Digging Deeper Into the K–8 Mathematics Common Core State Standards

Bradford R. Findell, PhDFebruary 8, 2011

forEducation Northwest

Association of State Supervisors of Mathematics

Overview

• Two views of math education today• A look inside the system• A look inside the CCSS• Mathematical interlude• About curriculum• Implementation challenges and

suggestions• CCSS implementation resources

The Prevailing Story Line

• The system is broken• People often blame

– Administrators– Unions– Teachers of previous grades– Parents– Students– …

• Simple explanations lead to simple solutions that are unlikely to make a difference

• But let’s look at the data

TIMSS 2007 Results, Grades 4 and 8

• Students in a handful of countries perform significantly better than those in the U.S.

TIMSS Trends, 1995 to 2007

PISA Results, 2006

Mathematics Achievement Trends

• Achievement is up by many indicators – Significant growth in grades 4 and 8– High school diploma, math course taking– College attendance, college completion

• High school achievement is flat– U.S. 15-year-olds lag in applying math– Poor results on H.S. end-of-course exams– College remediation rates remain high

• Today’s world demands more

This Is a Challenge, not a Crisis!

• A crisis is a “turning point” – John Ewing, Math for America

• We have long-term structural problems– And many inadequate improvement initiatives

• We need to make fundamental long-term changes over decades

• The Common Core State Standards are a response to this challenge

Looking Inside the System

220

240

260

280

300

1995 2000 2005 2010

CaliforniaMinnesotaNational PublicMississippiSouth CarolinaDistrict of ColumbiaMassachusettsOhio

Main NAEP Trends, Grade 8

Digging Deeper

• Within the U.S., there are significant differences among states

• Within states, there are significant differences among districts

• Within districts, there are significant differences among schools

Within Schools ...

• How many of you work in a school where at least one teacher is phenomenal?

• How many of you work in a school where at least one teacher is less than effective?

• Conclusion: The variation within a school is usually greater than the variation between schools

• What matters is the interactions among the teacher, the students, and the mathematical ideas

It’s About Instruction

• We don’t have an achievement gap• We have an instruction gap

– Matt Larson

• So how can we reduce the discrepancies in instructional quality within and among schools?– Increase the mean and reduce the variance

• How can harness the expertise of our best teachers?

Instruction as Interaction

What matters are the interactions, in classrooms, among the teacher, the students, and the mathematical ideasSource: Cohen & Ball, 1999, 2000, as cited in NRC, 2001.

Myth: Teachers Are Born• Myth: Teaching ability is a natural

predisposition– Teaching is an art that cannot be learned– The system does not learn; we rarely refine

the wisdom of practice• Teaching is a mass profession

– Ordinary people doing extraordinary things (Japan)

• Teaching is a skill, with a knowledge base

A Look Inside the CCSS for K–8 Mathematics

CCSS Principles

• Focus– Identifies key ideas, understandings and skills for

each grade or course– Stresses deep learning, which means applying

concepts and skills within the same grade or course• Coherence

– Articulates a progression of topics across grades and connects to other topics

– Vertical growth that reflects the nature of the discipline

CCSS Mathematical Practices

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

Grade Level Overview

Critical Area of Focus

Cross-cutting themes

Format of K–8 StandardsGrade Level

Domain

Standard

Cluster

CCSS Domain ProgressionK 1 2 3 4 5 6 7 8 HS

Counting & Cardinality

Number and Operations in Base Ten Ratios and Proportional Relationships Number &

QuantityNumber and Operations –Fractions

The Number System

Operations and Algebraic ThinkingExpressions and Equations Algebra

Functions Functions

Geometry Geometry

Measurement and Data Statistics and Probability Statistics & Probability

Mathematical Interlude

Underlying Principle

• “Everyone is good at mathematics because everyone can think. And mathematics is about thinking.” – Yeap Ban Har, National Institute of Education,

Singapore. • Corollary 1: Strategies that attempt to remove thinking

from learning are bound to fail in the long run.• Corollary 2: When learning is effective, “getting the right

answer” is but a small piece of the work.

The Division Problem

• The question was, “What is 42 ÷ 7?”• Here is a second grader’s solution. He did not write

anything. This is what he said:

I don’t know how many 7s are in 42, but I do know:40 ÷ 10 = 44 × 3 = 1212 + 2 = 1414 ÷ 7 = 22 + 4 = 6

The answer is 6.

40 = 4×10 = 4×(7 + 3)= 4×7 + 4×3

So 4 7s is less than 40 by 4×3, or 12So 4 7s is less than 42 by 14… which is 2 more 7s

The Chicken and Dog Problem

• Chickens and dogs are running around outdoors. Together they have 40 heads and 96 feet. How many dogs are there?

• Here is the second grader’s solution:

Imagine all the chickens are standing on one leg and all the dogs are standing on their hind legs.

Now there are 96 ÷ 2, or 48 feet on the ground.There are 8 extra legs. The chickens are all on one leg. The extra legs belong to dogs. There are 8 dogs.

x + y = 402x + 4y = 96

x + 2y = 48x + y = 40

y = 8

The Square and Triangle Problem

+ + + = 47 – = 1

• From the second equation, I know that square is one more than triangle.

• Then I imagined that the triangle in the first equation was a square.

• Then the sum would be one more, so the 4 squares would equal 48.

• The square is 12 and the triangle is 11.

= + 1

Add the two equations …

Divide by 4 …

Implications for Practice

• Instruction should take advantage of what students are thinking – So that symbol manipulations more often represent

ideas in their heads• Instruction should emphasize

– Standards for Mathematical Practice (CCSS)– Mathematical Process Standards (NCTM)– Strands of Mathematical Proficiency (NRC)

About Curriculum

Meanings of Curriculum

• Curriculum is a coherent progression of experiences that organize learning

• Curriculum is not – Standards– Curriculum frameworks– Collections of lesson plans– Instructional materials

• Curriculum is your plan for using these things to guide instruction

Predicted Probability of a Lesson Being Highly Rated

Source: Horizon Research, Inc.

Teachers as Curriculum Developers?

• Teacher-developed lessons often lack focus and rigor

• Teacher-collected lessons often lack coherence and a trajectory of learning

• Quality curricular materials require years of research, development, testing, and revision

• Teachers are unlikely to do much of this well in their spare time

Improving Curriculum and Instruction

• Teachers and administrators should work collaboratively to choose high-quality curricular materials

• Teachers should be supported, through professional learning communities, to develop shared understandings of how to use those materials effectively to meet the Standards

• Ohio’s Model Curriculum provides support for this kind of work

Implementation Suggestions, Challenges, and Resources

Research-Based Principles

• Implementation matters– Variation within a model is greater than the variation

between models– Adoption of standards, programs, or textbooks merely

opens the door• High-quality professional development

– Focuses on the content the teachers are teaching – Draws on curricular materials teachers are using– Involves analyzing student work– Takes time

Maintain Focus and Coherence

• Focus and coherence are not in the CCSS document

• Focus and coherence are in interpretationsof the document

• The goal is coherence in the heads of teachers and students– Think in chapters, not lessons (Daro)– Standards are taken as atoms, but the power is

in the bonds (Zimba)

What is needed …

• To build new practices around new content?

• Short-term: Awareness– Clarify and elaborating the standards – Tasks and student work

• Medium-term: Capacity building– Shared resources– Transitional materials, moving toward focus and coherence– Professional development infrastructure

• Long-term: New systems– Ongoing, focused professional development practices – Digital curricula? Open access materials?

Challenges

• Crosswalk documents may encourage rearrangements of low-quality curricular materials and frameworks– Aim for focused, forward-thinking crosswalk documents at the

level of clusters or big ideas• Unpacking standards (a la backwards design) may

perpetuate the atomized check-list mentality– Unpack clusters of standards via descriptive paragraphs

• Response to Intervention may be misused to sort students into groups that receive fundamentally different instruction – Begin with high-quality, Tier 1 instruction for all students

• Data-driven decision making may remain only about numbers– Use data to provoke targeted discussions about instruction

Challenges

• Formative assessment may be misconstrued as a task bank– Formative assessment must provide insight into student thinking

• Professional development may be largely generic and unfocused– Develop strategies for content-based professional learning

communities• Publishers may add chapters to existing materials

– Insist on materials with focus and coherence• Local control and limited resources may create excuses

– Share and borrow materials– Leverage resources– Take advantage of the assessment consortia

Special High School Challenges

• Deeply entrenched practices tied to particular course names

• Algebra 1: Does it belong in eighth or ninth grade? – This is the wrong question

• New content and perspectives– Geometry, algebra, statistics, modeling

• Career readiness and connecting to college readiness• Belief in teachers’ content expertise

– Hard to get the conversations started

What Should Districts Do Now?

• Get to know the CCSSM through Professional Learning Communities– Use the “critical areas” – Take a “progressions view”

• Begin developing the Mathematical Practices • Develop support structures for struggling students

– Use previous mathematics in service of new ideas– All students need access to the regular curriculum, a la

Response to Intervention• Watch for new opportunities and resources• Be skeptical of easy alignment and quick fixes

What Should States Do Now?

• Establish a sensible timeline for implementation – Some states plan early phase-in for some grades (e.g., K-2)

• Develop awareness of the CCSS and the timeline– Some states plan “road shows” emphasizing the critical areas,

the mathematical practices, and a “progressions view”• Be modest in solo development

– Exploit the assessment consortia and other partnerships– Create new opportunities to collaborate– Anticipate resources developed by other groups

• Build within-state leadership networks– Identify individuals for leadership roles– Identify schools to pilot new materials and strategies

Implementation Questions for You

• Can we get the incentives right? – So that teachers will regularly work together to reach

more students more of the time– So that we all learn from and with our best teachers

• Can we bring mathematics leadership to the decision-making table?– So that school-improvement efforts focus on long-

term improvements not short-term fixes• Can we learn to share strategies and resources?

– So that we create economies of scale

Reporting Progress

• Can we develop district habits of tracking the impact of innovations, interventions, including the level of implementation?

Resources for Implementation

Supplemental Grants to Assessment Consortia• SBAC plans to develop

– Curriculum Materials– Formative Processes and Tools/Professional Development– Multi-state Collaborative Supporting Implementation of Common

Core Systems– Communications

• PARCC plans to develop– Collaborative efforts to develop the highest priority curricular

and instructional tools– Multi-state support to build leadership cadres of educators who

are deeply engaged in the use of those tools, the CCSS and the PARCC assessments.

Activities of AMTE, ASSM, NCSM, NCTM

• Major outreach initiative supporting implementation of the CCSS (proposed)– Toolkit and regional leadership meetings

• Assessment Working Group drawing on lessons from curriculum development

• Joint Committee for ongoing collaborations supporting CCSS implementation

• Conceptual framework for teacher professional development systems

• Research agenda focused on implementation of the CCSS

NCSM Activities

• Tool for analyzing instructional materials and related professional development activities– Bill Bush, chair, and CCSSO– Target release date: Spring, 2011

• Illustrating the Standards for Mathematical Practiceprofessional development materials– Drawing on http://www.insidemathematics.org

• Webinars on the CCSS– Available at http://www.mathedleadership.org/

NCTM Activities

• CCSS PowerPoints posted on nctm.org• Making it Happen: A Guide to Implementing and

Interpreting the Common Core Standards– See http://www.nctm.org/mih

• Leadership Conference: Focus in High School Mathematics—Reasoning, Sense Making & Connections, late July, 2011

• Interactive Institutes: Algebra Readiness & the Common Core, early August, 2011 and following

CCSS Companion Resources

• Standards Progressions (McCallum)– Describes how ideas connect and grow across

grades– Technical appendix (Zimba) highlights structural

features that are not highly visible in the document• Illustrative Mathematics (McCallum)

– Review board and task vetting process– See http://www.illustrativemathematics.org/