Las competencias comunicativas

Teacher educators are key mediators in shaping and enacting the curriculum of teacher education and it is, therefore, crucial to know what the philosophical underpinnings of their pedagogy are. In this vein, Flores (2016, pg. 214), argued that, in addition to gaining a better understanding of initial teacher education within its political, social and economic context, the following questions should be asked: “Who are the teacher educators?” “What are their views about teaching, teacher education and learning?” and “How do they connect key components of ITE curriculum in their practice explicitly?” Moreover, the philosophy of training held by teacher educators as well as how these philosophies are articulated in policy documents and expressed in practice should be thoroughly scrutinized (Flores, 2016). Loughran (2006) argued that there is an assumption that good teachers will make good teacher educators, but that this might not be the case because the pedagogy of teacher education has some distinct features and there is not yet consensus on what constitutes a good teacher. Indeed, the pedagogy of teacher education has distinct features and therefore unique expectations. One of the main expectations

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is to model and illustrate innovative mathematics teaching practices, especially mathematical problem-solving practices.

This study’s perspective on the modelling of problem-solving in teacher preparation programmes is similar to the notion of Collins, Brown, and Holum (1991), whose cognitive apprenticeship model of learning was first published by Collins, Brown and Newman (1989). In the cognitive apprenticeship model, the ‘master’ or expert plays an important role and the teaching activities include – in some sequence, modelling, scaffolding, coaching and fading.

Modelling as educator of apprentices refers to students who observe while the expert performs the task; while the students observe they need to construct a conceptual model of the processes needed to complete the task. Coaching: the aim of coaching is to bring students closer to expert performance through specific guidance, giving clues and feedback. Scaffolding refers to the support that is provided by the teacher to assist students completing the task. Fading refers to the process where the support is slowly removed until students are able to complete the task on their own.

However, the pedagogy of initial teacher education has been widely critiqued as not seeing student teachers as apprentices. Some labelled it as a ‘weak intervention’ (Feiman-Nemser, 2001) because teacher educators’ pedagogical practices do not illustrate and model the reform teaching practices about which they ‘preach’. Teacher educators’ pedagogy is one of the main challenges faced by ITE, specifically also in South Africa (Taylor, & Bowie, 2014). Feiman- Nemser (2001) argue that the pedagogy of ITE “mirrors the pedagogy of higher education where lectures, discussions, and seat-based learning are coins of the realm. Too often teacher educators do not practice what they preach. Classes are either too abstract to challenge deeply held beliefs or too superficial to foster deep understanding” (Feiman - Nemser, 2001, pg. 1020). Similarly, Korthagen et al (2006), critiqued teacher educators for not practicing what they preach by asserting that teacher educators should not merely promote innovative teaching practices but they should model and illustrate these innovative teaching practices in their own lecture rooms. However, strides have been recently made in mathematics content courses to model mathematical problem-solving practices.

2.7.1 Mathematical problem-solving courses in teacher preparation programmes

The study conducted by Guberman and Leikan (2013) was aimed at developing Israeli elementary student teachers’ mathematical thinking through a problem-solving course. The mathematical problem-solving course was structured around mathematical tasks, specifically

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multi-solution tasks because the study was underpinned by the NCTM’s (2000) notion of problem-solving. The NCTM (2000, p.52) view problem-solving as a process requiring of students to engage in “tasks for which the solution method is not known.” The multi-solution tasks selected in Guberman and Leikin’s (2013) study required various solution paths, solvers could solve the tasks in multiple ways using different problem-solving heuristics or strategies. Thus, solutions for the mathematical tasks could be illustrated with different mathematical representations, properties, tools or theorems.

The structure of the course was such that the students were encouraged to solve as many mathematical problems as possible from the textbook in many different ways. These problems were used as ‘lead-in-activities’ and student teachers had 30 minutes to work on these activities. The purpose of these activities was for student teachers to develop mathematical connections between representations of the mathematical concepts, different mathematical tools and concepts from the same content strand, and different mathematical topics. In addition to these ‘lead-in-activities, students had to engage with thirty multi-solution tasks, spanned across the course, as well as solve three pre-test problems and two post-test mathematical problems which were unconventional in terms of the Israeli textbooks. The pre-and post-test problems involved multi-solutions across different concepts and topics.

Findings of Guberman and Leikin’s (2013) study suggested the mathematical problem-solving course was successful in developing student teachers’ problem-solving competences. The student teachers became more competent in solving mathematical problems, as evident in the significant changes in their solution strategies. Their solution strategies included more advanced mathematics concepts and different representations. The student teachers adapted to mathematical problem-solving process and they started to make connections between mathematical problems, concepts and properties.

In a study conducted at a Singapore university, Lam et al. (2013) redesigned a mathematics content course for secondary school student teachers. The mathematics content course was an introductory Differential Equations course. Lam and his colleagues (2013) decided that it would be best to incorporate the pedagogy of problem-solving into the traditional lecture- model, by modelling how to teach mathematical problem-solving. They were motivated by mathematical problem-solving at the centre of the Singapore mathematics curriculum for primary school up until pre-university. Polya’s (1945) problem-solving framework was equally promoted in the Singapore mathematics curriculum at school level. The study was based on

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two questions; first whether the student teachers would be able to apply the problem-solving heuristics learnt in the methods course to solve mathematical problems; and second, to what extent the secondary school student teachers would adopt the “the look back and check” phase in Polya’s (1945) problem-solving framework. Thus, Polya’s (1945) stages of problem-solving was a guiding principle for the redesign of the mathematics content course. Polya’s (1945) framework together with Schoenfeld’s (1945) framework found expression in the mathematical practical lessons (Lam et al., 2013; Toh et al., 2011b). The mathematical practical lessons were designed to assist the student teachers “develop the mental habit of following through with a problem-solving model”, especially when they ‘get stuck’ in the problem-solving process (Lam et al., 2013, p. 103).

The sequence of the practical lessons was that the tutor would start by highlighting Polya’s (1945) and Schoenfeld’s (1985) problem-solving framework. This was done so that the student teachers could understand that problem-solving does not only require subject matter knowledge but also heuristics, useful beliefs and metacognitive control (Toh et al., 2011b). Thereafter the tutor demonstrated and discussed how different problem-solving strategies could be implemented to solve problems (Toh et al., 2011b). This was followed by an illustration of how to use the practical worksheets by solving a specific problem. After the discussion and demonstration students had to solve reasonably challenging problems referred to as the problem of the day in 40 minutes. The tutor then went over the problem while the students engaged in peer-marking (Lam et al, 2013). All in all, student teachers had to solve six reasonably difficult problems, which were assessed.

Lam et al. (2013) explained that it was important for them to assess the practical worksheets because if they did not assess it the students would not be motivated to learn through problem- solving because they were of the view that student teachers only work hard on what is formally assessed. Furthermore, the assessment of the practical worksheets served as an indication to student teachers that mathematical problem-solving is valued in mathematics education. Taking these aspects about assessment into consideration the assessment of the practical worksheets played a crucial role in the redesign of the mathematics content course. Thus, student teachers’ performance in the practical lessons constituted a significant percentage of the continuous assessment of the course. The assessment instrument, a rubric consisted of four main components: (1) applying Polya’s 4-stage approach to solving mathematical problems, (2) making use of heuristics, (3) exhibiting ‘control’ during problem-solving, (4) checking and expanding the problem (Toh, et al., 2011b, p. 105).

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The overall findings suggested all the student teachers, except one, were generally able to respond with relevant heuristics. The one student teacher presented an incorrect solution with incorrect heuristics for one of the two most difficult problems, which required proofs that are more challenging. Similarly, many student teachers could not complete stage four (check and expand) for these difficult problems but they could solve the other problems. Lam et al. (2013) concluded that the redesign of the mathematics content course was beneficial for the student teachers’ learning of mathematics. Students were able to learn the same amount of mathematics content as with the traditional lecture mode but they also learnt to use problem-solving heuristics to solve university level mathematics problems. An important point to take into consideration is that these student teachers were able to apply various heuristics even though they were not always able to get the correct solution. They were also able to expand on the problem once they found the solution to the problem. Lam et al.,’s (2013) study was the beginning of a vision to prepare student teachers who will be able to teach mathematical problem-solving in Singapore. More specifically prepare teachers who as confident problem solvers.

2.8 TEACHER EDUCATORS: PERSONIFYING MATHEMATICAL THINKING AND