CCSSI FOR MATHEMATICS“STANDARDS OF

PRACTICE”Collegial Conversations

GRADES 2 - 3

Today’s GoalTo explore the Standards for Content and

Practice for MathematicsBegin to consider how these new CCSS

Standards are likely to impact your classroom practices

What are the Common Core State Standards?

Aligned with college and work expectationsFocused and coherentIncluded rigorous content and application of

knowledge through high-order skillsBuild upon strengths and lessons of current state

standardsInternationally benchmarked so that all students are

prepared to succeed in our global economy and society

Research and evidence basedState led- coordinated by NGA Center and CCSSO

Focus

• Key ideas, understandings, and skills are identified

• Deep learning of concepts is emphasized– That is, time is spent on a topic and on

learning it well. This counters the “mile wide, inch deep” criticism leveled at most current U.S. standards.

Benefits for States and Districts• Allows collaborative professional development based on

best practices• Allows development of common assessments and other

tools• Enables comparison of policies and achievement

across states and districts• Creates potential for collaborative groups to get more

economical mileage for:– Curriculum development, assessment, and

professional development

Common Core Development• Initially 48 states and three territories

signed on

• As of November 29, 2010, 42 states have officially adopted

• Final Standards released June 2, 2010, at www.corestandards.org

• Adoption required for Race to the Top funds

Michigan’s Implementation Timeline

• Held October and November of 2010 rollouts• District curricula and assessments that provide a

K-12 progression for meeting the MMC requirements will require minimal adjustments to meet CCSS

• Curriculum and assessment alignment in SY10-11• Implementation SY11-12• New assessment 2014-15 (Smarter Balanced

Assessment Consortium or SBAC – replaces MEAP and MME)

Background

Each State that is a member of the Consortium in 2014–2015 also agrees to do the following:

Adopt common achievement standards no later than the 2014–2015 school year,

Fully implement the Consortium summative assessment in grades 3–8 and high school for both mathematics and English language arts no later than the 2014–2015 school year,

Adhere to the governance requirements, Agree to support the decisions of the Consortium, Agree to follow agreed-upon timelines, Be willing to participate in the decision-making process and, if a Governing

State, final decisions, and Identify and implement a plan to address barriers in State law, statute,

regulation, or policy to implementing the proposed assessment system and address any such barriers prior to full implementation of the summative assessment components of the system.

Responsibilities of States in the Consortium

 

Technology Approach

SBAC Item Bank

• Partitioned into a secure item bank for summative assessments and a non-secure bank for the interim/benchmark assessments:

• Traditional selected-response items• Constructed-response items• Curriculum-embedded performance events• Technology-enhanced items (modeled after

assessments in use by the U.S. military, the architecture licensure exam, and NAEP)

Domains are large groups of related standards. Standards from different domains may sometimes be closely related. Look for the name with the code number on it for a Domain.

HOW TO READ THE GRADE LEVEL STANDARDS

Common Core Format

Clusters are groups of related standards. Standards from different clusters may sometimes be closely related, because mathematics is a connected subject.• Clusters appear inside domains.

Common Core FormatStandards define what students should be able to understand and be able to do – part of a cluster.

•They are content statements. An example content statement is: “Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number can be decomposed into two equal addends”, 3.OA.9. The “OA” stands for “Operations and Algebraic Thinking”. Please refer to page three in your grade level appropriate Common Core document.

•Progressions of increasing complexity from grade to grade

Common Core - Clusters

• May appear in multiple grade levels in the K-8 Common Core. There is increasing development as the grade levels progress

• What students should know and be able to do at each grade level

• Reflect both mathematical understandings and skills, which are equally important

Common Core Format

High School

Conceptual Category

Domain

Cluster

Standards

K-8

Grade

Domain

Cluster

Standards

(There are no preK Common Core Standards)

Format of K-8 Standards Grade Grade LevelLevel

DomainDomain

Format of K-8 Standards

ClusterCluster

ClusterCluster

StandardStandard

StandardStandard

Mathematics » Grade 2 » Introduction

In Grade 2, instructional time should focus on four critical areas: (1) extending understanding of base-ten notation; (2) building fluency with addition and subtraction; (3) using standard units of measure; and (4) describing and analyzing shapes.

1.Students extend their understanding of the base-ten system. This includes ideas of counting in fives, tens, and multiples of hundreds, tens, and ones, as well as number relationships involving these units, including comparing. Students understand multi-digit numbers (up to 1000) written in base-ten notation, recognizing that the digits in each place represent amounts of thousands, hundreds, tens, or ones (e.g., 853 is 8 hundreds + 5 tens + 3 ones).

2.Students use their understanding of addition to develop fluency with addition and subtraction within 100. They solve problems within 1000 by applying their understanding of models for addition and subtraction, and they develop, discuss, and use efficient, accurate, and generalizable methods to compute sums and differences of whole numbers in base-ten notation, using their understanding of place value and the properties of operations. They select and accurately apply methods that are appropriate for the context and the numbers involved to mentally calculate sums and differences for numbers with only tens or only hundreds.

3.Students recognize the need for standard units of measure (centimeter and inch) and they use rulers and other measurement tools with the understanding that linear measure involves an iteration of units. They recognize that the smaller the unit, the more iterations they need to cover a given length.

4.Students describe and analyze shapes by examining their sides and angles. Students investigate, describe, and reason about decomposing and combining shapes to make other shapes. Through building, drawing, and analyzing two- and three-dimensional shapes, students develop a foundation for understanding area, volume, congruence, similarity, and symmetry in later grades.

Mathematics » Grade 3 » Introduction

In Grade 3, instructional time should focus on four critical areas: (1) developing understanding of multiplication and division and strategies for multiplication and division within 100; (2) developing understanding of fractions, especially unit fractions (fractions with numerator 1); (3) developing understanding of the structure of rectangular arrays and of area; and (4) describing and analyzing two-dimensional shapes.

1.Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown factor in these situations. For equal-sized group situations, division can require finding the unknown number of groups or the unknown group size. Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving single-digit factors. By comparing a variety of solution strategies, students learn the relationship between multiplication and division.

2.Students develop an understanding of fractions, beginning with unit fractions. Students view fractions in general as being built out of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole. Students understand that the size of a fractional part is relative to the size of the whole. For example, 1/2 of the paint in a small bucket could be less paint than 1/3 of the paint in a larger bucket, but 1/3 of a ribbon is longer than 1/5 of the same ribbon because when the ribbon is divided into 3 equal parts, the parts are longer than when the ribbon is divided into 5 equal parts. Students are able to use fractions to represent numbers equal to, less than, and greater than one. They solve problems that involve comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators.

3.Students recognize area as an attribute of two-dimensional regions. They measure the area of a shape by finding the total number of same-size units of area required to cover the shape without gaps or overlaps, a square with sides of unit length being the standard unit for measuring area. Students understand that rectangular arrays can be decomposed into identical rows or into identical columns. By decomposing rectangles into rectangular arrays of squares, students connect area to multiplication, and justify using multiplication to determine the area of a rectangle.

4.Students describe, analyze, and compare properties of two-dimensional shapes. They compare and classify shapes by their sides and angles, and connect these with definitions of shapes. Students also relate their fraction work to geometry by expressing the area of part of a shape as a unit fraction of the whole.

Fractions, Grades 3–6

3. Develop an understanding of fractions as numbers. 4. Extend understanding of fraction equivalence and ordering. 4. Build fractions from unit fractions by applying and extending

previous understandings of operations on whole numbers. 4. Understand decimal notation for fractions, and compare

decimal fractions. 5. Use equivalent fractions as a strategy to add and subtract

fractions. 5. Apply and extend previous understandings of multiplication

and division to multiply and divide fractions. 6. Apply and extend previous understandings of multiplication

and division to divide fractions by fractions.

Domains Grade Levels

Counting and Cardinality K only

Operations and Algebraic Thinking

1-5

Number and Operations in Base Ten

1-5

Number and Operations - Fractions

3-5

Measurement and Data 1-5

Geometry 1-5

K – 5 DOMAINS

Domains Grade Levels

Ratio and Proportional Relationships

6-7

The Number System 6-8

Expressions and Equations 6-8

Functions 8

Geometry 6-8

Statistics and Probability 6-8

MIDDLE GRADES DOMAINS

Michigan GLCE vs. CCSS Topic 1 2 3 4 5 6 7 8

Whole Number: Meaning l l l l lWhole Number: Operations l l l l lMeasurement Units l l l l l l l lCommon Fractions l l l l lEquations & Formulas l l l l lData Representation & Analysis l l l l l l l2-D Geometry: Basics l l l l l l2-D Geometry: Polygons & Circles l l l l l l l lMeasurement: Perimeter, Area & Volume l l l l l l lRounding & Significant Figures lEstimating Computations l l l l l l l lWhole Numbers: Properties of Operations lEstimating Quantity & SizeDecimal Fractions l l l l lRelation of Common & Decimal Fractions l l l lProperties of Common & Decimal Fractions lPercentages l l lProportionality Concepts l l lProportionality Problems l l2-D Geometry: Coordinate Geometry l l l lGeometry: Transformations l l l l lNegative Numbers, I ntegers, & Their Properties l lNumber Theory l l lExponents, Roots & Radicals l l l lExponents and Orders of Magnitude l lMeasurement: Estimation & Errors l lConstructions Using Straightedge & Compass l3-D Geometry l l l l l l lGeometry: Congruence & Similarity l lRational Numbers & Their Properties l l lPatterns, Relations & Functions l l lProportionality: Slope & Trigonometry lUncertainty & Probability l lReal Numbers: Their Subsets & Properties l l

Topic intended in Michigan GLCE lTopic intended in CCSS

Grade

MAJOR SHIFTS K - 5Numeration and operation intensified, and introduced earlier•Early place value foundations in Kindergarten•Regrouping as composing/decomposing in Grade 2•Decimals to hundredths in Grade 4

All three types of measurement simultaneously•Non-standard, English and metric

Emphasis on fractions as numbers

Emphasis on number line as visualization/structure

Observations About Place Value and Base Ten in the Early Grades

•KindergartenFoundation in bundlingEmphasis on the teen numbers

•Grade 1•Extends to 10, 20, 30…•Learn to compare

•Grade 2•Extend to 100 as a bundle of ten 10s•Extend to 100, 200, 300…•Expanded notation and comparison

HOW IS THERE LESS?

•Backed off of algebraic patterns K – 5

•Backed off of statistics and probability in K – 5

•Delayed content like percent and ratios and proportions

CCSSM Mathematical Practices

The Common Core proposes a set of Mathematical Practices that all teachers should develop in their students. These practices are similar to NCTM’s Mathematical Processes from the Principles and Standards for School Mathematics.

THE REASON WHY WE ARE HERE TODAY!

Design and Organization

Mathematical Practice – expertise students should acquire: (Processes & proficiencies)

• NCTM five process standards: • Problem solving• Reasoning and Proof• Communication • Connections• Representations

NCTM Process Standards CCSS Mathematical Practices

Problem Solving Make sense of problems and persevere in solving them.Use appropriate tools strategically

Reasoning and Proof Reason abstractly and quantitatively.Critique the reasoning of others.Look for and express regularity in repeated reasoning

Communication Construct viable arguments

Connections Attend to precision.Look for and make use of structure

Representations Model with mathematics.

NCTM Process Standards and theCCSS Mathematical Practice Standards

Design and Organization

• Mathematical proficiency (National Research

Council’s report Adding It Up) – Adaptive reasoning– Strategic competence– Conceptual understanding (comprehension of

mathematical concepts, operations, relations)– Procedural fluency (skill in carrying out procedures

flexibly, accurately, efficiently, and appropriately)– Productive disposition (ability to see mathematics as

sensible, useful, and worthwhile

Mathematics/Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them2. Reason abstractly and quantitatively3. Construct viable arguments and critique the reasoning

of others4. Model with mathematics5. Use appropriate tools strategically6. Attend to precision7. Look for and make use of structure8. Look for and express regularity in repeated reasoning

Mathematics/Standards for Mathematical Practice

“The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students.

These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education.” CCSS, 2010

Standards for Mathematical Practice• Carry across all grade levels• Describe habits of a mathematically expert student

Standards for Mathematical Content• K-8 presented by grade level• Organized into domains that progress over several grades• Grade introductions give 2-4 focal points at each grade level• High school standards presented by conceptual theme (Number &

Quantity, Algebra, Functions, Modeling, Geometry, Statistics & Probability

Standards of Mathematical Practice1.Choose a partner at your table and “Pair Share” the Standards of Practice between you and your partner.

2. When you and your partner feel you understand generally each of the standards, discuss the following question:

What implications might the standards of practice have on your classroom?

Transition from Current State Standards & Assessments to New Common Core Standards

• Develop Awareness• Needs Assessment/Gap Analysis• Planning• Capacity Building• Job-embedded Professional Development

Transition PlanningNext Steps:• Alignment of CCSS with curriculum• Gap analysis (content and skills that vary from

the MEAP and MME)• What instructional practices will facilitate the

transition?• What new assessment strategies will be

needed?• Professional development needs?

Transition Planning• Gather in teams from your schools and discuss

– What are your immediate needs as a classroom teacher being asked to implement the CCSS?

– What professional development is needed?– What initial gaps come to mind and how do you address these

gaps?– As a school team, look at the eight Standards for Mathematical

Practice. What do they look like? Sound like? What will students need in order to implement them? What will teachers need? What are the implications for assessment and grading?

Select a recorder, time keeper and someone to report out for your group.

Questions? Please contact:

PUT YOUR INFORMATION HERE!

Have a great day!