Supporting Rigorous Mathematics Teaching and Learning Using Assessing and Advancing

1. Supporting Rigorous Mathematics Teaching and Learning Using Assessing and Advancing Questions to Target Essential Understandings Tennessee Department of Education Elementary School Mathematics Grade 3

2. 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.

3. 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.

4. 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.

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

6. Linking to Research/Literature: The QUASAR Project The Mathematical Tasks Framework TASKS as set up by the teachers TASKS as implemented by students TASKS as they appear in curricular/ instructional materials Student Learning Stein, Smith, Henningsen, & Silver, 2000 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®discussion Accountable Talk ®is a registered trademark of the University of Pittsburgh

7. Solving and Discussing Solutions to the Half of a Whole Task

8. The Structure and Routines of a Lesson • MONITOR: Teacher selects • examples for the Share, Discuss, • and Analyze Phase based on: • Different solution paths to the • same task • Different representations • Errors • Misconceptions Set Up of the Task Set Up the Task The Explore Phase/Private Work Time Generate Solutions The Explore Phase/ Small Group Problem Solving Generate and Compare Solutions Assess and advance Student Learning 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. Share Discuss and Analyze Phase of the Lesson 1. Share and Model 2. Compare Solutions Focus the Discussion on Key Mathematical Ideas 4. Engage in a Quick Write

9. Half of a Whole: Task Analysis • Solve the task. Write sentences to describe the mathematical relationships that you notice. • Anticipate possible student responses to the task.

10. Half of a Whole Task Identify all of the figures that have one half shaded. Be prepared to show and explain how you know that one half of a figure is shaded. If a figure does not show one half shaded, explain why. Make math statements about what is true about a half of a whole. Adapted from Watanabe, 1996

11. Half of a Whole: Task Analysis • Study the Grade 3 CCSS for Mathematical Content within the Number and Operations—Fractions domain. Which standards are students expected to demonstrate when solving the fraction task? • Identify the CCSS for Mathematical Practice required by the written task.

12. The CCSS for Mathematical Content: Grade 3 Common Core State Standards, 2010, p. 24, NGA Center/CCSSO

13. The CCSS for Mathematical Content: Grade 3 Common Core State Standards, 2010, p. 24, NGA Center/CCSSO

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

15. The Common Core State Standards

16. Mathematical Essential UnderstandingNot All Halves Are Created Equal 3.NF.A.3dCompare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. What is mathematically true about the figures shown above? Objective Students will discover, via the use of the fractional pieces, that not all halves are equal. Essential Understanding If the wholes differ, then a fractional piece from each of the wholes will not be equal because their initial whole was not the same (e.g., of a large pizza is not the same as of a small pizza).

17. Mathematical Essential UnderstandingA Whole Can Be Represented as a Fraction • 3.NF.A.3c Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram. Objective Students will recognize that fractions in the form , , , , etc., are fractional names for a whole, or 1. Essential Understanding

18. Mathematical Essential UnderstandingHalf of the Denominator Is Half of the Whole 3.NF.A.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. Objective Students recognize that two equal parts of the whole represent half of the figure. Essential Understanding

19. Mathematical Essential UnderstandingHalves Take Up the Same Amount of Space • 3.NF.A.3a Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. Objective Students recognize half of a figure as two spaces of equal size. Essential Understanding

20. Mathematical Essential UnderstandingContinuous and Discrete Figures Represent a Whole • 3.NF.A.3b Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model. Objective Students recognize equivalent fractions as those that have the same amount of space of a figure. Essential Understanding

21. Essential Understandings

22. ? Assess Target Mathematical Goal Students’ Mathematical Understandings

23. ? Advance Mathematical Trajectory A Student’s Current Understanding Target Mathematical Goal

24. Target Mathematical Understanding Illuminating Students’ Mathematical Understandings

25. 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. Essential Understandings

27. Supporting Students’ ExplorationAnalyzing Student Work Analyze the students’ responses. Analyze the 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?

28. Group A: Lauren and Austin

29. Group B: Jacquelyn, Alex, and Ethan

30. Group C: Tylor, Jessica, and Tim

31. Group D: Frank, Juan, and Kimberly

32. Group E: JT, Fiona, and Keisha

33. Reflecting on the Use of Essential Understandings How does knowing the essential understandings help you in writing advancing questions?