7th Grade Lesson Study – Equivalent Expressions with Subtracting Negative Integers
March 6, 2017
Research Question:
What are the most important factors in supporting students to connect student-generated strategies, methods
and reasoning to procedures or algorithms?
This lesson study team consisted of a 2nd grade teacher, 5th grade teacher, 7th grade teacher and a district
instructional coach. Given the vast roles in this team, there were many opportunities to make vertical
connections with mathematical content, and generalizations about how students advance reasoning and
conceptual strategies to more abstract and procedural representations. With that, the shared understandings are
broken into two components – important observations and implications for K-5 teachers.
Important observations
•
Studentstendtoovergeneralizerulessuchas“anegativeandanegativeisapositive”eveniftheir
representationsdonotsupporttheargument.
•
Thecontextwillsignificantlyinfluencewhichmodelorrepresentationstudentschoosetouse.A
distance/lengthcontextwillbemorelikelytobemodeledandunderstoodwithanumberline,whereas
atake-awaymodelwillmostlikelyresultinstudentsusingcounters.Bothareimportant,and,
therefore,makingconnectionsbetweenthosedifferentrepresentationsisalsoimportantinadvancing
student-generatedreasoningtoabstractandproceduralmethods.
•
Towritebothanadditionandsubtractionequation,somestudentsdisregardthemodelsandcontext,
andsolelyusewhattheyunderstandabouttherelationshipbetweenadditionandsubtractionto
determineanotherequationthatwouldmatchorhavethesamevalue.
•
Opportunitiesforstudentstoworkindependentlyfirst,thendiscussinsmallgroupsbeforethe
discussioninwholegroupallowsthemtocomparesimilaritiesanddifferencesintheirwork.Insome
cases,misconceptionsareconfirmedbyamoredominantmemberinasmallgroup.
•
Inthisgroupofstudents,itappearedthatstudentsassumedthattherepresentationsdisplayedwere
correct.Varyingwhat’sdisplayedthatincludesmisconceptionsisanimportantelementtoconsider.
ImplicationsforK-5
• Representationalfluencyimpactsstudents’flexibilitywiththeirmathematicalunderstanding.
Teachers’awarenessofeffectiverepresentationshasimplicationsfortheirinstructionaldecisionsand
teachingpractices.Inthislessonstudy,studentswhowereabletoexplainhowthedifferent
representationswereconnectedwerethemostadvancedintheirreasoningandbegantoconjecture
aboutpatternstheywerenoticing.Therefore,itiscriticalforstudentstohavemanyopportunitiesto
makeconnectionsamongdifferentrepresentationswithinthesamecontext.
• Thenumberlineshoulddefinitelybeamodelthatisusedforstudents’sense-makingandrepresenting
thecontextofaproblem.
• Studentsneedmanyopportunitiestoexplainandusetheirown,andotherstudents’visualmodelsto
buildunderstanding,priortofocusingonfluency(accuracy,efficiency,flexibility).
• Students’understandingofequivalence,thatavalueoranexpressioncanberepresentedinmany
differentways,andthatthosevalues/expressionswouldberepresentedasthesamedistancefrom
zero(8–6=7-5)andequaltoeachotherisacriticalwaytothinkaboutexpressionsandsubtraction
throughoutgradesKindergartenthrough8thgrade.
• Students’understandingofsubtractionastake-awayisimportant,butnotenough.Students’
experienceswiththinkingofsubtractionasadistancewillsupportthedevelopmentofunderstanding
ofrationalnumbers(particularlywithnegativenumbers)inlatergrades.
• Ifthenumberlineisusedtomodelsubtraction,theemphasisshouldbeonthedistancebetweenthe
twonumbers,orthedifference.
• Students,includingupperelementarystudents,needtounderstandthatzeroseparatespositiveand
negativenumbers.
• Theideathataunitofonelengthremainsconstantonthenumberlinetodeterminethepositionof
othernumberscouldbeassumedbyadults,butisnotreadilyapparentforstudents.
•
K-5instructionshouldemphasizethedifferencebetweenapositiononthenumberlineandthe
iteratedunitsof“one”(length).Thenumbersonthenumberlinearereferencepointsforhowmany
unitsoflengthof“one”therearebetweenthatnumberandzero.
•
Statementssuchas“twonegativesisapositive”,“subtractionalwaysresultsinasmalleranswer”,
“additionalwaysresultsinabiggeranswer”negativelyimpactstudents’understandingofnegative
numbersandoperationsandshouldnotbeusedwithstudents.