Overall Outcomes

1. CCSS-M in the Classroom: Grades 3-5 Number and Operations FractionsWeaving Content and Standards for Mathematical Practices

2. Overall Outcomes • Recognize the interconnectedness of the Standards for Mathematical Practice and content standards in developing student understanding and reasoning. • Illuminate practices that establish a culture where mistakes are a springboard for learning, risk-taking is the norm, and there is a belief that all students can learn. • Deeping content knowledge and pedagogy within an important focus area for our grade band: • Number and Operations - Fractions

3. Effective Classrooms

4. What research says about effective classrooms The activity centers on mathematical under-standing, invention, and sense-making by all students. The culture is one in which inquiry, wrong answers, personal challenge, collaboration, and disequilibrium provide opportunities for mathematics learning by all students. The tasks in which students engage are mathematically worthwhile for all students. A teacher’s deep knowledge of the mathematics content she/he teaches and the trajectory of that content enables the teacher to support important, long-lasting student understanding

5. What research says about effective classrooms The activity centers on mathematical understanding, invention, and sense-making by all students. The culture is one in which inquiry, wrong answers, personal challenge, collaboration, and disequilibrium provide opportunities for mathematics learning by all students. The tasks in which students engage are mathematically worthwhile for all students. A teacher’s deep knowledge of the mathematics content she/he teaches and the trajectory of that content enables the teacher to support important, long-lasting student understanding

6. What research says about effective classrooms The activity centers on mathematical understanding, invention, and sense-making by all students. The culture is one in which inquiry, wrong answers, personal challenge, collaboration, and disequilibrium provide opportunities for mathematics learning by all students. The tasks in which students engage are mathematically worthwhile for all students. A teacher’s deep knowledge of the mathematics content she/he teaches and the trajectory of that content enables the teacher to support important, long-lasting student understanding

7. What research says about effective classrooms The activity centers on mathematical understanding, invention, and sense-making by all students. The culture is one in which inquiry, wrong answers, personal challenge, collaboration, and disequilibrium provide opportunities for mathematics learning by all students. The tasks in which students engage are mathematically worthwhile for all students. A teacher’s deep knowledge of the mathematics content she/he teaches and the trajectory of that content enables the teacher to support important, long-lasting student understanding.

8. Effective implies: Students are engaged with important mathematics. Lessons are very likely to enhance student understanding and to develop students’ capacity to do math successfully. Students are engaged in ways of knowing and ways of working consistent with the nature of mathematicians ways of knowing and working.

9. Reflection What is your current reality around classroom culture? What can you do to enhance your current reality?

10. Outcomes: Day 1 Reflect on teaching practices that support the shifts in the Standards for Mathematical Practice and content standards. Understand how to analyze student work with the Standards for Mathematical Practice and content standards. Analyze, adapt and implement a task with the integrity of the Common Core State Standards.

11. A message from OSPI

12. WA CCSS Implementation Timeline

13. Transition Plan for Washington State

14. Why Shift? Almost half of eighth-graders in high achieving countries showed they could reach the “advanced” level in math, meaning they could relate fractions, decimals and percent to each other; understand algebra; and solve simple probability problems. In the U.S., 7 percent met that standard. • Results from the 2011 TIMMS

15. The Three Shifts in Mathematics Focus: Strongly where the standards focus Coherence: Think across grades and link to major topics within grades Rigor: Require conceptual understanding, fluency, and application

16. Focus on the Major Work of the Grade • Two levels of focus: • What’s in/What’s out • The standards at each grade level are interconnected allowing for coherence and rigor

17. Focus in International Comparisons – Ginsburg et al., 2005 TIMSS and other international comparisons suggest that the U.S. curriculum is ‘a mile wide and an inch deep.’ “…On average, the U.S. curriculum omits only 17 percent of the TIMSS grade 4 topics compared with an average omission rate of 40 percent for the 11 comparison countries. The United States covers all but 2 percent of the TIMSS topics through grade 8 compared with a 25 percent noncoverage rate in the other countries. High-scoring Hong Kong’s curriculum omits 48 percent of the TIMSS items through grade 4, and 18 percent through grade 8.”

19. Content Emphasis by Cluster—Grade 3

20. Grade 3(supporting cluster)

22. Content Emphasis by Cluster—Grade 4

24. Content Emphasis by Cluster—Grade 5

25. Focus on Major Work In any single grade, students and teachers spend the majority of their time, approximately 75% on the major work of the grade. The major work should also predominate the first half of the year.

26. Engaging with the 3-5 Content How would you summarize the major work of 3-5? What would you have expected to be a part of the major work that is not? Give an example of how you would approach something differently in your teaching if you thought of it as supporting the major work, instead of being a separate, discrete topic.

27. Focus on Fractions One of the Major Works of the 3-5 Grade Band Deeping Content Knowledge and

28. Shifts - Implications for Fractions http://www.illustrativemathematics.org/pages/fractions_progression Grade 3: Developing an understanding of fractions as numbers is essential for future work with the number system. It is critical that students at this grade are able to place fractions on a number line diagram and understand them as a related component of their ever expanding number system.

29. Shift Two: CoherenceThink across grades, and link to major topics within grades Carefully connect the learning within and across grades so that students can build new understanding onto foundations built in previous years. Begin to count on solid conceptual understanding of core content and build on it. Each standard is not a new event, but an extension of previous learning.

30. CoherenceAcross and Within Grades It’s about math making sense. The power and elegance of math comes out through carefully laid progressions and connections within grades.

31. CoherenceThink across grades, and link to major topics within grades Carefully connect the learning within and across grades so that students can build new understanding onto foundations built in previous years. Begin to count on solid conceptual understanding of core content and build on it. Each standard is not a new event, but an extension of previous learning.

32. How will it look different? Varied problem structures that build on the student’s work with whole numbers 5 = 1 + 1 + 1 + 1 +1 builds to 5/3 = 1/3 + 1/3 + 1/3 + 1/3 + 1/3 and 5/3 = 5 x 1/3 Conceptual development before procedural Use of rich tasks-applying mathematics to real world problems Effective use of group work Precision in the use of mathematical vocabulary

33. Coherence -Think Across Grades

34. Coherence -Think Across Domains Grade 4: • Operations and Algebraic Thinking: • Students use four operations with whole numbers to solve problems. • Students gain familiarity with factors and multiples which supports student work with fraction equivalency. • Number and Operations Fractions: • Students build fractions from unit fractions by applying and extending previous understandings of operations with whole numbers.

35. The Structure is the Standards

36. Rigor: Illustrations of Conceptual Understanding, Fluency, and Application Here rigor does not mean “hard problems.” It’s a balance of three fundamental components that result in deep mathematical understanding. There must be variety in what students are asked to produce.

37. Some Old Ways of Doing Business Lack of rigor • Reliance on rote learning at expense of concepts • Severe restriction to stereotyped problems lending themselves to mnemonics or tricks • Aversion to (or overuse) of repetitious practice • Lack of quality applied problems and real-world contexts • Lack of variety in what students produce • E.g., overwhelmingly only answers are produced, not arguments, diagrams, models, etc.

38. Some Old Ways of Doing Business Concrete Semi Concrete Abstract Unfortunately this model (Jerome Bruner, 1964) was interpreted as giving hierarchal value to the symbolic above the concrete or semi concrete… Which lead to: Abstract Semi Concrete (used to “prove” or show why the abstract worked) and an implication that the concrete was only for those who didn’t “get it”

39. Desired outcome is a balance that leads to flexible thinking about concepts and an ability to apply knowledge in novel situations Conceptual and Procedural Understanding

40. How do students currently perceive mathematics? Doing mathematics means following the rules laid down by the teacher. Knowing mathematics means remembering and applying the correct rule when the teacher asks a question. Mathematical truth is determined when the answer is ratified by the teacher. -Mathematical Education of Teachers report (2012)

41. How do students currently perceive mathematics? Students who have understood the mathematics they have studied will be able to solve any assigned problem in five minutes or less. Ordinary students cannot expect to understand mathematics: they expect simply to memorize it and apply what they have learned mechanically and without understanding. -Mathematical Education of Teachers report (2012)

42. Redefining what it means to be “good at math” • Expect math to make sense • wonder about relationships between numbers, shapes, functions • check their answers for reasonableness • make connections • want to know why • try to extend and generalize their results • Are persistent and resilient • are willing to try things out, experiment, take risks • contribute to group intelligence by asking good questions • Value mistakes as a learning tool (not something to be ashamed of)

43. 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.

45. Poster Activity Standards for Mathematical Practices

47. Let’s look at the Assessment Shifts in Focus, Coherence and Rigor in the assessment

48. “Students can demonstrate progress toward college and career readiness in mathematics.” Assessment Claims for Mathematics • “Students can demonstrate college and career readiness in mathematics.” Overall Claim (Gr. 3-8) • “Students can explain and apply mathematical concepts and interpret and carry out mathematical procedures with precision and fluency.” Overall Claim (High School) • “Students can solve a range of complex well-posed problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies.” Claim 1 Concepts and Procedures • “Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others.” Claim 2 Problem Solving • “Students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems.” Claim 3 Communicating Reasoning Claim 4 Modeling and Data Analysis

49. Claim 1Concepts and Procedures Students can explain and apply mathematical concepts and interpret and carry out mathematical procedures with precision and fluency.

50. Cognitive Rigor and Depth of Knowledge • The level of complexity of the cognitive demand. • Level 1: Recall and Reproduction • Requires eliciting information such as a fact, definition, term, or a simple procedure, as well as performing a simple algorithm or applying a formula. • Level 2: Basic Skills and Concepts • Requires the engagement of some mental processing beyond a recall of information. • Level 3: Strategic Thinking and Reasoning • Requires reasoning, planning, using evidence, and explanations of thinking. • Level 4: Extended Thinking • Requires complex reasoning, planning, developing, and thinking most likely over an extended period of time.