1. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Supporting Rigorous Mathematics Teaching and Learning Tennessee Department of Education Elementary School Mathematics, Grade 2 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 Standards for Mathematical Practice? What are you still wondering about? 7

8. Operations and Algebraic Thinking 2.OA Represent and solve problems involving addition and subtraction. 2.OA.1 Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. Add and subtract within 20. 2.OA.2 Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers. The CCSS for Mathematical Content: Grade 2 Common Core State Standards, 2010, p. 19, NGA Center/CCSSO 8

9. Operations and Algebraic Thinking 2.OA Work with equal groups of objects to gain foundations for multiplication. 2.OA.3 Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends. 2.OA.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends. Number and Operations in Base Ten 2.NBT Use place value understanding and properties of operations to add and subtract. 2.NBT.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. 2.NBT.6 Add up to four two-digit numbers using strategies based on place value and properties of operations. The CCSS for Mathematical Content: Grade 2 Common Core State Standards, 2010, p. 19, NGA Center/CCSSO 9

10. 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 10

11. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Analyzing a Performance-Based Assessment 11

12. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Analyzing Assessment Items (Private Think Time) 4 assessment items have been provided:  Three Problems  The Difference is 50  Forty-Five People  Animal Shelter For each assessment item: Solve the assessment item. Make connections between the standard(s) and the assessment item. 12

13. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 1. Three Problems Task Solve each of these 3 problems. Show your work by writing an equation or drawing a picture for each problem. Which problems can be solved in the same way? Show with a diagram or explain in words how the problems are the same or different from each other. 13 Problem AProblem BProblem C Mark sets up chairs for a party. He sets up 6 chairs on the right and 7 chairs on the left. How many chairs does he use? Ed has 6 apples and 7 oranges. How many pieces of fruit does he have in all? Sam has 6 bags with 7 apples in each bag. How many apples does he have in all?

14. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 2. The Difference is 50 Task The difference between 2 numbers is 50. One of the numbers is 75. a.Identify 2 possibilities for the missing number. b.Use diagrams, words, or numbers to show how you thought about the numbers. 14

15. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 3. Forty-Five People Task 15 How many children are at the park? Show a diagram, equation, or explain with words how you know the number of children that are at the park. There are 45 people at the park. 9 of them are men. 8 of them are women. The rest are children.

16. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 4. Animal Shelter Task Read about the animal shelter. Tommy and his mom went to the animal shelter to adopt a pet. There were 17 kittens and 9 puppies at the shelter. Tommy asked himself some questions about the kittens and the puppies. a.If Tommy writes 17 + 9 = 26, then what does 26 represent in the situation? b.If Tommy writes 17 – 9 = 8, then what does the 8 tell you about the animals at the animal shelter? 16

17. 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? 17

18. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Assessing Conceptual Understanding 18

19. 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. 19

20. 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 fraction equivalence and ordering, and of building fractions from unit fractions. 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? 20

21. 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? 21

22. 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? 22

23. 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 23

24. 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 24

25. 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 25

26. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Analyzing a Student’s Performance Analyze Josh’s performance on 4 tasks. What do you notice? What does Josh know? 26

27. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 1. Three Problems Task: Josh’s Work Solve each of these 3 problems. Show your work by writing an equation or drawing a picture for each problem. Which problems can be solved in the same way? Show with a diagram or explain in words how the problems are the same or different from each other. 27 Problem AProblem BProblem C Mark sets up chairs for a party. He sets up 6 chairs on the right and 7 chairs on the left. How many chairs does he use? Ed has 6 apples and 7 oranges. How many pieces of fruit does he have in all? Sam has 6 bags with 7 apples in each bag. How many apples does he have in all?

28. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 2. The Difference is 50 Task: Josh’s Work The difference between 2 numbers is 50. One of the numbers is 75. a.Identify 2 possibilities for the missing number. b.Use diagrams, words, or numbers to show how you thought about the numbers.. 28

29. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 3. Forty-Five People Task: Josh’s Work 29 How many children are at the park? Show a diagram, equation or explain with words how you know the number of children that are at the park. There are 45 people at the park. 9 of them are men. 8 of them are women. The rest are children.

30. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH 4. Animal Shelter Task: Josh's Work 30 Read about the animal shelter. Tommy and his mom went to the animal shelter to adopt a pet. There were 17 kittens and 9 puppies at the shelter. Tommy asked himself some questions about the kittens and the puppies. a.If Tommy writes 17 + 9 = 26, then what does 26 represent in the situation? b.If Tommy writes 17 – 9 = 8, then what does the 8 tell you about the animals at the animal shelter?

31. 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? 31

32. 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 the Mathematical Practice Standards? 32