M

,

&

:

:

2011

~ 2 ~

:

1)

( )

.

..

2)

.

....

3)

.

....

~ 3 ~

,

~ 4 ~

:

,

,

.

.

,

~ 5 ~

.................................................................................................................... 6 .................................................................................................................... 7 : 1.1

1.1.1 : ....................................................................8

1.1.2 Piaget .......................................... 9 1.1.3 Van Hieles ............... 10 1.1.4 Battista ................................. 14 1.1.5 Houdement & Kuzniak ................................... 16 1.1.6 Fischbein ..................... 18 1.1.7 Duval .................................................................. 19

1.2 1.2.1 : ....................... 23 1.2.2 ..................................................................... 25 1.2.3 .................................................. 27 1.2.4 ............................................. 31

1.3 1.3.1 : ........................................................... 36 1.3.2 ................................................ 37 1.3.3 .......................................... 40 1.3.4 :

......................................................................................... 45 :

2.1 & .................................. 49 2.2 ................................................................................. 49

2.2.1 .................................................................... 49 2.2.2 ................................................................. 54

: 3.1 .................................................................................... 61 3.2 1 ................... 63 3.3 2 .................. 89

: 4.1 - ..................................................................... 107

- ............................................................................................ 114 ................................................................................................................ 119

~ 6 ~

- . , . ( - ), , . - , .

: , ,

This research study was carried out in order to lead to the award of a Postgraduate Diploma in Didactics and Methodology in Mathematics. This research study focuses on the role of visualization as a component in the development of geometrical reasoning, the formulation of a conjecture and the construction of a proof. The six 10th grade students were selected considering their Mathematical Visuality Score (three visualizers and three non-visualizers) and the data were collected by the analysis of two task-based interviews. The findings indicate that students proving activity is enhanced by visualization in case that student can easily switch back and forth between visual and analytical reasoning, whilst neither the visual approach nor the analytical approach can ensure successful proofs on their own.

Key Words: Visualization, Reasoning, Proof

~ 7 ~

. , ,

, ,

, .

, ,

.

,

,

. ,

,

,

.

,

, .

-

,

, .

, .

.

,

.

,

. ,

.

~ 8 ~

1

1.1

1.1.1 :

,

(Battista, 2007).

, ,

.

,

,

, , . ,

.

,

.

. (2008)

(McGee, Linn & Peterson, Lohmann, Kimura)

.

.

,

(drawing), , (figure),

. Battista (2007)

.

: ,

, (

~ 9 ~

), .

.

.

.

,

Piaget Duval.

1.1.2 Piaget

Jean Piaget,

,

: , ,

.

,

.

J. Piaget B. Inhelder (1967)

.

Piaget,

,

(Clements & Battista,1992).

Piaget (Piaget & Inhelder,

1967, Battista & Clements, 1995):

1: 7 8

, .

,

,

.

.

~ 10 ~

2: 7-8 11-12 .

. ,

.

3: , 11-12

.

.

,

. Piaget

(Piaget 1924,

Battista & Clements,1995). ,

,

.

Piaget

-. ,

Van Hieles

.

1.1.3 Van Hieles

Van Hieles

(Van Hiele, 1986). Van

Hiele

,

(Battista, 2007). Usiskin (1982)

~ 11 ~

:

,

.

Van Hiele,

,

(Clements & Battista,1992 , , 2008):

- 0 ( -/ Pre-recognition): Clements & Battista

(1992) - 0,

1.

.

- 1 ( / Visual):

.

.

.

- 2 ( / Descriptive-Analytic):

.

,

,

.

.

- 3 ( / Abstract-Relational):

,

,

.

, .

- 4 ( / Formal deduction):

.

, ,

.

.

~ 12 ~

- 5 (/ Rigor-mathematical):

.

- ,

.

, Van

Hieles (Usiskin, 1982). ,

.

( /fixed sequence),

(/ adjacency). ,

(/ distinction)

(/separation).

. Van Hieles (1986)

,

Senk (1989). , Piaget,

,

. Usiskin (1982)

(/ attainment) .

P.M. Van Hiele (1959)

, (Usiskin 1982, Mason, 2009):

- (inquiry)

.

,

.

~ 13 ~

- (directed orientation)

.

- - (explanation)

,

.

- (free orientation)

,

,

.

-, (integration)

, .

Van Hieles

,

.

Van Hiele

,

.

,

(Battista, 2007). , ,

(Battista, 2001, Lehrer et al, 1998). ,

Gutierrez & Jaime (1998)

,

,

. ,

,

Van Hieles,

.

~ 14 ~

1.1.4 Battista

Battista (2007) Van Hiele

, ,

.

Van Hieles ,

.

,

(Battista 2007, Borrow 2000):

-1 - (Visual-Holistic Reasoning):

,

, ,

.

(

) (

).

:

1.1 (Pre-Recognition)

.

1.2 (Recognition)

.

-2 - (Analytic-Componential Reasoning):

,

.

, , ,

2.1

2.3, 2 Van Hiele.

:

2.1 (Visual-informal componential

reasoning)

. ,

( )

(

),

, .

~ 15 ~

2.2

(Informal and insufficient-formal componential reasoning)

.

,

.

,

( )

( ). ,

.

2.3 (Sufficient formal

property-based reasoning)

. ,

. ,

,

, .

-3 -

(Relational-Inferential Property-Based Reasoning):

,

, . ,

,

.

, 3

.

:

3.1 (Empirical relations)

, .

3.2 (Componential Analysis)

,

,

, . ,

~ 16 ~

.

3.3 (Logical Inference)

. ,

.

,

. ,

, .

3.4

(Hierarchical shape classification based on logical inference)

.

,

.

, .

-4 (Formal Deductive Proof):

.

,

.

, , .

1.1.5 Houdement & Kuzniak

Houdement & Kuzniak (2003) Van Hieles,

,

,

.

,

,

.

~ 17 ~

Houdement & Kuzniak (2003)

:

-I ( / Natural Geometry):

.

,

.

-II ( / Natural Axiomatic Geometry):

,

.

.

-III ( / Formalist Axiomatic Geometry):

.

.

( ).

Van

Hieles Houdement & Kuzniak.

( 1.1: Van Hiele

Houdement & Kuzniak )

Van Hiele

,

, .

,

~ 18 ~

. ,

. , ,

.

,

.

Houdement & Kuzniak

(1923)

. ,

,

(Jones, 1998). ,

Van Hieles

(Van Hiele,

1986).

1.1.6 Fischbein

. ,

(Duval, 1995).

, , ,

,

(Mariotti, 1997). (Duval, 1988, Fischbein, 1993, Laborde, 1994 )

, :

( ),

( ).

Fischbein (1993)

: (definition), (image)

(figural concept).

,

.

~ 19 ~

, ,

(, , )

, , , .

, ,

.

,

(Fischbein &

Nachlielli, 1998 , 2008).

( 1.2: Fischbein)

,

,

, .

,

(Mariotti, 1997).

1.1.7 Duval

Duval,

Van Hieles,

.

(Duval,

1995, Jones,1998, , 2008):

() (perceptual apprehension):

.

~ 20 ~

, .

() (sequential apprehension):

,

.

() (discursive apprehension):

.

() (operative apprehension):

.

(Duval, 1999):

- (mereologic)

.

- (optic) ,

.

.

- (place way)

, .

,

~ 21 ~

,

.

, , .

,

,

. Duval

. ,

.

,

:

(1) (visualization processes),

. -

(Duval, 2004 , 2008).

,

.

. , -

,

.

(2) (construction processes)

, ,

.

(3) (reasoning processes)

.

, ,

, .

~ 22 ~

,

.

, ,

. , Duval

. ,

,

(Jones,

1998).

. 2

,

5 5

.

( 1.3: )

~ 23 ~

1.2

1.2.1 :

. Rival (1987)

, .

,

,

.

.

.

,

.

, ,

(, 2005). ,

, Bernoulli

(Acta

Eruditorum, 1696).

17

,

. , ,

,

,

(Guzman, 2002). Descartes Regulae ad directionem ingenii,

,

. Isaak Newton,

,

,

(Rival, 1987).

Cauchy,

~ 24 ~

,

(, 2006). Davis (1993) Lagrange

, Gauss

.

18 ,

,

.

. Newton

,

. Weierstrass (1872),

,

,

.

Bolzano (1934)

,

, (Rival, 1987).

, ,

.

Jordan.

. ,

,

. ,

.

1832 1829, Bolyai Lobachevsky ,

,

. ,

,

.

19 20

~ 25 ~

,

.

Jean Dieudonne

(Guzman, 2002).

, ,

,

. 1988 Bishop

12 Conference of the International Group for the Psychology of

Mathematics Education (PME)

. ,

,

(Presmeg, 2006).

1.2.2

,

. ,

,

,

. Zimmermann & Cunningham (1991)

,

.

- ,

-

(Dreyfus, 1991).

~ 26 ~

, (mental image)

, -

- (Battista, 2007).

, , Presmeg (2006)

,

.

(imagery), ,

,

.

Presmeg (2006), 54

, ,

.

: () (concrete

image) ,

, () (memory images of formula)

, () (pattern images)

, ()

(kinaesthetic images)

() (dynamic

images)

.

,

Lakoff & Johnson Wheatley (Battista, 2007),

(rich images)

(image schemata). ,

,

. ,

Presmeg ,

.

, Bishop (1983, , 2008): ()

(visual processing)

~ 27 ~

, ,

()

(interpreting figural information)

,

.

,

. Del Grande (1987),

Frostig & Horne (1964) Hoffer (1977),

:

(figure-ground perception):

,

(perceptual constancy):

,

,

(position-in-space perception):

,

(perception of spatial relationship):

,

(mental rotation):

(visual discrimination):

, ,

(visual memory):

,

1.2.3

,

,

.

,

~ 28 ~

(Gutierrez, 1996).

.

Guzman (2002)

.

, : ,

, .

.

,

,

. ,

. , ,

,

.

Duval (1999)

.

, , :

,

,

. Bishop (1989)

-

. Presmeg

(2006)

. Arcavi (2003)

,

, , .

~ 29 ~

, ,

, Zimmermann & Cunningham (1991), Arcavi

(2003), Hershkowitz et al. (1989), :

, ,

, , ,

, ,

,

.

,

, Fischbein Duval. Zimmermann & Cunningham

(1991) (vision)

,

.

,

, ,

. Fujita, Jones & Yamamoto(2004)

, ,

,

.

:

, ,

,

. Fischbein

,

.

,

,

(Arcavi, 2003).

~ 30 ~

. Barwise & Etchemendy (1991)

:

,

. ,

,

- ,

,

.

,

. ,

,

,

(Dreyfus, 1991). Rival (1987)

. ,

, .

,

(Steen) (Thomas),

.

,

, ,

,

. (pattern)

.

. , ,

~ 31 ~

, (Presmeg,

1999).

1.2.4

,

, ,

. , ,

. Eisenberg &

Dreyfus (1991)

: , .

,

, .

,

,

Presmeg

,

.

Dreyfus (1991),

,

, .

Fischbein (1999)

,

. , ,

, ,

. ,

.

:

~ 32 ~

.

.

, ,

(Eisenberg & Dreyfus, 1991 , Rosken & Rolken, 2006).

,

. Goethe

,

.

. Whiteley (2004)

,

. ,

,

. , ,

,

(Guzman, 2002).

,

. Dreyfus (1991)

,

. Rosken &

Rolka (2006)

.

,

Duval (1999)

~ 33 ~

.

, - ,

.

. Zodik & Zaslavsky (2007)

. ,

. , ,

(geometrical

rigidity),

(Acuna & Larios, 2008).

,

. Arcavi (2003)

,

. , ,

.

Presmeg (1986)

, ,

.

Tall (1994, Rosken & Rolka, 2006):

.

. Eisenberg & Dreyfus (1991)

. , ,

, . , Presmeg (2006)

~ 34 ~

. ,

,

. , ,

.

, -

.

,

. ,

,

,

(Presmeg, 1986).

.

Gauss

(Presmeg, 1985). ,

, (Kruteskii, 1976 ,

Presmeg, 1986). ,

,

. , Kruteskii,

,

~ 35 ~

.

McKim (ambidextrous thinking) .

,

,

,

.

.

,

.

~ 36 ~

1.3 E

1.3.1 :

.

,

. ,

,

(

, ). ,

,

.

6-5 .. ,

,

, .

4 3 ..

. ,

,

,

(Szabo 1973, Lloyd 1979 ).

18 19 ,

. ,

Hilbert

. ,

,

.

Griffiths (2000, Weber, 2003)

,

. ,

~ 37 ~

.

Davis & Hersh (Hersh, 1993)

, Balacheff

, . , Davis & Hersh

,

,

, (Hanna, 1990).

1.3.2

.

(Hanna 1990 & 2000, Hersh 1993,

de Villiers, 1990, Weber,2003).

(verification-justification):

,

.

(explanation):

,

. Hersh (1993)

.

(systematization):

. , ,

.

(discovery):

.

~ 38 ~

(communication):

,

.

, .

(intellectual challenge):

.

(developing intuition):

.

(exploration)

.

(construction) .

(incorporation)

.

. Hanna (2000)

,

, ,

,

. , de Villiers (1999)

.

( 1.4: de Villiers)

~ 39 ~

, :

O (deductive argumentation)

.

.

(inductive argumentation)

. ,

,

.

(abductive argumentation) Pierce

.

.

: , C

, C

.

, :

: ,

,

.

: ,

" ",

" " ( )

. ,

,

.

:

N = { 1, 2, 3, 4, ... } P(n)

n N, (i) P(1)

, , P(n) n = 1 (ii) P(m + 1) P(m) ,

P(m) P(m + 1). , P(n)

n.

:

P.

~ 40 ~

.

, P

.

: ,

,

.

.

:

" P Q" " P

Q".

:

,

, .

/ :

,

.

, .

/ :

. ,

,

.

,

, ,

. Zazkis & Chernoff (2008)

,

.

1.3.3

(Balacheff 1987, Duval,1991, Boero et

all,2010 ).

~ 41 ~

,

,

.

.

Balacheff (1999)

. ,

,

. ,

(Balacheff, 1994 Pedemonte, 2000).

. ,

,

.

.

Duval

(Mariotti,

2006). Duval

.

.

,

(Balacheff, 1999).

,

.

,

(Garuti et all, 1996, Boero et al, 1996, Pedemonte,

2000).

~ 42 ~

, ,

, . ,

(Boero et al, 1996):

.

,

.

( 1.5: )

(Garuti et al, 1996 , Boero et

al, 1996 , Mariotti, 2001)

. , ,

,

. ,

Boero et al

,

.

Pedemonte (2007)

,

~ 43 ~

.

:

(referential system) (structure).

(, ,

) (, , )

.

, , .

,

. , ,

.

Toulmin.

, .

.

:

,

.

.

,

,

. ,

(Pedemonte, 2007).

.

~ 44 ~

. Harel & Sowder (

Mariotti, 2006)

:

(result pattern generalization).

,

.

(process pattern generalization). ,

.

,

. ,

.

, Boero et al (2010)

Habermas (rational behavior)

. Habermas

: (epistemic),

(teleological) (communicative). Boero et

al

,

.

Arzarello & Sabena (2011) Boero et al

: (ground level)

-

(meta-level)

~ 45 ~

. , -

, -

.

Pedemonte

,

-

. ,

.

1.3.4 :

,

. ,

,

.

. , ,

(Hanna, 2000).

,

.

Hanna (2000)

. , Haddas & Hershcovitz

(1998)

~ 46 ~

. , Raman & Weber (2006)

.

,

,

. ,

de Villiers (1999) Hanna (2000)

.

.

.

de Villiers Jones (2000)

.

. (Jones, 2000).

. Hoyles & Healy

(1999, Christou et al., 2004)

,

.

,

.

~ 47 ~

(Christou, 2004).

Mariotti (2006)

.

.

,

,

,

(de Villiers, 1999).

,

,

.

,

,

. ,

.

Hanna(2000)

:

Francis (1996)

,

Palais (1999)

,

Borwein & Jorgenson (1997)

,

.

~ 48 ~

.

, ,

.

,

.

.

,

.

~ 49 ~

2

2.1 &

, .

:

1.

;

2.

;

2.2

, (22)

.

,

, ,

(6) (Mathematical

Visuality Score)

.

2.2.1

:

,

,

(MVS) .

Norma Presmeg,

,

~ 50 ~

.

(Presmeg

1986, 2006) (MVS)

. , , :

,

.

.

.

.

:

,

,

.

.

,

,

,

.

.

,

.

, ,

.

:

.

~ 51 ~

:

,

.

,

.

. ,

.

.

,

.

55 .

.

.

:

,

.

, 2.

0.

2. ,

1. ,

0-24.

.5

. . ;

~ 52 ~

( 2.1: 2 MVS)

( 2.2: 0 MVS)

.

Presmeg Moses Suwarsono (Presmeg, 1986).

(visual method)

, ,

,

.

(nonvisual method)

.

.

~ 53 ~

(visualisers)

,

.

- (nonvisualisers)

.

:

.

:

MV SCORE

/

1 11 9 - 3

2 10 9 - 3

3 12 9 - 3

4 9 6 - 6

5 7 8 - 4

6 A 17 2 - 10

7 7 8 - 4

8 7 1 - 11

9 8 3 - 9

10* 14 3 - 9

11 9 5 - 7

12 11 6 - 6

13 5 6 - 6

14 11 1 - 11

15 13 2 - 10

~ 54 ~

16 B 16 5 - 7

17 14 5 - 7

18 16 2 - 10

19* 11 1 - 11

20 15 5 - 7

21 8 7 - 5

22* 12 2 - 10

( 2.3: )

:

1. *

,

.

2. (. )

.

MVS -

MVS.

,

.

2.2.2

:

-

- ,

:

~ 55 ~

1

,

. , ,

,

.

-

.

,

,

.

,

.

Geogebra, ,

.

, ,

.

~ 56 ~

( 2.4: 1

Geogebra)

: Geogebra

,

, ,

.

2

,

,

, .

,

,

.

,

.

~ 57 ~

.

.

:

(VS)

.

5-7

,

16-17

.

. ,

.

:

30-40 ,

.

:

,

.

.

,

.

~ 58 ~

; ;

;

;

; ;

; ;

;

;

/

;

&

/

;

; ;

/;

/

;

;

;

;

; ;

/ ;

; ;

.

( )

( )

.

.

~ 59 ~

:

;

;

;

;

;

;

;

; ;

;

/

;

;

;

;

;

:

.

.

.

,

,

.

.

Excel. ,

,

~ 60 ~

.

,

.

,

.

. Excel,

.

.

( 2.5: 1- )

~ 61 ~

3

3.1

,

(MVS) ,

.

( 3.1: )

: MVS

. (2-10)

.

.

( )

( )

( )

( )

( )

( )

~ 62 ~

: MVS

. (5-7)

.

.

: MVS

. (8-4)

.

.

: MVS

. (6-6)

.

.

: MVS

. (2-10)

.

,

.

: MVS

. (1-11)

.

, .

~ 63 ~

3.2 1

. :

,

.

.

.

(MVS 17)

.

,

.

:

,

.

.

1: ;

: ..

: ;

: ; ! !

( 3.2).

: . . , !

1 ,

, .

~ 64 ~

( 3.2: )

, ,

,

.

: (

) .. !

.

: ;

: 2 , ;

.

: ;

: . ().

, .

.

: ;

: . .

( ). .

( 3.3: )

2

. .

~ 65 ~

.

: ; ,

;

.

: .

: ;

: ! . ,

.

.

,

,

,

. ,

.

. , ,

.

: . ,

. .

.

: ,

.. ( )

.

~ 66 ~

,

:

: .

.

: , ,

.

,

.

: .. . .

: ; ;

: .

.

, MVS,

.

,

.

.

,

. ,

,

.

. ,

.

~ 67 ~

(MVS 16)

.

.

: , ..

;

: ,

: . ,

.

: ;

;

: .

( 3.4). , ,

.

: . .

. ( ) .

, .

: ;

: , .

: ;

: ! . ;

;

( 3.4: )

~ 68 ~

.

.

: (

) .

: , ..

: ;

.

: ;

: .

: . , . . ;

: !

: ( ) .

.

: ;

: .. ,

( 3.5: )

.

,

.

: . ( /) .

, .

.

: ;

:. ,

, 45, 45,45.

: ;

: .

: ;

~ 69 ~

: ( /),

() 90,

.

:

;

:. .

.

,

.

: ,

, ,

,

45.

:

. ;

: . ( ) ,

() .

: ;

:

,

.

.

,

.

.

,

,

,

.. ,

,

~ 70 ~

.

:

.

: . .. , . .

:

;

: ( ) ;

: ;

: ( ) .. ;

: ;

.

.

.

.

:

.

: ;

: ;

: ; ;

:

;

: ;

, .

: ;

: ( )..

;

: .

(

). , .

(

)

, .

~ 71 ~

.

:

.

:

. ,

( ) .

.

.

,

, .

:

; ,

.

- , .

.

.

,

.

,

.

. ,

.

.

~ 72 ~

(MVS 7)

.

: . ..

: ; ;

: .. .

:

;

: ..

.

. ,

( 3.6).

: , .

.

;

: ;

: ;

: ;

: . . .

.

( 3.6: )

.

: ;

~ 73 ~

: .

: . ;

: ; , .

() , .

.

.

,

.

: . . ,

; . ,

.

: ;

: ,

;

: ;

: . .. .. .

. , . ,

.

:

;

: . ..

: ;

: ,

. ,

.

,

.

,

, .

~ 74 ~

.

,

.

: ( ) , ..

.

.

:

. ;

: .

:

;

: .. .

.. .

: ;

: .. .

: ;

: ; !

.

: ;

: ( ) .

. .

.

,

.

,

.

: , ,

.

- MVS 7,

.

,

,

~ 75 ~

,

.

,

.

,

.

,

.

,

. ,

.

(MVS 5)

( 3.7).

: , .

: ;

:

.

: ;

: .

.

( 3.7: )

.

~ 76 ~

.

: . ;

: , .

: ;

:

;

: ; ;

: .

: ; ;

: . ( ) ( )

( )

.

: ;

: ; .

: ;

: .. ; . .

( 3.8: )

.

: ,

;

: , .

: ;

: .

: ;

: ( ) .

: ;

: , . . ;

. ;

~ 77 ~

( 3.9: / )

.

.

: , . ,

.

. ,

.

: ;

: ;

: ;

: ..

.

:. , . .

, ,

( ) .

: ;

:

,

.

.

,

.

~ 78 ~

.

-

.

, .

:

.

.

.

: ;

: , ,

. .

:

: ; ;

.

:

;

: ; ;

;..

: ! ;

: . .. 6 3; (

) , .

6 4 .

: ;

;

: . . ! .

.

.

.

,

.

,

.

: ,

. , ,

.

~ 79 ~

, - MVS,

.

.

.

. ,

,

.

(MVS 16)

.

, .

:

, .

. ,

: ;

:

.

: ;

: , .

.

, ,

. , ,

.

: . ; ;

: .

.

: ; ;

~ 80 ~

: . ( )

( ). . , , ,

, ( ),

( ), (

).

: ;

: ;

: ;

: , .

: ;

: . .

,

. ,

,

. ,

.

: - ;

: .. !

: ;

: .. .

: ;

.

: . ( ) (

).

: ; ( 3.10)

: . ; .

. , .

.

: , , ;

: , .

( 3.10: )

~ 81 ~

.

,

.

: ( ) ,

, ,

, .. .. .

: ( ) ;

: , ,

, ; , .

: ;

:

: ;

.

: . . .

.

.

: .

.

: .

;

: ( ) ,

.

: (. ) ;

: () () . (, )

( 3.11)

: ;

: ;

: ;

: . .

: ,

;

: . , ;

. , .

~ 82 ~

( 3.11: /)

.

.

: ;

: . (

)

: ;

: .

:

;

: ( , )

, .

: ;

: . ..

: ; ;

: ( ).

. .

: ;

: ( ) . .

.

,

.

: ;

;

: , .

~ 83 ~

:

;

: ,

.

, , ..

: ;

: .

: ;

:.

: ;

: , .

,

( )

: ;

: .

,

.

:

.

, ,

.

.

.

,

.

.

~ 84 ~

,

.

, ,

.

(MVS 7)

.

.

: .

: ;

: .. ;

: , ..

:

: ;

:

,

;

: ;

: .

.

.

: ;

: .

: , , ;

: , .

: ;

: ;

: ;

: , .

~ 85 ~

: ;

: , .

: ;

. ;

: !.. (

)..

: , .

: , ; ;

: ;

: ;

: .

( 3.12: )

. ,

,

.

: .

: ;

: .

: .

: , ()

.

: ;

: .

: . ;

:

.

~ 86 ~

: .

;

: . .. .

, .

: ;

: .. ; .

: . ;

: ( )

: .

: .

: ;

: . .

: ;

: ;

.

.

: , ,

. ;

: . ,

.

: ;

: ;

: ;

: ; . .

: ;

: ; .

: ;

: . .

: ( ) ;

: . .

: ;

;

: . . ;

~ 87 ~

. ,

.

: ( ) .

, .

: ;

: ,

, .

:

;

: ( )

.

, .

: ;

: ( . ) .

: . ;

:.

, .

:

. ;

:

.

: ;

: .

: ;

: ;

: ;

;

: () .. ( )

:

;

: ,

.

~ 88 ~

.

:

;

: .

: .

. ( )

.

;

: .

: ;

: . .

: ( ,

) ;

: ,

.

:

;

:

, .

:

.

.

- MVS 7

.

,

. ,

,

.

,

.

.

~ 89 ~

. ,

.

3.3 2

(MVS 17)

,

.

: . , , .. ;

: ;

: . , , , !

!... .. , .

: ;

: . . , , . ;

.

.

: ; , ,!

: ; ;

: . .

: . , ;

: . .

: ,

. ,

.. , .

,

..

: . ;

: , . , .

: .

~ 90 ~

: .

. , ,

.

: . .

. .

: . .

: . , .

. , ,

. () (

).

: , ..

: ; ;

: ( )

( ) , .

( )

( ) .

.

: ,

, .

: .

.

: , .

.

: . ;

, ,

. .

, . ! ;

: ;

: , .

: . - ;

: . , , - !

: ;

~ 91 ~

: , . 2 .. , ,

.. ;

: .

.. - -

.

: ;

: -2

,

.

: ;

: . .

.

: ;

: .. -2.

: ;

: . -2

: ;

: ;

: .

: ( ).

, -2. ,

. , .

,

,

. ,

,

.

.

.

,

, .

~ 92 ~

(MVS 16)

.

.

: ;

: , , . (

) . ( )

, 4 7, 28.

. ,

.

: ;

: . , .

. 6 (), 3 6 18 9.. ,

. ; (

).. ( )

: ;

: ( 3.13) , 3

. 5 6 4. ,

,

5 6 4, 6 7

5.

: ;

: , 14. 6, 20.

: . .

14,

28. ;

: 28 . ,

, .

( 3.13: )

~ 93 ~

,

.

: ,

, 7,

4. ;

: , .

;

: . 2;

: ;

: ..

.

: -

;

: . . 2.

.

: 6 2.

: ;

: .. . 2,5. . .

:

;

:

, .

: ;

: , . ;

. 5-2 (),

6 6-2 . . , ( ). .

. .. 4 7

3. 5 2 .

:

;

: . -3;

: ; ;

:

~ 94 ~

, ,

.

: -;

: -3./2

: , , 40;

. ,

, .

, .

, .

.

.

.

.

,

.

, .

(MVS 7)

.

.

: , .

. ( )

: ;

: . , ..

: ;

: , .

~ 95 ~

: ,

() . 6 ( ) 4, 11. ..

, . 19

: ;

: .

: ;

: 9. . , 9.

: .

: ;

: , . ,

;

: ;

: , , , ..(

) , , .

: ;

: , ,

.

: ;

: . .

.

, .

: ( )

, 28.

: ;

: . 4, 4 7 28.

: -3 .

: -3 ;

: 7-4=3, 6-3=3, 5-2=3

3

:

;

: .., .

;

: ;

: ,

:

-;

: .(-3)

~ 96 ~

.

.

: ; 10; ( 3.14)

: ;

: .

;

: .

:. ( ) 2!

: 2;

: .

( 3.14: )

.

,

.

: ;

: , . , .

: .(-3)/2. 3

.

.

-

.

.

.

~ 97 ~

.

,

.

(MVS 5)

..

.

: , , .

. ;

.

: ;

: , ( ), , ,

,.( 4,8,12)

: ;

: ,

... . 4 .

: ;

: , , , (

) 24 , 4 28.

: ;

: ,

.

,

.

: ; ,

: ;

: 8, .. 6.. ; 10, 10 .

: ;

~ 98 ~

: . ( ).

( ) ( )

( )

: ;

: , . , 14.

: ;

: 28 .

,

.

: ; . .

:

;

: ;

: .. , , . 5 3,

( ) 2. , 4

3, 2.

: -

: ;.. -3 .

:

;

: ! -3 .

: .

: ( ) 5-3=2 5, 10 6-3=3 6,

: ; ,

;

: . ,

.

.

.

: - .

. ,

3 . ,

,

.

~ 99 ~

.

(-3) .

- ,

MVS 5,

.

,

,

.

.

.

,

,

.

.

(MVS 16)

.

.

: , . . ,

;

:

;

:.. . .

: ;

: . . .

: ;

: .

.

~ 100 ~

,

.

: , .. .

.

: .

: . .

.

() .. ,

. , ..

,

(). .

: ;

: ; .. .. !

.

.

:

.

: ..( ) (

) .

: ;

: ; .. , , , ,.. .. (

) . ( )

, .

: ;

: ( )

. .

: ,

: 28

: 28;

: .

.

: ;

: .. .

: , . ;

: ; . , ,

: ; .

~ 101 ~

: .. , .

, .

, . ,

. ( 3.15)

( 3.15: )

.

. ,

.

: . ;

: .

: ;

: .

: ;

: ( )

: ;

: .... , . ,

. . ..

: ;

;

: ; .

.

: , .

;

: .

: .

: .

: ;

~ 102 ~

: ;

: ; ;

:.. . .

, -

.

: - ;

: ;

: .

6, 7, 8 , ,

.

: - , ;

. ,

, .

. ;

: - . -;

.. ..

.

,

,

.

,

.

,

.

,

.

,

. ,

-,

,

.

~ 103 ~

(MVS 7)

.

.

.

.

: .

: ;

: ; .

: ;

:

: ; ;

: ( )

: . ;

: .

: ;

: . ( )

: ( )

: ; ; .

.

: ;

: .

:

;

: .

: .

; ;

: . .

,

. , ,

.

: ; ;

: . .

: () . . .

~ 104 ~

: ;

: ( )

: ;

: .

: ;

: .

: . ;

: ;

: ;

: .

.

: ;

: . .

.

.

.

: ( ) . .

. , ;

: ;

: ()

.

: .

;

: .

: ; ( )

: . . . .

,

.

.

: 3 618 ;

: 16;

: () . 6

18

: .

;

: ! !

~ 105 ~

: .

: . 2 5 10, 2

5.

: ,

: () 7 28.. .. 13

.

, .

.

.

.

: .

;

: .

: ;

: . .

: ;

:

:

;

: . .

.

: ;

:

. 20 .

, -

.

: - ;

: -; ;

: - . 5

. 6 .

: .. .

:

;

: . ; ;

~ 106 ~

, - ,

.

,

, .

.

.

,

,

.

,

.

~ 107 ~

4

4.1 -

,

,

.

. -

.

,

,

2-10,

8-4. ,

,

,

. ,

,

.

.

.

,

. ,

, ,

.

,

,

, , .

,

,

. ,

.

~ 108 ~

,

.

,

.

,

. ,

, ,

.

.

,

. ,

.

.

,

.

.

,

,

.

,

.

,

.

,

.

.

, ,

.

.

~ 109 ~

.

.

,

.

,

. ,

,

.

,

,

. ,

.

,

. ,

, ,

,

.

. , ,

. , ,

,

. ,

.

.

,

~ 110 ~

. ,

-

,

.

. ,

,

.

,

.

,

,

.

,

,

.

.

,

. ,

.

.

,

, .

, -

.

.

,

cKim (ambidextrous thinking),

,

. ,

, Presmeg

(1986) Kruteskii (1976),

. ,

~ 111 ~

,

.

,

,

, isenberg & Dreyfus (1991)

Rosken & Rolken (2006).

.

. ,

.

,

Jones (2000).

.

,

,

.

,

Arcavi (2003) Presmeg (1986)

.

,

. , ,

, ,

.

Pedemonte (2007)

.

, .

, .

,

~ 112 ~

,

. ,

,

.

, Mariotti (2006).

,

.

. ,

,

.

, .

. ,

,

,

Ramon & Weber (2006),

,

. , -

,

. ,

. ,

. ,

.

,

~ 113 ~

.

,

.

, ,

,

.

,

,

,

. ,

.

.

,

,

,

. , ,

,

- .

. ,

. ,

,

. ,

. ,

.

~ 114 ~

-

, . . (2005). ( 11-45).

, .

Acuna, C. A. & Larios, V. O. (2008). Prototypes and learning of geometry: A reflection on its

pertinence and its causes. Topic Study Group 20 Visualization in Teaching and Learning

Mathematics at the 11th International Congress of Mathematical Education (ICME 11).

Mexico.

Arcavi, A. (2003). The role of visual represantations in the learning of mathematics.

Educational Studies in Mathematics, 53,pp 215-241.

Arzarello, F. & Sabena, C. (2011). Meta-Cognitive Unity in Indirect Proofs. CERME 7 Congress.

Rzeszw, Polonia.

Balacheff, N. (1999). Is argumentation an obstacle ? Invitation to a debate... International

Newsletter on the Teaching and Learning of the Mathematical Proof. May/June

Barwise, J. & Etchemendy J. (1991). Visual Information and Valid Reasoning. In

Zimmermann, W. & Cunningham, S. (Eds.), Visualization in Teaching and Learning

Mathematics (pp.9-24). Mathematical Association of America.

Battista, M. & Clements, D. (1995). Geometry and Proof. Mathematics Teacher,88(1),pp. 48-

54.

Battista, M. T. (2007). The development of Geometrical and Spatial Thinking. In NCTM

Second Handbook of Research on Mathematics Teaching and Learning, vol. 2, pp.843-908.

Boero, P., Garuti, R. & Mariotti, M.A. (1996). Some Dynamic Mental Processes Underlying

Producing and Proving Conjectures. Proceedings of the 20th Conference of the International

Group for the Psychology of Mathematics Education, Vol. 2, pp. 121-128. Valencia, Spain.

Boero, P. , Douek, N. , Morselli, F. , Pedemonte, B. (2010). Argumentation and Proof: A

contribution to theoretical perspectives and their classroom implementation. Proceedings of

the 34th Conference Of the International Group for the Psychology of Mathematics

Education, Vol. 1, pp. 179-209. Belo Horizonte, Brazil.

, .,, .,, .,, . & , . (2008).

.

:-

-, . 315-365.

~ 115 ~

, .,, .,, .,, . & , . (2008).

.

:--,. 77-88.

, . (2006). : 5 .

19 . .

Christou, C., Mousoulides, N., Pittalis, M. & Pitta-Pantazi, D. (2004). Proofs Through

Exploration in Dynamic Geometry Environments. Proceedings of the 28th Conference of the

International Group for the Psychology of Mathematics Education, Vol. 2, pp. 215-222.

Bergen, Norway.

Clements, D. & Battista, M (1992). Geometry and Spatial Reasoning. In D. Grouws, ed.

Handbook of Research on Mathematics Teaching and Learning,(pp. 420-464). New York:

Macmillan Publishing Co.

, .

. .

de Villiers, M. D. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17-

24.

de Villiers, M. D. (1999). The role and function of proof with Sketchpad. Rethinking Proof with

Sketchpad. Key Curriculum Press.

Del Grante, J. J. (1987). Spatial Perception and Primary Geometry. In M. M. Lindquist (Ed.)

Learning and Teaching Geometry, K-12, pp. 127-135. Reston, VA: National Council of

Teachers of Mathematics.

Dreyfus, T. (1991). On the status of visual reasoning in mathematics and mathematics

education. In F. Furinghetti (Ed.), Proceedings of the 15th PME International Conference, 1,

33-48.

Duval, R. (1995). Geometrical Pictures: kinds of representation and specific processings. In R.

Suttherland & J. Mason (eds), Exploiting Mental Imagery with Computers in Mathematics

Education (pp. 142-157), Berlin: Springer.

Duval, R. (1999). Representation, Vision and Visualization: Cognitive Functions in

Mathematical Thinking. Basic Issues for Learning. In F. Hitt & M. Santos (Eds.) Proceedings of

the 21st Annual Meeting of the North American Chapter of the International Group for the

Psychology of Mathematics Education, I, 3-26. Mexico.

Eisenberg, T., & Dreyfus T. (1991). On the reluctance to visualize in mathematics. In W.

Zimmermann, & S. Cunningham S. (Eds.) Visualization in Teaching and Learning

Mathematics. Washington, DC: Mathematical Association of America.

~ 116 ~

Fishbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics, 24,

pp. 139 -162.

Fischbein E., (1999). Intuitions and Schemata in Mathematical Reasoning. Educational

Studies in Mathematics 38, 11-50.

Fujita, T., Jones, K. and Yamamoto, S. (2004). Geometrical intuition and the learning and

teaching of geometry. Topic Study Group on the teaching of geometry at the 10th

International Congress on Mathematical Education, Copenhagen, Denmark, 4-11, July.

Garuti, R., Boero, P., Lemut, E. & Mariotti, M.A. (1996). Challenging the traditional school

approach to theorems: a hypothesis about the cognitive unity of theorems. Proceedings of

the 20th Conference of the International Group for the Psychology of Mathematics

Education, Vol. 2, pp. 113-120. Valencia, Spain.

Gutierrez, A. & Jaime, A. (1998). On the assessment of the van Hiele levels of reasoning.

Focus on Learning Problems in Mathematics, 20 (2/3), 27-47.

Guzman M. (2002). The Role of Visualization in the Teaching and Learning of Mathematical

Analysis. Proceedings of the International Conference on theTeaching of Mathematics at the

Undergraduate Level (2nd) SE066909.

Haddas N. & Hershkowitz R. (1998). Proof in geometry as an explanatory and convincing

tool. Proceedings of the 22th Conference of the International Group for the Psychology of

Mathematics Education, Stellenbosh, South Africa. Vol 3, pp. 25 - 32.

Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6-13.

Hanna, G. (2000). Proof, Explanation and Exploration: An Overview. Educational Studies in

Mathematics, 44 (1-2),5-23.

Hersh, R. (1993). Proving is convincing and explaining. Educational Studies in Mathematics,

24(4), 389-399.

Houdement, C. & Kuzniak, A. (2003). Elementary geometry split into different geometrical

paradigms. In M. Mariotti (Ed.), Proceedings of CERME 3, Bellaria Italy.

Jones, K. (1998). Theoretical Frameworks for the Learning of Geometrical Reasoning.

Proceedings of the British Society for Research into Learning Mathematics, 18(1&2), 29-34.

Jones, K. & Bills, C. (1998). Visualisation, Imagery and the Development of Geometrical

Reasoning. Proceedings of the British Society for Research into Learning Mathematics, 18(1-

2), 123-128.

~ 117 ~

Jones, K. (2000). Providing a Foundation for Deductive Reasoning: students interpretations

when using Dynamic Geometry Software and their Evolving Mathematical Explanations.

Educational Studies in Mathematics, 44 (1-2), pp.55-85.

Marriotti, M. A. (1995). Images and concepts in Geometrical Reasoning. In R. Sutherland & J.

Mason (Eds), Exploiting Mental Imagery with Computers in Mathematics Education, 97-116.

Berlin: Springer.

Mariotti, M. A. (2006). Proof And Proving In Mathematics Education Introduction. In

Gutierrez & Boero (eds.). Handbook of Research on the Psychology of Mathematics

Education: Past, Present and Future, pp. 173-204.

Mason, M. (2009). The van Hiele Levels of Geometric Understanding. Professional Handbook

for Teachers, Geometry: Explorations and applications, pp. 1-5.

. & .(2008). Hoffer

t van Hiele

. 25 .., . 689-709.

Pedemonte, B. (2000). Some Cognitive Aspects of the Relationship between Argumentation

and Proof. International Newsletter on the Teaching and Learning of the Mathematical

Proof. November/ December

Pedemonte, B. (2007). How can the relationship between Argumentation and Proof be

analysed? Educational Studies, 66(1), 23-41.

Piaget, J. & Inhelder, B. (1967). The Child's Conception of Space, pp. 375-418. New York:

W.W. Norton

Presmeg N. C. (1986). Visualization and Mathematical Giftedness. Educational Studies in

Mathematics, 17, 297-311.

Presmeg, N. C. (1999). On Visualization and Generalization in Mathematics. In F. Hitt & M.

Santos (Eds.), Proceedings of the Twenty First Annual Meeting of the North American

Chapter of the International Group for the Psychology of Mathematics Education, 1, pp. 151-

155.

Presmeg, N. C. (2006). Research on Visualization in Learning and Teaching Mathematics. In

A. Gutierrez & P. Boero (Eds.), Handbook of Research on the Psychology of Mathematics

Education: Past, Present and Future, pp. 205-235.

Raman, M. & Weber, K. (2006). Key Ideas and Insights in the context of Three High School

Geometry Proofs. Mathematics Teacher, Vol. 99, No. 9.

~ 118 ~

Rival, I. (1987). Picture Puzzling: Mathematicians are discovering the Power of Pictorial

Reasoning. The Sciences, 27, pp. 41-46.

Rosken, B. & Rolka, K. (2006). A picture is worth a 1000 words-The Role of Visualization in

Mathematics Learning. Proceedings of the 30th International Conference on the Psychology

of Mathematics, Vol. 4, pp. 457-464. Czech Republic.

Usiskin, Z. (1982). Van Hiele levels and achievement in secondary school geometry. Final

report of the Cognitive Development and Achievement in Secondary School Geometry

Project. Chicago: University of Chicago, Department of Education.

Van Hiele, P. M. (1986). Structure and Insight: A Theory of Mathematics Education. New

York: Academic Press.

Weber, K. (2003). Students Difficulties with Proof. Mathematical Association of America.

.

Whiteley, W. (2004). Visualization in mathematics: Claims and questions towards a research

program. Paper presented at the 10th International Congress of Mathematical Education,

Copenhagen, Denmark.

Zazkis, R. & Chernoff, E. (2008). What makes a Counterexample Exemplary? Educational

Studies in Mathematics, 68, pp. 195-208.

Zimmermann, W. & Cunningham, S. (1991). Editor's Introduction: What is Mathematical

Visualization?. In Zimmermann, W. & Cunningham, S. (Eds.), Visualization in Teaching and

Learning Mathematics (pp.1-8). Mathematical Association of America.

Zodik, I. & Zaslavsky, O. (2007). Is a Visual Example in Geometry always helpful? Proceedings

of the 31th International Conference on the Psychology of Mathematics, Vol. 4, pp. 265-272.

Korea.

~ 119 ~

~ 120 ~

(1)

A

: .................................................................................................. : ............................................................................................................

: .........................................................................................

(1)

.

.

(2) .

(3) .

~ 121 ~

.1

.

450.

350.

250. ;

.2

10 .

4 3

. ;

.3

16

2. ,

;

.4

8 .

4,5 .

;

.5

. 20 ,

. ;

.6

.

. ;

~ 122 ~

(2)

A

: .................................................................................................. : ............................................................................................................

: .........................................................................................

(1)

.

.

(2) .

(3) .

~ 123 ~

.1

200 ,

100 100. 50

.

;

.2

.

24 . ;

.3

,

. ,

, .

30 , ;

.4

. .

;

.5

.

.

;

.6

, . ;

~ 124 ~

(1)

: .................................................................................................. : ............................................................................................................

: .........................................................................................

.

.

!!!

.

(1)

.

.

(2)

.

.

~ 125 ~

.1

1 :

.

=450-350=100

=450-250=200

, =150

___________________________________________________________________________

2 :

.

200, 150

100 .

___________________________________________________________________________

3 :

.

:

450 m x+y+z = 450

350m x+y = 350 : =450-350=100m z = 100

250m y+z = 250 : =450-250=200m x = 200

, = 450-200-100=150m y = 150 ___________________________________________________________________________

.

_____________________________________________________________________

~ 126 ~

.2

1 :

. 3 .

___________________________________________________________________________

2 :

.

3 .

___________________________________________________________________________

3 :

:

4m 3m

.

: 7m .

10m .

: 3m .

___________________________________________________________________________

.

_____________________________________________________________________

~ 127 ~

.3

1 :

8 .

7 8

. , 72=14

___________________________________________________________________________

2 :

1

.

___________________________________________________________________________

3 :

.

16m, 8 8 .

, 7 .

72=14 .

__________________________________________________________________________

.

____________________________________________________________________

~ 128 ~

.4

1 :

:

x .

(8-x) .

21 (8-x) .

, x + 21 (8-x)=4

21 . x=1.

, 1 .

___________________________________________________________________________

2 :

.

, 8-4 21 =3 2

1 .

, 7 , 1 .

( , 4 21 -3 2

1 =1 )

___________________________________________________________________________

3 :

2 .

___________________________________________________________________________

4 :

2 ,

.

___________________________________________________________________________

.

____________________________________________________________________

~ 129 ~

.5

1 :

.

x 2x

.

(x-20) (2x-20)

.

3(x-20)=2x-20x=40

, 40 80 .

___________________________________________________________________________

2 :

.

, ,

.

20 . , 40 80 .

___________________________________________________________________________

3 :

2 ,

.

___________________________________________________________________________

.

____________________________________________________________________

~ 130 ~

.6

1 :

:

x y .

(6x+y)

10x=3y + (6x+y), x=y

, 6x +x = 7x 7 .

___________________________________________________________________________

2 :

.

6 . 4 4

. 1 1 . 1 6

1 , 7 .

___________________________________________________________________________

3 :

2 ,

.

___________________________________________________________________________

4 :

, .

6 1 .

, 10 3 , 6 1

.

, 4 4 .

, 7 .

___________________________________________________________________________

.

_____________________________________________________________________

~ 131 ~

(2)

: .................................................................................................. : ............................................................................................................

: .........................................................................................

.

.

!!!

.

(1)

.

.

(2)

.

.

~ 132 ~

.1

1 : . 100+50=150.

___________________________________________________________________________

2 :

.

=100+50=150m

___________________________________________________________________________

3 : ( ). 100m+50m=150m.

___________________________________________________________________________

.

~ 133 ~

.2

1 : : 2 26

3 27

4 28

4 28 .

___________________________________________________________________________

2 :

:

x .

7x .

24 7x-x=24. 6x=24 x=4.

4 28 .

___________________________________________________________________________

3 :

.

,

6

, 24 .

, 4 . 4

28 .

___________________________________________________________________________

4 :

3 6

24 , 4 (

). 4 28 .

__________________________________________________________________________

.

_____________________________________________________________________

~ 134 ~

.3

1 :

.

1

1

(30 )

,

.

, 60.

___________________________________________________________________________

2 :

1 ,

.

___________________________________________________________________________

3 :

.

30. 1 ,

20 .

3X20=60.

___________________________________________________________________________

4 :

:

x .

3x .

1,5x .

, ,

230=60 .

_________________________________________________________________________

.

____________________________________________________________________

~ 135 ~

.4

1 :

.

, ,

.

, .

___________________________________________________________________________

2 :

:

.

___________________________________________________________________________

3 :

2 ,

.

___________________________________________________________________________

4 :

. ..

8 .

, 8 10. ,

4 .

___________________________________________________________________________

5 :

:

x .

,

(x-2) , (x+2) .

(x+2)-(x-2)=4 .

___________________________________________________________________________

.

____________________________________________________________________

~ 136 ~

.5

1 :

.

, 6 , 2

, 1/3 .

___________________________________________________________________________

2 :

1

.

___________________________________________________________________________

3 :

:

x .

, 21 x .

, x+ 21 x

2(x+ 21 x)=3x .

1/3 .

___________________________________________________________________________

.

____________________________________________________________________

~ 137 ~

.6

1 :

.

,

. , 3 .

___________________________________________________________________________

2 :

1 ,

.

___________________________________________________________________________

3 :

:

x .

x=43 x +

43 . , x=3.

, 3 .

___________________________________________________________________________

4 :

:

. ,

3 .

___________________________________________________________________________

.

~ 138 ~

:..

:.

( )

1

.

. . . .

, , . ;

;

;

~ 139 ~

:..

:.

( )

2

, , . ) ; . ) -; .