数学学习中问题解决、计算机技术以及学生动机研究(IJEME-V1-N1-1)

I.J.EducationandManagementEngineering2011,1,1-5PublishedOnlineJuly2011inMECS()DOI:10.5815/ijeme.2011.01.01Availableonlineat’MotivationinLearningMathematicsYixunShiDepartmentofMathematics,ComputerScienceandStatisticsBloomsburgUniversityofPennsylvaniaBloomsburg,PA17815,USAAbstractInrecentyears,moreandmoreattentionsaregiventodevelopingproblemsolvingskillsandusingcomputertechnologyintheteachingandlearningofmathematics.Casestudies,independentprojects,andexamplesofapplicationsofmathematicsareusedmoreandmorefrequentlyinmathematicsclassesinordertoenhancestudents’developmentinmathematicalthinkingandproblemsolvingskills.Twoexamplesofsuchstudiesarepresentedinthispaper.IndexTerms:Problemsolving;Computerapplications;Numbergames;Mathematicseducation;Studentprojects©2011PublishedbyMECSPublisher.Selectionand/orpeerreviewunderresponsibilityoftheInternationalConferenceonE-BusinessSystemandEducationTechnology1.IntroductionInrecentyears,mathematicseducatorsaregettingmoreandmoreawareoftheimportanceofdevelopingproblemsolvingskillsandusingcomputertechnologyintheteachingandlearningofmathematics.Casestudies,independentprojects,andexamplesofapplicationsofmathematicsareusedmoreandmorefrequentlyinmathematicsclassesinordertoenhancestudents’developmentinmathematicalthinkingandproblemsolvingskills.Inthispaper,Iwillpresenttwoapplicationexamplesthatcanbeusedinteachingvariousmathematicssubjectsatcollege,secondary,orelementarylevels.Theseexamplesmaynotonlybeusedtodemonstrateorexplainmathematicsconcepts,butwillalsodisplayvariousstrategiesforsolvingthesameproblemandexploretheconnectionsbetweendifferentmathematicalprocedures.Thustheymayhelpstudentsgainadeeperinsightofmathematicalpatterns,andenhancetheirskillsofproblemsolving.Theseexamplesalsorequiretheuseofcomputertechnologyinproblemsolving.Theexamplesgiveninthispaperaredevelopedfromsimplebutinterestingnumbergames.Sincemoststudentstendtoenjoygames,examplesofthiskindwillingeneralraisestudents’interestsandmotivationinlearningmathematics.Inthispaper,wewillalsoaddressthepracticalwaysofusingtheseapplicationexampleswithinamathematicsclassroom.TheideasCorrespondingauthor:E-mailaddress:[e-mail]2ProblemSolving,ComputerTechnology,andStudents’MotivationinLearningMathematicsandstrategiesofusinggamesinteachingmathematicsaremostbeneficialtomathematicseducatorsandstudents.ThefollowingsectionIIpresentstheapplicationofanumbergame“24-points”.TheninsectionIIItheapplicationofthe“containers”gameisdiscussed.Somefurthercommentsaregiveninthelastsectionofthepaper.2.ApplicationOfNumberGame24-Points24-pointsisasimplenumbergamethatyoungkidsplaytobecomecapableoffiguringnumbersoutrightandfast.Schoolteachershavealsofounditpopularbecauseofitseffectivenessinimprovingchildren'sabilityindoingmentalarithmetic.Thereexistavarietyofdifferentwaystoplaythegame,butessentiallytheyfollowthesefiverules:1)Fournumbersarerandomlychosenfromtheset{1,2,3,4,5,6,7,8,9,10}withreplacement;2)Onlytheoperationsaddition,subtraction,multiplication,anddivisionmaybeusedonthefournumbersandintermediateresults;3)Allintermediateresultsmustbenon-negativeintegers;4)Eachofthefourchosennumbersmustbeusedonceandonlyonce;5)Thegoalistogetexactly24.Forexample,ifthefourchosennumbersare2,2,5,7,thentheseoperationsgive24:2×5=10,2×7=14,10+14=24Orequivalently,2×5+2×7=14.Notethateachofthefournumbers2,2,5,and7isusedonceandonlyonce.Each2inthegroupisconsideredasanindividualnumber.Alsonoticethattheintermediateresults10and14arebothnon-negativeintegers,asrequiredbyrule3.Anotherexamplealsoillustratestherules.Ifthefourchosennumbersare1,2,3,5,then24canbereachedby5+1=6,2+6=8,8×3=24.Here,24isgivenby(5+1+2)×3.Again,eachofthefournumbersisusedexactlyonce,andtheintermediateresults6and8arenon-negativeintegers.Thearithmeticcombinationsusedinthetwoexamplesrepresenttwodifferenttypesofprocedures.Inthefirstexample,thefournumbersarepaired.Arithmeticoperationsareappliedtoeachpairandthenontheresultingnumbers.Thatrepresentsthebasicconceptofparallelcomputation.Inthesecondexample,thearithmeticoperationsareappliedtotwoofthenumbers,thenthethirdnumber,andfinallythelastnumber.Thatrepresentsthebasicconceptofsequentialcomputation.Intheclassroom,theteachershouldmentiontostudentsthatthesetwoaretheonlytypesofproceduresinvolvedinthisgame.Thisisanimportantfacttobeusedbystudentswhentheywritecomputerprogramsorusecomputersoftwaretosolvethegame.Thisgameismostconvenientlyplayedwithadeckof40playingcards.Theface-cardsandthetwojokersarenotused.Onlynumber-cards(Aceusedas1)areused.Fourcardsarepickedatrandomfromthedeck.Becausethedeckcontainedfourofeachnumberin{1,2,3,4,5,6,7,8,9,10},asinglenumbercanappearuptofourtimes.Studentsshouldbeawarethatgivenfournumberstheremaybeafewalternativewaystoget24.Theyshouldalsoberemindedthatsomecombinationoffournumberswillneveryield24.Forexample,ifthefournumbersare2,2,2,2,thenthenumber24cannotbeobtainedundertherulesofthegame.Followingquestionsnowcanbeaskedandstudiedintheclassroom:1)Howm