teaching with the core a journey toward mathematical understanding, focus, and coherence dr. deann...
TRANSCRIPT
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
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
Progression
Progressio
n
Progressio
n
Understanding
Focus
Coherence
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
Great
Moderate
Strong
Magnitude
Major
SmallMinor
Not Felt
How much of a shift is theMath Common Core for …
District
School
Curriculum
Teaching
Students
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
Mathematics content Teaching of mathematics Student learning of mathematics
Digging in…
Begin to unearth some discoveries:
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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…
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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…
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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…
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
Which is larger?
or34
67
Find a common
numerator!
68
67
Rename
or
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
Focus and
Coherence
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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.
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
CCSS “design principles”
Focus Coherence
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
Unifying Themes Details
Domains Clusters Standards
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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
Unifying Themes Details
Grade Domains Clusters Standards
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
Conceptual Category
Domains Clusters StandardsAll
StandardsAdvanced
Number & Quantity
Algebra
Functions
Geometry
Statistics & Probability
Modeling
Unifying Themes Details
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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 * * * *
Unifying Themes Details
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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.
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
“Ways of Knowing”the mathematics
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
The Hunt Institute Video SeriesCommon Core State Standards: A New Foundation for Student Success
www.youtube.com/user/TheHuntInstitute#p
Operations and Algebraic Thinking Dr. Jason Zimba
Professor of Physics and Mathematics
Bennington College, Vermont
Lead Writer, Common Core Standards for Mathematics
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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.”
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
Properties of the Operations
ContextualSituations
Meanings of the Operations
The foundation for algebra!
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
72 – 29 = ?
Mental Math Solve in your head. No pencil or paper!
24 x 25 = ?
Nor calculators, cell phones computers or iPads or ....
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
72 – 29 = ? 24 x 25 = ?
Turn and share your reasoning.
Discuss how you used:
“Composing and decomposing”
a.k.a Properties of the operations
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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.
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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.”
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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.
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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….
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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……
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
Develop and use strategies
based on properties of the operations
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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.
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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).
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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.)
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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)
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
Strategies first!Develop and use strategies
for learning basic facts before any expectation of
knowing facts from memory.
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
Strategies first!Develop and use strategies to calculate with whole numbers, fractions, decimals …. before use of standard algorithms.
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
Properties of the Operations
ContextualSituations
Meanings of the Operations
The foundation for algebra!
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
Lots of real-world contexts!
Proficient students make sense of quantities and their relationships
in problem situations. (MP2)
decontexualize & contextualize
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
Properties of the Operations
Algorithms Real-world Contexts
Strategies
Walk Away
Message
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
Great
Moderate
Major
SmallMinor
Not Felt
Strong
Shifts in Classroom Practice
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
And so in closing …
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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.
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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.
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
Thank you!
Progression
Progressio
n
Progressio
n
Understanding
Focus
Coherence
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
Resources
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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.youtube.com/user/TheHuntInstitute#p
www.corestandards.org
Dr. DeAnn HuinkerUniversity of Wisconsin-Milwaukee
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
www.youtube.com/user/TheHuntInstitute#p