teaching and learning mathematics with technology.discovery.ucl.ac.uk/1474515/2/clark-wilson... ·...
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Citeas:Clark-Wilson,A.,&Mostert,I.(2016).Teachingandlearningmathematicswithtechnology.InC.Hopkins,J.Anghileri,&J.Gage(Eds.),AIMSSECMathsTeacherSupportSeries:MathematicalThinkingintheLowerSecondaryClassroom.Cambridge:CambridgeUniversityPress.
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Teachingandlearningmathematicswithtechnology.
AlisonClark-Wilson
UCLInstituteofEducation,UniversityCollegeLondon,UK
IngridMostert
AfricanInstituteofMathematicalSciences,Muizenberg,SouthAfrica
Introduction
Thearrivalofdigitaltechnologiessuchascomputers,mobilephones,tablets(likeiPads
andAndroids)isrevolutionisingtheopportunitiesforbothteachersandlearnersalike
withineducationingeneralandwithinmathematicseducationinparticular.Alloverthe
worldteachersarelookingtodeveloptheirpersonalskillsandexpertisewith
technology.‘Technology’includesmanythings–fromcomputerstomobiledevices,
fromsoftwareandapplicationstoweb-sites.Thereisalotoftechnologyoutthereandit
keepschanging.
Inthischapterwearehopingtopointyouintherightdirectionsothatyoucanbeginto
explorearangeoftechnologiesthatyoucanuseinyourmathematicsteachingcareer
andbeyond.Technologycansupportbothteachersandlearners.
Forteachersofmathematics,technologyenablesustocreate(anddisplayorprint)our
ownteachingresourcesthatincludeaccuratemathematicaltextanddiagrams-and
sharethesewithotherteachers.Wearenolongerwhollyreliantonbuyingpublished
textbooksandworksheets-wecancreateourownandadaptthemyear-by-yearto
buildourownbankofteachingresources.Technologycanalsohelpustokeepadigital
recordoflearners’testscoresandhelpustoanalyselearners’progressovertime–we
canusetheinformationtoinformwhattopicswemightneedtorevisitorwhentobegin
toteachthenexttopic.
Forlearnersofmathematics,technologyoffersanopportunityforthemtobeintroduced
mathematicalideasandconceptsincompletelynewways.Mathematicalsoftwareand
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Citeas:Clark-Wilson,A.,&Mostert,I.(2016).Teachingandlearningmathematicswithtechnology.InC.Hopkins,J.Anghileri,&J.Gage(Eds.),AIMSSECMathsTeacherSupportSeries:MathematicalThinkingintheLowerSecondaryClassroom.Cambridge:CambridgeUniversityPress.
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applicationsareavailablethatenablelearnerstoplaywithmathematicalobjectsina
waythatissimplynotpossibleusingtraditionalpaperandpencilmethods.
Inthischapteryouwilllearnaboutawiderangeoftechnologiesthatareavailableto
supportteachingandlearningmathematics.
Fig.1MakiNekhavhambeusingadataprojectorforthefirsttimeinherclassroom.
IMAGE1.jpg
Ofcourse,itmaynotbepossibleforyoutotrysomeoftheseideasrightaway.However
asyourpersonalaccesstotechnologyincreases,youcanreturntothischapterwhen
youareabletotrysomeoftheapproachesinyourownclassroom.
Differenttypesoftechnology
Technologyincludesaverywiderangeofdevices,software(orapplicationsor‘apps’),
websitesetc.thathaveallbeendesignedformanydifferentpurposesandaudiences.To
helpyouwithsomeoftheterminology,wehaveincludedaGlossaryattheendofthe
chapter-soifyoucomeacrosssomethingthatisnewtoyou,lookitup!
Also,becausetherearemanydifferenttechnologyproductsthatareavailable,we
cannotprovidethespecificinstructionsforalloftheideasweoffer–youwillneedto
usetheindividual‘Help’resourcesandproductmanualsthataccompanythedifferent
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Citeas:Clark-Wilson,A.,&Mostert,I.(2016).Teachingandlearningmathematicswithtechnology.InC.Hopkins,J.Anghileri,&J.Gage(Eds.),AIMSSECMathsTeacherSupportSeries:MathematicalThinkingintheLowerSecondaryClassroom.Cambridge:CambridgeUniversityPress.
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technologyproducts-butwehopethatyouwillseethevalueinmakingthistime
investmentforyourself.
Inaddition,therearemanyonlinecommunitiesandaquickInternetsearchwilloften
takeyoutothehelpyouarelookingfor.ThemillionsofInternetusersaroundtheworld
meansthatusuallysomeoneelsehasalreadyasked(andpublishedananswer)tothe
exactquestionyouareaskingyourself!
Wehaveorganizedthechapterbydividingtechnologyintothetechnologiesthatmight
beusefulinsidetheclassroomwithlearnersandtechnologiesthatcanbeusedbyyou
and/oryourlearnersawayfromtheclassroom.
Technologyintheclassroomlooksattechnologyfor
1) exploringmathematicsina)number,b)algebra,c)geometryandd)statistics
andprobability;
2) projectingmathematicssuchasa)dataprojectors,b)visualisersandc)
webcams.
Technologyawayfromtheclassroomlooksatcreating
1) worksheetsbyinsertinga)symbolsandequations,b)tables,c)graphsandd)
geometricimages.
2) resourcestoprojectusinga)presentationsoftwareandb)interactive
whiteboardsoftware
3) electronicmarkbooksbya)enteringdataandthenb)makingsenseof
assessmentdata
andatonline
4) communitiesforteachersandlearnershostedbya)Ministryofeducationsites
b)AimingHighandc)theAfricanMathematicsInitiative
5) resourcessuchasa)MicrosoftMaths,b)EverythingMathsandScience,c)Khan
Academy,d)MathsExcellenceande)Math-Aid.
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Citeas:Clark-Wilson,A.,&Mostert,I.(2016).Teachingandlearningmathematicswithtechnology.InC.Hopkins,J.Anghileri,&J.Gage(Eds.),AIMSSECMathsTeacherSupportSeries:MathematicalThinkingintheLowerSecondaryClassroom.Cambridge:CambridgeUniversityPress.
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Technologyintheclassroom
1) Technologyforexploringmathematics.
Thisistechnologythathasbeencreatedtochangethewaythatweengagein
mathematicsaltogether.Suchtechnologiesletusgettothe'nutsandbolts'of
mathematicsandhavethepotentialtoengagemanymorelearnersinthemore
challengingmathematicalconcepts.
a) Number
SpreadsheetsoftwaresuchasaGeoGebraspreadsheetorMicrosoftExcel(which
wasoriginallydesignedforbusinessandaccountancy)canbeusedvery
creativelytoexplorenumberproperties.Thebasicideaisthatthecolumnsand
rowsarelabeledwithlettersandnumbersandyouusethe‘Input’bartoenter
textornumbersintothespreadsheet‘cells’.Youcanthenenterdatamore
efficientlybysettingupsomemathematicalrulesorrelationships!Youarenot
expectedtoentereachnumberonebyone!
Forexample,inFig.2below,whichshowsa‘divisiontable’fornumeratorsand
denominatorsfrom1to10,onlythevaluesincellsA2andB1wereentered
manually.Alloftheothernumbersweremadebyenteringaformula,whichwas
thencopied–eitheracrossarowordownacolumnofdata.InFig.2,youcansee
thatthevalueof3÷4(incellD5)wascalculatedusingtheformula=$D$1/A5.
Inspreadsheetsoftwarethe‘$”notationiscalledabsolutereferencing–thisis
importantifyouwanttofixthevalueofanumberinaformula.
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Citeas:Clark-Wilson,A.,&Mostert,I.(2016).Teachingandlearningmathematicswithtechnology.InC.Hopkins,J.Anghileri,&J.Gage(Eds.),AIMSSECMathsTeacherSupportSeries:MathematicalThinkingintheLowerSecondaryClassroom.Cambridge:CambridgeUniversityPress.
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Fig.2AdivisiontablecreatedusingaGeoGebraspreadsheet
IMAGE2_Spreadsheet_Division.jpg
Thepowerofaspreadsheetisthat,bychangingjustonenumberinthe
spreadsheet(thenumberincellA2froma1toa2),awholenewsetofdecimal
numberscanbecreated.Thiscreatesaverymotivatingspaceforpupilsto
explorefractiontodecimalconversionsandanswerquestionssuchas:
• Howmanytimesdoesthedecimalnumber0.5appear?Why?
• Canyoupredictfivedifferentfractionsthatwouldhaveadecimal
equivalenceof0.33?
b) Algebra
Manyofthe‘bigideas’andkeyconceptsinmathematicsarerelatedtoimportant
generalisations,suchasthatallfunctionsofthetypef(x)=a*x+b,whenplottedon
aCartesian(x-y)graphplane,producegraphsthathaveaparticulargeometric
feature(astraightline).Mostofuswillhavelearnedthisbyroteandbelievedit
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Citeas:Clark-Wilson,A.,&Mostert,I.(2016).Teachingandlearningmathematicswithtechnology.InC.Hopkins,J.Anghileri,&J.Gage(Eds.),AIMSSECMathsTeacherSupportSeries:MathematicalThinkingintheLowerSecondaryClassroom.Cambridge:CambridgeUniversityPress.
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becauseourteachers(andthetextbook)toldusthatitwastrue.However,with
technology,itispossibleforthelearnerstoobservethegraphfeaturesfor
themselvesand,bychangingthevaluesofaandb,discovermuchmorebesides.
WithinasoftwarepackagessuchasGeoGebra,manylineargraphscanbe
exploredveryquickly,withouttheratherslowerprocessofplottingthe
individualfunctionsbyhandusingpaperandpencil.Thisdoesnotmeanthatthe
learners’graphplottingskillsarenolongerimportant–butitdoesmeanthatthe
moreimportantknowledgeabouttheeffectthatvaryingthevalesofaandbhas
ontheappearanceofthelineargraphscanbeaccessedbyalllearners.InFig.3
showsanexploratorytaskinGeoGebrathatuses‘sliders’tochangethevaluesof
aandb.
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Citeas:Clark-Wilson,A.,&Mostert,I.(2016).Teachingandlearningmathematicswithtechnology.InC.Hopkins,J.Anghileri,&J.Gage(Eds.),AIMSSECMathsTeacherSupportSeries:MathematicalThinkingintheLowerSecondaryClassroom.Cambridge:CambridgeUniversityPress.
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Fig.3AGeoGebrafilethatusessliderstoexploretheeffectofchangingthe
valueson‘a’and‘b’inlinearfunctionsgivenbyf(x)=a*x+b
IMAGE3_Geogebra_Linear_functions.jpg
Itisusuallynotenoughtoonlypresentthisfiletothelearnersand‘show’them
whathappens.Itisbettertoarousethelearners’curiositybyaskingquestions
suchas:
• Whatdoyounoticeaboutallgraphsforwhichthevalueofb=0?(or3?Or
5?Or…)
• Whatdoyounoticeaboutallgraphsforwhichthevalueofa=3?(or1.5?
or-2?Or…)
• Canyoupredictwhatthegraphofy=-1x+2willlooklike?Howdoyou
know?
c) Geometry
Dynamicgeometrysoftwarehasrevolutionisedthewayinwhichwecan
understandandexploreideasfromgeometry.Thisisbecauseanydiagramcan
be‘interactive’aswecanchangeitsappearancebydragging-andexplorewhat
happens.Wecancreatemathematically‘robust’2-Dshapesthatkeeptheir
mathematicalproperties–evenwhenitsverticesandsidearetransformedby
dragging.Figs.4and5showasetoftriangles,eachofwhichhasbeencreatedin
thesoftwaresuchthatitretainsaparticularsetofproperties.Canyouspot
whichtriangleiswhich(andwhy?).
Fig.4ThreetrianglescreatedinGeogebra
IMAGE4_Three_triangles1.jpg
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Fig.5Thethreetriangleshavebeentransformedbydraggingonlyoneofeach
triangle’svertices!
IMAGE5_Three_triangles2.jpg
Youreallydoneedtoplaywithconstructingthesetrianglesinthesoftwareto
appreciatehowdynamicgeometryenvironmentsaredifferent–theequilateral
andisoscelestriangleswerecreatingusingadigitalsetofgeometricconstruction
tools.The‘Circlebycenterandradius’replacestheprotractorandmid-points
andperpendicularbisectorsarebuilt-indigitaltools.Aswiththegraphing
example,dynamicgeometrysoftwaredoesnotreplacepaperandpencil
constructiontechniques.Howeveritdoesprovideastimulatingenvironmentin
whichtoexploregeometricconstructionsandtheorems.
ManycountrieshaveGeoGebraInstitutesthatofferface-to-face
meetings/coursesandonlinesupportforteachers.(see
www.geogebra.org/institutes)
d) Statisticsandprobability
AnydatathathasbeenimportedintoamathematicaltoolsuchasExcelor
Geogebracanbeanalysed.Agoodsourceofinternationaldataonlearners’
lifestylesandinterestisprovidedbytheCensusatSchoolwebsite
(http://www.censusatschool.org.uk/).Theinternationalsectionofthewebsite
showsthemanycountriesthathavetakenpart–anditispossibletorequesta
randomiseddatasetwithwhichtocontrastyourownlearners’data.
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Citeas:Clark-Wilson,A.,&Mostert,I.(2016).Teachingandlearningmathematicswithtechnology.InC.Hopkins,J.Anghileri,&J.Gage(Eds.),AIMSSECMathsTeacherSupportSeries:MathematicalThinkingintheLowerSecondaryClassroom.Cambridge:CambridgeUniversityPress.
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Fig.6TheinternationalpageoftheCensusatSchoolwebsite.
IMAGE6_Census_at_School.jpg
2) Technologyforprojectingmathematics
Thistechnologyisdesignedtoallowyoutoprojectimagesfromyourtabletor
computertotheclass.Itisnotlimitedtoprojectingmathematics.Youcanuseitto
projectanycontent.
a) Dataprojectors
Adataprojectorconnectstoyourcomputerortabletandprojectsitsdisplayontoa
wallorscreen.Ifyouhadaninteractivewhiteboard,itwouldbepossibletooperate
thesoftwareonyourcomputerortabletbytouchingthisprojectedimageand
annotateontopusing‘electronicwriting’.
b) Visualisers
Avisualiserisahighresolutioncameraonastandthatconnectstoadataprojector
and,whenyouplaceobjects,learners’workoratabletscreenunderneathit,this
imageisprojectedontothewalldisplay.Thisisparticularlyusefulwhenyouare
doingpracticaldemonstrationsasthelearnerscanseeyourhands!
c) Webcams
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Citeas:Clark-Wilson,A.,&Mostert,I.(2016).Teachingandlearningmathematicswithtechnology.InC.Hopkins,J.Anghileri,&J.Gage(Eds.),AIMSSECMathsTeacherSupportSeries:MathematicalThinkingintheLowerSecondaryClassroom.Cambridge:CambridgeUniversityPress.
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Somecomputersandmosttabletshaveabuilt-inwebcam,whichisatinypin-sized
camera,usuallyfoundonthesideofthescreen.Whenthecameraisswitchedonand
youareconnectedtotheInternet(i.e.usingSkype,FaceTimeorsimilar),thisimage
(usuallyofyourface!)canbeseenbyothers.However,ifyoupointthecamera
towardssomelearners’work,youcanprojectthisimagetoadataprojectorforallin
theclasstosee.Itisalsopossibletobuyalow-costwebcamwithabendyarmthat
plugsintoyourcomputer/tablet–andmakesiteasiertoplacelearnerswork
underneath.Thisisalowcostvisualiser!
Technologyawayfromtheclassroom
1) Creatingworksheetsandtestswithaccuratemathematicaltextanddiagramsin
wordprocessingsoftwarehasthedistinct(andtime-saving)advantageinthatyou
caneditthedocumentagainandagaintomakemanydifferentversions.
OneofthefirstITapplicationsthatmanypeopleuseisawordprocessingpackage
suchas‘Word’toproducetypedtextthatcanbesaved,editedandprintedasand
whenyouneedit.Thismeansyoucanproduceyourowntasksandtests,writeyour
ownhelpfulnotesforyourlearnersandeven,bychoosingalargecleartext(orfont),
produceposterstodisplayintheclassroom.
a) Insertingsymbolsandequations.
Thebasicletters,numeralsandsimplesymbolssuchas+and–areallonthe
keyboard.Whenyouneedtousemoremathematicalsymbolssuchas÷,×,and
√youwillusuallygotoadropdownmenusuchas‘Insert’andchoose‘Symbols’
togetapaletteofothercommonsymbols.
Fig.11AtypicalpaletteofmoremathematicalsymbolsIMAGE11a_Symbols.jpg
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Towritemorecomplexnumericalandalgebraicexpressionsandequations,and
writingfractionsverticallyyouwillneedtousean‘equationeditor’whichis
normallyanoptionalfeatureofawordprocessingapplication.Asbefore,you
wouldselectan‘Insert’menuandchoose‘Equation’,whichwouldreveala
toolbarofdifferenttypes.
Fig.12Atoolbarofchoicesofdifferenttypesofequationsmathematicalsymbols
IMAGE12_Equations.jpg
b) Insertingtables
Therewillbemanyoccasionswhenyouwanttoproducetablesand,withyour
imagination,thesecanbequitecreative!
AlloftheimagesinFig.13belowhavebeencreatedastableswithinaword
processingpackage–thesecretistolearnhowtocontrolthesize,positioning,
textandshadingofeachcellinthetable.
Fig.13Aselectionofdifferentgraphics–allcreatedusingtablesinword
processingsoftware.IMAGE13_Tables.jpg
Herearesometoptips:
• Whenyouchoosetoinsertatable,onceyouhavecreateditwiththe
numberofrowsandcolumnsthatyouneed,selectthe‘Tableproperties’
totakecontrolofthesizeoftherowsandcolumns.Choosingawidthand
heightof1cmisveryusefulforproducingaccurateimages!
• Useborderlinescreatively–for,exampletocreatethe‘fractioncards’
above,whichcanbecutoutforlearnerstomatchandsort.
c) Insertinggraphs
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Firstly,youmusthaveaWorddocumentopen.
InGeoGebra,usethePointertooltodragaboxaroundthegraphthatyouwant
tocopy.OpentheFilemenuandchoose‘ExportgraphicsviewtoClipboard’(See
Fig.14below).
(Nothingmuchwillseemtohappen)
OpenyourWorddocumentandclickonPaste.
Fig.14ExportingagraphfromGeoGebraImage14_Geogebra_export.jpg
Youcanthenwritethetextaroundyourpicture.
d) Insertinggeometricimages
Therearealimitedrangeofbasicgeometricshapesavailableinmostword
processingapplicationswhenyouchooseInsertandShape.However,although
youcanchangethesizeofthembydragging,youhaveverylittlecontroloverthe
sizesofperimeters,areasandangles.Itismuchbettertocreatetheaccurate
shapethatyouwantinamathematicalapplicationsuchasGeoGebra,andthen
dragaroundyourimagetoselectitandthenusetheEditmenutocopyitfrom
GeoGebra-andpasteintoyourwordprocessingapplication,usingthesame
stepsfor‘Insertinggraphs’.
2) Creatingresourcestoprojecttotheclass
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Citeas:Clark-Wilson,A.,&Mostert,I.(2016).Teachingandlearningmathematicswithtechnology.InC.Hopkins,J.Anghileri,&J.Gage(Eds.),AIMSSECMathsTeacherSupportSeries:MathematicalThinkingintheLowerSecondaryClassroom.Cambridge:CambridgeUniversityPress.
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Assoonasyoucanuseadataprojector,itislikelythatyouwillwanttomake
presentation‘slides’thatcanincludeimagesfromothersoftware(suchasgraphing
andgeometrysoftware)tosupportyoutointroduceyourlessonandprovide
stimulatingandmathematicallycorrectdiagrams,symbolsandtext.Youcancopy
andpasteimagesfromothersoftwareintosuchsoftwareinthesamewayas
describedearlierinthischapter.
a) Presentationsoftware
ThemostcommonlyusedpresentationsoftwareisMicrosoftPowerPointbutother
freesoftwarewithsimilarfeaturesisavailablefordownloadfromtheInternet.
Fig.15CreatingaslideinPowerPointIMAGE15_PowerPoint_Presentation.jpg
Asyouinserteachslideintoyourpresentation,youcanchoosethelayoutthatbest
suitsyourlesson.Acombinationofcleartextinalargefontsizeandimages(which
canbemovies)worksbest.
b) Interactivewhiteboardsoftware
Ifyouhaveaccesstoaninteractivewhiteboard,whichmustalwaysbeconnectedto
bothyourcomputerANDadataprojector,youcanusethespecialsoftwarethat
comeswiththeboardtocreatepresentationfiles.Thissoftwareisusuallyfreely
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Citeas:Clark-Wilson,A.,&Mostert,I.(2016).Teachingandlearningmathematicswithtechnology.InC.Hopkins,J.Anghileri,&J.Gage(Eds.),AIMSSECMathsTeacherSupportSeries:MathematicalThinkingintheLowerSecondaryClassroom.Cambridge:CambridgeUniversityPress.
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downloadedfromthemanufacturer’swebsite,accompaniedbytutorialsandhelp
resourcestohelpyoumakethemostofthe‘fingertouch’functionality.
3) Creatinganelectronicmarkbook
Manyteachershavefoundspreadsheetsoftwaretobemostusefulforcreatingan
electronicmarkbookinwhichtheykeeparecordoflearners’outcomes.The
electronicnatureofthisbookmeansthatitisquickandeasytosortanynumeric
datafromhighesttolowestvalues,calculatemean.Modeandmedianaveragesand
producegraphsandtablestoshowlearners’progressovertime.Ofcoursethisdoes
relyonhighqualityassessmentdatainthefirstplace!
a) Enteringdata(Names/dates/marks)
Thestartingpointistoenterthestudents’names,genderandtheassessmentmarks.
Fig.16EnteringassessmentdatainaGeoGebraspreadsheet
IMAGE16_Assessment_data.jpg
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Citeas:Clark-Wilson,A.,&Mostert,I.(2016).Teachingandlearningmathematicswithtechnology.InC.Hopkins,J.Anghileri,&J.Gage(Eds.),AIMSSECMathsTeacherSupportSeries:MathematicalThinkingintheLowerSecondaryClassroom.Cambridge:CambridgeUniversityPress.
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b) Makingsenseofassessmentdata
WithinGeoGebra,onceyourdataisentered,whenyouselectthecolumnofdatayou
cangenerateanumberofsummarystatisticsandgraphs.Soforthedataonthe
learners’achievementsinAssessment1,thisanalysiscouldbe:
Fig.17SummarystatisticsandBoxandWhiskerplotforassessment1.
IMAGE17_Box_whisker1.jpg
TheBoxandwhiskerplotisausefulgraphicthatshowstherangeofmarks(71–7=
64),andthat50%ofthestudentswereinthemarkrangefrom36to48.
IfthesummarystatisticsforAssessment2arealsogenerated,ASLONGASthescales
forthex-axisarethesame,yougetausefulgraphicthathelpsyoutoassesswhether
thestudentsaremakingprogressovertime.
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Fig.18SummarystatisticsandBoxandWhiskerplotforassessment2.
IMAGE18_Box_whisker1.jpg
Soitappearsthattheclassismakingsomeprogress!!
4) Onlinecommunitiesforteachersandlearners
a) Ministryofeducationsites
Manycountriesaredevelopingtheirwebsitestoincludeinformationaboutthe
curriculum,detailsofanyregionalandnationaltestingarrangementsandasa
gatewayor‘portal’forresourcestosupportteachingandlearning.
Forexample,
SouthAfrica:theThutongwebsitewww.thutong.doe.gov.za.
Zimbabwe:theMinistryofPrimaryandSecondaryeducationwebsite
www.mopse.gov.zw
b) Aiminghigh
TheAimingHighteachernetworkhasgrownfromthehundredsofteacherswho
haveattendedtheAfricanInstituteforMathematicalSciencesSchoolEnrichment
Centre’s(AIMSSEC)coursesinSouthAfricasince2005.
ThewebsiteincludesTeachingresources,anonlineForumandNewsaboutthe
AIMSSECcourses.AIMSSECworkcloselywiththeGeogebraInstituteinSouth
Africatooffercoursesandsupportfortechnologyuse.
aiminghigh.aimssec.ac.za
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Fig.20TheAimingHighTeacherNetworkhomepage
IMAGE20_Aiming_high.jpg
c) AfricanMathematicsInitiative
TheAfricanMathematicsInitiative(AMI)isaKenyanNGOformedby
mathematiciansandmathematicseducatorswhoareworkingtocreatea
strongermathematicalcommunityandcultureofmathematicsacrossAfrica.
AMIhasproducedresourcestosupporttheuseofdigitaltoolsformathematics
suchasGeoGebraandScratch,whichareavailableonitswebsite.
www.africanmathsinitiative.net
5) Onlineresourcesforteachersandlearners
a) MicrosoftMath
MicrosoftMathisafreeresourcethatcanbeaccessedbyhighschoollearnerson
theircellphones.Itincludestheorypagesandquizzes.Learnerscancompete
againsteachother.Teacherscancreateagroupfortheirclassandcanthenkeep
trackofthenumberofpointsearnedbyeachlearner.Tocreateanaccount,goto
math.microsoft.com.
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Fig.21MicrosoftMathmenuandexampleofapracticequestion
IMAGE21a_MicrosoftMath_Menu.pngIMAGE21b_MicrosoftMaths_Example.png
b) EverythingMathsandScience
EverythingMathsandSciencehasfreehighschooltextbookswithvideosand
simulations.ThecontentisspecificallycreatedfortheSouthAfricancurriculum
butcanbeusedbyanybody.Thetextbookscanbeviewedonlineordownloaded
andprinted.Thesitealsohasafreeonlinepracticeservicethatallowsteachers
tosetproblemsfortheirlearners.Teacherscancreateagroupfortheirclass,set
problemsandthenkeeptrackofthenumberofpointsearnedbyeachlearner.
www.everythingmaths.co.za.
c) KhanAcademy
TheKhanAcademyisanonlineresourcethatincludestestquestionsandvideos
tosupportlearnersawayfromtheclassroom.Youwillneedtowatchthevideos
carefullytoo-astheymayintroduceyourlearnerstoarangeofdifferent
problemsolvingapproaches,mathematicalvocabularyandnotation.The
resourcesareorganisedbymathematicaltopicsandtheyarenotalignedtoany
particularcountry’scurriculumorexaminationsystem.Iflearnerscreatean
account,thentheirprogressthroughthetopicsisrecordedandtheyareawarded
pointsandrewardsfortheirwork.www.khanacademy.org
d) MathsExcellence
MathsExcellenceisawebsitestartedbySouthAfricanteacherstoimprovethe
standardofMathseducationandtomakemathsmoreaccessible.Itcontains
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manydifferentfreeresourcessuchastextbooks,videos,tutorialsandexam
papers.
www.mathsexcellence.co.za
e) MathAids
MathAidsprovidesfreemathsworksheetsforteachersandlearners.Thereare
manydifferenttopicstochoosefromandbecausetheworksheetsarerandomly
theyarealldifferent.Theworksheetscanbedownloadedandprinted.
www.math-aids.com
Nextsteps
Anychangestoourteachingpracticescomeatapersonalcost.Ittakestimetolearnto
usenewtoolsandresources;weoftenmakeapersonalfinancialinvestmentin
technologythatwillsupportusinourwork;andintheclassroomthingsmightnot
alwaysgoasexpected.However,thereareveryfewteacherswhodisagreewiththe
needtodeveloppersonallyandprofessionallyasameanstosupportingourlearnersto
achievemathematicstotheverybestoftheirabilities.
Glossary
Dynamicgeometry:aninteractivemathematicsenvironmentthatenablesgeometric
constructionstobecreatedandmanipulated.Forexample,2-Dshapescanbecreated
andtheirangles,sidelengths,areasandperimeterscanbemeasured.Usingthe
software,theseshapescanbetransformedbydraggingandtheyretaintheirunderlying
mathematicalproperties,althoughthemeasurementsmaychange.Theaccurateimages
canbecopiedandpastedintoothersoftware,suchasMSWord.
Examples:GeoGebra,Cabri-geometry,TheGeometer’sSketchpad,Cinderella
Dynamicgraphing:aninteractivemathematicsenvironmentthatenablesCartesian
andpolargraphstobeplottedautomaticallybyenteringthefunction.The
accompanyingtablesofvaluescanbedisplayedandthefunctionscanbetransformed.
Examples:GeoGebra,Desmos,Autograph
Spreadsheet:aninteractiveenvironmentwhereyoucaninput,organiseanalyseand
storetextandnumericaldataasatable.Thisdatacanbedisplayedgraphicallyanda
numberofstatisticalanalysescanbeusedtoqueryandinterpretthedata.
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Citeas:Clark-Wilson,A.,&Mostert,I.(2016).Teachingandlearningmathematicswithtechnology.InC.Hopkins,J.Anghileri,&J.Gage(Eds.),AIMSSECMathsTeacherSupportSeries:MathematicalThinkingintheLowerSecondaryClassroom.Cambridge:CambridgeUniversityPress.
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Examples:Excel,GeoGebra,
ApplicationorApp:Theseareusuallydigitaltoolsthathavebeendesignedtoonlyrun
onSmartPhones,iPadsandAndroidtablets,althoughsomepeoplearenowusingthis
wordtodescribesoftware.However,manyAppshavelimitededitingandfile-saving
features.Forexample,GeoGebraisavailableasanApp,butthereislimited
functionalitywhencomparedtothesoftwareversion.
Software:Aparticulardigitaltoolthathasbeendesignedforaspecificpurposeiscalled
‘software’.Forexample‘word-processingsoftware’allowsyoutocreate,edit,saveand
sharefiles.Thistexthasbeenwritteninwordprocessingsoftware.
Web-based:Ifaparticularresourceisweb-based,youwillneedtohaveanInternet
connectiontobeabletoaccessandusetheresource.Sometimesyoucansavethe
resourceontoyourowncomputersothatyoucanuseit‘off-line’.
Searchengine:Whenyouhaveaninternetconnection,youuseasearchenginetolook
forresourcesandinformationontheinternet.Themostcommonsearchenginesare
Google,InternetExplorerandYahoo.
Internet(orWorldWideWeb):Thisisaglobalnetworkthatconnectseveryone’s
computerstotheresourcesthatdifferentindividualsandcommunitieshavepublished
ontheirwebsites.
Mobilemessagingapplication:Thisisaninstantmessagingservicethatcanbeused
onmobile‘phonesandcomputers.Itcanbeaquickwaytocommunicateinformationin
‘real-time’.
Dataprojector:Adataprojectorprojectstheimagefromyourcomputerscreenontoa
wallorscreensothateveryonecanseetheimage.Itwillusuallyneedapowersource
andaconnectioncablefromyourcomputerortablet.Themoreexpensiveversions
haveamorepowerfulbulb,whichmeanstheywillprojectabrighterimage.