a journey toward mathematical understanding, focus, and coherence

8/12/2019 A Journey Toward Mathematical Understanding, Focus, and Coherence

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TEA CHING W ITH THE

COREA Jo urn ey Tow ard

Mathem atical Und erstan din g ,

Focu s , and Coh erenc e

Dr. DeAnn HuinkerUniversity of [email protected]

Wisconsin Mathematics CouncilMathematical Proficiency for Every Student Conference

Wausau, WI 11 November 2011


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Jo u rney to the Core


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Understanding

Focus

Coherence


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Shared,the

same foreveryone

Essential,fundamental

knowledgeand skillsnecessary for

studentsuccess

Adoptedandmaintainedby States;

not afederalpolicy

Whatstudents are

expected tolearn in acontent area,benchmarksof learning

Common Core StateStandards

(CCSS)


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+

Make sense of problems

Reason quantitatively

Viable arguments & critique

Model with mathematics

Strategic use of tools

Attend to precision

Look for and use structure

Look for regularity in reasoning

K-8 Domains

HS Conceptual Categories

Standards forMathematical Practice

Standards forMathematics Content


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Goals

Understand Focus and Coherence

Examine specific Ways of Knowing the

mathematics along standards progressions Embrace Shifts

content topics

curriculum & assessmentinstructional approaches


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Great

Moderate

Strong

Magnitude

Major

SmallMinor

Not Felt

How much of a shift is theMath Common Core for Your

District/School

Curriculum

Teaching

Students


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But hope and change have arrived! Like the longawaited cavalry, the new Common Core StateStandards for Mathematics (CCSS) presents us at least those of us in the 44 states+ that have nowadopted them (representing over 80% of the nations

students) a once in a lifetime opportunity to rescueourselves and our students from the myriadcurriculum problems weve faced for years.

---Steve Leinwand, Principal Research Analyst American Institutes for Research in Washington, D.C


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Mathematics content Teaching of mathematics

Student learning of mathematics

Digging in

Consider the discoveries we canunearth in the standards in regards to:


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2NBT9 . Explain why addition and subtractionstrategies work, using place value and theproperties of operations.

3OA3 . Use multiplication and division within 100to solve word problems in situations involvingequal groups, arrays, and measurementquantities, e.g., by using drawings andequations with a symbol for the unknownnumber to represent the problem.

Reflecting MathTeaching Students


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4NF2 . Compare two fractions with differentnumerators and different denominators,e.g., by creating common denominators ornumerators, or by comparing to a benchmarkfraction such as 1/2. Recognize thatcomparisons are valid only when the two

fractions refer to the same whole. Record theresults of comparisons with symbols >, =, or


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Focusand

Coherence


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The Hunt Institute Video SeriesCommon Core State Standards: A New Foundation for Student Success

www.youtube.com/user/TheHuntInstitute#p

Helping Teachers: Coherence and Focus

Dr. William McCallumProfessor of Mathematics, University of Arizona

Lead Writer, Common Core Standards for Mathematics


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Discuss In what ways do thesefeatures support Focus and Coherence ?

See how ideas at a grade level fit with whatteachers in subsequent grade levels or previousgrade levels are doing.

The standards free up time to do fewer thingsmore deeply.

Another aspect was to try in certain places to givemore detail than teachers were used to seeing instandards.


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Domains and Clustersidentify unifying

themes within andacross grades

Detail in thestandards make

clear and giveguidance on ways

of knowing themathematics

Focus

Critical areas

indicate instructionaltime priorities


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Grade Domains Clusters Standards

K 5 9 221 4 11 21

2 4 10 26

3 5 11 254 5 12 28

5 5 11 26

6 5 10 297 5 9 24

8 5 10 28

9-12 3-6* 5-15* 15-37*

Unifying Themes Details


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Reflects hierarchicalnature and structure of

the discipline.--Progressions

--Ways of Knowing

Reflects how knowledgeis generated within thediscipline (Practices).

Reflects what is knownabout how students

develop mathematicalknowledge.

Reflects learners needto organize and connect

ideas.

Discipline of

mathematics

Research on students

mathematics learning

Coherence


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Progressions for the Common CoreState Standards in Mathematics (draft)

() The Common Core Standards Writing Team

ime.math.arizona.edu/progressions

Comprehensive discussions on theintent of specific standards,development within and acrossgrades, connections across

domains, and suggestedinstructional approaches.


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Ways of Knowing

the mathematics


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The Hunt Institute Video SeriesCommon Core State Standards: A New Foundation for Student Success


Operations and Algebraic Thinking

Dr. Jason ZimbaProfessor of Physics and Mathematics

Bennington College, VermontLead Writer, Common Core Standards for Mathematics


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Operations and AlgebraicThinking (OA)

Number andOperations

Fractions

(NF)

Number and Operations inBase Ten (NBT)

K 1 2 3 4 5

Algebra

High School

Expressionsand Equations

(EE)

NumberSystem (NS )

6 7 8


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Discuss insights and implications

The number strand has often been a single strandin elementary school, but in CCSS it is actuallythree domains.

Addition, subtraction, multiplication, and divisionhave meanings, mathematical properties, and usesthat transcend the particular sorts of objects thatone is operating on, whether those be multi-digitnumbers or fractions or variables or variablesexpressions.


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Properties of theOperations

ContextualSituations

Meanings of the

Operations

The foundation for algebra!


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Meanings of the

Operations



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72 29 = ?

Mental MathSolve in your head.No pencil or paper!

24 x 25 = ?


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72 29 = ? 24 x 25 = ?

Turn and share your reasoning.

Discuss how you used: Composing and decomposing Place value in base ten Properties of the operations


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24 x 25 = ?

I thought24 x 100 = 2400,

and 2400 4 = 600.

I thought 25 x 25 = 625 and then Isubtracted 25. 625 25 = 600.

I figured that there are4 twenty-fives in 100,and there are 6 fours

in 24, so 100 x 6 =600.


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24 x 25 = ?

25 x 4 = 100,6 x 100 = 600,600 + 100 = 700.

Well, 10 x 25 = 250,2(10 x 25) = 500,

500 x 4 = 2000.

I would try tomultiply in my head,but I can't do that.


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The properties of operations.

Associative property of addition (a + b) + c = a + (b + c)Commutative property of addition a + b = b + a

Additive identity property of 0 a + 0 = 0 + a = a

Existence of additive inverses For every a there exists a so

that a + ( a) = ( a) + a = 0 Associative property of multiplication (a b) c = a (b c)

Commutative property ofmultiplication

a b = b a

Multiplicative identity property of 1 a 1 = 1 a = a

Existence of multiplicative inverses For every a 0 there exists1/a so that a 1/a = 1/a a = 1

Distributive property of multiplicationover addition

a (b + c) = a b + a c


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And in the domain of Operations and AlgebraicThinking, it is those meanings, properties, and useswhich are the focus; and it is those meanings,properties, and uses that will remain when studentsbegin doing algebra in middle grades [and beyond].

--Jason Zimba


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In Grades K-8, how many standardsreference properties of the operations ?

28standards

Grade 1: OA, NBTGrade 2: NBT

Grade 3: OA, NBT

Grade 4: NBT, NF

Grade 5: NBT

Grade 6: NS, EE

Grade 7: NS, EE

Grade 8: NS

12% of K-8 standards


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Using properties of operations1OA3 . Apply properties of operations as strategies toadd and subtract .3OA5 . Apply properties of operations as strategies tomultiply and divide .

4NBT5 . Multiply two two-digit numbers using strategiesbased on place value and the properties of operations.

5NBT6 . Find whole-number quotients and remainders with using strategies based on place value, properties of

operations . 5NBT7 . Add, subtract, multiply, and divide decimals tohundredths, using concrete models or drawings andstrategies based on place value, properties of operations.


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6EE3 . Apply the properties of operations togenerate equivalent expressions .

7NS2c : Apply properties of operations asstrategies to multiply and divide rational numbers .

7EE1 . Apply properties of operations asstrategies to add, subtract, factor, and expandlinear expressions with rational coefficients.

and into high school


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Develop and

use strategiesbased on propertiesof the operations


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CCSS Glossary

Computation strategy

Purposeful manipulations that may be chosen forspecific problems, may not have a fixed order,and may be aimed at converting one probleminto another.

Computation algorithm

A set of predefined steps applicable to a class ofproblems that gives the correct result in everycase when the steps are carried out correctly.

I G d K 8 h d d


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In Grades K-8, how many standardsreference using strategies ?

26standards

Grade K: CC

Grade 1: OA, NBT

Grade 2: OA, NBTGrade 3: OA, NBT

Grade 4: NBT, NF

Grade 5: NBT

Grade 7: NS, EE11% of K-8 standards


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Standard 1OA6: Basic Facts

Add and subtract within 20, demonstrating fluency foraddition and subtraction within 10.

Use strategies such as counting on; making ten (e.g.,

8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing anumber leading to a ten (e.g., 13 4 = 13 3 1 = 10

1 = 9); using the relationship between addition andsubtraction (e.g., knowing that 8 + 4 = 12, one knows12 8 = 4); and creating equivalent but easier orknown sums (e.g., adding 6 + 7 by creating the knownequivalent 6 + 6 + 1 = 12 + 1 = 13).


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Standard 3OA5: Basic Facts

Apply properties of operations as strategies tomultiply and divide .Examples:

If 6 4 = 24 is known, then 4 6 = 24 is also known.(Commutative property of multiplication.)

3 5 2 can be found by 3 5 = 15, then 15 2 = 30, orby 5 2 = 10, then 3 10 = 30.(Associative property of multiplication.)

Knowing that 8 5 = 40 and 8 2 = 16, one can find 8 7as 8 (5 + 2) = (8 5) + (8 2) = 40 + 16 = 56.(Distributive property.)

I G d K 8 h t d d


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In Grades K-8, how many standardsreference using algorithms ?

5standards

Grade 3: NBT2

Grade 4: NBT4

Grade 5: NBT5

Grade 6: NS2, NS3

2% of K-8 standards


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Algorithms

Grade 3 use strategies and algorithms to add andsubtract within 1000. (Footnote: A range of algorithmsmay be used.) (3NBT2)

Grade 4 use the standard algorithm to add andsubtract multi-digit whole numbers. (4NBT4)

Grade 5 use the standard algorithm to multiply multi -digit whole numbers. (5NBT4)

Grade 6 use the standard algorithm to divide multi -digit numbers and to divide multi-digit decimals.(6NS2, 6NS3)


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Meanings of the

Operations


I G d K 8 h d d f


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In Grades K-8, how many standards referencereal-world contexts or word problems ?

54standards

Grade K: OAGrade 1: OA

Grade 2: OA, MD

Grade 3: OA, MD

Grade 4: OA, NF, MD

Grade 5: NF, MD, G

Grade 6: RP, EE, NS, G

Grade 7: RP, EE, NS, G

Grade 8: EE, G24% of K-8 standards


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Talk it over.

Considering the emphasis onstrategies and algorithms,as well as on properties and contexts,

speculate on some messages thestandards are giving us.


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Develop and use strategiesfor learning basic facts before anyexpectation of knowing facts from memory.

Develop and use strategies to add, subtract,multiply, and divide multi-digit whole

numbers, fractions, decimals. before useof standard algorithms.

Strategies first!


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Great

Moderate

Major

SmallMinor

Not Felt

Strong

Shifts in

Classroom Practice

h f


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Shifts . . . Content

Less data analysis and probability in K-5 More statistics in 6-8 and lots more in HS Much more emphasis on statistical variability

Less algebraic patterns in K-5

Much more algebraic thinking in K-5 More algebra in 7-8 and functions in 8 th

More geometry in K-HS Much more transformational geometry in HS.

More focus on Ratio and Proportion beginning in 6th Percents in 6-7, not in K-5

Shif C i l & A


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Shifts Curriculum & Assessment

HS standards as

conceptual categories

not courses .

supports either integrated or traditional approachor new models that synthesize both approaches.

Real-world applications, contexts, and problem solving Strong emphasis on contexts and word problems from K-HS

Use of measurement contexts across domains,especially linear and liquid contexts

Multi-step Word Problems beginning in Grade 2

Mathematical modeling interwoven throughout HS

Shif T hi


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Shifts . . . Teaching

Using a unit fraction approach Understand and use unit fraction reasoning and

language and expect it of our students

Increased emphasis on specific models Number line model Area model

Strategies and sense-making before algorithms Strategies based on properties of the operations Algorithms culminate years of prior work


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Jot down two discoveriesthat you want to discuss

with colleagues.


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And so in closing

Focus : Unifying themes


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Focus : Unifying themesand guidance on ways ofknowing the mathematics.

Coherence : Progressionsbased on mathematics

and student learning.

Understanding : Deep, genuineunderstanding of mathematicsand ability to use that knowledgein real-world situations.


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So I really hope these standards will helpteachers be more creative in the classroom,

engender the mathematical practices,

and free up time to really focus onteaching mathematics.

--Bill McCallum


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Please keep digging, there are morediscoveries in the Core to unearth andwe know that the work we are all doing

is important for Wisconsin students, fortheir learning and understanding ofmathematics, and for their futures.


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Thank you!

Understanding

Focus

Coherence


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Resources


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CCSSM Resources

www.dpi.wi.gov/standards/ccss.htmlwww.mmp.uwm.edu

Quick link: CCSS Resources

www.tinyurl.com/CCSSresourcescommoncoretools.wordpress.com

ime.math.arizona.edu/progressions


www.corestandards.org

Video Series: William McCallum and Jason Zimba


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Video Series: William McCallum and Jason Zimbalead writers of the CCSSM (The Hunt Institute)

The Mathematics Standards: How They Were Developed and Who Was Involved

The Mathematics Standards: Key Changes in Their EvidenceThe Importance of Coherence in MathematicsThe Importance of Focus in MathematicsThe Importance of Mathematical PracticesMathematical Practices, Focus and Coherence in the ClassroomWhole Numbers to Fractions in Grades 3-6Operations and Algebraic ThinkingThe Importance of Mathematics ProgressionsThe Importance of Mathematics Progressions from the Student PerspectiveGathering Momentum for AlgebraMathematics Fluency: A Balanced Approach

Ratio and Proportion in Grades 6-8Shifts in Math Practice: The Balance Between Skills and UnderstandingThe Mathematics Standards and the Shifts They RequireHelping Teachers: Coherence and FocusHigh School Math Courses


R f


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Reference

Watanabe, T. (2007). Initial treatment of fractions inJapanese textbooks. Focus on Learning Problems inMathematics, 29 (2), 41-60.


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The End

a journey toward mathematical understanding, focus, and coherence

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