Práticas de ensino exploratório da Matemática: Ações e intenções de uma professora

9. Práticas de ensino exploratório

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de uma professora

Ana Paula Canavarro

Universidade de Évora e UIDEF,

Instituto de Educação, Universidade de Lisboa

[email protected]

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Instituto de Educação, Universidade de Lisboa

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Introdução

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Perspetivas teóricas

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A prática de ensino exploratório da professora Célia

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XP HVIRUoR GH LQWHUSUHWDomR &pOLD WHQWD DVVHJXUDU TXH RV DOXQRV FRPSUHHQGDP EHPRFRQWH[WRHRVREMHWLYRVGDWDUHIDRXYLQGRDWHQWDPHQWHRVVHXVFRPHQWiULRV e perguntas: SRUWDQWRHXDFKRTXHGHYHVHUXPDDSUHVHQWDomRGHXPGHVDÀR HGHYHVHUXPDLQWHUSUHWDomRHXPDFRPSUHHQVmRGRTXHpTXH DTXHOHGHVDÀRSUHWHQGH«3HQVRTXHVLPTXHVHUmRHVVHVRVGRLV objetivos, o interpretar, o compreender o que se pretende. (…) (XWHQKRDSUHRFXSDomRGHGHGLFDURWHPSRVXÀFLHQWHSDUDHOHV SHUFHEHUHPHGDUWDPEpPWHPSRSDUDHOHVFRORFDUHPDVTXHVW}HV H DV G~YLGDV« 2 HVWDU D TXHVWLRQiORV H R HVWDU j HVSHUD TXH SHUJXQWHPRXTXHWHQKDPG~YLGDVpWDPEpPSDUDTXHHXSUySULD perceber se eles perceberam.

1D SHUVSHWLYD GH &pOLD VH RV DOXQRV QmR HQWHQGHP D WDUHID ´HOHV YmR FULDU FRQIXV}HV H HQWmR jV YH]HV p WDUGH GHPDLV QD DXOD SDUD HVFODUHFHU H SDUD RV UHFXSHUDUµ$OpPGHJDUDQWLUDFRPSUHHQVmRGDWDUHIDSHORVDOXQRV&pOLDWDPEpP TXHUTXHHOHVVHHQYROYDPFRPHVWDHDVVXPDPRGHVDÀRGHDUHVROYHU

e LPSHOLORV SDUD D WDUHID SUHGLVS{ORV SDUD D WDUHID 1mR p Vy FRPSUHHQGHUPDVDVVXPLUDWDUHIDFRPRVXD«QRIXQGRDTXLHQWUD WDPEpPXPERFDGRDSDUWHGRGHVDÀRDVVXPLUFRPRXPDTXHVWmR minha que eu quero resolver.

Desenvolvimento da tarefa. &pOLD FHQWUD D VXD UHÁH[mR VREUH HVWD IDVH GD

DXODQRVHXDSRLRDRVDOXQRVSDUDDUHVROXomRDXWyQRPDGDWDUHID&RQIHVVDDV VXDVGLÀFXOGDGHVHPPRQLWRUDURSURJUHVVRGRWUDEDOKRGRVDOXQRV,VWRH[LJHQD VXD RSLQLmR ´XP HTXLOtEULRµ HQWUH GHL[iORV SRU FRQWD SUySULD H GDUOKHV DOJXPD RULHQWDomR0DVDFLPDGHWXGRTXHUGDUOKHVDRSRUWXQLGDGHGHUHVROYHUDWDUHID SHODVVXDVSUySULDVHVWUDWpJLDVDLQGDTXHLVVRUHTXHLUDGHVLHVIRUoRSDUDWHQWDU compreender o pensamento dos alunos:

4XDQGRHXSURFXURLUGHSDUHPSDURXGHJUXSRHPJUXSRSURFXUR SHUFHEHUFRPRpTXHHOHVHVWmRRTXHpTXHHOHVSHUFHEHUDPFRPR pTXHHOHVHVWmRDWUDEDOKDU²HSURFXURID]HULVWRFRPSHUJXQWDV HjVYH]HVQmRpIiFLO«7HQKRXPERFDGRHVWDSUHRFXSDomRGH

WHQWDUVHJXLURVUDFLRFtQLRVGHOHVTXHQmRpIiFLOSRUTXHjVYH]HV há situações em que a gente está muito longe e estamos a ver que pFRUUHWR«PDV´FRPRpTXHHOHSHQVRX"µ

2XWUDSUHRFXSDomRGH&pOLDpQmRYDOLGDUDFRUUHomRGDVHVWUDWpJLDVRXUHVSRVWDV dos alunos, nem quando estão certos, nem quando estão enganados, embora saiba TXHjVYH]HVDVXDH[SUHVVmRIDFLDOUHYHODPDLVGRTXHJRVWDULD4XDQGRLGHQWLÀFD DOJXPJUXSRFXMRWUDEDOKRQmRHYROXLGHIRUPDSURGXWLYDFRORFDOKHSHUJXQWDVFRP DLQWHQomRGHTXHUHÁLWDPVREUHDDERUGDJHPjWDUHIDHRVVHXVHUURVHDGRWHPXPD RXWUDHVWUDWpJLDGHUHVROXomRRXFRUULMDPRVHUURV 6HHXHVWRXDYHUTXHHOHVHVWmRDLUSRUXPFDPLQKRFRPSOHWDPHQWH HUUDGRWHQWRQmRGL]HU´LVWRQmRHVWiEHPµ«7HQWRGL]HU´PDVFRPRp TXHFRPHoDVWH"(QWmRYROWDOiDWUiV«6HUiTXHLVWRVHYHULÀFD"µ3RU exemplo, se eles pensaram mal para os dez cubos, pergunto “essa IRUPD GH SHQVDU YHULÀFDVH GD IRUPD FRPR SHQVDVWH SDUD RV WUrV FXERV"µ«7HQWDUID]HUFRPTXHHOHVFKHJXHPDRFDPLQKRFRUUHWR

&pOLD WDPEpP VH UHIHUH j GLItFLO IXQomR TXH WHP GH GHVHPSHQKDU HQTXDQWR DFRPSDQKDRWUDEDOKRGRVJUXSRVUHODWLYDjRUJDQL]DomRGDGLVFXVVmRGDWDUHID$ SURIHVVRUDHVFROKHFXLGDGRVDPHQWHDVUHVROXo}HVGRVJUXSRVTXHLGHQWLÀFDFRPR relevantes para a discussão matemática coletiva que quer promover com o objetivo GHFXPSULURVHXSURSyVLWRPDWHPiWLFRHVSHFtÀFRGDDXOD

escolher quais daquelas resoluções são importantes para a GLVFXVVmR FROHWLYD ² H LVVR p PXLWR GLItFLO SRUTXH« jV WDQWDV p TXDVHXPGLVWDQFLDPHQWRTXHpFRPSOLFDGRJHULUQDDXOD«FRP WRGDVDTXHODVVROLFLWDo}HV«'HVWDVRLWRUHVROXo}HVGLIHUHQWHV porque não podem ser todas, qual aquela que me interessa discutir QHVWDDXODHPSDUWLFXODU"

&pOLDHVWDEHOHFHWDPEpPDRUGHPGHGLVFXVVmRGDVUHVROXo}HVTXHVHOHFLRQDHP IXQomRGRSRWHQFLDOTXHOKHUHFRQKHFHFRPYLVWDDSURPRYHUXPDPDLRUFODULÀFDomR H FRPSUHHQVmR GD WDUHID H WDPEpP SDUD H[SORUDU UHSUHVHQWDo}HV PDWHPiWLFDV produtivas e esclarecedoras:

7HU HVFROKLGR HVWD SULPHLUR WHP D YHU FRP D WDO TXHVWmR GD visualização… era a mais clara para eles entenderem as construções HDUHJUD(VWDRXWUDHUDPHVPRSRUFDXVDGDLPDJHPHGDIRUPD FRPRHOHVUHSUHVHQWDUDPHOHVIRUDPDTXLPXLWRFODURV« SRUWDQWRHUDPRVRLWRFXERVDUHSUHVHQWDUDVGLIHUHQWHVIDFHV« Esta [outra] era das representações mais claras desta questão da YLVXDOL]DomRFRPRHOHVYLVXDOL]DYDPDVFRQVWUXo}HV²HXDFKHLTXH VHDLQGDKDYLDG~YLGDVRXVHDLQGDKDYLDSUREOHPDVHVWDHUDPDLV uma tentativa de resolver.

Discussão da tarefa.&pOLDH[SOLFLWDPHQWHYDORUL]DHVWDIDVHGDDXODSRLVSDUDVL

constitui uma importante oportunidade de aprendizagem matemática para todos os alunos em sala de aula:

pH[WUHPDPHQWHLPSRUWDQWHSRUTXHQmRpVyHOHVWHUHPHVWDGRD SDUHVDWUDEDOKDUPDVp«'HSRLVRTXHpTXHLVVRpHPWHUPRV GDDXODGH0DWHPiWLFDRTXHpTXHÀFDQRFROHWLYRRTXHpTXHp GLVFXWLGR" $SURIHVVRUDSHGHDRVDXWRUHVGDVUHVROXo}HVVHOHFLRQDGDVTXHH[SRQKDPDVVXDV HVWUDWpJLDVHH[SOLTXHPRVHXUDFLRFtQLRDRVFROHJDVEHPFRPRTXHUHVSRQGDPjV TXHVW}HVXQVGRVRXWURV(YLWDLQWHUIHULUQDGLVFXVVmRSRLVSUHWHQGHTXHRVDOXQRV sejam protagonistas e eles parecem aceitar bem esse papel:

D WHQWDWLYD p TXH R PRPHQWR VHMD GHOHV D DSUHVHQWDomR VHMD deles, que as questões sejam colocadas ao grupo que apresenta (…) E eles estão constantemente a questionar os colegas e estão interessados…

1HVWDIDVHDVSULQFLSDLVSUHRFXSDo}HVGDSURIHVVRUDFHQWUDPVHQDRUTXHVWUDomR das intervenções dos alunos, de modo a promover o esclarecimento das ideias PDWHPiWLFDV HPHUJHQWHV 3DUD HVWD SURIHVVRUD D GLVFXVVmR YDL PXLWR DOpP GD FRUUHomRGDWDUHIDHRFRQIURQWRGHGLIHUHQWHVHVWUDWpJLDVHUHVROXo}HVFRQVWLWXLXP enriquecimento que quer proporcionar aos alunos, mesmo quando eles discutem resoluções com erros:

(XDFKRTXHVHDSUHVHQWDUGLIHUHQWHV>UHVROXo}HV@VHHOHVWLYHUHP FDSDFLGDGHGHDSUHVHQWDUGLIHUHQWHVUHSUHVHQWDo}HVHHXGHSRLV SXGHUOLJiODVHHVWDEHOHFHUDVFRQH[}HVHQWUHHODVpPXLWRPDLV ULFRGRTXHDSUHVHQWDUXPDVy«$LGHLDWDPEpPpGHL[DUVXUJLU DTXLORTXHVDLGHOHV«HjVYH]HVDWpSHJDUHPVLWXDo}HVTXHQmR estão certas. $SHVDUGHQDDXODREVHUYDGDPDQWHUQHVWDIDVHXPSDSHOGLVFUHWRGHRULHQWDomR DSURIHVVRUDFRQGX]LXXPDGLVFXVVmRHVSHFtÀFDSRUHODSURSRVWDFRQIURQWDQGRWUrV GDVGLIHUHQWHVUHVROXo}HVMiDSUHVHQWDGDVSHORVDOXQRVTXHH[S{VHPVLPXOWkQHRQR retroprojetor, recorrendo aos acetatos dos grupos. A sua explicação para esta ação DSDUHQWHPHQWHFRQWUDGLWyULDEDVHLDVHQDVXDYLVmRGRSDSHOGDGLVFXVVmRGDWDUHID +iXPDDOWXUDHPTXHHXFRQIURQWRWUrVUHVROXo}HV$LGHLDpHVVD GHKDYHURFRQIURQWR>$GLVFXVVmR@pXPPRPHQWRGHDSUHQGL]DJHP SRUWDQWR«1mRSRGHVHUXPDDSUHVHQWDomRQHPpXPDFRUUHomR SRUTXHQmRpLVVR«RPRPHQWRpGHS{UHPFRQIURQWRSHQVDUPRVHP FRQMXQWRQDVGLIHUHQWHVUHVROXo}HVQDVGLIHUHQWHVUHSUHVHQWDo}HV LVVRWHPGHVDLUGDOLGDDSUHVHQWDomR«pXPREMHWLYR«eFRORFi ORVDHOHVDWHUHVVHVHQWLGRFUtWLFR´QHVWDVLWXDomRTXDOpDTXHOD UHSUHVHQWDomRDTXHODUHVROXomR«TXDOpDIRUPDGHSHQVDUTXHGH IDFWRPHDMXGDPDLV"3RUTXHSRVVRWHUGX]HQWDVPDVVHFDOKDUKi DOLXPDTXHHXSRVVRHVFROKHU«µ

Sistematização da aprendizagem matemática. $SyV D GLVFXVVmR TXH FRQFOXLX

sobre uma regra que permite determinar quantos autocolantes são utilizados numa FRQVWUXomRFRPXPGHWHUPLQDGRQ~PHURGHFXERV&pOLDDGRWRXXPSDSHOPDLVGLUHWLYR SDUDUHDOL]DUXP´PRPHQWRGHVLVWHPDWL]DomRµ(ODPRVWUDDRVDOXQRVXPDWDEHOD LQFRPSOHWDSURMHWDGDQXPDWUDQVSDUrQFLDTXHSUHYLDPHQWHSUHSDURXTXHUHODFLRQDR Q~PHURGHFXERVHRQ~PHURGHDXWRFRODQWHVHGHVDÀDRVDXPDRXWUDTXHVWmRTXH HVWHQGHDWDUHIDDQWHULRU$SURIHVVRUDTXHUWLUDUSURYHLWRGDGLVFXVVmRGDWDUHIDSDUD UHIRUoDUDLPSRUWkQFLDHRSRGHUGDVUHJUDVFRPOHWUDVDRH[SORUDUVLWXDo}HVHPTXH VHOLGDFRPXPDYDULiYHO5HFRUGDWDPEpPDVRSHUDo}HVLQYHUVDVSURFXUDQGRTXHD UHJUDVHMDOLGD´DRFRQWUiULRµHVWDEHOHFHQGRDVVLPFRQH[}HVFRPFRQKHFLPHQWRVH procedimentos já abordados com os alunos em situações anteriores.

Depois deles trabalharem em pares e de discutir coletivamente o TXHÀ]HUDPKiXPDSDUWHGHVLVWHPDWL]DomRRQGHHXFRQIURQWR« &RPHoRSRUHVWDWDEHOD²HVWHQ~PHURGHFXERV>@DWpIRLXPD FRLVDTXHHOHVWUDEDOKDUDP²>HSHUJXQWR@´&RPRpTXHGHVFREULPRV RQ~PHURGHDXWRFRODQWHV"µ(QWmRÀ]HPRVHHVFUHYHPRVDUHJUD« ´(DJRUDVDEHQGRRQ~PHURGHDXWRFRODQWHVFRPRpTXHVDEHPRV RQ~PHURGHFXERV"µ(GHSRLVKiRFRQIURQWRGDVUHJUDVQRVHQWLGR GHVHYHUDVRSHUDo}HVLQYHUVDVTXHIRUDPWUDEDOKDGDV(LVVRIRL IHLWRQDVLVWHPDWL]DomRQRPRPHQWRGHVLVWHPDWL]DomR

&RQVLGHUDo}HVÀQDLV

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$SURIHVVRUDWDPEpPUHYHORXXPIRUWHHQWHQGLPHQWRGRSDSHOGDVDo}HVUHDOL]DGDV HPFDGDIDVHGDDXODHGHVXDLQÁXrQFLDQRVXFHVVR3RUH[HPSOR&pOLDYDORUL]DTXHRV DOXQRVWUDEDOKHPUHDOPHQWHGHIRUPDDXWyQRPDSRUTXHQDIDVHGHGLVFXVVmRSUHWHQGH WHU XP FRQIURQWR H FRPSDUDomR GH GLIHUHQWHV UHVROXo}HV H QmR DSHQDV XP GHVÀODU GHUHVSRVWDVLJXDLVHFRUUHWDV$JLUGHIRUPDDQmRFRPSURPHWHURVXFHVVRGHXPD DERUGDJHPH[SORUDWyULDGD0DWHPiWLFDpFHUWDPHQWHGLItFLOHH[LJHFRQWUDULDUPRGRV GHID]HUGHFRUUHQWHVGRHQVLQRPDLVWUDGLFLRQDO&DQDYDUUR6WHLQHWDO

Quadro 1 – $o}HVHLQWHQo}HVGH&pOLDQDVXDSUiWLFDGHHQVLQRH[SORUDWyULRGD0DWHPiWLFD

Promoção da aprendizagem matemática Gestão dos alunos e da turma

In tr o d u ç ã o d a t a re

fa Garantir a apropriação da tarefa pelos alunos:

)DPLOLDUL]DUFRPRFRQWH[WRGDWDUHIDPDWHULDO cubos e autocolantes para apresentação) (VFODUHFHUDLQWHUSUHWDomRGDWDUHIDFRPR" (VWDEHOHFHUREMHWLYRVRTXHVHTXHUVDEHU"

Promover a adesão dos alunos à tarefa:

(VWDEHOHFHUFRQH[}HVFRPH[SHULrQFLDDQWHULRU 'HVDÀDUSDUDRWUDEDOKR

Organizar o trabalho dos alunos:

'HÀQLUIRUPDVGHRUJDQL]DomRGRWUDEDOKR (grupos de dois alunos para o trabalho DXWyQRPRHWXUPDWRGDSDUDDGLVFXVVmR coletiva)

2UJDQL]DUPDWHULDLVGDDXODIROKDVFRP HQXQFLDGRGDWDUHIDHFXERVHDXWRFRODQWHV para todos os grupos)

R e a li za ç ã o d a t a re fa

Garantir o desenvolvimento da tarefa pelos alunos: &RORFDUTXHVW}HVHGDUSLVWDV 6XJHULUUHSUHVHQWDo}HV )RFDULGHLDVSURGXWLYDV 3HGLUFODULÀFDo}HVHMXVWLÀFDo}HV 0DQWHURGHVDÀRFRJQLWLYRHDXWRQRPLDGRV alunos: &XLGDUGHSURPRYHURUDFLRFtQLRGRVDOXQRV &XLGDUGHQmRYDOLGDUDFRUUHomRPDWHPiWLFD das respostas dos alunos (nem respostas, nem H[SUHVV}HVIDFLDLV

3URPRYHURWUDEDOKRGHSDUHVJUXSRV

5HJXODUDVLQWHUDo}HVHQWUHDOXQRV 3URYLGHQFLDUPDWHULDLVSDUDRJUXSR

Garantir a produção de materiais para a apresentação pelos alunos:

3HGLUUHJLVWRVHVFULWRV

)RUQHFHUPDWHULDLVDXVDUDFHWDWRVHFDQHWDV

Organizar a discussão a fazer:

,GHQWLÀFDUHVHOHFLRQDUUHVROXo}HVYDULDGDV FODULÀFDGRUDVFRPHUURDH[SORUDUHFRP representações relevantes) 6HTXHQFLDUDVUHVROXo}HVVHOHFLRQDGDV D is c u ss ã o d a t a re fa

Promover a qualidade matemática das apresentações dos alunos:

3HGLUH[SOLFDo}HVFODUDVGDVUHVROXo}HV 3RUTXr"

3HGLUMXVWLÀFDo}HVVREUHRVUHVXOWDGRVHDV IRUPDVGHUHSUHVHQWDomRXWLOL]DGDV

'LVFXWLUDGLIHUHQoDHHÀFiFLDPDWHPiWLFD das resoluções apresentadas (tabelas e regras escritas como expressões com letras)

Regular as interações entre os alunos na discussão:

,QFHQWLYDURTXHVWLRQDPHQWRSDUDFODULÀFDomR de ideias apresentadas ou esclarecimento de G~YLGDV

,QFHQWLYDUDUHVSRVWDjVTXHVW}HVFRORFDGDV

Criar ambiente propício à apresentação e discussão:

'DUSRUWHUPLQDGRRWHPSRGHUHVROXomRGD WDUHIDSHORVDOXQRV

3URYLGHQFLDUDUHRUJDQL]DomRGRVOXJDUHV espaço para a discussão

3URPRYHUDWLWXGHGHUHVSHLWRHLQWHUHVVH JHQXtQRSHORVGLIHUHQWHVWUDEDOKRV apresentados

Gerir relações entre os alunos:

'HÀQLUDRUGHPGDVDSUHVHQWDo}HV 3URPRYHUHJHULUDVSDUWLFLSDo}HVGRVDOXQRV na discussão S is te m a ti za ç ã o d a s a p re n d iz a g e n s m a te m á ti c a

s Institucionalizar ideias ou procedimentos relativos ao desenvolvimento do pensamento algébrico suscitado pela exploração da tarefa:

,GHQWLÀFDUUHSUHVHQWDo}HVSURGXWLYDVSDUD obter generalizações (tabela)

5HFRQKHFHURYDORUGHXPDUHJUDFRPOHWUDV

Estabelecer conexões com aprendizagens anteriores:

(YLGHQFLDUOLJDo}HVFRPFRQFHLWRVPDWHPiWLFRV e procedimentos anteriormente trabalhados (ideia de regra com letras; ideia de operação inversa).

Criar ambiente adequado à sistematização:

)RFDURVDOXQRVQRPRPHQWRGHVLVWHPDWL]DomR coletiva

3URPRYHURUHFRQKHFLPHQWRGDLPSRUWkQFLD de apurar conhecimento matemático a partir da WDUHIDUHDOL]DGD

Garantir o registo escrito das ideias resultantes da sistematização:

5HJLVWRSHODSURIHVVRUDHPDFHWDWRTXH previamente estruturou

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research to practice. Mathematics Teaching in the Middle School, 3,²

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* 7DUHIDDGDSWDGDGH0RVV-%HDW\50F1DE6/ (LVHQEDQG-The potential of

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