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Focus, Coherence, and Rigor May 2013 Common Core Training for Administrators Middle Grades Mathematics Division of Academics, Accountability, and School Improvement

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Mathematical Shifts of the Common Core State Standards: Focus, Coherence and Rigor AGENDA Purpose and Vision of CCSSM Implementation Timeline Six Shifts in Mathematics Design and Organization Instructional Implications: Classroom Look-fors Expectations of Student Performance CCSSM Resources: Websites Reflections / Questions and Answers

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Community Norms We are all learners today We are respectful of each other We welcome questions We share discussion time We turn off all electronic devices __________________

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Why Common Standards? Previously, every state had its own set of academic standards and different expectations of student performance. Consistency Common standards can help create more equal access to an excellent education. Equity Students need the knowledge and skills that will prepare them for college and career in our global economy. Opportunity These new standards are clear and coherent in order to help students, parents, and teachers understand what is expected. Clarity

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College and Career Readiness: Anchor for the Common Core The Common Core State Standards were back-mapped from the anchor of college and career readiness because governors and state school chiefs realized there was a significant gap between high school expectations for students and what students are expected to do in college/career. – Among high school graduates, about only half are academically prepared for postsecondary education.

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Remediation rates and costs are staggering As much as 40% of all students entering 4-year colleges need remediation in one or more courses As much as 63% in 2-year colleges Degree attainment rates are disappointing Fewer than 42% of adults aged hold college degrees Source: The College Completion Agenda 2010 Progress Report, The College Board College Remediation and Graduation Rates

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Are We Mathematically Ready for College and Careers?

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The Common Core State Standards provide a consistent, clear understanding of what students are expected to learn, so teachers and parents know what they need to do to help them. The standards are designed to be robust and relevant to the real world, reflecting the knowledge and skills that our young people need for success in college and careers. With American students fully prepared for the future, our communities will be best positioned to compete successfully in the global economy. Common Core State Standards Mission

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F – Full Implementation of CCSSM L – Full implementation of content area literacy standards including: text complexity, quality and range in all grades (K-12) B – Blended instruction of CCSS with NGSSS; last year of NGSSS assessed on FCAT 2.0 (Grades 3-8); 4th quarter will focus on NGSSS/CCSSM grade level content gaps Florida’s Common Core State Standards Implementation Timeline M-DCPS Year / Grade level K123 – 89 – F LLLLL F L LLL CCSS fully implemented F L B L CCSS fully implemented and assessed F L

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FCAT FCAT FCAT FCAT PARCC PARCC PARCC PARCC PARCC K K K123 (4 th 9 weeks Common Core Lockdown) (4 th 9 weeks Common Core Lockdown) (4 th 9 weeks Common Core Lockdown) (4 th 9 weeks Common Core Lockdown) (4 th 9 weeks Common Core Lockdown) (4 th 9 weeks Common Core Lockdown) NGSSS CCSSM NGSSS

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FOCUS deeply on what is emphasized in the Standards COHERENCE: Think across grades, and link to major topics within grades RIGOR: Require- Fluency Dual Intensity Deep Understanding Model/Apply Mathematical Shifts

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Shift 1: Focus Teachers use the power of the eraser and significantly narrow and deepen the scope of how time and energy is spent in the math classroom. They do so in order to focus deeply on only the concepts that are prioritized in the standards so that students reach strong foundational knowledge and deep conceptual understanding and are able to transfer mathematical skills and understanding across concepts and grades. Students are able to transfer mathematical skills and understanding across concepts and grades. – Spend more time on Fewer Concepts Achievethecore.org

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What the Student Does…What the Teacher Does… Spend more time on fewer concepts Extract content from the curriculum Focus instructional time on priority concepts Give students the gift of time Mathematics Shift 1: Focus Spend more time on Fewer Concepts

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- Ministry of Education, Singapore

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K-8 Priorities in Math Priorities in Support of Rich Instruction and Expectations of Fluency and Conceptual Understanding K–2Addition and subtraction, measurement using whole number quantities 3–5Multiplication and division of whole numbers and fractions 6Ratios and proportional reasoning; early expressions and equations 7Ratios and proportional reasoning; arithmetic of rational numbers 8Linear algebra Achievethecore.org

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Shift 2: Coherence Principals and teachers carefully connect the learning within and across grades so that students can build new understanding onto foundations built in previous years. Teachers can 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. A student’s understanding of learning progressions can help them recognize if they are on track. Keep Building on learning year after year Achievethecore.org

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What the Student Does…What the Teacher Does… Build on knowledge from year to year, in a coherent learning progression Connect the threads of critical areas across grade levels Connect to the way content was taught the year before and will be taught the following years Focus on priority progressions Mathematics Shift 2: Coherence Keep Building on learning year after year

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Shift 3: Fluency Teachers help students to study algorithms as “general procedures” so they can gain insights to the structure of mathematics (e.g. organization, patterns, predictability). Students are able to apply a variety of appropriate procedures flexibly as they solve problems. Students are expected to have speed and accuracy with simple calculations and procedures so that they are more able to understand and manipulate more complex concepts. Spend time Practicing Achievethecore.org Rigor (First Component of Rigor )

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What the Student Does…What the Teacher Does… Spend time practicing, with intensity, skills (in high volume) Push students to know basic skills at a greater level of fluency Focus on the listed fluencies by grade level Uses high quality problem sets, in high volume Mathematics Shift 3: Fluency Spend time Practicing

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GradeRequired Fluency KAdd/subtract within 5 1Add/subtract within 10 2 Add/subtract within 20 Add/subtract within 100 (pencil and paper) 3 Multiply/divide within 100 Add/subtract within Add/subtract within 1,000,000 5Multi-digit multiplication 6 Multi-digit division Multi-digit decimal operations 7Solve px + q = r, p(x + q) = r 8Solve simple 2 2 systems by inspection K-8 Key Fluencies Achievethecore.org

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Shift 4: Deep Conceptual Understanding Students demonstrate deep conceptual understanding of core math concepts by applying them to new situations as well as writing and speaking about their understanding. Understand Math, Do Math, and Prove it Teachers teach more than “how to get the answer;” they support students’ ability to access concepts from a number of perspectives so that students are able to see math as more than a set of mnemonics or discrete procedures. Achievethecore.org Rigor (Second Component of Rigor )

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What the Student Does…What the Teacher Does… Show mastery of material at a meaningful level Articulate mathematical reasoning Demonstrate deep conceptual understanding of priority concepts Explain and justify their thinking Create opportunities for students to understand the “answer” from a variety of access points Ensure that students understand WHY they are doing what they’re doing-ASK PROBING QUESTIONS Guide student thinking instead of telling the next step Continuously self reflect and build knowledge of concepts being taught Mathematics Shift 4: Deep Understanding Understand Math, Do Math, and Prove it

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Shift 5: Applications (Modeling) Students are expected to use math and choose the appropriate concept for application even when they are not prompted to do so. Apply math in Real World situations Teachers provide opportunities to apply math concepts in “real world” situations. Teachers in content areas outside of math ensure that students are using math to make meaning of and access content. Rigor (Third Component of Rigor )

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What the Student Does…What the Teacher Does… Utilize math in other content areas and situations, as relevant Choose the right math concept to solve a problem when not necessarily prompted to do so Apply math including areas/ courses where it is not directly required (i.e. in science) Provide students with real world experiences and opportunities to apply what they have learned Mathematics Shift 5: Application (Modeling) Apply math in Real World situations

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Shift 6: Dual Intensity There is a balance between practice and understanding; both are occurring with intensity. Teachers create opportunities for students to participate in “drills” and make use of those skills through extended application of math concepts. Think fast and Solve problems Achievethecore.org Rigor (Fourth Component of Rigor )

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What the Student Does…What the Teacher Does… Practice math skills with an intensity that results in fluency Practice math concepts with an intensity that forces application in novel situations Find the balance between conceptual understanding and practice within different periods or different units Be ambitious in demands for fluency and practice, as well as the range of application Mathematics Shift 6: Dual Intensity Think fast and Solve problems

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Design and Organization

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Design and Organization Standards for Mathematical Content KK-8 standards presented by grade level OOrganized into domains that progress over several grades GGrade introductions give 2–4 focal points at each grade level Standards for Mathematical Practice CCarry across all grade levels DDescribe habits of mind of a mathematically expert student

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Mathematical Practices 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

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Learning Experiences It matters how students learn Learning mathematics is more than just learning concepts and skills. Equally important are the cognitive and metacognitive process skills. These processes are learned through carefully constructed learning experiences. For example, to encourage students to be inquisitive and have a deeper understanding of mathematics, the learning experiences must include carefully structured opportunities where students discover mathematical relationships and principles on their own.

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Reasoning and Explaining Seeing Structure and Generalizing Overarching Habits of Mind of a Productive Mathematical Thinker 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 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning 1. Make sense of problems and persevere in solving them 6. Attend to precision

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Mathematically proficient students can… explain the meaning of the problem and look for entry points to its solution monitor and evaluate their progress and change course if necessary use a variety of strategies to solve problems Overarching Habits of Mind of a Productive Mathematical Thinker MP 1: Make sense of problems and persevere in solving them. Gather Information Make a plan Anticipate possible solutions Continuously evaluate progress Check results Question sense of solutions Mathematically proficient students can… use mathematical vocabulary to communicate reasoning and formulate precise explanations calculate accurately and efficiently and specify units of measure and labels within the context of the situation MP 6: Attend to precision

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Mathematically proficient students can… have the ability to contextualize and decontextualize problems involving quantitative relationships: decontextualize - to abstract a given situation and represent it symbolically and manipulate the representing symbols contextualize - to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Reasoning and Explaining MP 2: Reason abstractly and quantitatively. Mathematically proficient students can… make a mathematical statement (conjecture) and justify it listen, compare, and critique conjectures and statements MP 3: Construct viable arguments and critique the reasoning of others

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Mathematically proficient students can… apply mathematics to solve problems that arise in everyday life reflect and make revisions to improve their model as necessary map mathematical relationships using tools such as diagrams, two-way tables, graphs, flowcharts and formulas. Modeling and Using Tools MP 4: Model with Mathematics. Mathematically proficient students can… consider the available tools when solving a problem (i.e. ruler, calculator, protractor, manipulatives, software) use technological tools to explore and deepen their understanding of concepts MP 5: Use appropriate tools strategically -In early grades, this might be as simple as writing an addition equation to describe a situation. -In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. -In early grades, this might be as simple as writing an addition equation to describe a situation. -In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community.

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Mathematically proficient students can… look closely to determine possible patterns and structure (properties) within a problem analyze a complex problem by breaking it down into smaller parts Seeing Structure and Generalizing MP 7: Look for and make use of structure Mathematically proficient students can… notice repeating calculations and look for efficient methods/ representations to solve a problem generalize the process to create a shortcut which may lead to developing rules or creating a formula MP 8: Look for and express regularity in repeated reasoning

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GROUP ACTIVITY Each group will receive: An envelope containing 8 small cards and a handout of the listed mathematical practices. Instructions: Match each of the 8 small cards with its mathematical practice (using the handout of the listed mathematical practices).

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Reasoning and Explaining Seeing Structure and Generalizing Overarching Habits of Mind of a Productive Mathematical Thinker 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 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning 1. Make sense of problems and persevere in solving them 6. Attend to precision

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Content Standards and Progressions

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Domains Domains are larger groups/categories of standards that progress across grades Clusters Clusters are groups of related standards Content standards Content standards define what students should understand and be able to do Cluster Standard Domain

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New Florida Coding for CCSSMNew Florida Coding for CCSSM MACC.2.OA.1.1 Math Common Core Grade Level Domain Cluster MACC.7.EE.1.2 Standard Note: In the state of Florida, clusters will be numbered Note: In the state of Florida, clusters will be numbered.

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Florida Coding Scheme for Common Core State Standards MACC.8.EE.3.7 Identify the cluster Identify the grade level Identify the standard Identify the domain Find the standard in your CCSSM binder

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Common Core Progressions

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coherent progressions The Standards are designed around coherent progressions from grade to grade. across grades Teachers carefully connect the learning across grades so that students can build new understanding onto foundations built in previous years. Each standard is not a new event, but an extension to previous learning. Each standard is not a new event, but an extension to previous learning. Think across grades and link to major topics within grades

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6 3 Understand that shapes in different categories may share attributes. 6 Find the area of right triangles, other triangles, special quadrilaterals, and polygons. GeometryGeometry

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Progression- Activity In your groups, use the geometry progression handout to identify the grade level corresponding to each bullet.

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HS 6 3 Understand that shapes in different categories may share attributes. 6 Find the area of right triangles, other triangles, special quadrilaterals, and polygons. K Correctly name shapes regardless of their orientations and overall size. 1 Distinguish between defining attributes versus non‐defining attributes. 2 Recognize and draw shapes having special attributes. 4 Classify two-dimensional figures based on the presence or absence of parallel and perpendicular lines. 5 Understand that attributes belonging to a category of two‐dimensional figures also belong to all subcategories of that category. 7 Solve real‐world and mathematical problems involving area, volume and surface area. 8 Know the formulas for the volumes of cones, cylinders, and spheres. GeometryGeometry

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Common Core Progressions

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Grades 3-5 Elementary grades work with part to whole relationships in fraction form. Grade 6 In 6 th grade, the concept of ratio is introduced, and students look at tables of equivalent ratios and their representations on a graph. Grade 7 In 7th grade, students focus more on the use of multiple representations to solve proportion problems, including ratio tables, graphs, and equations, with a special focus on the constant of proportionality or unit rate. Grade 8 Finally in 8th grade, students extend the proportional reasoning developed in 6th and 7th grade to examine linear relationships with multiple representations. Ratio and Proportionality Grade Level Progression

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PARCC Sample Items vs. FCAT 2.0 Sample Items

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Next generation assessment system Technology-based Assesses at a conceptually DEEP level The Standards & The Assessment Define what students should understand and be able to do in their study of mathematics These standards are “focused” and “coherent” (i.e., conceptually DEEP)

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PARCC Assessments WHAT PROBLEMS WORTH DOING Multi-step problems, conceptual questions, applications, and substantial procedures will be common, as in an excellent classroom. BETTER STANDARDS DEMAND BETTER QUESTIONS Instead of reusing existing items, PARCC will develop custom items to the Standards. FOCUS PARCC assessments will focus strongly on where the Standards focus. Students will have more time to master concepts at a deeper level. HOW DRAG & DROP FILL-IN RESPONSES COMPARISONS RADIO BUTTONS / MC CHECK BOXES WRITTEN RESPONSES Transformative Formats FOCUS PARCC assessments will focus strongly on where the Standards focus. Students will have more time to master concepts at a deeper level. PROBLEMS WORTH DOING Multi-step problems, conceptual questions, applications, and substantial procedures will be common, as in an excellent classroom. BETTER STANDARDS DEMAND BETTER QUESTIONS Instead of reusing existing items, PARCC will develop custom items to the Standards.

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Overview of PARCC Mathematics Task Types Task TypeDescription of Task Type I. Tasks assessing concepts, skills and procedures Balance of conceptual understanding, fluency, and application Can involve any or all mathematical practice standards Machine scorable including innovative, computer-based formats Will appear on the End of Year and Performance Based Assessment components II. Tasks assessing expressing mathematical reasoning Each task calls for written arguments / justifications, critique of reasoning, or precision in mathematical statements (MP.3, 6). Can involve other mathematical practice standards May include a mix of machine scored and hand scored responses Included on the Performance Based Assessment component III. Tasks assessing modeling / applications Each task calls for modeling/application in a real-world context or scenario (MP.4) Can involve other mathematical practice standards May include a mix of machine scored and hand scored responses Included on the Performance Based Assessment component

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Design of PARCC Math Summative Assessments Performance Based Assessment (PBA) – Type I items (Machine-scorable) – Type II items (Mathematical Reasoning/Hand-Scored – scoring rubrics are drafted) – Type III items (Mathematical Modeling/Hand-Scored and/or Machine-scored - scoring rubrics are drafted) End-of-Year Assessment (EOY) – Type I items only (All Machine-scorable)

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Grade 6 Example Numbering / Ordering Numbers / Absolute Value

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FCAT 2.0 – Grade 6

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PARCC – Grade 6 Part a

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PARCC – Grade 6 Part b

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PARCC – Grade 6 Part c

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PARCC – Grade 6 Part d

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Office of Academics and Transformation Division of Academics, Accountability, & School Improvement Questions/Concerns: Department of Mathematics and Science Middle Grades Mathematics 1501 N.E. 2nd Avenue, Suite 326 Miami, FL Office: Fax:

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