1. TEACHERS’ KNOWLEDGE AND PEDAGOGICAL CONTENT KNOWLEDGE

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

3. “To teach is first to understand “ (Shulman, 1986)
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. Knowledge Base of 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
The amount and organisation of knowledge per se in the mind of the teacher.
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.

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 programmes that 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.
Erti kurikulum
Satu program aktiviti yang disusun
oleh guru untuk membolehkan pelajar mencapai objektif yang telah dirancangkannya.
(Hirst, 1975)

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.
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)
Comprehension
Transformation
-Preparation
-Representation
-Selection
-Adaptation
Instruction
Evaluation
Reflection
New Comprehension

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 of Self
Knowledge of Educational Ends

25. Model of Knowledge Base for Teaching (Shulman, 1987)
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

26. Understanding of purposes for teaching subject matter
Components of PCK
Understanding of purposes for teaching subject matter
Knowledge of student understanding in a subject
Curricular Knowledge
Knowledge of instructional practices
Grossman (1990) and Shulman (1986)

27. RELATIONSHIP BETWEEN CONTENT KNOWLEDGE, CURRICULUM KNOWLEDGE, PEDAGOGICAL KNOWLEDGE, AND KNOWLEDGE ABOUT THE STUDENTS AND INSTRCUTIONAL STRATEGIES
Pedagogical knowledge
Content knowledge
PCK
Orientation to
mathematics teaching )
Mathematical content knowledge
Specific mathematic curriculum
Goal and outcomes of T & L mathematics
Knowledge about evaluation
Dimension of learning mathematics
Methods of mathematics learning
Knowledge on how students understand mathematics
The need of learning mathematics
Difficult topic to learn
Knowledge about instructional strategies
Specific teaching methods
Specific strategies for topic