GOMES, A. S. ; VERGNAUD, Gérard . On the Learning of geometric concepts using Dynamic Geometry...

7/30/2019 GOMES, A. S. ; VERGNAUD, Grard . On the Learning of geometric concepts using Dynamic Geometry Software. R

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naafbfne} fe }`n oni|efek ha knhcn}|v% Boncne}r iem Li}}fr}i (966=&% ]`n ifc ha

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thfe} ha qfn{/ r}~mvfek }`n ib}fher ch|n mnntov/ }|vfek }h }|ibn }`n ncn|knebn ha knhcn}|fbio

jeh{onmkn m~|fek }`n ~rn ha }`n fe}n|aibn f}rnoa%

T|i}} iem Ifeonv (966;& ny~ioov mncher}|i}nm `h{ }`n ani}~|nr ha Bil|f feao~nebn

bhct|n`nerfhe ha knhcn}|fb bher}|~b}fher% Mf|nb} cieft~oi}fhe ha bn|}ife noncne}r ha }`n

fe}n|aibn iooh{r r}~mne}r }h bhebnfqn i bher}|~b}fhe ir i rn} ha m|i{fekr/ bohrn| }h }`n }`nh|n}fbio

mnafef}fhe ha i knhcn}|fbio afk~|n%

M~qio (9660& irrn|}r }ì} }`n mveicfb knhcn}|v rvr}ncr i|n r~tn|fh| }h titn| ! tnebfo: `n

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thrrflfof}fnr ha qfr~iofri}fhe/ ti|}fb~oi|ov }`|h~k` }`n fe}|hm~b}fhe ha }`n irtnb} ha chqncne}

(&% Iem lnbi~rn }`fr }hho mfrrhbfi}nr }`n chcne} ha fe}ne}fhe (}`n r~lgnb} ìr heov }h b`hhrn ie

fer}|~b}fhe fe t|fcf}fqnr ha i knhcn}|fbio rha}{i|n& iem }`n chcne} ha t|hm~b}fhe {`fb` fr

npnb~}nm lv }`n bhct~}n|/ f} htner ofjn ie nptn|fcne}io itt|hib` fe knhcn}|v% (M~qio/ t%

=;&%

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ci}`nci}fbio bhebnt}r fe rf}~i}nm iem cnmfi}nm bhe}np}r% Rnb}fhe 2 t|nrne}r i }`nh|n}fbio

a|icn{h|j l~fo} ~the bher}|~b}fqfr} ah~emi}fher ah| }`n ieiovrfr ha nm~bi}fheio fe}n|aibnr

ahb~rnm he ib}fhe iem cnmfi}nm oni|efek ha rtnbfafb bhebnt}r% Rnb}fhe = mfrb~rrnr ie npicton fe

{`fb` }`fr a|icn{h|j fr nctohvnm% Fe }`fr rnb}fhe/ {n mfrb~rr r}~mne}r imit}i}fhe }h }`n

fe}n|aibn iem cnmfi}nm oni|efek% Rnb}fhe > kfqnr i bhebo~rfhe ilh~} i bhebnt}~iofwi}fhe thfe} ha

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ricn bhe}np}/ Rb`hneanom (9601& r}i}nr }ì} }`n fe}n|ib}fhe ln}{nne t|ib}fbio (h| nctf|fbio&/ iem

mnm~b}fqn ib}fqf}fnr onimr }h c~}~io |nfeah|bncne}% @n bherfmn|r }ì} }`n mfb`h}hcv ln}{nne

nctf|fbfrc iem mnm~b}fhe fe Knhcn}|v fr aiorn iem hlrn|qnr }ì} }`n cigh|f}v ha r}~mne}r

bherfmn| }`n nctf|fbio iem mnm~b}fqn itt|hib`nr ir rnti|i}n mhcifer/ {f}` }`nf| h{e

rtnbfafbi}fher }h nr}ilofr` }`n qiofmf}v ha i |nr~o}% I} }`n nem ha }`n af|r} }n|c ha rnbhemi|v

rb`hho/ ciev ha }`nc i|n fenptn|fnebnm nctf|fbfr}r/ ah| {`fb` }`n |nrho~}fhe ha }`n |~on| iem

tif| ha bhctirrnr t|hloncr i|n g~r} i ci}}n| ha }|fio iem n||h|% ]`n |noi}fhe ln}{nne ib}fqf}fnr

{f}` tnmikhkfb ci}n|fio iem }`n bher}|~b}fhe ha knhcn}|fb jeh{onmkn |nr}r ~the }`n

ibjeh{onmkncne} lv }`n r}~mne} ha }`n npfr}nebn ha i mhcife ha |nan|nebn iem ha rhcn |noi}fhe

ln}{nne }`fr |nan|nebn iem }`n ci}`nci}fbio hlgnb} bheqnvnm lv }`n ~rn ha }`n fer}|~cne}% ]`fr

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ti|}fb~oi| ci}`nci}fbio ne}f}fnr/ iem }`n htn|i}fher ittofnm }h }`nrn hlgnb}r i|n t|hbnm~|nr iem|nr~o}r }ì} bh||nrthem }h }`nrn ne}f}fnr% Lnrfmnr/ }`n|n fr i rvclhofb rvr}nc ({&/ {`fb` fr i} }`n

ricn }fcn i rvclhofb |nt|nrne}i}fhe ha rhcn ci}`nci}fbio ilr}|ib}fher rnonb}nm a|hc [% ]`n

rvclhor fe { |nt|nrne} ne}f}fnr fe [%

]`fr ieiohkv ln}{nne |niof}v iem }`h~k`} fctofnr }`n r~tthrf}fhe }ì} femfqfm~ior ìqn i aif|

bhct|n`nerfhe ha }`n cniefek ha }`n hlgnb}r iem t|hbnm~|nr fe }`n mhcife ha |nan|nebn [/ iem

ha }`nf| fe}n||noi}fher% ]h ~emn|r}iem }`n ei}~|n ha }`n rvclhofb ilr}|ib}fhe {`fb` cijnr {

a~eb}fheio/ femfqfm~ior r`h~om ìqn i boni| bhct|n`nerfhe ha }`n rvclhor iem t|hbnm~|nr fe }`n

rvclhofb rvr}nc f}rnoa/ iem ~emn|r}iem }`n mfaan|ne} r}|~b}~|nr ha mfaan|ne} rvclhofb rvr}ncr

}ì} {h|j ir |nt|nrne}i}fher ha }`n ricn fmni rh ir }h tn|bnfqn }`n r}|~b}~|io ch|t`frcr ln}{nne

}`nrn |nt|nrne}i}fher iem }`n|nah|n ln ilon }h bhebnfqn }`n ~emn|ovfek ci}`nci}fbio fmni% Cnf|i

b|f}fbfwnr }`fr chmno (Cnf|i/ 966>/ 9660&% @fr b|f}fy~nr i|n mf|nb}nm cifeov ikifer} }`n np}n|fh|

|nt|nrne}i}fhe a~eb}fher7 ieiovwfek }`n t|hm~b}fhe ha |nt|nrne}i}fher (af|r} mnk|nn a~eb}fher& `n

ah~em h~} }ì} }`n t|hm~b}fhe t|hbnrr cn|knm {f}` }`n ib}fhe ha t|hlonc rhoqfek f}rnoa%

@n hlrn|qnm }ì} }`n |nt|nrne}i}fher t~} mh{e he titn| ìm }`n a~eb}fhe ha r~tth|}fek }`n

r~lgnb}r chmnofek ib}fqf}v i} }`n ricn }fcn }ì} }`nrn |nt|nrne}i}fher {n|n chmnonm lv }`nib}fqf}v% ]`n|nah|n/ he }`n hen ìem/ }`n |noi}fhe ln}{nne }`n t|hm~b}fhe iem }`n ~rn ha

|nt|nrne}i}fher/ iem }`n ib}fqf}v he }`n h}`n| ìem/ ìr i mfionb}fbio ei}~|n% Cnf|i t~}r rfmn lv

rfmn }`n a~eb}fhe ha }`n fe}n|eio iem np}n|eio |nt|nrne}i}fher% F} fr eh} thrrflon }h rtnij ha i

lf~efqhbio |noi}fher`ft/ ir lh}` |nt|nrne}i}fher i|n ti|}r ha hen ricn toie%

2% I bher}|~b}fqfr} a|icn{h|j ah| }`n ieiovrfr ha nm~bi}fheio fe}n|aibnr


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Q% < E 9/ Ci|h/ %9% Ieiovrfr ha fer}|~cne}io knenrfr

]`n ah|c ha }`n r~bbnrrfqn fer}|~cne}r {f}` Bil|f t|fcf}fqnr nqhoqnm% ]`n r}~mne}r kh a|hc i

bheafk~|i}fhe {f}` }`|nn bf|bonr iem i tn|tnemfb~oi| ofen }h m (rnbhem i}}nct}& }h i bheafk~|i}fhe

{f}` }{h en{ bf|bonr iem i tn|tnemfb~oi| ofen (}`f|m iem ah~|}` i}}nct}r& }h afeioov |nib` i

qn|rfhe {f}` heov hen en{ bf|bon iem i tn|tnemfb~oi| ofen% ]`n ~rn ha Bil|f iorh iooh{nm }`n

ibbhcchmi}fhe ha }`n rb`ncn ha |nth|}fek i onek}` }h ohbiofwn }`n rvccn}|fbio thfe} ha ieh}`n|

thfe}% ]`n ~rn ha }`n |~on| iem i tif| ha bhctirrnr ìr t|hlilov feao~nebnm lv }`n ah|c r}~mne}r

mnqnoht r}|i}nkfnr }h bher}|~b} knhcn}|fbio afk~|nr {f}` Bil|f/ r}nt lv r}nt% ]`fr bhebo~rfhe frbheqn|kne} {f}` }`n hen haan|nm lv h}`n| i~}`h|r (Oilh|mn ! Bitheef/ 966=: T|i}} ! Ifeonv/

966;: @ow/ 966;&% L~} {n {h~om ofjn }h kh hen r}nt a~|}`n|%

[n bie hlrn|qn }ì} r}~mne}r ib}nm ir fa }`nv lnofnqnm }ì} i bf|bon fr bne}n|nm he i rnbhem bf|bon

iem ir fa }`n af|r} |imf~r {ir rcioon| }ìe }`n rnbhem mficn}n|% ]`n|nah|n/ }`nv i|n rnbie} iem

}`n }{h rnbie}r thfe}r {f}` }`n bne}n| ha }`n af|r} bf|bon mnafen ie frhrbnonr }|fiekon ferb|flnm fe

}`n rnbhem bf|bon% ]`n bhcthrf}fhe ha }`nf| i|}faib} |naonb}r }`fr jeh{onmkn lnbi~rn {n ìqn i

bf|bon bne}n|nm fe H tirrfek }`|h~k` I iem i rnbhem bf|bon bne}n|nm fe +i+ tirrfek }`|h~k` L% ]`n

fe}n|rnb}fhe ha lh}` bf|bonr {ir toibnm he }`n Bil|f m|i{fek (Afk~|n 1&% F} fr boni| }ì}/ fe h|mn|

}h bi||v h~} }`fr bher}|~b}fhe/ }`n r}~mne}r ennmnm }h mnio {f}` rfkefafbie}ov ch|n knhcn}|fbio

jeh{onmkn }ìe fe iev fe}n|cnmfi}n r}iknr%

[n eh}fbnm }ì} ir rhhe ir }`n i|}faib}r bhcthrf}fhe bìeknr/ }`n|n fr enbnrri|fov i

}|ierah|ci}fhe ha iev |~onr r~lgibne} }h }`n i|}faib}r h|kiefwnm ib}fher% Bherny~ne}ov/ }`n

rb`ncn ha ~rn irrhbfi}nm {f}` }`fr fer}|~cne} bìeknr% [`ne r}~mne}r tirr a|hc ie nctf|fbio

r}|i}nkv }h i bh||nb} knhcn}|fbio hen/ }`n {`hon r~lgibne} bhebnt}~iofwi}fhe bìeknr

m|ici}fbioov% ]`n rb`ncnr i|n eh} h|kiefwnm rfctov lirnm he jeh{onmkn |nki|mfek }`n frhrbnonr}|fiekonr }hthohkv iem he onek}` qi|fi}fher% M~|fek i bh||nb} bher}|~b}fhe/ he }`nf| }`f|m

i}}nct}/ }`nv cnir~|nm }`n rnkcne} ZILS/ iem }`ne }`nv m|n{ ieh}`n| rnkcne} {f}` npib}ov }`n

ricn cnir~|ncne}r/ {`nenqn| }`nv }~|enm f}r {|f}fek nem }`n h}`n| {iv/ lv ~rfek }`n

RNKCNE] MNA < T]R }hho% ]`fr ib}fhe fr h|kiefwnm r}i|}fek a|hc }`n fmni }ì} }`n bf|bon fr }`n

ohb~r ha }`n ny~fmfr}ie} thfe}r ha i thfe} iem }ì}/ fe ti|}fb~oi|/ }`n thfe}r ha fe}n|rnb}fhe

ln}{nne i bf|bon iem ieh}`n| bf|bon bne}n|nm he }`n af|r} mnafen ie frhrbnonr }|fiekon ferb|flnm


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Ehqir ]nbehohkfir ei Nm~bihBFE]NM#Â\KR

fean|r iem ieiovrn }`n rf}~i}fher/ {n bie nqio~i}n `h{ ie nm~bi}fheio fe}n|aibnr/ i ti|}fb~oi|

rvr}nc ha fer}|~cne}r/ aiqh~| }`n ncn|knebn ha rtnbfafb bhebnt}~iofwi}fher%

Fe ]ilon 2/ {n toibn }{h rf}~i}fher fe }`n ricn bho~ce/ iem ~rn rvclhor }h femfbi}n }`n rvr}nc

ha fer}|~cne}r {f}` {`fb` }`n feqi|fie}r ittni|7 \B ah| |~on| iem bhctirrnr/ iem B ah| Bil|f% Fa

}`n ricn feqi|fie} ittni| fe lh}`/ {n ~rn }`n eh}i}fhe B#\B% Fe }`n af|r} bho~ce/ {n femfbi}n }`n

bhebnt}r iem }`nh|ncr fe ib}fhe }ì} ittni| m~|fek }`n ~rn ha i} onir} hen rvr}nc ha fer}|~cne}r7

cifeov ipfio rvccn}|v (IR&/ bf|bon (B\&/ }|fiekon (]K&/ frhrbnonr }|fiekon (F]&/ iem cnmfi}|fp

(C]&%

]ijfek i ohhj i} }`fr }ilon/ hen bie fccnmfi}nov rnn }ì} }`n oni|efek t|hqfmnm lv Bil|f iem lv

}`n |~on| iem bhctirrnr fr ti|}fb~oi|% ]`n bhebnt}~iofwi}fhe ncn|kfek a|hc }`n ~rn ha nib`

rvr}nc ha fer}|~cne}r fr mfaan|ne}% Fa f} {n|n h}`n|{frn/ }`n|n {h~om ln i k|ni}n| e~cln| ha B#\Br t|nrne}nm he }`n }ilon% Ah| nib` rvr}nc/ }`n ~rn|r ~rnm mfaan|ne} t|htn|}fnr ha }`n ricn

ci}`nci}fbio bhebnt}r% Nib` fe}n|aibn ìr f}r h{e rbhtn ha qio~n ah| }`n }nib`fek iem oni|efek

ha i rtnbfafb bhebnt}% [|hek htn|i}fheio feqi|fie}r {n|n hlrn|qnm heov {`ne r}~mne}r ~rnm

|~onr iem i tif| ha bhctirrnr (C]#4;/ C]#40 iem IR#92&% ]`fr rvr}nc rnncr ch|n aonpflon fe

~rn/ fem~b}fek febo~mfek {|hek ~rikn% He }`n htthrf}n iem/ Bil|fr fe}n|aibn fr ch|n |nr}|fb}fqn


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GOMES, A. S. ; VERGNAUD, Gérard . On the Learning of geometric concepts using Dynamic Geometry...

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Transcript of GOMES, A. S. ; VERGNAUD, Gérard . On the Learning of geometric concepts using Dynamic Geometry...