1. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Supporting Rigorous Mathematics Teaching and Learning Tennessee Department of Education Elementary School Mathematics, Grade 1 December 7, 2012 A Performance-Based Assessment: A Means to High-Level Thinking and Reasoning

2. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Forms of Assessment Assessment for LearningAssessment of Learning Assessment as Learning 2

3. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Session Goals Deepen understanding of the Common Core State Standards (CCSS) for Mathematical Content and Mathematical Practice. Understand how Performance-Based Assessments (PBAs) assess the CCSS for both Mathematical Content and Practice. Understand the ways in which PBAs assess students’ conceptual understanding. 3

4. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Overview of Activities Analyze and discuss the CCSS for Mathematical Content and Mathematical Practice. Analyze PBAs in order to determine the way the assessments are assessing the CCSSM. Discuss the CCSS related to the tasks and the implications for instruction and learning. Discuss what it means to develop and assess conceptual understanding. 4

5. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH The Common Core State Standards The standards consist of:  The CCSS for Mathematical Content.  The CCSS for Mathematical Practice. 5

6. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Discussing Content Standards (Small Group Time) For each assessment item: With your small group, discuss the connections between the Content Standard(s), the Mathematical Practice Standards, and the assessment item. 6

7. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Deepening Understanding of the Standards via the Assessment Items (Whole Group) As a result of looking at the assessment items, what do you better understand about the specifics of the Content Standards and the Standards for Mathematical Practice? What are you still wondering about? 7

8. Operations and Algebraic Thinking 1.OA Represent and solve problems involving addition and subtraction. 1.OA.1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. 1.OA.2 Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. Understand and apply properties of operations and the relationship between addition and subtraction. 1.OA.3 Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.) The CCSS for Mathematical Content: Grade 1 Common Core State Standards, 2010, p. 15, NGA Center/CCSSO 8

9. Operations and Algebraic Thinking 1.OA Understand and apply properties of operations and the relationship between addition and subtraction. 1.OA.4 Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8. Add and subtract within 20. 1.OA.5 Relate counting to addition and subtraction (e.g., by counting on 2 to add 2). 1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13). The CCSS for Mathematical Content: Grade 1 Common Core State Standards, 2010, p. 15, NGA Center/CCSSO 9

10. Operations and Algebraic Thinking 1.OA Work with addition and subtraction equations. 1.OA.7 Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 3 = 5 + 2. 1.OA.8 Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of these equations: 8 + ? = 11, 5 = ? – 3, 6 + 6 = ? The CCSS for Mathematical Content: Grade 1 Common Core State Standards, 2010, p. 15, NGA Center/CCSSO 10

11. The CCSS for Mathematical Practice 1.Make sense of problems and persevere in solving them. 2.Reason abstractly and quantitatively. 3.Construct viable arguments and critique the reasoning of others. 4.Model with mathematics. 5.Use appropriate tools strategically. 6.Attend to precision. 7.Look for and make use of structure. 8.Look for and express regularity in repeated reasoning. Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO 11

12. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Analyzing a Performance-Based Assessment 12

13. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 2012 – 2013 Tennessee Focus Clusters Grade 1 Extend understanding of Operations and Algebraic Thinking. Represent and solve problems involving addition and subtraction. Understand and apply properties of operations and the relationship between addition and subtraction. Add and subtract within 20. Work with addition and subtraction equations. 13

14. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Analyzing Assessment Items (Private Think Time) Four assessment items have been provided:  Story Problems Task  Sue and Janice’s Solutions Task  Does Order Matter? Task  How Do You Solve It? Task For each assessment item: Solve the assessment item. Make connections between the standard(s) and the assessment item. 14

15. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 1. Story Problems Task Solve each situational problem. Make a diagram and write a number sentence to show how you are thinking about each problem. a.John has 4 red marbles and 5 green marbles. How many marbles does John have? b.Tina has 6 dolls. For her birthday she gets 5 more dolls. How many dolls does Tina have now? c.Mary has 6 pencils. Her mother buys her some more. Now she has 10 pencils. How many more pencils did her mother buy her? 15

16. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 2. Sue and Janice’s Solutions Task Sue and Janice tried to solve Henry’s problem. Read the problem and study their answers. Henry had some cookies. He ate 17 of them. Now he has 3 cookies. How many cookies did Henry have at the beginning? a.Can Sue use the number sentence above to solve for the number of cookies Henry had in the beginning? Why or why not? b.Can Janice use the number sentence above to solve for the number of cookies Henry had in the beginning? Why or why not? 16 Here is Sue’s solution: 17 – 3 = 14 Here is Janice’s solution: 17 + 3 = 20

17. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 3. Does Order Matter? Task 17 Ramon is solving the problem 6 + 9. He makes this claim. Do you agree with his claim? a.Is Ramon right or wrong? Use diagrams and words to explain if you agree or disagree that 6 + 9 and 9 + 6 both equal 15. b.Is Ramon’s claim true for the problem 4 + 3? Show a diagram or explain how Ramon’s claim is true for this problem. c.Can you write 2 other new problems for which the claim is true? When you add 2 numbers, it doesn’t matter what order you add them in. Either way, the answer will be the same. I’ll show you: 6 + 9 = 15 and 9 + 6 = 15 When you add 2 numbers, it doesn’t matter what order you add them in. Either way, the answer will be the same. I’ll show you: 6 + 9 = 15 and 9 + 6 = 15

18. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 4. How Do You Solve It? Task Keisha and Frank said they can solve 8 + 7. Each student has a different way of solving the problem. Do you agree with each of their ways? Keisha’s Make Ten Strategy: a.Make a diagram and write a number sentence that shows if you agree with Keisha’s way of solving 8 + 7. Frank’s Doubling Strategy: b. Make a diagram and write a number sentence that shows if you agree with Frank’s way of solving 8 + 7 by using doubles. 18 I solve 8 + 7 by making a group of 10 and then adding the rest that are left off. I solve 8 + 7 by using doubles to solve the problem. I solve 8 + 7 by using doubles to solve the problem.

19. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Linking Standards to Assessment Items (Whole Group) As a result of looking at the assessment items, what do you better understand about the specifics of the Content Standards and the Standards for Mathematical Practice? What are you still wondering about? 19

20. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Assessing Conceptual Understanding 20

21. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Rationale We have now examined assessment items and discussed their connection to the CCSS for Mathematical Content and Practice. A question that needs considering, however, is if and how these assessments will give us a good means of measuring the conceptual understandings our students have acquired. In this activity, you will have an opportunity to consider what it means to develop conceptual understanding as described in the CCSS for Mathematics and what it takes to assess for it. 21

22. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Assessing for Conceptual Understanding The set of PBA items are designed to assess student understanding of addition and subtraction situational word problems. Look across the set of related items. What might a teacher learn about a student’s understanding by looking at the student’s performance across the set of items as a whole? What is the variance from one item to the next? 22

23. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Developing and Assessing Understanding Why is it important, when assessing a student’s conceptual understanding, to vary items in these ways? 23

24. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Conceptual Understanding What do the authors mean by conceptual understanding? How might analyzing student performance on this set of assessments help us determine if students have a deep understanding of Operations and Algebraic Thinking? 24

25. Developing Conceptual Understanding Knowledge that has been learned with understanding provides the basis of generating new knowledge and for solving new and unfamiliar problems. When students have acquired conceptual understanding in an area of mathematics, they see connections among concepts and procedures and can give arguments to explain why some facts are consequences of others. They gain confidence, which then provides a base from which they can move to another level of understanding. Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press 25

26. The CCSS on Conceptual Understanding In this respect, those content standards which set an expectation of understanding are potential “points of intersection” between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics. Common Core State Standards for Mathematics, 2010, p. 8, NGA Center/CCSSO 26

27. Assessing Concept Image Tall (1992) differentiates between the mathematical definition of a concept and the concept image, which is the entire cognitive structure that a person has formed related to the concept. This concept image is made up of pictures, examples and non-examples, processes, and properties. A strong concept image is a rich, integrated, mental representation that allows the student to flexibly move between multiple formulations and representations of an idea. A student who has connected mathematical ideas in this way can create and use a model to analyze a situation, uncover patterns and synthesize them to form an integrated picture. They can also use symbols meaningfully to describe generalizations which then provides a base from which they can move to another level of understanding. Brown, Seidelmann, & Zimmermann. In the trenches: Three teachers’ perspectives on moving beyond the math wars. http://mathematicallysane.com/analysis/trenches.asp 27

28. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Analyzing a Student’s Performance Analyze Sarah’s performance on 4 tasks. What do you notice? What does Sarah know? 28

29. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 1. Story Problems Task: Sarah’s Work 29 Solve each situational problem. Make a diagram and write a number sentence to show how you are thinking about each problem. a.John has 4 red marbles and 5 green marbles. How many marbles does John have? b.Tina has 6 dolls. For her birthday she gets 5 more dolls. How many dolls does Tina have now? c.Mary has 6 pencils. Her mother buys her some more. Now she has 10 pencils. How many more pencils did her mother buy her?

30. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 2. Sue and Janice’s Solutions Task: Sarah’s Work 30 a.Can Sue use the number sentence above to solve for the number of cookies Henry had in the beginning? Why or why not? b.Can Janice use the number sentence above to solve for the number of cookies Henry had in the beginning? Why or why not?

31. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 3. Does Order Matter? Task: Sarah's Work 31 a.Is Ramon right or wrong? Use diagrams and words to explain if you agree or disagree that 6 + 9 and 9 + 6 both equal 15. b.Is Ramon’s claim true for the problem 4 + 3? Show a diagram or explain how Ramon’s claim is true for this problem. c.Can you write 2 other new problems for which the claim is true?

32. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 4. How Do You Solve It? Task: Sarah’s Work 32 a.Make a diagram and write a number sentence that shows if you agree with Keisha’s way of solving 8 + 7. b.Make a diagram and write a number sentence that shows if you agree with Frank’s way of solving 8 + 7 by using doubles.

33. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Using the Assessment to Think About Instruction In order for students to perform well on the PBA, what are the implications for instruction? What kinds of instructional tasks will need to be used in the classroom? What will teaching and learning look like and sound like in the classroom? 33

34. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Step Back What have you learned about the CCSS for Mathematical Content that surprised you? What is the difference between the CCSS for Mathematical Content and the CCSS for Mathematical Practice? Why do we say that students must work on both Mathematical Content and Mathematical Practice? 34