这就要求我们针对初中阶段的学生,就其数学观察能力要加以培养

EARCOME3,TSG6Shanghai5–12August2005AssessingtheapplicationofOpen-endedproblemsinSMSEE1AssessingOpen-endedProblemsinMathematicsTestsofShanghaiSeniorMiddleSchoolEntranceExaminationWangDingShanghaiMunicipalEducationalExaminationAuthorityAbstractThefailuresofthe“NewMathematics”movementinthe1960sand“BacktoBasics’Movementinthe1970sinAmericahadledtotheexplorationofalternativesforbettermentinmathematicsinstruction.Analternative,Open-endedproblemsapproachisnowgainingincreasingattentionfromtheeducationalcircleintheworld.Hopingtochangetheusualexercise-immersionapproach,China’sMathematicseducationalcommunityintroducedtheapproachofopen-endedquestionsaimingtocultivatestudents’abilitiestosolveproblemswhilepayingattentiontobasicexercisepractice.AfterreviewingChina’sresearchontheopen-endedquestions,andexaminingsomeoftheseOpen-endedquestionsintheShanghaiMiddleSchoolEntranceExamination(SMSEE)since1998,thispaperpointsoutthedifficultyinestablishingastandardevaluationsystemforsuchquestions.Thedifficultyleadstotheauthor’squestioningoftheireffectivenessinteaching,anditalsoposesasaseriouschallengetothevalidityandeffectivenessinpreparingtestpapers.Keywords:open-endedproblems,assessment,validity,levelsofdifficultyBackgroundThefailuresoftheNewMathematicsMovementinthe1960sand“ReturntotheBasic”Movementinthe1970sintheUnitedStatesofAmericaforcedpeopletoexplorethedeeperstructuresoftheMathematicseducation,thusleadingtotheproposalofandresearchonproblemsolutionapproach.Open-endedProblem,whichisonetypeofproblemsintheproblemsolutionapproach,isgainingincreasingattentionfromtheeducationalcircleintheworld.Inthepasttwentyyears,duetodifferentkindsoftestsforselectionpurposes,suchasentranceexamsforthemiddleschoolandcollege,andMathematicsOlympiccontests,studentsatalllevelswereeducatedinthe“exercise-immersionapproach”.Thestudentsaretrainedtobeonly“tests-handlingmachines”thatcangiveremarkablyquickresponsestoalltypesoftestquestions.InordertosaveMathematicseducationinChinafromthemistakeof“Exercise-immersion”teachingapproach,anincreasingnumberofscholarsinChinaputforwardanideaofusing“questions”tosupplement,alter,andinfluencethetestpaperssoastofurtherthereformofMathematicseducationinChinasothatChina’sMathematicseducationcanbeproblem-solvingorientedandwithonlythenecessaryamountofexercise.Duringthe“MathematicsEducationCurriculumReformSeminar”heldatTeachingMaterialResearchCenterofChinaPrimaryEducationinChinaStateEducationCommitteein1993,scholarsalsoagreed“Problemsolutionapproachisateachingapproachthathasbeeninternationallypopularforyears.Theproblems,whichcanberelatedtoreallifeorproductionapplications,andsomeofwhichareopen-endedproblemswithmorethanonesolution,requirestudentstoanalyzetheproblem,establishamodel,andthensolvetheproblem.”Thisapproachshouldbeadvocatedbecauseitisanimportantwaytocultivatestudents’abilities,improvetheirsenseofapplicationandcooperation.In1971,withtheleadofDaoTian-maoinJapan,Open-endedproblemswerestudied(citedinDai,2000).Andin1980,ZeTian-LifufirstintroducedtoChina’sMathematicseducatorstheresearchdoneonOpen-endedEARCOME3,TSG6Shanghai5–12August2005AssessingtheapplicationofOpen-endedproblemsinSMSEE2problemsinJapan(citedinZe,1999),creatinganinterestinChina’seducationalcircle.From1980,DaiZai-pingwasthefirstscholarwhoappliedopen-endedproblemsintestsinChina.Aftertheteachingexperimentdoneonasmallscalein1993,thenewprojectnamed“Open-endedproblems-aNewModelinMathematicsEducation”wasestablishedasoneofthekeyprojectsintheNinthFiveYearPlanofChinaNationalTeachingScience(Pedagogies).Fromtheperspectiveofthetestingprocess,reports,ortheses(Silver,1995;Pehkonen,1995;Qiao,1997;Dai,1999;Zhen,2001a,2001b,2001c),itmarkedthebeginningofwidespreadadoptionofopen-endedproblemsintesting.However,experimentsandresearchinthisfieldarenotsoestablished.Atthesametime,thevalidityandeffectivenessoftheapproachhavenotbeenyetconducted.Themainreasonisthelackofaneasilyoperationalevaluationsystem,thusraisingthequestionontheeffectivenessandvalidityofOpen-endedproblems.Meanwhile,duetothecomplexityanddifficultyincontrollingthementalprocesses,open-endedproblemsfacedaseriouschallengeintheirdefinition,applications,andevaluations.Atleast,thistypeofproblemsfollowslessrules,regulations,andmethodsinthepreparationoftestsandevaluationusedinconventionaltestsorquestions.AssessingtheCurrentStatusofOpen-EndedProblemsintheShanghaiSeniorMiddleSchoolEntranceExamination(SMSEE)Realizingthatthelearners’psychologicalprocesshasbeenignoredintheMathematicsteaching,suchasneglectingtheconceptualizing,problem-solving,andmethod-selectingandmodifying,manygoodattemptshavebeenmadeinpreparingSMSEEpaperstoguidetheMathematicsteachingsoastofocusonthelearningprocess.AccordingtotherelevantdocumentationofSMSEEpapersinShanghai,Open-endedproblemsmadetheirfirstappearancein1999.However,towhatextenttheseopen-endedproblemshaveassistedthelearnerstoachievethishigherorderthinkingprocess.Letusconsidersomeexamplesoftheopen-endedproblemsformtheSMSEEasgivenbelow:Example1(ShanghaiMunicipal,1999)IfaquadrilateralABCDsatisfiesthecondition___