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A Model for Mathematical Argumentation
Making It Work in Your Classroom
NCTM 2017
Teresa Lara-Meloy | Jennifer Knudsen | Harriette Stevens
Bridging.sri.com
Imagesby:VerenaLoewensberg
Session107Bridging.sri.com
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This presentation
• I’ll give you a bit of background on our project
• We’ll explore our model while doing math
• We’ll do a table read of a vignette
• There’ll be time for questions
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CCSS-M: “Construct viable arguments and critique the reasoning of others”
Most powerful mathematics
practice
All students should have access
Aids in conceptual understanding
Joyful
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Bridging is professional development for mathematical argumentation in middle school.
Varietyofschoolanddistrictse>ngs,withculturallyandlinguisAcallydiversestudentpopulaAons
InteracAvedigitalcurriculumunitsANDadaptaAonsofadoptedcurriculum
SummerinsAtutesand
Schoolyear:3hrs/month
alternaAngvirtualandfacetoface
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Results: twice as much classroom argumentation
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Results: gain of 10 pts in student learning
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One situation, four practice standards
Problem Solving
Create a trip with 3 segments that ends at 200 feet.
Modeling The bus travels at 3 different speeds for city, country and highway. Design a route to get home in less than 3 hours.
Precision An object’s time and position are noted at (0,0) (2,3), (3,5) and (10,14). What line best fits this data?
Argumentation Raj says that if a line is steeper than another, then it represents a faster motion. Is this always true?
We focus here
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What math argumentation is / is not
It is a social practice to establish the truth about a mathematical idea. (Thurston, 1998)
Collaborative
Not the use of mathematics to justify a non-mathematical claim:
i.e., which team is going to win the game.
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Argumentation in four parts
Generating cases
Making and finding patterns
Conjecturing making informed
guesses about mathematical truth
Justifying creating a logical chain of statements to support or
disprove a conjecture
Concluding deciding on the
truth of a conjecture
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Argumentation requires new norms
makeboldconjectures
it’sOKtobewrong
findoutthetruth
together
buildoffother
people’sideas
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We use improv games to help students learn norms.
Games come from improvisational theater
Rules structure freedom to participate
Freestyling as improv
Lin-ManuelMirandaattheWhiteHouse
OnTV
Inmovies
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Let’s play: Gift Giving
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• Standfacingyourpartner.• There’sahugeclosetofunlimitedgiWsbehindyou.• Onepersonisthegiver,anotherpersonisthereceiver.
• ThegiveroffersthereceiverawrappedgiWfromthecloset.
• ThereceiveropensthegiWandgratefullydescribes(andnames)thegiW.
• ThegiverrespondswithanexplanaAonofhowandwhythegiWwasselectedandwhyitwouldbeenjoyed.
• Thenswitchroles.
Let’splay:Gi#Giving
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How didwedowiththenorms?
makeboldconjectures
it’sOKtobewrong
findoutthetruth
together
buildoffother
people’sideas
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Let’s do argumentation together
Handoutoronline:h\p://www.geogebra.org/m/3141695
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Generating Cases
Createexamples(numericexpressionsorgeometricshapes)thatyoucanusetolookforandexaminepa\erns.
Norms• Makemul3plecases.• Becrea3ve:Trysimplenumbersorshapes.Trydifficultnumbersorshapes.Try“weird”numbersorshapes.
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Generate cases
Handoutoronline:h\p://www.geogebra.org/m/3141695
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Conjecturing
AconjectureisamathemaAcalstatementthatyouthinkmightbetrue.
…inourcase,allrectangles.
Norms:• UsepaDernstomakestatementsaboutwhatwillalwaysbetrue.
• Makeboldconjecturesaboutwhatmightbetrue.
• Don’tjudgeotherpeople’sconjectures.
DoesthisrelatetootherMPs?
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Your conjectures
All rectangles whose sides are parallel to the axis must have whole number coordinates.
The y-coordinates are the same, and the x-coordinates are the same. The difference in the x-values represent the width, and the difference in the y-values represent the height.
All horizontal lines have the same y-coordinates. All vertical lines have the same x-coordinates.
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Justifying
AgoodjusAficaAonisaconnectedchainofstatementsthatconvincesothersofthetruthorfalsityofastatement;itgoesbeyondapersonalexploraAonofanidea.
Norms:• Lookforreasonswhyaconjectureistrueorfalse.
• Buildoffofotherpeople’sideas.
• Trytoconvinceothersofyourideas,butkeepinmindthatyoucouldbewrong—whichisOK.
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Conjecture:
Handoutoronline:h\p://www.geogebra.org/m/3141695
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Concluding
Concludingmeansdecidingwhetheraconjectureistrueorfalse,basedonajusAficaAon.
Norms• Knowwhentostop.• Retelltheargumentfrombeginningtoend.
• Baseyourconclusionsonwhatissaid,notwhosaidit.
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Let’s examine a classroom example
In groups of 3-4, enact this vignette (i.e., do a table read)
As you do so, pay attention to the teaching moves.
§ A move is the smallest piece of behavior that can be aimed at a purpose.
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Teaching moves for argumentation
• WhatpaDernsdoyousee?DescribethepaDernsinasentence.
elicitconjectures
• Howdoyouknowthat?• Howdoweknowitistrue?• Whatmakesyouthinkso?• Showhowyouknow.• Explainwhythismustbetrue.• What’sthemathema3calreasonit’strue?
askwhy
• whatconjecturing,jus3fyingandconcludingare.explaintostudents
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Includes4-partmodeland
acAviAesfromthispresentaAon
New book for teachers
September 2017