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Teacher education research has recently focused on the correlation between teachers’ beliefs and their teaching practices (Fang, 1996). The relationship between the two aspects, however, is not directly one of cause-and-effect (A. Thompson, 1992, p. 140). There are different ways of teaching and learning mathematics as introduced first, and each area will have its unique view of the effective way, or the expected teaching approach in the second.

With regards to what teachers believed, there are two models of mathematics teaching in teacher education research which closely correspond with each other (A. Thompson, 1992). The first model of mathematics teaching identified by Kuhs and Ball (1986) which describes four distinctive categories of teacher beliefs: learner-focused; content-focused with an emphasis on conceptual understanding; content-focused with an emphasis on performance; and classroom-focused. In the second model of mathematics teaching, Ernest (1989)

categorised three key elements on which the practice of teaching depended: (1) teachers’ role in the intended outcome; (2) the use of curricular materials; and (3) the enacted model of learning mathematics (see Table 4).

Table 4 Ernest’s Model Key elements of mathematics Patterns Teacher’s role in intended outcome

1. Instructor: skills mastery with correct performance 2. Explainer: conceptual understanding with unified

knowledge

3. Facilitator: confident problem posing and solving The use of curricular

materials

1. The strict following of a text or scheme

2. Modification of the textbook approach, enriched with additional problems and activities

3. Teacher or school construction of the mathematics curriculum

Enacted model of learning mathematics

1. Mastery of skills model

2. Reception of knowledge model

3. Active construction of understanding model

4. Exploration and autonomous pursuit of own interests model In line with the work of A. Thompson (1992), I linked the two models with

philosophical backgrounds as shown in Table 5. Table 5

Models of Mathematics Teaching

View on teacher beliefs Philosophy background Teacher’s role Learning Learner-focused Constructivist Facilitator

Active construction of understanding model Content-focused with an emphasis on conceptual understanding

Platonist (Ernest, 1989) Explainer Reception of knowledge model Content-focused with an emphasis on performance Instrumentalist view of the nature of mathematics

Instructor Mastery of skills model Classroom-focused from

teaching effectiveness studies

Not from any learning theory Classroom activities Active construction of understanding model In the rise of constructivism in Western education community, it is well accepted that knowledge is constructed by learners themselves rather than transmitted from teachers (Hoy, Hughes, & Walkup, 2008). Along with these two main learning theories, the learning

classroom instruction. Later, Swan (2005) summarised two extreme teaching approaches: transmission and challenging (see Figure 3). In the transmission approach, methods will be explained step by step and the teacher plays key role on the learning direction. This teaching approach fit with the belief of mathematics as pure maths. Swan (2005) also argued that the transmission approach can only be effective in short-term. The short-term learning outcome from students of the teacher-centred lesson would achieve significantly better than student- centred where the motivation was higher (Sturm & Bogner, 2008). Furthermore, Swan (2006) proposed to develop a collaborative orientation towards teaching instead of teacher-centred or transmission pedagogic practice for England through five teaching activities: classifying mathematical objects, interpreting multiple representations, evaluating mathematical statements, creating problems, and analysing reasoning and solution.

Figure 3.Two views of teaching (Swan, 2005, p. 5)

Between the extreme teacher-centred teaching approach and the learner-centred one, Land, Hannafin, and Oliver (2012, p. 8) suggested four core values and assumptions: ‘(a) centrality of the learner in defining meaning; (b) scaffold participation in authentic tasks and sociocultural practices; (c) importance of prior and everyday experiences in meaning

construction; and (d) access to multiple perspective, resources, and representations’. Pampaka et al. (2012) divided the middle into three levels: Level 1, students-centered connectionist practice; Level 2, medium – teaching practices from both ends; Level 3, teacher-centred, transmissions, fast pace, exam orientated. Here, connectionism has two aspects: (a)

connecting teaching to students’ mathematical understanding and productions; and (b) connecting teaching and learning across mathematics’ topics, and between mathematics and other subject knowledge. Although the distinct teaching style cannot easily be identified by English large scale, qualitative studies (Askew, 2001), the connectionism is regarded as the desirable teaching approach in England.

In China, effective teaching and learning mathematics has been summarised by a theory named teaching with variation (Gu, Huang, & Marton, 2004) which explains the paradox of Chinese learners (discussed in the first chapter) from mathematics classroom perspective. The abstract mathematics concepts are built upon concrete and perceptual experience, while teaching with variation connects the experience and the concept.

Teaching with variation, this pedagogy has two forms: conceptual variation and procedural variation. The former one focuses on mastering the essential features of the

mathematics concept by two forms: concept variation and non-concept variation. The concept variation varies the multiple perspectives of the concept (see Figure 4), standard figures and non-standard figures. Particularly, the non-standard figures provide different orientations to enhance the understanding of key characteristics of the concept. Another set of examples are related with the figures which do not belong to the concept, as called non-concept variation. Then, the essential of the concept could be understood by comparing with the non-concept examples (see Figure 5) or counterexamples. Teaching with non-concept variation can help student build upon the relationships between related concepts and clarify the confusion students might have. In case of linear function in the Shanghai textbook, a set of exercises requires the students to distinguish which algebraic expression belongs to linear function: , , , and . These samples are mainly of non-concept variation.

The conceptual variation, however, regards the concept as a static object during the teaching and learning process. Each concept might also have its own development process which the procedural variation emphasises on.

Figure 4. Examples of the concept variation(Gu et al., 2004, p. 317)

Figure 5. Non-concept example in terms of the concept of vertical opposite angles The assumption that teaching with procedural variation will be effective is that experiencing the formation process of the concept helps students understand the concept step by step. For example, teaching the concept of equation can be formed by three scaffoldings: ‘representing the unknown by concrete things’; ‘symbolizing the unknowns’; and ‘replacing unknown x with symbolic “□”’ (Gu et al., 2004, p. 321). Meanwhile, Gu et al. proposed a framework to explain how procedural variation is applied for problem-solving (see Figure 6). For example, after learning alternate angles, corresponding angles and co-interior angles and angles in a triangle, students are expecting to solve problems from the first known-problem in Figure 7 to the rest of three unknown-problems through procedural variation for problem- solving teaching method.

Figure 6. Procedural variation for problem-solving (Gu et. al., 2004, p.322)

Figure 7. Examples of procedural variation for problem-solving

In sum, the teaching belief or effective teaching approach in England priorities students, and different activities; while Shanghai or China tends to focus on the mathematics itself.