1. Common Core State Standards for Mathematics: Shifts and Implications for Math Instruction
Common Core State Standards for Mathematics: Shifts and Implications for Mathematics Instruction
Please do the math problem now
K-5 Common Core Lead Teachers
6-8 Math Department Heads
Spring 2012
Common Core Lead Teachers, Spring 2012

2. The Three Shifts in Mathematics
Common Core State Standards for Mathematics: Shifts and Implications for Math Instruction
The Three Shifts in Mathematics
Focus strongly where the standards focus
Coherence: Think across grades and link to major topics within grades
Rigor: Require conceptual understanding, fluency, and application
Common Core Lead Teachers, Spring 2012

3. Shift One: Focus strongly where the Standards focus
Common Core State Standards for Mathematics: Shifts and Implications for Math Instruction
Shift One: Focus strongly where the Standards focus
Significantly narrow the scope of content and deepen how time and energy is spent in the math classroom
Focus deeply only on what is emphasized in the standards, so that students gain strong foundations
Common Core Lead Teachers, Spring 2012

4. Focus in International Comparisons
Common Core State Standards for Mathematics: Shifts and Implications for Math Instruction
Focus in International Comparisons
U.S. curriculum is “a mile wide and an inch deep”
TIMSS – Trends in International Mathematics and Science Study
Highest performing countries omit more material
U.S. omits 17% of TIMSS items through grade 4, and 2% through grade 8
Hong Kong omits 48% of TIMSS items through grade 4, and 18% through grade 8
Average omission rate is 40% for 11 comparison countries
Less topic coverage is associated with higher scores
Students have more time to master the content that is taught
TIMSS and other international comparisons suggest that the U.S. curriculum is ‘a mile wide and an inch deep.’
“Further evidence of the broad topic coverage characterizing U.S. mathematics curriculum is indicated by the low percentage of TIMSS mathematics topics not included in the U.S. curriculum through grades 4 and 8 compared with the percentage of TIMSS topics that other countries do not include…. On average, the U.S. curriculum omits only 17 percent of the TIMSS grade 4 topics compared with an average omission rate of 40 percent for the 11 comparison countries. The United States covers all but 2 percent of the TIMSS topics through grade 8 compared with a 25 percent noncoverage rate in the other countries. High-scoring Hong Kong’s curriculum omits 48 percent of the TIMSS items through grade 4, and 18 percent through grade 8. Less topic coverage can be associated with higher scores on those topics covered because students have more time to master the content that is taught.”
– Ginsburg et al., 2005
– Ginsburg et al., 2005
Common Core Lead Teachers, Spring 2012

5. The shape of math in A+ countries
Mathematics topics intended at each grade by at least
two-thirds of A+ countries
Mathematics topics intended at each grade by at least two-thirds of 21 U.S. states
This graphic is included to emphasize the difference in the shape of these two charts.
Along the vertical axis, each row represents a different piece of math content.
In the high performing countries, note that topics are taught, mastered, and not re-addressed yearly.
In the U.S., topics tend to be taught every year.
1 Schmidt, Houang, & Cogan, “A Coherent Curriculum: The Case of Mathematics.” (2002).

6. Higher Demands for Similar Content…

7. But Much Sharper Focus
Note that estimation, patterns, and pattern-related number sentence items are not assessed in Hong Kong.
More time is spent on essential content.

8. Traditional U.S. Approach
Common Core State Standards for Mathematics: Shifts and Implications for Math Instruction
Traditional U.S. Approach K
Number and Operations
Measurement and Geometry
Algebra and Functions
Statistics and Probability
How U.S standards used to be arranged, giving equal importance to all four areas - like “shopping aisles.” Each grade goes up and down the aisles, tossing topics into the cart, losing focus. The CCSS domain structure communicates the changing emphases throughout the elementary years (e.g., Ratios and Proportional Relationships in grades 6 and 7).
Common Core Lead Teachers, Spring 2012

9. Focusing attention within Number and Operations
Common Core State Standards for Mathematics: Shifts and Implications for Math Instruction
Focusing attention within Number and Operations
Operations and Algebraic Thinking
Expressions and Equations
Algebra 
Number and Operations— Base Ten
The Number System
Number and Operations—Fractions K 1 2 3 4 5 6 7 8
High School
Note how the emphasis changes in middle school and in high school, no longer addressing the same number sense skills which should be mastered in elementary school.
Common Core Lead Teachers, Spring 2012

10. Shift One: Focus Find the Fib – Which is not true?
Common Core State Standards for Mathematics: Shifts and Implications for Math Instruction
Shift One: Focus Find the Fib – Which is not true?
Focus means we will narrow the scope of content in each grade level.
Focus means we will deepen how time and energy is spent in the math classroom.
Focus means some standards will be emphasized more than others.
Focus means we will not teach the less important standards in the common core.
This is the Fib!
Common Core Lead Teachers, Spring 2012

11. Common Core State Standards for Mathematics: Shifts and Implications for Math Instruction
Shift Two: Coherence Think across grades, and link to major topics within grades
Carefully connect the learning within and across grades so that students can build new understanding onto foundations built in previous years.
Begin to count on solid conceptual understanding of core content and build on it. Each standard is not a new event, but an extension of previous learning.
Common Core Lead Teachers, Spring 2012

12. Coherence example: Progression across grades
“The coherence and sequential nature of mathematics dictate the foundational skills that are necessary for the learning of algebra. The most important foundational skill not presently developed appears to be proficiency with fractions (including decimals, percents, and negative fractions). The teaching of fractions must be acknowledged as critically important and improved before an increase in student achievement in algebra can be expected.”
Final Report of the National Mathematics Advisory Panel (2008, p. 18)
Carefully laid progressions of conceptual development, not just moving topics earlier in the grade sequence.

13. Coherence example: Progression across grades
Informing Grades 1-6 Mathematics Standards Development: What Can Be Learned from High-Performing Hong Kong, Singapore, and Korea?
American Institutes for Research (2009, p. 13)
Hong Kong
CCSS
5.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
5.NF.7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
Grade 5
Grade 4
4.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
Grade 6
6.NS. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
6.NS.1. Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.

14. Coherence example: Grade 3
Common Core State Standards for Mathematics: Shifts and Implications for Math Instruction
Coherence example: Grade 3
The standards make explicit connections at a single grade
Multiplication and Division
Properties of Operations
3.OA.5
3.MD.7a
3.MD.7c
Area
Identify specific standards which are connected in this way.
3.NBT.3, 3.OA.5 (Mult one-digit numbers by 10, apply prop of operations to mult and div)
3.MD.7, 3.OA1, 3.OA.3 (Relate area to mult and addition, interpret products, use mult to solve measurement word problems)
Common Core Lead Teachers, Spring 2012

15. Shift Two: Coherence Find the Fib – Which is not true?
Common Core State Standards for Mathematics: Shifts and Implications for Math Instruction
Shift Two: Coherence Find the Fib – Which is not true?
Coherence means the standards within a grade level are related and the relationships are used to deepen understanding.
Coherence means the standards are the same at different grade levels.
Coherence means the standards at one grade level are built upon in the next grade level.
Coherence means authors made great efforts to illustrate the connectedness of math ideas within and between grade levels.
This is the Fib!
Common Core Lead Teachers, Spring 2012

16. Common Core State Standards for Mathematics: Shifts and Implications for Math Instruction
Shift Three: Rigor Equal intensity in conceptual understanding, procedural skill/fluency, and application
The CCSSM require a balance of:
Solid conceptual understanding
Procedural skill and fluency
Application of skills in problem solving situations
This requires equal intensity in time, activities, and resources in pursuit of all three
This shift is about the depth of what is expected in the standards, and also about what one should expect to see happening in the classroom, in curricular materials, and so on.
Common Core Lead Teachers, Spring 2012

17. (a) Solid Conceptual Understanding
Common Core State Standards for Mathematics: Shifts and Implications for Math Instruction
(a) Solid Conceptual Understanding
Teach more than “how to get the answer” and instead support students’ ability to access concepts from a number of perspectives
Students are able to see math as more than a set of mnemonics or discrete procedures
Conceptual understanding supports the other aspects of rigor (fluency and application)
Common Core Lead Teachers, Spring 2012

18. (b) Procedural Skill and Fluency
Common Core State Standards for Mathematics: Shifts and Implications for Math Instruction
(b) Procedural Skill and Fluency
Fluency Standards: The standards require speed and accuracy in calculation.
Teachers structure class time and/or homework time for students to practice core functions such as single-digit multiplication so that they are more able to understand and manipulate more complex concepts
Note that this is not memorization absent understanding. This is the outcome of a carefully laid out learning progression.
At the same time, we can’t expect fluency to be a natural outcome without addressing it specifically in the classroom and in our materials. Some students might require more practice than others, and that should be attended to.
Calculator point: So, while you might have creative ways to use calculators in the K-5 classroom, you should know that your students aren’t meeting the standards when they use them.
Common Core Lead Teachers, Spring 2012

19. Required Fluencies in K-6
Common Core State Standards for Mathematics: Shifts and Implications for Math Instruction
Required Fluencies in K-6
Grade
Standard
Required Fluency K
K.OA.5
Add/subtract within 5 1
1.OA.6
Add/subtract within 10 2
2.OA.2
2.NBT.5
Add/subtract within 20 (know single-digit sums from memory)
Add/subtract within 100 3
3.OA.7
3.NBT.2
Multiply/divide within 100 (know single-digit products from memory)
Add/subtract within 1000 4
4.NBT.4
Add/subtract within 1,000,000 5
5.NBT.5
Multi-digit multiplication 6
6.NS.2,3
Multi-digit division
Multi-digit decimal operations
Fluent in the particular Standards cited here means “fast and accurate.” It might also help to think of fluency as meaning the same thing as when we say that somebody is fluent in a foreign language: when you’re fluent, you flow. Fluent isn’t halting, stumbling, or reversing oneself.
The word fluency was used judiciously in the Standards to mark the endpoints of progressions of learning that begin with solid underpinnings and then pass upward through stages of growing maturity.
Some of these fluency expectations are meant to be mental and others with pencil and paper. But for each of them, there should be no hesitation about how to proceed in getting the answer with accuracy.
Common Core Lead Teachers, Spring 2012

20. Common Core State Standards for Mathematics: Shifts and Implications for Math Instruction
(c) Application
Students can use appropriate concepts and procedures for application even when not prompted to do so
Provide opportunities at all grade levels for students to apply math concepts in “real world” situations, recognizing this means different things in K-5, 6-8, and HS
Teachers in content areas outside of math, particularly science, ensure that students are using grade-level-appropriate math to make meaning of and access science content
Using mathematics in problem solving contexts is the third leg of the stool supporting the learning that is going on in the math classroom.
This is the “why we learn math” piece, right? We learn it so we can apply it in situations that require mathematical knowledge. There are requirements for application all the way throughout the grades in the CCSS. But again, we can’t just focus solely on application—we need also to give students opportunities to gain deep insight into the mathematical concepts they are using and also develop fluency with the procedures that will be applied in these situations. The problem solving aspect of application is what’s at stake here—if we attempt this with a lack of conceptual knowledge and procedural fluency, the problem just becomes three times harder.
At the same time, we don’t want to save all the application for the end of the learning progression. Application can be motivational and interesting, and there is a need for students at all levels to connect the mathematics they are learning to the world around them.
Common Core Lead Teachers, Spring 2012

21. Shift Three: Rigor Find the Fib – Which is not true?
Common Core State Standards for Mathematics: Shifts and Implications for Math Instruction
Shift Three: Rigor Find the Fib – Which is not true?
Rigor means teachers should emphasize conceptual understanding, application, and procedural skills/fluency equally.
Rigor calls for solid conceptual understanding.
Rigor means teachers should no longer emphasize procedural skills.
Rigor calls for problem solving and explaining skills.
This is the Fib!
Common Core Lead Teachers, Spring 2012

22. Next Steps in our transition
Common Core State Standards for Mathematics: Shifts and Implications for Math Instruction
Next Steps in our transition
Incremental changes to the pacing each year for the next 3 years
Gradually increase time for major work of each grade (focus)
Begin to include problem solving and conceptual development lessons (focus)
Modify SBA’s to include open ended item(s) which require explanation (rigor)
Development teams (grades 3-5, and 6-7) will create instructional materials emphasizing problem solving and conceptual development (rigor)
Common Core Lead Teachers, Spring 2012

23. Can you name and describe the 3 shifts?
Common Core State Standards for Mathematics: Shifts and Implications for Math Instruction
And finally…
Can you name and describe the 3 shifts?
Common Core Lead Teachers, Spring 2012