sustaining high cognive demand in intermediate algebra · • work 1me creang a shared resource to...
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JasonSlowbeGreatOakHighSchool
SanDiego,CA
@[email protected]#NCTMinst
SustainingHighCogniCveDemandinIntermediateAlgebra
• SlidesandhandoutwillbeonIns1tutewebsite:h3p://nctm.org/hs16
• Everythingwillbeonmyownwebsite:h3p://theslowbe.weebly.com
• Pleasetweetthroughouttoday’ssession
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@theslowbe #NCTMinst
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Goalsforthisworkshop
• Avoidingfocuson“answer-geFng”• WhatisCogni1veDemand?• Modifyingexis1ngproblemsandtasks,not“reinven1ngthewheel”
• Work1mecrea1ngasharedresourcetotakebacktoourclassroomsandcolleagues
4 @theslowbe #NCTMinst
TowardBeJerPD…
Ar1cleonmywebsite,h3p://theslowbe.weebly.com
Johnson,Karen.“5ThingsTeachersWantfromPD,andHowCoachingandCollabora1onCanDeliverThem–IfImplementa1onImproves”.EdSurge.com,June28,2016h3ps://www.edsurge.com/news/2016-06-28-5-things-teachers-want-from-pd-and-how-coaching-and-collabora1on-can-deliver-them-if-implementa1on-improves
5 @theslowbe #NCTMinst
6 @theslowbe #NCTMinst
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7 @theslowbe #NCTMinst
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OurGoogleDoc…
• Gotoh3p://theslowbe.weebly.com,gotothisworkshop’spage,opentheGoogleDoc
• Populatethedocumentwithsome“ordinary”Algebra2problemsfromeachunit
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Iftherewasone“big”thingyoucouldchangeaboutAlgebra2,whatwoulditbe?
PhilDaroon“answer-geSng”
• h3ps://www.youtube.com/watch?v=DgTnmRyV9bc
• Calltoac1on:
Delayanswer-ge2ng
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Swoopingin
• h3p://www.1mssvideo.com/97
12 @theslowbe #NCTMinst
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Goalsforthisworkshop
• Avoidingfocuson“answer-geFng”• WhatisCogni1veDemand?• Modifyingexis1ngproblemsandtasks,not“reinven1ngthewheel”
• Work1mecrea1ngsharedresourcetotakebacktoourclassrooms
13 @theslowbe #NCTMinst
WhatisCogniCveDemand?
PegSmith,JuliDixon,others:• Taskselec1onandimplementa1onareimportant…
• …butwhataboutAlgebra2contentspecifically?
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WhatisCogniCveDemand?
Thisisabridgeoflengththree.Determinethenumberofbeamsinabridgeofanylength.
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DixonNolanAdamsMathema1cs2016
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HighCogniCveDemandTask
Thisisabridgeoflengththree.Describehowthefollowinggeneraliza1onswerevisualized:
16 @theslowbe #NCTMinst
DixonNolanAdamsMathema1cs2016
3+4 n−1( ) n+2n+ n−1( ) 3n+ n−1( )
LevelsofCogniCveDemand
• Memoriza1on• Procedureswithoutconnec1ons• Procedureswithconnec1ons• Doingmathema1cs
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Lower-Level
Demands
Higher-Level
Demands
Smith&Stein,“Selec1ngandCrea1ngMathema1calTasks:FromResearchtoPrac1ce”,2012
What“HighCogniCveDemand”LooksLike
• Messy!• Students“grapplewithcomplexity”• Studentsresponddirectlytootherstudents• Studentsdothesense-making• LeverageStandardsforMathema9calPrac9ce
18 @theslowbe #NCTMinst
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Rethinking“I-We-You”
• Not“I-we-you”=gradualrelease• Instead“You-we-I”=produc1vestruggle
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What“HighCogniCveDemand”LooksLike–Teachermoves
• Facilita1ngdiscourseamongstudents– “Whatdidshesay?”– “Isthatwhatshesaid?”– “Youseemtodisagree.Why?”– Summarizedifferentresponses,thensay“Itsoundslikeyouhavemoretotalkabout”andwalkaway
– S:“Idon’tgetit”T:“Thenyouhaveaques1ontoask(otherstudent)”
– Teacherrarelygivesauthorita1veanswertoques1on
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What“HighCogniCveDemand”LooksLike–Teachermoves
• Ques1onstudents’correctanswers!Givenoindica1onofcorrectness
• “Isawastudentdo______.Whatdoyouthink?”• Have[student2]use[student1]’sapproachtosolvetheproblem
• Wait1me
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CogniCveDemandinAlgebra2
• Promp1ngstudentstounderstandeachothers’thinkingkeepsstudentsengagedlongerandatahighercogni1velevel
• LeveragingtheMathema9csTeachingPrac9cestofocusonprocesses,notjustanswers
• Givestudentsmorecredit:givethemtheopportunitythenletthempursueit
22 @theslowbe #NCTMinst
Goalsforthisworkshop
• Avoidingfocuson“answer-geFng”• WhatisCogni1veDemand?• Modifyingexis1ngproblemsandtasks,not“reinven1ngthewheel”
• Work1mecrea1ngsharedresourcetotakebacktoourclassrooms
24 @theslowbe #NCTMinst
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PosePurposefulQuesCons
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FromNCTM’sPrinciplestoAc9ons(pp35-36):“Purposefulques9onsallowteacherstodiscernwhatstudentsknowandadaptlessonstomeetvariedlevelsofunderstanding,helpstudentsmakeimportantmathema9calconnec9ons,andsupportstudentsinposingtheirownques9ons.”
ModifyingProblemstoIncreaseCogniCveDemand
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Apply a Context
Multiple Representations
Open-Middle
Generalize
Reversibility
Uniqueness
Flexibility Defend/Dispute a Claim
Generalize
Then:Writeanequa1onforanellipsethatcontainsthepoint(6,0).Now:Write2moreequa1onsofellipsesthatallcontainthepoint(6,0).Writeageneralruleforanyellipsecontaining(6,0)Writeageneralruleforanyellipsecontaining(a,0)
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Generalize
Then:Determinetheendbehaviorofy=2x–3Now:Writearuletodeterminetheendbehaviorofanyexponen1alfunc1on:
y=ab(x–c)+d
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Reversibility
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Then:Findthefirstfouriteratesoff(n)=n2+2wheren0=-1.Now:Given-1,3,11,123,…
Explainthepa3ernanddeterminea(recursive)formula
Reversibility
30 @theslowbe #NCTMinst
Then:Solvethissystemofequa1ons:
Now:Createasystemofequa1onswhose
solu1onis(3,0,1)
x +2y + z = 4y − z = −1
−2x + z = −5
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Reversibility
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Then:Findcsc(A):Now:5/4isthecosecantofwhichangle?
Y
E
A
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Uniqueness
“Istheremorethanonesolu1on?”“Howdoyouknow?”
Notes:• Askthisques9onregularly,evenwhenthereisonly1uniquesolu9on
• Useinteres9ngques9onsregularlythathavenon-uniquesolu9ons
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Flexibility
NCTMon“proceduralfluency”:• Applyproceduresaccurately,efficiently,andflexibly
• Transferprocedurestodifferentproblemsandcontexts
• Buildormodifyproceduresfromotherprocedures
• Recognizewhenonestrategy/procedureismoreappropriatetoapplythananother
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Flexibility
Then:Factorcompletely:y=x3–5x2+3x–15
Now:Factorcompletely:
y=(x3–6x2+5x)+(x2–2x–15)Nowfactoragainusingadifferentapproach.Whichdoyouprefer?Writeanexpressionforwhichyouprefertheotherapproach.
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Flexibility
Then:Writeanequa1onfor:Now:Writetwoequa1onsusingtwodifferentfunc1ons.Whichdoyouprefer?Why?Sketchagraphthatwouldmakeyouprefertheotherfunc1on
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ApplyaContext
• Recall:
• Createyourowncontextfor:B=4n+(n-2)
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ApplyaContext
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http://www.nba.com/gametracker/#/20160508/CLEATL/lp/analysis
How is “root multiplicity” represented in this graph? What do they tell us about the game?
MulCpleRepresentaCons
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www.nctm.org
MulCpleRepresentaCons
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Then:Theheightofahotairballooncanbemodeledbyh(t)=–0.08t4+4.7t3–84.3t2+539t.Findalllocalextrema.Now:Createabe3errepresenta1onforthecompanytodisplayintheiradver1sementthatgivesmorecleardetailsabouttheflight.
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Open-Middle
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Open-Middle
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Open-Middle
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Defend/DisputeaClaim
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y=asin(b(x–c))+dEsme:“Thereisonlyonevalueforbthatmakesthisfunc1onhaveperiod3π.”DoyouagreeordisagreewithEsme?Writeaparagraphtoexplainyourreasoning.
Defend/DisputeaClaim
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Esme:“Tofindthecomplexrootsofaquadra1c,Icanreflecttheparabolaver1callyoverits
vertexthenusetherealx-intercepts.”DoyouagreeordisagreewithEsme?Supportyouranswermathema1cally.
Defend/DisputeaClaim
45 @theslowbe #NCTMinst
Esme:“Tofindthecomplexrootsofaquadra1c,Icanreflecttheparabolaver1callyoverits
vertexthenusetherealx-intercepts.”
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Defend/DisputeaClaim
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Esme:“Everypolynomialhasauniquefactoriza1on.”
DoyouagreeordisagreewithEsme?Supportyouranswermathema1cally.
PhillipsExeterAcademy
h3p://www.businessinsider.com/what-its-like-to-a3end-phillips-exeter-academy-2014-11?op=1
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ModifyingProblemstoIncreaseCogniCveDemand
49 @theslowbe #NCTMinst
Apply a Context
Multiple Representations
Open-Middle
Generalize
Reversibility
Uniqueness
Flexibility Defend/Dispute a Claim
ModificaConandReflecCon
Designate1personfromyourgrouptoshare:1)oneofyourfavoritemodifica1ons2)onereflec1ononthestrategies
50 @theslowbe #NCTMinst
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52 @theslowbe #NCTMinst
https://www.edsurge.com/news/2016-06-28-5-things-teachers-want-from-pd-and-how-coaching-and-collaboration-can-deliver-them-if-implementation-improves
Inthefigure[below],arcBDiscenteredatA,andithasthesamelengthastangentsegmentBC.ExplainwhysectorABDhasthesameareaastriangleABC.
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Leavewithaproblem…
@dynamic_math
h3ps://www.geogebra.org/m/npZNjN2Kh3ps://www.geogebra.org/tbrzezinski
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JasonSlowbeGreatOakHighSchool
SanDiego,CA
@[email protected]#NCTMinst
SustainingHighCogniCveDemandinIntermediateAlgebra
Disclaimer The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring equitable mathematics learning of the highest quality for all students. NCTM’s Institutes, an official professional development offering of the National Council of Teachers of Mathematics, supports the improvement of pre-K-6 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. It is a forum for the exchange of mathematics ideas, activities, and pedagogical strategies, and for sharing and interpreting research. The Institutes presented by the Council present a variety of viewpoints. The views expressed or implied in the Institutes, unless otherwise noted, should not be interpreted as official positions of the Council.
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