August 13, 2011 Elaine Watson, Ed.D . elaine.watson0729@gmail

1. Common Core State Standards in MathematicsHow They Impact CTE Programs in the State of New York August 13, 2011 Elaine Watson, Ed.D. elaine.watson0729@gmail.com

2. Agenda Introductions Timeline for Transition to Common Core State Standards Overview of CCSS for Grades 6 – 8 and High School Math Content Standards Standards for Mathematical Practice Next Generation Assessments Six Shifts in Mathematics What Does This Mean for CTE Administrators and Teachers? Tools for a Smooth Transition Next Steps

3. Introductions Your name and role Any specific thing that you would like to discuss that is NOT on the agenda

4. “Math is more than numbers and equations; it is logical and intellectually totally honest, an equity that is useful in ANY field in life.” Dan Douer, MD Leader, Acute Lymphoblastic Leukemia Program Memorial Sloan-Kettering Cancer Center

5. Timeline for Transition to Common Core State Standards 2014 - 2015 2011- 2012 2012- 2013 2013- 2014 We are here First CCSS Assessment

6. Overview of CCSS Standards for Mathematical Content Standards for Mathematical Practice Specific to Grade Level K – 8 High School Same for All Grade Levels

8. Think of the K – 8 Math Content Standards as forming the trunk of a tree…the core …giving students the foundation for Mathematical Modeling in High School. See the “Math Tree” handout

9. Structure of High School Mathematics Content Standards High School Content Standards are listed in conceptual categories Number and Quantity Algebra Functions Modeling Geometry Statistics and Probability

10. Number and Quantity Overview Structure of High School Mathematics Content Standards • The Real Number System • Quantities • The Complex Number System • Vector and Matrix Quantities

11. Algebra Overview Structure of High School Mathematics Content Standards • Seeing Structures in Expressions • Arithmetic with Polynomials and Rational Expressions • Creating Equations • Reasoning with Equations and Inequalities

12. Functions Overview Structure of High School Mathematics Content Standards • Interpreting Functions • Building Functions • Linear, Quadratic, and Exponential Models • Trigonometric Functions

13. Geometry Overview Structure of High School Mathematics Content Standards • Congruence • Similarity, Right Triangles, and Trigonometry • Circles • Expressing Geometric Properties with Equations • Geometric Measurement and Dimension • Modeling with Geometry

14. Statistics and Probability Overview Structure of High School Mathematics Content Standards • Interpreting Categorical and Quantitative Data • Making Inferences and Justifying Conclusions • Conditional Probability and the Rules of Probability • Using Probability to Make Decisions

15. Standards for Mathematical Practice Describe ways in which student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity

16. Standards for Mathematical Practice Provide a balanced combination of Procedure and Understanding Shift the focus to ensure mathematical understanding over computation skills

17. Eight Standards for Mathematical Practice Students will be able to: 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.

18. Modeling with Mathematics Modeling is the processof choosing and using appropriate mathematics toanalyzeempirical situations tounderstand them better, and to improve decisions.

19. Modeling with Mathematics Modeling a situation is a creative process that involves making choices. The Next Generation Assessments associated with the CCSS will expect students to apply mathematical knowledge to model real world situations when they are not prompted as to exactly HOWto apply the mathematics.

20. Modeling with Mathematics Example of problem situations that need to be modeled mathematically in order to solve: Estimate how much water and food is needed for emergency relief in a devastated city of 3 million people, and how it might be distributed

21. Modeling with Mathematics Example of problem situations that need to be modeled mathematically in order to solve: Plan a table tennis tournament for 7 players at a club with 4 tables, where each player plays against each other player

22. Modeling with Mathematics Example of problem situations that need to be modeled mathematically in order to solve: Analyze the stopping distance for a car Analyze the growth of a savings account balance or of a bacterial colony

23. Modeling with Mathematics Models devised depend upon a number of factors: How precise do we need to be? What aspects do we most need to understand, control, or optimize? What resources of time and tools do we have?

24. Modeling with Mathematics Models we devise are also constrained by: Limitations of our mathematical, statistical, and technical skills Limitations of our ability to recognize significant variables and relationships among them

25. Modeling with Mathematics Powerful tools for modeling: Diagrams of various kinds Spreadsheets Graphing technology Algebra CAD technology

26. Modeling with Mathematics Problem Formulate Validate Report Compute Basic Modeling Cycle Interpret

27. Modeling with Mathematics • Problem • Identify variables in the situation • Select those that represent essential features Basic Modeling Cycle

28. Modeling with Mathematics • Formulate • Select or create a geometrical, tabular, algebraic, or statistical representation that describes the relationships between the variables Basic Modeling Cycle

29. Modeling with Mathematics • Compute • Analyze and perform operations on these relationships to draw conclusions Basic Modeling Cycle

30. Modeling with Mathematics • Interpret • Interpret the result of the mathematics in terms of the original situation Basic Modeling Cycle

31. Modeling with Mathematics Validate Validate the conclusions by comparing them with the situation… Basic Modeling Cycle

32. Modeling with Mathematics Validate Re - Formulate Report on conclusions and reasoning behind them Basic Modeling Cycle EITHER OR

33. Next Generation Assessments http://www.nextnavigator.com/ This site has examples of NGAs that have been developed to assess students’ ability to synthesize the mathematical content they have learned and flexibly apply it to develop a mathematical model of real world situations.

34. Next Generation Assessments See Handout for “Blown Away” What standards for mathematical content does this task assess? What standards for mathematical practice does this task assess? How does this assessment differ from traditional assessments?

35. Six Instructional Shifts for the Common Core in Mathematics Focus Coherence Fluency Deep Understanding Applications Dual Intensity

36. Shift 1: Focus In mathematics instruction, the eraser means as much as the pen. Teachers must make a conscious effort to focus on fewer things. Decide what areas will get intensive focus (70 %) rethink and link focus (20 %) sampling focus (10%)

37. Shift 1: Focus Look at handout “Priorities for Focus in Mathematics K - 8” 70% of focus on procedural fluency, application, and solving diverse problems 30% of focus on important material to cover, but which can be done in such a way as to reinforce primary areas of focus 10% of focus on useful knowledge, but not body of material that needs to be mastered in itself

38. Shift 2: Coherence Each concept builds on previous learning. Teachers are explicit about making connections between new learning and previous learning. Certain themes arise over and over again (both within a school year and across years)… but in more complexity as the students mature mathematically. (Think back to the Math Tree Handout)

39. Shift 3: Fluency Fluency with Math Facts: Students are expected to memorize math facts and develop automaticity in retrieving the facts. “Know times tables like you know your name…” Students may learn at different rates, but we must insist that all students acquire fluency. Short, frequent practice…in and out of school.

40. Shift 3: Fluency Computational Fluency: As students mature mathematically, they develop the skills to compute using increasingly complex operations on increasingly larger numbers without using calculators. See handout: Mathematical Fluency by Grade Level

41. Shift 3: Fluency An aside about calculators: Calculators have their place, but should be used sparingly in the elementary grades. As students grow in mathematical maturity and deal with more complex numbers and larger data sets in middle and high school, calculators should be used in an artful way.

42. Shift 4: DeepUnderstanding Hong Kong only covers 50% of items on the TIMMS test, while U.S. students cover 100% of the items. Hong Kong students outperform U.S. students because they have an understanding of the core math concepts that allows them to apply their knowledge to solve unfamiliar, complex problems.

43. Shift 4: DeepUnderstanding Students develop a deep understanding of math concepts that allows them to apply their knowledge to new situations. Students are able to express their understanding through oral and written explanations.

44. Shift 5: Applications Students need to learn to apply math concepts even when not told specifically how to solve a problem. Teachers need to regularly provide students opportunities to solve diverse, unpredictable real world problems. This shift is a key one for CTE teachers!

45. Shift 6: Dual Intensity Practice Understanding High Achievement in Mathematics

46. What Does this Mean for CTE Administrators and Teachers? How does the transition to Common Core Affect unit planning? Affect lesson planning? Affect instruction? Affect assessment practices?

47. What Does this Mean for CTE Administrators and Teachers? It will take time for the benefits of the fewer and more focused Common Core Content Standards to come through the pipeline. However, we can immediately start using the best practices highlighted in the Standards for Mathematical Practice to transition to developing students as mathematicians.

48. What Does this Mean for CTE Administrators and Teachers? The real world content of CTE programs provides the perfect “lab setting” for the Standards for Mathematical Practice.

49. What Does this Mean for CTE Administrators and Teachers? CTE programs offer relevant, real-world content and contexts that naturally foster curiosity, creative problem solving and intellectual risk taking. CTE programs require students to bring together their knowledge and skills from many content areas to think through problems and accomplish their work.

50. What Does this Mean for CTE Administrators and Teachers? Mathematics is naturally embedded in all CTE content areas… Therefore all CTE teachers not only teach their subject area, but also must teach students to apply mathematics in many diverse situations.