How Can We Enhance Students Mathematical Thinking Through Discourse

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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
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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
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• 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
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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.
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Aug. 16-20, 2007
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• (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?

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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
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• 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
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Need to Change

• A conjecturing oriented teaching approach
• The case of Pythagorean Theorem

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Aug. 16-20, 2007
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• 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
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• 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
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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
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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.
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Aug. 16-20, 2007
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2-dim
Finding the square as big as the sum of two
given squares with area p and q.
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Aug. 16-20, 2007
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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?
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Aug. 16-20, 2007
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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
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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
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Aug. 16-20, 2007
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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)
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Aug. 16-20, 2007
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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
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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
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Phase 1 Formulation of the statement (by
students)
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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.

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Aug. 16-20, 2007
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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 .

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Aug. 16-20, 2007
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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

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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)
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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
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Phase 3Making and selecting arguments Phase 4
Chaining arguments (pictorically)

• Comment
• Perceptual apprehension
• Operative apprehension
• Discursive apprehension

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

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

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Aug. 16-20, 2007
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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.

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Aug. 16-20, 2007
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• 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.
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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.

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Aug. 16-20, 2007
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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
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  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
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  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