an introduction to the common core state standards for mathematics presented at the hawaii...

An Introduction toThe Common Core State Standards

for MathematicsPresented at the Hawaii Department of EducationCommon Core State Standards Training Sessions

January – March 2011

Dewey GottliebEducational Specialist for Mathematics

Office of Curriculum, Instruction and Student Support

Desired Outcomes

Increased understanding of the

• CCSS domain progressions

• Standards for Mathematical Practice

• CCSS-HCPS III Crosswalk documents

A Shift in Perspective

The CCSS for Mathematics compel a change in the culture of traditional mathematics classroom.

In the typical mathematics classroom students are “too busy covering content” to be engaged with mathematics.

CCSS for Mathematics

The emphasis on teaching and learningThe CCSS attempts to tell teachers when to slow down and emphasize student understanding of significant mathematical ideas.

“To say, ‘It was a good lesson but the students didn’t get it’,

is like saying, ‘The operation was a success, but the patient died.’”

(Lewis, 2002)

But what does “higher standards” mean?

More topics?

No. The U.S. curriculum is already cluttered with too many topics

Teaching topics in earlier grades?

No. Analyses of the standards of high-performing countries suggest otherwise.

In Singapore, division of fractions is a 6th grade expectation; in the U.S. it is typically a 4th or 5th grade expectation.

In Japan, probability is introduced in the 7th grade; in the U.S., it can be found anywhere throughout grades 3-6, depending on the state.

Standards are “high” for what students take away from cumulative learning experiences


Current U.S. curricula (“mile wide, inch deep”) coupled with high-stakes testing pressures teachers to

“cover” at “pace”

turn the page regardless of student needs

However, the study of mathematics should not be reduced to merely “a list of topics to cover”

Singapore preaches, “Teach less, learn more”

The Domains in the CCSS

• Groups of related standards are organized into domains. Domains are overarching big ideas that connect topics across grades.

• Standards from different domains may be closely related, conveying an internal coherence among the domains.

• In HCPS III, the benchmarks were organized into “strands.” With the transition to CCSS, what we used to call a strand will now be referred to as a domain.

Each grade level focuses upon particular DOMAINS

• Counting and Cardinality (K)

• Operations and Algebraic Thinking (K-

5)

• Number and Operations in Base Ten

(K-5)

• Number and Operations: Fractions

(3-5)

• Measurement and Data (K-5)

• Geometry (K-5)

Grades K – 5 Mathematics

Each grade level focuses upon particular DOMAINS

• Ratios & Proportional Relationships (6 -

7)

• The Number System (6-8)

• Expressions and Equations (6-8)

• Functions (8)

• Geometry (6-8)

• Statistics and Probability (6-8)

Grades 6 – 8 Mathematics

Learning Expectations are Organized by Conceptual Categories

• Number and Quantity

• Algebra

• Functions

• Modeling*

• Geometry

• Statistics and Probability

High School Mathematics

The Clusters in the CCSS

• Within a domain, smaller groups of related standards are organized into clusters.

• The clusters help to inform teachers’ decision-making regarding their instructional design and the learning and assessment opportunities provided to students in the mathematics classroom.


For example, in grade 4, the standards in the Fractions domain are organized into three clusters:

• Extend understanding of fraction equivalence and ordering.

• Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.

• Understand decimal notation for fractions, and compare decimal fractions.


For example, in grade 6, the standards in “The Number System” domain are organized into three clusters:

• Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

• Compute fluently with multi-digit numbers and find common factors and multiples.

• Apply and extend previous understandings of numbers to the system of rational numbers.


• We don’t want to simply teach to the standards (i.e., as if checking off a to-do list).

• Rather, we want to teach THROUGH the standards, using the specific learning expectations (i.e., the standards) as building blocks for student understanding of significant mathematical ideas (i.e., the clusters) that will prepare them for the mathematics they will be engaging with in subsequent grades.

Getting to the Clusters: Teaching THROUGH the Standards

Grade 2 Cluster: Use place value understanding and properties of operations to add and subtract.2.NBT.5: Fluently add and subtract within 100 using strategies based on place

value, properties of operations, and/or the relationship between addition and subtraction.

2.NBT.6: Add up to four two-digit numbers using strategies based on place value and properties of operations.

2.NBT.7: Add and subtract within 1000, using … strategies based on place value, properties of operations, ….

2.NBT.8: Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.

2.NBT.9: Explain why addition and subtraction strategies work, ….

Getting to the Clusters: Teaching THROUGH the Standards

Grade 6 Cluster: Apply and extend previous understandings of numbers to the system of rational numbers.

6.NS.5: Understand that positive and negative numbers are used together to describe quantities having opposite directions or values …

6.NS.6: Understand a rational number as a point on the number line …

6.NS.7: Understand ordering and absolute value of rational numbers …

6.NS.8: Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane …

Learning Progressions: Developing Expertise

The brain is a sense-making machine ... it does not store what doesn’t make sense.

If we want to make information meaningful to students, we have two options:

Find the prior experience they’ve had and hook the new information to it.

ORCreate the experience with them (i.e., build a new network).

The “understand” standardsThe “understand” standards interact with the “skills”

standards to support the development of expertise

Students who understand a concept can(a) Explain it(b) Demonstrate or illustrate it(c) Use it into their own arguments and critique someone

else’s explanation of it(d) Show an example of how to apply it (make connections to

other mathematical ideas and/or to real-world contexts)

CCSS for Mathematics

Small Group Task:• Select one domain that goes across grades

6-8.• Review the clusters and standards for that

domain for each grade level.• Discuss how the clusters and standards are

organized into learning progressions that develop student expertise over time.

Domain Progressions


Too often, students view mathematics as a trivial exercise because they are rarely given the opportunity to grapple with and come to appreciate the intrinsic complexity of the mathematics.

Despite our instincts and best intentions, we need to stop “GPS-ing” our students to death.

Source: Shannon, A. (2010). Common Core: Two Perspectives on Tasks and Assessments. Presentation at the Urban Mathematics Leadership Network Retreat, June 2010.

“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)

The Standards for Mathematical Practice


Source: Briars, D. & Mitchell, S. (2010). Getting Started with the Common Core State Standards. A National Council of Supervisors of Mathematics webinar. November 2010.



Strategic Competence

AdaptiveReasoning

ProductiveDisposition

ProceduralFluency

Conceptual Understanding

“Encouraging these practices in students of all ages should be as much a goal of the mathematics curriculum as the learning of specific content” (CCSS, 2010).

1.Make sense of problems and persevere in solving them.2.Reason abstractly and quantitatively.3.Construct viable arguments and critique the reasoning of others.4.Model with mathematics.5.Use appropriate tools strategically.6.Attend to precision.7.Look for and make use of structure.8.Look for and express regularity in repeated reasoning.


The description of each Mathematical Practice begins with the same first three words:

Mathematically proficient students …



The Mathematical Practices “describe the

thinking processes, habits of mind and

dispositions that students need to develop a

deep, flexible, and enduring understanding of

mathematics; in this sense they are also a

means to an end.”



MP #1: Make sense of problems and persevere in solving them.Mathematically proficient students … analyze givens, constraints, relationships and goals … they monitor and evaluate their progress and change course if necessary … and continually ask themselves, “Does this make sense?”



MP #3: Construct viable arguments and critique the reasoning of others

Consider the following subtraction algorithm: • How could I demonstrate the idea that

the algorithm always works?

400 – 139 399 – 138

43 – 17 46 – 20

Points of Intersection: Content and Practices

MP #7: Look for and make use of structure

Partitioning• 8 x 7• 33 + 58


MP #7: Look for and make use of structure

Example:

Understanding and interpreting the equation of a line expressed in “Point-Slope Form”

y – y1 = m(x – x1)


MP #4: Model with Mathematics

• Model Drawing (“Singapore Math”)

– Mrs. Obama has 28 students in her fifth grade class and 2/7 of the class is girls. How many of her students are boys?


MP #4: Model with mathematics

• Double number lines

Today 40% of the 370 sixth graders at Barack Obama Middle School are on a field trip.


0% 100%

0 sixth graders

370 sixth graders

MP #4: Model with mathematics

MP #5: Use appropriate tools strategically

• Compare and contrast directly and inversely proportional relationships


MP #2: Reason abstractly and quantitatively.

Consider :

• x2 – 1 = (x + 1)(x – 1)

• (a + b)2 = a2 + 2ab + b2


Small Group Task:• Select two of the Standards for

Mathematical Practice • Identify apparent “points of intersection”

between the content standards (in CCSS) and the Standards for Mathematical Practice.


CCSS-HCPS III Crosswalks

Crosswalk Documents posted at

http://standardstoolkit.k12.hi.us/index.html

(Click on the “Document Library” link near

the top of the webpage)

Crosswalk Documents posted at

http://standardstoolkit.k12.hi.us/index.html

(Click on the “Document Library” link near

the top of the webpage)

37

Crosswalk Documents posted at http://standardstoolkit.k12.hi.us/index.html

(Click on the “Document Library” link near the top of the webpage)

Crosswalk Documents posted at http://standardstoolkit.k12.hi.us/index.html

(Click on the “Document Library” link near the top of the webpage)

• Grade level overview• Mapping of CCSS to HCPS III benchmarks

Standards matched to benchmarks

Degree of Match

Comments• Mapping of HCPS III benchmarks to CCSS


Small Group Task:• Gather in groups according to your grade

level of interest • Analyze the crosswalk document for your

selected grade level to respond to the prompts in the handout


"These Standards are not intended to be new names for old ways of doing business. They are a call to take the next step. It is time for states to work together to build on lessons learned from two decades of standards based reforms. It is time to recognize that these standards are not just promises to our children, but promises we intend to keep."

Final Thoughts

Video: “Math Class Needs a Makeover”

Dan Meyer (a high school mathematics teacher)

http://www.ted.com/talks/dan_meyer_math_curriculum_makeover.html

After the video: “Think-Pair-Share”

•One idea that resonated with you
