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Transcript of Διδακτικη Αξιοποιηση Της Ιστοριας Των Μαθηματικων Η...
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M
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2011
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27 2011
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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.
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, Fasanelli (Fasanelli et al 2000) -
, 1772
Jos Monteiro Da Rocha. ,
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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",
-
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(,
1987) , ,
. -
, . -
(, 1987).
1820, Niels Henrik
Abel, Fasanelli,
(Fasanelli et al 2000):
"
-
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1853 R. Baltzer "Die Ele-
ment der Mathematic", -
,
(, 1987).
16 1865, Augustus de Morgan
London Mathematical Society, -
-
(, -
1987). Fasanelli, (Fasanelli et al
2000):
"
.
, -
. ()
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- 18 -
1873 Eugenio Beltrami,
, "Giornale di matematiche" (
) , (Fasanelli et al 2000):
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()
."
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):
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.
."
1896, Florian Cajori,
"A History of Elementary Mathematics with Hints of Methods of Teaching"
(Cajori 1896, Fasanelli et al 2000):
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Pestalozzi Froebel,
,
-
- 19 -
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1897, Hermann Schubert,
"Mathematical Essays and Recreations", (Fasanelli et al
2000):
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1907, Felix Klein,
"Elementary mathematics from an advanced
standpoint", . ,
(, -
1987):
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1922, . . "-
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1930, Vera Sanford, "A Short History of Mathematics", -
(Fasanelli et al 2000):
" Augustus De Morgan
,
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1931, Mark Yakovlevich Vygodskii, "Foundations
of Infinitesimal Calculus", (Fasanelli et al 2000):
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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" -
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(Toeplitz 2007):
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, -
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, 1919, Mathematical Association
Committee, (Fasanelli et al 2000):
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( 2009). -
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(Tzanakis,
Thomaidis 2000). -
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,
(Jankvist 2009).
,
,
,
(Jahnke 2001, Kleiner
2001).
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(Jankvist 2009).
,
,
Bachelard. O Brousseau -
-
(Brousseau 1997):
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, (Brousseau 1997):
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Brousseau
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Thomaidis, Tzanakis 2007). -
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(in-issues) ( 2009, Jank-
vist 2009).
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- 28 -
.
(
2009). -
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, - (Philippou, Christou
1998).
(Gulikers, Blom 2001, Thomaidis, Tzanakis 2007).
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,
(Tzanakis, Thomaidis
2000).
(Jankvist 2009).
(2009).
, -
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(Jankvist 2009).
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- (meta-issues) ( 2009, Jankvist 2009).
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- 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), -
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. (a posteriori)
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2009):
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, F. Klein, o G. Plya
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1979). Jankvist (Jankvist 2009) 1 ("
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. (Siu 2006, 2009).
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(Tzanakis, Arcavi et al 2000, Siu 2006, 2009).
. (Siu 2006, 2009).
( 2009). -
(Tzanakis, Arcavi et al 2000). -
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(,
2006, 2009).
Jankvist 1.. 1.. -
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(Jankvist 2009).
Tzanakis, Arcavi et
al " " (isolated factual in-
formation) " " (historical snippets) -
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Siu (Siu
2000) . Jankvist
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T. Lindstrm "Kalkulus" 1995, ,
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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), -
-
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(Tzanakis, Arcavi et al 2000, Jankvist 2009).
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10 20 . -
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et al 2000).
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Roskilde (Tzanakis, Arcavi et al
2000, Niss 2001, Jankvist 2009).
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2000, 2009):
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Tzanakis,
Arcavi et al , -
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2009):
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al 2000, 2009):
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( 2009):
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1. "" -
(historical snippets)
, Tzanakis, Arcavi et al "-
" (historical snippets) (Tzanakis, Arcavi et al 2000)
(format) (content) .
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(Tzanakis, Arcavi
et al 2000):
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( 3 -
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).
3. , videos , (films and other visual means)
-
,
'
(Tzanakis, Arcavi et al 2000). -
, "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).
,
-
(Tzanakis, Arcavi et al 2000).
-
- 67 -
posters
,
, -
, ..
(Tzanakis, Arcavi et al 2000).
.
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):
. -
,
. ,
. -
(
,
),
. , , -
.
-
. ,
-
-
-
. , -
, .
,
(Jahnke et al 2000) ( 1
).
-
- 71 -
. -
"Geometry"
Descartes.
Descartes , -
.
-
.
,
' (Jahnke
et al 2000).
:
-
. ,
.
,
. -
,
.
:
-
,
, -
.
,
-
,
(Barbin 1994). -
-
- 72 -
-
,
(Jahnke et al 2000) ( 2
).
-
(representational systems) -
.
,
-
. , -
,
(Jahnke et al 2000).
,
-
"" (Jahnke et al
2000).
:
-
, -
-
.
, -
.
-
,
, ,
.
-
-
- 73 -
" ", "" ,
(Jahnke et al 2000) (
2 ).
:
,
. -
' ""
, -
, / .
"" - -
(Jahnke et al 2000).
:
,
, -
. Jahnke et al -
, Dedekind "Essays on the theory of num-
bers" 1872 -
.
, , -
-
. -
(Jahnke et al 2000).
-
,
,
-
- 74 -
-
,
.
( 2009). Franois Vite
-
Leonhard Euler -
(-
, ).
-
( 3 -
).
:
.
, -
-
.
, -
.
, -
. ,
, -
, , -
, .. (Jahnke et al 2000).
-
- 75 -
:
-
. -
.
(Jahnke et al 2000) ( -
5 ).
, :
(synchronous) (diachronous) -
. -
,
, ,
, ,
. -
-
, -
(Jahnke et al 2000).
. -
,
. -
-
.
,
.
-
- 76 -
() .
-
.
:
, -
(Jahnke et al 2000) ( 3
).
. -
:
,
.
-
.
,
. -
-
(Jahnke et al 2000).
, :
,
-.
, -
. ,
. -
-
- 77 -
,
( )
, -
. -
:
( 1 -
).
,
( , ,
, -
).
,
(Jahnke et al 2000).
:
, Jahnke et al, -
: "" (direct)
, "" (indirect) -
(Jahnke et al 2000).
-
. -
,
. -
.
,
-
-
- 78 -
(Jahnke et al 2000) ( -
2 ).
, ,
-
.
-
,
(Jahnke et al 2000) (
3 ).
-
( 5 -
). (
) ( ) -
( ).
,
(Jahnke et al 2000).
.
-
' .
,
.
, .
-
(Jahnke et al 2000).
-
, -
-
- 79 -
. -
, -
.
. -
(Jahnke et al 2000).
:
-
.
.
. -
.
,
. -
-
.
(Jahnke et al 2000).
-
(debates) . -
, . -
. -
, -
.
,
-
- 80 -
, .
, -
, -
, (Jahnke et
al 2000).
:
-
-
.
.
,
-
, -
. , -
,
, -
(Jahnke et al 2000).
:
-
,
. -
-
, -
. -
-
- 81 -
, ' . -
,
-
-
(Jahnke et al 2000).
:
,
' .
: -
( -
)
( -
,
, ) (Jahnke et al
2000).
:
, -
-
. -
.
-
, -
, -
.
, -
-
- 82 -
, -
.
-
(Jahnke et al 2000).
:
-
, .
, .
, -
.
(Jahnke et al 2000) ( 2 -
).
:
, , -
(Jahnke et al 2000).
-
-
.
,
, -
, -
.
-
- 83 -
.
, -
, . -
-
. -
.
3. " " (worksheets)
. -
, -
, .
, ,
. ,
, ,
( ),
(Tzanakis, Arcavi et
al 2000).
-
, -
,
( / -
) -
( 1 -
).
,
-
- 84 -
( 2 2 -
), -
( 1 -
).
, "-
" .
, -
-
( -
3 ). -
-
( 1
). -
. -
( -
2 ).
,
, / -
( -
3, 3 3 )
(Tzanakis, Arcavi et al 2000).
4. " " (historical packages)
Bruckheimer Arcavi " " (historical hap-
penings) ( " " (historical packages))
, 2 3
(Bruckheimer, Arcavi 2000).
-
- 85 -
,
.
-
(Bruckheimer, Arcavi 2000). ""
, 3 4 ,
'
,
( -
4 2 )
(Tzanakis, Arcavi et al 2000).
,
-
.
. -
,
, , -
( -
), -
( -
-
, , ,
, ,
..) (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.)
-
-
, ,
,
, , -
,
-
, , -
(Tzanakis, Arcavi et al 2000).
-
( 2
).
:
: , 2000 ..,
-
. , -
, , , 2
2
++ .
-
- 87 -
, , ,
2
2 + .
. -
-
(Eves 1969).
:
-
. ""
.
, -
. 16
, Vite " ",
,
Fermat Descartes.
( -
) ( -
) -
(Eves 1969).
: Bertrand 1889
"Calcul des probabilits" ( ).
, Bertrand ,
,
.
Bertrand
, ,
(Tzanakis, Arcavi et al 2000).
-
- 88 -
2. / - (historical problems)
-
. , -
(Tzanakis, Arcavi et al 2000):
. :
( , -
)
,
.
. :
Fermat ( xn+yn=zn
n 2)
. -
Goldbach (
)
, 100.
. , :
,
.
. /
:
Galois '
Weierstrass
.
. :
, Lagrange
-
- 89 -
(
).
. ""
1.
(mechanical in-struments)
: -
( -
5, 5 5 )
(
2 )
(Tzanakis, Arcavi et al 2000). -
-
, -
.
, -
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
) (Tzanakis, Arcavi et al 2000).
: -
( -
-
- 91 -
, ) -
. -
-
. , -
, -
"" ,
( 1
) (Tzanakis, Arcavi et
al 2000).
: -
.
. -
.
Al-
Khwarizmi Descartes, -
(Tzanakis,
Arcavi et al 2000).
: -
Fermat
Pascal : "
,
.
.
, ;"
, -
-
(Tzanakis, Arcavi et al 2000).
-
- 92 -
3. (plays)
,
/ .
, -
-
. Tzanakis, Arcavi et al -
(Tzanakis,
Arcavi et al 2000):
. -
, -
(, , )
. Tzanakis, Arcavi et al -
,
Galois (Tzanakis, Arcavi et al 2000). -
' ,
" -
".
.
.
-
,
( 4 -
) (Tzanakis, Arcavi et al 2000).
. (the WorldWide Web).
(World Wide Web Internet)
-
- 93 -
, : -
.
, Tzanakis, Arcavi et al
-
, , Zehavi
1999 Weizmann Institute.
" " (face to face) -
.
(hypertexts)
. -
, -
site ,
. -
forum -
, -
.
-
-
' ,
-
(Tzanakis, Arcavi et al 2000).
, -
. -
, -
,
, . , (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).
-
. , -
, -
.
(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 -
, -
(Radford, Boero, Vasco 2000).
, 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)
-
(Radford, Boero, Vasco 2000).
"
", -
,
, .
, -
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,
(Gulikers, Blom 2001).
Eagle 1998 ( "A typical slice", Mathematics in School, 27(4), pp. 37-39)
-
.
-
- 111 -
,
( 3r34V = ).
(Gulikers, Blom 2001).
MacKinnon 1989 ( "What do you doabout 4r2?", The Mathematical Gazette, 73(464), pp. 107 110) -
-
, -
(Gulikers, Blom 2001).
Fhrer 1991 ( "Historical stories in themathematics classroom", For the Learning of Mathematics, 11(2), pp. 24
31)
, -
.
" -".
Fhrer , Vite Des-
cartes -
(Gulikers, Blom 2001).
1987, Paris VIII.R.E.M., " "
. ,
-
, -
( 2000)
Corris 1990 ( "Experimental pi", Mathe-matics in School, 19(1), pp. 18 21)
,
(Gulikers, Blom 2001).
-
- 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, -
-
,
(Gulikers, Blom 2001).
1996 Makowski Strong ( "Sizing up earth:
A universal method for applying Eratosthenes' earth measurement", Jour-
nal of Geography, 95(4), pp. 174 179) ,
. -
-
. -
(Gulikers, Blom 2001).
1990 ( " ", ', 7(25), pp. 27 41) -
( -
). -
, "" (2
..), -
"" ,
16 .. -
-
-
- 113 -
, -
, -
-
( 1990).
Hogendijk 1996 ( "Een workshop overIraanse mozaken", Nieuwe Wiskrat, 16(2), pp. 38 42) -
,
.
-
.
. Hogendijk
(Gulikers, Blom 2001).
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
.
(Gulikers, Blom 2001).
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 -
.
,
. -
'
-
(Gulikers, Blom 2001).
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
(Gulikers, Blom 2001).
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). -
,
-
(Gulikers, Blom 2001).
Lumpkin 1978 ( "A mathematics club pro-ject from Omar Khayyam", Mathematics Teacher 71(9), pp. 740 744) -
.
-
- 116 -
-
,
. : -
. -
,
(Gulikers, Blom 2001).
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) -
, -
.
,
(Gulikers, Blom 2001).
1991 Artmann ( "Quadratische Probleme in Euklids'Elementem' und ihre Behandlung im Mathematikunterricht", Didaktik der
Mathematik, 19(2), pp. 94 110)
,
,
, , (iso-
morphic). -
-
- 118 -
,
(Gulikers, Blom 2001).
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, -
: , -
-
,
(Gulikers, Blom 2001).
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')
, -
(Gulikers, Blom 2001).
-
- 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)
1679 2929161 289
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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