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Transitioning to the Common Core: Changing the Definition of Mathematical Proficiency Patrick Callahan Statewide Co-Director, California Mathematics Project UCLA OCMC Feb 24, 2014

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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.

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“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.

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“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. You have experienced about 7.692% Common Core!

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“Fully implementing” and text books

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

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And bought new Common Core textbooks!

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Implementation and Textbooks

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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.

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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?

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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.”

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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.

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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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CCSS Mathematical Practices OVERARCHING HABITS OF MIND 1. Make sense of problems and persevere in solving them 6. Attend to precision REASONING AND EXPLAINING 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others MODELING AND USING TOOLS 4. Model with mathematics 5. Use appropriate tools strategically SEEING STRUCTURE AND GENERALIZING 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning

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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

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Sample Algebra Worksheet This should look familiar. What do you notice? What is the mathematical goal? What is the expectation of the student?

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A sample Algebra Exam

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I typed #16 into Mathematica

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Look at the circled answers. What do you notice?

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“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

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What are these assessing?

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SBAC Claims

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SBAC Item

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Real-life?

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So what is a vision for common core algebra?

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Grade K-5: A Story of Units Grades 6-8: A Story of Ratios Grades 9-12: A Story of Functions

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A narrative arc for algebra CONTEXTS (identifying quantities in meaningful situations) CONTEXTS (identifying quantities in meaningful situations) FUNCTIONS (modeling, relationships between quantities) FUNCTIONS (modeling, relationships between quantities) EQUATIONS (solving, manipulation, symbolism) EQUATIONS (solving, manipulation, symbolism) GENERALIZATIO N (structure, precision, abstracting) GENERALIZATIO N (structure, precision, abstracting) Families of Functions: Linear (one variable) Linear (two variables) Quadratic Polynomial and Rational Exponential Trigonometric Families of Functions: Linear (one variable) Linear (two variables) Quadratic Polynomial and Rational Exponential Trigonometric

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What evidence does this item support?

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New expectations require new Pathways

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Changing expectations The 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 8 th 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

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Changing expectations The 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 8 th 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

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An important equation: Algebra 1 ≠ Algebra 1

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Previous 8 th grade CA standards

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Crosswalks are not the answer

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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.

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

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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.

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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

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Common Core Grade 8 Curriculum Plan Common Core is much more rigorous than previous middle school expectations.

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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.

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Framework Suggested Pathways Better than accelerating Middle School. But doubling up is not necessary!

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Framework Suggested Pathways Better than accelerating Middle School. But doubling up is not necessary! “Pre-calculus” is not necessary!

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

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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.

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Units have many types of lessons that have different purposes INTRO LESSON Purpose: Engage students, spark curiosity, “hook” and necessitate CONCEPT LESSON CONCEPT LESSON GETTING PRECISE LESSON 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 GENERAL LESSON Purpose: use concepts across contexts, generalize via variables and parameters and different types of numbers, operations, functions, structures CONCEPT LESSON FORMATIVE ASSEMENT LESSON ROBUSTNESS AND DIFFEREN- TIATION LESSONS ROBUSTNESS AND DIFFEREN- TIATION LESSONS SUMMATIVE ASSESSMENT LESSON 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. CONCEPT LESSON … … CLOSURE LESSON Purpose: Revisit and organize the unit goals and outcomes

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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)

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Course Plans and Unit Blueprints With support from the

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Project Personnel Illustrative Mathematics Bill McCallum, University of Arizona Kristin Umland, University of New Mexico Patrick Callahan, University of California Los Angeles High Tech High Jade White, 9 th Grade Sarah Strong, 10 th Grade Amy Callahan, 11 th Grade Mathalicious Kate Nowak Karim Kai Ani Contact: callahan.web@gmail.com

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