Slide 1:1 Issues of Integrating Mathematics into Pedagogical Knowledge Fou-Lai Lin National Science Council, Taiwan Department of Mathematics, National Taiwan Normal University e-mail: linfl@math.ntnu.edu.tw Plenary Lecture, Korea Conference in Mathematics Education,22nd,Feb.2003


Slide 2:2 To say what is know as known, and what is not know as not known. This is to know. (The Analects) Confucious (B.C.551~479)


Outline :3 Outline §1.Learning Mathematics Teaching Reflective Thinking §2.Entry of Maths Ed. Courses Content Methods §3.Integrating Mathematics into Pedagogy Two orders of the Activity Example §4. Internalization of Pedagogical concepts §5. Discussion


§1. Learning Maths Teaching :4 §1. Learning Maths Teaching 1.1. A Challenging Issue in MTE Using X as a learning strategy within MTE. Lack of Well-Developed Teachers’ Learning Theories. Bridging the Two Issues.


1.2 Making Sense of Learning Maths Teaching :5 1.2 Making Sense of Learning Maths Teaching Reflective Thinking


§2. Entry of Maths Ed. Courses :6 §2. Entry of Maths Ed. Courses 2.1 A program for pre-service 2nd Maths Teacher.


A Program for Pre-service 2nd Math. Teacher :7 A Program for Pre-service 2nd Math. Teacher General Literacy Course: 28(credits) Maths. Course:69(up to 100) Education Courses Maths. Education:18 Education:8 (cf. Department of Maths, NTNU)


Maths. Education Courses:(*:optional) :8 Maths. Education Courses:(*:optional) Mathematics learning(2,2) Mathematics Instruction and Assessment(2) Content and Methods(2,2) Mathematics Problem Solving(2) Mathematics and Computer(2) Teaching Practice(2,2) *Mathematics Curriculum(3) *Mathematics Education Research(3)


Intended Goal for Learning Maths. Teaching :9 Intended Goal for Learning Maths. Teaching Developing Mathematical Power and Pedagogical Power (c.f. Cooney & Shealy, 1997)


2.2 Entry of Pedagogy :10 2.2 Entry of Pedagogy The Content Generic example in learning concepts Cognitive confliction Arithmetic thinking and algebraic thinking Advance mathematics thinking van Hiele levels of geometric thinking Concept image Concept development, The intuitive rules theory,


The Methods :11 The Methods Research process Learning Activities Goal Context – school maths.


Preservice Teachers’ Learning Characteristic :12 Preservice Teachers’ Learning Characteristic Attitude Understanding of Maths Views about Maths Preconception of Maths Learning &Teaching.


Phenomenal Understanding :13 Phenomenal Understanding iff , story: = iff , story:


The Best Entry :14 The Best Entry “The best entry into their belief systems about mathematics and the teaching of mathematics is through the study of school mathematics. It is here that doubt (reflective thinking) can become a commonplace with respect to mathematics and pedagogy” (Cooney, 2001).


Integrating Maths into Pedagogy :15 Integrating Maths into Pedagogy 1st order reveal multiple ways of thinking and different levels of understanding, aware the nature of their views of mathematics, e.g. computational or conceptual, 2nd order analyze their views pedagogically, conceptualize pedagogical conceptions, reconceptualize their beliefs about mathematics, mathematics learning, and mathematics teaching.


3.2 Examples of Integrating Maths into Pedagogy :16 3.2 Examples of Integrating Maths into Pedagogy Examples: (1) Arithmetic thinking vs. Algebraic thinking (2) Concept Image & van Hiele’s models of Thinking (3) Generic example (4) Cognitive confliction (5) Algebraic strategy


3.2(1) Arithmetic Thinking vs. Algebraic Thinking :17 3.2(1) Arithmetic Thinking vs. Algebraic Thinking Task1: What is an odd number? Task2: Applying your answer on Task1 to show that If N2 is even. Then N is even. Computational vs. Structural Thinking


3.2(2) Concept Image & The van Hiele Model of Geometric Thinking :18 3.2(2) Concept Image & The van Hiele Model of Geometric Thinking Task1: Which of the following figure represent a portion of a Conic? Task2: Defining a Conic.


3.2(3) Generic Exampe(Tall,1986) of ‘Ratio’& C=(S,I,R) (Vergnaud,1983) :19 3.2(3) Generic Exampe(Tall,1986) of ‘Ratio’& C=(S,I,R) (Vergnaud,1983) Task1:Analyzing the Content of Ratio in SMP 11~16, England Mathematics1, Singapore Mathematics3, Taiwan Task2: What is a ‘Ratio’? Task3: Differentiating 2:5, 2/5,40%,0.4


3.2(4) Cognitive Confliction :20 Mr.B Mr.A 13 11 3 3.2(4) Cognitive Confliction Task1 To find the length of Mr.B’s neck? Task2 Mr.C’s body length is 10 units, How long is his neck? Metaphor: Neck Disappear


2.Investigating Children’s Mathematics Understanding :21 2.Investigating Children’s Mathematics Understanding Carrying out 3 ‘mini’ research projects Questionnaire used in MUT-program or others. 3.3


§4 Internalization of Pedagogical Concepts :22 §4 Internalization of Pedagogical Concepts 4.1 Understanding Semi-Structured Interviews“Would you describe what you have learned during this year about geometry (algebra,ratio)learning” Responds Ⅰ:Term with Meaning described Ⅱ:Meaning without mentioning the Term Ⅲ:Term without Meaning U :Uncertain of one’s understanding


Slide 23:23 Ⅰ Ⅱ Ⅲ U Understanding Pedagogy Generic Example Cognitive Confliction van Hiele’s Theory Concept Image Error Analysis Arith. Vs. Alg. Thinking Diagnostic Asses. …. 14/15 0/15 1/15 5(Alg.) 0/3 3/3 3/3 3(ratio) 5/6 0/6 1/6 1 0/5 5/5 0/5 2 13/17 1/17 3/17 3(ratio) 0/1 1/1 0/1 6 0/3 3/3 0/3 4 Understanding


4.2 Internalization—The case of generic example :24 4.2 Internalization—The case of generic example Ethnographic approach (4 student teachers) Vygotsky’s Scientific Concept Formation model Individual Conflicts ex.of parabola Criticizing others Modifying peer’s work Developing own “Example” for practicingex. Of beef –noodle price. Bonus and tickets


§ 5 Discussion :25 § 5 Discussion 5.1 Knowing vs.Practicing 5.2 Theorizing


Slide 26:26 THE END