Educar em Revista
ISSN: 0104-4060
[email protected]
Universidade Federal do Paraná
Brasil
Almouloud, Saddo Ag
As transformações do saber científico ao saber ensinado: o caso do logaritmo
Educar em Revista, núm. 1, 2011, pp. 191-210
Universidade Federal do Paraná
Paraná, Brasil
Disponível em: http://www.redalyc.org/articulo.oa?id=155019936013
Como citar este artigo
Número completo
Mais artigos
Home da revista no Redalyc
Sistema de Informação Científica
Rede de Revistas Científicas da América Latina, Caribe , Espanha e Portugal
Projeto acadêmico sem fins lucrativos desenvolvido no âmbito da iniciativa Acesso Aberto
$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDR
saber ensinado: o caso do logaritmo
7KHWUDQVIRUPDWLRQRIVFLHQWL¿FNQRZOHGJH
LQWRWDXJKWNQRZOHGJHWKHFDVHRI
ORJDULWKPV
Saddo Ag Almouloud1
RESUMO
3DUDHQVLQDUXPDQRomRFLHQWt¿FDHPXPGDGRQtYHOGHHVFRODULGDGHpQHFHVViULRTXHHODVHMDDFHVVtYHODRVDOXQRV3RUWDQWRSUHFLVDVHWUDQVIRUPiOD
DSDUWLUGHXPVDEHUGHUHIHUrQFLDTXHpHPJHUDORVDEHUGRVHVSHFLDOLVWDV
GDGLVFLSOLQDRVDEHUViELR1HVWHWUDEDOKRVmRDSUHVHQWDGDVIHUUDPHQWDV
WHyULFDVGDGLGiWLFDGDPDWHPiWLFDSDUDHVWXGDUDVWUDQVIRUPDo}HVGRVDEHU
FLHQWt¿FRHPVDEHUHQVLQDGReWRPDGRFRPRREMHWRPDWHPiWLFRGHHVWXGR
R³ORJDULWPR´$IXQomRORJDULWPRpXPDGDVQRo}HVPDLVLPSRUWDQWHV
GDVTXHLQWHJUDPRFXUUtFXORGR(QVLQR0pGLR(ODWHPYiULDVDSOLFDo}HV
HPGLYHUVDViUHDVGHFRQKHFLPHQWRWDLVFRPRItVLFDTXtPLFDHFRQRPLD
DVWURQRPLDRTXHMXVWL¿FDVXDPDQXWHQomRQDVSURSRVWDVFXUULFXODUHVGH
YiULRVSDtVHV
Palavras-chaveWUDQVSRVLomRGLGiWLFDIXQomRORJDULWPRHQVLQRDSUHQGL]DJHPFXUUtFXOR
ABSTRACT
,QRUGHUWRWHDFKDVFLHQWL¿FFRQFHSWDWDJLYHQVFKRROOHYHOWKLVFRQFHSW
PXVWEHDFFHVVLEOHWRWKHVWXGHQWVWKHUHIRUHLWPXVWEHWUDQVIRUPHGEDVHG
RQDUHIHUHQFHNQRZOHGJHZKLFKLVLQJHQHUDOWKHNQRZOHGJHIURPWKH
GLVFLSOLQH¶VVSHFLDOLVWVVFKRODUNQRZOHGJH,QWKLVSDSHUWKHRUHWLFDOWRROV
'RXWRUDGRHP(GXFDomR0DWHPiWLFD±8QLYHUVLGDGHGH5HQQHV,±)UDQoD3URIHVVRUGD
3RQWLItFLD8QLYHUVLGDGH&DWyOLFDGH6mR3DXOR38&63%UDVLOVDGGRDJ#JPDLOFRP ou saddoag@
SXFVSEU
Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR
191
ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR
IURPWKHGLGDFWLFRI0DWKHPDWLFVDUHSUHVHQWHGWRVWXG\WKHWUDQVIRUPDWLRQ
RIVFLHQWL¿FNQRZOHGJHLQWRWDXJKWNQRZOHGJH7KHPDWKHPDWLFDOREMHFW
RQIRFXVLVWKH³ORJDULWKP´7KHORJDULWKPIXQFWLRQLVRQHRIWKHPRVW
LPSRUWDQWFRQFHSWVLQWKH+LJK6FKRROFXUULFXOXP,WKDVYDULRXVDSSOLFDWLRQVLQDZLGHUDQJHRINQRZOHGJHDUHDVVXFKDVSK\VLFVFKHPLVWU\
HFRQRPLFVDVWURQRP\MXVWLI\LQJLWVFRQWLQXHGLQFOXVLRQLQWKHFXUULFXOD
SURSRVDOVRIYDULRXVFRXQWULHV
KeywordsGLGDFWLFWUDQVSRVLWLRQORJDULWKPIXQFWLRQWHDFKLQJOHDUQLQJ
FXUULFXOXP
Introdução
3DUD*X\%URXVVHDX
XPSURFHVVRGHDSUHQGL]DJHPSRGHVHUFDUDFWHUL]DGRGHPRGRJHUDOVH
QmRGHWHUPLQDGRSRUXPFRQMXQWRGHVLWXDo}HVLGHQWL¿FiYHLVQDWXUDLV
RXGLGiWLFDVUHSURGXWtYHLVFRQGX]LQGRIUHTXHQWHPHQWHjPRGL¿FDomRGH
XPFRQMXQWRGHFRPSRUWDPHQWRVGHDOXQRVPRGL¿FDomRFDUDFWHUtVWLFDGD
DTXLVLomRGHXPGHWHUPLQDGRFRQMXQWRGHFRQKHFLPHQWRV %52866($8
SWUDGXomRQRVVD
(PGLGiWLFDGDPDWHPiWLFDSURFXUDVHWHRUL]DURVIHQ{PHQRVOLJDGRVj
DWLYLGDGHPDWHPiWLFDYLVDQGRjHVSHFL¿FLGDGHGRFRQKHFLPHQWRHQVLQDGR3DUD
LVVRRVLVWHPDPtQLPRDOHYDUHPFRQVLGHUDomRpRVLVWHPDGLGiWLFRstricto
sensu)LJXUDRXVHMDDVLQWHUDo}HVHQWUHSURIHVVRUHDOXQRVPHGLDGDVSHOR
VDEHUQDVVLWXDo}HVGRHQVLQR
192
Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR
ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR
),*85$75,Æ1*8/2','È7,&2
3DUDHQVLQDUXPDQRomRFLHQWt¿FDHPXPGDGRQtYHOGHHVFRODULGDGHp
QHFHVViULRWRUQiODDFHVVtYHODRVDOXQRV3RUWDQWRSUHFLVDVHWUDQVIRUPiODD
SDUWLUGHXPVDEHUGHUHIHUrQFLDTXHHPJHUDOpRVDEHUGRVHVSHFLDOLVWDVGD
GLVFLSOLQDRVDEHUViELR1HVWHWUDEDOKRDSUHVHQWDPRVIHUUDPHQWDVWHyULFDV
GDGLGiWLFDGD0DWHPiWLFDSDUDUHVSRQGHUDVVHJXLQWHVTXHVW}HV &RPRDQDOLVDU
RFXUUtFXORGHXPQtYHOGHHQVLQR" 3DUDXPDGDGDQRomRTXDLVDVSHFWRVVmR
SULYLOHJLDGRVQRHQVLQRQDVSURSRVWDVFXUULFXODUHVHQDVSUiWLFDVGHFODVVHV"
4XDLVVmRRVDVSHFWRVLPSRUWDQWHVGDQRomRTXHHVWmRDXVHQWHVQRVSURFHVVRV
GHWUDQVIRUPDomR"4XDLVHVFROKDVGLGiWLFDVSRGHPVHUIHLWDV" 7RPDUHPRVFRPR
REMHWRPDWHPiWLFRGHHVWXGRR³ORJDULWPR´$IXQomRORJDULWPRpXPDGDVQRo}HVPDLVLPSRUWDQWHVGHQWUHDVTXHLQWHJUDPRFXUUtFXORGR(QVLQR0pGLR(OD
WHPYiULDVDSOLFDo}HVHPGLYHUVDViUHDVGHFRQKHFLPHQWRWDLVFRPRDItVLFDD
TXtPLFDDHFRQRPLDDDVWURQRPLDRTXHMXVWL¿FDVXDPDQXWHQomRQDVSURSRVWDV
FXUULFXODUHVGHYiULRVSDtVHV
Origem do conceito de transposição didática
9HUUHWLQWURGX]LXRFRQFHLWRGHWUDQVSRVLomRGLGiWLFD,QWHUHVVRX
VHSHODDomRKXPDQDTXHYLVDjWUDQVPLVVmRGHVDEHUHVWRUQDQGRRVSURQWRV
SDUDTXHVHMDP³HQVLQiYHLV´HDSUHQGLGRV6HJXQGRHVVHSRQWRGHYLVWDFRQ-
Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR
193
ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR
YpPVHPG~YLGDWRUQDURVVDEHUHVDFHVVtYHLVDRVDSUHQGL]HVPHGLDQWHXPD
VLPSOL¿FDomRHXPDYXOJDUL]DomROHYDQGRHPFRQWDDLGDGHGHVVHVDSUHQGL]HV
HVHXVFRQKHFLPHQWRVSUpYLRV$WUDQVSRVLomRGLGiWLFDSDVVDVHJXQGR9HUUHW
SRUWUDQVIRUPDo}HVUDGLFDLV(OHDSRQWDFLQFR
x
$HVWUXWXUDomRGRVDEHUHPFDPSRVHGRPtQLRVGLVWLQWRV2VVDEHUHV
FLHQWt¿FRVMiVmRRUJDQL]DGRVHPGLVFLSOLQDV
x
$GHVSHUVRQDOL]DomRGRVDEHURXVHMDRVDEHUViELRQmRHVWiDWUHODGRDLQGLYtGXRVHJUXSRVGHLQGLYtGXRVTXHRSURGX]HPRXRXVDP
x
8PDSURJUDPDomRQHFHVViULDSRLVXPVDEHUDHQVLQDUQmRSRGHVHU
DVVLPLODGRHPXPDVyYH]RVDEHUHQVLQDGRSDVVDHQWmRSRUFDPLQKRVGHIRUPDWDomREDOL]DGD
x
8PDSXEOLFDomRGRVDEHUHPOLYURVUHYLVWDVHSURSRVWDVFXUULFXODUHVTXHSHUPLWDPDFDGDVXMHLWRFRQKHFHURVDEHUDHQVLQDUHDV
FRPSHWrQFLDVHKDELOLGDGHVDDOFDQoDU
x
8PFRQWUROHGDVDTXLVLo}HV
1D HVFROD HVVDV WUDQVIRUPDo}HV FRPHoDP FRP D WUDQVSRVLomR H[WHUQD
WUDQVIRUPDomRGHVDEHUHVHSUiWLFDVHPSURSRVWDVFXUULFXODUHVHSURVVHJXHP
SHODHIHWLYDomRGDVSURSRVWDVWUDQVSRVLomRLQWHUQD(VVDVWUDQVIRUPDo}HVOHYDP
HPFRQVLGHUDomRDVFRQGLo}HVGHWUDEDOKRGRSURIHVVRUHGRDOXQRQDHVFROD
Transposição didática segundo Chevallard
2WHUPR³WUDQVSRVLomRGLGiWLFD´VHJXQGR&KHYDOODUG-RVKXD designa
RFRQMXQWRGDVWUDQVIRUPDo}HVTXHVRIUHXPVDEHUGLWRViELRSDUDVHUHQVLQDGR2X
VHMDUHIHUHVHjVWUDQVIRUPDo}HVTXHVRIUHPDVWHRULDVGRVPDWHPiWLFRVTXDQGRVH
WRUQDPVDEHUHVHVFRODUHVHPSULPHLUROXJDUQDVSURSRVWDVFXUULFXODUHVGHSRLVQRV
OLYURVGLGiWLFRVHHPVDODGHDXOD2VDEHUViELRpFRQVWUXtGRHID]SDUWHGRSDWULP{QLRFXOWXUDOGRSHVTXLVDGRU$VRFLHGDGHVROLFLWDRHQVLQRGHXPDSDUWHGHVVHVDEHU
SRUUD]}HVSXUDPHQWHVRFLDLVIRUPDomRSUR¿VVLRQDOSRUQHFHVVLGDGHVHFRQ{PLFDV
e QHFHVViULR HQWmR WUDQVIRUPDU HVVHV VDEHUHV SDUD TXH SRVVDP VHU HQVLQDGRV H
FRQVHTXHQWHPHQWHHQWHQGLGRVHPGDGRQtYHOeHQWmRLQGLVSHQViYHOH[DPLQDUDV
FDUDFWHUtVWLFDVGRREMHWRGHVDEHUGRSRQWRGHYLVWDHSLVWHPROyJLFRHGDVKLSyWHVHVGH
194
Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR
ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR
DSUHQGL]DJHPHVFROKLGDV
$FRQVWUXomRGRVDEHUSHORSHVTXLVDGRU
7RGRSHVTXLVDGRUGHYHSRUPHLRGHXPDSXEOLFDomRHPXPDUHYLVWDFLHQWt¿FD
WRUQDUDFHVVtYHODWRGRVRVUHVXOWDGRVGHVXDSHVTXLVD3DUDLVVRHOHGHYHWUDQVIRUPDU
WRGDDKLVWyULDGHVHXVWUDEDOKRV(OHHOLPLQDWRGDVDVUHÀH[}HVLQ~WHLVRVHUURVDV
HVWUDWpJLDVXWLOL]DGDVHTXHQmRGHUDPFHUWRRXTXHQmRWLQKDPLPSRUWkQFLDQDUHVROXomRGRSUREOHPD$OpPGLVVRHOHQmRDSUHVHQWDVXDVSUySULDVPRWLYDo}HVQHPVXD
FRQFHSomRGDFLrQFLD(OHdespersonaliza o saber que comunica.(OLPLQDWDPEpP
WRGDDKLVWyULDSDVVDGDGHVVHVDEHUHHYHQWXDOPHQWHQmRRUHODFLRQDFRPRSUREOHPD
SDUWLFXODUTXHTXHULDUHVROYHULQLFLDOPHQWH
Ele desconstextualiza e destemporaliza o saber comunicado'H¿QHSUHYLDPHQWHRYRFDEXOiULRQRYRQDIRUPDGHGH¿QLomRTXHVypFRPSUHHQGLGDSHOROHLWRU
TXHSRVVXLRVFRQKHFLPHQWRVLQGLVSHQViYHLVSDUDDVVLPLODURQRYRVDEHUREMHWRGD
FRPXQLFDomR Ele contribui para o enriquecimento da língua matemática, trazendo um
YRFDEXOiULRHVSHFt¿FR. 2XWURVSHVTXLVDGRUHVUHWRPDUmRHVVHQRYRVDEHUSDUDDSOLFiOR
QDUHVROXomRGHSUREOHPDV1HVWDRFDVLmRHOHVRWUDQVIRUPDUmRRJHQHUDOL]DUmRHP
FDVRGHQHFHVVLGDGH
2HQVLQRGHXPVDEHU
2WUDEDOKRGRSURIHVVRUVXS}HHYLGHQWHPHQWHXPFRQKHFLPHQWRGRREMHWRGH
VDEHUPDVWDPEpPGRPRGRSHORTXDORVDOXQRVFRQVWUXDPVHXVFRQKHFLPHQWRV
(PGLGiWLFDGDPDWHPiWLFDFRQVLGHUDVHTXHDDSUHQGL]DJHPLGHDOFRQVLVWH
HPFRORFDURDOXQRHPVLWXDo}HVSUREOHPiWLFDVFXMDVROXomROHYDULDjFRQVWUXomRGR
FRQKHFLPHQWRYLVDGR2FRQKHFLPHQWRpHQWmRrecontextualizado, é conhecimento que
DSDUHFHHQWmRFRPRVROXomRDXPSUREOHPDHVSHFt¿FR$OpPGRPDLVHVVHQRYRFRQKHFLPHQWRVHQGRFRQVWUXtGRSHORDOXQRSRUVXDLQLFLDWLYDSUySULDprepersonalizado.
Esta recontextualização e repersonalização constituem o trabalho do SURIHVVRU1mR
VHWUDWDGHUHFRQVWLWXLUDRULJHPKLVWyULFDGDGHVFREHUWDGHVVHVDEHUEHPFRPRGDV
GL¿FXOGDGHVTXHSRVVLYHOPHQWHRDFRPSDQKDUDPPDVFULDUXPFDPLQKRPDLVFXUWR
SDUDRDOXQRSDUWLUGDFRQVWUXomRGHVHXVFRQKHFLPHQWRVO trabalho do professor
seria semelhante ao inverso do trabalho do pesquisador. 2SURIHVVRUGHYHFRQVWUXLU
VLWXDo}HVSUREOHPDHPTXHRFRQKHFLPHQWRPDWHPiWLFRDSRQWDGRVHMDUHFRQWH[WXDOL]DGRHUHSHUVRQDOL]DGRHPYLVWDGHVHWRUQDUXPFRQKHFLPHQWRGRDOXQRRXVHMD
XPDUHVSRVWDPDLVQDWXUDOjVFRQGLo}HVLQGLVSHQViYHLVSDUDTXHHVVHFRQKHFLPHQWR
WHQKDXPVHQWLGR3DUD%URXVVHDXKiWUrVFRQGLo}HVSDUDDWLQJLUHVVHREMHWLYR
x
VLPXODUQDVDODGHDXODXPD³PLFURVVRFLHGDGH´PDWHPiWLFDSDUD
SURYRFDUXPGHEDWHFLHQWt¿FRGRPLQDQGRDVVLWXDo}HVGHIRUPXODomRHGHYDOLGDomR
x
LQVWLWXFLRQDOL]DURVDEHUFXOWXUDOHFRPXQLFiYHOTXHVHTXHLUDHQVLQDU
Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR
195
ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR
DRVDOXQRVSRLVHVWHVWDPEpPGHYHPdescontextualizar e recontextualizarVHXVDEHU
&RQVWUXomRGDIHUUDPHQWDPDWHPiWLFD
3DUD TXH D IHUUDPHQWD PDWHPiWLFD FRQVWUXtGD SHOR DOXQR VH WRUQH XP
REMHWRGHVDEHUpQHFHVViULRID]rODIXQFLRQDUQDUHVROXomRGHRXWURVWLSRVGH
SUREOHPDV'HSRLVGHVFRQWH[WXDOL]iODHGHVSHUVRQDOL]iODD¿PGHTXHHODVHMD
UHLQYHVWLGDHPGLIHUHQWHVVLWXDo}HVSUR¿VVLRQDLVSRUH[HPSOR
(VVHWUDEDOKRGHGHVFRQVWH[WXDOL]DomRHGHGHVSHUVRQDOL]DomRGHYHQHFHVVDULDPHQWHVHUIHLWRSHORDOXQRSDUDTXHSRVVDLQWHJUDUHVVHQRYRFRQKHFLPHQWRDRVHX
SDWULP{QLRFXOWXUDO(OHGHYHHQWmRFRPRRSHVTXLVDGRUID]HUXPDFRPXQLFDomR
HVFULWDDRFRQMXQWRGHVHXVFROHJDVGHFODVVHQRLQWXLWRGHSURYRFDUGHEDWHVTXH
HQULTXHFHULDPHVFODUHFHULDPHVVHQRYRVDEHURXGHOLPLWDULDPDVFRQGLo}HVGH
VXDVDSOLFDo}HV2WUDEDOKRGRDOXQRGHYHVHUGRPHVPRWLSRTXHRGRSHVTXLVDGRU
$VFRQGLo}HVFRQFUHWDVGHHQVLQR
$VWUDQVIRUPDo}HVGRREMHWRGHVDEHUHPREMHWRGHHQVLQRGHYHPVHUQHFHVVDULDPHQWHDFRPSDQKDGDVGHXPDDQiOLVHHSLVWHPROyJLFDGDVKLSyWHVHVGHDSUHQGL]DJHP
HGRFRQWH[WRVRFLDO2SURIHVVRUQmRWUDQVIRUPDSRULQLFLDWLYDSUySULDRVDEHUViELR
HPREMHWRGHHQVLQR$HVFROKDGRVREMHWRVDHQVLQDUpGH¿QLGDLQVWLWXFLRQDOPHQWH
SRUPHLRGHSURSRVWDVFXUULFXODUHVHpFRQWURODGDGHDOJXPDIRUPDSHODVRFLHGDGH
DXWRULGDGHVORFDLVSDLVGHDOXQRVDXWRULGDGHVDGPLQLVWUDWLYDVGDHGXFDomR
1D)LJXUDHVWiUHSUHVHQWDGDDLGHLDGHTXHRSURIHVVRUQmRWHPLQÀXrQFLD
GLUHWDQDHODERUDomRGDVSURSRVWDVFXUULFXODUHVR¿FLDLV
),*85$5(7Æ1*8/2','È7,&2
196
Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR
ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR
7HPRVXPFRQMXQWRGHLQÀXrQFLDVGHTXHGHSHQGHDDomRGRSURIHVVRUR
H[HPSORPDLVPDUFDQWHpRGRSDSHOGRVOLYURVGLGiWLFRVQRVSURFHVVRVGHHQVLQR
HDSUHQGL]DJHP2WUDEDOKRGHWUDQVSRVLomRDQWHULRUDRWUDEDOKRGRSURIHVVRU
OHYDDGHFRPSRUDWUDQVSRVLomRHPGRLVQtYHLVFRQIRUPHD)LJXUDLOXVWUD
),*85$(7$3$6'$75$16326,d­2','È7,&$
*HUDOPHQWHRSURIHVVRUVyLQWHUYpPQRQtYHOGRVDEHUHQVLQDGR2VDEHU
DHQVLQDUQmRVHOLPLWDjVSURSRVWDVFXUULFXODUHVVHXHQVLQRQHFHVVLWDGHVXD
LQWHUSUHWDomR2VDEHUDHQVLQDUpRTXHRSURIHVVRUDFKDTXHGHYHHQVLQDUD
SDUWLUGDOHLWXUDGHOLYURVGLGiWLFRVGROLYURGRSURIHVVRURXDSDUWLUGHSUiWLFDV
WLGDVDQWHULRUPHQWH2WH[WRGRVDEHUDHQVLQDUQmRHVWiFRPSOHWDPHQWHHVFULWR
HPOXJDUDOJXPeLQGLVSHQViYHOH[DPLQDUVHDGLVWkQFLDDGHIRUPDomRHQWUHR
REMHWRGHVDEHUHRREMHWRGHHQVLQRQmRpQDSLRUGDVKLSyWHVHVXPDOLQJXDJHP
SVHXGRFLHQWt¿FD
$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGRRFDVRGR
logaritmo
$QRomRGHORJDULWPRpXPDGDVQRo}HVPDLVLPSRUWDQWHVTXHHQFRQWUDPRV
HPPDWHPiWLFDHQRFXUUtFXORGHHQVLQRPpGLR&RPRDVVLQDODGRDQWHVHODWHP
YiULDVDSOLFDo}HVHPGLYHUVDViUHDVGHFRQKHFLPHQWR(VVDVDSOLFDo}HVOKHVFRQ-
Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR
197
ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR
IHUHPXPSDSHOLPSRUWDQWHQRHQVLQR$QWHVGH1pSHURIXQGDPHQWRGD
WHRULDGRVORJDULWPRVHUDUHODFLRQDUXPDVHTXrQFLDGHSRWrQFLDVGHXPQ~PHUR
GHVGHDpSRFDSDOHREDELO{QLFD
2PDWHPiWLFR-RKQ1pSHUapud*5283(0$7+LQYHQWRX
DSDODYUDHRFRQFHLWRGHORJDULWPRHP6HXREMHWLYRHUDVLPSOL¿FDUR
FiOFXORGHXPSURGXWRSRURGHXPDVRPD1DpSRFDDQRomRGHIXQomRQmR
H[LVWLDHRVORJDULWPRVVHUYLUDPGHIHUUDPHQWDSDUDRGHVHQYROYLPHQWRGHVVH
FRQFHLWRGH¿QLGRSRU(XOHUVRPHQWHHP7RUQRXVHXPPHLRSDUDID]HU
FRUUHVSRQGHUXPORJDULWPRDFDGDQ~PHURSRVLWLYRWUDWDYDVHHQWmRGHXPD
IXQomR3DUD1pSHURVORJDULWPRVVmRQ~PHURVTXHFRUUHVSRQGHPDQ~PHURV
SURSRUFLRQDLV 8VD XPD SURSRUomR JHRPpWULFD RX VHMD D E F G VmR HP
SURSRUomRJHRPpWULFDVHHVRPHQWHDG EFDVVLPDVRPDORJDORJG ORJEORJF
6HJXQGR -DFTXHV 2]DQDP apud *5283( 0$7+ ³RV
ORJDULWPRVVmRQ~PHURVHPSURSRUomRDULWPpWLFDTXHFRUUHVSRQGHPDRXWURV
Q~PHURVHPSURSRUomRJHRPpWULFD´(OHHVFROKHXXPDIXQomRIFRQWtQXDHHV*
tritamente crescente em IIR + TXHYHUL¿FDDVHJXLQWHSURSULHGDGH³ORJDUtWPLFD´
f transforma toda proporção geométrica em uma proporção aritmética e tal
que f(10)=1 f(1)=0.
Um breve estudo da transposição didática nos livros didáticos
(PDOJXQVOLYURVGLGiWLFRVGR%UDVLO log a x ORJDULWPREDVHa de x é o
\
número a que temos de elevar a para obter x, ou seja, log a x = y œ a = [ D ([LVWHXPDIXQomRORJDUtWPLFDTXHpLQYHUVtYHOHFXMDLQYHUVDpDIXQomR
H[SRQHQFLDOGH¿QLGDSDUDWRGRQ~PHURUHDO
E 1DPDLRULDGRVOLYURVGLGiWLFRVGH(QVLQR0pGLRGD)UDQoDHGRVSDtVHV
DIULFDQRVGHOtQJXDIUDQFHVDRORJDULWPRQHSHULDQRpGH¿QLGRSRUOQt œt1 1 dx G[
x
Se admitirmos o seguinte teorema “Toda função f contínua em um intervalo I
tem primitivas´HQWmRGHPRQVWUDVHTXHH[LVWHXPD~QLFDIXQomR)TXHYHUL¿FD
SDUDHSRUWDQWRDIXQomRIGH¿QLGDSRUI[ WHPXPDSULPLWLYD)HPWDOTXH
) (VWDSUREOHPiWLFDGHEXVFDGHSULPLWLYDVHVWiQDRULJHPGDLQWURGXomRGR
FRQFHLWRGHORJDULWPR(VWDFRQFHSomRQmRSRGHVHUHQVLQDGDQRWHUFHLURDQR
GR(QVLQR0pGLR$SUREOHPiWLFDGHFiOFXORRXVHMDDUHODomRHQWUHPXOWLSOLFDomRHDGLomRHVWiQDRULJHPGHQRVVDSURSRVWDVHUYLQGRQRVGDVSURSRUo}HV
JHRPpWULFDVHDULWPpWLFDVFRPRIHUUDPHQWD(VWHWLSRGHDERUGDJHPRULJLQDGD
198
Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR
ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR
SHORXVRGDUpJXDGHFiOFXORSDUHFHEHPDIDVWDGRGRXVRGDVFDOFXODGRUDV
PDVJXDUGDVHXLQWHUHVVHQDPHGLGDHPTXHFRORFDHPHYLGrQFLDXPDSUiWLFD
GHFiOFXORTXHGHVHPSHQKRXXPSDSHOLPSRUWDQWHQDKLVWyULDGDPDWHPiWLFD
1RVVDSURSRVWDHVWiDSRLDGDQRXVRGHXPDSURSULHGDGHGRFRQMXQWRGRVQ~PHURVUHDLVPDLVHVSHFL¿FDPHQWHDGRLVRPRU¿VPRGRJUXSRDGLWLYRGRFRQMXQWR
GRVQ~PHURVUHDLVVREUHRJUXSRPXOWLSOLFDWLYRGRFRQMXQWRGRVQ~PHURVUHDLV
HVWULWDPHQWHSRVLWLYRV
f IIR + o IIR +* u
f a + b = f a u f b
3HUFHEHPRV TXH D PDLRULD GRV OLYURV GLGiWLFRV GD )UDQoD H GRV SDtVHV
DIULFDQRVGHOtQJXDIUDQFHVDDSUHVHQWDVLWXDo}HVHPTXHVHSURFXUDPIXQo}HV
TXHWrPDVHJXLQWHSURSULHGDGH
f IIR +* oIIR tal que f xy y = f x + f y 2REMHWLYRGHQRVVDSURSRVWDSDUDLQWURGX]LURFRQFHLWRGHORJDULWPRp
SUHYDOHFHUVH GD SURSULHGDGH ORJDUtWPLFD GH XPD IXQomR HVWULWDPHQWH FRQWtQXDQRFRQMXQWRGRVQ~PHURVUHDLVSRVLWLYRVHQmRQXORV³f transforma toda
proporção geométrica em uma proporção aritmética´'RSRQWRGHYLVWDGD
WUDQVSRVLomRGLGiWLFDKiXPDGLIHUHQoDQRWiYHOHQWUHDVDERUGDJHQVSURSRVWDV
QRVOLYURVGLGiWLFRVEUDVLOHLURVDQDOLVDGRVHDVSURSRVWDVGDPDLRULDGRVOLYURV
GLGiWLFRVIUDQFHVHVHGDÈIULFDGHOtQJXDIUDQFHVD&RPRGHVWDFDPRVRVOLYURV
GHOtQJXDIUDQFHVD)UDQoDHÈIULFDGHOtQJXDIUDQFHVDGH¿QHPDIXQomRORJD1
ULWPRQHSHULDQRFRPRDIXQomR)SULPLWLYDGDIXQomRIGH¿QLGDSRU I [a I[ =
[
[RXVHMD
VHQGR ) VHQGR ) QRV OLYURV EUDVLOHLURV
\
RORJDULWPRpGH¿QLGRFRPRQ~PHUR\WDOTXH log a x = y œ a = [ 3URSRPRVRXWUDPDQHLUDGHLQWURGX]LUHHVWXGDUDQRomRGHIXQomRORJDULWPRXWLOL]DQGRVHGDSURSULHGDGHDFLPDFLWDGDDSDUWLUGHXPDVLWXDomRTXH
SHUPLWHHVWXGDUWRGDVDVSURSULHGDGHVGHVVDIXQomRUHVROYHUHTXDo}HVORJDUtWPLFDVHVERoDUDFXUYDHDUHWDWDQJHQWHHPXPSRQWRGDFXUYDGHVVDIXQomR
$SRLDPRQRVQRWUDEDOKRSURSRVWRSHOR*URXSH0$7+,5(0GH3DULV9,,
$GDSWDPRVDVLWXDomRSURSRVWDSRUHVWHJUXSRGHSHVTXLVD
Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR
199
ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR
Enunciado da situação-problema estudada2
$SUHVHQWDPRVGXDVGH¿QLo}HVLPSRUWDQWHVSDUDDUHVROXomRGDVLWXDomR
1. Quatro números reais positivos e não nulos a, b, c, d dizem-se em
proporção geométrica se, e somente se ad = bc
2.
Quatro números reais a, b, c, d são em proporção aritmética se, e
somente se a + d = E + c .
4XHVWmR6HMDIXPDIXQomRHP50RVWUHTXHVHIWUDQVIRUPDWRGDSURSRUomRJHRPpWULFDHPXPDSURSRUomRDULWPpWLFDHQWmRIWUDQVIRUPDWRGDVHTXrQFLD
JHRPpWULFDHPXPDVHTXrQFLDDULWPpWLFDSRGHVHYHUL¿FDUTXHVHXnpXPDVHTXrQFLD
JHRPpWULFDGHUD]mRq z 0HQWmR1, q, un , un+1VmRHPSURSRUomRJHRPpWULFD
$JRUDTXHUHPRVWUDoDUDFXUYDUHSUHVHQWDWLYDGHXPDIXQomRIFRQWtQXDH
estritamente crescente em IR+* TXHYHUL¿FD DVHJXLQWHSURSULHGDGH³ORJDUtWPLFD´
f transforma toda proporção geométrica em uma proporção aritmética. Para
ID]HUHVVDUHSUHVHQWDomRHVWXGDUHPRVDOJXPDVGDVFDUDFWHUtVWLFDVGHVWDIXQomR
&RPR-2]DQDPHVFROKHPRVHVWXGDUDIXQomRIWDOTXHI HI $JRUDTXHWHPHVVDVLQIRUPDo}HVUHVSRQGDjVVHJXLQWHVTXHVW}HV
4XHVWmR Calcule :
n
f(100,
f(1000) f(10.000)
f(10 )
4XHVWmR a) Mostre que f(15) = f(3) + f(5),
HTXHSDUD[H\HOHPHQWRVGH5+* I[\ I[I\
E&DOFXOHSDUD[HOHPHQWRGH5+* RYDORUGH
em
IXQomRGHI[
F'HWHUPLQHSDUDDHEHOHPHQWRVGH5+* f ( ab ) HPIXQomRGH
IDHIE
4XHVWmR 5HVROYDQR5+* DVHTXDo}HV
DI[ ò
EI[ FI[ 8PDVLWXDomRSUREOHPDpDHVFROKDGHTXHVW}HVDEHUWDVRXIHFKDGDVQXPDVLWXDomRPDLV
RX PHQRV PDWHPDWL]DGD HQYROYHQGR XP FDPSR GH SUREOHPDV FRORFDGRV HP XP RX HP YiULRV
FDPSRVGHFRQKHFLPHQWRVPDWHPiWLFRV$IXQomRSULQFLSDOGHXPDVLWXDomRSUREOHPDpDXWLOL]DomR
LPSOtFLWDGHSRLVH[SOtFLWDGHQRYDVIHUUDPHQWDVPDWHPiWLFDVHPQRVVRFDVRDVSURSULHGDGHVGD
IXQomRISRUPHLRGHTXHVW}HVTXHRDOXQRVHFRORFDQRPRPHQWRGHVXDSHVTXLVD
200
Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR
ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR
4XHVWmR&RPSOHWHDVHJXLQWHWDEHOD
x
1/10
f(x)
-1
1
-3/4
-1/2
-1/4
0
10
10
1/4
1/2
3/4
1
5/4
3/2
7/4
4XHVWmR(PXPVLVWHPDFDUWHVLDQRRUWRJRQDOGHXQLGDGHFPWUDFHD
FXUYDUHSUHVHQWDWLYDGDIXQomRI
4XHVWmR &RQVLGHUHRVSRQWRV0iGDFXUYDGHRUGHQDGDV
D 5HSUHVHQWHRVSRQWRV0i¶GHFRRUGHQDGDV \ M i - &XLGDGR
$XQLGDGHpFP
(
)
l
E &RQVWUXDDVUHWDV M i M i ' l
F ,GHQWL¿TXHDUHWD M i M i ' TXHDSUR[LPDPHOKRUDFXUYD4XDOpR
FRH¿FLHQWHDQJXODUGHVVDUHWD"(QFRQWUDI[
1. Análise matemática da situação-problema
'H¿QLPRVDVQRo}HVGHSURSRUomRJHRPpWULFDHGHSURSRUomRDULWPpWLFD
(VWDVGXDVQRo}HVVmRIXQGDPHQWDLVQDUHVROXomRGDVLWXDomRDVVLPFRPRQD
XWLOL]DomRGDSURSULHGDGHORJDUtWPLFDGDIXQomRI
$ SULPHLUD TXHVWmR WHP SRU REMHWLYR GHPRQVWUDU TXH D IXQomR I
WUDQVIRUPD WRGD SURJUHVVmR JHRPpWULFD HP XPD SURJUHVVmR DULWPpWLFD
3DUWLQGR GH XPD SURJUHVVmR JHRPpWULFD Xn GH UD]mR T WHPRV TXH T
u n u n +1 HVWmR HP SURSRUomR JHRPpWULFD SRLV 1u n +1 = qu
qu n /RJR
p g 1 q un un +1 Ÿ f 1 + f un +1 = f q + f un 6HMDt n = f u n e t n +1 = f u n +1 SRUWDQWR t n +1 = r + t n RQGH r = f q f 1 1DVHJXQGDTXHVWmRSHGHVHFDOFXODUDVLPDJHQVGHFHUWRVQ~PHURVSRU
I OHYDQGR HP FRQVLGHUDomR DV VHJXLQWHV FRQGLo}HV
­ f 1 = 1D
®
¯ f = 1
UHVROXomRGRVGLIHUHQWHVLWHQVGHVWDSUHFLVDPVHUXWLOL]DGDVSURSRUo}HVJHRPpWULFDVHDSURSULHGDGHDOJRUtWPLFDGDIXQomRI
a)&DOFXOH f f f e f n
p g 1
œ f = f f 1 + f = f = p g 1 œ f = f f 1 + f = 1 + = p g œ f = f f 1 + f = 1 + = 3RULQGXomRFRPSOHWDGHPRQVWUDVHTXH n  IN
IN f nn = n
3DUDQ I I Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR
201
ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR
+1
6XS}HVHTXH n  IN
IN f nn = n HGHPRQVWUDVHTXH f nn+1
= n + 1
+1
p g 1
n n +1 œ f nn+1
= f f 1 + f nn = 1 + n
3RUWDQWRSDUDWRGR n  IN
nn = n SULQFtSLRGDLQGXomRFRPSOHWD
IN f b)
D
Demonstrar que f = f + f p g œ f = f f 1 + f = f + f SRLV f 1 = *
'HPRQVWUDUTXHSDUDWRGRV[H\GHI IR + WHPVH f xy
xy = f x + f y x, y,\ [\
xy œ f xy
xy = f x f 1 + f y = f x + f y S g [
f xy
xy = f x + f y 3URSULHGDGHIXQGDPHQWDOGHI
E 'HGX]DSDUDWRGR[GH IR
IR +* f 1 e f x HPIXQomRGH f x
x
1
1
1
f 1 = f x = f x + f = o œ f = f x
x
x
x
1
f x = f x x = f x + f x = f x œ f x = f x
IR + f J &DOFXOHSDUDWRGRDHEGH IR
*
abab HPIXQomRGH f a e f b
1
1
1
f a + f b = f a + f b
$WHUFHLUDTXHVWmRWHPSRUREMHWLYRHVERoDURJUi¿FRGDFXUYDGDIXQomRI
f ab
ab = f a b = f a + f b =
x
f x
202
1
-1
1
1
1
1
1
1
1
Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR
ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR
&XUYDUHSUHVHQWDWLYDGHI
),*85$*5È),&2'$)81d­2)('$65(7$60i0¶i
D0¶10¶0¶0¶0¶
0¶0¶0¶
EFI)LJXUD
F$UHWDTXHPHOKRUDSUR[LPDDFXUYDGHIpDUHWDTXHSDVVDSHORVSRQWRV
6HXFRH¿FLHQWHDQJXODUpLJXDOD
$QiOLVHGLGiWLFDGDVLWXDomRSUREOHPD
$VLWXDomRSUREOHPDIRLHODERUDGDOHYDQGRHPFRQVLGHUDomRDVVHJXLQWHV
FRQGLo}HV
2VDOXQRVFRPSUHHQGHPIDFLOPHQWHRVGDGRVHSRGHPHQJDMDUVHQD
H[SORUDomRGHVVHVGDGRVFRPRVFRQKHFLPHQWRVGLVSRQtYHLV3RGHP
FRQFHEHUFODUDPHQWHRTXHpXPDUHVSRVWDSRVVtYHOHSHUWLQHQWHj
TXHVWmRFRORFDGD
$VLWXDomRSUREOHPDHQYROYHXPFDPSRFRQFHLWXDORFDPSRFRQFHLWXDOGRORJDULWPRTXHGHVHMDPRVHIHWLYDPHQWHH[SORUDUHHPTXH
Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR
203
ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR
VHVLWXDPDVDSUHQGL]DJHQVYLVDGDV
2VFRQKHFLPHQWRVDQWLJRVGRVDOXQRVVmRLQVX¿FLHQWHVSDUDDUHVROXomRLPHGLDWDGRSUREOHPD
2VFRQKHFLPHQWRVREMHWRVGDDSUHQGL]DJHPIRUQHFHPDVIHUUDPHQWDVDGHTXDGDVSDUDREWHUDVROXomR
$VLWXDomRSUREOHPDHQYROYHRVVHJXLQWHVTXDGURVDOJpEULFRJHRPpWULFRQXPpULFR
2IDWRGHVXSRUTXHDIXQomRIpGH¿QLGDHPIR
IR +* UHPRQWDjKLVWyULDGRV
ORJDULWPRV$IXQomRIpFRQWtQXDHHVWULWDPHQWHFUHVFHQWH(VWDSURSULHGDGH
JDUDQWHDXQLFLGDGHGDVVROXo}HVGDVHTXDo}HVORJDUtWPLFDV6DEHVHTXHXPD
IXQomRFRQWtQXDHFUHVFHQWHHPLQWHUYDORpELMHWRUD3RUWDQWRDHTXDomR
IR e x  IR
f x = m m  IR
IR +* DGPLWHXPDVROXomR~QLFDRTXHSHUPLWHFRPSOHWDU
DWDEHOD
As igualdades f 1 = HI VmRGXDVFRQGLo}HVUHODFLRQDGDVFRP
DSURSULHGDGHGDIXQomRHVWXGDGD2VQ~PHURVHVmRSDUWLFXODUPHQWHLPSRUWDQWHVQDHVFROKDGDVSURSRUo}HVJHRPpWULFDVFXMDVLPDJHQVSRUISHUPLWHP
UHVROYHUSUDWLFDPHQWHWRGDVDVTXHVW}HVGDVLWXDomR
3DUDMXVWL¿FDUDHVFROKDGHXVDVHDHTXDomRGDWDQJHQWHjFXUYD
GHIHPXPSRQWR[ RXVHMD\ I¶[[±[I[6HI[ ORJ[HQWmR
DHTXDomRGDWDQJHQWH7jFXUYDGHIHPXP
SRQWRGHDEVFLVVD[p7
PDV
VH[ WHPVH
SRUWDQWRy=ymo-0,43RQGH \ M = I[ 4XDLVFRQKHFLPHQWRVRVDOXQRVSRGHPPRELOL]DUQDUHVROXomRGDVLWXDomR"
'L]HPRVTXHXPDOXQRSRVVXLXPFRQKHFLPHQWRPDWHPiWLFRVHHOHIRUFDSD]
GHXWLOL]iORGHIRUPDH[SOtFLWDQDUHVROXomRGHSUREOHPDV1DVLWXDomRSURSRVWDR
DOXQRSUHFLVDPLQLPDPHQWHPRELOL]DURVVHJXLQWHVFRQKHFLPHQWRVDSURYDSRULQGXomR
FRPSOHWDTXHSHUPLWHGHPRQVWUDUTXHf(10n)=nDSURSULHGDGHORJDUtWPLFDGDIXQomRI
HDVFDUDFWHUL]Do}HVGDVSURJUHVV}HVJHRPpWULFDVHDULWPpWLFDVTXHSHUPLWHPGHVYHQGDU
DVSLVWDVTXHOHYDULDPjVROXomRGDVLWXDomRSURSRVWDDVSURSULHGDGHVGDUDL]TXDGUDGD
GHXPQ~PHURSRVLWLYRTXHVmRLPSUHVFLQGtYHLVDRFiOFXOR f x e
1RWDPRVDLPSRUWkQFLDHRSDSHOGDVPXGDQoDVGHTXDGURQRWUDWDPHQWRGDV
TXHVW}HVGDVLWXDomR(VVDVPXGDQoDVSHUPLWHPPXGDUGHSRQWRGHYLVWDHWUDGX]LU
DOJXPDVGDVTXHVW}HVGDVLWXDomRGRTXDGUR'28$'<DOJpEULFRDRTXDGURGD
JHRPHWULDDQDOtWLFDFRPD¿QDOLGDGHHVSHFt¿FDGHPRELOL]DUDVIHUUDPHQWDVDGHTXDGDV
204
Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR
ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR
jUHVROXomRGHVVDVTXHVW}HV$VLQWHUDo}HVHQWUHHVVHVTXDGURVFRQVWLWXHPXPSRGHURVR
LQVWUXPHQWRSDUDDFRQVWUXomRSRUSDUWHGRDOXQRGRFRQKHFLPHQWRVDEHUHPMRJRQD
VLWXDomRSUREOHPDDSDUWLUGHVHXVFRQKHFLPHQWRVGLVSRQtYHLV
Uma breve análise da fase experimental
$H[SHULrQFLDTXHUHODWDPRVIRLUHDOL]DGDHPXPDWXUPDGHVpULHGR(QVLQR
0pGLR&LrQFLDV([DWDVGH0DOL(VWDH[SHULrQFLDIRLUHDOL]DGDSRU6LDND.RQDWp
3DUWLFLSDUDPGDSHVTXLVDGDVLWXDomRSUREOHPDDOXQRVTXHDLQGDQmR
HVWXGDUDPRFRQFHLWRGHORJDULWPR2VDOXQRVWUDEDOKDUDPHPJUXSRJUXSRVGH
TXDWURDOXQRVHJUXSRVGHDOXQRVTXHGHQRPLQDPRVGH*1* GG**
GDOXQRV *e GFRPSRVWRVSRUDOXQRV2WHPSRSUHYLVWRSDUDDUHVROXomR
GDVLWXDomRIRLGHK
1DPDLRULDGRVJUXSRVKRXYHFRQIXVmRHQWUHDSURSRUomRDULWPpWLFDHDSURJUHVVmRDULWPpWLFDH[FHWRQR*TXHMXVWL¿FRXVXDUHVSRVWDGDVHJXLQWHIRUPDYQ
IXQIT±I UHWIXn YnSRUWDQWRYQ YnU
3DUDRFiOFXORIVDEHQGRTXHI HI IRLFRQVWDWDGRTXHRV
DOXQRVQmRVDELDPSRURQGHFRPHoDUDVROXomRGRSUREOHPD3UHFLVRXVHLQGDJiORV
VREUHDSRVVLELOLGDGHGHHQFRQWUDUXPQ~PHUR[WDOTXHRVQ~PHURV[H
HVWLYHVVHPHPSURSRUomRJHRPpWULFD(VWDSHUJXQWDSHUPLWLXDRVDOXQRVLQLFLDUD
UHVROXomRGDTXHVWmR3DUDRFiOFXORGHInDPDLRULDGRVJUXSRVIH]XPDFRQMHFWXUDPDVVHPGHPRQVWUiOD3DUDLOXVWUDUHVWHIDWRpDSUHVHQWDGDDVHJXLU)LJXUD
H)LJXUDDSURGXomRGRJUXSR*
),*85$5(62/8d­2$35(6(17$'$3(/2*5832.21$7eS
Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR
205
ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR
),*85$5(62/8d­2$35(6(17$'$3(/2*5832.21$7eS
2VDOXQRVQmRWLYHUDPTXDOTXHUSUREOHPDDRSUHHQFKHUDWDEHODQHPQD
GHWHUPLQDomRGDUHWDWDQJHQWHjFXUYDeDSUHVHQWDGRQD)LJXUDXPH[HPSOR
GDVSURGXo}HVGRVDOXQRV
),*85$5(62/8d­2$35(6(17$'$3(/2*5832.21$7eS
206
Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR
ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR
),*85$352'8d­2'2*5832.21$7eS
&RPUHODomRjUHSUHVHQWDomRJUi¿FDGDFXUYDGHIRXVRGRSDSHOPLOLPHWUDGRHDHVFROKDGHFPFRPXQLGDGHQRVHL[RVGRSODQRFDUWHVLDQRIRUDP
GHWHUPLQDQWHVQDFRQVWUXomRGRJUi¿FRFRPRSRGHVHUREVHUYDGRQDSURGXomR
GRJUXSR)LJXUDVH
Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR
207
ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR
),*85$352'8d­2'2*5832.21$7eS
208
Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR
ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR
),*85$352'8d­2'2*5832.21$7eS
Conclusão
)RLSURSRVWRGRSRQWRGHYLVWDGDWUDQVSRVLomRGLGiWLFDXPPpWRGRSDUD
LQWURGX]LUHHVWXGDURORJDULWPRDSDUWLUGDVQRo}HVGHSURSRUomRDULWPpWLFDH
JHRPpWULFDVHPH[SOLFLWDUDOHLGDIXQomR$OpPGLVVRFRQVHJXLXVHDQDOLVDU
GHWHUPLQDGRVFRPSRUWDPHQWRVGRVHVWXGDQWHVEHPFRPRDVGL¿FXOGDGHVTXHHQFRQWUDUDPQDVHTXrQFLDGH.RQDWp(YLGHQWHPHQWHDVLWXDomRSUREOHPD
H[DPLQDGD QmR SHUPLWH HVWXGDU WRGRV RV HOHPHQWRV GR FDPSR FRQFHLWXDO GR
ORJDULWPR&RPRSRGHVHUSHUFHELGRHODIRFDOL]DRORJDULWPRGHEDVHGHFLPDO
+iSRUWDQWRDQHFHVVLGDGHGHFRQVWUXomRGHRXWUDVVLWXDo}HVFXMRSURSyVLWR
VHULDRGHHVWXGDUDIXQomRORJDULWPRGHEDVHGLIHUHQWHGH$OpPGLVVRKiD
QHFHVVLGDGHGHHVWXGDUDUHODomRTXHH[LVWHHQWUHDIXQomRORJDULWPRHDIXQomR
H[SRQHQFLDO
Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR
209
ALMOULOUD6$$VWUDQVIRUPDo}HVGRVDEHUFLHQWt¿FRDRVDEHUHQVLQDGR
REFERÊNCIAS
$/028/28' 6 Fundamentos da didática da matemática &XULWLED (GLWRUD GD
8)35
%52866($8 * )RQGHPHQWV HW PpWKRGHV GH OD GLGDFWLTXH GHV PDWKpPDWLTXHV
Recherche en didactique des mathématiques*UHQREOHYQS
&+(9$//$5'<-2+68$0$La transposition didactique*UHQREOH/D3HQVpH
6DXYDJHeGLWLRQV
'28$'<5-HX[GHFDGUHVHWGLDOHFWLTXHRXWLOREMHWRecherche en didactique des
mathématiques*UHQREOHYQS
*5283( 0$7+ ,5(0 GH 3DULV 9,, 0DWKpPDWLTXHV DSSURFKH SDU GHV WH[WHV
KLVWRULTXHVRepères-IREM3RQWj0RXVVRQ7RSLTXHV(GLWLRQVYS
.21$7(6L’enseignement et l’apprentissage des logarithmiques dans nos classes
de terminales0pPRLUHGH'($GH'LGDFWLTXHGHV0DWKpPDWLTXHV%DPDNR0DOL
8QLYHUVLWpGH%DPDNR
9(55(70Le temps des études3DULV+RQRUp&KDPSLRQ
210
Educar em Revista, Curitiba, Brasil, n. Especial 1/2011, p. 191-210, 2011. Editora UFPR
Download

Full screen - Red de Revistas Científicas de América Latina y el