2. CHAPTER 2: Literature Review and Theoretical Framework 13
2.3. Theoretical Framework
2.3.2. Pedagogic Content Knowledge
For purposes of this study, PCK is used interchangeably with content specific pedagogy.
Whilst the two are two separate concepts, PCK is seen as the ability by the teacher, to present the content known by him/herself to learners, in a way that will make such content to be understood and apprehended by the learners. It involves the ability of a teacher to become a reflective practitioner, who can look back into his/her own teaching and identify the strongpoints and weakpoints. Content specific pedagogy also allows for the integration of ideas in the content with real life situations that will be understood by the learners.
An important aspect of PCK in ML is task design. Tasks in ML serve a somewhat different purpose than Maths tasks. This is noticed by learners as well, as noted by Debba (2011) in his study when he stated that “in terms of the nature of classroom tasks, learners viewed contextual tasks as more accessible, practical, „visualise-able‟ and providing opportunities for communication, participation and sense-making inside and outside school (Debba, 2011, p.11).
Peressini et al..assert that the selection of tasks given to learners is “situated in particular classrooms filled with students who bring with them different experiences and backgrounds”
(Peressini et al.. 2004, p.78). Bansilal & Wallace (2008) identified six issues of concern which should be considered in the design of tasks by both mathematics and ML teachers, if the tasks are meant to deal with real-life contexts. Three of these six concerns, which seem to be most relevant to ML are:
a) Knowledge Gaps: this refers to “a deficit in basic skills and foundational knowledge concepts for reasons largely beyond the control of the learners”
(Bansilal & Wallace, 2008, p.84). Such „gaps‟ are usually a result of an earlier misunderstanding by the learner and usually result in them failing to or struggling with any given task.
33 b) Task Language: the poor language skills of the learners may interfere with their efforts to solve a given task and may result in learners having a „skewed‟
understanding of the task requirements (Bansilal,& Wallace, 2008).
c) Information Overload: Learners usually struggle in trying to sift „context information‟ from „crucial information‟ (Bansilal & Wallace, 2008). The teacher must try and present tasks that are not overloaded with information that will not be useful in solving the mathematical problem.
The concerns raised by Bansilal and Wallace (2008) show the enormity of the task at hand for teachers who train to become ML teachers, where this one element of task design becomes crucial in their professional development. Task design forms a huge part of the PCK of ML teachers.
As alluded to earlier, ML strives to produce learners who are “self-managing individuals, contributing workers and participating citizens” (DoE, 2008, p.7). This is quite distinct from the way in which most of South African mathematics teachers have been trained. Bansilal (2012) confirms this view when she says that “the subject ML is not about learning more mathematics but about developing skills that will enable them (learners) to participate in (and not be excluded from) situations which use numerically based arguments” (Bansilal, 2012, p.2).
As opposed to Mathematics, tasks in ML provide learners with the ability to „see‟ what is being discussed and this leads to better understanding, as compared to mathematics tasks, which were “hard to visualise and the selection of procedures (was) often described in terms of random guessing rather than any notion of sense-making” (Venkat& Graven, 2008, p.37).
It is for this reason that Hechter (2011) noted that teachers of ML should be familiar with the GET curriculum in order to facilitate the „visibility‟ of the problem in a task in ML. This concern was also echoed indirectly by Pillay (2006), who stated that textbooks in ML always assumed that the learner was familiar with the work that was supposed to be done in the previous grades. Therefore task design in ML requires careful planning on the side of teachers, whose content knowledge and PCK should be well developed.
It is important to note that pedagogical content knowledge in ML is much more complex than it is in Mathematics. The way in which Mathematics is taught as well as the purpose of teaching that Mathematics, is quite different from the purpose and philosophy of ML.
34 A context can be seen as a contextual domain within which particular rules of engagement are recognized. Each context has specific attributes which need to be recognised and used in specific ways which may be different from those encountered in the traditional mathematics classroom. A learner must be able to understand these attributes in order to make meaningful decisions around the context (Bansilal & Debba, 2012).
Thus the teacher needs a sound understanding of the contexts that are specified in the curriculum. Furthermore the teacher will need to find ways of representing these attributes in ways which make it easier for their learners to understand.
Venkat& Graven (2008) point out that internationally, as well as in South Africa, there has been “evidence of low levels of confidence, disaffection and lack of engagement” (Venkat&
Graven, 2008, p.31) in the teaching and learning of Mathematics (and hence, ML). They further assert that even with critiques of the ML curriculum being very pessimistic about the introduction of the subject, the “negative experiences of mathematics learning to positive perceptions of ML learning seemed to us to merit attention” (Venkat& Graven, 2008, p.31). It is for this reason and others that focus should be made towards why ML teachers need to be developed professionally in their content knowledge, their PCK and their identity and beliefs about the subject ML. Learners in the Venkat& Graven study indicated that there is “more room for sharing ideas and discussing a range of alternative solution strategies” within an ML class, this due to a “more collaborative learning environment in ML” (Venkat& Graven, 2008, p.35). In line with the situative learning view, two aspects identified by Venkat&
Graven (2008) in a ML classroom were that “the nature of classroom tasks and the nature of classroom interactions in ML were in contrast to their (the learners‟) prior experience in mathematics” (Venkat& Graven, 2008, p.36).
For purposes of this study, I will consider PCK for ML teachers as the awareness of common learner misconceptions and knowledge gaps in elementary mathematics; and consider how teachers may be able to address these misconceptions. PCK also involves the knowledge of various contextual settings and rules that operate within those settings as well as having the skill to design tasks and the ability to set up classrooms which foster collaborative learning environments.
In conclusion, it is important to reiterate that with PCK being a complex phenomenon in ML, due to the nature of the subject, a strong need to explore the professional development of teachers along such lines is imperative. The complexity of this PCK lies in the relative ability
35 (or lack thereof) of the ML teacher to realise what is important in the lives of ML learners, as this facilitates better choices in designing tasks suitable for ML. Barriers such as language and decoding of complex situations are better handled when the teacher identifies with the subject. The intention of this study is precisely to find out how the professional development of teachers of ML was enhanced by their participation in the ACEML programme.