(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 1

STUDENT-LEDINQUIRY

Howmightstudentsbeencouragedtoaskandfollow-uptheirownquestions?

Introduction

Atitsmostfundamental,inquiry-basedlearningisaboutengagingstudents'curiosityintheworldandtheideasthatsurroundthem.Asscientistsandmathematicians,theyobserveandposequestionsaboutsituations;iftheirquestionsaretoocomplex,theymaytrytosimplifyormodelthesituation;theymaythentrytoanswertheirquestionsbycollectingandanalyzingdata,makingrepresentations,andbymakingconnectionswithwhattheyalreadyknow.Theytrytointerprettheirfindings,checkthattheyareaccurateandsensibleandthensharetheirfindingswithothers.

Thisprocessisoftenmissingintheschoolclassroom.There,theteacherusuallypointsoutwhatmustbeobserved,sheprovidesthequestions,demonstratesthemethodstobeusedandcheckstheresults.Studentsaremerelyaskedtofollowtheinstructions.

Inthismodule,teacherswillbeencouragedtoexperiencewhatitfeelsliketothinklikeamathematicianorscientist,andreflectontheroleshiftsthatarenecessaryforstudentstosharethisexperienceintheclassroom.Teachersareshownphenomenaandsituationsandareinvitedtoposeandpursuetheirownquestions.Thisexperienceisthentransferredtotheclassroom.

ActivitiesActivityA: Askquestionsaboutphenomena.....................................................................................1ActivityB: Makingobservationsfromphotographs..........................................................................4ActivityC: Observeandanalysealesson...........................................................................................6ActivityD: Planalesson,teachitandreflectontheoutcomes.........................................................7Suggestedfurtherreading....................................................................................................................10Acknowledgement:ThismaterialisadaptedforPRIMASfrom:Swan,M;Pead,D(2008).Professionaldevelopmentresources.BowlandMathsKeyStage3,BowlandTrust/DepartmentforChildren,SchoolsandFamilies.AvailableonlineintheUKat:http://www.bowlandmaths.org.ukItisusedherebypermissionoftheBowlandTrust.

(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 2

ACTIVITYA: ASKQUESTIONSABOUTPHENOMENA

Timeneeded:30minutes.

Inthisactivity,youmayliketoofferteachersachoiceoftwopossiblestartingpoints:

• Oneinvolvesrollingapapercup• Theotherinvolvesapieceofcomputersoftware:Spirolaterals

Forthefirstactivity,youwillneedtoprovideeachgroupofteacherswithatleastthreedifferentpapercups.Trytoincludea'shortandfat'oneanda'longandthinone'and'oneinbetween'Forthesecond,wehaveprovidedacomputermicroworld.Teacherswillneedtoworkinpairsusingalaptop.Forthesituationyouchoosetoexplore:

• Makealistofthingsyounoticeaboutthesituation.• Whatquestionsoccurtoyou?• Youmightbeginbyaskingquestionsthatstart:

o Whatwouldhappenif....?o WhatcanIvary...?o Whateffectwilleachvariablehaveon...?

• Nowsetyourselfaproblemandattempttotackleit.Whenyouhaveexperimentedwiththesituationtrytoanalyzeyourfindings.

• Whatdatahaveyoucollected?• Howhaveyouorganizedyourdata?• Howcanyouexplainyourfindings?

Afterteachershaveexploredbothsituationsaskthemtoreflectontheprocesstheyhavebeenthrough.Handout2willhelpthemdothis.Didthey:Formulateproblems?

• listvariables?• simplifyingandrepresent?

Analyseandsolve?• visualise;drawdiagrams?• systematicallychangevariables?• lookforpatternsandrelationships?• makecalculationsandkeeprecords?• makeconjecturesandgeneralisations?• uselogical,deductivereasoning?

Interpretandevaluate?• formconclusions,argumentsandgeneralisations• considerappropriatenessandaccuracy• relatebacktotheoriginalsituation

Communicateandreflect?• communicateanddiscussfindingseffectively• consideralternativesolutions• considerelegance,efficiencyandequivalence• Makeconnectionstootherproblems?

(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 3

Handout1: Phenomenatoexplore

Handout2: Themodelingcycle

(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 4

ACTIVITYB: MAKINGOBSERVATIONSFROMPHOTOGRAPHS

Timeneeded:20minutes.

Itisnotalwayseasyforpupilstoseeanyconnectionbetweentherealworldandtheirlessonsinschool.Asaresult,theydon’tusewhattheylearnedinsecondaryschool,eventhoughthinkingscientificallycouldhelpthemunderstandtheworldbetter–andmakebetterdecisions.LookattheselectionofphotographsonHandout1.

• Makealistofthingsyounoticeaboutthesituation.• Whatquestionsoccurtoyou?• Youmightbeginbyaskingquestionsthatstart:

o Howmany...?o Whatwouldhappenif....?

• Nowsetyourselfaproblemattempttotackleit.

Afterteachershaveexploredbothsituationsaskthemtosharesomeofthequestionstheyhavecreated.Forexample,thefollowingselectioncamefromonegroup:Dominoes:

• Whichdominoismissing?• Howcanyouorganizethedominoessystematically?• Canyoumakeachainoraringwiththecompleteset?• Howmanyspotsarethereinacompleteset?Whatisaquickwayofcountingthem?• Howmanydominoesarethereinacompletesetfrom(1,1)to(n,n)?

Calendar: • Howarethenumbersarrangedonthecubes?• Canyoudrawnetsandmakethecubes?• Whatimpossibledatescanbemadefromthesecubes?

Stackofbarrels• Howmanybarrelsareinthestack?• Ifyoumakeatallerstack4,5,...barrelshigh,howmanybarrelswillyouneed?Generalize?• Howelsecouldyoustackthesebarrels?Whatotherpyramidsarepossible?

ApavementinGermany• Areallthepavingslabsidentical?Whatshapearethey?Canyouworkoutanyangles?• Canyoudrawoneoftheslabsaccurately?• Canyoufindotherpentagonsthattessellate?• Whatothershapescanpavingslabsbe?

Trikewithsquarewheels• Doesthetrikerunsmoothly?Canyoumakeasimplemodel?• Whatistheheightofeach'bump'onthetrack?• Canyoudrawtheshapeofthe'bumpyroad'accurately?• Whatwouldhappenifyouhadtriangularwheelsorhexagonalwheels?

Russiandolls• Dothetopsoftheheadslieonastraightline?Whatdoesthistellyou?• Ifyouweretomakesomebiggerdollsinthisset-howbigwouldtheyhavetobe?

Askteacherstobringtheirownphotographstoafollow-upsessionanddevelopquestionsaboutthem.Generatingquestionsisanactivitythatisessentialforinquiry-basedlearning.Wewillseeinalatersessionhowstudentsmaybeencouragedtodeveloptheirownquestioning.

(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 5

Handout3. Photographstoexplore

Dominoes

Calendar

Barrels

PavementinGermany

Trikewithsquarewheels

RussianDolls

(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 6

ACTIVITYC: OBSERVEANDANALYSEALESSON

Timeneeded:30minutes.

Wehaveprovidedtwovideosoflessonsforteacherstowatch.Choosejustoneofthese.OnevideousestheSpirolateralsproblemfromActivityAOnevideousestheBuildingaSchoolphotographsshownopposite.Eachvideolastsabout10minutes.Asyouwatcheachlesson,askyourself:

• Whichprocessescanyouseeintheworkofthesepupils?• Canyouseethem:

o Simplifyingandrepresentingthesituation?§ Whatquestionsdidtheyformulate?§ Whatsimplificationsandrepresentationsdidtheycreate?§ Whatchoicesdidtheymakeofinformation,methodsandtools?

o Analysingandsolvingthemodelthey’vemade?§ Whichvariablesdidtheyconsider?§ Whatinformationdidtheycollect,orguess?§ Whatrelationshipsdidtheyformulate?§ Whatcalculationsdidtheymake?

o Interpretingandevaluatingtheresults?§ Whatdidtheylearnaboutthesituation?§ Weretheirresultsplausible?

o Communicatingandreflectingonthefindings?§ Howdidtheyexplaintheiranalyses?§ Whatconnectionsdidtheyseetootherproblems?

(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 7

Handout4: BuildingaschoolwithbottlesinHonduras

Handout5.Themodelingcycleappliedtothe“buildingaschool”task.

ACTIVITYD:PLANALESSON,TEACHITANDREFLECTONTHEOUTCOMES

(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 8

Timeneeded:

• 15minutesdiscussionbeforethelesson• 1hourforthelesson• 15minutesafterthelesson

NowitisyourturntoplanalessonusingtheBuildingaSchoolortheSpirolateralssituationfromActivityC.Discusshowyouwill:

• introducethesituationtopupils;• introducetheideaofthemodelingcycle;• organisetheclassroomandtheresourcesneeded;• answerthequestion"Whyarewedoingthisinmaths?";• concludethelessoninawaythatgivespupilsabetterunderstandingofthenatureof

scientificprocessesinvolved.Afteryouhavedesignedyourlesson,compareyourplanwiththelessonplansuppliedonHandout6.Discussthedifferences.

Itishelpfultopresentthelessonusingadataprojector.Inaddition,itishelpfultohaveasupplyofthefollowingresourcesavailableforworkingontheproblemsthatarise:

• Somesample1litreplasticbottles• Rulersortapemeasures,• Circularcountersorcoins(forworkingouthowbottlespacktogether),• Isometricdottedpaper(tohelpwithdrawingandcounting).• SomecopiesofHandout3forpupilstouseanddiscuss.

Afteryouhavetaughtthelesson,takesometimetoreflectonwhathappenedandtheprocessesthatwereinevidence.

• Whatquestionswereidentified?• Didpupilsusearangeofrepresentations?• Whatrelationshipsdidtheyfindinthesituation?• Whatcalculationsdidtheydo?Couldtheyinterpretthemeaningofthese?• Weretheyabletocommunicatetheirconclusionseffectively?• Didyourpupilsfeelthatthiswasdifferentfromanormallesson?• Aretheynowbeginningtoappreciatehowthetechniquestheyhavestudiedinschool

maybelinkedtounfamiliarsituations?

(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 9

Handout6 Asamplelessonplan

(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 10

SUGGESTEDFURTHERREADING

Learningmathematicsthroughcontextualisedsituations.BoalerJ.(1993)‘TheRoleofContextsintheMathematicsClassroom’,FortheLearningofMathematics13(2)Lookingattheapprenticeshipmodeloflearning.Brown,J.S.,Collins,A.andDuguid,P.(1989)‘SituatedcognitionandtheCultureofLearning’,EducationalResearcher,18(1),pp32-42.LookingatadifferentwaytoorganisetheYear9curriculumCarter,C.(2008)‘Adifferentway’,MathematicsTeaching,207,pp38-40http://www.atm.org.uk/mt/archive/mt207files/ATM-MT207-38-40-mo.pdfWhatdopupilsseeasmathematical?Doesithavetohavenumbers?Mendick,H.,Moreau,M.andEpsteinD.(2007)‘Lookingformathematics’inD.Kuchemann(Ed.)ProceedingsoftheBritishSocietyforResearchintoLearningMathematics27(1)pp60–65http://www.bsrlm.org.uk/IPs/ip27-1/BSRLM-IP-27-1-11.pdfAcomparisonofthemathematicspeopleuseinschoolandoutofschool.Nunes,T.,Schliemann,A.D.,Carraher,D.W.(1993),Streetmathematicsandschoolmathematics,CambridgeUniversityPressWhatisimportantinmathematicseducation?PolyaG(2002)‘Thegoalsofmathematicaleducation:part1andpart2’MathematicsTeaching,181,pp6-7and42-44http://www.atm.org.uk/mt/archive/mt181files/ATM-MT181-06-07.pdfhttp://www.atm.org.uk/mt/archive/mt181files/ATM-MT181-42-44-mo.pdf