# Andy Clark [email protected][email protected]

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2012NCTMRegionalConferenceAndy aicThinkingWednesday, October 31, 12

Top Scoring Nationfor 15 years!Wednesday, October 31, 12

67778992Low9390959898% of students whoreached at least thislevelSample problemsTIMSS–4thGradeProblemTrends in International Math and Science StudyWednesday, October 31, 12

Example of 8th Grade TIMSSProblemJoe knows that a pen costs 1 zed %Receivedfullmore than a pencil. His friendcreditbought 2 pens and 3 pencils for17 zeds. How many zeds will Joe Singapore:need to buy 1 pen and 2 pencils?59%UnitedStates37%Wednesday, October 31, 12

Example of 8th Grade TIMSSAdvanced ProblemJoe knows that a pen costs 1 zed more than apencil. His friend bought 2 pens and 3 pencils for17 zeds. How many zeds will Joe need to buy 1 penand 2 pencils?Pencil 3 zedsPen115Wednesday, October 31, 121 pen and 2 pencils 4 6 10 4 zeds1 15} 17

Example of 8th Grade TIMSSProblemThe length of a rectangle is 6 cm, and itsperimeter is 16 cm. What is the area of therectangle in square centimeters?AnswerMeasurement: Singapore, 86%; Korea, 66%; Japan,65%;Canada, 41%; France, 36%;U.S., 22%Wednesday, October 31, 12

hina(600)US487 Singapore Math, A World LeaderPedagogy based upon 20 yearsof research and successWednesday, October 31, 12

Teach to Mastery“ mathematics curriculum in the United Statesmust address the problem of a curriculum thatis a “mile wide and an inch deep.”Common Core State StandardsWhat can we learn?Wednesday, October 31, 12

Singapore: The Model for CCSSWhat can we learn?Wednesday, October 31, 12

Mathematics Standardsfor Mathematical Practice Able:– tomakesenseofproblemsandpersevere– toreasonabstractlyandquanVtaVvely– toconstructviablearguments– tomodelwithmathemaVcs– tousetoolsappropriately– toaXendtoprecision– tolooksforandmakeuseofstructure– tolookforregularityWednesday, October 31, 12

PreparingforCommonCoreWednesday, October 31, 12

PreparingforCommonCoreWednesday, October 31, 12

PARCCAssessmentWednesday, October 31, 12

PreparingforCommonCoreWednesday, October 31, 12

PreparingforCCSSAssessmentsWednesday, October 31, 12

Wednesday, October 31, 12

CommonCoreInstrucVonalemphases Focus:PrioriVzetopicsandfocusonthem ldsonthepreviousones derstandhow,why,andwhentheywork withproceduresandfactsWednesday, October 31, 12

What can we learn from Singapore? Teach math as a vehicle for teaching thinking Use an effective pedagogy – concrete to pictorial toabstract - that develops understanding and fluency Recognize the importance of visualization Teach fewer topics to mastery and greaterproficiency in each gradeWednesday, October 31, 12

Teach math as a vehicle forteaching thinkingMake sense of problems and persevere insolving them.Mathematically proficient students start by explaining tothemselves the meaning of a problem and looking forentry points to its solution. They analyze givens,constraints, relationships, and goals.Wednesday, October 31, 12

Mathematics Frameworkcreated by the SingaporeMinistry of EducationWednesday, October 31, 12

e sense of problems and persevereReason abstractly and quantitativelyConstruct viable argumentsCCSSWednesday, October 31, 12

Math As ThinkingWednesday, October 31, 12

MathSyllabusWednesday, October 31, 12

UseaneﬀecHveinstrucHonalstrategy:Concrete- ‐Pictorial- ay, October 31, 12

UseaneﬀecHveinstrucHonalstrategy:Concrete- ‐Pictorial- ‐AbstractHow many fourths in 1 whole? In 2How many fourths in 6 wholes? In 20?If you know one whole, how can you find the numberin any number of wholes#wholesx4 numberoffourthsWednesday, October 31, 12

Concrete- ‐Pictorial- yin1,howcanweﬁndhowmany2/3in5wholes?5x3/2 5 2/37x3/2 7 2/3Wednesday, October 31, 12

Mine is notto reasonwhy Wednesday, October 31, 12

Concrete- ‐Pictorial- ‐AbstractPlank1/10Wednesday, October 31, 12Tententhsin5/5,so4/5x10in4/5ofayard

Concrete- ‐Pictorial- ‐AbstractBook A Pg. 70Wednesday, October 31, 12

Concrete- ‐Pictorial- ‐Abstract1/2 1/41/2Wednesday, October 31, 124fourthsin1,so1/2x4in1/2ofawhole

Concrete- ‐Pictorial- ‐Abstract1/4 1/21/2Wednesday, October 31, 12Twohalvesin1,so1/4Vmes2 1/2inaquarter

Why visualization? Lilian’spresentageis2/3VmesMay’sage. a)FindtheraVoofMaysagetoLilian’sage. ge? y5Wednesday, October 31, 12 25 Lilian is 10 and May is 15

nesday, October 31, 12

Multiplicative ll?32AhmadJustin25Wednesday, October 31, 124units 321unit 8(32 4 8)Justinequals8

GRADE 5: Try This¡ Ben took a test with 3 sections A,B,and C. Ben spent 1/5 of his time onSection A and 1/3 of the remainingtime on Section B. He spent 48minutes on section C. How muchtime did Ben take to complete thewhole test.Wednesday, October 31, 12

GRADE 5: Fifth GradeFraction Model Ben took a test with 3 sections A,B, and C. Ben spent 1/5 ofhis time on Section A and 1/3 of the remaining time onSection B. He spent 48 minutes on section C. How muchtime did Ben take to complete the whole test.A1/5BC1/31/3241/324482472 4/5, so 72 4 1/51/5 18, 5/5 90 He spent 90 minutes on testWednesday, October 31, 12

GRADE 5: Fifth Grade Ben took a test with 3 sections A,B, and C. Ben spent 1/5 ofhis time on Section A and 1/3 of the remaining time onSection B. He spent 48 minutes on section C. How muchtime did Ben take to complete the whole test.A1/51- 1/5 4/5 12/151/3 of 12/15 4/1512/15 - 4/15 8/15Wednesday, October 31, 12B1/3 rest? what fraction?488/15 48 so1/15 6 minutes15/15 90 minutes

Ricardo spends 8/9 hour reading the newspaper. He spends1/4 of the time reading the world news and splits the remainingtime equally between the sports and the comics.How much time does he spend reading the Wednesday, October 31, 12

Grade 6: From fractions to ratio15Elinor is 15Mother will be 5515Wednesday, October 31, 12

Grade 6: ratio problems enPat75Karen 60Patricia had 150Wednesday, October 31, 1275 5 15 15

Grade 6: From ratios to rates“Solve unit rates problems includingthose involving unit pricing andconstant speed.”Wednesday, October 31, 12

Grade 6: From ratios to farcantheracingcartravelin3hours?1751h3hWednesday, October 31, 12unitrate3x175 525km

Grade 6: From ratios to ratesormethod2:Distance RatexTimeDistance 175km/hourx3hours 525kmWednesday, October 31, 12

Grade 6: From ratios to ratesWednesday, October 31, 12

Grade 6: Variable sday, October 31, 12

Grade 7: Variable expressionsWednesday, October 31, 12

Grade 7: Variable expressionsWednesday, October 31, 12

Grade 7: Solving woposiVvenumbersormethod2Large#x- ‐1/3x 483/3x- ‐1/3x 48Small#2/3x 4848x 72,otheris24Wednesday, October 31, 12

Two apples and a mango cost 4.Two apples and three mangos cost 9.Find the cost of a mango. 4? 9Wednesday, October 31, 12

x y 8x 2y 108XY10XY Y(x 2 y) - (x y) 10 -8 2Wednesday, October 31, 12Solve algebraically,first with elimination,then withsubstitution

TrajectoryAlgebra8thGrade7XX2x 3y 7x 6y 8Y Y Y8X Y Y Y Y Y ract? 16 - 7 9y 1, x 2XXY Y YXXY Y Y Y Y Y Y Y Y Y Y Y16Wednesday, October 31, 12

Grade 8: Systems of equationsThengraphicalsoluVonsforproblemsWhatdoesd 60tmean?Whatdoesd 50t 20mean?Wednesday, October 31, 12

NAEP 2011 8th GradeWednesday, October 31, 12

Teach fewer topics to masteryand greater proficiency in each grade§ Concepts covered in one grade arecovered in later grades, but only at amore advanced level§ Concepts are covered thoroughly whenintroduced and each grade builds on theprevious gradeWednesday, October 31, 12

Teach to mastery and proficiencyWednesday, October 31, 12

Teach to mastery and proficiency- ‐35Wednesday, October 31, 12

Teach to mastery and proficiencyx 2/5x - ‐357/5x - ‐35dividebothsidesby7/5x - ‐25Wednesday, October 31, 12

What can we learn from Singapore? Teach math as a vehicle for teaching thinking Use an effective pedagogy – concrete to pictorial toabstract - that develops understanding and fluency Recognize the importance of visualization Teach fewer topics to mastery and greaterproficiency in each gradeWednesday, October 31, 12

Singapore*MOE*Math*Syllabus Wednesday, October 31, 12. w Grade 6 Concrete Pictorial . ms& and&numberrelationships&that&is& inning& insecondgrade andextendingallthe way&to&algebra& . Grade 6: ratio problems

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