Variation als fundamentale Idee

Preprint Nr. 361
Variation — eine fundamentale Idee
Hans Schupp zu seinem 80. Geburtstag gewidmet
Horst Hischer
Saarbrücken 2015
Fachrichtung 6.1 – Mathematik
Universität des Saarlandes
Preprint No. 361
submitted: June 10, 2015
Variation — eine fundamentale Idee
Hans Schupp zu seinem 80. Geburtstag gewidmet
Horst Hischer
Saarland University
Department of Mathematics
P.O. Box 15 11 50
66041 Saarbrücken
Germany
[email protected]
Edited by
FR 6.1 — Mathematik
Universität des Saarlandes
Postfach 15 11 50
66041 Saarbrücken
Germany
Fax:
e-Mail:
WWW:
+ 49 681 302 4443
[email protected]
http://www.math.uni-sb.de/
HORST HISCHER
9DULDWLRQ²HLQHIXQGDPHQWDOH,GHH
Hans Schupp zu seinem 80. Geburtstag gewidmet
3URORJ
'DVXPIDQJUHLFKHXQGYLHOVHLWLJH¯XYUHYRQ+$16 6&+833XPIDVVWDNWXHOOLQVJHVDPW
3XEOLNDWLRQHQGDUXQWHUGUHL]HKQ0RQRJUDSKLHQXQGHLQ6FKXOEXFKZHUNZREHLDQ]XPHU
NHQLVWGDVVHUDQVWHOOHYRQÄ'LGDNWLNGHU0DWKHPDWLN³IUGLH%H]HLFKQXQJÄ'LGDNWLNGHV
0DWKHPDWLNXQWHUULFKWV³SOlGLHUW±VRKDWWHVHLQ/HKUVWXKODQGHU8QLYHUVLWlWGHV6DDUODQGHV
GLH'HQRPLQDWLRQÄ0DWKHPDWLNXQG'LGDNWLNGHV0DWKHPDWLNXQWHUULFKWV³*DQ]LQGLHVHP
6LQQH VFKULHE PLU +DQV 6FKXSS LP 5DKPHQ XQVHUHU IDFKOLFKHQ .RUUHVSRQGHQ] „Es geht
mir immer nur um Mathematikunterricht, um Mathematik nur in diesem Zusammenhang.“
=ZHLIU6FKXSSZLFKWLJH$VSHNWHVHLQHVGLGDNWLVFKHQ6FKDIIHQVVHLHQLPYRUOLHJHQGHQ
%HLWUDJKHUDXVJHJULIIHQXQGEH]JOLFKP|JOLFKHU=XVDPPHQKlQJHEHWUDFKWHWÄVariation“
XQG„fundamentale Ideen“ +HUYRUKHEHQVZHUWLVWGDEHLGDVVIULKQÄ2SWLPLHUHQ³SDUV
SURWRWRHLQHUVHLWVHLQH„fundamentale Idee der Mathematik“LVWMHGRFKQLFKWGHV0DWKH
PDWLNXQWHUULFKWV]XJOHLFKDEHUDOV„Leitlinie des Mathematikunterrichts“ IXQJLHUW±HLQH
VRZRKO IHLQVLQQLJH DOV DXFK ZLFKWLJH XQG YRU DOOHP EHDFKWHQVZHUWH 8QWHUVFKHLGXQJ 'LH
9LHOIDOWGHULQGHU/LWHUDWXU]XILQGHQGHQ'HXWXQJHQXQG%HGHXWXQJHQYRQÄIXQGDPHQWDOH
,GHH³ PDJ LKUHQ *UXQG ]7 DXFK LQ HLQHU GDQQ VR QLFKW PLWJHGDFKWHQ 8QWHUVFKHLGXQJ
KDEHQ GLH P|JOLFKHUZHLVH GDV $XIWUHWHQ XQWHUVFKLHGOLFKHU 7HUPLQL ZLH Ä]HQWUDOH ,GHHQ³
Ä/HLWLGHHQ³XVZHUNOlUWZDVXQWHUVXFKHQVZHUWZlUHKLHUDEHUQLFKWXQWHUVXFKWZLUG
1DFKIROJHQGJHKWHVGDKHU±JDQ]LP6LQQHYRQ+DQV6FKXSS±XPfundamentale Ideen
der Mathematik GHUDUW GDVV GDQQ JHZLVVH VROFKHU ,GHHQ GHQ 0DWKHPDWLNXQWHUULFKW leiten
N|QQHQXQGLKQstrukturierenKHOIHQ'RFKZDVNDQQDOVÄIXQGDPHQWDOH,GHH³JHOWHQ"
)XQGDPHQWDOH,GHHQGHU0DWKHPDWLN².ULWHULHQ
'LH 'LVNXVVLRQ XP IXQGDPHQWDOH ,GHHQ HQWIDFKWH EHNDQQWOLFK GHU 3V\FKRORJH -HURPH
6H\PRXU%UXQHULQVHLQHPIDFKEHUJUHLIHQGHQ%XFKÄ'HU3UR]H‰GHU(U]LHKXQJ³PLWGHU
7KHVHGDVV
GLH*UXQGODJHQHLQHVMHGHQ)DFKHVMHGHP0HQVFKHQLQMHGHP$OWHULQLUJHQGHLQHU)RUPEHL
JHEUDFKWZHUGHQN|QQHQ>XQGGDVV@GLHEDVDOHQ,GHHQGLHGHQ.HUQDOOHU1DWXUZLVVHQVFKDIW
XQG0DWKHPDWLNELOGHQXQGGLHJUXQGOHJHQGHQ7KHPHQGLHGHP/HEHQXQGGHU'LFKWXQJ
LKUH)RUPYHUOHLKHQHEHQVRHLQIDFKZLHGXUFKVFKODJHQGVLQG
6HLW6WDQG0DLYJOKWWSPDGLSHGLDGHZLNL+DQVB6FKXSS
9JOEHWUÄIXQGDPHQWDOH,GHHQ³>[email protected]Ä2SWLPLHUHQDOVIXQGDPHQWDOH,GHHGHU0D
WKHPDWLNXQGDOV/HLWOLQLHLP0DWKHPDWLNXQWHUULFKW³XQGEHWUÄ9DULDWLRQ³YRUDOOHP>6FKXSS@
>%UXQHU@GLHHQJOLVFKH2ULJLQDOIDVVXQJÄ7KH3URFHVVRI(GXFDWLRQ³HUVFKLHQ
+RUVW+LVFKHU9DULDWLRQ±HLQHIXQGDPHQWDOH,GHH
6R JHKW HV DOVR EHUHLWV EHL %UXQHU XP grundlegende Ideen und Themen der WissenschaftenGLHIUGHQIDFKOLFKHQ8QWHUULFKWÄHEHQVRHLQIDFKZLHGXUFKVFKODJHQG³VLQGXQG
GLHDOVRLQGLHVHP6LQQJHPl‰6FKXSS]XLeitlinien des UnterrichtsZHUGHQN|QQHQ'DV
HUIRUGHUW MHGRFK LQ %H]XJ DXI GHQ 0DWKHPDWLNXQWHUULFKW HLQH GLVNXUVLY HUDUEHLWHWH 9HU
VWlQGLJXQJ GDUEHU ZDV ]X ÄIXQGDPHQWDOHQ ,GHHQ GHU 0DWKHPDWLN³ ]lKOHQ VROO JHQDXHU
REPDQDN]HSWLHUEDUH.ULWHULHQILQGHQNDQQGLHGLH(UVWHOOXQJHLQHV.DWDORJVVROFKIXQGD
PHQWDOHU,GHHQHUP|JOLFKHQZREHLVROFKHLQ.DWDORJRIIHQXQGYHUlQGHUEDUGHQNEDULVW
*UXQGVlW]OLFKH%HLWUlJH]XU3Ul]LVLHUXQJGHVVHQZDVXQWHUÄIXQGDPHQWDOHQ,GHHQ³YHU
VWDQGHQZHUGHQNDQQILQGHW PDQXDEHL>6FKZHLJHU@HUJlQ]WXP ZHVHQWOLFKH%H
WUDFKWXQJHQYRQ+DQV6FKXSS'DVOlVVWVLFK]XVDPPHQIDVVHQGZLHIROJWGDUVWHOOHQ
Fundamentale Ideen der Mathematik ...
« VLQGDXI]HLJEDU
LQGHU historischen Entwicklung GHU0DWKHPDWLN
« VLQGJHZLVVHUPD‰HQDOVArchetypen des Handelns und Denkens,
DXFKaußerhalb der MathematikXQG
B
vor der wissenschaftlichen Aufnahme DXIILQGEDU
« JHEHQ]XPLQGHVWSDUWLHOO
6
$XIVFKOXVVEHU das Wesen der Mathematik
« VLQGWUDJIlKLJXPFXUULFXODUH(QWZUIHGHV
Ú
0DWKHPDWLNXQWHUULFKWVvertikal ]XJOLHGHUQ
« VLQGJHHLJQHWGHQ0DWKHPDWLNXQWHUULFKW
EHZHJOLFKHUXQGdurchsichtiger ]XJHVWDOWHQ
+LVWRUL]LWlW
$UFKHW\SL]LWlW
:HVHQWOLFKNHLW
'XUFKJlQJLJNHLW
7UDQVSDUHQ]
'LHersten drei KriterienVLQGdeskriptiv±VLHN|QQHQKLOIUHLFKVHLQEHLGHUSucheQDFKIXQ
GDPHQWDOHQ,GHHQ'LHVH.ULWHULHQJHKHQQDKH]XZ|UWOLFKDXI>6FKXSS@]XUFN
'LH letzten beiden Kriterien KLQJHJHQ VLQG präskriptiv RGHU JDU normativ GHQQ VLH EH
VFKUHLEHQ EHUSUIEDUH Erwartungen DQ HLQHQ JHPl‰ VROFKHQ ,GHHQ NRQ]LSLHUWHQ 8QWHU
ULFKW'LHGHVNULSWLYHQ.ULWHULHQNHQQ]HLFKQHQfundamentale Ideen der MathematikLPYRU
VHLWLJJHQDQQWHQ6LQQQLFKWDEHUGLHEHLGHQOHW]WHQ.ULWHULHQGLHGDUOHJHQLQZLHIHUQVROFKH
IXQGDPHQWDOHQ,GHHQJJI]XLeitlinien des MathematikunterrichtsZHUGHQN|QQHQ
'DUEHU KLQDXV LVW ]X EHUFNVLFKWLJHQ GDVV GLH Bedeutsamkeit von Ideen PLW ]XQHK
PHQGHU3Ul]LVLRQDEQLPPWZDV]XHLQHPZHLWHUHQYLHUWHQGHVNULSWLYHQ.ULWHULXPIKUW
Fundamentale Ideen der Mathematik ...
|
« VLQGHKHUYDJHDOVSUl]LVH
9DJKHLW
6FKZHLJHUKHEWGLHVHÄ9DJKHLW³EH]JOLFKGHV3UREOHPVGHU.HQQ]HLFKQXQJXQGGHV$XI
VXFKHQV NRQNUHWHU IXQGDPHQWDOHU ,GHHQ GXUFKDXV SRVLWLY KHUYRU GHQQ Ideen VHLHQ YDJH
XQGVLHEUlXFKWHQNHLQH'HWDLOOLHUXQJZDVVLHHUVWJHZLFKWLJPDFKH'DVOlVVWVLFKGDQQ
DOVHLQH UnschärferelationIUIXQGDPHQWDOH,GHHQIRUPXOLHUHQ
x :HUGHQIdeen zunehmend fundamentalerVRZLUGLKUH%HVFKUHLEXQJzunehmend vager
GKVLHZHUGHQ]XQHKPHQG allgemeiner und unschärfer±XQGXPJHNHKUW
$XV>+LVFKHU@EDVLHUHQGDXI>+LVFKHU@XQG>+LVFKHU@
'LH (UJlQ]XQJHQ „Handeln“ XQG „wissenschaftliche Aufnahme“ LP ]ZHLWHQ .ULWHULXP ZHUGHQ EHL
>6FKZHLJHU@QRFKQLFKW]LWLHUWDEHU>6FKXSS@IKUWVLHDXIXQGEH]HLFKQHWKLHU„Handeln
und Denken“DOV„(… höchst sublimierte) Tätigkeiten“
>6FKZHLJHU@YJODXFK>6FKZHLJHU@
+RUVW+LVFKHU9DULDWLRQ±HLQHIXQGDPHQWDOH,GHH
>+H\PDQQ@VWHOOWVHFKVYRQLKPVRJHQDQQWHÄ]HQWUDOH,GHHQ³YRUGLHIdee GHU Zahl
GHV MessensGHV funktionalen ZusammenhangsGHV räumlichen StrukturierensGHV AlgorithmusXQGGHV mathematischen Modellierens6LHVLQGVSlWHUPRGLIL]LHUWDOVIQIÄ/HLW
LGHHQ³LQGLHVRJÄ%LOGXQJVVWDQGDUGV³GHU.0.PLWHLQJHJDQJHQZahl, Messen, Raum
und Form, funktionaler Zusammenhang, Daten und Zufall+LHUIlOOW]XQlFKVWDXIGDVVGLH
VHMHZHLOLJHQÄ,GHHQ³YHUVFKLHGHQHVSUDFKOLFKH(EHQHQEHWUHIIHQ%HLVSLHOVZHLVHZLUGPLW
Ä=DKO³HLQBegriffEH]HLFKQHWPLWÄ0HVVHQ³KLQJHJHQHLQHHandlung=ZDUNDQQPDQHLQ
ZHQGHQGDVVHWZDGLHÄ,GHHGHU=DKO³NXOWXUKLVWRULVFKNDXPRKQHGLHÄ,GHHGHV=lKOHQV³
HQWVWHKHQNRQQWHXQGGHQQRFKEOHLEWGLH)UDJHEHVWHKHQZHVKDOEGLHVHÄ,GHHQ³VSUDFKOLFK
VR XQJOHLFKDUWLJ IRUPXOLHUW ZXUGHQ +LOIUHLFK XQG EHU]HXJHQG LVW LQ GLHVHP =XVDPPHQ
KDQJIROJHQGH(LQJUHQ]XQJYRQ>6FKZHLJHU@
(LQHIXQGDPHQWDOH,GHHLVWHLQ%QGHOYRQ+DQGOXQJHQ6WUDWHJLHQXQG7HFKQLNHQ>@
'LHVHU Aspekt der Handlung XQWHU (LQVFKOXVV YRQ Strategien XQG Techniken LVW GDKHU ±
6FKZHLJHU IROJHQG ± PLW DOV wesentlich IU IXQGDPHQWDOH ,GHHQ DQ]XVHKHQ'DV EHWULIIW
GDQQIXQGDPHQWDOH,GHHQVRZRKOLPGHVNULSWLYHQDOVDXFKLPQRUPDWLYHQE]ZSUlVNULSWL
YHQ6LQQ'LHVH$XIIDVVXQJZLUGGDGXUFKJHVWW]WGDVVGLH(QWVWHKXQJE]Z%LOGXQJHLQHV
%HJULIIV±HLQHVVXEMHNWLYHQ.RQVWUXNWV±DQHLQHLQGLYLGXHOOH+DQGOXQJJHEXQGHQLVWQlP
OLFK DQ GDV Begreifen 8QG GDV %HJUHLIHQ EH]HLFKQHW DOV +DQGOXQJ GHQ 3UR]HVV GHU (QW
ZLFNOXQJ GHV %HJULIIV im ,QGLYLGXXP XQG durch GDV ,QGLYLGXXP DOVR GLH ontogenetische
Begriffsentwicklung LP .RQWUDVW ]XU NXOWXUKLVWRULVFKHQ %HJULIIVHQWZLFNOXQJ'DPLW VLQG
GDVBegreifenDOV+DQGOXQJXQGGHUBegriffLP3UR]HVVGHU%HJULIIVHQWZLFNOXQJXQWUHQQ
EDUDQHLQDQGHUJHNRSSHOW±ZREHLGLHVHU3UR]HVVP|JOLFKHUZHLVHÄKDQGHOQG³EHJLQQWDQ
GHUHUVHLWV ZRKO SULQ]LSLHOO OHEHQVODQJ XQDEJHVFKORVVHQ EOHLEW 6FKOLH‰W PDQ VLFK GLHVHU
$XIIDVVXQJDQVREHGHXWHWGDVDOOHUGLQJVGDVVEHLVSLHOVZHLVHÄ=DKO³IUVLFKJHQRPPHQ
QRFK NHLQH IXQGDPHQWDOH ,GHH EH]HLFKQHW DEHU DXFK QLFKW GDV Ä=lKOHQ³ IU VLFK JHQRP
PHQVRQGHUQGDVVYLHOPHKUHUVWEHLGHJHPHLQVDPGLHIXQGDPHQWDOH,GHHDXVPDFKHQDOVR
Ä=DKO³DOV%H]HLFKQXQJIUHLQHQgrundlegenden BegriffXQGÄ=lKOHQ³DOV%H]HLFKQXQJIU
GLH]XJHK|ULJHgrundlegende HandlungNXU]XP
x (LQH NRQNUHWH fundamentale Idee RIIHQEDUW VLFK VRZRKO DOV grundlegende Handlung DOV
DXFKDOVgrundlegender BegriffLQVJHVDPWLQGHUHQÄ6\PELRVH³DOVHLQHDyas
6RN|QQWHPDQGDQQ]%HLQLJHGHUYRQ+H\PDQQHU|UWHUWHQÄ]HQWUDOHQ,GHHQ³ZLHIROJW
NHQQ]HLFKQHQZahl & ZählenMaß & MessenAlgorithmus & Algorithmieren.'HQQRFK
ZLUGPDQJHQHLJWVHLQLQ.XU]NRPPXQLNDWLRQYRQÄ=DKO³DOVHLQHUIXQGDPHQWDOHQ,GHH]X
VSUHFKHQZREHLPDQGDEHLLP%OLFNEHKDOWHQVROOWHGDVVGLHKLHUPLWYHUEXQGHQH,GHHXQ
WUHQQEDUDXFKGDVÄ=lKOHQ³EHWULIIW±XQGYLFHYHUVD
,QGLHVHP6LQQHUZHLVWVLFKGDVVEHLVSLHOVZHLVHDXFKGLHMittelwertbildung±JHQDXHU
0LWWHOZHUW0LWWHOZHUWELOGXQJ±GLHJHQDQQWHQ.ULWHULHQIUIXQGDPHQWDOH,GHHQHUIOOW (VIlOOWDEHUDXIGDVVGLHVH,GHHGHUMittelwertbildungQLFKWLQGHQ5DKPHQGHUVHFKVYRQ
+H\PDQQJHQDQQWHQ,GHHQKLQHLQSDVVW
$OV:LGHUVSUXFKLQVLFKHLQEHGDXHUOLFKHUVSUDFKOLFKHU/DSVXVÄ%LOGXQJ³LVWQLFKWVWDQGDUGLVLHUEDU
'LHVKDW+DQV6FKXSSPLUJHJHQEHULQYLHOHQ*HVSUlFKHQEHWRQWXQGHVLVWGHU7HQRULQ>6FKXSS@
XQG6FKXSS>@
9JO>+LVFKHU@>+LVFKHU@>+LVFKHU@]XVDPPHQJHIDVVWLQ>+LVFKHU±@
+RUVW+LVFKHU9DULDWLRQ±HLQHIXQGDPHQWDOH,GHH
'HQQRIIHQEDULVWGLH,GHHGHU0LWWHOZHUWELOGXQJWURW]GHULKUDQKDIWHQGHQ9DJKHLWHV
JLEWHLQH)OOHYRQÄ0LWWHOZHUWHQ³YLHONRQNUHWHUDOVGLHVHFKV+H\PDQQVFKHQ,GHHQ$Q
GHUHUVHLWVZLUNWVLHJHUDGHZHJHQLKUHU.RQNUHWKHLWYLHOXQWHUULFKWVQlKHUDOVGLHHKHU]XDOO
JHPHLQHQ+H\PDQQVFKHQ,GHHQGLHZLHGHUXPZHJHQLKUHU$OOJHPHLQKHLWXQGGDPLWJHULQ
JHQ $Q]DKO JHHLJQHW HUVFKHLQHQ LQ NQDSSHU )RUP JUXQGVlW]OLFKH LQKDOWOLFKH $VSHNWH GHV
0DWKHPDWLNXQWHUULFKWVIHVW]XPDFKHQ(LQH/|VXQJDXVGHP'LOHPPDEHVWHKWLQGHU.RQ
]HSWLRQfundamentale Ideen zumindest in zwei verschiedenen, prinzipiell offen zu denkenden Ebenen mit unterschiedlichem KonkretisierungsgradDQ]XRUGQHQZLHHV$EE]HLJW
$EE0|JOLFKH(EHQHQXQG%H]LHKXQJHQIXQGDPHQWDOHU,GHHQ
=ZLVFKHQGHQLQEHLGHQ(EHQHQDXIJHIKUWHQ,GHHQEHVWHKHQLQKDOWOLFKH=XVDPPHQKlQJH
ZLHVLHXQYROOVWlQGLJDQJHGHXWHWVLQG,P8QWHUULFKWZLUGHVGDQQDXIZHLWHUHQQDFKJHRUG
QHWHQ(EHQHQXQWHU%HUFNVLFKWLJXQJGHUQRUPDWLYHQ.ULWHULHQXPGDVWecken von Grundvorstellungen zuBegriffen, Verfahren XVZJHKHQGLHPLWGLHVHQ,GHHQ]XVDPPHQKlQJHQ
Ä9DULDWLRQ³DOVIXQGDPHQWDOH,GHHGHU0DWKHPDWLN
+LHUVROOQXQGDUJHOHJWZHUGHQGDVVGDVYRQ+DQV6FKXSSYLHOIDFKSURSDJLHUWHGLGDNWLVFKH
3ULQ]LSGHUÄ9DULDWLRQ³±XQG]ZDUDOVÄ$XIJDEHQYDULDWLRQ³±DXFKLP=XVDPPHQKDQJPLW
GHPYRQLKPIUZLFKWLJHUDFKWHWHQ.RQ]HSWIXQGDPHQWDOHU,GHHQLP.RQWH[WYRQÄ/HLWOL
QLHQGHV0DWKHPDWLNXQWHUULFKWV³]XVHKHQLVW
=XQlFKVW LVW DQ]XPHUNHQ GDVV GHU 7HUPLQXV Ä9DULDWLRQ³ SHU VH EHUHLWV GLH HUZlKQWH
'\DVYRQÄ%HJULII³XQGÄ+DQGOXQJ³YHUHLQWGHQQGLH%HGHXWXQJYRQÄ9DULDWLRQ³LVWÄ9HU
lQGHUXQJ³ZDVZHJHQGHU(QGXQJÄXQJ³YHUGHXWOLFKGDVVKLHU]XJOHLFK3URGXNWXQG3UR
]HVVJHPHLQWVLQGÄ9DULDWLRQ³LVWDOVRHLQH+DQGOXQJGHUHQ(UJHEQLVHLQHÄ9DULDWLRQ³LVW
hEHUSUIHQZLUQXQNXU]GLHGHVNULSWLYHQ.ULWHULHQIUIXQGDPHQWDOH,GHHQ
+LVWRUL]LWlW'UHL%HLVSLHOHVHLHQVNL]]LHUW%HUHLWVLQGHUgriechischen AntikeZDU
GLHÄ9DULDWLRQ³DOV+DQGOXQJHLQH0|JOLFKNHLW]XU3UREOHPO|VXQJVREHLGHQÄ(LQVFKLHEH
O|VXQJHQ³ GHU GUHL EHUKPWHQ NODVVLVFKHQ 3UREOHPH .UHLVTXDGUDWXU :LQNHOGUHLWHLOXQJ
:UIHOYHUGRSSHOXQJ'LHVH/|VXQJHQE]Z/|VXQJVYHUVXFKHEDVLHUWHQDXIHLQHU9DULDWLRQ
YRQ 3XQNWHQ RGHU DQGHUHQ 2EMHNWHQ LP 6LQQH HLQHU Bewegungsgeometrie ]XPLQGHVW DXI
HLQHUJHGDFKWHQE]ZYRUJHVWHOOWHQQLFKWXQEHGLQJWSK\VLVFKHQ%HZHJXQJ,QGLHVHP6LQQ
HQWVWDQGHQDXFKGLYHUVH.XUYHQDOVÄ2UWVNXUYHQ³GXUFKUHDOHRGHUJHGDFKWH%HZHJXQJYRQ
3XQNWHQ 6WUHFNHQ RGHU DQGHUHQ 2EMHNWHQ 'LH Ä9DULDWLRQ³ ZDU DOVR VFKRQ ]X %HJLQQ GHU
0DWKHPDWLN DOV :LVVHQVFKDIW HLQ ZHVHQWOLFKHV 3ULQ]LS GHU (UNHQQWQLVJHZLQQXQJ 'LH
/|VXQJ YRQ ([WUHPDOSUREOHPHQ LVW RKQH Ä9DULDWLRQ³ DOV HLQHP JHGDFKWHQ PHWKRGLVFKHQ
:HUN]HXJDOVRHLQHPÄ'HQN]HXJ³NDXPYRUVWHOOEDUEHJLQQHQGHWZDEHLFermatXQGZHL
WHUEHLJakob I BernoulliXQGJohann I BernoulliPLWGHUÄ(UILQGXQJ³GHU9DULDWLRQVUHFK
+RUVW+LVFKHU9DULDWLRQ±HLQHIXQGDPHQWDOH,GHH
QXQJELVKLQ]XUKHXWLJHQ$QDO\VLV6RLVWGLH(SVLORQWLNGHUUHHOOHQ$QDO\VLVDOVJHGDFKWH
9DULDWLRQDXIIDVVEDUÄ]XMHGHPeJLEWHVHLQdVRGDVV«³XQGLQGHU)XQNWLRQDODQDO\VLV
ZHUGHQ)XQNWLRQHQPLWEHVWLPPWHQ=LHOVHW]XQJHQYDULLHUW%HUHLWVLP-KKDWWHGHU
JHQLDOHJakob SteinerHLQHOHGLJOLFKYRUJHVWHOOWHBewegungsgeometrieHQWZLFNHOWZDVZLU
KHXWHPLWJUR‰DUWLJHU6RIWZDUHQDFKYROO]LHKHQXQGIRUWVHW]HQN|QQHQ
$UFKHW\SL]LWlWÄ9DULDWLRQ³LVWNHLQHVZHJVQXUIUGLH0DWKHPDWLNW\SLVFKVRQGHUQVLH
LVWDXFKDX‰HUKDOEEHGHXWVDP6RVSLHOWVLHLQGHU0XVLNEHL.RPSRVLWLRQHQDOVÄ9DULDWLRQ
HLQHV7KHPDV³HLQHJUR‰H5ROOHZHLWHUKLQWULWWVLH]%LQGHU/LQJXLVWLNXQGLQGHU%LROR
JLHDXI$EHUQLFKWQXULP:LVVHQVFKDIWVEHUHLFKJLEWHVÄ9DULDWLRQHQ³LQXQWHUVFKLHGOLFKHQ
.RQWH[WHQDXFKLP$OOWDJLVWÄ9DULDWLRQ³DOVÄ9HUlQGHUXQJ³LP.RQWUDVW]XUÄ%HKDUUXQJ³
HLQ ZHVHQWOLFKHV 0HUNPDO YRQ /HEHQ XQG *HVWDOWXQJ XQG GDPLW YRQ .UHDWLYLWlW 8QG LQ
GLHVHP 6LQQH LVW 9DULDWLRQ VFKRQ LQ YRUJHVFKLFKWOLFKHU =HLW W\SLVFK IU GLH YLHOVFKLFKWLJH
(QWZLFNOXQJGHV0HQVFKHQ
:HVHQWOLFKNHLW%HUHLWVHLQ %OLFNDXI HLQLJHDXVJHZlKOWH PDWKHPDWLVFKH7HUPLQLGLH
LP=XVDPPHQKDQJPLWÄ9DULDWLRQ³JHELOGHWVLQGPDFKWGHXWOLFKGDVVÄ9DULDWLRQ³ZHVHQW
OLFK IU GLH 0DWKHPDWLN LVW LQ DOSKDEHWLVFKHU 5HLKHQIROJH 9DULDEOH YDULDEOH 0HWULNHQ
9DULDEOHQWUDQVIRUPDWLRQ9DULDEOHQWUHQQXQJ9DULDQ]9DULDQ]DQDO\VH9DULDWLRQGHU.RQ
VWDQWHQ9DULDWLRQVUHFKQXQJ9DULHWlW9DULVROYHQ],QVEHVRQGHUHLVWKLHUEHLGLHÄ9DULDEOH³
KHUYRU]XKHEHQGLH]ZDULP.RQWUDVW]XU1DPHQVJHEXQJQLFKWVHOEHUYDULLHUWZLUGVRQ
GHUQEHLGHUQXULKUHÄ%HOHJXQJPLW:HUWHQ³YDULDEHOLVW'HQQGDV.RQ]HSWHLQHUÄ9DULDE
OHQ³E]ZJOHLFKEHGHXWHQGHLQHUÄ9HUlQGHUOLFKHQ³XQGGLHGDPLWYHUEXQGHQHZHVHQWOLFKH
9DULDWLRQVP|JOLFKNHLWLVWXQ]ZHLIHOKDIW VRIXQGDPHQWDOGDVV GLH 0DWKHPDWLNDOV:LVVHQ
VFKDIWKHXWHRKQHHLQHQ%HJULIIYRQÄ9DULDEOH³QLFKWYRUVWHOOEDULVW
9DJKHLW Ä9DULDWLRQ³ QXU DOV Ä9HUlQGHUXQJ³ GHILQLHUHQ ]X ZROOHQ ZUGH GHP 5HLFK
WXPLKUHV$XIWUHWHQVLQGHU0DWKHPDWLNXQGLKUHU%HGHXWXQJIUGLH0DWKHPDWLNQLFKWJH
UHFKW ZHUGHQ *OHLFKZRKO LVW GLHVH Ä9HUlQGHUXQJ³ HLQ ZLFKWLJHV± LP MHZHLOLJHQ .RQWH[W
NRQNUHW ]X GHILQLHUHQGHV ± 3ULQ]LS IU GLH 0DWKHPDWLN ,Q GLHVHP 6LQQ LVW Ä9DULDWLRQ³ LQ
GHU0DWKHPDWLNPLWHLQHUGXUFKDXVNRQVWUXNWLYHQ9DJKHLWYHUEXQGHQ
'LH EHLGHQ SUlVNULSWLYHQ .ULWHULHQ Ä'XUFKJlQJLJNHLW³ XQG Ä7UDQVSDUHQ]³ KDW +DQV
6FKXSSDQGHPYRQLKPYRUJHVWHOOWHQGLGDNWLVFKHQ3ULQ]LSGHUAufgabenvariationDXVIKU
OLFKYHUGHXWOLFKWÄ9DULDWLRQ³LVWDEHUQLFKWQXUHLQZLFKWLJHVGLGDNWLVFKHV3ULQ]LSVRQGHUQ
Ä9DULDWLRQ³LVWDXFKW\SLVFKIUGLH0DWKHPDWLNDOV:LVVHQVFKDIWVRZHQQDXFKQLFKW
DOV Ä$XIJDEHQYDULDWLRQ³ LP XQWHUULFKWOLFKHQ 6LQQ VRQGHUQ ] % EHL GHU HLQVFKUlQNHQGHQ
RGHUHUZHLWHUQGHQ9DULDWLRQYRQ9RUDXVVHW]XQJHQLQ'HILQLWLRQHQXQG7KHRUHPHQDOVHL
QHPNUHDWLYHQ:HUN]HXJ]XU*HZLQQXQJQHXHU$XVVDJHQXQG(LQVLFKWHQ'DUEHUKLQDXV
]HLJHQ GLH DNWXHOOHQ 0|JOLFKNHLWHQ GHV H[SHULPHQWLHUHQGHQ 9RUJHKHQV LQ HLQHU BewegungsgeometriePLW+LOIHGHU9HUZHQGXQJYRQ'\QDPLVFKH*HRPHWULH6\VWHPHQ'*6
HLQGULQJOLFKMöglichkeiten entdeckenden Lernens durch Variation±VRZRKOIU6FKOHULQ
QHQXQG6FKOHUDOVDXFK(UZDFKVHQHLQXQWHUVFKLHGOLFKHQ5ROOHQ
+LHUZLUGEHZXVVWQLFKWGLHLQNRUUHNWH%H]HLFKQXQJÄ'\QDPLVFKHV*HRPHWULH6\VWHP³JHZlKOWGHQQ
QLFKWGLHVHV6\VWHPGLH6RIWZDUHLVWG\QDPLVFKVRQGHUQDOOHQIDOOVGLHVREHWUDFKWHWHE]ZHUOHEWH*HR
PHWULH(VOLHJWDOVRHLQ6\VWHPIUHLQHÄ'\QDPLVFKH*HRPHWULH³ÄXQPLWWHOEDU³LP6LQQHGHVJULHFKL
VFKHQ:RUWXUVSUXQJVYRU9JOKWWSPDGLSHGLDGHLQGH[SKS"WLWOH '*6ROGLG +RUVW+LVFKHU9DULDWLRQ±HLQHIXQGDPHQWDOH,GHH
(LQ%HLVSLHO:LQNHOGUHLWHLOXQJ
3UREOHPODJH
'LH ÄNRQVWUXNWLYH³ :LQNHOKDOELHUXQJ PLW =LUNHO XQG /LQHDO LVW NHLQ 3UREOHP Aufgabenvariation gemäß Schupp IKUW ]XU )UDJH ZLH PDQ HLQHQ JHJHEHQHQ :LQNHO LQ PHKU DOV
]ZHLJOHLFKJUR‰H:LQNHOWHLOHQNDQQZREHLGLH7HLOXQJLQ n :LQNHOXQSUREOHPDWLVFKLVW
)ROJHQGH (UZDUWXQJVKDOWXQJ LP 8QWHUULFKW LVW GHQNEDU (LQHQ
JHJHEHQHQ:LQNHOPLWWHOV=LUNHOXQG/LQHDO]XGULWWHOQGUIWHHEHQ
IDOOVSUREOHPORVVHLQLVWGLHVGRFKEHLP$XIWHLOHQHLQHUJHJHEHQHQ
6WUHFNH LQ n JOHLFK ODQJH $EVFKQLWWH PLWWHOV 6WUDKOHQVDW] HLQIDFK
XQGGLH+DOELHUXQJHLQHV:LQNHOVPLW=LUNHOXQG/LQHDOVFKHLQWGD
IU HLQH %ODXSDXVH ]X OLHIHUQ %HL JHJHEHQHP :LQNHO D GULWWHOH
PDQ GLH 6HKQH ]ZLVFKHQ GHQ (QGSXQNWHQ GHU EHLGHQ JOHLFK ODQJHQ $EE:LQNHOGUHLWHLOXQJ
6FKHQNHOPLWWHOV6WUDKOHQVDW]XQGGLHEHLGHQHQWVWDQGHQHQ6WUHFNHQ PLWWHOV6WUDKOHQVDW]"
YHUOlQJHUHPDQELV]XGHP.UHLVERJHQ$EE-HGRFK'LHGUHLVR
HQWVWDQGHQHQ 'UHLHFNH KDEHQ ]ZDU GHQVHOEHQ )OlFKHQLQKDOW JOHLFK ODQJH %DVLV JOHLFKH
+|KH DEHU GLH GDUDXV HQWVWDQGHQHQ GUHL .UHLVERJHQGUHLHFNH VLQG GDPLW GDQQ RIIHQVLFKW
OLFKQLFKWJOHLFKJUR‰6RJHKW¶VDOVRQLFKWDEHUHVZDUHLQHQ9HUVXFKZHUW
%HUHLWVLQGHUJULHFKLVFKHQ$QWLNHZDUGLH:LQNHOGUHLWHLOXQJHLQHVGHUEHUKPWHQNODV
VLVFKHQ3UREOHPHQHEHQGHU4XDGUDWXUGHV.UHLVHVXQGGHU:UIHOYHUGRSSHOXQJPLWGHP
VLFK0DWKHPDWLNHUEHIDVVWHQ(UVWPLW%HJUQGXQJGHUPRGHUQHQ$OJHEUDLP-DKUKXQ
GHUW *DORLV7KHRULH NRQQWH EHNDQQWOLFK EHZLHVHQ ZHUGHQ GDVV GLH Winkeldreiteilung LQ
HLQHPVWUHQJHQ6LQQPLW=LUNHOXQGHLQHPQLFKWVNDOLHUWHP/LQHDOnicht durchweg möglichLVWZHQQJOHLFKLQVSH]LHOOHQ)lOOHQ'HQQRFK VLQG VHLW GHU$QWLNHYLHOIlOWLJHDQGHUH
:HUN]HXJH HQWZLFNHOW ZRUGHQ GLH GLH :LQNHOGUHLWHLOXQJ ]XPLQGHVW QlKHUXQJVZHLVH HU
P|JOLFKHQ'UHLVROFKHU:HUN]HXJHVHLHQNXU]YRUJHVWHOOW
/|VXQJVZHUN]HXJGLH7ULVHFWUL[GHV+LSSLDVYRQ(OLV
'LH HUVWH .XUYH MHQVHLWV YRQ .UHLV XQG *HUDGH YHUGDQNHQ ZLU JHPl‰ HLQHP %HULFKW YRQ
3URNORV±GHP6RSKLVWHQ+LSSLDVDXV(OLVJHEY&KUGHUGLHVHXP
Y&KU]XU:LQNHOGUHLWHLOXQJHUVRQQHQKDEHQVROO+LSSLDVEHGLHQWVLFKGD]XHLQHUkinematischen DefinitionlKQOLFKZLHVSlWHUDXFK$UFKLPHGHVEHLGHUQDFKLKPEHQDQQWHQ6SLUDOH
,P4XDGUDW ABCD YRQ$EEZHUGHGLH6WUHFNH >DC @ SDUDOOHO
]XVLFKPLWNRQVWDQWHU*HVFKZLQGLJNHLWELVLQGLH/DJH>AB @ YHU
VFKREHQXQG>AD @URWLHUHXP APLWNRQVWDQWHU:LQNHOJHVFKZLQ
GLJNHLW ELV LQ GLH /DJH >AB @ %HLGH %HZHJXQJHQ VWDUWHQ
JOHLFK]HLWLJXQGK|UHQJOHLFK]HLWLJDXI'HUJHRPHWULVFKH2UWGHU
6FKQLWWSXQNWHLVWGLH]XGHILQLHUHQGH7ULVHFWUL[$EE]HLJWHLQH
punktweise Konstruktion GLHVHU 7ULVHFWUL[ LQGHP GLH GLVNUHWHQ
6WW]VWHOOHQYRQ+DQGÄJODWW³YHUEXQGHQZHUGHQZLHPDQHVDXFK
GDPDOVJHPDFKWKDEHQN|QQWHZREHLVLFKGLHDXIWUHWHQGHQ:LQ $EESXQNWZHLVH
NHOVFKHQNHOGXUFKIRUWJHVHW]WHWinkelhalbierungJHZLQQHQODVVHQ .RQVWUXNWLRQGHU7ULVHFWUL[
>AB@EH]HLFKQHWQDFKIROJHQGGLH6WUHFNHPLWGHQ(QGSXQNWHQAXQGB_AB_GHUHQ/lQJHXQG ¢ AB ² LVW
GLH*HUDGHGXUFKAXQGB.
+RUVW+LVFKHU9DULDWLRQ±HLQHIXQGDPHQWDOH,GHH
$QVWHOOH GHU YHUEDOHQ 'HILQLWLRQ NDQQ PDQ DXFK HLQH KDQG
OXQJVRULHQWLHUWHZlKOHQLQGHPPDQVLFKZLHLQ$EEHLQHQ„Trisectrix-Zirkel“ vorstellt GHU GLH NLQHPDWLVFKH (U]HXJXQJ GHU 7UL
VHFWL[symbolisierenVROO0LWHLQHP'*6LVWGLH(QWVWHKXQJGHU
7ULVHFWUL[ GXUFK GLH DQLPLHUWH (U]HXJXQJ HLQHU 2UWVOLQLH simulierbar/LHJWGLH7ULVHFWUL[ZLHHLQH3DUDEHOVFKDEORQHDOVÄ.XUYHQ
OLQHDO³ YRU VR NDQQ PDQ PLW LKU LP 5DKPHQ GHU ]HLFKQHULVFKHQ $EEÄ7ULVHFWUL[=LUNHO³
*HQDXLJNHLW VSLW]H :LQNHO XQWHU $QZHQGXQJ GHV 6WUDKOHQVDW]HV DOV,GHH
ZLHLQ$EEGUHLWHLOHQGLH7ULVHFWUL[WULWWGDQQDOV+LOIVPLWWHODXI
,QGHU$OJHEUDLVFKHQ*HRPHWULHQHQQW PDQHLQH.XUYH„organisch erzeugbar“ZHQQVLHLP*DQ]HQDOVRQLFKWQXUSXQNWZHL
VHGDUVWHOOEDULVWZLH]%HLQH(OOLSVHEHUGLHÄ*lUWQHU.RQVWUXN
WLRQ³'LHTrisectrixLVWKLHUDOVRnicht organisch erzeugt(VHQW
VWHKWDEHUGLH)UDJHREGLH7ULVHFWUL[organisch erzeugbarLVW
'LH 9DULDWLRQ GHU $XIJDEH Ä:LQNHOKDOELHUXQJ³ IKUW VR ]XP
NODVVLVFKHQ3UREOHPGHU:LQNHOGULWWHOXQJGDVLQGHU$QWLNHPLWWHOV $EE:LQNHOGUHLWHLOXQJ
PLW+LOIHGHU7ULVHFWUL[
HLQHU]XPLQGHVWJHGDFKWHQ9DULDWLRQYRQ3XQNWHQHLQH2UWVNXUYH
OLHIHUWH QlPOLFK GLH 7ULVHFWUL[ GLH DOV :HUN]HXJ ]XU 3UREOHPO|VXQJ YHUZHQGHW ZHUGHQ
NRQQWH'LQRVWUDWRV±YHUZHQGHWHVSlWHUGLH7ULVHFWUL[]XU4XDGUDWXUGHV.UHLVHV
ZHVKDOEVLHDXFKÄ4XDGUDWUL[³KHL‰W
/|VXQJVZHUN]HXJGLH$UFKLPHGLVFKH6SLUDOH
8QWHUGHQDXI$UFKLPHGHVYRQ6\UDNXV±Y&KU]XUFNJHKHQGHQ:HUNHQWUlJW
HLQHGHQ7LWHOÄDELINEISSPIRALIBUS³>&DQWRU@QHQQWVLH
Ä'LH 6FKQHFNHQOLQLHQ RGHU 6SLUDOHQ³ (LQH DXVIKUOLFKH 'DUVWHOOXQJ
GHU.RPPHQWDUHYRQ(XWRFLRV]XUGHQ:HUNHQYRQ$UFKLPHGHVILQ
GHW VLFK EHL >+HDWK II@ LQVEHVRQGHUH LP JULHFKLVFKODWHL
QLVFKHQ2ULJLQDONRPPHQWDUEHL>+HLEHUJII@'LHKLHUEH
KDQGHOWH XQG VSlWHU QDFK LKP EHQDQQWH $UFKLPHGLVFKH 6SLUDOH LVW
ZLHGLH7ULVHFWUL[HLQHNLQHPDWLVFKGHILQLHUWH.XUYH$EEGHUHQ $EE$UFKLPHGLVFKH
(U]HXJXQJ$UFKLPHGHVLQVHLQHP%XFKZLHIROJWEHVFKUHLEW)RUPX 6SLUDOHLQNLQHPDWLVFKHU
(QWVWHKXQJ
OLHUXQJDXV>&DQWRU@
:HQQ HLQH JHUDGH /LQLH LQ HLQHU (EHQH XP HLQHQ LKUHU (QGSXQNWH ZHOFKHU XQEHZHJOLFK
EOHLEW PLW JOHLFKI|UPLJHU *HVFKZLQGLJNHLW VLFK EHZHJW ELV VLH ZLHGHU GDKLQ JHODQJW YRQ
ZRGLH%HZHJXQJDXVJLQJXQGZHQQ]XJOHLFKLQGHUEHZHJWHQ/LQLHHLQ3XQNWPLWJOHLFK
I|UPLJHU *HVFKZLQGLJNHLW YRQGHP XQEHZHJWHQ (QGSXQNWH DQIDQJHQG VLFK EHZHJW VR EH
VFKUHLEWGLHVHU3XQNWHLQH6FKQHFNHQOLQLHLQGHU(EHQH
$XIHLQHPPLWNRQVWDQWHU:LQNHOJHVFKZLQGLJNHLWURWLHUHQGHQ5DGLDOVWUDKOEHZHJWVLFKDOVR
PLWNRQVWDQWHU%DKQJHVFKZLQGLJNHLWHLQ3XQNWYRP0LWWHOSXQNWQDFKDX‰HQ'LH%DKQNXU
YHGLHVHV3XQNWHVDOVRGHVVHQÄ6SXU³HUJLEWGDQQGLHVH6SLUDOHDOVHLQHQLFKWEHJUHQ]W]X
GHQNHQGH Ä6FKQHFNHQOLQLH³ $XFK GLHVH 6SLUDOH NDQQ PDQ ZLH GLH 7ULVHFWUL[ SXQNWZHLVH
NRQVWUXLHUHQLQGHPPDQ]%3RODUNRRUGLQDWHQSDSLHUYHUZHQGHWGDVPDQNDXIHQRGHUPLW
9JO>+LVFKHU@
9JO>%ULHVNRUQ.Q|UUHU@
+RUVW+LVFKHU9DULDWLRQ±HLQHIXQGDPHQWDOH,GHH
HLQHP9HNWRUJUDSKLNSURJUDPPOHLFKWVHOEVWKHUVWHOOHQNDQQ$EE
'LH (QWVWHKXQJ GLHVHU .XUYH LVW ZLH EHL GHU 7ULVHFWUL[ PLW HLQHP
'*6 GXUFK (U]HXJXQJ HLQHU 2UWVOLQLH VLPXOLHUEDU $XFK KLHU VWHOOW
VLFK GLH )UDJH RE GLH $UFKLPHGLVFKH 6SLUDOH organisch erzeugbar
LVW 6RGDQQ LVW RIIHQVLFKWOLFK ZLH HLQH $UFKLPHGLVFKH 6SLUDOH ]XU
:LQNHOGUHLWHLOXQJYHUZHQGHWZHUGHQNDQQ'HU5DGLDOVWUDKODOVHLQ
6FKHQNHO GHV JHJHEHQHQ :LQNHOV VFKQHLGHW GLH 6SLUDOH LQ HLQHP $EE$UFKLPHGLVFKH
3XQNW GHU VLFK GDGXUFK HUJHEHQGH 5DGLXV ZLUG PLWWHOV 6WUDKOHQVDW] 6SLUDOHPLW3RODUNRRU
GLQDWHQSDSLHU
JHGULWWHOWPLWGHPQHXHQ5DGLXVZLUGHLQQHXHU.UHLVXPGHQVHOEHQ
0LWWHOSXQNW JH]HLFKQHW XQG GHVVHQ 6FKQLWWSXQNW PLW GHU 6SLUDOH OLHIHUW GHQ 6FKHQNHO IU
GHQJHVXFKWHQ:LQNHO(VVHLDQJHPHUNWGDVVPDQDXVGHQ3DUDPHWHUJOHLFKXQJHQ r a ˜ t IUGHQ5DGLXVXQGM b ˜ t IUGHQ:LQNHOGLH6SLUDOHQJOHLFKXQJ r c ˜ M HUKlOW
/|VXQJVZHUN]HXJGDV(LQVFKLHEHOLQHDOQDFK$UFKLPHGHV
$XI $UFKLPHGHV JHKW GDV VRJ Ä(LQVFKLHEHOLQHDO³ GDV HOHJDQWHVWHV /|VXQJVZHUN]HXJ IU
GDV3UREOHPGHU:LQNHOGUHLWHLOXQJ]XUFN'DV]%LQ>&DQWRUI@EHVFKULHEHQH
9HUIDKUHQZLUGPLWGHQGUHLIROJHQGHQ$EELOGXQJHQYHUHLQIDFKWYRUJHVWHOOW,Q$EEVHL
LP .UHLV XP M HLQ EHOLHELJHU VSLW]HU :LQNHO
)QMP JHJHEHQ)HUQHUVHLYRQP DXVHLQH6WUHFNH
>PA@]XHLQHP3XQNW A DXIGHU*HUDGHQ ¢QM ² JH
]RJHQGLHGHQ.UHLVLQ B VFKQHLGHW A VROOQXQVR
DXI ¢QM ² YHUVFKREHQ ZHUGHQ ELV |AB| |PM | LVW
ZDV DXFK RIIHQVLFKWOLFK P|JOLFK LVW (V JLOW GDQQ
|AB| |PM | |QM |
$EE(LQVFKLHEHOLQHDO±*UXQGVLWXDWLRQ
Behauptung:'DQQLVW)QMP 3)MAB Beweis:'LHLQ$EEPLW DEH]HLFKQHWHQ:LQ
NHO VLQG JOHLFK JUR‰ WHLOV DOV :HFKVHOZLQNHO E]Z
6WXIHQZLQNHO DQ GHQ DOV +LOIVJHUDGHQ HLQJH]HLFKQH
WHQ 3DUDOOHOHQ XQG GDV 'UHLHFN AMB LVW JOHLFK
VFKHQNOLJ 'DV 'UHLHFN PBM LVW HEHQIDOOV JOHLFK $EE=XP%HZHLVYRQ$UFKLPHGHV¶
9HUIDKUHQ
VFKHQNOLJDOVRJLOWE 2D XQGGDPLWIROJW
G S(DJ) S(D(S2DE)) S(D(S 4D)) 3D ‹
'LHELVKHULJH'DUVWHOOXQJOlVVWVLFKDXFKkonstruktiv inversVHKHQ0DQJHKWYRQGHP
.UHLV XP M DXV ZlKOW HLQHQ 3HULSKHULHSXQNW Q IHUQHU DXI GHU *HUDGHQ ¢QM ² HLQHQ
3XQNW A VRDX‰HUKDOEGHV.UHLVHVGDVVGHU.UHLVXP A PLWGHP5DGLXV|MQ|GHQ.UHLV
XP M LQHLQHPQHXHQ3XQNW B VFKQHLGHW'LH*HUDGH ¢AB ² VFKQHLGHWGDQQGHQ.UHLVXP
M LQHLQHPZHLWHUHQ3XQNWP GHUDUWGDVV)QMP 3)MAB JLOW
'LHVZDUHQQXU9RUEHUHLWXQJHQIUHLQJHVXFKWHVWHFKQLVFKHV9HUIDKUHQ]XU'UHLWHLOXQJ
GHV:LQNHOV )QMP PLWGHPQXQGHU3XQNW ASDVVHQGJHIXQGHQZHUGHQPXVV$EE
]HLJWGD]XPRGHOOKDIWHLQÄ(LQVFKLHEHOLQHDO³LP6LQQHGHUDUFKLPHGLVFKHQ,GHH/LQNVYRP
.UHLVXPM EHILQGHWVLFKHLQDXIGHU*HUDGHQ ¢QM ² IHVWLQVWDOOLHUWHV/LQHDOPLWHLQHU)K
UXQJVULOOH IU HLQHQ LQ LKU JOHLWHQGHQ 6WLIW GHU YHUWLNDO ]XU=HLFKHQHEHQH LP 3XQNW A GHV
]ZHLWHQGXQNHOJUDXGDUJHVWHOOWHQ/LQHDOVPRQWLHUWLVW
+RUVW+LVFKHU9DULDWLRQ±HLQHIXQGDPHQWDOH,GHH
'LHVHV ]ZHLWH /LQHDO ± HV LVW GDV Ä(LQ
VFKLHEHOLQHDO³±LVWPLWWHOVGHV6WLIWVGUHKEDU
XP AJHODJHUWXQGHVEHVLW]WHEHQIDOOVHLQH
)KUXQJVULOOH GLH HLQHQ LP JHJHEHQHQ
3XQNW P GHV .UHLVHV PRQWLHUWHQ ZHLWHUHQ
6WLIW XPJUHLIW 'DV (LQVFKLHEHOLQHDO NDQQ
GDPLWGXUFKKRUL]RQWDOH%HZHJXQJYRQALQ
GHU )KUXQJVULOOH GHV HUVWHQ IL[LHUWHQ /LQH
DOVXPGHQ6WLIWYRQP KHUXPJOHLWHQZREHL $EE(LQVFKLHEHOLQHDOQDFK$UFKLPHGHV
VLFKGHU$EVWDQGYRQ A]XU.UHLVSHULSKHULHlQGHUW$XIGHP(LQVFKLHEHOLQHDOZXUGH]XYRU
HLQ3XQNWB JHPl‰|AB | |PM |PDUNLHUW1XQPXVV AQXUQRFKVRvariierendEHZHJWZHU
GHQGDVVGLHVH0DUNLHUXQJPLWGHU.UHLVSHULSKHULH]XU'HFNXQJNRPPW'DPLWOlVVWVLFK
GXUFK PDQXHOOHV Ä(LQVFKLHEHQ³ MHGHU VSLW]H :LQNHO DQJHQlKHUW GULWWHOQ 0DQ EHUOHJH
VLFKREGDV9HUIDKUHQDXFKIUVWXPSIH:LQNHOIXQNWLRQLHUW
=XUSUDNWLVFKHQ'XUFKIKUXQJEHQ|WLJWPDQGHQLQ$EEGDUJHVWHOOWHQ$SSDUDWDEHU
QLFKWZLUNOLFKGHQQHVJHQJHQRIIHQEDUÄ=LUNHOXQG/LQHDO³ZHQQPDQZLHIROJWYRUJHKW
0DQ]HLFKQHWHLQH*HUDGH ¢QM ² HLQHQ.UHLV k XP M GXUFKQ XQGJLEWPLW P  k HLQHQ
:LQNHO)QMP YRU$XIGHP/LQHDOPDUNLHUWPDQHLQH6WUHFNH>AB @ GHU/lQJH|PM |XQG
SRVLWLRQLHUWQXQGDV/LQHDO PDQXHOOVRLQGHU=HLFKHQHEHQHGDVVGLHOLQNH 0DUNLHUXQJ A
YRQ>AB @ DXIGHU*HUDGHQ ¢QM ² OLHJWGLH/LQHDONDQWHGXUFKGHQ3XQNWP YHUOlXIWXQGGLH
UHFKWH0DUNLHUXQJB YRQ>AB @ DXIGHP.UHLVUDQGOLHJW'LHVRSRVLWLRQLHUWH6WUHFNH>AB @ LVWGDQQQHEHQ>AM @GHU]ZHLWH6FKHQNHOGHVJHVXFKWHQJHGULWWHOWHQ:LQNHOV)MAB 'DVN|QQWHQXQ]XGHU$QVLFKWYHUOHLWHQGDVVGDV:LQNHOGUHLWHLOXQJVSUREOHPPLW=LU
NHOXQG/LQHDOO|VEDUVHLMHGRFKZlUHGLHVHLQFehlschlussZHLOHWZDV:HVHQWOLFKHVQLFKW
EHDFKWHWZXUGH'DVÄ(LQVFKLHEHOLQHDO³EHVLW]WHLQHLängenmarkierung,P6LQQHGHV9HU
VWlQGQLVVHVYRQÄ.RQVWUXNWLRQPLW=LUNHOXQG/LQHDO³DOVHLQHPalgebraischen Problem
nicht aber einem praktischen Problem GDUI DEHU QXU GLH Ä=HLFKHQNDQWH³ HLQHV /LQHDOV
DOVPDWHULHOOH6\PEROLVLHUXQJHLQHV*HUDGHQDEVFKQLWWVYHUZHQGHWZHUGHQQLFKWDEHUGLH]X
HLQHPKDQGHOVEOLFKHQÄ/LQHDO³JHK|UHQGHPD‰VWlEOLFKH6NDOLHUXQJ
$EJHVHKHQ YRQ GLHVHP Ä/LQHDOSUREOHP³ LVW GLH VR GXUFK Variation der Position des
EinschiebelinealsJHQHULHUWHÄ/|VXQJ³QLFKWH[DNWVRQGHUQQXUPDQXHOODSSUR[LPLHUW
1DFKZRUW
'DVNODVVLVFKH3UREOHPGHU:LQNHOGUHLWHLOXQJHUZHLVWVLFKLQPHKUIDFKHU+LQVLFKWDOVHLQ
%HLVSLHO]XP$VSHNWÄ9DULDWLRQ±HLQHIXQGDPHQWDOH,GHH³
,P8QWHUULFKWNDQQHVEHUGDV6FKXSSVFKH.RQ]HSWGHUAufgabenvariationHQWVWHKHQ
XQG]ZDUGXUFKHLQHQXUJHULQJIJLJHUVFKHLQHQGH0RGLILNDWLRQGHUEHUHLWVEHNDQQWHQ.RQ
VWUXNWLRQVDXIJDEHGHU:LQNHOKDOELHUXQJPLWWHOV=LUNHOXQG/LQHDO±DOOHUGLQJVPLWXQJHDKQ
WHQ)ROJHQ:lKUHQGVLFKHLQH7HLOXQJLQ«:LQNHOPLWWHOVIRUWJHVHW]WHU+DOELHUXQJ
QRFK IDVW YRQ DOOHLQH HUJLEW OlXIW PDQ LP )DOOH GHU 'UHLWHLOXQJ HLQHV EHOLHELJHQ :LQNHOV
JHJHQGLH:DQGZHQQJOHLFKGLHNRQVWUXNWLYH'ULWWHOXQJJHZLVVHUVSH]LHOOHU:LQNHOOHLFKW
P|JOLFKLVW'LH0LWWHLOXQJGDVVGDV3UREOHPGHUÄ:LQNHOGUHLWHLOXQJPLW=LUNHOXQG/LQH
DO³GLH0DWKHPDWLNHUVFKRQVHLWIDVW-DKUHQEHVFKlIWLJWKDWXQGHUVWLP-DKUKXQ
+RUVW+LVFKHU9DULDWLRQ±HLQHIXQGDPHQWDOH,GHH
GHUW HQGJOWLJ QHJDWLY HQWVFKLHGHQ ZHUGHQ NRQQWHQ VROOWH 9HUZXQGHUXQJ EHZLUNHQ QRFK
GD]XZHQQPLWJHWHLOWZLUGGDVVVHLWGHPHLQH)OOHYRQÄ/|VXQJHQ³PLWDQGHUHQ:HUN]HX
JHQ HQWZLFNHOW ZRUGHQ LVW 'LH GUHL GDUJHVWHOOWHQ %HLVSLHOH VLQG LP 8QWHUULFKW HOHPHQWDU
EHKDQGHOEDU (V LVW HLQH VFK|QH hEXQJ GLH 7ULVHFWUL[ XQG DXFK GLH 6SLUDOH VRUJIlOWLJ
SXQNWZHLVH ]X NRQVWUXLHUHQ XQG GHU (LQVFKLHEHDSSDUDW NDQQ WDWVlFKOLFK JHEDXW ZHUGHQ
'DUEHUKLQDXVN|QQHQDOOHGUHL9HUIDKUHQPLWWHOV'*6VLPXOLHUHQGGDUJHVWHOOWZHUGHQ(LQ
ZHLWHUHV VHKU LQWHUHVVDQWHV :HUN]HXJ LVW EULJHQV GLH .RQFKRwGH Ä0XVFKHOOLQLH³ DOOHU
GLQJVLVWHVNRPSOH[HUXQGNDQQGDKHULQGLHVHP5DKPHQQLFKWGDUJHVWHOOWZHUGHQ
=XJOHLFK LVW GDV 3UREOHP GHU :LQNHOGUHLWHLOXQJ HLQ ,QGL] GDIU GDVV Ä9DULDWLRQ³ LP
6LQQH GHU HLQJDQJV DXIJHIKUWHQ GHVNULSWLYHQ .ULWHULHQ ]X GHQ fundamentalen Ideen der
Mathematik JHK|UW 'XUFK 9DULDWLRQ GHU =LHOVHW]XQJ HQWVWDQG GDV 3UREOHP EHUHLWV LQ GHU
0DWKHPDWLNGHU$QWLNHXQGGLHLQGHQIROJHQGHQ-DKUKXQGHUWHQHQWZLFNHOWHQ/|VXQJVYHU
VXFKHEDVLHUWHQDXIGHU9DULDWLRQJHZLVVHUJHRPHWULVFKHU2EMHNWH3XQNWH6WUHFNHQ*HUD
GHQ«Ä9DULDWLRQ³HUZHLVWVLFKDOVVRIXQGDPHQWDOIUGLH0DWKHPDWLNGDVVHLQH3ODW]LH
UXQJLQ(EHQHYRQ$EEQDKHOLHJW8QGVFKOLH‰OLFKOLHJWQRFKHLQZHLWHUHU$VSHNWGHU
9DULDWLRQ YRU 'LHVH GUHL %HLVSLHOH ]HLJHQ GDVV HLQ NRQNUHWHV 3UREOHP DXI UHFKW XQWHU
VFKLHGOLFKH $UWHQ ÄYDULLHUHQG³ JHO|VW ZHUGHQ NDQQ XQG VROFK HLQH 9DULDWLRQ HUP|JOLFKW
]XJOHLFKHLQYHUWLHIHQGHV9HUVWlQGQLVHLQHV3UREOHPVXQGPDWKHPDWLVFKHU0HWKRGHQ
,FK GDQNH 3URI 'U 8OULFK )HOJQHU 8QLY 7ELQJHQ IU GHQ *HGDQNHQDXVWDXVFK XQG
GLH+LQZHLVHLQHLQHPXPIDVVHQGHUHQJU|‰HUHQPDWKHPDWLNKLVWRULVFKHQ=XVDPPHQKDQJ
/LWHUDWXU
%ULHVNRUQ(JEHUW.Q|UUHU+RUVW>@(EHQHDOJHEUDLVFKH.XUYHQ%RVWRQ%DVHO6WXWWJDUW%LUNKlXVHU
%UXQHU-HURPH6H\PRXU>@Der Prozeß der Erziehung%HUOLQ6FKZDQQ'VVHOGRUI%HUOLQ9HUODJ$XI
ODJHGHUGHXWVFKHQ$XVJDEH
&DQWRU0RULW]>@9RUOHVXQJHQEHU*HVFKLFKWHGHU0DWKHPDWLN(UVWHU%DQG/HLS]LJ7HXEQHU]ZHLWH
EHUDUEHLWHWHXQGDNWXDOLVLHUWH$XIODJH$XIODJH$XIODJH
+HDWK7KRPDV/LWWOH>@7KH:RUNVRI$UFKLPHGHV(GLWHGLQ0RGHUQ1RWDWLRQZLWK,QWURGXFWRU\&KDSWHUV
&DPEULGJH$WWKH8QLYHUVLW\3UHVV
+HLEHUJ-RKDQ/XGYLJ+UVJ>@$UFKLPHGHV±2SHUD2PQLD&XP&RPPHQWDULLV(XWRFLL9RO,,6WXWWJDUW
7HXEQHU'RSSHOVHLWLJLQJULHFKLVFKODWHLQLVFK
+H\PDQQ+DQV:HUQHU>@Allgemeinbildung und Mathematik. Studien zur Schulpädagogik%DQG5HLKH
3lGDJRJLN:HLQKHLP%DVHO%HOW]
+LVFKHU+RUVW>@*HVFKLFKWHGHU0DWKHPDWLNDOVGLGDNWLVFKHU$VSHNW/|VXQJNODVVLVFKHU3UREOHPH(LQ
%HLVSLHOIUGLHJ\PQDVLDOH2EHUVWXIH,QMathematik in der Schule±
²>@Ä)XQGDPHQWDOH,GHHQ³XQGÄ+LVWRULVFKH9HUDQNHUXQJ³±GDUJHVWHOOWDP%HLVSLHOGHU0LWWHOZHUWELOGXQJ
,Qmathematica didactica±
² >@0LWWHOZHUWELOGXQJ±HLQHGHUlOWHVWHQPDWKHPDWLVFKHQ,GHHQ,Qmathematik lehren+HIW
±
²>@0LWWHQELOGXQJDOVIXQGDPHQWDOH,GHH,QDer Mathematikunterricht±
²>@Grundlegende Begriffe der Mathematik: Entstehung und Entwicklung. Struktur — Funktion — Zahl.
:LHVEDGHQ6SULQJHU6SHNWUXP
6FKXSS+DQV>@Optimieren – Extremwertbestimmung im Mathematikunterricht.0DQQKHLP%,:LVVHQ
VFKDIWVYHUODJ
²>@2SWLPLHUHQLVWIXQGDPHQWDO,Qmathematik lehren+HIW
²>@Thema mit Variationen. Aufgabenvariation im Mathematikunterricht+LOGHVKHLP)UDQ]EHFNHU
6FKZHLJHU)ULW]>@„Fundamentale Ideen“ der Analysis und handlungsorientierter Unterricht,Q%HLWUlJH
]XP0DWKHPDWLNXQWHUULFKW+DQQRYHU6FKURHGHO±
²>@)XQGDPHQWDOH,GHHQ±(LQHJHLVWHVZLVVHQVFKDIWOLFKH6WXGLH]XU0DWKHPDWLNGLGDNWLN,QJournal für
Mathematikdidaktik±
²>@Fundamentale Ideen,Q6FKULIWHQ]XU'LGDNWLNGHU0DWKHPDWLNXQG,QIRUPDWLNDQGHU8QLYHUVLWlW
6DO]EXUJ%DQG$DFKHQ6KDNHU9HUODJ