1. Common Core and the Community College May 20, 2014

2. Teaching and Learning Mathematics Ways of doing Ways of thinking Habits of thinking

3. Ways of Doing?

4. The Broomsticks

5. The RED broomstick is three feet long The YELLOW broomstick is four feet long The GREEN broomstick is six feet long The Broomsticks Source: http://tedcoe.com/math/wp-content/uploads/2013/10/broomsticks-for-nctm.doc

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11. Ways of Thinking? Learning Progressions in the Common Core

12. From the CCSS: Grade 3 12Source: CCSS Math Standards, Grade 3, p. 24 (screen capture)

13. 3.OA.1: Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7. From the CCSS: Grade 3 13 Soucre: CCSS Grade 3. See: Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction. Daro, et al., 2011. pp.48-49

14. 4.OA.1, 4.OA.2 1.Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. 2.Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. From the CCSS: Grade 4 14Source: CCSS Grade 4

15. 4.OA.1, 4.OA.2 1.Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. 2.Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. From the CCSS: Grade 4 15Source: CCSS Grade 4

16. 5.NF.5a Interpret multiplication as scaling (resizing), by: Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. From the CCSS: Grade 5 16Source: CCSS Grade 5

17. “In Grades 6 and 7, rate, proportional relationships and linearity build upon this scalar extension of multiplication. Students who engage these concepts with the unextended version of multiplication (a groups of b things) will have prior knowledge that does not support the required mathematical coherences.” Source: Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction. Daro, et al., 2011. p.49

18. What do we mean when we talk about “measurement”? Measurement

19. “Technically, a measurement is a number that indicates a comparison between the attribute of an object being measured and the same attribute of a given unit of measure.” – Van de Walle (2001) But what does he mean by “comparison”? Measurement

20. How about this? Determine the attribute you want to measure Find something else with the same attribute. Use it as the measuring unit. Compare the two: multiplicatively. Measurement

21. Source: Fractions and Multiplicative Reasoning, Thompson and Saldanha, 2003. (pdf p. 22)

22. %

23. The circumference is three and a bit times as large as the diameter. http://tedcoe.com/math/circumference

24. Similar Figures

25. http://tedcoe.com/math/algebra/constant-rate

26. CCSS: Grade 8 (8.EE.6, p.54) 26Source: CCSS Grade 8

27. Assume http://tedcoe.com/math/geometry/similar-triangles

28. CCSS: Geometry (G-SRT.6, p. 77) 28Source: CCSS High School Geometry (screen capture)

29. Teaching and Learning Mathematics Ways of doing Ways of thinking Habits of thinking

30. Standards for Mathematical Practice 30 Eight Standards for Mathematical Practice Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and critique the understanding 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 Source: CCSS

31. Assessment? 31

32. The error of excessively literal reading “Reading individual standards as individual ingredients of a test. When the explicit goal is to have the ingredients cook into a cake, tasting the uncooked ingredients is a poor measure of how the cake tastes (although it is related). The goal, as stated in the grade-level introductions and the practices standards, is for the students to cook.” Source: Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction. Daro, et al., 2011. p.50-51

33. Grade 5: AIMS (Arizona) Sample Test 33 Source: http://www.azed.gov/standards-development-assessment/aims/sample-for-students/ (April 2014)

34. Grade 5: AIMS (Arizona) Sample Test 34 Source: http://www.azed.gov/standards-development-assessment/aims/sample-for-students/ (April 2014)

35. Grade 6: AIMS (Arizona) Sample Test 35 Around 96 o Source: http://www.azed.gov/standards-development-assessment/aims/sample-for-students/ (April 2014)

36. High School: AIMS (Arizona) Sample Test 36 Source: http://www.azed.gov/standards-development-assessment/aims/sample-for-students/ (April 2014)

37. Next Generation Mathematics Assessments: Beyond “doing”… 37

38. Next-Gen Assessment Claims

39. Smarter Balanced Claims http://www.smarterbalanced.org/wordpress/wp-content/uploads/2012/09/Smarter-Balanced-Mathematics-Claims.pdf

40. Smarter Balanced Claims http://www.smarterbalanced.org/wordpress/wp-content/uploads/2012/09/Smarter-Balanced-Mathematics-Claims.pdf

41. PARCC Mathematics Claims Source: PARCC

42. Standards for Mathematical Practice 42 Eight Standards for Mathematical Practice Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and critique the understanding 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 Source: CCSS

43. Type I Tasks Type II Tasks Type III Tasks PARCC Design 43

44. Type I Tasks: 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 Sub-claims A, B and E PARCC Design 44Source: PARCC

45. Type II Tasks: 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 Sub-claim C PARCC Design 45Source: PARCC

46. Type III Tasks: 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 Sub-claim D PARCC Design 46Source: PARCC

47. PARCC Sample Items 47 Source: http://www.parcconline.org/samples/math

48. PARCC Sample Items 48 Source: http://www.parcconline.org/samples/math

49. PARCC Sample Items 49 Source: http://www.parcconline.org/samples/math

50. 50Source: Source: http://www.parcconline.org/samples/math PARCC Sample Items

51. 51 Source: http://www.parcconline.org/samples/math PARCC Sample Items

52. 52 Source: http://www.parcconline.org/samples/math

53. PARCC Sample Items 53 Source: http://www.parcconline.org/samples/math

54. PARCC Sample Items 54 Source: http://www.parcconline.org/samples/math

55. PARCC Sample Items 55 Source: http://www.parcconline.org/samples/math

56. PARCC Sample Items 56 Source: http://www.parcconline.org/samples/math

57. Algebra I PARCC Sample Items 57 Source: http://www.parcconline.org/samples/math

58. PARCC Sample Items 58 Source: http://www.parcconline.org/samples/math

59. PARCC Sample Items 59 Source: http://www.parcconline.org/samples/math

60. Materials? 60Source: http://ime.math.arizona.edu/progressions/

61. Materials? 61Source: EngageNY

62. Materials? 62Source: EngageNY

63. EQuIP 63Source: www.achieve.org/equip

64. Community College Impact? Placement Assessment Content Instruction Preservice

65. Ted Coe Director of Mathematics Achieve, Inc. tcoe@achieve.org 65