Practice Community Identity Meaning
2.3 Dimensions of teacher knowledge
2.3.2 Pedagogic Content Knowledge (PCK)
Shulman (1986) stated that subject knowledge and the actual skill of teaching are dependent on each other which he called ‘pedagogical content knowledge’. The diagram in figure 2.3.2 illustrates the relationship between PCK, content knowledge and curricular knowledge as exemplified by Shulman (1986).
Figure 2.3.2 Shulman’s component of knowledge
As expected content knowledge is very closely linked to PCK and it is inevitable that a discussion of one aspect, will entail a discussion of the other, which is the case in the description of PCK that is presented. The subject matter content knowledge involves knowing more than knowledge of the facts, but also to understand the structure of the subject matter. Subject matter knowledge also refers to the teacher’s professional knowledge.
Similarly, PCK refers to an in depth understanding of how the subject matter is to be taught.
“The most useful forms of representation of ideas, the most powerful analogies, illustrations, examples, explanations, and demonstrations -in a word- the ways of representing and formulating the subject that makes it comprehensible to others.”
(Shulman,1986b, p.7).
Pedagogical content knowledge also includes an understanding of what makes the learning of specific topics easy or difficult: the conceptions and preconceptions that students of different ages and backgrounds bring with them to the learning of those most frequently taught topics and lessons. One will find that all this expertise come with years of experience and a wealth of knowledge. The teacher must therefore know what misconceptions learners have and be able to use different skill and strategies in imparting the subject matter and giving meaning to
CONTENT KNOWLEDGE
SUBJECT MATTER
CONTENT KNOWLEDGE PCK CURRICULAR KNOWLEDGE
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it. Shulman (1986, p.15) describes “PCK as the capacity of a teacher to transform the content knowledge he or she possesses into forms that are pedagogically powerful and yet adaptive to the variations in ability and background presented by students.” In relation to curricular knowledge teachers must be able to teach the content at different levels and be able to build up the content from the preceding year to the next according to the level of difficulty that is required at that level. Shulman (1986, p.9) also emphasises the importance of knowing the content but then making it comprehensible to others.
In the following year Shulman (1987) characterized professional knowledge into the following categories:
• general pedagogical knowledge, with special reference to those broad principles and strategies of classroom management and organisation that appear to transcend subject matter.
• knowledge of learners and their characteristics.
• knowledge of educational contexts, ranging workings of the group or classroom, the governance and financing of school districts, to the character of communities and culture.
• knowledge of educational ends, purposes, and values and their philosophical and historical grounds.
• content knowledge.
• curriculum knowledge, with particular grasp of the materials and programs that serve as ‘tools of the trade’ for teachers.
• pedagogical content knowledge, that special amalgam of content and pedagogy that is uniquely the province of teachers, their special form of
Professional development. (Shulman, 1987, p.8)
These categories emphasise the importance of content knowledge and situates it in terms of all the knowledge that a teacher needs to know for effective teaching in the classroom.
Shulman also made it clear that all categories could not exist without each other, they were interdependent when he said “mere content knowledge is likely to be useless pedagogically as content free skill.” (Shulman, 1987, p.8). Content knowledge consists of knowledge of the subject and its organising structures (Grossman, Wilson, & Shulman, 1986b, 1987).
Curricular knowledge is “represented by the full range of program designed for the teaching of particular subjects and topics at a given level, the variety of instructional materials available in relation to those programs, and the set of characteristics that serve as both the indications and contraindications for the use of particular curriculum or program materials in particular circumstances.” (Shulman, 1986, p.10). “A teacher requires proper subject matter knowledge and a high level of pedagogic content knowledge to assure effective teaching.”
(Shulman,1986, Ma,1999). This theory is also supported by (Kilpatrick, 2001; Taylor, 2008) 27
when they say that “the most fundamental aspect in effective and proficient teaching of mathematics is a high level of knowledge.”
Ball (1991) states that teachers subject matter knowledge interacts with their assumptions and explicit beliefs about teaching and learning, about students and about context to shape the ways in which they teach their subject to the students. Therefore one can see the relevance of pedagogic content knowledge and its importance for quality education in the classroom. This can only be achieved when the teacher has mastered the art of knowing the subject content, and the manner in which the students can learn the subject content.This implies that the teacher must use strategies that are applicable to that particular content and that particular group of community of students or class. Hill, Rowan and Ball (2005) investigated both specialized content knowledge and skills used in teaching and found that “teachers’
mathematical knowledge was significantly related to student achievement gains in both first and third grades.” (Hill, Rowan and Ball, 2005, p.1). Given the extensive research supporting the importance of instructor knowledge, it is clear that the professional development plan must address the issue of content and pedagogical knowledge for all mathematics teachers.
Shulman (1986, p.9) describes Pedagogical content knowledge as “...subject matter knowledge for teaching...the ways of representing and formulating the subject that makes it comprehensible to others.” This theory is also reinforced by Johnson & Hodges and Monks (2000, p.185-186) who says “the actual classroom practice the teacher uses for a particular group of students on a particular day with a topic can only be selected from the teachers stock of pedagogical content knowledge.”
Grossman (1990) classifies PCK into four components as depicted in figure 2.3.3 by the diagram below
Figure 2.3.3 Grossmans classification of PCK
In the first component the teacher has to map ideas that will enable him to make decisions with regards to classroom objectives, the strategies he is going to use, assignments /work that is going to be given to the learners, curricular materials and assessment of learners. This comprises the teacher’s professional knowledge base, since it comprises the knowledge about the subject as well as what is important and necessary for the learners. The second component concerns the teachers’ ability and knowledge about what mathematics the learners will understand, how he will teach it in order for them to understand and through his expertise, what misconceptions the learners would have. Like Grossman, Leach & Moon (2008) stated
PCK
PURPOSE FOR TEACHING MATHEMATICS
LEARNERS UNDERSTANDING, CONCEPTIONS AND
MISUNDERSTANDINS
CURRICULUM AND CURRICULAR MATERIALS
INSTRUCTIONAL STRATEGIES AND REPRESENTATIONS FOR TEACHING TOPICS
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that pedagogy encompasses many facets namely strategies of teaching, arrangement of classrooms, formulating questions and developing schemes of work. These authors mention that content ispecific pedagogy does not mean that teachers acquire a possession that they keep forever. It is ever changing. Leach & Moon (2008) argue that a pedagogic setting has a past, a present and a future.
This knowledge is clearly evident in the research by Sowder, (2007, p.165) “... plan more effectively because they can anticipate learners difficulties. They know what prior knowledge must be present to understand... how to scaffold knowledge to assist learners in developing understanding....how to listen to students. Much of this knowledge comes from practice...” In the third component teachers are expected to use all resources according to their strengths and weaknesses. They need to be able to integrate the knowledge of what they have to teach across the grades as well as within a grade and also be able to link the subject matter to other subjects. In the subject ML the issue of designing activities that are linked to the curriculum is important. Contexts which help learners develop decision skills in real life, are crucial tools in trying to achieve complexities in ML. Thus a key competence of an ML teacher is the ability to design suitable, relevant and meaningful tasks which can contribute to this end.
Hence ML teachers need to be able to produce clear and unambiguous tasks which allow learners to investigate issues that they encounter. Lastly the fourth component consists of the teacher’s ability to adapt representations according to the specific needs and goals of individual learners.
Teacher content knowledge and pedagogical knowledge have also been shown to have a profound effect on the manner in which teaching and learning takes place. Not only is a teacher’s deep understanding of mathematical content important, but his/her pedagogical knowledge also plays a key role in student learning. Koency and Swanson (2000) found that studies in classrooms with high expectations and challenging mathematics suggest that
“teacher knowledge of mathematical content is a key factor that underlies the quality of classroom instruction.” (Koency and Swanson, 2000, p.3).
Adler and Reed (2002, p.25) indicate that subject knowledge on its own is insufficient but
“the issue is how to integrate further learning of the subject with learning about how students in school acquire subject teaching.” A teacher must be able to take the knowledge he or she has learnt and be able to deliver that content to the type of community of learners he teaches.
Benita Nel’s theory is closely allied to Wenger’s belief that “knowledge is about competencies related to valued enterprises is reflected in the inclusion of knowledge that is considered to be essential in a mathematics community.” In her explanation of this statement she goes onto say that teachers are able to take what they have learnt in the four modules of ML and are able to integrate the content within a context that pertains to a community of participants and thus engaging them in real life activities. Similarly Graven wrote that
“mastery of the profession of mathematics teaching involves mastery of particular epistemic demands relating to mathematics and pedagogy.” (Graven, 2002, p.165)
Sfard and Cole(2003) also asserts that in order for a teacher to be regarded as being mathematically literate, the teacher has to be able to use the two types of mathematical discourses both when the initiative comes from others or from their own accord in any situation in which the discourse can be helpful. This describes the ‘how’ and then ‘when’ of
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literate mathematical discourse. They assert that “.... when it comes to poor results in developing mathematical literacy, the school is found to be the culprit.” (Sfard and Cole, 2003, p.10). They also stated that successful teaching and learning of mathematics for the purpose of promoting literate discourses can best be achieved by changing school practices such as “discontinuing the practice of using mathematics as a tool for measuring human potential.” (Sfard and Cole, 2003, p.10). Hence the development of the competency mathematical literacy is dependent on the teachers’ skill and practices and this requires sound content knowledge and PCK.
Taylors (2008) through his research found that most teachers in literacy and mathematics did not have the knowledge that the curricula demanded to teach the learners proficiently. Hence there was a need to emphasise expert knowledge as the teaching of mathematics requires content knowledge, knowledge of the curriculum and the methodology in terms of how learners learn.
Ball, Thames & Phelps (2008) also emphasise the fact that pedagogical content knowledge bridges “content knowledge with the practice of teaching ,assuring that discussions of content are relevant to teaching and that discussion of teaching retain attention to content.” (Ball, Thames, Phelps, 2008, p.3).
A fundamental difference between the teaching of mathematics and the teaching of ML is the role of the context. In ML, engagement with one context, requires the application of more than one type of subject matter content for example area of a rectangle and costing. An activity set around finding the cost to tile a rectangular surface would involve knowledge of area of a rectangle, as well as working out conversions based on units of measurements.
Context forms an important part of ML. From a South African context, in a recent study by Thembela T.E. (2012), found that “teachers noted the difference between teaching and learning of mathematics and ML with regards to their teaching practice … context form a huge part of ML problems, and therefore teachers need to be creative and spontaneous in designing problems whose context fits with their respective learners.” (Thembela T.E., 2012, p.103). Venkat (2010) remarks that “a life-preparation orientation, in which contextualisation in everyday life situations is central is a prevalent feature of the ML curriculum.” (Venkat, 2010, p.55). Similarly Christiansen (2006) critiques the tension between the contextual teaching approach and the mathematically organized Mathematical Literacy curriculum in the following quote: By claiming that it is about life-related topics, the curriculum renders the underlying organizing principles of the content invisible to the learners (and possibly to some teachers, too), who therefore will not learn mathematics, unless the teacher is in a position to ensure coherence and progression of mathematical concepts. The learner who thinks that AIDS is the topic, when generally it is about reading graphs, will get it wrong, yet the curriculum does not encourage that the learner is given the necessary guidance to develop the mathematical concept of graphs. (Christiansen, 2006)
Hence, a curriculum based on these objectives and competencies would assist in developing the learner’s skills in interpreting, assessing, using and applying numerical information in real life contexts. Hector (2011) also stated that “the training with respect to teaching practice needs to include training with respect to curriculum implementation, lesson planning, lesson presentation and the setting of assessment tasks”. (Hector, 2011, p.149). Nel (2009) also noted in her study that “a definite shift took place from being unable to introduce a concept to a more definite approach” (Nel, 2009, p.51).
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Content knowledge and PCK are dependent on each other and are very closely linked. ML is a complex subject which deals with the real world. Hence teachers, with the knowledge that they possess, have to develop contexts and have the capacity to transform the content knowledge and present it to the learners. Innovative teaching, keeping in mind the multicultural and diverse society within which teachers’ work, the language barriers faced by teachers and learners and the misconceptions that occur while reading a context is a challenge for the teacher within the South African context. These are some of the aspects researched.
The formation of tasks for class based exercises and assessments are all part of PCK that a teacher needs to know and the task is to find how the teacher has developed professionally in terms of PCK.