TEACHING WITH THE CORE A Journey Toward Mathematical Understanding, Focus, and Coherence Dr. DeAnn Huinker Milwaukee Mathematics Partnership University

TEACHING WITH THE CORE

A Journey Toward Mathematical Understanding,

Focus, and Coherence

Dr. DeAnn HuinkerMilwaukee Mathematics PartnershipUniversity of [email protected]

Mathematics Teacher Leader Seminar

Milwaukee, Wisconsin

6 December 2011

Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee

Journey to the Core


Progression

Progressio

n

Progressio

n

Understanding

Focus

Coherence


Shared, the

same for everyone

Essential, fundamental knowledge and skills

necessary for student success

Adopted and

maintained by States;

not a federal policy

Benchmarks of what

students are expected to learn in a

content area

Common Core State Standards


We are learning to...

Understand “Focus” and “Coherence”

Consider how the standards detail or specify “Ways of Knowing” mathematics

Embrace “Shifts” content topics curriculum & assessment instructional approaches


Great

Moderate

Strong

Magnitude

Major

SmallMinor

Not Felt

How much of a shift is theMath Common Core for …

District

School

Curriculum

Teaching

Students


A Long Overdue Shifting of the Foundation

For as long as most of us can remember, the K-12 mathematics program in the U.S. has been aptly characterized in many rather uncomplimentary ways: underperforming, incoherent, fragmented, poorly aligned, narrow in focus, skill-based, and, of course, “a mile wide and an inch deep.”

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



But hope and change have arrived! Like the long awaited cavalry, the new Common Core State Standards for Mathematics (CCSS) presents us – at least those of us in the 44 states+ that have now adopted them (representing over 80% of the nation’s students) – a once in a lifetime opportunity to rescue ourselves and our students from the myriad curriculum problems we’ve faced for years.

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

Make no mistake,

for K-12 math in the

United States, this IS a brave

new world.

--Steve Leinwand


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 for Mathematical Practice

Standards for Mathematics Content

Standards

Domains

Clusters



Mathematics content Teaching of mathematics Student learning of mathematics

Digging in…

Begin to unearth some discoveries:


2NBT9. Explain why addition and subtraction strategies work, using place value and the properties of operations.

3OA3. Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Reflecting…


4NF2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Reflecting…


4NF2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Reflecting…


Which is larger?

or34

67

Find a common

numerator!

68

67

Rename

or


Focus and

Coherence


For over a decade, research of mathematics education in high-performing countries have pointed to the conclusion that the math curriculum in the United States must become substantially more focused and coherent in order to improve mathematics achievement in this country.


CCSS “design principles”

Focus Coherence


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 McCallum

Professor of Mathematics, University of Arizona

Lead Writer, Common Core Standards for Mathematics


Features of Focus and Coherence

“Give more detail than teachers were used to seeing in standards.”

Fewer Topics

Progressions

More Detail

Show how ideas fit with what subsequent or previous grade levels.

“Free up time” to do fewer things more deeply.

Discuss


Unifying Themes Details

Domains Clusters Standards


Grade Domains Clusters Standards

K 5 9 22

1 4 11 21

2 4 10 26

3 5 11 25

4 5 12 28

5 5 11 26

6 5 10 29

7 5 9 24

8 5 10 28


Grade Domains Clusters Standards


Conceptual Category

Domains Clusters StandardsAll

StandardsAdvanced

Number & Quantity

Algebra

Functions

Geometry

Statistics & Probability

Modeling



Conceptual Category

Domains Clusters StandardsAll

StandardsAdvanced

Number & Quantity 4 9 9 18

Algebra 4 11 23 4

Functions 4 10 22 6

Geometry 6 15 37 6

Statistics & Probability 4 9 22 9

Modeling * * * *



Domains and Clusters identify unifying

themes within and across grades

Domains and Clusters identify unifying

themes within and across grades

Detail in the standards make clear and give

guidance on “ways of knowing” the mathematics

Detail in the standards make clear and give

guidance on “ways of knowing” the mathematics

Focus

Critical areas indicate instructional “time”

priorities


Reflects hierarchical nature and structure of

the discipline.

--Progressions

--Ways of Knowing

Reflects how knowledge is generated within the discipline (“Practices”).

Reflects how students develop mathematical

knowledge.

Reflects learners’ need to organize and connect

ideas.

Discipline of mathematics

Research on students’ mathematics learning

Coherence


Progressions for the Common CoreState Standards in Mathematics (draft)

(©) The Common Core Standards Writing Team

ime.math.arizona.edu/progressions

Required

Professional Reading

& Discussion

Comprehensive discussions on the “intent” of specific standards, development within and across grades, connections across domains, and suggested instructional approaches.



“Ways of Knowing”the mathematics


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


Operations and Algebraic Thinking Dr. Jason Zimba

Professor of Physics and Mathematics

Bennington College, Vermont

Lead Writer, Common Core Standards for Mathematics


Operations and Algebraic Thinking (OA)

Number and Operations –

Fractions (NF)

Number and Operations in Base Ten (NBT)

K 1 2 3 4 5

Algebra

High School

Expressions and Equations

(EE)

Number System (NS)

6 7 8


Operations & Algebraic Thinking (OA)

The number strand “has often been a single strand in elementary school, but in CCSS it is actually three domains.”

‘“Addition, subtraction, multiplication, & division have meanings, mathematical properties, and uses that transcend the particular sorts of objects that one is operating on, whether those be multi-digit numbers or fractions or variables or variables expressions.”


Properties of the Operations

ContextualSituations

Meanings of the Operations

The foundation for algebra!


72 – 29 = ?

Mental Math Solve in your head. No pencil or paper!

24 x 25 = ?

Nor calculators, cell phones computers or iPads or ....


72 – 29 = ? 24 x 25 = ?

Turn and share your reasoning.

Discuss how you used:

“Composing and decomposing”

a.k.a Properties of the operations


24 x 25 = ?

I thought

24 x 100 = 2400,

and 2400 ÷ 4 = 600.

I thought 25 x 25 = 625 and then I subtracted 25. 625 – 25 = 600.

I figured that there are 4 twenty-fives in 100,

and there are 6 fours in 24, so 100 x 6 = 600.


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 to multiply in my head, but I can't do that.”


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 of multiplication

a × b = b × a

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

Existence of multiplicative inverses For every a ≠ 0 there exists 1/a so that a × 1/a = 1/a × a = 1

Distributive property of multiplication over addition

a × (b + c) = a × b + a × c

Not just learning them,

but learning to use them.


And in the domain of Operations and Algebraic Thinking, it is those meanings, properties, and uses which are the focus; and it is those meanings, properties, and uses that will remain when students begin doing algebra in middle grades [and beyond].

--Jason Zimba


In Grades K-8, how many standards reference “properties of the operations”?

28 standards

Grade 1: OA, NBT

Grade 2: NBT

Grade 3: OA, NBT

Grade 4: NBT, NF

Grade 5: NBT

Grade 6: NS, EE

Grade 7: NS, EE

Grade 8: NS12% of K-8 standards


Using properties of operations 1OA3. Apply properties of operations as strategies to

add and subtract.

3OA5. Apply properties of operations as strategies to multiply and divide.

4NBT5. Multiply two two-digit numbers using strategies based 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 to hundredths, using concrete models or drawings and strategies based on place value, properties of operations….


6EE3. Apply the properties of operations to generate equivalent expressions.

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

7EE1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

and into high school……


Develop and use strategies

based on properties of the operations


CCSS Glossary

Computation strategy

Purposeful manipulations that may be chosen for specific problems, may not have a fixed order, and may be aimed at converting one problem into another.

Computation algorithm

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


In Grades K-8, how many standards reference using “strategies”?

26 standards

Grade K: CC

Grade 1: OA, NBT

Grade 2: OA, NBT

Grade 3: OA, NBT

Grade 4: NBT, NF

Grade 5: NBT

Grade 7: NS, EE

11% of K-8 standards


Standard 1OA6: “Basic Facts”

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.

Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).


Standard 3OA5: Basic Facts

Apply properties of operations as strategies to multiply 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, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.)

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


In Grades K-8, how many standards reference using “algorithms”?

5 standards

Grade 3: NBT2

Grade 4: NBT4

Grade 5: NBT5

Grade 6: NS2, NS3

2% of K-8 standards


Algorithms

Grade 3 “use strategies and algorithms” to add and subtract within 1000. (Footnote: A range of algorithms may be used.) (3NBT2)

Grade 4 “use the standard algorithm” to add and subtract 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)


Strategies first!Develop and use strategies

for learning basic facts before any expectation of

knowing facts from memory.


Strategies first!Develop and use strategies to calculate with whole numbers, fractions, decimals …. before use of standard algorithms.



ContextualSituations

Meanings of the Operations

The foundation for algebra!


In Grades K-8, how many standards reference “real-world contexts” or “word problems”?

54 standards

Grade K: OA

Grade 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, G

24% of K-8 standards


Lots of real-world contexts!

Proficient students make sense of quantities and their relationships

in problem situations. (MP2)

decontexualize & contextualize



Algorithms Real-world Contexts

Strategies

Walk Away

Message


Great

Moderate

Major

SmallMinor

Not Felt

Strong

Shifts in Classroom Practice


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 8th

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


Shifts… Curriculum & Assessment

HS standards as “conceptual categories” not courses ….

supports either integrated or traditional approach or 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


Shifts . . . Teaching

Using a “unit fraction” approach Understand and use unit fraction reasoning and language and expect it of our students

Increased emphasis on visual 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

Discrete to

continuous quantities


And so in closing …


Focus: Unifying themes and guidance on “ways of knowing” the mathematics.

Coherence: Progressions across grades based on discipline of mathematics and on student learning.

Understanding: Deep, genuine understanding of mathematics and ability to use that knowledge in real-world situations.


I really hope these standards will help teachers be more creative in the classroom,

engender the mathematical practices, and free up time to really focus on

teaching mathematics.

--Bill McCallum


Please keep digging, there are many more discoveries in the Core to unearth and we know that the work we are all doing is important for Milwaukee students, for their learning and understanding of mathematics, and for their futures.


Thank you!

Progression

Progressio

n

Progressio

n

Understanding

Focus

Coherence


Resources


CCSSM Resources

www.dpi.wi.gov/standards/ccss.html

www.mmp.uwm.edu Quick link: CCSS Resources

www.tinyurl.com/CCSSresources

commoncoretools.wordpress.com

ime.math.arizona.edu/progressions


www.corestandards.org


Video Series: William McCallum and Jason Zimba lead 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 Evidence The Importance of Coherence in Mathematics The Importance of Focus in Mathematics The Importance of Mathematical Practices Mathematical Practices, Focus and Coherence in the Classroom Whole Numbers to Fractions in Grades 3-6 Operations and Algebraic Thinking The Importance of Mathematics Progressions The Importance of Mathematics Progressions from the Student Perspective Gathering Momentum for Algebra Mathematics Fluency: A Balanced Approach Ratio and Proportion in Grades 6-8 Shifts in Math Practice: The Balance Between Skills and Understanding The Mathematics Standards and the Shifts They Require Helping Teachers: Coherence and Focus High School Math Courses


Documents

TEACHING WITH THE CORE A Journey Toward Mathematical Understanding, Focus, and Coherence Dr. DeAnn Huinker Milwaukee Mathematics Partnership University