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This thesis is making a case for prospective teachers’ mathematical problem-solving proficiency to be foundational in their education and training as teachers. As with much of learning, what one believes about a topic holds sway in the construction of new knowledge. This is evident in most theories of learning. Feiman-Nemser (2001) has argued that beliefs play a pivotal role in making sense of new knowledge. She notes that the knowledge and experiences, encountered by prospective teachers, is also cumulative. It is further documented, by authors such as Thompson (1992) and Pajares, (1992) that beliefs greatly influence prospective teachers’ assumptions of learning as well as their pedagogy. Either these beliefs could serve as an aid in the learning process, or as a barrier, depending on the exposure to

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mathematics teaching they received during their primary and secondary school career. Prior experiences, such as being a learner in a primary- and secondary mathematics classroom and the influence of teacher preparation programmes are key in the development of mathematics beliefs (Raymond, 1997). Beliefs could thus be viewed as part of the “apprenticeship of observation,” of which Lortie (1975) writes. Pedagogical practice and, with it beliefs, build up though the many years that student teachers have spent in school as learner. To that, their teacher education studies will add.

As with definitions of mathematical problem-solving, there are various terms used to describe beliefs (Pajares, 1992) and consensus is rare (McLeod & McLeod, 2002). Phillips (2007) suggested that the notion of ‘beliefs’ is such a widely used term in mathematics education research, that researchers do not deem it necessary to define it explicitly. Researchers mostly assume that readers know what ‘beliefs’ refer to (Thompson, 1992). Further to that, according to Pajares (1992), one cannot separate knowledge from beliefs. He proposes that there is a confluence of a learner’s epistemology (view of knowledge) and ontology (worldview). Ribiero (2006) would refer to this idea as ‘footing.’ Raymond (1997) defines mathematical beliefs as “personal judgments about mathematics, formulated from experiences in mathematics, including beliefs about the nature of mathematics, mathematics learning and teaching mathematics.” Phillip (2007), defined beliefs as:

Psychologically held understanding, premises, or propositions about the world that are thought to be true. Beliefs are more cognitive, are felt less intensely, and are harder to change than attitudes. Beliefs might be thought of as lenses that affect one’s view of some aspect of the world or as dispositions toward action. Beliefs, unlike knowledge, may be held with varying degrees of conviction and are not consensual. Beliefs are more cognitive than emotions and attitudes (Phillip, 2007, p. 259).

Beliefs are ‘more cognitive’ than emotions and attitudes, for it is impossible to separate beliefs and knowledge, which suggests that beliefs are epistemological. (Fennema & Franke, 1992). There are many aspects that influences how student teachers view the nature of mathematics, and more specifically mathematical problem-solving, which, of course, is based on what they know and how competent they are in problem-solving itself. I am of the view that someone who is an ‘ace’ in problem-solving has strong beliefs and that these may differ from someone

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who is anxious about problem-solving and whose self-esteem is low. This invokes psychological components too – which goes beyond the scope of this discussion. But, I would argue, beliefs (with self-concept) originates in students’ many years at school, where they were inundated with content in the primary school and did not build a strong foundation of early mathematics conceptual knowledge and when they were in such dire need to construct such a foundation of knowledge (Fritz, et al., 2013). Pre-conceived ideas were formed through primary- and secondary school experiences, personal life, and by the teacher education practicum and theoretical knowledge gained from university courses.

The greatest influences on how student teachers interpret and construct mathematical knowledge for teaching has been student teachers’ beliefs formed through school experiences with mathematics. Lortie (1975) pointed out that teaching is the only profession that children witness for 12 years or more and that it influences their thinking and their pedagogy. They were ‘apprentices of observation’. Petker (2018) pointed to the fossilized notions student teachers hold, and which they find hard to revise, despite much practice and theory learning. By now it is well documented that preservice teachers do not enter teacher preparation programmes as ‘pedagogical blank slates’ but with preconceived beliefs or understandings of what teaching is (Darling – Hammond, 2006) and more specifically mathematics teaching (Kennedy, 1991; Ball, 1988, Brown & Cooney, 1982). As already mentioned, such beliefs are a consequence of observing and experiencing teaching throughout childhood and adolescence. The ‘dosage’ of this apprenticeship of observation (Lortie, 1975) has thus been intense.

As learners in primary and secondary schools, preservice teachers do not only learn mathematics content knowledge and teaching strategies (Handal, 2003) but also images of teachers’ behaviour and beliefs about teaching (Korthagen & Lagerwerf, 1996). The apprenticeship of observation is the first ‘informal method’ of learning to teach on the continuum of teacher learning and it also frames how student teachers understand what and how they learn to teach in teacher preparation programmes. Feiman -Nemser (2001), introduced the ‘continuum of teacher learning perspective’, meaning when one looks at teacher learning in one phase, the whole spectrum of teacher learning should be considered. According to her, the spectrum of teacher learning consists of three phases; (1) learning opportunities encountered as learners in both primary and secondary school, (2) as students in preservice teacher education programmes and then (3) the ongoing learning in and through practice as a professional teacher. This perspective can be compared to similar models of Snow, Griffin and Burns (2005), Darling Hammond, and Bransford (2005). Feiman-Nemser suggests that

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designers of teacher learning programmes should remain mindful of not only the long-term professional outcomes, but also should take cognisance of student teachers’ learning histories. For, whatever happens in learning in one phase, such as the initial teacher education phase, should be informed and will be influenced by what has been learnt before. Students come to university with a personal learning history. It has to be taken into account.

This was the origin of my thesis study: Students had to unlearn fixed examples of maths pedagogy and, with that; they had to learn much maths content knowledge, the bulk of which, as I have argued so far in this thesis, would be based on problem-solving.

I agree with Darling-Hammond (2006), referring to the apprenticeship of observation as one of three persistent challenges in learning to teach, because learning to teach requires from new teachers to firstly “understand teaching quite different from their own experience as students” (p. 305) at school. In terms of learning how to teach mathematics, the apprenticeship of observation influences more than just student teachers’ understanding of teaching, but also how they understand mathematical concepts, procedures and the nature of mathematics (Ball, 1988). This means that preservice teachers’ understanding of teaching is directly influenced by their school teachers’ beliefs about the nature of mathematics, their beliefs about teaching, as well as their beliefs about learning.

In a study of teachers’’ philosophical views, Ernest (1989) found that teachers held three main philosophical views about the nature of mathematics, namely, an instrumentalist-, a Platonist- and a problem-solving view. According to Ernest (1989), teachers with an instrumentalist view of mathematics see mathematics as a set of rules and skills utilised to reach a specific goal. In this view, sub-strands that make up mathematics curricula are approached in silos - teachers do not emphasise the interconnectedness of topics. The ‘Platonist’ view of mathematics sees maths as a “static but unified body of certain rules” (p. 250) that should be ‘discovered’. These teachers do not approach various mathematics topics in silos, but view the interconnectedness of various mathematics topics as key in understanding mathematics. The teachers who hold a problem-solving view of mathematics perceive maths as a body of knowledge that is open for autonomous investigation and discovery. I believe that teachers’ beliefs about the nature of mathematics have a direct influence on how teaching is viewed and how maths is learned. The teacher is therefore seen as chiefly an instructor in the instrumentalist view, as largely an explainer in the Platonist view and as a facilitator for learning in the problem-solving view (Ernest, 1989). I would argue that this distinction may have held true for teachers 30 years ago,

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but that maths may be perceived differently by current teachers, or, at least that there may be elements of all three types in one teacher and in her practice. Teachers cannot teach if they do not engage with instructional explanation and devising interaction that gives freedom to explore. One sometimes instructs, sometimes explains and sometimes leads learners to make discoveries. Beswick (2005), illustrated showed how the beliefs about the nature of mathematics (Ernest, 1989), beliefs of mathematics learning (Ernest, 1989) and categories about the beliefs of teaching mathematics (Van Zoest, Jones & Thorton, 1994) are connected (Table 4). The table is, to my mind, somewhat reductionist, however.

Table 4:

Categories of teacher

Beliefs about the nature of mathematics (Ernest, 1989)

Beliefs about mathematics teaching (Van Zoest et al, 1994)

Beliefs about mathematics learning (Ernest, 1989)

Instrumentalist Content focussed with an emphasis on performance

Skill mastery, passive reception of knowledge

Platonist Content focussed with an emphasis on understanding

Active construction of understanding

Problem-solving Learner focussed Autonomous exploration of own interest

According to Beswick’s (2005) summary, the three different views have implications for teaching of mathematics in the following ways:

1. From an instrumentalist perspective, the teaching of mathematics by the instructor is primarily focused on content coverage with an emphasis on performance. Learning means the mastery of skills and knowledge is passively acquired.

2. Similarly, to the instrumentalist view the Platonist view implies that the teaching of mathematics is focused on content coverage but also emphasising the importance of understanding. Learning of mathematics means the active construction of understanding.

3. The problem-solving view implies the teaching of mathematics is learner centred and not content centred. Learning is an autonomous activity of one’s own interest.

Beswick (2005) further explained that teachers may not neatly fit into one category but could be bordering on more than one category. Not only do practicing teachers border on these categories of beliefs but student teachers do as well, because they are products of the teaching

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methods and beliefs of their schoolteachers. I do, surmise, though, that these categories are too narrow.

On this view, however, Ball (1990) found that most student teachers enter teacher education with an instrumentalist view of mathematics. Ball (1990, p. 460), stated that the student teachers’ main beliefs about mathematics was that “doing mathematics means following set procedures step-by-step to arrive at answers; knowing mathematics means knowing "how to do it"; and mathematics is a largely arbitrary collection of facts”. Consequently, these beliefs influenced how the student teachers approached and solved the mathematical tasks (Ball, 1990).

Student teachers’ beliefs about the nature of mathematics may also influence the learning of mathematics and their dispositions towards problem-solving (Kloosterman & Stage, 1992). Studies capturing student teachers’ mathematical problem-solving beliefs indicated that there are student teachers who hold positive beliefs about problem-solving and there are also those who perceive of problem-solving as following the correct procedures and doing calculations (Hart, 2002).