Supporting Rigorous Mathematics Teaching and Learning

1. Supporting Rigorous Mathematics Teaching and Learning Using Assessing and Advancing Questions to Target Essential Understandings Tennessee Department of Education Elementary School Mathematics Grade 4

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® discussions Accountable Talk®is a registered trademark of the University of Pittsburgh

7. Solving and Discussing Solutions to the Thirds and Sixths 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. Thirds and Sixths: Task Analysis • Solve the task. Write sentences to describe the mathematical relationships that you notice. • Anticipate possible student responses to the task.

10. Thirds and Sixths Joel looks at the picture below and says, “I see of the picture is shaded.” Sammy says, “No, of the picture is shaded.” Who is correct? Write addition and multiplication equations to prove your answer.

11. Thirds and Sixths: Task Analysis • Study the Grade 4 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 4 Common Core State Standards, 2010, p. 30, NGA Center/CCSSO

13. The CCSS for Mathematical Content − Grade 4 Common Core State Standards, 2010, p. 30, 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 Understanding(Fractional Equivalence) • 4.NF.A.1Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. Common Core State Standards, 2010, p. 30, NGA Center/CCSSO

17. Mathematical Essential Understanding(Adding Iterations of a Unit Fraction) 4.NF.B.3Understand a fraction a/b with a > 1 as a sum of fractions 1/b. Common Core State Standards, 2010, p. 30, NGA Center/CCSSO

18. Mathematical Essential Understanding(Multiplying Iterations of a Unit Fraction) • 4.NF.B.4aUnderstand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4). Common Core State Standards, 2010, p. 30, NGA Center/CCSSO

19. Mathematical Essential Understanding(Equivalent Unit Fraction Expressions) • 4.NF.B.4bUnderstand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.) Common Core State Standards, 2010, p. 30, NGA Center/CCSSO

20. Essential Understandings

21. Asking Advancing Questions that Target the Essential Understanding

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. 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. Essential Understandings

28. Group A

29. Group B

30. Group C

31. Group D

32. Group E

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