Common Core and the Community College

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

6. Source: http://tedcoe.com/math/wp-content/uploads/2013/10/broomsticks-for-nctm.doc

7. Source: http://tedcoe.com/math/wp-content/uploads/2013/10/broomsticks-for-nctm.doc

8. Source: http://tedcoe.com/math/wp-content/uploads/2013/10/broomsticks-for-nctm.doc

9. Source: http://tedcoe.com/math/wp-content/uploads/2013/10/broomsticks-for-nctm.doc

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

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

13. From the CCSS: Grade 3 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. 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. From the CCSS: Grade 4 • 4.OA.1, 4.OA.2 • 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. • 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. Source: CCSS Grade 4

15. From the CCSS: Grade 4 • 4.OA.1, 4.OA.2 • 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. • 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. Source: CCSS Grade 4

16. From the CCSS: Grade 5 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. Source: 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. Measurement What do we mean when we talk about “measurement”?

19. Measurement • “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”?

20. Measurement 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.

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

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

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

28. CCSS: Geometry (G-SRT.6, p. 77) Source: 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 Source: CCSS 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

31. Assessment?

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

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

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

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

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

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 Source: CCSS 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

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

44. PARCC Design • 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 Source: PARCC

45. PARCC Design • 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 Source: PARCC

46. PARCC Design • 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 Source: PARCC

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

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

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

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