Designing Mathematics Conjecturing Activities to Foster Thinking and Constructing Actively : Designing Mathematics Conjecturing Activities to Foster Thinking and Constructing Actively Fou-Lai Lin
Mathematics Department
National Taiwan Normal University
Taipei, Taiwan
linfl@math.ntnu.edu.tw 2006 Mathematical Meeting and annual Meeting of the Mathematical Society of ROC
Dec. 8~10, 2006, Taiwan
Aims : Aims Based on the perspective that
a good lesson must provide opportunities
for learner’s to think and construct actively.
The aims of this address are
to present a framework of designing conjecturing (FDC) with examples;
to show that conjecturing is an avenue towards all phases of mathematics learning - conceptualizing, procedural operating, problem solving and proving, and
to argue that conjecturing is to encourage thinking and constructing actively, hence to drive innovation.
Slide3 : Rationale:
Why is thinking and constructing actively very crucial for APEC’s learners?
A Good Lesson Must Provide Opportunities for Learners to Think and Construct Actively.
Dilemma between Students’ Achievement and Self-Confidence of Mathematics (TIMSS-2003) : Dilemma between Students’ Achievement and Self-Confidence of Mathematics (TIMSS-2003) § Data of TIMSS-2003 Presents A Dilemma
Slide5 : Students’ Self-Confidence in Learning Math-Grade 4 (TIMSS 2003)：
Slide6 : Students’ Self-Confidence in Learning Math-Grade 4 (TIMSS 2003)：
Slide7 : Students’ Self-Confidence in Learning Math-Grade 8 (TIMSS 2003)：
Slide8 : Students’ Self-Confidence in Learning Math-Grade 8 (TIMSS 2003)：
A Conjecture on the Phenomenon of High Achievement and Low Self-Confidence in Mathematics : A Conjecture on the Phenomenon of High Achievement and Low Self-Confidence in Mathematics Competitive examination system drives passive and rote learning.
Resolution?
Slide10 : A Good Lesson Must Provide Opportunities for Learners to Think and Construct Actively.
Ⅰ.Thinking: Experiential and Behavioral Point of View : Ⅰ.Thinking: Experiential and Behavioral Point of View Experiential/Phenomenological point of view
Thinking consists in
-envisaging, realizing structural features and structural requirements; proceeding in accordance with, and determined by, these requirements; and thereby changing the situation in the direction of structural improvements, which involves:
Slide12 : -that gaps, trouble-regions, disturbances, superficialities, etc., be viewed and dealt with structurally;
-that inner structural relations – fitting or not fitting – be sought among such disturbances and the given situation as a whole and among its various parts;
-that there be operations of structural grouping and segregation, of centering, etc.;
-that operations be viewed and treated in their structural place, role, dynamic meaning, including realization of the changes which thus involves.
~Productive Thinking (Wertheimer, 1961)
Metaphor : Metaphor
Slide14 : Behavioral point of view
Comparison and discrimination (identification of similarities and differences)
Analysis (looking at parts)
Induction (generalisation, both empricial and structural)
Experience (gathering facts or vividly grasping structure)
Experimentation (seeking to decide between possible hypotheses)
Slide15 : Expressing one variable is a function of another variable
Associating (items together and recognising structural relationships)
Repeating
Trial and error
Learning on the basis of success (with or without appreciating structural significance )
(based on Wertheimer, 1961,pp.248-51)
Learner’s powers : Learner’s powers Discerning similarities and differences
to distinguish
to discern
to make distinctions
Mental imagery and imagination
the power that we are able to be
simultaneously present and
yet “somewhere else”
Slide17 : Generalising and abstracting
generalising and abstraction are the foundations of language
Generalising and specialising
two sides of the coin
seeing the particular in the general
seeing the general through the particular
Conjecturing and convincing
making an assertion about a pattern detected
justifying it so that others are convinced
(Mason & Johnston-Wilder, 2004)
Slide18 : Components of Thinking (behavior point of view/learner’s power) & Meta-Cognition (thinking about thinking) will be used to analyze Conjecturing Activities.
Ⅱ. Designing Conjecturing Activity: Examples and a Framework of Conjecturing : Ⅱ. Designing Conjecturing Activity: Examples and a Framework of Conjecturing Three Entries of Conjecturing
Starting with:
- A False Statement
- A True Statement
- A Conjecture of Learners
Slide20 : 1-1 False statement as starting point
ex.(1) Using students’ misconception
e.g. - a×b＞a and a×b＞b
- 4/9＞2/3 (if a＞c and b＞d, then b/a＞d/c)
- a multiple must be an integer or a half
◎ the additive strategy on ratio task
◎ a quadrilateral with one pair of opposite right
angle is a rectangle
◎ the sum of a multiple of 3 and 6 is a multiple of 9
◎ the square of a given number is even
…
Slide21 : ex.(2) Applying a proceduralized refutation model
A procedualized refutation model (PRM) : A procedualized refutation model (PRM)
Table1. PRM Item-Thinking Analysis : Table1. PRM Item-Thinking Analysis
Table2. PRM Item-Metacognition Analysis : Table2. PRM Item-Metacognition Analysis
1-2 True statement as starting point : 1-2 True statement as starting point
ex1. Heron’s Formula as Starting Point:
A= , where s = (a+b+c)
Slide26 : (i) Making your own sense of the formula:
Convincing yourself that A do represent the area of a triangle with three sides a, b, and c.
Observing it's beauty.
Slide27 : (ii) A model of conjecturing: A triad of mathematics thinking
Symmetry
Degree of Expression
Specializing/Extreme Cases
Slide28 : (iii) Application of the Triad
e.g. What can you say about the formula B:
B= , where s= (a+b+c+d)
Slide29 : (iv) Your conjecture about B will be:
(v) Convincing yourself and peers about your conjecture.
Slide30 : (vi) Conjecturing the volume of
V = ?
V = ?
Slide31 : 1-3 Starting with students’ own conjecture
(1) Defining activity
ex. Swimming Pool
Conan is going to move to a new home，he has a rectangular swimming pool built in the backyard. When he checked the pool，he said，” Is it really a rectangular swimming pool? ” If you were Conan，what places and what properties would you ask the workers to measure so that you can be sure it is rectangular?（It costs NT$1000 to check each item. ）
Be sure，the payment is the less the better.
(Lin & Yang, 2002)
Slide32 : (2) Perceiving from an exploration process
ex. Triangle and Tetrahedron
(i) Demo:
Folding out a tetrahedron from a given regular triangle:
Slide33 : (ii) Could you folding out a tetrahedron from a given isosceles triangle?
(iii) Would some kind of isosceles triangles work?
(iv) Could an isosceles right triangle work?
Slide34 : (v) How would you classify triangles?
(vi) According to your classification, which kind of triangle would work?
(vii) Making your conjectures
(viii) Un-folding a tetrahedron, which kind of polygon you can obtain?
Slide35 : (3) Constructing Premise/Conclusion
ex.
If…, then the sum (product) of two numbers is even
If the sum (product) of two numbers is even, then …
If…, then their product is bigger than each of them
If their product is bigger than each of them, then …
If…, then the line L bisects the area of the quadrilateral
If a is an intersection point of two diagonal lines of a quadrilateral, then …
2. A Frame for Designing Conjecturing (FDC) : 2. A Frame for Designing Conjecturing (FDC)
Ⅲ. Mathematics Learning Processes : Ⅲ. Mathematics Learning Processes What is Mathematics?
Mathematics viewed as concepts and patterns with their underlying situations
Slide38 : Mathematics Learning Processes (mathematizing):
Conceptualizing for Conceptual Understanding Procedural Operating for Procedural Fluency
Problem Solving for Strategic Competence
Proving for Adaptive Reasoning
Productive Disposition associates with all phases of learning processes
Mathematical Proficiency -to learn mathematics successfully : Mathematical Proficiency -to learn mathematics successfully Conceptual Understanding
Procedural Fluency
Strategic Competence
Adaptive Reasoning
Productive Disposition
(NRC, 2001)
Slide40 : 3. To Show: Conjecturing is the Core of Mathematizing
3-1 Conjecturing to Enhance Conceptual Understanding : 3-1 Conjecturing to Enhance Conceptual Understanding ex.(1) Using students’ misconceptions as the starting statement in PRM.
ex.(2) Inviting students to make conjecture of fraction addition after they have learned the meaning of fractions. Using the error pattern a/b + c/d = (a+c)/(b+d) as the starting statement in PRM.
3-2 Conjecturing to Facilitate Procedural Operating : 3-2 Conjecturing to Facilitate Procedural Operating ex.(1) Using “the sum of a multiple of 3 and a multiple of 6 is a multiple of 9” as the starting statement is PRM.
ex.(2) Focusing on the Thinking Triad to make conjecture of the volume of a conical shape.
3-3 Conjecturing to Develop Competency of Proving : 3-3 Conjecturing to Develop Competency of Proving Learning strategy: Constructing Premise/Conclusion
ex. Refer to 1-3:(3)
(2) Learning strategy: Defining
ex. the Swimming Pool Task
3-4 Conjecturing is a Necessary Process of Problem Solving : 3-4 Conjecturing is a Necessary Process of Problem Solving Mathematical Discovery (Polya, 1962)
-Mathematics thinking as problem solving: the first and foremost duty of the high school in teaching mathematics is to emphasize mathematical…problem solving.
-Specialising and generalising as an ascent and descent, in an ongoing process of conjecturing.
Slide45 : Thinking Mathematically (Mason, Burton, &Stacey, 1985)
Specialising
Generalising
Conjecturing
Convincing
4. Conjecturing Approach : 4. Conjecturing Approach Participating in a conjecturing designed with FDC in which everyone is encouraged
to construct extreme and paradigmatic
examples,
to construct and test with different kind of
examples,
to organize and classify all kinds of examples,
to realize structural features of supporting
examples
to find counter-examples when realizing a
falsehood,
Slide47 : to experiment
to adapt conceptually
to evaluate one’s own doing-thinking
to formalize a mathematical statement
to image/extrapolate/explore a statement
to grasp fundamental principles of mathematics
involves learners in thinking and constructing
actively.
Conjecturing Involves Learners in Thinking & Constructing Actively : Conjecturing Involves Learners in Thinking & Constructing Actively Participating in a conjecturing atmosphere in which everyone is encouraged to construct extreme and paradigmatic examples, and to try to find counter-examples (through exploring previously unnoticed dimensions-of-possible-variation) involves learners in thinking and constructing actively. This involves learners in, for example, generalising and specialising.
(Mason, J. & Johnstone-Wilder, S., 2004, p.142 )
This extract have been extrapolated in the above synthesis.
Conjecturing as a Strategy for Innovation : Conjecturing as a Strategy for Innovation Since
Conjecturing encourage learners to think and to construct actively.
And
Thinking & constructing actively is the foundation of innovation.
Conjecturing is an adequate learning strategy for innovation.
Features of FDC - unlike modeling which share the same core status with mathematizing as conjecturing : Features of FDC - unlike modeling which share the same core status with mathematizing as conjecturing FDC is easy to implement
Some case studies have shown its effectiveness
Inviting all of you to experience FDC’s power!
Slide51 : Let’s Do It Together!
Slide52 : Appendix:
Slide53 :
Slide54 :
Slide55 : References:
Mason, J. and Johnston-Wilder, S., 2004. Fundamental Construct in Mathematics Education. RoutledgeFalmer.
Wertheimer, M., 1961. Productive Thinking (enlarged edition, Wertheimer, E. ed.) Social Science Paperbacks with Tavistock Publications, London.
Lin, F.L. and Yang, K.L., 2002. Defining a Rectangle under a Social and Practical Setting by two Seventh Graders, ZDM, 34(1), 17-28.
Kilpatrick, J., Swafford, J, and Findell, B., 2001. Adding It Up. National Research Council.