1. DDI and Assessments in Mathematics: Designing Assessments that Provide Meaningful Data At the 6-12 Level February 4, 2014

2. Session Objectives Know the elements of a high-quality Common Core-aligned mathematics assessment Know how to create and choose assessment tasks that reflect a balance of rigor and give comprehensive information about students at all levels

3. Agenda Introduction Sorting Task: Why Do We Assess? Balance of Rigor: 7.RP.1 Evaluate and Modify Your Assessment, Part I Students At All Levels: 7.RP.1 Evaluate and Modify Your Assessment, Part II

4. Introduction

5. Why Assess At All? To learn information about students

6. Why Assess At All? 1. Compare the assessments (Documents A, B, and C). What kinds of information would we learn about students from each one? 2. What are the strengths and weaknesses of each?

10. So… What kinds of information do we want to learn about our students?

11. Two Big Ideas We’ll Talk About Today 1. The assessment should reflect a balance of rigor, helping us understand students’ procedural abilities, conceptual understanding, and abilities to apply. 2. The assessment should give us information about students at all levels.

12. Other Considerations (Not Our Focus Today) Focus: Strong majority of questions assess the Major Work of the grade/course Coherence: Supporting Content is assessed in ways that engage students in the Major Work Practices: Items signal the Standards for Mathematical Practice (not necessarily all items)

13. What Not To Do:

14. 1. Balance of Rigor Procedural Fluency Conceptual Understanding Application

15. From the Publishers’ Criteria:

16. Procedural Fluency $64,000 Question: Are there items that ask or imply that students “do” math procedures? Some good verbs: solve, add, subtract, divide, multiply, graph, compute, find Practice Connections MP.7 Look for and make use of structure MP.8 Look for and express regularity in repeated reasoning *Note fluency expectations named at each grade level. While these are not named for HS level courses there should still be procedural assessment.

17. “But what does mathematical understanding look like? One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. There is a world of difference between a student who can summon a mnemonic device to expand a product such as (a + b)(x + y) and a student who can explain where the mnemonic comes from. The student who can explain the rule understands the mathematics, and may have a better chance to succeed at a less familiar task such as expanding (a + b + c)(x + y). Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness.” From the New York State Common Core Learning Standards

18. Conceptual Understanding $64,000 Question: Are there items that ask students to explain their thinking? Some good verbs: Explain, determine, prove, show, compare, justify Practice Connections MP.3 Construct viable arguments and critique the reasoning of others. MP.6 Attend to precision. *More important in standards that begin with “understand.”

19. Application $64,000 Question: Does the item have a real-world context? Some good examples: Single-step word problems, multi- step word problems, model-building Practice Connections MP.1 Make sense of problems and persevere in solving them. MP.2: Reason abstractly and quantitatively. MP.4: Model with mathematics. *Application Paradox: The more directed the task, the more we can be sure it aligns to the standard, but the less authentic it becomes.

20. 7.RP.1 Let’s look at a standard together. What would assessment items based on procedural, conceptual, and application aspects look like?

21. Three Free, Public On-Line Resources To Support A Balance of Rigor 1. www.illustrativemathematics.org 2. EngageNY: Curriculum Modules 3. EngageNY: Annotated Items

22. Illustrative Math Task

25. Annotated Item

26. Module Assessment

27. Evaluate and Modify 1.Does your assessment reflect a balance of rigor? 2.What needs to be added or modified in your assessment? 3.Can you generate an item or two that would improve the rigor of your assessment?

28. 2. Students at All Levels

29. From the K-8 Publishers’ Criteria: The natural distribution of prior knowledge in classrooms should not prompt abandoning instruction in grade level content, but should prompt explicit attention to connecting grade level content to content from prior learning. To do this, instruction should reflect the progressions on which the CCSSM are built. For example, the development of fluency with division using the standard algorithm in grade 6 is the occasion to surface and deal with unfinished learning with respect to place value. Much unfinished learning from earlier grades can be managed best inside grade level work when the progressions are used to understand student thinking.

30. Three Free, Public On-Line Resources To Support Learning About Students at All Levels 1.EngageNY: Performance Level Descriptions 2.Common Core Learning Standards 3.Progressions

31. Performance Level Descriptions Describe the trajectory of knowledge and skills at each grade level. Provide rich information about what students in each performance category know and can do. Serve as basis of discussion among educators about the specific knowledge and skills that distinguish the “just barely” Level 3 proficient student, or the “just barely” Level 2 student.

32. Performance Level Descriptions

34. Important Questions What performance levels does the item tell us about? How could the item be modified to tell us information about different performance levels?

35. Annotated Item

36. Use Cluster and Domain Headings

37. Illustrative Math Task

39. Evaluate and Modify 1.Does your assessment give information about students at a variety of levels? 2.What needs to be added or modified in your assessment? 3.Can you generate or modify an item or two that would better differentiate your assessment?

40. To Consider: Cluster Level Alignment Solve the “paradox” / More authentic tasks Standards are not written at uniform grain size Avoid “micro-sizing” (using twigs to build a fire that burns down our tree) Easier data management PLDs are written at the cluster level

41. AET

43. Two Big Ideas Revisited 1. The assessment should reflect a balance of rigor, helping us understand students’ procedural abilities, conceptual understanding, and abilities to apply. 2. The assessment should give us information about students at all levels.

44. Thanks! Q & A