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, 2005

4

................................................................................................................. 7 1 ............................................................................................................ 8 ........................................................................... 8

1.1 8 1.2 .................................. 14 1.2.1 HANS FREUDENTAL.................................................... 14 1.2.1.1 ........................................................................................ 15 1.2.1.2 ( ) ........................................................................ 17

2 .......................................................................................................... 21 ...................................... 21

2.1 ..................................................................................................... 21 2.2 ............................................................................. 22 2.3 SIERPINSKA ............................. 25 2.3.1 ........... 26

3 .......................................................................................................... 31 ....................................... 31

3.1 ..................................................................................................... 32 3.2 .................. 35 3.3 , ............................................................................................... 37 3.3.1 .................................................................................................. 37 3.3.2 ......................................................................... 38 3.4 .................................................... 41 3.4.2 ................... 43 3.4.3 ........................................................................................... 44 3.5 ............. 44 3.5.1 ............................................................................................... 45 3.5.2 ................................................................................................... 45 3.5.3 .................................................................................................. 46 3.6 ( )................................................................................ 47 3.7 ............................................... 48 3.8 .................................................. 50 3.8.1 ........ 50

4 .......................................................................................................... 53 ........................................................ 53

4.1 .. ................................. 53 4.2 ......................................................................................................... 57 4.2.1 .................................................................................................. 58 4.2.1.1 ..................................................................... 58 4.2.1.2 ........................................................................................ 59 4.3 ............................. 61

5 .......................................................................................................... 64

5

............................................................................................................... 64 5.1 ..................................................................................................... 64 5.2 ..................................................................... 65 5.3 ........................................................................................... 66 5.4 ................................................................................ 66 5.5 ................................................. 68 5.6 ............................................................................. 73 5.7 ................................................. 74 5.8 ....................................................................................... 82 .................................................................................................. 83

6

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17 .

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1 Smith, D.E.A Sourse Book in Mathematics, London, 1929, Dover, 2 VOLS., 1959.

8

(

). Descartes La Geometrie (1637),

x y (

- ),

: y, x , .

function ( fungor ,

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3

Fourier

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1822).

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11

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uler . (-)

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18 .

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20 ,

4 The Nature and Meaning of Numbers (1887), Dover, 1963, . 50.

12

.

19

.

Dirichlet , , .

, y, x

. ,

.

A. Mostowski :

Dirichlet , , . Sierpinski, 1911 Lvon, x, f(x) . ;

( ) (notion).

,

(concept).

J. Von Neumann,

(notion).

,

Godel Bernays,

.

Giuseppe Peano (1911)

(Sulla definitione di funzione, Atti dei Linzei, 1911)

: ,

. Peano

.

, ,

. Hausdorff

, Grundzuge der Mengenlehre, 1913.

13

Peano Russel Whitehead

Principia Mathematica,

. ,

.

Hausdorff. Hausdorff

.

a b,

. 1920 Kuratowski

[(a,b):={a,{a,b}}]

,

( A. Mostowski, W. Guzicki

, 1973).

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x , y .

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1.2.1 HANS FREUDENTAL

ans Freudental (1904 - 1990)

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14

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, Husserl5 , eideger , -

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ans Freudental 6: ()

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ans Freudental

7 .

1.2.1.1

5 E. Husserl, ( Jacques Derrida). 6 Didactical Phenomenology of Mathematical Structures, . 2, . 28. 7 Didactical Phenomenology of Mathematical Structures, . 17, . 491.

15

, Freudental,

(polyvalent) .

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.

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> 0 n0 nx < n no.

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Piaget

,

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.

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20

2

2.1 ,

(Sierpinska, 1992). Freudental (1983)

, , .

Piaget,

,

8.

Piaget

,

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,

Piaget.

,

.

,

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8 , , . Gutenberg, 1995, . 15.

21

. ,

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2.2

(constructivism)

iaget 1920 1930

. iaget

.

. ishop (1967)

.

1975 Von

Glaserferd.

, .

,

-

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, .

() :

1.

.

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.

22

3.

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23

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1.2

,

3 .

24

,

,

Anna Sierpinska

.

2.3 SIERPINSKA

.

: , , ,

: ,

.

...

.. f(x)

.

.

Sierpinska:

),sin(,),( txyxxf +

2 3 :

x(2)=3. : 2 x

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.

25

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Sierpinska ,

.

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.

2.3.1

,

,

,

.

Willem Kooky (1982),

:

:

,

() .

2.1

26

Piaget

,

.

Sierpinska

.

Lakatos (1996).

, ,

.

.

,

. , , ,

,

Sierpinska (1992).

Sierpinska.

1.

, ,

.

.

2. ,

,

.

3. x y.

x y,

27

.

.

x y,

.

4. .

5. .

.

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7.

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8. .

9. 1-1.

Sierpinska

:

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Sierpinska

. .

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28

Anna Sierpinska ,

.

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all (all 1959), ,

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29

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30

3

31

3.1

Hilbert (1902)

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( & , 1995). ,

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32

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33

non-trivial (

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27

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304 42 = xx34)( xxxf = 4

Davis (1988),

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36

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Mathematica.

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3.3 ,

3.3.1

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Olson & Campbell,

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37

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3.3.2

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(2001)

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Kaput

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38

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1996). (von Glasersfeld,

1987)

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39

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Schoenfeld (1992)

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40

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3.4

.

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.

.

41

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82).

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43

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44

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45

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46

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Thompson (1985a) Harel&Tall.

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47

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Lesh et al (1987a)

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48

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49

3.8

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3.8.1

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Hadamard ,

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50

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,

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19 20

.

9..:Guerson Harel and Java Trgalova, Purdue University U.S.A and Universite Joseph Fourier, France

53

.

. Robert (1987),

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54

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Robert (1987).

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1) ( )

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1986).

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55

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56

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4.2

, Michele Artigue,

.

.

57

. :

i.

ii.

iii.

: , ,

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.

.

4.2.1

,

.

, (y = x) (y = x + )

8 9 10,

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(Munyazikwie, 1995).

.

.

58

()

: 0,999

1 40%

n>0 | A- B |

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(1992) Dubinsky & Harel (1992).

.

1. :

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2- 1)

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62

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63

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5.1 .

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67

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5.5

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69

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FUNCTIONS

D1

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Pa

Pb

Vla

Vlb

Vlc

D2

G2br

Ga

Gb

Gc

D3

D4

A rbre de sim ilarite : E :\functionsgreece.csv

5.1

70

D3 D4. D1 D2.

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D1 .

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D 2

D 3

D 4

V la

V lb

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71

105

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D1: 8 7

D2:

41 39

D3: 22 21

D4: 34 33

105 100%

Vla 63 60

Vlb 81 77

Vlc 70 67

Pa: 52 50

Pb: 17 16

(46%), 39%

x y,

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72

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D3:

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79

.

75

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D1: 15 18

D2:

30 37

D3: 19 26

D4: / 15 19

79 100%

;

V1: 13 17

V2: 29 37

V3: / 37 46

..

F1: 30 37

F2: 49 63

Pa: 45 56

Pb: 16 20

Pc: 35 44

5.2 :

.

= 79 %

76

Q4a 57 72

Q4b 27 34

Q4c 37 46

Q4d 43 54

Q4e 51 64

Q4f 50 63

Gf Q5 18 22

Q6a 42 53

Q6b 53 67

Q6c 45 56

Q6d 30 37

Q6e 55 69

5.3:

.

.

.

77

,

.

.

7

39

21

33

18

37

26

19

0

5

10

15

20

25

30

35

40

45

D1: D2:

D3: D4:

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5.3

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50

16

56

20

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10

20

30

40

50

60

Pa: Pb:

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5.4

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79

0

10

20

30

40

50

60

70

80

Q4a Q4b Q4c Q4d Q4e Q4f Q5:Gf

%

Q4aQ4bQ4cQ4dQ4eQ4fQ5:Gf

5.5

. x y,

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80

53

67

56

37

69

0

10

20

30

40

50

60

70

80

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5.8

,

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:

a)

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c)

.

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.

.

.

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D. Tall,

(embodied)

(perceptual) .

.

(interactive teaching)

,

.

82

1. , ., .. (1997). , , .

2. , . (1985). ,

, .

3. , ., , ., .. (1997). ,

, .

4. Artigue, M., (1989 a). Ingnierie didactique, Recherches en Didactique des

Mathematiques, 9(3).

5. Artigue, M., (1997) Teaching and learning elementary analysis: What can we

learn from didactical research and curriculum evolution?, First Mediterranean

Conference on Mathematics.

6. Bachelard, G. (1938). La formation de 1 esprit sceintifique, Paris : Editions

J. Vrin.

7. Baron, M. (1969). The origins of infmitesimal calculus, New York :

Pergamon Press.

8. Bernstein, B. (1971). Class, codes and control, London : Routledge and

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