AN EXAMPLE OF THE FRAGILITY OF A PROCEDURALAPPROACH TO SOLVING EQUATIONS程序脆弱性的一个实例 求解方程的方法

《AN EXAMPLE OF THE FRAGILITY OF A PROCEDURALAPPROACH TO SOLVING EQUATIONS程序脆弱性的一个实例 求解方程的方法》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、ANEXAMPLEOFTHEFRAGILITYOFAPROCEDURALAPPROACHTOSOLVINGEQUATIONSDraft:March2010RosanaNogueiradeLima(UNIBAN–Brazil)DavidTall(UniversityofWarwick–UK)Inthispaperweconsidertheperformanceofstudentsonquadraticequationsaftertheyhaddevelopedproceduralmethodsofsolvinglinearequationsbyshiftingsymbolsusingr

2、ulessuchas‘changesides,changesigns’(Lima&Tall,2008).Knowingtheirdifficultieswithalgebraicmanipulation,theteachershadfocusedfirstonsimplequadraticswithonlytwonon-zerotermsandthenontheuseoftheformulawhichtheyconsideredasauniversalmethodtosolveallquadraticequations.Theconsequencewasthatlonger-term

3、onlyafewstudentscouldsolveaquadraticequationusingtheformula,andafurtherproceduralembodimentwasadoptedbya2minorityofstudentstosolveanequationsuchasx=9by‘passingthepowerover2theothersideandchangingittoaroot’togetthesinglesolutionx=9=3.Otherthanthisallothermethodsinvolvedproceduralsymbolmanipulati

4、on,oftenleadingtoerror.Whileitisthedutyofmathematicseducatorstoimprovestudentlearning,itisalsoaresponsibilitytounderstandwhysomanystudentsendupperforming‘ruleswithoutreason’thatleadtofailure.Wediscussthisphenomenonintermsofembodiedcognitioninwhichtheproceduralembodimentofsymbol-shiftingisprefer

5、redtoanyembodiedmodel(suchasabalancetorepresentanequation)andthemanipulationofsymbolsfailstohavetherichermeaningproposedbytheoriesofprocess-objectencapsulation.Thislong-termfailureisisplacedwithinawidertheoreticalframeworkincorporatinghumanembodimentandsymbolicmanipulation.INTRODUCTIONInthispap

6、erweconsiderdatacollectedfromstudentswhohadlearnedproceduralwaysofsolvinglinearequations(LimaandTall,2008)andwerenowdealingwithquadraticequations.Theirresponsestothesolutionoflinearequationswasdescribedasa‘proceduralembodiment’inwhichthesymbolswereshiftedaroundaccompaniedbyrulessuchas‘changesid

7、es,changesigns.’Theirteacherswereconcernedwiththestudents’difficultiesandfocusedonteachingthestudentstoperformproceduresthatfocusedonbeingabletosolvequadraticequationsandtoperformwellonthetests.Thesyllabusspecifiedthatthestudentss