grace kelemanik · · 2016-07-12learn how the capturing quan--es instruc@onal ... capturing...

AnInstruc*onalRou*netoFosterMathPrac*ce2

[email protected]

@GraceKelemanik

CapturingQuan**es

DoNow

Juanhad3redmarbles.Hefound12moremarblesthatwereblue.HowmanymorebluemarblesthanredmarblesdidJuan

have?

1.   Solvebothtasks.2.   An*cipatehowstudentsmightsolvethetasks.

Aschoolboughtsomemathbooksand4@mesasmanysciencebooks.Thecostofamathbookwas$12whileasciencebookcost$8.Altogethertheschoolspent$528.Howmanysciencebooksdid

theschoolbuy?

Deepenyourunderstandingofreasoningabstractlyandquan@ta@vely(MP2).Youwillknowyourlearningisontrackifyoucan:

– Describehowyouand/oracolleagueisreasoningabstractlyandquan@ta@vely

– Ar@culateabstractandquan@ta@vereasoningthatwillbehelpfulinfutureproblemsolving

Goal#1

LearnhowtheCapturingQuan--esinstruc@onalrou@neembodiesNCTMMathema@[email protected]@fywhenandhowtherou@ne:•  EstablishesaMathGoaltoFocusLearning•  UsesandConnectsMathema@cal

Representa@ons•  SupportsMeaningfulMathema@calDiscourse•  PosesPurposefulQues@ons

Goal#2

Agenda

Opening:Intro,Goals,Framing,andDoNow

UnpackMP2ReasonAbstractlyandQuan@ta@vely

DoMath:CapturingQuan--esRou@ne

Reflect:NCTMTeachingPrinciples

AnalyzeStudentWork

5

DoNowRevisited

6

DoNowRevisited

7

StandardsforMathema*calPrac*ce

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning

of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

Structure

MP2

MP4 MP5

MP6MP3 MP7

MP4 MP5

MP6MP3

MP8

MP4 MP5

MP6MP3

RoutinesforReasoning

MP2ReasonAbstractlyandQuan*ta*vely

AMendto…

Quan@@es

andRela@onships

Askyourself…

•  WhatcanIcountormeasureinthisproblemsitua@on?

•  Howdothequan@@esrelatetoeachother?

•  HowcanIrepresentthisproblem?

•  Whatdoesthis(expression,variable,number,shadedregion,etc.)representintheproblemcontext?

EnteringthetaskfromaReasoningAbstractlyandQuan*ta*velyAvenuePaidaMen*onto

•  NumberofredmarblesJuanhad

•  NumberofbluemarblesJuanfound

•  Rela@onshipbetweenthenumberofredmarblesandthenumberofbluemarbles

Askedyourself…

•  WhatcanIcountinthissitua@on?

•  Howarethenumberofredmarblesrelatedtothenumberofbluemarbles?

•  HowcanIrepresentJuan’smarblessothatIcanseearela@onshipsbetweenthenumberofblueandredmarbles?

PutintoAc*on…

•  Iden@fyquan@@esexplicitlymen@onedintheproblemsitua@on

•  Surfaceimpliedquan@@es•  Abstractproblemsitua@ons•  Userepresenta@onstoseequan@@esandrela@onships

•  Recallandconsiderreferents

Quan*ta*veReasoningShiPs

Thenumbersandkeywordsina

problemstatement

Thequan@@esandrela@onshipsthosenumbers

describe

Routines for Reasoning Kelemanik, Lucenta, Janssen Creighton

Howdoweteachstudentsto“reasonabstractlyand

quan*ta*vely?”

14

Instruc*onalrou*nesdesignedtodevelopthehabitof

aMendingtoquan**esandrela*onships

15

ROUTINES FOR REASONING


CapturingQuan**es

Aninstruc*onalrou*nethatdevelopsthequan*ta*vereasoning

avenueofthinking

16Routines for Reasoning

Kelemanik, Lucenta, Janssen Creighton

WHAT?

DiagramaProblemSitua*on

ThinklikeaMathema*cian!Tobuildthehabitoflooking

forquan**esandrela*onshipsinproblem

situa*ons

WHY?

How?

1.  Iden@fyQuan@@es&Rela@onships

2.  CreateDiagrams

3.  DiscussDiagrams

4.  ReflectonLearning

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Pennyhadabagofcandies.Shegave⅓ofhercandiestoRebecca.ThenPennygave¼ofthecandiesshehadlegtoJohn.AgergivingcandiestoRebeccaandJohn,Pennyhad24candiesleginherbag.

Iden*fyQuan**esandRela*onships

WhatcanIcountinthissitua@on?

Pennyhadabagofcandies.Shegave⅓ofhercandiestoRebecca.ThenPennygave¼ofthe

candiesshehadlegtoJohn.AgergivingcandiestoRebeccaandJohn,Pennyhad24candiesleginherbag.


Aquan@tyIfoundwasthenumberof…

Arela@onshipIfoundwas…


CreateaDiagram

HowcanIshowthe

rela@onshipsbetweenthequan@@es?

✎

1.  Sharehowyoustartedtorepresentquan@@esandrela@onships

2.  Togethercreateadiagram

Ishowed...by...Howdidyourepresent…?

CreateaDiagram

Wherearethequan@@esinthediagram?Wherearetherela@onshipsinthediagram?

Therela@onshipis…SoIlooked

for...

[email protected]@@esthat...

DiscussDiagrams

•  Whenlookingforquan@@esinawordproblem,Ilearnedto.

•  Whenanalyzingadiagram,Ilearnedtopayaien@ontobecause.

✎ReflectonLearning


What’s the Question?

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LaunchtheRou*ne:Introducethinkinggoal,reviewrou@ne’ssteps1

ThinkingG

oal:Bu

ildingan

Avenu

eofThink

ing

5Pairs Share&RecordIndividualWriteTime

ReflectonYourThinking

2Pairs Share:

Discuss&AnnotateIndividualThinkTime

Iden*fyQuan**es&Rela*onships ✍

3PairsIndividualThinkTime

CreateDiagrams

4Pairs Share:

Discuss&AnnotateIndividualThinkTimeDiscussDiagrams ✍

✎

✍


CAPTURING QUANTITIES ROUTINE

1.  Identify Q & R? (T-P-S)

2. Create Diagram (T-P)

3. Discuss Diagrams (Full group)

4. Reflect on Learning (W-P-S)

Reflec*ngontheCapturingQuan**esRou*neandtheNCTMMathema*csTeachingPrac*ces

ATYOURTABLESdiscusshowtheCapturingQuan--esrou@ne:•  EstablishesaMathGoaltoFocusLearning•  PosesPurposefulQues@ons•  SupportsMeaningfulMathema@calDiscourse•  UsesandConnectsMathema@cal

Representa@ons

29


ATYOURTABLESdiscusshowtheCapturingQuan--esrou@ne:•  EstablishesaMathGoaltoFocusLearning•  PosesPurposefulQues@ons•  SupportsMeaningfulMathema@calDiscourse•  UsesandConnectsMathema@cal

Representa@ons

30

ThinklikeaMathema*cian!Tobuildthehabitoflooking

forquan**esandrela*onshipsinproblem

situa*ons

WHY?

CapturingQuan**esMathema*calGoalstoFocusLearning



•  Whenlookingforquan@@esinawordproblem,Ilearnedto.

•  Whenanalyzingadiagram,Ilearnedtopayaien@ontobecause.

✎ReflectonLearning

CapturingQuan**esMathema*calGoalstoFocusLearning



Reflecting on Learning Prompts


ATYOURTABLESdiscusshowtheCapturingQuan--esrou@ne:•  EstablishesaMathGoaltoFocusLearning•  PosesPurposefulQues*ons•  SupportsMeaningfulMathema@calDiscourse•  UsesandConnectsMathema@cal

Representa@ons

35



WhatcanIcountinthissitua@on?


CreateaDiagram

HowcanIshowthe

rela@onshipsbetweenthequan@@es?

✎

CapturingQuan**esPosePurposeful“askyourself”Ques*ons




ATYOURTABLESdiscusshowtheCapturingQuan--esrou@ne:•  EstablishesaMathGoaltoFocusLearning•  PosesPurposefulQues@ons•  SupportsMeaningfulMathema*calDiscourse•  UsesandConnectsMathema@cal

Representa@ons

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CapturingQuan**esSupportsMeaningfulMathDiscourse



✍


ATYOURTABLESdiscusshowtheCapturingQuan--esrou@ne:•  EstablishesaMathGoaltoFocusLearning•  PosesPurposefulQues@ons•  SupportsMeaningfulMathema@calDiscourse•  UsesandConnectsMathema*cal

Representa*ons

41

StudentWorkAnalysis

LookforevidencethatstudentsareaBendingtoquan**esand

rela*onshipsbetweenquan**es

42

Ul*mately…



HSConceptualCategoryNumberandQuan*ty

Quan**es.Inrealworldproblems,theanswersareusuallynotnumbersbutquan@@es:numberswithunits,whichinvolvesmeasurement.IntheirworkinmeasurementupthroughGrade8,studentsprimarilymeasurecommonlyusedaiributessuchaslength,area,andvolume.Inhighschool,studentsencounterawidervarietyofunitsinmodeling,e.g.,accelera@on,currencyconversions,derivedquan@@essuchasperson-hours....Theyalsoencounternovelsitua@onsinwhichtheythemselvesmustconceivetheaiributesofinterest.Forexample,tofindagoodmeasureofoverallhighwaysafety,theymightproposemeasuressuchasfatali@esperyear,fatali@esperyearperdriver,[email protected]@mescalledquan@[email protected]@fica@onisimportantforscience,aswhensurfaceareasuddenly“standsout”[email protected]@fica@onisalsoimportantforcompanies,whichmustconceptualizerelevantaiributesandcreateorchoosesuitablemeasuresforthem.

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Rou*nesforReasoning:FosteringtheMathema*calPrac*cesinAllStudents

-  Kelemanik,Lucenta,Janssen-Creighton

[email protected]

@gracekelemanik

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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grace kelemanik · · 2016-07-12learn how the capturing quan--es instruc@onal ... capturing...

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