Sustaining High Cognive Demand in Intermediate Algebra · • Work 1me creang a shared resource to...

7/15/16

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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



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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


Swoopingin

•  h3p://www.1mssvideo.com/97


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•  Work1mecrea1ngsharedresourcetotakebacktoourclassrooms


WhatisCogniCveDemand?

PegSmith,JuliDixon,others:•  Taskselec1onandimplementa1onareimportant…

•  …butwhataboutAlgebra2contentspecifically?


WhatisCogniCveDemand?

Thisisabridgeoflengththree.Determinethenumberofbeamsinabridgeofanylength.


DixonNolanAdamsMathema1cs2016

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HighCogniCveDemandTask

Thisisabridgeoflengththree.Describehowthefollowinggeneraliza1onswerevisualized:


DixonNolanAdamsMathema1cs2016

3+4 n−1( ) n+2n+ n−1( ) 3n+ n−1( )

LevelsofCogniCveDemand

•  Memoriza1on•  Procedureswithoutconnec1ons•  Procedureswithconnec1ons•  Doingmathema1cs


Lower-Level

Demands

Higher-Level

Demands

Smith&Stein,“Selec1ngandCrea1ngMathema1calTasks:FromResearchtoPrac1ce”,2012

What“HighCogniCveDemand”LooksLike

•  Messy!•  Students“grapplewithcomplexity”•  Studentsresponddirectlytootherstudents•  Studentsdothesense-making•  LeverageStandardsforMathema9calPrac9ce


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Rethinking“I-We-You”

•  Not“I-we-you”=gradualrelease•  Instead“You-we-I”=produc1vestruggle


What“HighCogniCveDemand”LooksLike–Teachermoves

•  Facilita1ngdiscourseamongstudents– “Whatdidshesay?”– “Isthatwhatshesaid?”– “Youseemtodisagree.Why?”– Summarizedifferentresponses,thensay“Itsoundslikeyouhavemoretotalkabout”andwalkaway

– S:“Idon’tgetit”T:“Thenyouhaveaques1ontoask(otherstudent)”

– Teacherrarelygivesauthorita1veanswertoques1on


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




•  Work1mecrea1ngsharedresourcetotakebacktoourclassrooms


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PosePurposefulQuesCons


FromNCTM’sPrinciplestoAc9ons(pp35-36):“Purposefulques9onsallowteacherstodiscernwhatstudentsknowandadaptlessonstomeetvariedlevelsofunderstanding,helpstudentsmakeimportantmathema9calconnec9ons,andsupportstudentsinposingtheirownques9ons.”

ModifyingProblemstoIncreaseCogniCveDemand


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


Reversibility


Then:Findthefirstfouriteratesoff(n)=n2+2wheren0=-1.Now:Given-1,3,11,123,…

Explainthepa3ernanddeterminea(recursive)formula

Reversibility


Then:Solvethissystemofequa1ons:

Now:Createasystemofequa1onswhose

solu1onis(3,0,1)

x +2y + z = 4y − z = −1

−2x + z = −5

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Reversibility


Then:Findcsc(A):Now:5/4isthecosecantofwhichangle?

Y

E

A

10 8

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Uniqueness

“Istheremorethanonesolu1on?”“Howdoyouknow?”

Notes:•  Askthisques9onregularly,evenwhenthereisonly1uniquesolu9on

•  Useinteres9ngques9onsregularlythathavenon-uniquesolu9ons


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.


Flexibility

Then:Writeanequa1onfor:Now:Writetwoequa1onsusingtwodifferentfunc1ons.Whichdoyouprefer?Why?Sketchagraphthatwouldmakeyouprefertheotherfunc1on


ApplyaContext

•  Recall:

•  Createyourowncontextfor:B=4n+(n-2)


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ApplyaContext


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


www.nctm.org

MulCpleRepresentaCons


Then:Theheightofahotairballooncanbemodeledbyh(t)=–0.08t4+4.7t3–84.3t2+539t.Findalllocalextrema.Now:Createabe3errepresenta1onforthecompanytodisplayintheiradver1sementthatgivesmorecleardetailsabouttheflight.

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Open-Middle


Open-Middle


Open-Middle

42

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Defend/DisputeaClaim


y=asin(b(x–c))+dEsme:“Thereisonlyonevalueforbthatmakesthisfunc1onhaveperiod3π.”DoyouagreeordisagreewithEsme?Writeaparagraphtoexplainyourreasoning.



Esme:“Tofindthecomplexrootsofaquadra1c,Icanreflecttheparabolaver1callyoverits

vertexthenusetherealx-intercepts.”DoyouagreeordisagreewithEsme?Supportyouranswermathema1cally.



Esme:“Tofindthecomplexrootsofaquadra1c,Icanreflecttheparabolaver1callyoverits

vertexthenusetherealx-intercepts.”

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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


Apply a Context

Multiple Representations

Open-Middle

Generalize

Reversibility

Uniqueness

Flexibility Defend/Dispute a Claim

ModificaConandReflecCon

Designate1personfromyourgrouptoshare:1)oneofyourfavoritemodifica1ons2)onereflec1ononthestrategies


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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.


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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