1. Common Core State Standards for Mathematics: Review Focus and Coherence A Closer look at Rigor

2. www.achievethecore.org 2 The CCSS Requires Three Shifts in Mathematics 1.Focus: Focus strongly where the standards focus. 2.Coherence: Think across grades, and link to major topics 3.Rigor: In major topics, pursue conceptual understanding, procedural skill and fluency, and application

3. www.achievethecore.org Engaging with the shift: What do you think belongs in the major work of each grade? Grade Which two of the following represent areas of major focus for the indicated grade? K Compare numbersUse tally marksUnderstand meaning of addition and subtraction 1 Add and subtract within 20 Measure lengths indirectly and by iterating length units Create and extend patterns and sequences 2 Work with equal groups of objects to gain foundations for multiplication Understand place value Identify line of symmetry in two dimensional figures 3 Multiply and divide within 100 Identify the measures of central tendency and distribution Develop understanding of fractions as numbers 4 Examine transformations on the coordinate plane Generalize place value understanding for multi-digit whole numbers Extend understanding of fraction equivalence and ordering 5 Understand and calculate probability of single events Understand the place value system Apply and extend previous understandings of multiplication and division to multiply and divide fractions 6 Understand ratio concepts and use ratio reasoning to solve problems Identify and utilize rules of divisibility Apply and extend previous understandings of arithmetic to algebraic expressions 7 Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers Use properties of operations to generate equivalent expressions Generate the prime factorization of numbers to solve problems 8 Standard form of a linear equation Define, evaluate, and compare functions Understand and apply the Pythagorean Theorem Alg.1 Quadratic inequalitiesLinear and quadratic functionsCreating equations to model situations Alg.2 Exponential and logarithmic functionsPolar coordinatesUsing functions to model situations 3

4. www.achievethecore.org 4 Shift #2: 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 on 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.

5. www.achievethecore.org 5 Coherence: Think Across Grades Example: Fractions “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)

6. www.achievethecore.org 4.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 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. 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. Grade 4 Grade 5 Grade 6 CCSS 6 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)

7. www.achievethecore.org One of several staircases to algebra designed in the OA domain. Alignment in Context: Neighboring Grades and Progressions 7

8. www.achievethecore.org 8 Coherence: Link to Major Topics Within Grades Example: Data Representation Standard 3.MD.3

9. www.achievethecore.org 9 Example: Geometric Measurement 3.MD, third cluster Coherence: Link to Major Topics Within Grades

10. www.achievethecore.org Rigor 10 The CCSSM require a balance of:  Solid conceptual understanding  Procedural skill and fluency  Application of skills in problem solving situations Pursuit of all threes requires equal intensity in time, activities, and resources.

11. www.achievethecore.org 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) 11

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14. www.achievethecore.org Fluency 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 14

15. www.achievethecore.org 15 Required Fluencies in K-6 GradeStandardRequired Fluency KK.OA.5Add/subtract within 5 11.OA.6Add/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 44.NBT.4Add/subtract within 1,000,000 55.NBT.5Multi-digit multiplication 66.NS.2,3 Multi-digit division Multi-digit decimal operations

16. www.achievethecore.org Fluency in High School 16

17. www.achievethecore.org Application Students can use appropriate concepts and procedures for application even when not prompted to do so. Teachers 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. 17

18. www.achievethecore.org Group Discussion Shift #3: Rigor: Expect fluency, deep understanding, and application In your groups, discuss ways to respond to one of the following comments: “These standards expect that we just teach rote memorization. Seems like a step backwards to me.” Or “I’m not going to spend time on fluency—it should just be a natural outcome of conceptual understanding.” 18

19. www.achievethecore.org Engaging with the shift: Making a True Statement This shift requires a balance of three discrete components in math instruction. This is not a pedagogical option, but is required by the standards. 19 Rigor = ______ + ________ + _______

20. www.achievethecore.org Cautions: Implementing the CCSS is... Not about “gap analysis” Not about buying a text series Not a march through the standards Not about breaking apart each standard 20