Unwrapping the Common Core State Standards for Administrators

Objectives• Increase participant’s knowledge of the CCSS for

Mathematics

• Increase participant’s knowledge of the shifts of the CCSS for Mathematics

• Increase participant’s ability to unpack content and mathematical practice standards

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Outcomes• Knowledge to lead implementation of the Common Core

State Standards.

• Vision to integrate the implementation of the Common Core State Standards into broad education improvement efforts.

• Metrics to clearly describe what successful progress in implementation looks like and facilitates a flexible cycle of change.

• Build capacity so that all members of the education landscape are learning together.

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Rationale for the CCSS• Declining US competitiveness with other developed countries

• NAEP performance that is largely flat over the past 40 years in 8th grade

• Slight improvement at the 4th grade level

• Slight decline at the high school level

• High rates of college remediation

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Principles of the CCSS

• Aligned to requirements for college and career readiness

• Based on evidence

• Honest about time

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Activity

Digging into the Common Core

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Common Core State Standards for Mathematics: Key Shifts

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Mathematics: 3 Shifts

1. Focus: Focus strongly where the standards focus.

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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.

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K 12

Number and Operations

Measurement and Geometry

Algebra and Functions

Statistics and Probability

Traditional U.S. Approach

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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 17

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 numbers Use tally marksUnderstand meaning of addition and subtraction

1 Add and subtract within 20Measure 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 valueIdentify line of symmetry in two dimensional figures

3 Multiply and divide within 100Identify 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 systemApply 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 divisibilityApply and extend previous understandings of arithmetic to algebraic expressions

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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 equationDefine, evaluate, and compare functions

Understand and apply the Pythagorean Theorem

Alg.1 Quadratic inequalities Linear and quadratic functions Creating equations to model situations

Alg.2 Exponential and logarithmic functions Polar coordinates Using functions to model situations18

GradeFocus Areas in Support of Rich Instruction and Expectations of Fluency and Conceptual Understanding

K–2Addition and subtraction – concepts, skills, and problem solving and place value

3–5Multiplication and division of whole numbers and fractions – concepts, skills, and problem solving

6Ratios and proportional reasoning; early expressions and equations

7Ratios and proportional reasoning; arithmetic of rational numbers

8 Linear algebra

Key Areas of Focus in Mathematics

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Mathematics: 3 Shifts

1. Focus: Focus strongly where the standards focus.

2. Coherence: Think across grades, and link to major topics

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.

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Activity

Coherence

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Coherence: Link to Major Topics Within Grades

Example: Data Representation

Standard MACC.3.MD.2.3

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Example: Geometric Measurement

MACC.3.MD.3 (cluster)

Coherence: Link to Major Topics Within Grades

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

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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

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Algebra: Reasoning with Equations and Inequalities (A-REI.1-12)• Understand solving equations as a process of reasoning and explain the reasoning• Solve equations and inequalities in one variable• Solve systems of equations• Represent and solve equations and inequalities graphically

8.EE.7-8 Analyze and solve linear equations and pairs of simultaneous linear equations.

7.EE.3-4 Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

6.EE.5-8 Reason about and solve one-variable equations and inequalities.

5.OA.1-2 Write and interpret numerical expressions.

4.OA.1-3 Use the four operations with whole numbers to solve problems.

3.OA.1-4 Represent and solve problems involving multiplication and division.

2.OA.1 Represent and solve problems involving addition and subtraction.

1.OA.7-8 Work with addition and subtraction equations.

K.OA.1-5 Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

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Mathematics: 3 Shifts

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

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

• 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.

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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)

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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

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Required Fluencies in K-6

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Grade Standard Required FluencyK K.OA.5 Add/subtract within 5

1 1.OA.6 Add/subtract within 10

2 2.OA.22.NBT.5

Add/subtract within 20 (know single-digit sums from memory)Add/subtract within 100

3 3.OA.73.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 divisionMulti-digit decimal operations

Fluency in High School

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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.

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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.

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Rigor = ______ + ________ + _______

Discussion

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Have you observed any of these shifts in your schools with the implementation of NGSSS?

What have you seen?

Common Core in Action?

Teaching the Pythagorean Theorem

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Observe Mr. McKinney’s class.

Do you see the shifts of CCSS incorporated into his teaching?

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Activity

Reflecting on the Shifts for Mathematics

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Structure of CCSS

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Standards for Mathematical Practice

Grade level or High School conceptual category

Domain

ClusterStandard

Standards for Mathematical Practice

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Activity

Standards of Mathematical Practice

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Metrics: What it Looks Like

• Everyone in the system needs clarity around the goals – what it will look like when implemented.

• Metrics let us know what progress we are making in meeting goals.

• The system must be set up to collect progress data, and also monitor and adjust.

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ActivityUnpacking Practice Standards

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ActivityUnpacking Content Standards

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Unpacking Content Standards

Answers 2 questions:

•“What does that mean?”

•“What should I do to help my students demonstrate that?”

Product

•Clear explanations, definitions, and background research

•Ideas for tasks

•Guides differentiation and planning

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Unpacking Process

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Essential Element of Unpacking Process

1. Read the standard.

2. Identify and discuss the technical meaning of important words in the language of the standard.

3. Explore research pertaining to the content in the standard, including common misconceptions related to the topic.

4. Determine what the standard calls for students to know and be able to do.

5. Determine how the standard relates to learning progressions, standards progressions, and big ideas.

6. Determine how students will demonstrate proficiency.

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MACC.6.RP.1.3a

MACC.6.RP.1.3

Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations

MACC.6.RP.1.3a

Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

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MACC.6.RP.1.3a

MACC.6.RP.1.3

Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations

MACC.6.RP.1.3a

Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

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Assessable Instructional Objectives

• Students will relate quantities in a table of equivalent ratios (ratio table) by identifying how many times greater or less a quantity in one ratio is compared to a corresponding quantity in an equivalent ratio.

• Students will construct a table of equivalent ratios (ratio table) either in columns or rows to find a missing value in a real-world or mathematical problem. An additional column or row may be added for totaling quantities in cases where units are the same.

• Students will plot ratio pair values from a (ratio table) on a coordinate plane to solve problems. This method may be used to find missing values or to make a multiplicative comparison between equivalent ratios.

• Students will compare ratios from two different ratio tables to solve problems by finding corresponding quantities that have the same value on both tables.

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Elementary Middle High

MACC.5.NF.2.7c MACC.7.EE.2.3 MACC.912.A-CED.1.2Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem.

Solve multi step real life and ‐ ‐mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations as strategies to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

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Build Capacity

• You are acting as lead learner – this content is new to everyone.

• Not an issue of compliance.

• Teachers need opportunities to learn and process these expectations – not just a new scope and sequence.

• Everyone in the system needs to appreciate this initiative for what it is, an opportunity to reform education.

• Recognize this as hard work, worth doing.

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Stages of ChangeLook for people to go through the stages of awareness, application and experimentation, and ownership.

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ActivityBuilding Capacity for the Work

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You have just purchased an expensive Grecian urn and asked the dealer to ship it to your house. He picks up a hammer, shatters it into pieces, and explains that he will send one piece a day in an envelope for the next year. You object; he says “don’t worry, I’ll make sure that you get every single piece, and the markings are clear, so you’ll be able to glue them all back together. I’ve got it covered.” Absurd, no? But this is the way many school systems require teachers to deliver mathematics to their students; one piece (i.e. one standard) at a time. They promise their customers (the taxpayers) that by the end of the year they will have “covered” the standards.

~Excerpt from The Structure is the StandardsPhil Daro, Bill McCallum, Jason Zimba

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References

www.achievethecore.org

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