Research questions on the relationship between beliefs and instructional practices fall most often within one or more of the following specific questions (Ernest 1989):
(i) Are there differences between a teacher’s professed beliefs and his/her instructional practices?
(ii) How can these differences be explained?
(iii) Are different instructional practices related to different beliefs?
This study relates to (i) and (iii) as it investigates the teachers' espoused beliefs and their alignment with the aims and objectives of the national curriculum.
The research questions in the reviewed studies, and in this study, relate to (i) and (iii) above. The participants in the reviewed studies are predominantly in-service and/or secondary school Mathematics teachers. This study involved secondary school Mathematics teachers as participants.
Research by Cross (2009) on five in-service teachers revealed that in general, beliefs were very influential in their day-to-day teaching strategies and that their beliefs about the nature of Mathematics were the primary source for beliefs about the teaching and learning of Mathematics. The study went further and investigated how teachers’ beliefs hindered or supported the implementation of reformed-orientated teaching practices. The study concluded that the participating teachers all held strong beliefs that stemmed from their own schooling experience. In addition, a number of the participants held beliefs that did not align with the aims and objectives of their requisite curriculum, that is, the aims of the National Council of
Teachers of Mathematics (NCTM). These teachers emphasised the use of practice and not the use of reasoning and problem-solving in their teaching as required by their curriculum. Although the study concluded that the beliefs held by the teachers shaped their instructional practices, it is noted that research has been inconclusive regarding the degree of influence that beliefs have on instructional practices. There is no linear relationship between beliefs and practices and other factors do influence classroom practices. Additionally, contradictions between beliefs and practices are often noted where teachers espouse reformed-orientated beliefs which are not evident in their teaching practices (Thompson 1984; Stipek et al. 2001; Beswick 2007; Cross 2009). Cross (2009) cautions that studies into the relationship between beliefs and instructional practices should not only consider the verbalised beliefs of the teacher but should also seek evidence in classroom practices. Further, Cross (2009) notes in her study that beliefs about the teaching and learning of Mathematics differ according to the ability of the group being taught and also differ according to the section of the syllabus being covered. These contradictory views are evident in other studies (Fuchs & Fuchs & Hamlett & Karns 1998; Torff & Warburton 2005). Such conflicting beliefs are often upheld by a third belief, as discussed in section 2.3. Similar contradictory beliefs were evident in this study and are discussed further in Chapter 5.
Cross (2009) concluded that for learners to become active, creative, think critically and able to solve problems, it is necessary for teachers to possess beliefs that support a learning environment which is learner-centred and would develop problem-solving skills. Cross concludes that teachers who do not hold these beliefs should be enrolled in programmes that can develop these beliefs, more specifically their beliefs about the nature of Mathematics. Belief change should be an ongoing process of reflection, confrontation and awareness of one’s beliefs. Pre-service and in-service training can begin the process of changing beliefs but teachers must be confronted continuously with experiences that challenge their beliefs as only then is belief change likely to become permanent. Further, targeting pre-service and in-
service teachers’ Mathematics content knowledge rather than teaching methodologies is more likely to enact the desired belief change, as courses focusing on Mathematics methodology have not had the desired effect (Cross 2009).
An investigation of secondary Mathematics teachers who held beliefs that were consistent with the tenets of constructivism was conducted by Beswick (2007). Again, the study investigated the nature and relationship between beliefs and whether these beliefs were a predictor of classroom practice. Beswick acknowledges the complex nature and relationship
between beliefs and instructional practices. In addition, even the direction of influence is disputed by a number of studies, as some studies suggest that teaching practice influences the beliefs of the teacher and vice versa (Handel 2003; Beswick 2005; Beswick 2007).
Beswick notes that very few studies into teachers’ beliefs and the relationship between beliefs and practice have focused on improving Mathematics education. Her study identifies nine beliefs which are broadly held by teachers who create constructivist learning environments promoting problem solving. The study hypothesises that enacting these nine beliefs would directly improve the learning environment in Mathematics classrooms, and in so doing bring about the required reforms. The nine beliefs were grouped by Beswick (2007: 114) within the three belief dimensions as described previously in this chapter. A summary of this grouping is given below.
(i) Beliefs about the nature of Mathematics:
1. Mathematics is about connecting ideas and sense making: a problem-solving view of Mathematics;
2. Mathematics is fun. This leads to a confident and genuine interest in Mathematics.
(ii) Beliefs about the teaching of Mathematics:
3. It is the teacher’s responsibility to control the classroom discourse. In other words, it is the teacher’s responsibility to make sure that the construction of knowledge takes place in the classroom;
4. It is the teacher’s responsibility to facilitate the construction of mathematical knowledge;
5. It is the teacher’s responsibility to induct learners into the accepted ways of thinking and communicating in Mathematics;
6. The teacher is the authority with respect to acceptable behaviour that is expected of a learner (social norms) in the classroom;
7. It is the teacher’s responsibility to engage in continued professional development.
(iii) Beliefs about the learning of Mathematics:
8. Students’ learning is unpredictable. This belief is required to create an environment in which knowledge is individually and socially constructed; 9. All learners can learn Mathematics.
Beswick argues that teachers holding these nine beliefs are more likely to create a constructivist-learning environment. The nine beliefs can be categorised under Ernest’s (1989) problem-solving philosophical beliefs about the nature of Mathematics as cited by Beswick (2005; 2012). This again links the constructivist learning theory with the problem- solving view about the nature of Mathematics.
Research of teachers’ beliefs most often investigates the beliefs held on the continuum between a traditional beliefs and a reformed problem-solving beliefs about instructional practices. In theory, traditional instruction is associated with a behaviourist learning theory, while a progressive/reformed instruction is associated with a socio-constructivist theory of learning. In traditional instructional practices, rote learning of formula, rules and procedures is emphasised. Learning is an independent and isolated event with knowledge transmitted from teacher to learner. In contrast, socio-constructivist instructional practices emphasise problem-solving, metacognition and discovery. Learning happens collaboratively (socially) with the learners actively involved in their learning with teachers facilitating the discussions and guiding the construction of knowledge (Handel 2003).
Handel (2003) argues in his study that teachers’ beliefs originate from their own traditional schooling and that the beliefs reproduced in the classroom environment are due to the conservative or traditional nature of schools which in turn reinforces the traditional beliefs. Teachers’ beliefs affect their teaching. Therefore, a teacher’s beliefs will shape the way a teacher thinks and feels about Mathematics and the teaching and learning of Mathematics (Stipek et al. 2001; Oksanen & Hannula 2013). If a teacher’s beliefs do not support the aims and beliefs of curriculum reforms, then this might hinder the implementation of these
reforms. Handel (2003), however, adds that even if teachers’ beliefs match those of the reform curricula, the traditional nature of the schooling system will make it difficult for the teachers to enact their espoused beliefs. The relationship between beliefs and practice is complex and a number of external factors also influence practices. These include pressure from school and parents, lack of preparatory time to cover content and the challenges posed by different learner abilities (Handel 2003).
A number of studies involving pre-service teachers found that they largely hold beliefs which are traditional in nature. For example:
Mathematics is either right or wrong (Benbow 1993; Nisbet & Warren 2000); Mathematics requires neatness and speed (Civil 1990);
Mathematics requires logic and not intuition (Frank 1990);
The learning of Mathematics is based on the memorisation of facts and rules (Lappan & Evan 1989; Wood & Floden 1990; Southwell & Khamis 1989; Benbow 1993; Foss & Kleinsasser 1996);
Mathematical ability is innate (Frank 1990; Foss & Kleinsasser 1996).
Further, a study by Howard, Perry and Lindsay (1997) with 249 secondary in-service teachers in Sydney, Australia, showed two distinct belief groupings of teachers. The larger group could be associated with an instrumentalist/traditionalist view of the teaching and learning of Mathematics while the second smaller group held more constructivist views associated with the problem-solving view of the teaching and learning of Mathematics. A similar grouping in many of the beliefs held by the participants in this study was also noted. In addition, other studies involving in-service teachers also concluded that the participants largely hold traditionalist beliefs about the teaching and learning of Mathematics (Handel & Herrington 2003; Handal & Bobis 2004).
In addition to these influencing factors, researchers have suggested that professional
development programmes designed to reform teachers’ beliefs have largely been ineffective, as teachers filter what they learn through their existing beliefs (Cohen 1990; Stipek et al. 2001).
In conclusion, the reviewed studies on the relationship between beliefs and classroom practices found that the relationship is complex in nature, the participating teachers largely hold traditional beliefs, and traditionalist views could be associated with traditional practices. However, all studies concluded that teachers’ espoused beliefs were not always evident in observations of their classroom practices because of other prioritised beliefs and/or external factors.