AlgebraILessonStudy–IntroductiontoFactoringQuadraticExpressionsDecember12,2016

ResearchQuestionWhatcharacteristicsofinitialtasksandteacheractionsallowstudentstodevelopconceptualunderstandingoffactoringquadraticexpressions?

Teachersinthisresearchlessonconcludedthatteachersshouldconsiderthefollowingregardinginitialtasksforfactoringquadraticexpressions:

• Effectivetasksshouldbeintroducedwithsomeopportunitiesforstudentmisconceptionstoarisethatareimportantinmakingprogresstowardsthemathematicalgoalsuchasmisapplyingproceduresforaddingliketerms,misapplyingpropertiesofexponents,etc.Limitingthenumberofalgebratilestoasmallnumbertostart,allowsstudentstofocusonimportantmathematicalconceptsthatdefineeachoftheterms.

• Effectivetasksshouldnotprescribeforstudentshowtothinkorrepresenttheirideas.• Algebratilesallowstudentstobuildontheirunderstandingofareatohaveavisual

representationofquadraticexpressions.• Effectivetasksshouldallowforopportunitiestoseetheconnectionbetweenvisual

representationsofquadraticexpressionsinstandardformandfactoredform.• Requiringthevisualandconcreterepresentationsinthetaskallowsforastartingpoint

forstudents,butalsoactsasacatalystforallowingmisconceptionstomanifestwhen

studentsareaskedtoproduceasymbolicrepresentation(quadraticexpression)ofthevisual.Thoserepresentationsthenserveasasupportforstudentstoreasonaboutboththegradelevelcontent,andpreviousyears’content(propertiesofexponents).

• Thefocusonthetasksshouldemphasizeequivalentexpressions,butnotsimplifyingsincesomeexpressionsaremoreusefulinfactoredform,standardform,vertexform,etc.

Theteachershouldalsoconsiderthefollowingteacheractions:

• Havingstudentstalkabouttheirideasdeepenstheirunderstanding.Ifwehavestudentswritewithouttalkingabouttheirthinking,reflectiononwhattheydoanddon’tunderstandwouldbelimited.(SMP3andSMP6)

• Whenaskingaquestionfortheclasstothinkabout,ithelpstohavestudentswritetheirownthoughtsfirst,beforediscussingwiththeirclassmates.Thisseemstoallowthemtogatherthoughtsbeforeverbalizing,whilegivingtheteacherabetterideaofwhateachstudentunderstands.

• Afteraskingthestudentstothink,write,thendiscusstheirideaswiththeirpeers,awholeclassdiscussioncanensue.However,whenstudentsdiscusstheirideasasawholeclass,itcanbehelpfultore-phrasewhatstudentssay,andaskfollow-upquestionssuchas“whatdoyoumean,whatdoeshe/shemean,whydoyouthink___,whatelsecouldwedo,howwouldwerepresent___”.Thesethenleadtoanotheropportunityforstudentstodiscusstheirthoughtswitheachother.Thiscyclethenproducesthebuildingofsharedunderstandingsthatoriginatesinthestudents’reasoning.

• Whentheteacherasksquestionsthatfocusstudentsoneachothers’thinking,thestudents’developadeeperlevelofunderstanding.Byhavingstudentsdiscusswhattheirpeersarethinking,theyaremorelikelytotrytounderstandthecontentbetterastheycanseeotherstudentsdoingthethinking,whichimpliesthattheythemselvescanalsodothemathematicalthinking.(SMP1)