Transitioning to the Common Core: Changing the Definition of Mathematical

Proficiency

Patrick CallahanStatewide Co-Director, California Mathematics Project

UCLA

OCMC Feb 24, 2014

What do we mean by implementing the Common Core?

Many districts and even states are claiming or planning to fully implement the Common Core by 2014 or 2015.

“fully implemented?”

From a student’s perspective the first time the Common Core could be fully implemented is a student graduating in 2024.

Before that time every student will experience a hybrid of Common Core and previous mathematics.

“fully implemented?”

From a student’s perspective the first time the Common Core could be fully implemented is a student graduating in 2024.

Before that time every student will experience a hybrid of Common Core and previous mathematics.

CONGRATS CLASS OF 2014 !

You have experienced about 7.692% Common Core!

“Fully implementing” and text books

With a little $$$ we took our old textbook…

And bought new Common Core

textbooks!

Implementation and Textbooks

Implementation and Textbooks

Implementation vs Transition

The word “implementation” tends to refer to the policy aspects of adopting the Common Core.

In a policy sense you can be “fully implemented” right away.

Another, more student-centric, approach is to think in terms of “transition” rather than “implementation”.

This is a pragmatic approach that acknowledges that student, parents, teachers, and systems are where they are now and that it will take time to move the system to the Common Core.

Transition to What?

We use the phrase “implement the Common Core”

or “transition to the Common Core”

but what does that mean?

What exactly are the Common Core Standards?

Common Core Standards, what they are NOT and what they ARE:

The Common Core standards are not a list of topics to be covered or taught.

The Common Core State Standards are a description of the mathematics students are expected to understand and use, not a curriculum. The standards are not the building blocks of curriculum, they are the achievements we want students to attain as the result of curriculum. To quote page 5 of the Common Core State Standards for Mathematics (Common Core): “Just because topic A appears before topic B in the standards for a given grade, it does not necessarily mean that topic A must be taught before topic B. A teacher might prefer to teach topic B before topic A, or might choose to highlight connections by teaching topic A and topic B at the same time. Or, a teacher might prefer to teach a topic of his or her own choosing that leads, as a byproduct, to students reaching the standards for topics A and B.”

How are the CCSS different?The CCSS are reverse engineered from an analysis of what students need to be college and career ready.

The design principals were focus and coherence. (No more mile-wide inch deep laundry lists of standards)

The CCSS in Mathematics have two sections:CONTENT and PRACTICES

The Mathematical Content is what students should know.The Mathematical Practices are what students should do.

Real life applications and mathematical modeling are essential.

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.

Mathematical Practice

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REASONING AND EXPLAINING2. Reason abstractly and quantitatively3. Construct viable arguments and critique the reasoning of others

MODELING AND USING TOOLS4. Model with mathematics5. Use appropriate tools strategically

SEEING STRUCTURE AND GENERALIZING7. Look for and make use of structure8. Look for and express regularity in repeated reasoning

Shifts in Content

Because the Common Core were reverse engineered from a definition of Career and College Ready, there were shifts in content.

How is Algebra different?

More applications, modeling, equivalence

Less algorithms, answer-getting, simplifying

Sample Algebra Worksheet

This should look familiar.What do you notice?

What is the mathematical goal?

What is the expectation of the student?

A sample Algebra Exam

A sample Algebra Exam

I typed #16 into Mathematica

Look at the circled answers.What do you notice?

“Answer Getting”As Phil Daro has mentioned:There is a difference between

using problems to “get answers” and to learn mathematics.

This algebra exam sends a clear message to students:

Math is about getting answers.

Note also that there is no context, just numbers and expressions

What are these assessing?

SBAC Claims

SBAC Item

Real-life?

So what is a vision for common core algebra?

Grade K-5: A Story of UnitsGrades 6-8: A Story of RatiosGrades 9-12: A Story of Functions

A narrative arc for algebra

CONTEXTS(identifying quantities in meaningful situations)

FUNCTIONS(modeling, relationships between quantities)

EQUATIONS(solving, manipulation, symbolism)

GENERALIZATION(structure, precision, abstracting)

Families of Functions: Linear (one variable) Linear (two variables) Quadratic Polynomial and Rational Exponential Trigonometric

What evidence does this item support?

New expectations require new Pathways

Changing expectationsThe trouble with course names

In the particular case of mathematics, there is a “vocabulary” around the names of mathematics courses that is likely to cause confusion not only for educators, but also for parents. “Algebra 1” is a course that, prior to CA CCSSM, has been taught in 8th grade to an increasing number of students. That same course name will be the default for ninth grade for most students who moving forward will complete the CA CCSSM for grade eight – a course that is more rigorous and more demanding than the earlier versions of “Algebra 1.” Even so, we expect the changes to cause confusion. The single most practical solution is to describe detailed course contents, in addition to course names, as a way of clearing up confusion until “Algebra I” as commonly used, refers to a ninth grade and not an eighth grade course

Changing expectationsThe trouble with course names

In the particular case of mathematics, there is a “vocabulary” around the names of mathematics courses that is likely to cause confusion not only for educators, but also for parents. “Algebra 1” is a course that, prior to CA CCSSM, has been taught in 8th grade to an increasing number of students. That same course name will be the default for ninth grade for most students who moving forward will complete the CA CCSSM for grade eight – a course that is more rigorous and more demanding than the earlier versions of “Algebra 1.” Even so, we expect the changes to cause confusion. The single most practical solution is to describe detailed course contents, in addition to course names, as a way of clearing up confusion until “Algebra I” as commonly used, refers to a ninth grade and not an eighth grade course

An important equation:

Algebra 1 ≠ Algebra 1

Previous 8th grade CA standards

Crosswalks are not the answer

Changing expectations:Middle School is key

When the expectations for middles school mathematics were about speed and accuracy of computations it made sense to accelerate in middle school, and even skip grades.

This no longer makes sense.

Middle school mathematics is the key to success for all students. Rushing or skipping is a bad idea for almost all students.

NCEE Report (May, 2013)http://www.ncee.org/college-and-work-ready/

NCEE Summary Findings: Career and College Ready

1. Many community college career programs demand little or no use of mathematics. To the extent that they do use mathematics, the mathematics needed by first year students in these courses is almost exclusively middle school mathematics. But the failure rates in our community colleges suggest that many of them do not know that math very well. A very high priority should be given to the improvement of the teaching of proportional relationships including percent, graphical representations, functions, and expressions and equations in our schools, including their application to concrete practical problems.

NCEE Summary Findings: Career and College Ready

3. It makes no sense to rush through the middle school mathematics curriculum in order to get to advanced algebra as rapidly as possible. Given the strong evidence that mastery of middle school mathematics plays a very important role in college and career success,  strong consideration should be given to spending more time, not less, on the mastery of middle school mathematics, and requiring students to master Algebra I no later than the end of their sophomore year in high school, rather than by the end of middle school. This recommendation should be read in combination with the preceding one. Spending more time on middle school mathematics is in fact a recommendation to spend more time making sure that students understand the concepts on which all subsequent mathematics is based. It does little good to push for teaching more advanced topics at lower grade levels if the students’ grasp of the underlying concepts is so weak that they cannot do the mathematics. Once students understand the basic concepts thoroughly, they should be able to learn whatever mathematics they need for the path they subsequently want to pursue more quickly and easily than they can now

Common Core Grade 8 Curriculum Plan

Common Core is much more rigorous than previous middle school expectations.

CA Framework on Acceleration

1. Decisions to accelerate students into the Common Core State Standards for higher mathematics before ninth grade should not be rushed.

Placing students into an accelerated pathway too early should be avoided at all costs. It is not recommended to compact the standards before grade seven to ensure that students are developmentally ready for accelerated content. In this document, compaction begins in seventh grade for both the traditional and integrated sequences.

Framework Suggested Pathways

Better than accelerating Middle School.

But doubling up is not necessary!

Framework Suggested Pathways

Better than accelerating Middle School.

But doubling up is not necessary! “Pre-calculus” is not necessary!

A better pathway:

Enhanced means: Include the (+) standards, go deeper, more rigorous, not skim faster!

Unit Blueprints

We believe that the optimal way to achieve design curriculum is not to shuffle standards, but to identifying multi-week curricular units that form a natural pedagogical flow based on the mathematical coherence. This approach will lead students to achieving the standards. To be clear, these units are not simply groups of standards. Achieving a given standard will be the result of a sequence of experiences over many units throughout the academic year. The content in specific standards will be woven into multiple units and within certain units connections will be made between standards from different domains.

Unit Blueprints will provide structure and coherence that, over time, could be fleshed out to become a full curriculum. We see this as a gap solution that could become permanent.

Units have many types of lessons that have different purposes

INTRO LESSON

Purpose: Engage students, spark curiosity, “hook” and necessitate

CONCEPTLESSON

CONCEPTLESSON

GETTING PRECISELESSON

Sequence of problems or activities, purpose to develop specific concepts, designed to scaffold, outcome is a delicate (fragile) understanding

Purpose: attend to precision, pin down definitions, conventions, symbolism

GETTING GENERALLESSON

Purpose: use concepts across contexts, generalize via variables and parameters and different types of numbers, operations, functions, structures

CONCEPTLESSON

FORMATIVE ASSEMENT

LESSON

ROBUSTNESSAND

DIFFEREN-TIATIONLESSONS

ROBUSTNESSAND

DIFFEREN-TIATIONLESSONS

SUMMATIVE ASSESSMENT

LESSON

Different students work on different things, goals of both moving from a fragile to robust understanding via a variety on problems

Some possible examples:

Designing for opportunities for SMPs happens at the unit level.

CONCEPTLESSON

……CLOSURELESSON

Purpose: Revisit and organize the unit goals and outcomes

Design criteria for selecting and sequencing activities within a Unit

What is the task/lesson/project/activity and its purpose?

Where does it fit within the sequence of the unit?

How does it accomplish its purpose (including instructional strategies)?

Why is this coherent mathematically and pedagogically? (i.e. what is the rationale)

Course Plans and Unit Blueprints

With support from the

Project Personnel

Illustrative MathematicsBill McCallum, University of ArizonaKristin Umland, University of New MexicoPatrick Callahan, University of California Los Angeles

High Tech HighJade White, 9th GradeSarah Strong, 10th GradeAmy Callahan, 11th Grade

MathaliciousKate NowakKarim Kai Ani

Contact: callahan.web@gmail.com