How Can We Enhance Students’ Mathematical Thinking Through Discourse Fou-Lai Lin Mathematics Department National Taiwan Normal University Taipei, Taiwan

How Can We Enhance Students’ Mathematical

Thinking Through Discourse

Fou-Lai LinMathematics Department

National Taiwan Normal UniversityTaipei, Taiwan

[email protected]

Keynote Address on the APEC-Khon Kaen International SymposiumAug. 16~20, 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 a

nd learning video 1 video 2



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 Internationa

l Symposium


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 teacher’s implemented level of geometrical thinking.

(b) conceptual difficulties.

(c) perceptual difficulties.APEC-Khon Kaen Internationa

l Symposium


6

Are students thinking actively in typical maths classroom?



7

1. Left behind students very often are silent during intellectual interactions.

2. Teachers are either not awared or awared but can’t cope with behinders’ silence. Behinders’ responds usually are not able to carry on.



8

3. 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.

9

Need to Change:

A conjecturing oriented teaching approach –

The case of Pythagorean The case of Pythagorean TheoremTheorem



10

Conjecturing as the pivotal mathematical activities which are

ConceptualizingConceptualizing Procedural operatingProcedural operating Problem solvingProblem solving Convincing & ProofConvincing & Proof



11

Fig.1 The relation between conjecturing and mathematical activities

& Proof



12

ConjecturingConjecturing

Phase 0: Making sense of the problem situationPhase 0: Making sense of the problem situation

Phase1: Formulation of the statementPhase1: Formulation of the statement

Phase 2: Exploration of the content of the statementPhase 2: Exploration of the content of the statement

Phase 3: Making and selecting argumentsPhase 3: Making and selecting arguments

Phase 4: Chaining argumentsPhase 4: Chaining arguments

Phase 5: Writing proofPhase 5: Writing proof



13APEC-Khon Kaen Internationa

l Symposium


Interaction Frame for Conjecturing (Lin et al.,2004)

Students

Teacher Intervention

Exploration

Communication

Phase 0

Phase 1

Phase 2

Phase 3

Phase 4

Phase 5

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.



15APEC-Khon Kaen Internationa

l Symposium


Phase 0

Phase 1

Phase 2

Phase 3

Phase 4

Phase 5

The flow of teaching experiment (Hao, 2005)

1

2

3

4

5

56

16

Pythagorean Theorem

Phase 0: Making sense of the problem situation 　 Historic-genetic approach (Woo, 2007)

•Teacher’s Intended Intervention (embedded in students’ activities)

Given p and q, to find p ＋ q



1-dim:

＋p ＋ qqp

＝

Finding the line segment as long as the sum of two given segments with length a and b.

17

Finding the square as big as the sum of two given squares with area p and q.

p q p ＋ q＋＝

2-dim:



18

ex.1: 1+1=2, how to find a square of area 2?





Phase 0: Making sense of the problem situation

• approach 1: measuring with ruler• approach 2: matching directly

T: Using only compass and ruler to draw.

1. 1-dim:

2-dim:

19

Episode1Episode1 Group1: 1 ＋ 1 ＝2

O D

C

B

A

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



Phase 1: Formulation of the statement (Specializing)

20

Episode2Episode2 Group2: 1 ＋ 4 ＝ 5

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:

D

A

C

B



D

A

C

B

(adding)


21

Episode2Episode2 Group2: 1 ＋ 4 ＝ 5T: Any different approaches?

C:

E:

D:

T: How do you know it is a square?

F:

D

A

C

B

D

C

B

A

H D

AF G

C

E B

H D

A

MF G

C

E B

A

B

C

D

1

2 34

E F

GH



(taking away)

(rotating)(rotating & insight: 2+3=5; 1+4=5)


22

Episode2Episode2 Group2: 1 ＋ 4 ＝ 5

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




23

T: Given squares with area 9 & 16, 9+16=25. 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.

Phase 2: Exploration of the content of the statement(before the statement)



Comment

• Teacher is guiding to deriving the relation of sides from areas.

24

Phase 1: Formulation of the statement (by students)

Group 1 Group 2 Group 3

We want to get:a2+b2=c2

c2-a2=b2

If the sum of the squares of two sides is equal to the square of the third side, it must be a right triangle.

If ABC

This must be a right triangle with two sides and .

Group 4 Group 5 Group 6If the sum of squares of two sides is equal to the square of the third side, this triangle must be a right triangle.

Adding the areas of two squares to get new square, the sides of these three squares can form a right triangle.

Conclusion: A ‘right triangle’, it’s sum of squares of two sides is equal to the square of its hypotenuse.

cb

a ABBC

222 )()()( ACBCAB



25

If three sides of a triangle a, b, c satisfied

a2+b2=c2 , this triangle must be a right

triangle.

Phase 1: Formulation of the statement 1 (whole class)



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 a2+b2=c2 .



27

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

Given many copies of these figures to each group: c

c b+a

b+a

b

b

b

a

cb-a

b-aa

a



The statement: Given a right with sides a, b, and c, c is the △hypotenuse, then a2+b2=c2 .

28

Cooperating based on different student’s worksheets in a group

ac

c

c

b-a

c cc

c

c

b

a

a a

a

b

+1 b

b

1a

a = 1c

c

c

c




29

Cooperating based on different student’s worksheets in a group

c

c

b

a

a

b

c

b

a c

ab

a+b

a+b

+1 b

b

1a

a = 1c

c

c

c




30

Comment

• Perceptual apprehension

• Operative apprehension

• Discursive apprehension




31

A: I can get “a2+b2+2ab= c2+2ab” according to these two figures, and get “a2+b2=c2”.

c

c

b

a

a

b

c

b

a c

ab

Phase 4: Chaining arguments (symbolically)



32

C: According to the figure, I can get “c2=4×ab/2+(b-a)2”, and get “a2+b2=c2”.

ac

c

c

b-a

c




33

c

b

a c

ab

E: (a+b)2-2ab=c2

F: c2+2ab= (a+b)2

and get a2+b2=c2




34

b

a

a a

a

b




35

Are students thinking actively in the Pythagorean Thm. lesson?

How & Why?



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.



37

A lesson to learn1. About teacher Hao, she has shown i) good mathematics competencies in her teachin

g.

ii) well understanding on geometry learning theory, such as Duval’s theory of figural apprehensions: perceptual, sequential, operative and discursive apprehension.

iii) open minded on encouraging different approaches from students.



38

2. Hao’s 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.

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.



40

References:Boero, P. (1999). Argumentation and mathematical proof: A complex, productive, unav

oidable 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 Internationa

l Symposium


Documents

How Can We Enhance Students’ Mathematical Thinking Through Discourse Fou-Lai Lin Mathematics Department National Taiwan Normal University Taipei, Taiwan