2. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Supporting Rigorous Mathematics Teaching and Learning Using Assessing and Advancing Questions to Target Essential Understandings Tennessee Department of Education Middle School Mathematics Grade 7

3. Rationale There is wide agreement regarding the value of teachers attending to and basing their instructional decisions on the mathematical thinking of their students (Warfield, 2001). By engaging in an analysis of a lesson planning process, teachers will have the opportunity to consider the ways in which the process can be used to help them plan and reflect, both individually and collectively, on instructional activities that are based on student thinking and understanding. 2

4. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Session Goals Participants will: learn to set clear goals for a lesson; learn to write essential understandings and consider the relationship to the CCSS; and learn the importance of essential understandings (EUs) in writing focused advancing questions. 3

5. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Overview of Activities Participants will: engage in a lesson and identify the mathematical goals of the lesson; write essential understandings (EUs) to further articulate a standard; analyze student work to determine where there is evidence of student understanding; and write advancing questions to further student understanding of EUs. 4

6. TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning The Mathematical Tasks Framework Stein, Smith, Henningsen, & Silver, 2000 Linking to Research/Literature: The QUASAR Project 5

7. TASKS as they appear in curricular/ instructional materials TASKS as set up by the teachers TASKS as implemented by students Student Learning The Mathematical Tasks Framework Stein, Smith, Henningsen, & Silver, 2000 Linking to Research/Literature: The QUASAR Project Setting Goals Selecting Tasks Anticipating Student Responses Orchestrating Productive Discussion Monitoring students as they work Asking assessing and advancing questions Selecting solution paths Sequencing student responses Connecting student responses via Accountable Talk ® discussions Accountable Talk ® is a registered trademark of the University of Pittsburgh 6

8. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Solving and Discussing Solutions to the Positive, Negative, or Neither Task 7

9. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH The Structure and Routines of a Lesson The Explore Phase/Private Work Time Generate Solutions The Explore Phase/ Small-Group Problem Solving 1.Generate and Compare Solutions 2.Assess and Advance Student Learning Share, Discuss, and Analyze Phase of the Lesson 1. Share and Model 2. Compare Solutions 3.Focus the Discussion on Key Mathematical Ideas 4. Engage in a Quick Write MONITOR: Teacher selects examples for the Share, Discuss, and Analyze phase based on: Different solution paths to the same task Different representations Errors Misconceptions SHARE: Students explain their methods, repeat others’ ideas, put ideas into their own words, add on to ideas and ask for clarification. REPEAT THE CYCLE FOR EACH SOLUTION PATH COMPARE: Students discuss similarities and difference between solution paths. FOCUS: Discuss the meaning of mathematical ideas in each representation REFLECT: Engage students in a Quick Write or a discussion of the process. Set Up the Task Set Up of the Task 8

10. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Positive, Negative, or Neither: Task Analysis Solve the task. Write sentences to describe the mathematical relationships that you notice. Anticipate possible student responses to the task. 9

11. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Positive, Negative, or Neither Consider the following expressions. Use the number line above to determine whether each of the expressions has a value that is positive, negative, or equal to 0. Explain your reasoning. a.a + 1 b.b – b c.a + a d.a – b e.b – a f.-a g.ab + 1 10

12. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Positive, Negative, or Neither: Task Analysis Study the Grade 7 CCSS for Mathematical Content within the Number System domain. Which standards are students expected to demonstrate when solving the Positive, Negative, or Neither Task? Identify the CCSS for Mathematical Practice required by the written task. 11

13. The CCSS for Mathematical Content: Grade 7 Common Core State Standards, 2010, p. 48, NGA Center/CCSSO The Number System 7.NS Apply and extend previous understandings of operations and fractions to add, subtract, multiply, and divide rational numbers. 7.NS.A.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. 7.NS.A.1a Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. 7.NS.A.1b Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. 12

14. The CCSS for Mathematical Content: Grade 7 Common Core State Standards, 2010, p. 48, NGA Center/CCSSO The Number System 7.NS Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. 7.NS.A.1c Understand subtraction of rational numbers as adding the additive inverse, p – q = p +(-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. 7.NS.A.1d Apply properties of operations as strategies to add and subtract rational numbers. 7.NS.A.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. 7.NS.A.2a Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. 13

15. The CCSS for Mathematical Content: Grade 7 Common Core State Standards, 2010, p. 49, NGA Center/CCSSO The Number System 7.NS Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. 7.NS.A.2b Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then -(p/q) = (-p)/q = p/(-q). Interpret quotients of rational numbers by describing real-world contexts. 7.NS.A.2c Apply properties of operations as strategies to multiply and divide rational numbers. 7.NS.A.2d Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. 7.NS.A.3 Solve real-world and mathematical problems involving the four operations with rational numbers. 14

16. 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 15

17. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH The Common Core State Standards Common Core Content Standards and Mathematical Practice Standards Essential Understandings 16

18. Mathematical Essential Understanding (Addition and Subtraction Can be Modeled Using a Number Line) 7.NS.A.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. ObjectiveEssential Understanding Add and subtract rational numbers using a number line. Addition (and subtraction) of rational numbers can be represented by movement on a number line, because adding (or subtracting) two rational numbers has predictable effects on the magnitude and direction of the sum (or difference). 17

19. Mathematical Essential Understanding (The Sum of Two Rational Numbers Can be Located on a Number Line) 7.NS.A.1b Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. ObjectiveEssential Understanding Add rational numbers. 18

20. Mathematical Essential Understanding (The Sum of a Number and its Opposite is Zero) 7.NS.A.1b Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. ObjectiveEssential Understanding Identify zero pairs. 19

21. Mathematical Essential Understanding (Subtraction is Adding the Inverse) 7.NS.A.1c Understand subtraction of rational numbers as adding the additive inverse, p – q = p +(-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. ObjectiveEssential Understanding Subtract positive and negative rational numbers. 20

22. Essential Understandings Essential Understanding CCSS Addition and Subtraction Can be Modeled Using a Number Line Addition (and subtraction) of rational numbers can be represented by movement on a number line, because adding (or subtracting) two rational numbers has predictable effects on the magnitude and direction of the sum (or difference). 7.NS.A.1 The Sum of Two Rational Numbers Can be Located on a Number Line The sum of two numbers p and q is located q units from p on the number line. When q is a positive number, p + q is to the right of p. When q is a negative number, p + q is to the left of p. 7.NS.A.1b The Sum of a Number and its Opposite is Zero The sum of a number and its opposite, p + -p, is equal to zero because p and –p are the same distance from 0 in opposite directions. 7.NS.A.1b Subtraction is Adding the Inverse p – q = p + -q because they are modeled by the same movement along the number line. 7.NS.A.1c 21

23. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Asking Advancing Questions that Target the Essential Understandings 22

24. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Target Mathematical Goal Students’ Mathematical Understandings Assess 23

25. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Target Mathematical Goal A Student’s Current Understanding Advance MathematicalTrajectory 24

26. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Target Target Mathematical Understanding Mathematical Understanding Illuminating Students’ Mathematical Understandings 25

27. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Characteristics of Questions that Support Students’ Exploration Assessing Questions Based closely on the work the student has produced. Clarify what the student has done and what the student understands about what s/he has done. Provide information to the teacher about what the student understands. Advancing Questions Use what students have produced as a basis for making progress toward the target goal. Move students beyond their current thinking by pressing students to extend what they know to a new situation. Press students to think about something they are not currently thinking about. 26

28. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Supporting Students’ Exploration (Analyzing Student Work) Analyze the students’ group work to determine where there is evidence of student understanding. What advancing questions would you ask the students to further their understanding of an EU? 27

29. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Essential Understandings Essential Understanding CCSS Addition and Subtraction Can be Modeled Using a Number Line Addition (and subtraction) of rational numbers can be represented by movement on a number line, because adding (or subtracting) two rational numbers has predictable effects on the magnitude and direction of the sum (or difference) 7.NS.A.1 The Sum of Two Rational Numbers Can be Located on a Number Line The sum of two numbers p and q is located q units from p on the number line. When q is a positive number, p + q is to the right of p. When q is a negative number, p + q is to the left of p. 7.NS.A.1b The Sum of a Number and its Opposite is Zero The sum of a number and its opposite, p + -p, is equal to zero because p and –p are the same distance from 0 in opposite directions. 7.NS.A.1b Subtraction is Adding the Inverse p – q = p + -q because they are modeled by the same movement along the number line. 7.NS.A.1c 28

30. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Group A: Lauren and Austin 29

31. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Group B: Jacquelyn, Alex, and Ethan 30

32. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Group C: Tylor, Jessica, and Tim 31

33. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Group D: Frank, Juan, and Kimberly 32

34. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Group E: JT, Fiona, and Keisha 33

35. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2013 UNIVERSITY OF PITTSBURGH Reflecting on the Use of Essential Understandings How does knowing the essential understandings help you in writing advancing questions? 34