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We are currently preparingstudents for jobs that don’t yetexist . . . using technologies thathaven’t yet been invented . . . inorder to solve problems we don’teven know are problems yet.—The Jobs Revolution Richard Riley 2020fosteringmathpractices

Students need ways into and throughprickly problemsWe must teach them to think likemathematicians! 2020fosteringmathpractices

Standards for Mathematical Practice MP1 Make Sense of Problems and Persevere in Solving Them MP2 Reason Abstractly and Quantitatively MP3 Construct Viable Arguments andCritique the Reasoning of Others MP4 Model with MathematicsMP5 Use Appropriate of Tools StrategicallyMP6 Precision in MathematicsMP7 Look for and Make Use of StructureMP8 Look for and Express Regularity in Repeated Reasoning 2020fosteringmathpractices

Not All Math Math Practices Are Equal!87REPETITION6STRUCTUREPRECISION35CONSTRUCT& CRITIQUETOOLS2QUANTITIES& RELATIONS 2020fosteringmathpractices14MODELPROBLEMSOLVING

P3MP7MP6MP4MP5 Page 3

Avenues of ThinkingWhat do they look like? 2020fosteringmathpractices

Fish TankInstructions:Solve the task onyour own.Discuss yourapproach with apartner. 2020fosteringmathpracticesA 20.5 gallon fish tank is4/5 full. How many moregallons will it take to fillthe tank?

Avenues of eatedReasoningAttend toAsk YourselfActions to Take 2020fosteringmathpracticesPage 179

Three Avenues of Thinking in dReasoning 2020fosteringmathpracticesChapter 3

Quantitative Reasoning Avenue of ThinkingAsk yourself Attend to What can I count or measureQuantitiesandRelationshipsin this problem situation? How do the quantities relate 2020fosteringmathpracticesto each other?How can I represent thisproblem?What does this (variable,number, shaded region, etc.)represent in the problemcontext?

Students Typically Attend to Number7 2020fosteringmathpractices½

When Students Only Attend to Numbers 2020fosteringmathpractices

Students Typically Attend to Number7½Does the number tell mesomething about aquantity or is it describinga relationship? 2020fosteringmathpractices

Quantity or Relationship? Grace has 7 cookies Grace ran ½ mile Amy has 7 more cookies Amy ran ½ as far asthan Grace 2020fosteringmathpracticesGrace

What is a Quantity?A quantity issomething youcan count ormeasure:A quantityanswers one ofthe questions: The number How many? How much?of The amount of. 2020fosteringmathpracticesA quantity has 3parts: Value (7, ½ ,2π, x, etc) Unit (cookies,miles, etc ) Sign ( / -)

What are the Quantities in Gina’s Garden?Gina planted 24 flowers in heryard. Some of them were red andsome of them were purple. Thereare twice as many purple flowersas red flowers.What can Icount in thissituation?The number of 2020fosteringmathpractices

QUANTITIES in Gina’s GardenI can count The total number of flowersThe number of red flowersThe number of purple flowersThe number of flowers that are neither red nor purpleThe number of colors of flowers 2020fosteringmathpractices

Why Develop Quantitative Reasoning?in High School 2020fosteringmathpracticesCommon Core State Standards,HS Number and Quantity Conceptual Category, p58

Entering the Fish Tank via the QuantitativeAvenue of ThinkingPay attention to The amount of water:The tank holdsAlready in the tankAdded to the tank Relationships between theamount of water the tank holdsand the amount already in thetank 2020fosteringmathpracticesA 20.5 gallon fish tank is 4/5 full.How many more gallons will ittake to fill the tank?

Entering the Fish Tank via the QuantitativeAvenue of ThinkingAsk yourself. Is the 20.5 a value for a quantityor does it describe arelationship? What quantity has a value of20.5? Is the 4/5 describing arelationship or a quantity? What two quantities have a 4/5relationship? Are there quantities here thatdon’t have a value given? How can I represent thequantities so I can see therelationship between them? 2020fosteringmathpracticesA 20.5 gallon fish tank is 4/5 full. Howmany more gallons will it take to fill thetank?

Quantitative Reasoning Avenue of Thinking 2020fosteringmathpractices

Quantitative Reasoning Actions Identify quantities explicitly mentioned in the problem statementSurface hidden or implied quantitiesNote relationships between quantitiesAbstract problem situationsUse representations to see quantities and relationshipsRecall and consider referents 2020fosteringmathpractices

20.5is4/5 moreNumbersKey wordsAmt. of H2O to addedAmt. of H2O already in tankQuantitiesTotal amt. of H2O the tank holdsAmt. of H2O in tank is 4/5 ofamt. of H2O tank holds 2020fosteringmathpracticesRelationships

Quantitative Reasoning ShiftsLook beyond thenumbers and key wordsin a problem statement 2020fosteringmathpracticesTo the quantities andrelationships thosenumbers and key wordsdescribe

How Could Quantitative ReasoningSupport Students?Number GrabbingUnderstandingQuantitiesBlindly OperatingWorking withrelationships 2020fosteringmathpractices

Quantitative Reasoning Supports ALLStudents .Especially Students who don’t know where to begin to solve a word problemStudents who struggle with multi-step problemsStudents who benefit from working within contextsStudents who benefit from drawing/using visual representations 2020fosteringmathpractices

Three Avenues of Thinking in dReasoning 2020fosteringmathpracticesChapter 4

Structural Reasoning Avenue of ThinkingAsk yourself Attend to Is this behaving likeOrganization and PropertiesOf Number and Space 2020fosteringmathpracticessomething else I know?How can I use properties touncover structure?How can I change the form tomake it easier to work with?How can I “chunk” this tomake sense of it?How can I connect this tomath I know?

Entering the Fish Tank via theStructural Avenue of ThinkingPay attention to Types of numbers Composition of fractions 2020fosteringmathpracticesA 20.5 gallon fish tank is 4/5 full.How many more gallons will ittake to fill the tank?

Entering the Fish Tank via theStructural Avenue of ThinkingAsk yourself Is there another way I canthink about 4/5 full? How can I change the form of20.5 and 4/5 to make themeasier to work with? 2020fosteringmathpracticesA 20.5 gallon fish tank is 4/5 full.How many more gallons will ittake to fill the tank?

Entering the Fish Tank via theStructural Avenue of Thinking 2020fosteringmathpractices

When Students Don’t Think Structurally 2020fosteringmathpractices

Structural Thinking Actions Chunk complicated objects Connect math ideas and representations Change the form of objects Recall and use properties, rules of operations, and geometricrelationships Shift perspective 2020fosteringmathpractices

Find the total numberof circles quickly “inyour head” (i.e.without counting eachcircle individually) s.org/

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Structural Thinking ShiftsA collection ofunrelatedresults andproceduresto know 2020fosteringmathpracticesA set of interconnectedideas that build on eachother and make sense

Structural Thinking Supports ALLStudents .Especially Students who lose track of their work and/or calculations Students who see the ‘big picture’ Students who benefit from multiple representations 2020fosteringmathpractices

Three Avenues of Thinking in dReasoning 2020fosteringmathpracticesChapter 5

Repeated Reasoning Avenue of ThinkingAttend to Repetition inProcessesCountingCalculatingConstructingAsk yourself Do I keep doing the samething over and over again? What about the process isrepeating? How can I generalize therepetition? 2020fosteringmathpractices Have I included every step?

Repeated Reasoning (MP8)Actions You Take Count in an organized way Draw or build several figures Try several numbers and observe the process Record and track calculations Generalize the repetition Simultaneously maintain oversight of the process whileattending to details Monitor & evaluate reasonableness of intermediate results 2020fosteringmathpractices

Decompressing Repeated Reasoning Pay attention to the process Sense the regularity “Shortcut” the process Connect the process to an “input” value Generalize the process to a rule 2020fosteringmathpractices

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Repetition in Constructing 2020fosteringmathpractices

Repetition in Constructing 2020fosteringmathpractices

Repeated Reasoning (MP8)Attend to Repetition in ProcessesCountingConstructingCalculating 2020fosteringmathpractices

Repeated Reasoning (MP8)Adam has a 20.5 gallon fish tank that is ⅘ full.How many gallons will it take to fill the tank?GUESSAndCHECK 2020fosteringmathpractices

Repeated Reasoning (MP8)Adam has a 20.5 gallonfish tank that is ⅘ full.How many gallons will ittake to fill the tank? 2020fosteringmathpractices

When Students Focus on Number Patternvs. Repetition in Process 2020fosteringmathpractices

WE HAVE TO PRIVILEGE THEPROCESS! 2020fosteringmathpractices

Repeated Reasoning ShiftsPatternsinNumbersandResults 2020fosteringmathpracticesThe counting,calculating andconstructingprocesses thatgenerated thosenumbers and results

How Could Repeated ReasoningSupport Students?‘Magic’ Rules 2020fosteringmathpracticesGeneralizations thatare rooted inconcrete processes

Repeated Reasoning Supports ALLStudents .Especially Students who benefit from multiple modalities Students who struggle to abstract and generalize Students who work in organized and/or systematic ways Students who benefit from seeing how rules are developed 2020fosteringmathpractices

Math Thinking and Reasoning3 Key Messages Not all Math Practices are Equal Start with the Three Avenues of ThinkingQuantitative reasoning (MP2)Structural Thinking (MP7)Repeated Reasoning (MP8)MathThinkingandReasoning Rather than being barriers, the math practices support strugglingmath learners. 2020fosteringmathpractices

How do we develop avenues of thinking ineach and every student? 2020fosteringmathpractices

Instructional Routines: Repeatable & Predictable"Designs forinteractionthat organizeclassroom activities”Teachers– Magdalene Lampert,NCSM 2015Content 2020fosteringmathpracticesStudents

Instructional Routines that Develop MathematicalPractices Capturing Quantities (MP2) Connecting Representations (MP7) Recognizing Repetition (MP8) Three Reads (MP1) Decide and Defend (MP3) Contemplate then Calculate (MP7)www.FosteringMathPractices.com 2020fosteringmathpractices

Aug 03, 2021 · Math Thinking and Reasoning 3 Key Messages Not all Math Practices are Equal Start with the Three Avenues of Thinking Quantitative reasoning (MP2) Structural Thinking (MP7) Repeated Reasoning (MP8) Rather than being barriers, the math practices support struggling math learners. Math