TEACHERS’ KNOWLEDGE AND PEDAGOGICAL CONTENT KNOWLEDGE

1. TEACHERS’ KNOWLEDGE AND PEDAGOGICAL CONTENT KNOWLEDGE

2. LEARNING OUTCOMES: • State teachers knowledge, and • Describe PCK orientation to mathematics teaching

3. TeacherS’ Knowledge “To teach is first to understand “ (Shulman, 1986) “If the promise of the teaching profession is to achieve, we must attend to the processes by which its knowledge base is developed and transmitted” (Howsam et al. (1976)

4. KnowledgeBaseof Teaching • What knowledge base? • Is enough known about teaching to support a knowledge base? • Isn’t teaching little more than personal style, artful communication, knowing some subject matter, and applying the results of recent research on teaching effectiveness? • Policymakers and Teacher Educators : • Basic skills • Content knowledge • General pedagogical skills

5. Models of Teacher Knowledge • Elbaz (1983) – 5 components: a) knowledge of self b) knowledge of the milieu of teaching c) knowledge of subject matter d) knowledge of curriculum development e) knowledge of instruction • Leinhardt & smith (1985)- 2 components a) knowledge of subject’s matter b) knowledge of lesson’s structure

6. Models of Teacher Knowledge Shulman (1986) – 7 categories of knowledge • Content knowledge • General pedagogical knowledge • Pedagogical contents knowledge (PCK) • Curriculum knowledge • Knowledge of learners and their characteristics • Knowledge of educational context • Knowledge of educational ends, purpose, and value and their philosophical and historical grounds

7. Content knowledge (Subject matter knowledge • Substantive structure: Knowledge of the major facts, concepts, principles within a field and the relationships among them. • Syntactic structure: Knowledge regarding methods, rules of evidence and proofs in that domain and into how knowledge is being evaluated by the discipline’s experts. The amount and organisation of knowledge per se in the mind of the teacher.

8. General pedagogical knowledge Principle and strategy of classroom management as well as its organisation that arises in the delivery of the subject matter. - Example: understand how students learn, theories of learning, child psychology, teaching strategies, classroom management, assessment, etc....

9. Knowledge of Curriculum, aims and objectives • Curriculum knowledge: -Particular grasp of the materials and programmesthat serve as ‘tool of the trade’ of teachers. - CDC provides Mathematics teachers with syllabuses of the KBSM Mathematics and Additional Mathematics along with the curriculum specification. • Aims and Objectives: - Need to understand the aims and objectives of the mathematics curriculum – the planned teaching activities are in tandem.

10. Knowledge of learners and their characteristics • The needs of learning basic mathematical concepts • Learners difficulties • Learners misunderstanding • Learners misconception These knowledge involve conceptual and procedural knowledge, conceptual errors and level of understanding. • Need to know techniques in evaluating learners’ understanding and diagnosing misconception/appropriate learning strategies • Need to know students’ learning style (imaginative, analytical, practical and dynamic learner)

11. Knowledge of Educational Context • Knowledge of school, classrooms and all setting where learning takes place ( districts, school, communities and cultures). • Grossman (1990) – Contextual knowledge includes knowledge of the area where the teacher teaches like the area’s aspiration, expectations and limitations.

12. Knowledge of Educational ends, purposes, and values, and their philosophical and historical grounds. • Educational ends • Purposes of teaching and learning • Values • Philosophy of teaching mathematics • Historical ground

13. Pedagogical contents knowledge (PCK • Domain of knowledge that was different from both knowledge of the content and general knowledge of teaching (Shulman,1986). • Knowledge formed through the synthesis of three knowledge bases: content knowledge, pedagogical knowledge and contextual knowledge – unique mixture of pedagogy and content.

14. Pedagogical contents knowledge (PCK) Teacher’s deep understanding of a subject area she/he must also be able to foster understanding of subject or concepts for students ) Pedagogical knowledge Content knowledge PCK PCK also include: the most useful forms of representation of those ideas, the most powerful analogies, illustrations, examples, explanations, and demonstration.

15. Pedagogical Content Knowledge (PCK) The most regularly taught topics in one’s subject area, the most useful forms of representation of those ideas, the most powerful analogies, illustrations, examples, explanations, the ways of representation and formulating the subject that make it comprehensible to others (Shulman, 1986) Bridge that a teacher builds to link his or her understanding of the content to that of the students understanding of the same content (Grossman, et al. 1989) PCK A unique knowledge to the teacher and is the fundamental knowledge to have in enabling him or her to connect the pedagogical knowledge (how to teach) with the content knowledge (what to teach) (Pesno, 2002) The teachers organise the new knowledge related to the discipline into content that can be easily understood by the students during instructions (Tamir, 1987)

16. Components of Pedagogical Content Knowledge (PCK) • Knowledge about content • Knowledge about students • Knowledge of instructional strategies • Climate of T & L processes • Purpose of the instruction • Fernandez-Balboa (1995) • Knowledge related to teacher’s belief • Comprehension of the concepts & misconceptions • Knowledge about curriculum • Knowledge about instructional strategies & methods of delivery of the topic. • (Shulman, 1986 & Grossman, 1990), COMPONENTS OF PCK • Subject matter for instructional purposes • Students’ understanding of the subject matter • The use of media to teach the subject • The instructional process for the subject • (Marks, 1990)

17. Developing PCK in Mathematics of Pre-service Education • Subject Matter Knowledge • General Pedagogical Knowledge • Specific Pedagogical Knowledge (Mathematical Pedagogy Knowledge – MPK) • Contextual Knowledge

18. Knowledge of Subject Matter • Substantive Structure knowledge - knowledge of facts, concepts, principles of the discipline are organized to incorporate its facts • Syntactic structures knowledge - rules of evidence, generated and validate in the subject, proofs, history of the discipline • Knowledge of substantive and syntactic structures has implication for what teachers choose to teach, and how they teach. • The amount of the degree of substantive & syntactic structures that a teacher possesses concerning his/her respective field would certainly influence the delivery of the subject content to the students

19. PCK and Contextual Knowledge • Practical experience is necessary for the development of classroom experience and can be usefully supplemented by analysis of cases that provide realistic, contextualized exemplars of research-based principles of effective teaching. • Current practice: • School Orientation Programmes (2 weeks) • Micro and macro teaching sessions • Practicum in school (10 – 14 weeks) This practice is insufficient in helping future mathematics teachers to build up their contextual and pedagogical knowledge.

20. Each micro and macro teaching sessions should be recorded - critically analyse - carry out reflection • Macro teaching session should involve school pupils rather than the peers as what is being currently practiced. • School-based concept for macro teaching sessions and make a presentation of the teaching report as a ‘problem-based learning’ outcome • Writing journals and reflections after each instructional session in the classroom

21. Model of Teacher Knowledge(Shulman, 1987)

22. Model of Teacher Knowledge(Grossman 1990) Subject Matter Knowledge General Pedagogical Knowledge Syntactic structure, content, substantive structure Learners learning, classroom management, curriculum instruction, other subjects Pedagogical Content Knowledge Knowledge of students’ understanding, curricular, instructional strategies Knowledge of Educational Context Community, district, school

23. Model of Teacher Knowledge(Noor Shah, 2006) Developed teacher knowledge model based on Grossman (1990) – four components involved: - The content knowledge - The general pedagogical knowledge - The specific pedagogical knowledge - The contextual knowledge.

24. Model of Knowledge Base for Teaching (Turner-Bisset, 1997) • Subject matter knowledge • Substantive knowledge • Syntactic knowledge • Beliefs about the subject • Curriculum knowledge • General pedagogical knowledge • Knowledge/Models of Teaching • Knowledge of learner, empirical and cognitive, knowledge of contexts • Knowledge/Models of Teaching • Knowledge of Self • Knowledge of Educational Ends

25. Content knowledge • General pedagogical knowledge • Curriculum knowledge • PCK • Knowledge of learner, empirical and their characteristics • Knowledge of educational context • Knowledge of educational ends, purpose, values, and philosophical and historical grounds Model of Knowledge Base for Teaching (Shulman, 1987)

26. Grossman (1990) and Shulman (1986)

27. RELATIONSHIP BETWEEN CONTENT KNOWLEDGE, CURRICULUM KNOWLEDGE, PEDAGOGICAL KNOWLEDGE, AND KNOWLEDGE ABOUT THE STUDENTS AND INSTRCUTIONAL STRATEGIES Content knowledge Pedagogical knowledge PCK Orientation to mathematics teaching )