Title: How Can We Enhance Students Mathematical Thinking Through Discourse
1How 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
2Discourse
- 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
4Typical 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
6- Are students thinking actively in typical maths
classroom?
APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
7- 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
9Need to Change
- A conjecturing oriented teaching approach
- The case of Pythagorean Theorem
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10- Conjecturing as the pivotal mathematical
activities which are - Conceptualizing
- Procedural operating
- Problem solving
- Convincing Proof
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11APEC-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
13APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
14Background 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
15APEC-Khon Kaen International Symposium Thailand,
Aug. 16-20, 2007
16Pythagorean 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
172-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
18Phase 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
19Episode1 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
20Episode2 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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21Episode2 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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22Episode2 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
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23Phase 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)
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Aug. 16-20, 2007
25Phase 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
26Phase 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
27Phase 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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28Cooperating 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
29Cooperating 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
30Phase 3Making and selecting arguments Phase 4
Chaining arguments (pictorically)
- Comment
- Perceptual apprehension
- Operative apprehension
- Discursive apprehension
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31Phase 4 Chaining arguments (symbolically)
- A I can get a2b22ab c22ab according to
these two figures, and get a2b2c2.
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32Phase 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
33Phase 4 Chaining arguments (symbolically)
E (ab)2-2abc2 F c22ab (ab)2 and get
a2b2c2
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34Phase 4 Chaining arguments (symbolically)
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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
37A 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
39- 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
40References
- 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