wood, Cobb, & Yackel, 1991). 

Leder (1994) stated that in the behaviourist movement “the mind was regarded as a muscle that needed to be exercised for it to grow stronger” (p. 35). The study of teachers’ instructional beliefs and their influence on instructional practice gained momentum in the last decade. Some research on teachers’ thinking reveals that teachers hold wellarticulated educational beliefs that in turn shape instructional practice (Buzeika, 1996; Frykholm, 1995; McClain, 2002; Stipek, Givvin, Salmon, & MacGyvers, 2001; Thompson, 1992). Examples of research, as reviewed in this paper, have also shown that each teacher holds a particular belief system comprising a wide range of beliefs about learners, teachers, teaching, learning, schooling, resources,

knowledge, and curriculum (Gudmundsdottir & Shulman, 1987; Lovat & Smith, 1995). These beliefs act as a filter through which teachers make their decisions rather than just relying on their pedagogical knowledge or curriculum guidelines (Clark & Peterson, 1986). In fact, these beliefs appear to be cogent enough to either facilitate or slow down educational reform, whichever is the case (Handal & Herrington, 1993, in press). The literature also shows that there are internal and external factors mediating beliefs and practice (Pajares, 1992). This dissonance bears serious implications for the implementation of curricular innovations since teachers’ beliefs may not match the belief system underpinning educational reform. Even if teachers’ beliefs match curricular reform, very often the traditional nature of educational systems make it difficult for teachers to enact their espoused progressive beliefs. In contrast to linear and static approaches to curriculum implementation, modern perspectives look at how teachers make sense of educational innovations in order to re-appraise an ongoing and always flexible process of implementation (Handal & Herrington, 2003).

How do teachers’ mathematical beliefs originate? In part, teachers acquire these beliefs symbiotically from their former mathematics school teachers after sitting and observing classroom lessons for literally thousands of hours throughout their past schooling (Carroll, 1995; Thompson, 1984). This process parallels in many respects the apprenticeship style of learning that takes place while learning a trade. Traditionally, tradesmen learn by observing a master doing a particular job (Buchmann, 1987; Lortie, 1975). In the schooling process, students learn not only content-based knowledge but also instructional strategies as well as other dispositions. By the time the aspirant is admitted to a teacher education program, these beliefs about how to teach and learn are deeply embedded in the individual, and very often are reinforced by the traditional nature of some teacher education institutions which may not have positive effects on preservice teachers’ mathematical beliefs (Brown & Rose, 1995; Day, 1996; Foss & Kleinsasser, 1996; Kagan, 1992; McGinnis & Parker, 2001).

There is evidence that, in some cases, teacher education programs are so busy concentrating on imparting pedagogical knowledge that little consideration is given to modifying these beliefs (Tillema, 1995). Consequently, teacher education programs might have little effect in producing teachers with beliefs consistent with curriculum innovation and  research (Kennedy, 1991). For example, Marland (1994) found that reasons given by inservice teachers regarding their classroom strategies were not related to what was actually taught in their college training. There is also some evidence confirming that teachers’ decision making does not rely solely on their pedagogical knowledge but also on what they believe the subject-matter is and how it should be taught (Brown & Baird, 1993; Laurenson, 1995; Prawat, 1990). These beliefs are also difficult to change (Borko, Flory, & Cumbo, 1993) and very often conflict with educational innovations, threatening educational change (Brown & Rose, 1995; Fullan, 1993). As discussed in the next sections, there are also a number of external factors influencing teachers’ beliefs.

The context of school instruction obliges practicing elementary and secondary teachers to teach traditional mathematics even when they may hold alternative views about mathematics and about mathematics teaching and learning. Parents and professional colleagues, for example, expect teachers to teach in a traditional way. Teachers are also expected to focus on external examinations, to adhere to a textbook, and to keep a low level of noise and movement in their classrooms. In such environments, even teachers with progressive educational beliefs are forced to compromise and conform to traditional instructional styles (Handal, 2002; Perry, Howard, & Tracey, 1999; Sosniak, Ethington, & Varelas, 1991). Other accountable factors are ethnic background, social class origins, experience living in other cultures, gender issues, and prior styles of teaching experience (Butt & Raymond, 1989; Raymond, Butt & Towsend, 1991).

Thompson (1984) argued that teachers, in the exercise of their practice, and because of the large number and diversity of interactions, tend to develop quick responses to types of episodes, which in time become patterns in their instructional repertoire. McAninch (1993) reviewed a body of literature showing that teachers are very practical in their approach to pedagogical tasks. Jackson’s (1968) interviews revealed that teachers tend to be “confident, subjective, and individualistic in their professional views” (cited by McAninch, 1993, p. 7). In addition, Doyle and Ponder (1977) and Lortie (1975), both cited by McAninch (1993), described “teachers as pragmatic in their decision making…and intuitive in their approach to problem solving” (p. 7). Moreover, teaching is seen as a highly practical and utilitarian profession where teachers quickly label innovations as practical or impractical, depending on whether the teacher considers that the proposal will work for him or her. Success of innovations was also found to be related to a teacher’s personality and teachers were found to emphasize the peculiarities of their classroom over the generalizations of innovations.

Nespor (1987) adds that, given the unpredictability and uniqueness of classroom events, teachers have to resort to their own beliefs, particularly in pedagogical situations when formal knowledge is not available, is disconnected, or cannot be retrieved. In Nespor’s words, “When people encounter entangled domains or ill-structured problems, many standard cognitive processing strategies such as schema-abstraction or analytical reduction are no longer viable” (p. 325). This type of situation is characteristic of classroom teaching. In general, teaching is a decision-making based activity in which teachers have to make an interactive decision every two minutes (Brown and Rose, 1995; Clark & Peterson, 1986; Lovat and Smith, 1995).

In brief, the teaching job places great external demands on decisions that teachers have to make rapidly, in isolation, and in widely varied circumstances. These demands put teachers in the position of resorting to practicability and intuition as indispensable resources for survival in the profession. These demands in turn favour the development of beliefs about what works and what does not in a classroom. At the same time, it seems that teachers generate their own beliefs about how to teach in their school years and these beliefs are perpetuated in their teaching practice. Thus, educational beliefs are passed on to the students.