Assessing the Common Core: Mathematics Practices and Content Standards

1. Assessing the Common Core: Mathematics Practices and Content Standards Ted Coe, Ph.D., Grand Canyon University December 3, 2013

2. “Too much math never killed anyone”

3. Ways of doing • Ways of thinking • Habits of thinking Teaching and Learning Mathematics http://www.youtube.com/watch?v=ZffZvSH285c

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

5. http://commoncoretools.me/2012/02/16/the-structure-is-the-standards/http://commoncoretools.me/2012/02/16/the-structure-is-the-standards/

6. 3 4 6

7. http://ime.math.arizona.edu/progressions/

8. http://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdfhttp://www.cpre.org/ccii/images/stories/ccii_pdfs/learning%20trajectories%20in%20math_ccii%20report.pdf

9. Image from Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction. Daro, et al., 2011. Pp.48-49

10. Image from Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction. Daro, et al., 2011. Pp.48-49

11. Example of focus and coherence error: Excessively literal reading. Image from Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction. Daro, et al., 2011. Pp.50-51

12. Ways of Thinking Multiplicative comparison? Where does “copies of” work in the curriculum?

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

14. 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”?

15. Measurement • Determine the attribute you want to measure • Find something else with the same attribute. Use it as the measuring unit. • Compare the two: multiplicatively.

16. Measurement Image from Fractions and Multiplicative Reasoning, Thompson and Saldanha, 2003. (pdf p. 22)

17. http://tedcoe.com/math/circumference

18. Geometry • Area:

19. Geometry • Similarity

20. Pythagorean Theorem • Pythagorean Theorem http://tedcoe.com/math/geometry/pythagorean-and-similar-triangles

21. Constant Rate http://tedcoe.com/math/algebra/constant-rate

22. Trigonometry • Right Triangle Trigonometry http://tedcoe.com/math/geometry/similar-triangles

23. Irrational? • Irrational Numbers The first proof of the existence of irrational numbers is usually attributed to a Pythagorean (possibly Hippasus of Metapontum),who probably discovered them while identifying sides of the pentagram.Thethen-current Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other. However, Hippasus, in the 5th century BC, was able to deduce that there was in fact no common unit of measure, and that the assertion of such an existence was in fact a contradiction.  http://en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012

24. Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, http://en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012

25. Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans http://en.wikipedia.org/wiki/Irrational_numbers. 11/2/2012

26. Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans “…for having produced an element in the universe which denied the…doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.”

27. “Too much math never killed anyone” …except Hippasus

28. Archimedes died c. 212 BC during the Second Punic War, when Roman forces under General Marcus Claudius Marcellus captured the city of Syracuse after a two-year-long siege. According to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, saying that he had to finish working on the problem. The soldier was enraged by this, and killed Archimedes with his sword. http://en.wikipedia.org/wiki/Archimedes. 11/2/2012

29. The last words attributed to Archimedes are "Do not disturb my circles"  http://en.wikipedia.org/wiki/Archimedes. 11/2/2012

30. “Too much math never killed anyone” …except Hippasus …and Archimedes.

31. Ways of doing • Ways of thinking • Habits of thinking Looking at Learning Mathematics

32. 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 appropriately tools strategically. • Attend to precision. • Look for and make use of structure. • Look for and express regularity in repeated reasoning. • Mathematic Practices from the CCSS/ACCRS Habits of Thinking?

33. OK, so math is bigger than just “doing” How do we assess these things?

34. Next-Gen Assessment “Claims”

35. PARCC Mathematics Update May 2013

36. Assessment DesignMathematics, Grades 3-8 and High School End-of-Course 2 Optional Assessments/Flexible Administration • End-of-Year • Assessment • Innovative, computer-based items • Required • Performance-Based • Assessment (PBA) • Extended tasks • Applications of concepts and skills • Required • Mid-Year Assessment • Performance-based • Emphasis on hard-to-measure standards • Potentially summative • Diagnostic Assessment • Early indicator of student knowledge and skills to inform instruction, supports, and PD • Non-summative PBA EOY MYA 4

37. PARCC Model Content Frameworks Approach of the Model Content Frameworks for Mathematics • PARCC Model Content Frameworks provide a deep analysis of the CCSS, leading to more guidance on how focus, coherence, content and practices all work together. • They focus on framing the critical advances in the standards: • Focus and coherence • Content knowledge, conceptual understanding, and expertise • Content and mathematical practices • Model Content Frameworks for grades 3-8, Algebra I, Geometry, Algebra II, Mathematics I, Mathematics II, Mathematics III

38. Model Content Frameworks Grade 3 Example

39. How PARCC has been presenting Evidence-Centered Design (ECD) ECD is a deliberate and systematic approach to assessment development that will help to establish the validityof the assessments, increase the comparability of year-to year results, and increase efficiencies/reduce costs.

40. Claims Driving Design: Mathematics Students are on-track or ready for college and careers MP: 3,6 MP: 4

41. How PARCC has been presenting Evidence-Centered Design (ECD) ECD is a deliberate and systematic approach to assessment development that will help to establish the validityof the assessments, increase the comparability of year-to year results, and increase efficiencies/reduce costs.

42. Overview of Evidence Statements: Types of Evidence Statements Several types of evidence statements are being used to describe what a task should be assessing, including: • Those using exact standards language • Those transparently derived from exact standards language, e.g., by splitting a content standard • Integrative evidence statements that express plausible direct implications of the standards without going beyond the standards to create new requirements • Sub-claim C and D evidence statements, which put MP.3, 4, 6 as primary with connections to content

43. Overview of Evidence Statements: Examples Several types of evidence statements are being used to describe what a task should be assessing, including: • Those using exact standards language

44. Overview of Evidence Statements: Examples Several types of evidence statements are being used to describe what a task should be assessing, including: • Those transparently derived from exact standards language, e.g., by splitting a content standard

45. Overview of Evidence Statements: Examples Several types of evidence statements are being used to describe what a task should be assessing, including: • Integrative evidence statements that express plausible direct implications of the standards without going beyond the standards to create new requirements

46. Overview of Evidence Statements: Examples Several types of evidence statements are being used to describe what a task should be assessing, including: • Sub-claim C & Sub-claim D Evidence Statements, which put MP.3, 4, 6 as primary with connections to content

47. How PARCC has been presenting Evidence-Centered Design (ECD) ECD is a deliberate and systematic approach to assessment development that will help to establish the validityof the assessments, increase the comparability of year-to year results, and increase efficiencies/reduce costs.

48. Overview of Task Types • The PARCC assessments for mathematics will involve three primary types of tasks: Type I, II, and III. • Each task type is described on the basis of several factors, principally the purpose of the task in generating evidence for certain sub-claims. Source: Appendix D of the PARCC Task Development ITN on page 17

49. Overview of PARCC Mathematics Task Types For more information see PARCC Task Development ITN Appendix D.

50. Design of PARCC Math Summative Assessment • Performance Based Assessment (PBA) • Type I items (Machine-scorable) • Type II items (Mathematical Reasoning/Hand-Scored – scoring rubrics are drafted but PLD development will inform final rubrics) • Type III items (Mathematical Modeling/Hand-Scored and/or Machine-scored - scoring rubrics are drafted but PLD development will inform final rubrics) • End-of-Year Assessment (EOY) • Type I items only (All Machine-scorable)