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How Can We Enhance Students Mathematical Thinking Through Discourse

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... squares of two sides is equal to the square of its hypotenuse. ... The statement: Given a right with sides a, b, and c, c is the hypotenuse, then a2 b2=c2. ... – PowerPoint PPT presentation

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Title: How Can We Enhance Students Mathematical Thinking Through Discourse


1
How Can We Enhance Students Mathematical
Thinking Through Discourse
  • Fou-Lai Lin
  • Mathematics Department
  • National Taiwan Normal University
  • Taipei, Taiwan
  • linfl_at_math.ntnu.edu.tw

Keynote Address on the APEC-Khon Kaen
International Symposium Aug. 1620, 2007, Khon
Kaen, Thailand
2
Discourse
  • The discourse perspective makes explicit the
    integration of talking and thinking. Sfard
    Kieran (2001) refer to discourse as any specific
    instance of communication, whether diachronic or
    synchronic, whether with others or with oneself,
    whether predominantly verbal or with the help of
    any other symbolic system.

APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
3
  • Zooming in on classroom for making sense of maths
    teaching and learning
  • video 1
  • video 2

APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
4
Typical mathematics classroom phenomenon in
Taiwan junior high school
  • (1) Whole-class approach within classroom
    interactions dominated by choral answers.
  • a loud vocal recall of learnt phase by the whole
    class
  • A choral answer to teachers question

(2) The tasks given by teachers placed high
expectations on students. Teachers lead the
problem-solving-memorizing learning cycle.
APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
5
  • (3) Equity of leaning opportunity is forced to
    give up.
  • my teaching only takes care of the front
    half, not the middle or back.

Left behind students due to (a) mismatch of
students thinking level and teachers
implemented level of geometrical thinking. (b)
conceptual difficulties. (c) perceptual
difficulties.
APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
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  • Are students thinking actively in typical maths
    classroom?

APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
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  • Left behind students very often are silent during
    intellectual interactions.
  • Teachers are either not awared or awared but
    cant cope with behinders silence. Behinders
    responds usually are not able to carry on.

APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
8
  • Rather hard tasks presented in classroom stop
    many students involvement, and implicitly
    encourage teachers to demo and lecture by
    themselves.

Developing mathematical thinking through
discourse seemed more difficult to enhance when
one of the partners (particularly the teacher)
seemed the primary source of the utterance.
APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
9
Need to Change
  • A conjecturing oriented teaching approach
  • The case of Pythagorean Theorem

APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
10
  • Conjecturing as the pivotal mathematical
    activities which are
  • Conceptualizing
  • Procedural operating
  • Problem solving
  • Convincing Proof

APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
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APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
12
  • Conjecturing
  • Phase 0 Making sense of the problem situation
  • Phase1 Formulation of the statement
  • Phase 2 Exploration of the content of the
    statement
  • Phase 3 Making and selecting arguments
  • Phase 4 Chaining arguments
  • Phase 5 Writing proof

APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
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APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
14
Background of a teaching experiment
  • Subjects An eighth typical grade of 41 students
    with 20 males and 21 females, grouping into 6.
  • Periods 45(min)4
  • Equipments video camera, recorder, designed
    colour papers, grid papers
  • ?Formally, participating students have not learnt
    Pythagorean Thm. yet.

APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
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APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
16
Pythagorean Theorem
  • Teachers Intended Intervention (embedded in
    students activities)
  • Phase 0 Making sense of the problem situation
  • Historic-genetic approach (Woo, 2007)

Given p and q, to find pq
1-dim
Finding the line segment as long as the sum of
two given segments with length a and b.
APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
17
2-dim
Finding the square as big as the sum of two
given squares with area p and q.
APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
18
Phase 0 Making sense of the problem situation
1. 1-dim
  • approach 1 measuring with ruler
  • approach 2 matching directly
  • T Using only compass and ruler to draw.

2-dim
ex.1 112, how to find a square of area
2? ex.2 145, how to find a square of area
5? ex.3 4913, how to find a square of area 13?
APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
19
Episode1 Group1 112
Phase 1 Formulation of the statement
(Specializing)
T How do you know the area is 2? A The area of
triangle ABC is 1. There are two such triangles.
The area of the square is 2. T How do you know
the quadrilateral you draw is a square? A
Diagonals bisect equally each other! T Is a
quadrilateral which two diagonals bisect equally
a square? A No! It might be a rhombus. T How do
you know it is a square? Any explanation? M All
squares are the same, so their diagonals are the
same. Four sides with equal length. T Does four
equal sides imply a square? L Each angle is a
right triangle.
  • Comment
  • mathematics competency (conceptual understanding)
  • enhancing alternative thinking

APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
20
Episode2 Group2 145
Phase 1 Formulation of the statement
(Specializing)
T How do you know to draw a square standing like
this? B The side of a square with area 5 must be
a non-integer, so I draw sides neither
horizontally nor vertically. T How do you know
its area is 5? B
APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
21
Episode2 Group2 145
Phase 1 Formulation of the statement
(Specializing)
T Any different approaches? C

E D T How do you know it is a square? F
(taking away)
(rotating insight 235 145)
(rotating)
APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
22
Episode2 Group2 145
Phase 1 Formulation of the statement
(Specializing)
  • Comment
  • applying the equivalence relation of p ?q q
    ?p
  • encouraging operative apprehension
  • getting the insight of relation of squares with
    area 1, 4, 5

APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
23
Phase 2 Exploration of the content of the
statement(before the statement)
  • T Given squares with area 9 16, 91625. With
    such a good relation among these three squares
    sides (3, 4, 5), can we find the side relation
    of these three squares?
  • (Discussion, some students stare at their sheet,
    some are in trances)
  • T You may try to think about it, with such side
    relation like the above square, what kind of
    triangle can be formed? Hint You may diagram it
    with the skill we have learnt last semester.
  • Comment
  • Teacher is guiding to deriving the relation of
    sides from areas.

APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
24
Phase 1 Formulation of the statement (by
students)
APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
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Phase 1 Formulation of the statement 1 (whole
class)
  • If three sides of a triangle a, b, c satisfied
    a2b2c2 , this triangle must be a right
    triangle.

APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
26
Phase 1 Formulation of the statement 2 (by
teacher)
  • T Will it be true
  • If there is a right triangle with three sides
    a, b, and c, c is the hypotenuse, then a2b2c2 .

APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
27
Phase 3Making and selecting arguments Phase 4
Chaining arguments (pictorically)
The statement Given a right ? with sides a, b,
and c, c is the hypotenuse, then a2b2c2 .
  • Given many copies of these figures to each group

APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
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Cooperating based on different students
worksheets in a group
Phase 3Making and selecting arguments Phase 4
Chaining arguments (pictorically)
APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
29
Cooperating based on different students
worksheets in a group
Phase 3Making and selecting arguments Phase 4
Chaining arguments (pictorically)
APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
30
Phase 3Making and selecting arguments Phase 4
Chaining arguments (pictorically)
  • Comment
  • Perceptual apprehension
  • Operative apprehension
  • Discursive apprehension

APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
31
Phase 4 Chaining arguments (symbolically)
  • A I can get a2b22ab c22ab according to
    these two figures, and get a2b2c2.

APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
32
Phase 4 Chaining arguments (symbolically)
C According to the figure, I can get
c24ab/2(b-a)2, and get a2b2c2.
APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
33
Phase 4 Chaining arguments (symbolically)
E (ab)2-2abc2 F c22ab (ab)2 and get
a2b2c2
APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
34
Phase 4 Chaining arguments (symbolically)
APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
35
  • Are students thinking actively in the Pythagorean
    Thm. lesson?
  • How Why?

APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
36
  • Students in the lesson are typical, the teacher
    Hao and the teaching approach have created a
    non-typical lesson.
  • Most of students are thinking actively, though
    the class sounds noisily.

APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
37
A lesson to learn
  • 1. About teacher Hao, she has shown
  • i) good mathematics competencies in her
    teaching.
  • ii) well understanding on geometry learning
    theory, such as Duvals theory of figural
    apprehensions perceptual, sequential, operative
    and discursive apprehension.
  • iii) open minded on encouraging different
    approaches from students.

APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
38
  • 2. Haos reflection on this teaching experiment
  • whole-class lecturing traditionally for about 20
    years
  • the first time to try different teaching approach
  • the impact is enormous

During the teaching process, the unexpected
students reaction and responds put a pressure on
me. concentrating on listening and observing
students reaction, I discovered great potential
of students, their learning capability,
innovation and performances are amazing, this can
not be observed in the past whole class
lecturing.
Teaching behaviour is changeable.
APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
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  • 3. The historic-genetic example together with
    conjecturing oriented teaching approach have
    shown the power of enhancing students thinking
    actively.
  • 4. The classification of phases of conjecturing
    becomes the sub-goals of learning activities and
    is helpful be used for analyzing classroom
    discourse, not between teacher and students but
    also among peers.

APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
40
References
  • Boero, P. (1999). Argumentation and mathematical
    proof A complex, productive, unavoidable
    relationship in mathematics and mathematics
    education. September/October Newsletter.
  • Duval, R. (1995). Geometrical pictures Kinds of
    representation and specific proccessings. In R.
    Suttherland J. Mason (Eds.), Exploiting mental
    imagery with computers in mathematics education,
    142-157. Berlin Springer.
  • Gal, H., Lin, F. L. Ying, J. M. (2006). The
    hidden side in Taiwanese classrooms Through the
    lens of PLS in geometry. Proceedings of the 30th
    International Conference for the Psychology of
    Mathematics Education.
  • Hao, T. (2005). The case study of inquiry and
    discovery teaching method On Pythagorean
    theorem. Unpublished M. ed thesis, Taiwan Normal
    University.
  • Heinze, A. (2004). The proving process in
    mathematics classroom-Method and results of a
    video study. Proceedings of the 28th
    International Conference for the Psychology of
    Mathematics Education.
  • Lin, F.L. (2004), Research on learning and
    instruction theory of mathematical argument for
    adolescents Main project(1/4). Research Report
    of National Science Council (In Chinese).
  • Sfard, A. Kieran, C. (2001). Cognition as
    communication Rethinking learning-by-talking
    through multi-faceted analysis of students'
    mathematical interactions. Mind, Culture and
    Activity, 8(1), 42-76.
  • Woo, J.H. (2007). School mathematics and
    cultivation of mind. Proceedings of the 31st
    International Conference for the Psychology of
    Mathematics Education.

APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
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