Differentiation in Mathematics - WordPress.com · and teaching mathematics. 2. I love to snow ski. 3. I once went 30 days without using a proper toilet. Cooperative Problem Solving

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DifferentiationinMathematics

Whatareyouinterestedinrethinking duringthisconference?

Whatdoyousupposewemightberethinkingheretoday?

Algebra Readiness Test Results7th Graders at Beginning of School Year (2008)

0%

10%

20%

30%

40%

50%

0 to 10 11 to 20 21 to 30 31 to 40 41 to 50

Score

Perc

ent o

f Stu

dent

s

Typical readiness spectrum

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“theonlygoodkindofinstruction isthatwhich marchesaheadofdevelopmentandleadsit” Vygotsky

“enjoymentappearsattheboundarybetweenboredom andanxiety, whenthechallengesarejustbalanced withaperson’s capacitytoact.”

(Csikszentmihalyi Flow: The Psychology of Optimal Experience)

Whatisthebestwaytogroupstudentsforlearningmathematics?

abilitygrouping

heterogeneousgrouping

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Percent of 8th Graders in Low Track Class

0%5%

10%15%20%25%

2000-2005Average

2006-2007 2007-2008 2008-2009 2009-2010

Year

Whatisthebestwaytogroupstudentsforlearningmathematics?

abilitygrouping

heterogeneousgrouping

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Let’scontinuetoestablishapositiveclimateforlearningtogether

BuildIt

9/16/16

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WelcomeJ

Getintogroupsofthreeorfour.Introduceyourselvesandsharewhatyou’remosteagertogetfromourtimetogether.

Establishingaclimateforlearning

• Buildit

BreakingtheIce:TwoTruthsandaLieWhichisthelie?

1. I’vealwaysenjoyedlearningandteachingmathematics.

2. Ilovetosnowski.3. Ioncewent30dayswithout

usingapropertoilet.

CooperativeProblemSolvingActivities:BuildIt

Emotionalattentioncomesbeforecognitiverecognition

(Sousa, Howthe Brain Learns , 43)

Data affecting Survival

DatageneratingEmotions

DataforNewLearning WorkingM

emory

LongTerm

Mem

ory

?

Effectivedifferentiatedinstructionincreasesprocessingtimewithkeylessonideas.

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Effectivedifferentiatedinstructionincreasesprocessingtimewithkeyideas.

JoBoaler – from“HowtoLearnMath”

Intermsofsupporting studentcollaboration, whathasworkedwellforyou?

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Whenpersonalized learningishappening, thelearner…

Feelsastrongsenseofcommunitybelonging.

Engagesinmeaningfullearning.

Perceivesagoodfitbetweeninstruction andtheirreadiness, interests, andlearningprofile.

Isintrinsically motivated.

Operatesfromaninternallocusofcontrol andresponsibility.

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ExponentialFunctionsPublicServiceAnnouncementProject

ComplexKnowledgeofStudentsandofSelf

LimitedKnowledgeofStudentsandSelf

BeginningKnowledgeofCurriculum

AdvancedKnowledgeofCurriculum

DifferentiatingTeacherRelationship

OrientedTeacher

BeginningTeacher

TaskOrientedTeacher

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Whatisadifferentiating teacher?

Differentiatingmathteachersaredeeplyknowledgeableaboutourselves, ourstudents, andourcurriculum.Committedtoengagingalllearners,weutilizeproactiveapproachesthataddressawiderangeofmathlearningneeds.

Differentiatingteachersarepragmaticproblem solvers.We’reconstantlyrecognizingandacknowledgingwhenapproachesaren’tworking, andweceaselesslyandcollaborativelypursue growth.

Whataredistinguishingcharacteristicsofhighquality,tier1lessons?

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Let’stryalearningsequence

Workshoplearninggoals:• Whataredistinguishingcharacteristicsof

brainfriendly,highqualitytier1lessons?(pgs.8– 9)

• Whatdoesitmeantobegoodatmath?

Mathlessonlearninggoal:• Recognizepatternsandgeneralize

relationships

+ Relativestrengths-- Relativeweaknesses

0.TheDotProblem

Without countingeverytriangulardot,determinehowmanydots

areinthefigure?

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1.TheBorderProblem

Withoutcountingeverysquareindividually,howmanysmallbluesquaresareintheborderofthe

figure?

Howmanywayscanyoudeterminetheanswer?

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!

Thistime,let’slookatapatternasitgrows.

2.Copyorbuildthispattern.Another(Experience(with(Structural(Algebra(!

!!!

1. How!many!squares!would!be!in!the!5th!figure?!!!!!

2. How!many!would!be!in!the!20th!figure?!!!!!



5. How!many!would!be!in!the!nth!figure?!Explain!your!thinking.!

Figure1

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!!!






Figure1

Figure2


!!!






Figure1

Figure2

Figure3

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!!!






Figure1

Figure2

Figure3

Figure4

Howmanysquareswillbeinthe11th figure?

Another(Experience(with(Structural(Algebra(!

!!!






Figure1

Figure2

Figure3

Figure4

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Howmanysquareswillbeinthe100th figure?

Drawandlabelasketchthatillustrateshowyou’rethinkingaboutthe100th figure.


!!!






Figure1

Figure2

Figure3

Figure4

Howmanysquareswillbeinthenth figure?


!!!






Figure1

Figure2

Figure3

Figure4

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3a.i.Howmanysquareswilltherebensecondslater?Labelasketchthatshowswhyyourexpression makessense.

1secondlater

2secondslater

3secondslater

ii.Howmanysecondswillittakeforthefigure tohave100squares?

3c.Thencreateaninterestingstorythatincorporatesyourpattern.Whatquestionmightbeaskedtoprovokea

problemsolver togeneralizethepattern?

3b.Createauniquepatternofyourownwhichisgrowingataconstantrate.Howmanysquareswillbeinthenth figure?

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Whatquestions cometomind?Whatmightwewonderabout?

1.Whenwillthevirushittheborder?2.Whichcolorwillhittheborder?3.Howbigwillthevirusbewhenithitstheborder?

4.Whereexactlywillthevirushittheborder?

5. Whatwillbeits“aspectratio”ifitgrowsforever?

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Act2:Whatinformationwouldyoulikefromme?

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Sequel:Whatwillthe"aspectratio"ofthepixelpatternbecomeifthepatterngrowsforever?

http://threeacts.mrmeyer.com/pixelpattern/

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Asclearlyaspossible,explainyoursolution tothefirstquestion.Usedrawingstohelpillustrateyourthinking.

Act3– Gradeyourself (bottomofpage10)

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3ActMathTasks

https://gfletchy.com/3-act-lessons/

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(Sousa,HowtheBrainLearns,43)


DataforNewLearning

Workin

gMem

ory

LongTe

rm

Mem

ory

?

Willworkingmemorybeconvertedintolongtermmemory?

Theprobabilityofstoring informationinlongtermmemory


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WritinginMath

Theveryactofwritingallowsustoseeconceptualrelationships,toacquireinsights,andtounravelthe logicofwhatwaspreviouslymurkyorconfusing…whenstudents areaskedtoexplainorevaluateasolutionoralgorithminwriting,theycometoaclearer,deeperunderstanding.

(Schmoker,FOCUS:ElevatingtheEssentialstoRadicallyImproveStudentLearning,211)

Returntopg.8/9inyourreader.Reflectonourpatternlessonsequencebynotinginstanceswhenyouobserveddesiredbehaviors.Also,pleasenoteopportunitiesforimprovement.

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Meaning isenhancedthroughcognitiveclosure.Completethesethoughts inwriting:

• Aninsight I’vegainedorsomething I’velearnedtodayis:

• Thisconnects/addstowhatIknowabout:• Thiscanhelpmelaterwhen:• Aquestiononmymindis:

What does it mean to be good at math?

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“Mathematicsisaboutfiguringthingsoutandmakingsenseofthingswedon’tinitiallyunderstand.

It'snottedious,digit-orientedarithmeticoralgebraicmanipulations.It'snotaboutfollowingtherules…

Doingmathematicsrequiresthatkidsthink.”EverybodyCounts,MathematicalSciencesEducationBoard,1989andJohnVanDeWalle

FiveAspectsofMathematicalProficiencyConceptual Understanding– comprehensionofmathematical

concepts,operations,andrelationsProceduralFluency– skillincarryingoutprocedures flexibly,

accurately,efficiently,andappropriately

StrategicCompetence– abilitytoformulate,represent,andsolve[reallife]mathematicalproblems

AdaptiveReasoning– capacityforlogicalthought,reflection,explanation,andjustification

ProductiveDisposition– habitualinclinationtoseemathematicsassensible,useful,andworthwhile,coupledwithabeliefindiligenceandone’sownefficacy

(VandeWalle,24,andAddingitUp:HelpingChildrenLearnMathematics,2001)

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Learninggoal

Whatarespecificneedsofstrugglingandadvancedlearnersandwhatstrategiescanbeemployedtofacilitateproductivelearningexperiencesforall?

HowwillAbigailapproachhermathlearninginthefuture?

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UnderstandingandPerformanceNota2-waystreet

ConceptualUnderstanding

ProceduralFluency

Conceptual knowledgehasagreaterinfluenceontheacquisition ofprocedural

knowledgethanthereverse.

Studentswhodon’t engageinconceptualthinkingdon’t benefitfromcompression

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UnderstandingDoingLearning

ConceptualUnderstanding

ProceduralFluency

StrategicCompetence

AdaptiveReasoning

ProductiveDisposition

ConceptualUnderstandingisEssential



DataforNewLearning

Workin

gMem

ory

LongTe

rm

Mem

ory

?

Workingmemorycapacitysignificantlyimpactsmathlearningasdoapproaches

whichreducecognitiveloads

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Tohelpreducetheloadonworkingmemory

• Providehelp, cues,reminders• Useshort,simple sentences,picturesanddiagrams• Presentless information• Useonlyrelevantinformationandnodistracters.• Promptengagement(eg.totakenotesorcreatediagrams)

• Attachmeaningfromwhatisalreadyknown• Breakdowntasksintosmallchunks.Mastereachchunk sequentially, oneatatime

Whataspectsofthepatternlearningsequencehelpedlessondemandsonworkingmemory?Wheremight theloadstillbesignificant?

• Providehelp, cues,reminders• Useshort,simple sentences,picturesanddiagrams• Presentless information• Useonlyrelevantinformationandnodistracters.• Promptengagement(eg.totakenotesorcreatediagrams)

• Attachmeaningfromwhatisalreadyknown• Breakdowntasksintosmallchunks.Mastereachchunk sequentially, oneatatime

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Committedtoengagingalllearners,differentiatingteachersutilizeproactiveapproachesthataddressawiderangeofmathlearningneeds.

Effectivedifferentiatedinstructionincreasesprocessingtimewithkeyideas.

MultiplicationProperties ofExponentsLessonGoal

• Developadeeperunderstandingofthemultiplicationpropertiesofexponents.

• Beabletoexplain,withexamples,whythe“powerofapower”and“productofpowers”ruleswork.

BigIdeasaboutTeachingandLearning

• Transfertolongtermmemoryrequiresqualityrehearsal.• Duringrehearsal,learnersuseworkingmemorytoestablishsense

andmeaning.• Reducingloadsonworkingmemoryisessentialforsomestudents

andhelpfultoallstudents.• Promptingself-assessmentduringrehearsaldeepens

understandingwhilebuildingcapacityforindependent learning.

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ExponentPropertiesTieredLesson

Likes Concerns orQuestions

TieredInstructionandAssessment

Studentsengageanddemonstrateunderstandingofessentialconceptsatvaryinglevelsofdepthandbreadth

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ATieredLesson…

• isplannedaroundamajorconceptorbigidea• hasaclearobjective andmakesthegradelevel

proficiency standardcrystalclear• provides opportunities fordeeperandextended

learning• isflexiblypaced• likelyincludeswholegroupinstruction• mayincorporatesmallgroupinstruction

EffectiveMathLearnerBehaviors

(readerpages16and17)

Towhatextentdidtheexponentlessonsortheborderproblem,growthpatterns,andvirusproblemsequenceallowyoutodemonstratethedesiredstudentbehaviors?

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Effectivedifferentiatedinstructionincreasesprocessingtimewithkeyconceptsandideas.

Relationship between Teacher Knowledge and Teacher Effectiveness

ComplexKnowledgeofStudentsandSelf

LimitedKnowledgeofStudentsandSelf

BeginningKnowledgeofCurriculum

AdvancedKnowledgeofCurriculum

RelationshipOrientedTeacher

BeginningTeacher

TaskOrientedTeacher

DifferentiatingTeacher

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ChallengebyChoiceWhy?

“The way a child learns how to make decisions is by making decisions”

Alfie Kohn – “What to look for in a classroom”

“When students feel empowered to be responsible in school, they tend to prefer more challenging academic tasks, set higher academic goals, and persist when confronted with difficult tasks.”

Thomas Guskey, “Students at Bat,” Educational Leadership, Nov. 2008

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Challengeeachstudenttoreachtheirfullpotentialwhiledevelopingself-awareness,

self-confidence,andself-disciplinetotakeincreasingownershipfortheirlearning.

GuidingVision

StudentChoices

47% have tried more than one color53% have always chosen the same color

How much do individual students vary their choices?Summative Assessments HW Assignments

77% have tried more than one color23% have always chosen the same color

(415 student surveys)

Assessment Color Choices (Grades 6-8)

0%

10%

20%

30%

40%

50%

60%

70%

Green Blue Black

(13,468 choices)

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Students usuallyfeelappropriatelychallengedandcanadjustwhentheydon’t

…reported that the assessment was ________ as a destination

for their learning

Students whoselected ___

“too Simple” “Appropriate”

“too Challenging” Total

Reflections

Green 15 % 79 % 6 % 3400

Blue 7 % 83 % 10 % 2159

Black 7 % 78 % 15 % 690

4% of reflections represent a scenario that could not have been improved by a different choice

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Howmighttieredinstruction impactthelearningofstrugglingmathlearners?

Soundatallfamiliar?

“Ihavenoideawheretobegin...Ifeellikemybrainisn’tworking.”

Strategiescanbeusedtonegotiatelimitedworkingmemorycapacity.

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“Idon’tunderstand…couldyouhelpme?” yourstudent

“Whenakidishavingtrouble,inmyroom,duringalesson,Idon'tsayletmeshowyou,Isay‘tellmewhatyou'rethinking.’

JohnVandeWalle

Lowachieversareoftenthoughtofaslackingabilityorbeingslow,butthey’reoftenlowachievingbecausetheydon’tknowwhattheyshouldbepayingattentionto.

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Tosupportstrugglers,we’vediscussed:• waystoreducecognitiveloadsand

theimpactofanxiety• theimportanceofteachingfor

conceptualunderstanding• howtoscaffoldworkingmemory

processes• thevalueofhelpingstudentsknow

whattheyshouldbepayingattentiontothroughpeerandself-assessment

Whatelsecanwedo?

Asynthesisofrecommendationsforstrugglers (pages21– 24)

Lookforbigideasandnoteworthyhighlights.

Howmighttieredinstruction impactstrugglingmathlearners?

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Results

1. Student achievement, motivation, and responsibility

2. Student, teacher, and parent perspectives

Assessment Performance

40

50

60

70

80

90

100

Overall 2005 Green 2006 Blue 2006 Black 2006

Color

Ave

rage

Per

cent

Cor

rect

Whichstudents seemedtobelearningmoreaftertiering?

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StrugglingStudentPerspectives

Percent of 8th Graders in Low Track Class

0%5%

10%15%20%25%

2000-2005Average

2006-2007 2007-2008 2008-2009 2009-2010

YearTiering begins in 8thgrade

Tiering begins in 6th &

7th gr ade

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2015-2016Grades7to8ISAResults

75%ofemergingstudentsmoved toapproaching

50%ofapproachingstudentsmoved toproficient(notasinglestudentwentdown)

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Advancedstudentcharacteristics,needs,andrecommendations- TextRenderingProtocol

• Individually readandhighlightonesentence,onephrase,andoneword

• EachpersonsharestheirsentenceEachpersonsharestheirphrase(recordthese)Eachpersonsharestheirword(recordthese)

• Then,discuss wordsorideasthatemergedingroupsof3or4

• Finally, shareandnoteanynewinsights you’vegainedthroughthisreadingandsharingprocess

Howmighttieredinstruction impactthelearningofadvancedmathlearners?

Whatdoyouthinkadvancedstudentslike?

Whatmighttheydislike?

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AdvancedStudentPerspectives

Ifyouwereastudentinthisclass,whatthoughtswouldbegoingthroughyourmind?Whatquestionswouldyouwanttoask?

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GradingandReporting

“Measureableoutcomesmaybethe leastsignificantresultsoflearning.”

LindaMcNeil, RiceUniversity

“Reportcardformatsmustberesponsivetoourstudent’sexperience,andtheymustreflectthedifferentiatedpracticesprovided.”

LindaStarr,EducationWorld,1998

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Studentswhoareofteninflowcannotsustaintheireffortsinmostactivitiesthroughintrinsicmotivationalonebutalsoneedextrinsicrewardsandsocialrecognitionfortheiraccomplishments.

BecomingAdult– Csikszentmihalyi andSchneider

A

• Alllearninggoalsaremetwithinthetopicofstudy.• Accurate, clear, organized,andattentivetodetailatall times.• Sophisticatedunderstandingshownthroughcommunicationofhigherorder

thinking.

B

• Mostofthelearning goalsaremetwithinthe topicofstudy.• Accurate,clear, organized,andattentivetodetailmostofthetime.• Considerableunderstandingshownthroughcommunicationofhigherorder

thinking.

C

• Some learning goalsaremetwithinthetopicofstudy.• Accurate,clear, organized,andattentivetodetail someofthetime.• Someunderstandingshownthroughthecommunicationofhigherorder

thinking.

D

• Fewornoneofthelearninggoalsaremetwithinthetopicofstudy.• Rarelyaccurate,organized,andattentive todetail.• Limitedunderstandingshownthroughthecommunicationofhigherorder

thinking.

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5Excellent understanding of the required concepts, knowledge and skills. Able to apply them consistently and accurately in familiar and unfamiliar situations

4Good understanding of the required concepts, knowledge and skills. Able to apply them consistently and accurately in familiar situations

3 Fair understanding of the required concepts, knowledge and skills. Able to apply them in a reasonable number of situations

2 Limited understanding of the required concepts, knowledge and skills. Able to apply them with some assistance

1 Minimal understanding of the required concepts, knowledge and skills. Only able to apply these with high levels of assistance

GradingExamplewith aTraditional Reporting System

IndividualAssessmentGrades

Accuracy Green Blue Black

90-100% 3.5 4 4.5

80-89% 2.5 3 3.5

70-79% 1.5 2 2.5

60-69% .5 1 1.5

Lessthan60% 0 0 0

FinalMarkingPeriodGrades

WeightedAverage

FinalLetterGrade

4.3 A+

4 A

3.7 A-

3.3 B+

3 B

2.7 B-

2.3 C+

2 C

1.7 C-

1.3 D+

1 D

.7 D-

First,weassignweightedgrades toindividualassessments

Then,wecomputeanaverageoftheweightedgrades

Grade Intended Meaning

A Wow!

B Got it

C Getting there

D Struggling

BigPicture:We’dlikegradestoconvey…

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TomlinsononGradingandReporting

Among4approaches,Tomlinsonfavorsthefollowinginadifferentiatedclass:

Combinetraditionallettergradesonreportcardswithanadditionalpieceofinformation.Astillmeansexcellentperformance,Bmeansgoodperformance,andsoon.Buteachlettergradealsocarriesanumeric“superscript”:a1meansworkingabovegradelevel,a2meansworkingatgradelevel,anda3meansworkingbelowgradelevel.Thus,studentswhoachieveanA3 areclearlyworkinghardandprogressingwell,eventhough theirwork isnotyetatgradelevelnorms.

(Tomlinson, 93)

twodimensionalreportcard

9/16/2016 jispsa.jisedu.or.id/teachers/studentpages/report_card_hist_adm.html?studid=39385&schoolid=15&yearid=25&terms=2

http://jispsa.jisedu.or.id/teachers/studentpages/report_card_hist_adm.html?studid=39385&schoolid=15&yearid=25&terms=2 2/2

Science 8 (D'AMBROSIO, Allison) Q3 Q4 S2 Comment

Learning Goals Hyukjin is a friendly and cooperative student. Hyukjin has shown improvemntin his science skills and understanding of concepts this term. He is a quietstudent who could benefit from contributing more to class discussions andwork. Hyukjin should use group work to further extend his understanding.

Knowledge & Understanding B

Investigating & Communicating B

Thinking Critically & Problem Solving C

Theater Workshop 2 gr. 8 (WESTBERG,

Steven Michael)

Q3 Q4 S2 Comment

Learning Goals Creative Arts classes are experiential and performance based in nature.Assessment of the learning goals are ongoing throughout the class.Teachers give students direct feedback that is timely, ongoing and specific.This provides students with opportunities for reflection and improvement.

Understands how to create theatre AP

Demonstrates the principles of performance AP

Applies critical and creative thinking in a dramaticcontext

AP

Alg Geom I (SUAREZ, David Jorge) Q3 Q4 S2 Comment

Learning Goals Hyukjin has had a very successful second semester in AG1. He isextremely well prepared for success in 9th grade. Besides demonstratingstrong math understanding, Hyukjin was a pleasure to have in class, apositive contributor to our classroom learning community. I’m proud ofHyukjin for working hard to grow both academically and personally this year.I feel privileged to have been his teacher.

Note about math grades:STD 'Green Level' Problem solving tasks are foundational and appropriatefor the current grade level.ADV 'Blue Level' Problem solving tasks are advanced and complex.HADV 'Black Level' Problem solving tasks are extremely advanced andhighly complex.

Understanding and applying quadratic functions BHADV

Spatial reasoning and geometric relationships AADV

Solving and Applying Linear Systems ofEquations

BSTD

Understanding and applying exponentialfunctions

BHADV

Adv Strings 8 (SIMO, Gonzalo) Q3 Q4 S2 Comment

Learning Goals Creative Arts classes are experiential and performance based in nature.Assessment of the learning goals are ongoing throughout the class.Teachers give students direct feedback that is timely, ongoing and specific.This provides students with opportunities for reflection and improvement.

Technique P

Ensemble Skills P

Musical Concepts P

Social Studies 8 (ODDEN, Nevin) Q3 Q4 S2 Comment

Learning Goals Hyukjin met all conflict & sustainability unit goals in social studies. He did afine job with her expository paragraph writing. He can effectively organize 3main points, use transition words, and finish with an effective closing. Onething that he can work on improving is including more class vocabulary in hisexpository writing. This will show a better understanding of the class topic.Hyukjin was successful with the investigation and writing of his debate andESP film.

Knowledge and Understanding B

Investigating B

Reflecting and Deciding A

Communicating B

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NurturingProfessionalGrowthinOurselvesandOurTeams

“Allofusmustcrossthelinebetweenignoranceandinsightmanytimesbeforewetrulyunderstand.Notonlymustwecrossthatlinemanytimes,but…nobodyelsecancrossitforus,wemustcrossitbyourselves.Beingshovedordraggedacrossdoesnogood.”

JohnHolt– HowChildrenLearn

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Encouragementfromeducators

Whatwouldyousaytoteacherswhowerecontemplatingwhetherornottouseatieredapproach?

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Completethesethoughts inwriting:

• Thisafternoon, Ilearned:• Thisconnects/addstowhatIknowabout:• WhatIlearnedcanhelpmelaterwhen:• Aquestiononmymindis:

That’sallfolks.

StayintouchJ

[email protected]

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Differentiation in Mathematics - WordPress.com · and teaching mathematics. 2. I love to snow ski. 3. I once went 30 days without using a proper toilet. Cooperative Problem Solving