Understanding the Shifts in the Common Core State

StandardsA Focus on Mathematics

Wednesday, October 19th, 20112:00 pm – 3:30 pm

Doug Sovde, Senior Adviser, PARCC Instructional Supports and Educator Engagement, Achieve

Beth Cocuzza, Student Achievement Partners, LLC

The Six Shifts in Mathematics

Shift One: FocusShift Two:

CoherenceShift Three: Deep

Understanding Shift Four: FluencyShift Five:

ApplicationShift Six: Intensity

Shift One: Focus

Significantly narrow and deepen the scope and content of how time and energy is spent in the math classroom

Focus deeply on only the concepts that are prioritized in the standards so that students reach strong foundational knowledge and deep conceptual understanding

Students are able to transfer mathematical skills and understanding across concepts and grades

Shift Two: Coherence

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 deep conceptual understanding of core content and build on it. Each standard is not a new event, but an extension of previous learning.

The Importance of Focus

The current U.S. curriculum is ‘a mile wide and an inch deep.’

Focus allows each student to think, practice, and integrate each new idea into a growing knowledge structure.

K 12

Number and Operations

Measurement and Geometry

Algebra and Functions

Statistics and Probability

Traditional U.S. Approach

Domain Grades Major Work/Major Concerns (not a complete list)

Counting and Cardinality

K •Know number names and the count sequence •Count to tell the number of objects •Compare numbers

Operations and Algebraic Thinking

K-5 •Concrete use of the basic operations (word problems) •Mathematical meaning and formal properties of the basic operations •Prepare students to work with expressions and equations in middle school

Number and Operations—Base Ten

K-5 •Place value understanding •Develop base-ten algorithms using place value and properties of operations •Computation competencies (fluency, estimation)

Number and Operations—Fractions

3-5 •Enlarge concept of number beyond whole numbers, to include fractions •Use understanding of basic operations to extend arithmetic to fractions •Lay groundwork for solving equations in middle school

The Number System 6-8 •Build concepts of positive and negative numbers •Work with the rational numbers as a system governed by properties of operations •Begin work with irrational numbers

Expressions and Equations

6-8 •Understand expressions as objects (not as instructions to compute) •Transform expressions using properties of operations •Solve linear equations •Use variables and equations as techniques to solve word problems

Ratios and Proportional Relationships

6-7 •Consolidate multiplicative reasoning •Lay groundwork for functions in Grade 8 •Solve a wide variety of problems with ratios, rates, percents

Functions 8 •Extend and formalize understanding of quantitative relationships from Grades 3-7 •Lay groundwork for work with functions in High School

Measurement and Data K-5 •Emphasize the common nature of all measurement as iterating by a unit •Build understanding of linear spacing of numbers and support learning of the number line •Develop geometric measures •Work with data to prepare for Statistics and Probability in middle school

Geometry K-8 •Ascend through progressively higher levels of logical reasoning about shapes •Reason spatially with shapes, leading to logical reasoning about transformations •Connect geometry to number, operations, and measurement via notion of partitioning

Statistics and Probability

6-8 •Introduce concepts of central tendency, variability, and distribution •Connect randomness with statistical inference •Lay foundations for High School Statistics and Probability

CCSS

K-8

Dom

ain

Stru

ctur

e

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

The Importance of Coherence

Coherence provides the opportunity to make connections between mathematical ideas.

Coherence occurs both within a grade and across grades.

Coherence is necessary because mathematics instruction is not just a checklist of topics to cover, but a set of interrelated and powerful ideas.

Coherence example: Grade 3

Making connections at a single grade

Properties of Operations

Area

Multiplication and Division

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)

Content Emphases by Cluster Grade Four

Shift Three: Deep 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

Students demonstrate deep conceptual understanding of core math concepts by applying them to new situations

Shift Four: Fluency

Students are expected to have speed and accuracy with simple calculations

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

Required Fluencies

Grade Required Fluency

K Add/subtract within 5

1 Add/subtract within 10

2Add/subtract within 20Add/subtract within 100

3Multiply/divide within 100Add/subtract within 1000

4 Add/subtract within 1,000,000

5 Multi-digit multiplication

6Multi-digit divisionMulti-digit decimal operations

7 Solve px + q = r, p(x + q) = r

Shift Five: Application

Use math and choose the appropriate concept 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

Teachers in content areas outside of math, particularly science, ensure that students are using math – at all grade levels – to make meaning of and access content

Shift Six: Intensity

The standards call equally for conceptual understanding, procedural skill and fluency, and application of mathematics.

Meeting these standards requires intensity in the classroom.

Practice is intense: fluency is built and assessed through timed exercises. Solitary thinking and classroom discussion are intense, centered on thought-provoking problems that build conceptual understanding.

Applications are challenging and meaningful. The amount of time and energy spent practicing and understanding learning environments is driven by the specific mathematical concept and therefore, varies throughout the given school year.

The Shifts in Action—Two Examples

Place Value ◦Standards Progression◦Seeing the Six Shifts

Fractions ◦Standards Progression◦Seeing the Six Shifts

Place Value Problems for Deep Understanding

The Shifts in Action—Two Examples

Place Value ◦Standards Progression◦Seeing the Six Shifts

Fractions ◦Standards Progression◦Seeing the Six Shifts

Example: Fractions

4.NF

Fractions Problems for Deep Understanding

Questions?