Introduction to the Math Shifts of the Common Core State Standards

3 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.

4 Shift #1: 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 on what is emphasized in the standards, so that students gain strong foundations.

Focus Move away from "mile wide, inch deep" curricula identified in TIMSS. Learn from international comparisons. Teach less, learn more. “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.,

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 The shape of math in A+ countries 1 Schmidt, Houang, & Cogan, “A Coherent Curriculum: The Case of Mathematics.” (2002). 6

7 K 12 Number and Operations Measurement and Geometry Algebra and Functions Statistics and Probability Traditional U.S. Approach

8 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 High School

9 Grade Focus Areas in Support of Rich Instruction and Expectations of Fluency and Conceptual Understanding K–2 Addition and subtraction - concepts, skills, and problem solving and place value 3–5 Multiplication and division of whole numbers and fractions – concepts, skills, and problem solving 6 Ratios and proportional reasoning; early expressions and equations 7 Ratios and proportional reasoning; arithmetic of rational numbers 8 Linear algebra; linear functions Key Areas of Focus in Mathematics

Group Discussion Shift #1: Focus strongly where the Standards focus. In your groups, discuss ways to respond to the following question, “Why focus? There’s so much math that students could be learning, why limit them to just a few things?” 10

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 11

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 12

13 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.

14

15 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)

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 16 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)

One of several staircases to algebra designed in the OA domain. Alignment in Context: Neighboring Grades and Progressions 17

18 Coherence: Link to Major Topics Within Grades Example: Data Representation Standard 3.MD.3

19 Example: Geometric Measurement 3.MD, third cluster Coherence: Link to Major Topics Within Grades

Group Discussion Shift #2: Coherence: Think across grades, link to major topics within grades In your groups, discuss what coherence in the math curriculum means to you. Be sure to address both elements—coherence within the grade and coherence across grades. Cite specific examples. 20

Engaging with the Shift: Investigate Coherence in the Standards with Respect to Fractions In the space below, copy all of the standards related to multiplication and division of fractions and note how coherence is evident in these standards. Note also standards that are outside of the Number and Operations—Fractions domain but are related to, or in support of, fractions. 21

22 Shift #3: Rigor: In Major Topics, Pursue Conceptual Understanding, Procedural Skill and Fluency, and Application Understanding Fluency Application

Rigor 23 The CCSSM require a balance of:  Solid conceptual understanding  Procedural skill and fluency  Application of skills in problem solving situations Pursuit of all three requires equal intensity in time, activities, and resources.

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) 24

25

26

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 27

28 Required Fluencies in K-6 GradeStandardRequired Fluency KK.OA.5Add/subtract within 5 11.OA.6Add/subtract within OA.2 2.NBT.5 Add/subtract within 20 (know single-digit sums from memory) Add/subtract within OA.7 3.NBT.2 Multiply/divide within 100 (know single-digit products from memory) Add/subtract within NBT.4Add/subtract within 1,000, NBT.5Multi-digit multiplication 66.NS.2,3 Multi-digit division Multi-digit decimal operations

Fluency in High School 29

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. 30

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.” 31

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. Using grade __ as a sample, find and copy the standards which specifically set expectations for each component. 32 Rigor = ______ + ________ + _______

ASSESSMENT What are the three mathematical shifts? 33

Resources &RDtoken=51120&userID commoncoretools.me education/common-core-state-standards-tools- resources/ 34