Focus, Coherence, and Rigor

1. Mathematical Shifts of the Common Core State Standards May 2013 Common Core Training for Administrators High School Mathematics Division of Academics, Accountability, and School Improvement Focus, Coherence, and Rigor

2. Common Core State Standards for Mathematics: Focus, Coherence and Rigor AGENDA Purpose and Vision of CCSSM Shifts in Mathematics Design and Organization Expectations of Student Performance Instructional Implications: Engaging in Mathematical Practices’ Look-fors CCSSM Resources: Websites Reflections / Questions and Answers

3. 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 __________________

4. How do you know?

5. Good Mathematics is NOT how many answers you know… but how you behave when you don’t know.

6. M-DCPS Florida’s Common Core State Standards Implementation Timeline F L F L • 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

7. Purpose and Vision of the CCSSM

8. Common Core State Standards Mission 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.

9. Ministry of Education Singapore “Teach Less, Learn More…”

10. Mathematical Shifts of the Common Core State Standards Focus, Coherence, and Rigor Fluency, Deep Understanding, Application, Intensity

11. Mathematical Shifts FOCUSdeeply on what is emphasized in the Standards COHERENCE: Think across grades, and link to major topics within grades RIGOR:Requires Fluency Deep Understanding Model/Apply Dual Intensity

12. 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

13. Mathematics Shift 1: Focus Spend more time on Fewer Concepts http://www.fldoe.org/schools/ccc.asp

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

15. Pair-share activity: Describe what is and what is not FOCUS

16. 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. Eachstandard 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

17. Mathematics Shift 2: Coherence Keep Building on learning year after year http://www.fldoe.org/schools/ccc.asp

18. Pair-share activity: Describe what is and what is not COHERENCE

19. 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 expected to have speed and accuracy with simple calculations and procedures so that they are more able to understand and manipulate more complex concepts. Students are able to apply a variety of appropriateproceduresflexiblyas they solve problems. Spend time Practicing (First Component of Rigor) Achievethecore.org

20. Key Fluencies Achievethecore.org

21. Fluency in High School • Algebra I Fluency Recommendations • Analytic geometry of lines • Add, subtract, and multiply polynomials. • Transforming expressions and “chunking” (seeing parts of an expression as a single object) • Geometry Fluency Recommendations • Triangle congruency and similarity • Use coordinates to establish geometric results • Use construction tools • Algebra II Fluency Recommendations • Divide polynomials with remainders by inspection in simple cases. • Rewrite expressions • Translate between recursive definitions and closed forms

22. Mathematics Shift 3: Fluency Spend time Practicing http://www.fldoe.org/schools/ccc.asp

23. Pair-share activity: Describe what is and what is not FLUENCY

24. Shift 4: Deep Conceptual Understanding 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. Students demonstrate deep conceptual understanding of core math concepts by applying them to new situations as well as writing and speaking about their understanding. UnderstandMath, DoMath, andProveit (Second Component of Rigor) Achievethecore.org

25. Mathematics Shift 4: Deep Understanding UnderstandMath, DoMath, andProveit http://www.fldoe.org/schools/ccc.asp

26. Pair-share activity: Describe what is and what is not DEEP CONCEPTUAL UNDERSTANDING

27. Shift 5: Applications (Modeling) 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. 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 (Third Component of Rigor) Achievethecore.org

28. Mathematics Shift 5: Application Apply math in Real World situations http://www.fldoe.org/schools/ccc.asp

29. Pair-share activity: Describe what is and what is not APPLICATIONS /MODELING

30. 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. Thinkfast and Solveproblems (Fourth Component of Rigor) Achievethecore.org

31. Mathematics Shift 6: Dual Intensity Thinkfast and Solveproblems http://www.fldoe.org/schools/ccc.asp

32. Pair-share activity: Describe what is and what is not DUAL INTENSITY

33. Design and Organization

34. Design and Organization • Standards for Mathematical Practice • Carry across all grade levels • Connect with content standards in each grade • Describe habits of mind of a mathematically expert student Standards for Mathematical Content • K - 8 grade-by-grade standards organized by domains that progress over several grades. • 9 – 12 high school standards organized by conceptual categories

35. Standards for Mathematical Practice

36. Standards for Mathematical Practices “The Standards for Mathematical Practice are unique in that theydescribe how teachers need to teach to ensure their students become mathematically proficient. We were purposeful in calling them standards because then they won’t be ignored.” ~ Bill McCallum

37. Mathematical Practices • Make sense of problems and persevere in solving them • Reason abstractly and quantitatively • Construct viable arguments and critique the reasoning of others • Model with mathematics • Use appropriate tools strategically • Attend to precision • Look for and make use of structure • Look for and express regularity in repeated reasoning

38. Overarching Habits of Mind of a Productive Mathematical Thinker 1. Make sense of problems and persevere in solving them 6. Attend to precision Reasoning and Explaining Seeing Structure and Generalizing Modeling and Using Tools 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 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

39. Overarching Habits of Mind of a Productive Mathematical Thinker Gather Information Make a plan Anticipate possible solutions Continuously evaluate progress Check results Question sense of solutions MP 1: Make sense of problems and persevere in solving them. MP 6: Attend to precision Mathematically proficient students can… Mathematically proficient students can… • use clear definitions and mathematical vocabulary to communicate reasoning • carefully specify units of measure and labels to clarify the correspondence with quantities in a problem explain the meaning of the problem monitor and evaluate their progress “Does this make sense?” use a variety of strategies to solve problems

40. Reasoning and Explaining MP 2: Reason abstractly and quantitatively. MP 3: Construct viable arguments and critique the reasoning of others Mathematically proficient students can… Mathematically proficient students can… • have the ability to contextualize and decontextualize (navigate between the concrete and the abstract). • understand and explain the computation methods they use. • make a mathematical statement (conjecture) and justify it • listen, compare, and critique conjectures and statements • manipulatives pictures symbols

41. Modeling and Using Tools MP 4: Model with Mathematics. MP 5: Use appropriate tools strategically Mathematically proficient students can… 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 • apply mathematics to solve problems that arise in everyday life • reflect on their attempt to solve problems and make revisions to improve their model as necessary

42. Seeing Structure and Generalizing MP 7: Look for and make use of structure MP 8: Look for and express regularity in repeated reasoning Mathematically proficient students can… Mathematically proficient students can… • look closely to determine possible patterns and structure (properties) within a problem • analyze patterns and apply them in appropriate mathematical context • notice repeating calculations and look for efficient methods/ representations to solve a problem • evaluate the reasonableness of their results throughout the problem solving process.

43. Mathematical Practices Indicators a second look . . .

44. A second look . . . #. Mathematical Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Match each given set of student indicators to a Mathematical Practice Match each given set of teacher indicators to a Mathematical Practice Find another group so you can complete a set of Mathematical Practices Post your conclusions on chart paper for sharing out

45. Content Standards and Progressions

46. The High School Mathematics Standards • Expect students to practice applying mathematical ways of thinking to real world issues and challenges • Require students to develop a depth of understanding and ability to apply mathematics to novel situations, as college students and employees regularly are called to do • Emphasize mathematical modeling, the use of mathematics and statistics to analyze empirical situations, understand them better, and improve decisions • Identify the mathematics that all students should study in order to be college and career ready We need to shift our focus from H. S. completion to College and Career Readiness for All students.

47. Format of the High School Standards • The high school standards are organized around five conceptual categories: • Number and Quantity • Algebra • Functions • Geometry • Statistics and Probability • Modeling is a sixth category that is embedded within all other conceptual categories. Conceptual categories portray a coherent view of high school mathematics; a student’s work with functions, for example crosses a number of traditional course boundaries, potentially up through and including calculus.

48. Format of the High School Standards • Content categories: overarching ideas that describe strands of content in high school • Domains/Clusters: groups of standards that describe coherent aspects of the content category • Standards: define what students should know and be able to do at each grade level Notations within the Common Core State Standards for Mathematics • Modeling Standards are indicated with a • Standards indicated as (+) are beyond the college and career readiness level but are necessary for advanced mathematics courses, such as calculus, discrete mathematics, and advanced statistics. • Standards with a (+) may still be found in courses expected for all students.

49. Format of the High School Standards Domain Cluster Conceptual Category Overview

50. Format of the High School Standards Conceptual Category: Number and Quantity Overview Standards Cluster Domain Beyond the college and career readiness level