Διδακτικη Αξιοποιηση Της Ιστοριας Των Μαθηματικων Η Περιπτωση Της Γεωμετριας

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Toeplitz . . . 99

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Freudenthal . . . . . . . . 102

Brousseau . . . . . . . . 103

Radford . . . . . . . . 105

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Boero . . . . . . . . 106

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( )

Toeplitz, -

Freudenthal,

Brousseau. - -

Radford

Boero.

7

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, .

8 -

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- 15 -

1

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-

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,

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, -

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-

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, Fasanelli (Fasanelli et al 2000) -

, 1772

Jos Monteiro Da Rocha. ,

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, (prolegomena) (

) .

,

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.

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- 16 -

,

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. ,

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,

.

1790, Joseph Louis La-

grange,

,

,

, , -

-

.

(Fasanelli et al 2000).

Richard J. Gillings "Mathematics in the time

of Pharaohs" Thomas Cooper,

1811, Dickinson College Pennsyl-

vania (Gillings 1982):

" ( )

,

, -

.

, -

-

. -

."

1817 J. Leslie "The Philosophy of Arith-

metic exhibiting a progressive view to the Theory and Practice of Calculation",

- 17 -

(,

1987) , ,

. -

, . -

(, 1987).

1820, Niels Henrik

Abel, Fasanelli,

(Fasanelli et al 2000):

"

-

."

1853 R. Baltzer "Die Ele-

ment der Mathematic", -

,

(, 1987).

16 1865, Augustus de Morgan

London Mathematical Society, -

-

(, -

1987). Fasanelli, (Fasanelli et al

2000):

"

.

, -

. ()

-

."

- 18 -

1873 Eugenio Beltrami,

, "Giornale di matematiche" (

) , (Fasanelli et al 2000):

" , -

, ,

-

()

."

1877 P. Mansion,

(Ghent), "Note sur l' enseignament des Math-

mathiques dans les colleges". -

, 1883 S. Gnther,

, -

-

(, 1987).

J. W. L. Glaisher, 1890,

British Association for the Advancement of Science, (Cajori 1894,

Fasanelli et al 2000, Siu 2000):

" , , -

,

.

."

1896, Florian Cajori,

"A History of Elementary Mathematics with Hints of Methods of Teaching"

(Cajori 1896, Fasanelli et al 2000):

"

,

, . () ,

Pestalozzi Froebel,

,

- 19 -

-

. ,

, -

."

1897, Hermann Schubert,

"Mathematical Essays and Recreations", (Fasanelli et al

2000):

"

, .

' ,

,

."

1907, Felix Klein,

"Elementary mathematics from an advanced

standpoint", . ,

(, -

1987):

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-

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,

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-

."

- 20 -

1922, . . "-

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. (, -

1987):

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1930, Vera Sanford, "A Short History of Mathematics", -


" Augustus De Morgan

,

: -

,

.

."

1931, Mark Yakovlevich Vygodskii, "Foundations

of Infinitesimal Calculus", (Fasanelli et al 2000):

- 21 -

" -

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,

.

. ,

, -

, - . ()

,

, -

."

1962 L. Bers, M. Kline, G. Plya M. Schiffer, D.

Bressoud Otto Toeplitz "The Calculus.

A Genetic Approach" ( 2007), -

"The American Mathematical Monthly" -

" " -

(Toeplitz 2007):

" -

.

() -

' ."

, -

.

, 1919, Mathematical Association

Committee, (Fasanelli et al 2000):

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1923

. (, -

1987):

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1958 ,


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- 24 -

- 25 -

2

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. :

,

.

.

( 2009). -

. -

,

(Farmaki, Paschos 2007). ,

"" -

. -

, ,

, -

- 26 -

(Tzanakis,

Thomaidis 2000). -

-

,

(Jankvist 2009).

,

,

,

(Jahnke 2001, Kleiner

2001).

. -

, -

(Jankvist 2009).

,

,

Bachelard. O Brousseau -

-

(Brousseau 1997):

"

, -

. -

( )

."

,

, (Brousseau 1997):

"

,

, (clues)

-

- 27 -

."

Brousseau

.

,

, -

(Jankvist 2009).

-

"" (evolutionary) -

.

: " -

" (ontogenesis recapitulates phylogenesis). ( ,

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.) -

: " ,

".

,

. " ", -

-

,

(Farmaki, Paschos 2007, Tzanakis, Kourkoulos 2007,

Thomaidis, Tzanakis 2007). -

-

(Bartolini Bussi, Sierpinska et al 2000).

-

(in-issues) ( 2009, Jank-

vist 2009).

- 28 -

.

(

2009). -

' ,

.

, :

T , (Tzanakis, Thomaidis 2000).

, - (Philippou, Christou

1998).

(Gulikers, Blom 2001, Thomaidis, Tzanakis 2007).

-

,

(Tzanakis, Thomaidis

2000).

(Jankvist 2009).

(2009).

, -

,

, -

(Jankvist 2009).

-

, , -

- (meta-issues) ( 2009, Jankvist 2009).

- 29 -

(in-issues) -

(meta-issues), Jankvist (Jankvist 2009), -

" " (inner issues) "-

" (outer issues) Davis Hersh (Davis,

Hersh 1982) " " (knowl-

edge of mathematics from the inside) " "

(knowledge of mathematics from the outside) Niss (Niss

2001).

- 30 -

- 31 -

3

-

, -

,

. Tzanakis, Arcavi et al

(Tzanakis, Arcavi et al 2000), ( 2009) Jankvist (Jankvist

2009), -

17 -

, 5 :

1.

2.

3.

4. ,

5. -

.

-

:

1.

. .

-

. ,

-

- 32 -

. (a posteriori)

. Freudenthal

(Freudenthal 1983 Tzanakis, Arcavi et al 2000

2009):

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() -

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,

-

,

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(Tzanakis, Arcavi et al 2000).

-

.

( 2009).

,

, ,

,

(Freudenthal 1983 Tzanakis, Arcavi et al 2000).

,

.

- 33 -

, F. Klein, o G. Plya

I. Lakatos (Tzanakis, Arcavi et al 2000).

,

-

,

(" -

"). ,

-

, -

(ad hoc) ,

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-


. .

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,

-

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-

(Tzanakis, Arcavi et al

2000, 2009).

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, .

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.

-

(Tzanakis, Arcavi et al 2000, 2009).

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.. -

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" " (Tzanakis, Arcavi et al 2000,

2009).

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1 (" "),

-

. 1 ("

- 35 -

- ")

,

-

. -

(mathematics-as-an-end-product) -

(mathematics-in-the-making) (Siu, Siu

1979). Jankvist (Jankvist 2009) 1 ("

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2009) "-

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, ,

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(Tzanakis, Arcavi et al 2000,

2009).

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2000, 2009).

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2 (Tzanakis, Arcavi et al 2000):

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(Jankvist 2009).

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2009).

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2000, 2009).

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)

(for rote) (Tzanakis, Arcavi et al 2000, 2009).

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Arcavi et al 2000, 2009).

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2009).

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, (Tzanakis, Arcavi et al 2000, 2009).

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2009).

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cavi et al 2000, 2009).

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- 45 -

4

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. Tzanakis, Arcavi et al (Tzanakis, Ar-

cavi et al 2000), Siu (Siu 2006) ( 2009), -

-

16 -

2 :

1. ,

2.

-

:

1.

2 :

.

. .

.

:

(Tzanakis, Arcavi et al 2000, Siu 2006, -

2009).

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- 46 -

, -

( 2009).

. .


2000, Siu 2006, 2009). -

,

( 2009).

. .

, (Tzanakis, Arcavi

et al 2000, Siu 2006, 2009). ,

, ,

(-

2009).

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. . ,

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(Siu 2006, 2009). -

,

-

( 2009),

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. -

- 47 -

(Tzanakis, Arcavi et al 2000, Siu 2006, 2009). -

(-

2009).

. , (Tzanakis, Arcavi et al 2000,

2009). -

, -

. -

(Tzanakis, Arcavi et al 2000). -

,

, ( 2009).

2.

3 :

.

. (Tzanakis,

Arcavi et al 2000, Siu 2006, 2009).

-

-


,

,

(, 2006).

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- 48 -

( 2009).

. (Tzanakis, Ar-cavi et al 2000, Siu 2006, 2009). -

. ,

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,

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( 2009).

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(Tzanakis, Arcavi et al 2000, Siu 2006, 2009).

,


2000).

. (Tzanakis, Arcaviet al 2000, Siu 2006, 2009).

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-

( ,

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( 2009).

- 49 -

.

. test ;

(Tzanakis, Arcavi et al 2000, Siu 2006, 2009)

, -

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. ,

,

' (Tzanakis, Arcavi

et al 2000). -

,

( 2009).

. ;

(Tzanakis, Arcavi et al 2000, Siu 2006, 2009)

-

(Siu 2006).

. -

,

-

. -

, -

( 2009).

.

. (Tzanakis, Arcavi et al 2000, Siu 2006, 2009).

- 50 -

. (Siu 2006, 2009).

.

(Tzanakis, Arcavi et al 2000, Siu 2006, 2009).

. (Siu 2006, 2009).

( 2009). -


, -

(,

2006, 2009).

Jankvist 1.. 1.. -

-

. 1..,

1.., 2.., 2.., 2.. 2.., -

, ,

1.., 2.., 2.. 2..

(Jankvist 2009).

- 51 -

5

3

. -

. -

. .

-

.

. -

-

.

.

-

-

.

, -

. ,

. -

(Jankvist 2009):

1. (illumination approaches)

2. (modules approaches)

3. (history-based approaches)

-

:

- 52 -

1.

, -

, , -

, .

(Jankvist 2009).

Tzanakis, Arcavi et

al " " (isolated factual in-

formation) " " (historical snippets) -

, , -

, , , ,

, .. (Tzanakis, Arcavi et al 2000). , -

Siu (Siu

2000) . Jankvist

"" ""

(Jankvist 2009).

" " (historical epilogues or prologues). -

,

T. Lindstrm "Kalkulus" 1995, ,

Jankvist, -

(Jankvist 2009). -

, , -

, -

, ,

-

(Jankvist 2009). -

( "-

" ' ' , " -

" ' ..). -

, -

' (Jankvist 2009).

- 53 -

2.

. " " (modules)

Katz Michalowicz, 2004 -

"Historical modules for the teaching and learning of mathematics",

, Mathematical Association of

America CD (Jankvist 2009, 2009).

,

.

Tzanakis, Arcavi et al

" " (historical packages), -

-

, , 2 3 -

(Tzanakis, Arcavi et al 2000, Jankvist 2009).

, -

10 20 . -

, -

-

(Jankvist 2009). -

. Tzanakis, Arcavi et al :

, (projects)

, , ,

, , internet, .. (Tzanakis, Arcavi

et al 2000).

,

-

.

/

- 54 -


/ -

. ,

,

Roskilde (Tzanakis, Arcavi et al

2000, Niss 2001, Jankvist 2009).

3.

, ,

.

,

-

, (Jankvist 2009). , -

.

. ,

-

(Jankvist 2009). -

" " (genetic approach) -

.

.

,

,

, , -

.

. (Tzanakis, Arcavi et al

2000, 2009):

- 55 -

1.

2.

3. -

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:

1. -

" " :

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: , , , -

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.

.

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2009).

,

(

). -

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,

- 56 -


" " " " ( 2009).

, -

: , -

, -

, ,

, -

(WWW).

-

, ,

, ,

(Tzanakis, Ar-

cavi et al 2000).

, -

. -

(Jankvist 2009).

2.

-

. , ,

, ,

-

. -

"" -

-


" " (necessity of the

- 57 -

subject) -

, -

. Tzanakis, Ar-

cavi et al " "

'


,

, -

. ,

, (Tzanakis,

Arcavi et al 2000):

1. ,

,

.

2. ,

-

.

3. -

.

4. -

,

. -

-

,

(project) -

.

Tzanakis, Arcavi et al -

(Tzanakis, Arcavi et al 2000):

I. -

.

- 58 -

II. ( 2 ) -

. -

-

, ,

,

-

, .

, -

(patterns) -

.

III. ( 3 )

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IV. ,

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( 4 )

- 59 -

,

-

.

, -

-

,

-

.

(.. )

.

Tzanakis, Arcavi et al -

1 4 , -

, (Tzanakis, Arcavi et al

2000):

( 3 ) -

-

, , , -

( 1 3 -

).

, ' , (

2, 3 4 ).

- ( 2),

,

( 1

).

-

- 60 -

( 2 -

V).

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( 1 -

).

-

.

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( 1 1

).

- , ,

, .. ( 1 2).

( 2 -

).

- -

( 1 2) ( 3

).

-

, " " ( 2009,

Jankvist 2009).

- 61 -

3. '

Tzanakis,

Arcavi et al , -


. ,

. .

:

. ,

, (Tzanakis, Arcavi et al 2000,

2009):

I. " " (general conceptual

frameworks) , -

, -

( 4, 5 5

).

II.

, , ,

,

, -

, , -

( 2 2

).

III. , , , -

, -

, ,

( 4,

3 1 ).

- 62 -

. -

-

.

-

, (Tzanakis, Arcavi et

al 2000, 2009):

I.

,

, ,

, ( -

1, 5 5 -

).

II. , -

, -

( 5 5 -

).

III.

, -

( 5

).

IV. -

,

( 5 -

).

V. ' -

,

( -

5, 5 5 -

).

Jankvist

,

- 63 -

, -

(Jankvist 2009). -

, " " ( 2009).

.

.

( 2009):

.

.

. "" ,

. "" ,

. .

-

.

.

1. "" -

(historical snippets)

, Tzanakis, Arcavi et al "-

" (historical snippets) (Tzanakis, Arcavi et al 2000)

(format) (content) .

- 64 -

(Tzanakis, Arcavi

et al 2000):

I. -

. ,

, -

,

.

II. . -

, -

,

/ .

III. . -

.

IV. .

.

( -

, , ) .

, ,

(Tzanakis, Arcavi

et al 2000):

I. .

, -

, , , -

- ..

II. . -

, -

,

,

, ..

- 65 -

2. , , (outdoors experience)

-

, , -

, , . -

, -

.

Tzanakis, Arcavi et al (Tzanakis, Arcavi et al 2000).

, -

, 17

Wa-San ( ) -

. , -

,

. -

. -

, ,

.

,

.


, , ,

-

( 1

).

Wa-San -

- 66 -

( 3 -

).

,

, -

Wa-San

( 5 -

).

3. , videos , (films and other visual means)

-

,

'


, "The hill on the dark side of the

moon" - -

(Sofia-Sonya Kovalevskaya) "Agora" -

.

,

(Open Univer-

sity of UK),

,

(BBC).

"The Tunnel of Samos", -

30 , California Institute of Technol-

ogy (Caltech).

,

-


- 67 -

posters

,

, -

, ..


.

1. projects (research projects based on history texts)

projects -

, . -

.

projects -

Roskilde , Niss (Niss

2001),

-

projects. -

projects

(Niss 2001):

.

.

.

. (pure) ,

. .

project ( 5) ,

1 2 ,

2 8 . -

70

150 ,

- 68 -

(exhibition), . -

(

). projects

(" ")

,

,

,

(Tzanakis, Arcavi et al 2000, Niss 2001).

projects , Roskilde, -

(

5 )

( 1

). Tzanakis, Arcavi et al -

, projects

projects -

( 3, 2 1

) (Tzanakis, Arcavi et al 2000).

Tzanakis, Arcavi et al

( Tzanakis, Arcavi et al 2000) project "

Cayley fractals" (Cayley's problem and the early

development of what later became fractals), Niss

( Niss

2001) project " " (Proofs in Mathematics).

2. (primary sources)

, -

, .

- 69 -

-

, ,

. ( )

-

-

. ,

-

. Jahnke et al,

(suited) ,

(Jahnke et al 2000):

. (replacement): -

-

. ( )

, -

.

. (reorientation):

-

.

,

.

.

-

.

. (cultural understanding):

, -

, , -

.

.

- 70 -

-

, .

:

,

,

, -

-

. Jahnke et al, -

(

) (Jahnke et al 2000):

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,

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(

,

),

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. ,

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,

(Jahnke et al 2000) ( 1

).

- 71 -

. -

"Geometry"

Descartes.

Descartes , -

.

-

.

,

' (Jahnke

et al 2000).

:

-

. ,

.

,

. -

,

.

:

-

,

, -

.

,

-

,

(Barbin 1994). -

- 72 -

-

,


).

-

(representational systems) -

.

,

-

. , -

,

(Jahnke et al 2000).

,

-

"" (Jahnke et al

2000).

:

-

, -

-

.

, -

.

-

,

, ,

.

-

- 73 -

" ", "" ,

(Jahnke et al 2000) (

2 ).

:

,

. -

' ""

, -

, / .

"" - -


:

,

, -

. Jahnke et al -

, Dedekind "Essays on the theory of num-

bers" 1872 -

.

, , -

-

. -


-

,

,

- 74 -

-

,

.

( 2009). Franois Vite

-

Leonhard Euler -

(-

, ).

-

( 3 -

).

:

.

, -

-

.

, -

.

, -

. ,

, -

, , -

, .. (Jahnke et al 2000).

- 75 -

:

-

. -

.

(Jahnke et al 2000) ( -

5 ).

, :

(synchronous) (diachronous) -

. -

,

, ,

, ,

. -

-

, -


. -

,

. -

-

.

,

.

- 76 -

() .

-

.

:

, -


).

. -

:

,

.

-

.

,

. -

-


, :

,

-.

, -

. ,

. -

- 77 -

,

( )

, -

. -

:

( 1 -

).

,

( , ,

, -

).

,


:

, Jahnke et al, -

: "" (direct)

, "" (indirect) -


-

. -

,

. -

.

,

-

- 78 -

(Jahnke et al 2000) ( -

2 ).

, ,

-

.

-

,

(Jahnke et al 2000) (

3 ).

-

( 5 -

). (

) ( ) -

( ).

,


.

-

' .

,

.

, .

-


-

, -

- 79 -

. -

, -

.

. -


:

-

.

.

. -

.

,

. -

-

.


-

(debates) . -

, . -

. -

, -

.

,

- 80 -

, .

, -

, -

, (Jahnke et

al 2000).

:

-

-

.

.

,

-

, -

. , -

,

, -


:

-

,

. -

-

, -

. -

- 81 -

, ' . -

,

-

-


:

,

' .

: -

( -

)

( -

,

, ) (Jahnke et al

2000).

:

, -

-

. -

.

-

, -

, -

.

, -

- 82 -

, -

.

-


:

-

, .

, .

, -

.

(Jahnke et al 2000) ( 2 -

).

:

, , -


-

-

.

,

, -

, -

.

- 83 -

.

, -

, . -

-

. -

.

3. " " (worksheets)

. -

, -

, .

, ,

. ,

, ,

( ),

(Tzanakis, Arcavi et

al 2000).

-

, -

,

( / -

) -

( 1 -

).

,

- 84 -

( 2 2 -

), -

( 1 -

).

, "-

" .

, -

-

( -

3 ). -

-

( 1

). -

. -

( -

2 ).

,

, / -

( -

3, 3 3 )


4. " " (historical packages)

Bruckheimer Arcavi " " (historical hap-

penings) ( " " (historical packages))

, 2 3

(Bruckheimer, Arcavi 2000).

- 85 -

,

.

-

(Bruckheimer, Arcavi 2000). ""

, 3 4 ,

'

,

( -

4 2 )


,

-

.

. -

,

, , -

( -

), -

( -

-

, , ,

, ,

..) (Bruckheimer, Arcavi 2000, Tzanakis, Arcavi et al 2000).

-

( -

1 ),

( 1

)

( 4

).

,

- 86 -

,

(Bruckheimer, Arcavi 2000, Tzanakis, Arcavi et al 2000).

. "" , -

1. , , , / (-

) . (taking advantage of errors, alterna-tive conceptions, change of perspective, revision of implicit assump-tions, intuitive arguments etc.)

-

-

, ,

,

, , -

,

-

, , -


-

( 2

).

:

: , 2000 ..,

-

. , -

, , , 2

2

++ .

- 87 -

, , ,

2

2 + .

. -

-

(Eves 1969).

:

-

. ""

.

, -

. 16

, Vite " ",

,

Fermat Descartes.

( -

) ( -

) -

(Eves 1969).

: Bertrand 1889

"Calcul des probabilits" ( ).

, Bertrand ,

,

.

Bertrand

, ,


- 88 -

2. / - (historical problems)

-

. , -


. :

( , -

)

,

.

. :

Fermat ( xn+yn=zn

n 2)

. -

Goldbach (

)

, 100.

. , :

,

.

. /

:

Galois '

Weierstrass

.

. :

, Lagrange

- 89 -

(

).

. ""

1.

(mechanical in-struments)

: -

( -

5, 5 5 )

(

2 )


-

, -

.

, -

Descartes Gomtrie 1637.

(apparatus) D' Alembert

. -

, -

: http://museo.unimo.it/labmat/ http://kmoddl.library.cornell.edu/.

- 90 -

2. (experientialmathematical activities)

: , -

, ..,

. -

-

:

: -

-

.

,

, / -

( -

4 1 )

(Tzanakis, Arcavi et al 2000). Tzanakis, Arcavi et al -

: " -

, -

.

,

;". -

-

. -

-

, - ,

( 1, 5 2


: -

( -

- 91 -

, ) -

. -

-

. , -

, -

"" ,

( 1

) (Tzanakis, Arcavi et

al 2000).

: -

.

. -

.

Al-

Khwarizmi Descartes, -

(Tzanakis,

Arcavi et al 2000).

: -

Fermat

Pascal : "

,

.

.

, ;"

, -

-


- 92 -

3. (plays)

,

/ .

, -

-

. Tzanakis, Arcavi et al -

(Tzanakis,

Arcavi et al 2000):

. -

, -

(, , )

. Tzanakis, Arcavi et al -

,

Galois (Tzanakis, Arcavi et al 2000). -

' ,

" -

".

.

.

-

,

( 4 -


. (the WorldWide Web).

(World Wide Web Internet)

- 93 -

, : -

.

, Tzanakis, Arcavi et al

-

, , Zehavi

1999 Weizmann Institute.

" " (face to face) -

.

(hypertexts)

. -

, -

site ,

. -

forum -

, -

.

-

-

' ,

-


, -

. -

, -

,

, . , (links)

-

. -

, -

,

( ) -

- 94 -

.

, ,

(URL) (Van Brum-

melen 2000).

, Barrow-Green (Barrow-Green 2000):

1. Sites (General History of Mathematics

Sites). sites ,

site.

2. (Web Resources).

sites sites,

.

3. (Biography). -

. sites

/ -

sites.

4. (Regional Mathematics). sites

-

. sites -

.

5. (Web Exhibits). sites

, ..,

.

,

.

6. (Books on-line).

: ( -

)

- 95 -

( / )

.

7. (Student Presentations). -

-

.

, -

.

.

8. (Bibliography). sites

/

.

9. (Societies).

-

.

, -

.

10. (History of Computing). sites

-

, .

11. (Education). sites -

,

,

.

12. sites (Miscellaneous).

sites -

, (

-

).

- 96 -

- 97 -

6

-

,

,

. , -

.

-

. ,

-

. -

(Radford et al 2000):

- 98 -

-

. , , , -

, -

, .

"a

priori" .

- (norma-

tive) ,

-

(Radford et al 2000).

,

: -

.

(Radford, Boero, Vasco 2000).

-

. , -

, -

.


- 99 -

, ,

. :

1. - Toeplitz

2. - Freudenthal

3. Brousseau

4. Radford,

5. Boero.

, -

.

- Toeplitz

1874 Ernst Haeckel " -

" (recapitulation law): " -

" (ontogeny recapitulates phylogeny).

, Haeckel: "

,

,

" (Furinghetti, Radford 2002, Furinghetti,

Radford 2008).

-

Haeckel, -

, : "

-

" (Furinghetti, Radford 2002, Furinghetti, Radford 2008). -

,

.

(Gulikers,

Blom 2001).

- 100 -

Haeckel

20 Felix Klein, o Henry Poin-

car, o Gino Loria .. Henry Poincar -

(Furinghetti, Radford 2002, 2007, Furinghetti, Radford

2008):

"

, -

.

."

- Otto Toeplitz

1926. O Tzanakis

.

,

. -

/ .

,

(Tzanakis 2000).

Toeplitz 1926, Jankvist,

(Jankvist 2009):

" -

, ,

( ) -

. ()

,

. -

:

- 101 -

'

, -

. -

(direct genetic method).

-

-

,

.

(indirect genetic method)."

-

. Piaget Garcia -

-

. -

-

. - -

- , -

. , -

,

,

, ,

. -

,

, -

. -

: "-" (intra-subject) -

, "-" (inter-subject)

-

' , "-" (trans-subject)

. Vygotsky

-

- 102 -

.

:

, -

. Vygotsky

- .

,

-

. Piaget Garcia -

(Furinghetti, Radford 2002,

Furinghetti, Radford 2008).

- Freudenthal

Freudenthal 1973

( ,

""), "-

-". "Mathematics as an Educa-

tional Task", (Freudenthal 1973):

"

,

, -

.

()

,

,

. ()

-

, -

."

Freudenthal ,

,

- 103 -

Toeplitz.

,

,

. Freudenthal " ",

.

, -

" " (mathematizing)

. Freudenthal " "

.

" " -

, ,

(Freudenthal 1991).

Jankvist Schubring "-

- " (psychological-genetic principle),

- -

-

(Jankvist 2009).

Brousseau

Bachelard 1938

Brousseau -

1970 (Radford 1997). Brousseau

-

" " (system

of constrains). Brousseau,

,

.

. ' ,

- 104 -

, -


, Brousseau : "-

" (ontogenetic)

, "" (didactic)

-

"" (epis-

temological) -

-

. Brousseau (Brousseau 1997):

"

, -

. -

( )

.

-

."

, -

. -

,

. Brousseau -

. , -

,

,

,

(concep-

tualisations) (Radford, Boero, Vasco 2000).

- 105 -

Radford

Radford,

-

. Radford -

-

-

. , -

.

( ,

, ).

, , -

-

. , - ,

, , (Radford,

Boero, Vasco 2000). Radford,

, ,

(Radford 1997).

- Radford, -

.

-

, ,

, ,

, .

-

, "-

" (interiorisation). - ,

,

- 106 -

- (Radford, Boero,

Vasco 2000).

- Radford, -

- (micro-space) -

,

,

(Radford, Boero, Vasco 2000, Farmaki, Paschos 2007).

, -

(Radford, Boero, Vasco

2000).

, / -

. ,

. -

- (panculturally) -

.

, -

-

, - (Radford,

Boero, Vasco 2000).

Boero

Boero

, -

,

.

,

- 107 -

. Boero ""

(voices). -

, ,

, -

"" (echoes). , " -

" (voices and echoes game - VEG)

-


"

", -

,

, .

, -

Vygotsky . -

-

- , -

(

) (Radford, Boero, Vasco 2000).

,

( ), ,

(Radford, Boero, Vasco 2000):

- ( , , ..)

-

( ,

, ..)

,

- 108 -

(

, )

"" -

(

)

Boero , "

" -

. " "

(active imitation) " -

" (zone of proximal development) (Radford, Boero,

Vasco 2000).

- 109 -

7

.

-

. -

Gulikers Blom, -

(Gulikers, Blom 2001):

1. ( ,

, , -

).

2. (

).

3. ,

( ,

).

4. (Vision) ( -

, (nets)).

5. ( ,

).

6. (

).

7. ( ,

, , -

).

8. .

:

- 110 -

, Swetz 1997 ( "The 'piling up of squares' in ancient China", Mathematics Teacher, 70(1),

pp. 72 79), -

(piling up of

squares),

. Swetz, -

-

, (Gulikers,

Blom 2001).

Burns 1997 ( "The Babylonian clay tablet",Mathematics Teaching, 158, pp. 44 45) -

( Plimpton 322), : " -

;" , -

-

.

-

. ( )

,

(Gulikers, Blom 2001).

MacKinnon 1992 ( "Homage toBabylonia?", The Mathematical Gazette 76(475), pp. 158 178)

Plimpton 322,


Eagle 1998 ( "A typical slice", Mathematics in School, 27(4), pp. 37-39)

-

.

- 111 -

,

( 3r34V = ).


MacKinnon 1989 ( "What do you doabout 4r2?", The Mathematical Gazette, 73(464), pp. 107 110) -

-

, -


Fhrer 1991 ( "Historical stories in themathematics classroom", For the Learning of Mathematics, 11(2), pp. 24

31)

, -

.

" -".

Fhrer , Vite Des-

cartes -


1987, Paris VIII.R.E.M., " "

. ,

-

, -

( 2000)

Corris 1990 ( "Experimental pi", Mathe-matics in School, 19(1), pp. 18 21)

,


- 112 -

( " " ., ., .

(eds), -

. , -

: , pp. 269 298)

-

( 2009).

Ofiir 1991 ( "Historical happenings in themathematical classroom", For the Learning of Mathematics, 11(2), pp. 21

23) , 12

Maimonides, -

-

,


1996 Makowski Strong ( "Sizing up earth:

A universal method for applying Eratosthenes' earth measurement", Jour-

nal of Geography, 95(4), pp. 174 179) ,

. -

-

. -


1990 ( " ", ', 7(25), pp. 27 41) -

( -

). -

, "" (2

..), -

"" ,

16 .. -

-

- 113 -

, -

, -

-

( 1990).

Hogendijk 1996 ( "Een workshop overIraanse mozaken", Nieuwe Wiskrat, 16(2), pp. 38 42) -

,

.

-

.

. Hogendijk


Hischer 1994 ( :"Geschichte der Mathematik als didak-tisher Aspect (2). Lsung klassischer Probleme. Ein Beispiel fr die gym-

nasiale Oberstufe", Mathematik in der Schule, 32(5), pp. 279 291) -

(trisectrix).

, -

. Hischer

.


Ransom 1993 1995 ( "Astrolabes, cross staffsand dials", Mathematics in School, 22(4), pp. 2 8 "Navigation and

surveying: teaching through the use of old instruments", Histoire et

Epistmologie dans l' Education Mathmatique, IREM de Montpellier, pp.

227 239)

- 114 -

.

,

. -

'

-


1990 ( " ", ', 7(25), pp. 27 41) -

. -

,

,

-

, , -

-

( 1990).

2005 Taimina Henderson ( "How to Use Historyto Clarify Common Confusions in Geometry", A. Shell-Gellasch, D.

Jardine (eds), From Calculus to Computers: Using Recent History in the

Teaching of Mathematics, MAA Notes volume No.68, pp.57-73)

-

. "

;" Taimina Henderson

( Sarrus

Peaucellier-Lipkin) -

(Taimina, Henderson 2005).

Lumpkin 1978 ( "A mathematics club project fromOmar Khayyam", Mathematics Teacher 71(9), pp. 740 744)

- 115 -

Khayyam , -

11 .

. Lumpkin

, Khayyam


Proia Menghini 1984 ( "Conicsections in the sky and on earth", Educational Studies in Mathematics, 15,

pp. 191 210) -

: "

;".

, , -

-

(Gulikers,

Blom 2001).

Bartolini-Bussi Marioti ( "Semi-otic Meditation: from history to the mathematics classroom", For the Learn-

ing of Mathematics, 13(3), pp. 27 35)

(egg-shaped). -

,

-


Lumpkin 1978 ( "A mathematics club pro-ject from Omar Khayyam", Mathematics Teacher 71(9), pp. 740 744) -

.

- 116 -

-

,

. : -

. -

,


1988 Baptist Diener ( "Historische Aufgaben imMathematikunterricht", Die realschule, 96(6), pp. 231 234)

-

,

-

.

" " (two tower problem)

Leonardo of Pisa (Leonardo Pis-

ano Fibonacci) (Gulikers, Blom 2001).

1992 Rickey ( "How Columbus encoun-tered America", Mathematics Magazine, 65(4), pp. 219 225) -

, -

.

.

(Gulikers, Blom

2001).

Bartolus of Saxoferrato 1355, oVan Maanen 1992

. Bartolus

, -

- 117 -

,

. -

, -

(Gulikers, Blom

2001).

1987 Arsac ( "L' origine de la dmonstration: essai d'pistmologie didactique", Recherches en Didactique des Mathmatiques,

8(3), pp. 267 309)

,

500 .. : "

" (Gulikers, Blom 2001).

Horak Horak 1981 ( "Geometrical proofs of alge-braic identities", Mathematics Teacher, 74(3), pp. 212 216) -

, -

.

,


1991 Artmann ( "Quadratische Probleme in Euklids'Elementem' und ihre Behandlung im Mathematikunterricht", Didaktik der

Mathematik, 19(2), pp. 94 110)

,

,

, , (iso-

morphic). -

- 118 -

,


Deakin 1990 ( "From Pappus to today, thehistory of a proof", The Mathematical Gazette, 74(467), pp. 6 11) -

-

. ,

( 1 -

) (Gulikers, Blom 2001).

1996 Brodkey ( "Starting a Euclidclub", Mathematics Teacher, 89(5), pp. 386 388) -

1 -

. ' 20 ,

.

, 1 ,

. Brodkey, -

: , -

-

,


Laubenbacher Pengelley 1995 ( "Mathematical

masterpieces: teaching with original sources", R. Calinger (ed), Vita

Mathematica: Historical Research and Integration with Teaching, MAA, pp.

257 260). -

,

. Laubenbacher Pengelley -

,

(Laubenbacher, Pengelley

1996).

- 119 -

2006 ( "-

", . -

, . . (eds), -

, pp. 253 270)

-

. -

10 .

,

(300 .. ) (5

..),

.

,

(,

2006).

1999 Tzanakis ( "Unfolding interrelations betweenmathematics and physics, in a presentation motivated by history: two ex-

amples", International Journal of Mathematical Education in Science and

Technology, 30(1), pp. 103 118)

,

: , New-

ton Kepler ,

-

(matrix algebra) (Tzanakis 1999).

1998 Tzanakis Thomaids ( "Presuppositions ofa constructive role of the history of mathematics in understanding and

teaching mathematics", Luminy 'Reader')

, -


- 120 -

-

( ) -

:

-

.

- ,

.

-

, -

,

-

.

-

, :

;

-

;

- 121 -

- -

;

- , -

-

;

-

,

,

:

.

- -

.

, -

.

- 122 -

- 123 -

8

.

-

.

, -

( , , ,

..).

,

. -

(deduc-

tive reasoning) (Horn, Zakeri 1998).

( ) -

"" , 300 .. .

.47, -

:

"

" ( 1975).

(: "

.")

: " ,

" ( et al 2010).

- 124 -

-

(570 500 .. ).

1.000 .

Plimpton 322

( , ,

2+2=2). ,

, ,

.

,

"" , .

(500 300 ..) -

:

( ), -

( 2, 2 2

) ( 2

-

1) (Stillwell 2010).

(

, -

, -

) ( -

: ))()(( ---= , , 3 -

,

) ( et al 2010),

-

( -

).

,

-

- 125 -

, (600 ..

) (200

.. 200 ..). -

cut and

paste kou-ku (

kou=

ku=).

11 .. Bhaskara -

-

,

: "" (see) (Maor 2007).

1637 Ren Descartes

.

Pierre de Fermat

(

2007). ""

-

-

: -

P

Q , -

: 22 (dy)(dx)PTPQ += .

-

"" -

Newton Leibniz 1666 1676 (Maor 2007).

- 126 -

1731 Alexis Claude Clairaut "Recherches sur les

courbes a double courbure" ("Researches on curves of double curvature"

), , 18

, -

2 R(x1,y1)

P(x2,y2)

:2

122

12 )y(y)x(xPR -+-=

(Maor 2007).

-

Hilbert 1899, -

(Patronis, Thomaidis

1997, Stillwell 2010).

-

( -

) (). 1 ..

,

.

, (-

:" "), -

.

- 127 -

.

-

. -

18 .. , -

-

. 4 ..

,

,

(, -

). 18 ..

18 .., ,

, -

. ,

: ,

( )

.

( ' -

),

( ' ' ). -

:

, -

, ,

, - ,

, ,

( -

),

- 128 -

- , , -

,

, - -

(topic)

.

3 -

2 .

,

""

.

3 :

- 129 -

: ..

: ...

: .

..

: ..

: - 18 ..

1 -

, ' (

). '

5000 .. -

, -

. 3500 .. -

,

- 130 -

, ,

.

.

60 (

). 2300 ..

. ,

, 2100 .. , -

. 2000 .. -

, , -

,

, 1900 .. .

.

-

. -

: , ,

, ..

,

,

,

.

( ) , -

( ) .

-

, -

.

- 131 -

.

. 1

10 .

2 1 59

60 60,

10.

( 1,57,46,40)

o 1x603+57x602+46x60+40, 424.000

. -

0 ().

. -

,

.

6,9 6,9,0 6,0,9 , 0;6,9

0;6,0,9 6;9 6;0,9 6,0;9 ( ";"

). -

.

- 132 -

Plimpton 322.

Plimpton 322. -

' .

13 ., 9 . 2 .

.

,

.

,

George Plimpton 1922 10 , -

Larsa ( 1).

1800 .. . 1935 Plimpton -

Columbia University,

322.

. 2 -

Plimpton 322, -

. 3

. 4

Eleanor Robson

.

- 133 -

3 - Plimpton 322 ( )

4 - Plimpton 322 ( )

- 134 -

5 - Plimpton 322 Eleanor Robson

4 -

15 .

' . -

,

1:

- 135 -

B C119 169 1

3367 4825 24601 6649 3

12709 1854 465 97 5

319 481 62291 3541 7799 1249 8481 769 9

4961 8161 1045(2700) 75(4500) 11

1679 2929 12161 289 13

1771 3229 1456 106 15

1

1 -

-

Plimpton 322.

1.

.

1.

()

( 1).

- 136 -

1

. (-

: 13 2,

.)

B C119 169

3367 48254601 6649

12709 1854165 97

319 4812291 3541799 1249481 769

4961 816145(2700) 75(4500)

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1771 322956 106

2

2. A, B C; -

;

22 BC - 22 BCA -=

- 137 -

3. :

: ..

..

................

4.

; -

. , -

. -

3, .

B C C+B= C-B=56 10665 97

119 169161 289319 481481 769799 1249

1679 29291771 32292291 35412700 45003367 48254601 66494961 8161

12709 18541 3

- 138 -

5. ( ) ; -

C+B=22 C-B=22. ;

4.

B C C+B=22 C-B=22 56 10665 97

119 169161 289319 481481 769799 1249

1679 29291771 32292291 35412700 45003367 48254601 66494961 8161

12709 18541 4

6. C+B=22 C-B=22, B C

==+

2

2

2B-C2BC .

- 139 -

7. ,

5. -

.

B 2-2 C 2+2

9 5 56 90 1069 4 65 72 97

12 5 119 120 16915 8 161 240 28920 9 319 360 48125 12 481 600 76932 15 799 960 124948 25 1679 2400 292950 27 1771 2700 322954 25 2291 2700 354160 30 2700 3600 450064 27 3367 3456 482575 32 4601 4800 664981 40 4961 6480 8161

125 54 12709 13500 18541 5

8. =2 = 2-2, C= 2-2

2+B2=C2, : (2)2+(2-2)2=(2+2)2.

- 140 -

9. ,

, > -

=2, =2-2 C= 2-2 (

2+B2=C2). -

6.

=2 =2-2 C=2+2 2+2 C2

6

10. ;

;

- 141 -

: ..

: ...

: .

..

: ..

: 4 ..

( )

. ,

2 .., -

, -

. 600 .. , -

, -

. -

, -

, . -

. ,

.

4 ..

. -

, -

,

. 330 .. -

-

-

,

- 142 -

, -

. , -

. '

(, -

, , , ..) -

. ' (

),

300 .. 13 -

-

( , ,

, , , -

, , ..).

"" -

, .

2.

1. -

"".

.

-

.

.

.

.

-

. - ,

, , .

, , ,

, -

.

- 143 -

o ""

(82b9-82e3, 82e14-84a2, 84d3-85b7)

. Ep d moi, pa, gigns-

keij tetrgwnon cwron ti

toiotn stin;

, ,

;(

.)

. Egwge. .

. Estin on tetrgwnon

wron saj con tj

grammj tataj psaj, tt-

taraj osaj;

-

;

. Pnu ge. .

. O ka tautas tj di m-

sou stn saj con;

;(

-

.)

. Na. .

. Okon eh n toioton

cwron ka mezon ka

latton;

,

-

;

. Pnu ge. .

. E on eh ath pleur

duon podon ka ath duon,

pswn n eh podn t lon;

de d skpei e n tatV

duon podon, tatV d nj

podj mnon, llo ti pax n

n duon podon t cwron;

,

;

:

,

;

. Na. .

- 144 -

. 'Epeid d duon podon ka

tatV, llo ti dj duon

ggnetai;

-

,

;

. Ggnetai. .

. Duon ra dj ggnetai

podn;

;

. Na. .

. Psoi on esin o do dj

pdej; logismenoj ep.

; .

. Tttarej, Skratej. , .

. Okon gnoit n totou

to cwrou teron diplsion,

toioton d, saj con psaj

tj grammj sper toto;

,

,

;

. Na. .

. Pswn on stai podn; ;

. 'Okt. .

. Fre d, peir moi epen

phlkh tij stai kenou

gramm ksth. mn gr

tode duon podon t d

kenou to diplasou;

,

.

, -

;

. Dlon d, Skratej, ti

diplasa.

, , -

.

. S d moi lge p tj

diplasaj grammj fj t

diplsion cwron ggnesqai;

,

-

; -

- 145 -

toinde lgw, m tatV mn

makrn, t d brac, ll

son pantac stw sper

tout, diplsion d totou,

ktpounll ra e ti soi

p tj diplasaj doke

sesqai.

, ,

, -

, . -

,

;

. Emoige. , .

. Okon diplasa ath

tathj ggnetai, n tran

tosathn prosqmen

nqnde;

, -

' ,

;

. Pnu ge. .

. 'Ap tathj d, fj, stai t

ktpoun cwron, n ttta-

rej tosatai gnwntai;

-

, -

;

. Na. .

. 'Anagraymeqa d p atj

saj tttaraj. llo ti

tout n eh fj t kt-

poun enai;

.

;

. Pnu ge. .

. Okon n at stin taut

tttara, n kaston son

totJ stn t tetrpodi;

,

, ;

. Na. .

. Pson on ggnetai; o ; -

- 146 -

tetrkij tosoton; ;

. Pj d o; ;

. Diplsion on stin t

tetrkij tosoton;

-

;

. O m Da. , .

. 'All posaplsion; ;

. Tetraplsion. .

. 'Ap tj diplasaj ra,

pa, o diplsion ll

ttraplsion ggnetai

cwron.

,

, -

.

. 'Alhq lgeij. .

. Tettrwn gr tetrkij stn

kkadeka. oc;

, ;

. Na. .

. 'Oktpoun d p poaj

grammj; oc p mn

tathj tetraplsion;

;

.

. Fhm. .

. Tetrpoun d p tj

misaj tauths tout;

-

;

. Na. .

. Een t d ktpoun o

tode mn diplsin stin,

totou d misu;

( )

( );

. Na. .

- 147 -

. Ok p mn mezonoj stai

tosathj grammj, p

lttonoj d toshsd; o;

-

; ;

. Emoige doke otw. .

. Kaljt gr soi dokon

toto pokrnou. ka moi

lge oc de mn duon

podon n, d tettrwn;

' .

-

-

;

. Na. .

. De ra tn to ktpodoj

cwrou grammn mezw mn

enai tsde tj dpodoj,

lttw d tj tetrpodoj.

-

-

,

-

;

. De. .

. Peir d lgein phlkhn tin

fj atn enai.

-

;

. Trpoda. .

. Okon nper trpouj , t

misu tathj proslhymeqa

ka stai trpouj; do mn

gr ode, d ej ka

nqnde satwj do mn

ode, d ej ka ggnetai

toto t cwron fj.

, -

, -

;

.

. Na. .

. Okon n tde trin ka

tde trin, t lon cwron

trin trj podn ggnetai;

,

,

- 148 -

;

. Fanetai. .

. Trej d trj psoi es

pdej;

;

. 'Enna. .

. Edei d t diplsion pswn

enai podn;

-

;

. 'Okt. .

. Od r p tj trpodj pw

t ktpoun cwron ggnetai.

.

. O dta. .

. 'All p poaj; peir mn

epen kribj ka e m

bolei riqmen, ll dexon

p poaj.

;

-

. -

, .

. 'All m tn Da,

Skratej, gwge ok oda.

, , .

. Lge gr moi s o t mn

tetrpoun toto mn sti

cwron; manqneij;

; ;

. Egwge. .

. Eteron d at prosqemen

n tout son;

;

. Na. .

. Ka trton tde son katrJ

totwn;

;

- 149 -

. Na. .

. Okon

prosanaplhrwsameq n t

n t gwnv tde;

-

;( -

.)

. Pnu ge. .

. Allo ti on gnoit n

tttara sa cwra tde;

-

;

. Na. .

. T on; t lon tde

psaplsion tode ggnetai;

;

. Tetraplsion. .

. Edei d ge diplsion mn

gnsqai o mmnhsai;

-

;

. Pnu ge. .

. Okon stin ath gramm

k gwnaj ej gwnan [tin]

tmnousa dca kaston

totwn tn cwrwn;

;

. Na. .

. Okon tttarej atai

ggnontai gramma sai,

pericousai tout t cwron;

-

-

;

. Ggnontai gr. .

. Skpei d phlkon t stin

toto t cwron;

;

. O manqnw. .

. Oc tettrwn ntwn totwn

misu kstou ksth

- 150 -

gramm pottmhken ntj;

o;

, -

; ;

. Na. .

. Psa on thlikata n

totJ nestin;

-

' ;

. Tttara. .

. Psa d n tde; ;

. Do. .

. T d tttara ton duon t

stin;

;

. Diplsia. .

. Tde on pospoun ggnetai;

;

. 'Oktpoun. .

. 'Ap poaj grammj; ;

. 'Ap tathj. .

. 'Ap tj k gwnaj ej

gwnan teinoshj to

tetrpodoj;

-

;

. Na. .

. Kalosin d ge tathn

dimetron o sofista st

e tatV dimetroj noma,

p tj diamtrou n, j s

fj, pa Mnwnoj, ggnoit

n t diplsion cwron.

, -

, ,

,

.

. Pnu mn on, Skratej. , .

- 151 -

; -

.

2.

; -

.

- 152 -

3. 1:

1

a+b, -

-

. -

;

- 153 -

4. 2.

2

5. ( )

47 (.47):

"

".

: "

,

",

. -

3.

- 154 -

3

3:

( = 90 ). , . ^ .

, , , , , , .

:

- 155 -

. ; -

.

. () (),

.

. () (), .

. .

() (), .

- 156 -

. () (),

.

. () (),

.

. () (),

.

. () (),

.

- 157 -

. () (), .

. .

() (), .

. () (),

.

. () (),

.

- 158 -

. () (),

.

. . . -

: ()+()

6.

"" ;

- 159 -

7. ; ,

.

- 160 -

: ..

: ...

: .

..

: ..

3.

1. 1850 ,

(

350km ), -

1800 .. ,

:

"

50, 50 60.

."

-

.

==50 =60. -

.

R===.

.

-

. -

;

- 161 -

. -

.

. -

.

. R : ...

2. 1 .. ( 10

.. 75 ..).

. , -

"", "", "", "-

", "" ..

"". -

- 162 -

.

(12,1,1 12,1,16)

Estw trgwnon skalhnn xugnion,

o mn ttwn pleur scoinwn ig,

d bsij scoinwn id, d potenousa

scoinwn ie

( -

-

)

eren ato tn kqeton.

poei otwj

poluplasason t ig tj ttonoj

pleurj f' aut gnontai rxq

ka t id tj bsewj f' aut

gnontai r$

ka t ie tj poteinoshj f' aut

gnontai ske.

eta snqej tn tj bsewj polupla-

siasmn ka tn tj poteinoshj

goun t r$ ka t ske gnontai

uka

f' n farei tn poluplasiasmn

tj ttonoj pleurj goun t rxq

loip snb

n L' gnetai rk.

tata mrison par t id tj bsewj

gnontai q

tosotwn scoinwn potom.

13

( ),

14

(

) 15 .

() -

.

:

:

13x13=169.

: 14x14=196.

: 15x15=225.

,

, :

196+225=421.

: 421-169=252.

21 x252=126.

14 :

126:14=9.

-

(

-

).

- 163 -

tata f' aut gnontai pa

t pa farei p to kat tn po-

tenousan pleurn polupla siasmo,

toutsti tn ske loip rmd

n pleur tetragwnik ib

tosotwn st scoinwn kqetoj.

: 9x9=81.

,

: 225-81=144 (-

) .

144 =12 .

12 .

,

, -

,

. -

-

;

3. Leonardo of Pisa ( Fibonacci, 1170 ..

1240 ..) Pisa.

,

, .

- 164 -

, -

. -

( )

, .

1200 25 -

. 1202 -

"Liber Abaci" ( ).

(, , ) ,

.

, -

" ":

" , 30 40 ,

50 . .

, .

;"

-

Filippo Calandri,

1491.

- 165 -

:

30m 40m.

50m. .

.

- 166 -

4. 1261, , Yang Hui "Hsiang

Chieh Chiu Chang Suan Fa" (

" ").

24 . -

13 , -

" ":

" 10 ,

.

,

.

-

3 .

." (

.)

-

:

- 167 -

5. 1637 Descartes

. 1731 ( 100 -

!) Alexis Claude Clairaut, 18 ,

"Recherches sur les courbes double cour-

boure" ( ), :

.

,

.

, ,

R(x1,y1) P(x2,y2)

-

,

:

212

212 )y(y)x(xPR -+-= .

- 168 -

6. 5 -

;

7. 3 -

;

- 169 -

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