2. © 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 6

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. © 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. © 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. © 2013 UNIVERSITY OF PITTSBURGH Solving and Discussing Solutions to the Saving Money Task 7

9. © 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. © 2013 UNIVERSITY OF PITTSBURGH Saving Money: Task Analysis Solve the task. Write sentences to describe the mathematical relationships that you notice. Anticipate possible student responses to the task. 9

11. © 2013 UNIVERSITY OF PITTSBURGH Saving Money Task 5 friends are keeping track of their finances. Some of them have saved money. Others have borrowed money from their parents. 1.Plot points on the number line below representing the amount each friend has saved or owes. Explain why you located the points where you did. 2.Explain how you can use the points you placed on the number line to determine which friend has saved the most money and which has the greatest debt. 3.Brandon claims that the points representing 2 people have the same absolute value. Is he correct? Why or why not? NameMoney Saved or Owed AbbeySaved $12.10 BrandonSaved $5.50 ClaireOwes $12.10 DanteSaved $14 ElizabethOwes $8.25 10

12. © 2013 UNIVERSITY OF PITTSBURGH Saving Money: Task Analysis Study the Grade 6 CCSS for Mathematical Content within the Number System domain. Which standards are students expected to demonstrate when solving the Saving Money Task? Identify the CCSS for Mathematical Practice required by the written task. 11

13. The CCSS for Mathematical Content: Grade 6 Common Core State Standards, 2010, p. 43, NGA Center/CCSSO The Number System 6.NS Apply and extend previous understandings of numbers to the system of rational numbers. 6.NS.C.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation. 6.NS.C.6 Understand a rational number as a point on a number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. 6.NS.C.6a Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3, and that 0 is its own opposite. 12

14. The CCSS for Mathematical Content: Grade 6 Common Core State Standards, 2010, p. 43, NGA Center/CCSSO The Number System 6.NS Apply and extend previous understandings of numbers to the system of rational numbers. 6.NS.C.6b Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. 6.NS.C.6c Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. 6.NS.C.7 Understand ordering and absolute value of rational numbers. 6.NS.C.7a Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret -3 > -7 as a statement that -3 is located to the right of -7 on a number line oriented from left to right. 6.NS.C.7b Write, interpret and explain statements of order for rational numbers in real-world contexts. For example, write -3 o C > -7 o C to express the fact that -3 o C is warmer than -7 o C. 13

15. The CCSS for Mathematical Content: Grade 6 Common Core State Standards, 2010, p. 43, NGA Center/CCSSO The Number System 6.NS Apply and extend previous understandings of numbers to the system of rational numbers. 6.NS.C.7c Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of -30 dollars, write |-30| = 30 to describe the size of the debt in dollars. 6.NS.C.7d Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance of less than - 30 dollars represents a debt greater than 30 dollars. 6.NS.C.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. 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. © 2013 UNIVERSITY OF PITTSBURGH The Common Core State Standards Common Core Content Standards and Mathematical Practice Standards Essential Understandings 16

18. Mathematical Essential Understanding (Positive and Negative Numbers Can be Used to Represent Real-World Quantities) 6.NS.C.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation. ObjectiveEssential Understanding Represent real-world problems using positive and negative values. Many real-world situations can be modeled with both positive and negative values because it is possible to measure above and below a baseline value (often 0). 17

19. Mathematical Essential Understanding (Rational Numbers Can be Located on a Number Line) 6.NS.C.6 Understand a rational number as a point on a number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. ObjectiveEssential Understanding Plot points representing positive and negative integer values on a number line. 18

20. Mathematical Essential Understanding (The Sign of a Number Indicates Direction on a Number Line Relative to Zero) 6.NS.C.6a Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3, and that 0 is its own opposite. ObjectiveEssential Understanding Identify the opposite of a rational number. 19

21. Mathematical Essential Understanding (Rational Numbers are “Dense”) 6.NS.C.6c Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. ObjectiveEssential Understanding Plot points representing positive and negative rational, non-integer values on a number line. 20

22. Mathematical Essential Understanding (Rational Numbers Can be Ordered) 6.NS.C.7 Understand ordering and absolute value of rational numbers. ObjectiveEssential Understanding Order rational numbers from least to greatest. 21

23. Mathematical Essential Understanding (Absolute Value is a Measure of a Number’s Distance From Zero) 6.NS.C.7c Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real- world situation. For example, for an account balance of -30 dollars, write |-30| = 30 to describe the size of the debt in dollars. ObjectiveEssential Understanding Determine the absolute value of rational numbers. 22

24. Essential Understandings Essential UnderstandingCCSS Positive and Negative Numbers Can be Used to Represent Real-World Quantities Positive numbers represent values greater than 0 and negative numbers represent values less than 0. Many real-world situations can be modeled with both positive and negative values because it is possible to measure above and below a baseline value (often 0). 6.NS.C.5 Rational Numbers Can be Located on a Number Line The real number line extends infinitely in the positive and negative directions. 6.NS.C.6 The Sign of a Number Indicates Direction on a Number Line Relative to Zero If two rational numbers differ only by their signs, they are located equal distances from zero in opposite directions. 6.NS.C.6a Rational Numbers are “Dense” There are infinitely many rational numbers in between two integer values, because rational numbers cannot be listed. That is, there is no “next” number after. 6.NS.C.6c Rational Numbers Can be Ordered Rational numbers can be ordered according to their position on the real number line. Values increase from left to right on a horizontal number line and from bottom to top on a vertical number line. 6.NS.C.7 Absolute Value is a Measure of a Number’s Distance From 0 The absolute value of a number is the number’s magnitude or distance from 0. If two rational numbers differ only by their signs, they have the same absolute value because they are the same distance from zero. 6.NS.C.7c 23

25. © 2013 UNIVERSITY OF PITTSBURGH Asking Advancing Questions that Target the Essential Understandings

26. © 2013 UNIVERSITY OF PITTSBURGH Target Mathematical Goal Students’ Mathematical Understandings Assess 25

27. © 2013 UNIVERSITY OF PITTSBURGH Target Mathematical Goal A Student’s Current Understanding Advance MathematicalTrajectory 26

28. © 2013 UNIVERSITY OF PITTSBURGH Target Target Mathematical Mathematical Understanding Understanding Illuminating Students’ Mathematical Understandings 27

29. © 2013 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. 28

30. © 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? 29

31. © 2013 UNIVERSITY OF PITTSBURGH Essential Understandings Essential UnderstandingCCSS Positive and Negative Numbers Can be Used to Represent Real-World Quantities Positive numbers represent values greater than 0 and negative numbers represent values less than 0. Many real-world situations can be modeled with both positive and negative values because it is possible to measure above and below a baseline value (often 0). 6.NS.C.5 Rational Numbers Can be Located on a Number Line The real number line extends infinitely in the positive and negative directions. 6.NS.C.6 The Sign of a Number Indicates Direction on a Number Line Relative to Zero If two rational numbers differ only by their signs, they are located equal distances from zero in opposite directions. 6.NS.C.6a Rational Numbers are “Dense” There are infinitely many rational numbers in between two integer values, because rational numbers cannot be listed. That is, there is no “next” number after. 6.NS.C.6c Rational Numbers Can be Ordered Rational numbers can be ordered according to their position on the real number line. Values increase from left to right on a horizontal number line and from bottom to top on a vertical number line. 6.NS.C.7 Absolute Value is a Measure of a Number’s Distance From 0 The absolute value of a number is the number’s magnitude or distance from 0. If two rational numbers differ only by their signs, they have the same absolute value because they are the same distance from zero. 6.NS.C.7c 30

32. © 2013 UNIVERSITY OF PITTSBURGH Group A: Lauren and Austin 31

33. © 2013 UNIVERSITY OF PITTSBURGH Group B: Jacquelyn, Alex, and Ethan 32

34. © 2013 UNIVERSITY OF PITTSBURGH Group C: Tylor, Jessica, and Tim 33

35. © 2013 UNIVERSITY OF PITTSBURGH Group D: Frank, Juan, and Kimberly 34

36. © 2013 UNIVERSITY OF PITTSBURGH Group E: JT, Fiona, and Keisha 35

37. © 2013 UNIVERSITY OF PITTSBURGH Reflecting on the Use of Essential Understandings How does knowing the essential understandings help you in writing advancing questions? 36