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 8

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 Same or Different? 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 Same or Different? 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 Same or Different? 10

12. © 2013 UNIVERSITY OF PITTSBURGH Same or Different? Task Analysis Study the Grade 8 CCSS for Mathematical Content within the Expressions and Equations domain. Which standards are students expected to demonstrate when solving the Same or Different? Task? Identify the CCSS for Mathematical Practice required by the written task. 11

13. The CCSS for Mathematical Content − Grade 8 Common Core State Standards, 2010, p. 54, NGA Center/CCSSO Expressions and Equations 8.EE Analyze and solve linear equations and pairs of simultaneous linear equations. 8.EE.C.7 Solve linear equations in one variable. 8.EE.C.7a Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). 8.EE.C.7b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. 12

14. The CCSS for Mathematical Content − Grade 8 Common Core State Standards, 2010, p. 55, NGA Center/CCSSO Expressions and Equations 8.EE Analyze and solve linear equations and pairs of simultaneous linear equations. 8.EE.C.8 Analyze and solve pairs of simultaneous linear equations. 8.EE.C.8a Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. 8.EE.C.8b Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. 8.EE.C.8c Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. 13

15. 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 14

16. © 2013 UNIVERSITY OF PITTSBURGH The Common Core State Standards Common Core Content Standards and Mathematical Practice Standards Essential Understandings 15

17. Mathematical Essential Understanding (Some Equations Have One Solution) 8.EE.C.7a Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). ObjectiveEssential Understanding Determine if a linear equation has zero, one, or infinitely many solutions. A linear equation has one value that makes it true, if the equation can be simplified to the form x = a, where a is a real number, because there is no other complete simplification of the equation. 16 Common Core State Standards, 2010, p. 54, NGA Center/CCSSO

18. Mathematical Essential Understanding (Equivalent Equations Have the Same Solution(s)) 8.EE.C.7a Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). ObjectiveEssential Understanding Identify equations that have the same solution set. Common Core State Standards, 2010, p. 54, NGA Center/CCSSO 17

19. Mathematical Essential Understanding (Arithmetic Operations Preserve Equality) 8.EE.C.7b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. ObjectiveEssential Understanding Simplify equations using arithmetic operations. Common Core State Standards, 2010, p. 54, NGA Center/CCSSO 18

20. Mathematical Essential Understanding (Equations Can be Written in Different Forms) 8.EE.C.7b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. ObjectiveEssential Understanding Write equivalent equations. Common Core State Standards, 2010, p. 54, NGA Center/CCSSO 19

21. © 2013 UNIVERSITY OF PITTSBURGH Essential Understandings Essential UnderstandingCCSS Solutions Make the Equation True The solution(s) of an equation in one variable are the number(s) that make the equation a true statement. 8.EE.C.7 Some Equations Have One Solution A linear equation has one value that makes it true, if the equation can be simplified to the form x = a, where a is a real number, because there is no other complete simplification of the equation. 8.EE.C.7a Equivalent Equations Have the Same Solution(s) An equation is a statement that two expressions have the same value; performing the same operation(s) on each side of the equation changes the value of each expression in the same way, and so maintains the equality of the values of the expressions. 8.EE.C.7a Arithmetic Operations Preserve Equality If we perform the same arithmetic operation(s) to each side of an equation, we do not change the solutions of the equation. 8.EE.C.7b Equations Can be Written in Different Forms We can transform an equation into an equivalent equation using any operation that does not change the balance between the two sides. 8.EE.C.7b 20

22. © 2013 UNIVERSITY OF PITTSBURGH Asking Advancing Questions that Target the Essential Understandings 21

23. © 2013 UNIVERSITY OF PITTSBURGH Target Mathematical Goal Students’ Mathematical Understandings Assess 22

24. © 2013 UNIVERSITY OF PITTSBURGH Target Mathematical Goal A Student’s Current Understanding Advance MathematicalTrajectory 23

25. © 2013 UNIVERSITY OF PITTSBURGH Target Target Mathematical Understanding Mathematical Understanding Illuminating Students’ Mathematical Understandings 24

26. © 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. 25

27. © 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? 26

28. © 2013 UNIVERSITY OF PITTSBURGH Essential Understandings Essential UnderstandingCCSS Solutions Make the Equation True The solution(s) of an equation in one variable are the number(s) that make the equation a true statement. 8.EE.C.7 Some Equations Have One Solution A linear equation has one value that makes it true, if the equation can be simplified to the form x = a, where a is a real number, because there is no other complete simplification of the equation. 8.EE.C.7a Equivalent Equations Have the Same Solution(s) An equation is a statement that two expressions have the same value; performing the same operation(s) on each side of the equation changes the value of each expression in the same way, and so maintains the equality of the values of the expressions. 8.EE.C.7a Arithmetic Operations Preserve Equality If we perform the same arithmetic operation(s) to each side of an equation, we do not change the solutions of the equation. 8.EE.C.7b Equations Can be Written in Different Forms We can transform an equation into an equivalent equation using any operation that does not change the balance between the two sides. 8.EE.C.7b 27

29. © 2013 UNIVERSITY OF PITTSBURGH Group A: Lauren and Becca 28

30. © 2013 UNIVERSITY OF PITTSBURGH Group B: Meredith, Jasmine, and Frank 29

31. © 2013 UNIVERSITY OF PITTSBURGH Group C: Tim, Jessie, and Marco 30

32. © 2013 UNIVERSITY OF PITTSBURGH Group D: Louise, Adam, and Jayden 31

33. © 2013 UNIVERSITY OF PITTSBURGH Group E: Keisha and Mike 32

34. © 2013 UNIVERSITY OF PITTSBURGH Reflecting on the Use of Essential Understandings How does knowing the essential understandings help you in writing advancing questions? 33