SISTEMAS DE SEGURIDAD

He who knows does, and he who teaches understands (Shulman, 1986)

Shulman’s (1986, 1987) work focused on teachers’ knowledge base for teaching, and this was necessitated by the way teacher competencies were being assessed in that pedagogical

9 _{It is interesting to note how different the Zambian algebra school curriculum is from say the new South}

African curriculum and curriculum elsewhere. For example, in Zambia, the algebra curriculum at grades 10, 11 and 12 has not been impacted on in terms of functions as they emerge when technology such as the graphic calculators are used. The use of this technology illuminates functions and their transformations. In the South African algebra and functions curriculum, and curricular elsewhere, this is integrated.

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aspects were in focus while content knowledge had disappeared. He alluded to the fact that in the late 1800’s, 90-95% of teacher competence tests covered content knowledge and the remainder were pedagogical in nature. From the experiences reflected in the teachers’ autobiographies, it became evident that for a teacher to competently teach any subject matter to learners, they needed to demonstrate knowledge of that subject matter as a prerequisite to teaching. With the eroding of the curriculum content in evaluation items designed to assess teachers’ competencies in the 1980’s, Shulman (op cit) wondered where the subject matter had gone and what had happened to it. As a result, Shulman (op cit) argued that knowing some teaching procedures was not enough, one needed to know what it is that they were going to teach.

The strong alignment to pedagogical practices as a basis for assessing teachers’ competences was research driven in that research of that time focused on what was considered to be “good practice”, i.e. patterns of teacher behaviour that would bring about improved academic performance among learners. Shulman (1986) argued that while it was good to implement practices which are informed by research, it was equally important to realise that research has its own constraints in terms of narrowing the scope, focusing the view, and formulating questions in an effort to simplify a complex reality. Subject matter was used for subdividing data sets rather than considering it in its own right. There was no research that focused on how subject matter was transformed from the knowledge of the teacher into the content of instruction, and on how particular formulations of that content related to what learners came to know or misconstrue (although cognitive research on learning grappled with the idea).

Therefore, Shulman and his colleagues considered the missing link between subject matter and various studies that focused on teaching as the “missing paradigm” problem – hence problematic for both policy and research. It was problematic in the sense that policy makers were inclined to pedagogical practices rather than subject matter in formulating policy because that was what research had focused on. Sole focus on pedagogy was also problematic for research in that the studies about teaching did not focus on how the teachers organised content knowledge in terms of what questions are asked, and what explanations are given during the teaching and learning process.

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To substantiate his claim for the missing paradigm, Shulman (1986) made reference to the history of the academy, i.e. medieval university in the 1950’s, in that content and pedagogy were not separated, implying that what is known (content) was not separate from how to teach it (pedagogy). This indicates that understanding of content was related to how well one was able to teach it. Therefore, understanding was viewed as a demonstration of “what is known” (content) as well as “how it is taught” (pedagogy) – hence a connection between knowing and teaching. Going by Shulman’s (op cit) concern, what is it that teachers are able to teach if assessing their competencies is pedagogically inclined? How can one access that which they are teaching? I ask this because a competent teacher should be able to demonstrate that they understand what they are teaching in ways that would benefit their learners. I therefore agree with Shulman’s (op cit) argument that teacher competencies cannot be assessed pedagogically devoid of subject matter – the two complement each other. What this means for my study is that as student-teachers engage in the discourse of and about LMT, their understanding of school algebra in terms of content and pedagogy would come to the fore.

To address the imbalance that was realised in assessing teacher competencies, Shulman and his colleagues (1986) embarked on a longitudinal study entitled “Knowledge Growth in Teaching” – how teachers’ knowledge developed over time. They engaged with student- teachers that had finished their teaching programme three-quarter way and some were followed in their first year of full time teaching to track their “intellectual biography”, and their study was qualitative in nature. The researchers assumed that most teachers were experts in the subject matter they taught; as a result, they were concerned with the transition from expert student to novice teacher – and asked how that knowledge was transformed into a form which their learners could understand, hence the crux of PCK. Therefore, Shulman’s (op cit) focus of their study raises a question: How does knowing the subject matter and the teaching of it unfold in my study.

Krainer & Goffree (1999) as pointed out by Adler et al. (2005) identified four types of research related to teacher education. Firstly, research in the perspective of teacher education with a focus on teachers’ mathematical beliefs, teachers’ knowledge and aspects of teaching. Krainer and Goffree argue that although this kind of research is not directly linked to empirical studies in teacher education, the findings would inform the design of programmes

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in teacher education. Secondly, research in the context of teacher education that focuses on teachers’ professional development learning, establishing the gap between what is learned in pre-service training and the work they do in school, and tracking what changes in terms of teachers’ beliefs and practices. In this instance, the understanding is that such research could inform teacher education but that teacher education itself is not the object of focus in research. Thirdly, research on teacher education, where studies are directly linked to teacher education with a focus on the interaction practices going on in teacher education. Fourthly, research as teacher education in that research is the focus as a way of enhancing teacher development. Examples of such studies include all forms of action research and reflective practice.

From the four categories of research in teacher education identified, I argue that Shulman’s (1986) concept of PCK emerged from the study of teachers in the practice of teaching, and not from the study of teacher education. Shulman followed teachers’ knowledge growth while they were in practice. The difference with my study is that I want to establish the preparedness of final year mathematics education student-teachers. Thus locating my study in the first category of research in teacher education with a focus on what mathematical knowledge student-teachers know and enact when they engage in the discourse of and about LMT in general, and algebra in particular. The findings would illuminate mathematics teacher education in Zambia in relation to the discourse. This study, therefore, would make an original contribution by focusing on what and how the discourse of and about engaging with LMT is recognized and focused on in the mathematics education courses, and in the context of school algebra.