Supporting Rigorous Mathematics Teaching and Learning

1. Supporting Rigorous Mathematics Teaching and Learning Selecting and Sequencing Based on Essential Understandings Tennessee Department of Education Elementary School Mathematics Grade 5

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 about: • goal-setting and the relationship of goals to the CCSS and essential understandings; • essential understandings as they relate to selecting and sequencing student work; • Accountable Talk® moves related to essential understandings; and • prompts that problematize or “hook” students during the Share, Discuss, and Analyze phase of the lesson.

4. “The effectiveness of a lesson depends significantly on the care with which the lesson plan is prepared.” Brahier, 2000

5. “During the planning phase, teachers make decisions that affect instruction dramatically. They decide what to teach, how they are going to teach, how to organize the classroom, what routines to use, and how to adapt instruction for individuals.” Fennema & Franke, 1992, p. 156

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

7. 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

8. Identify Goals for Instructionand Select an Appropriate Task

9. 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

10. Contextualizing Our Work Together Imagine that you are working with a group of students who have the following understanding of the concepts. • 70% of the students need to multiply fractions. (5.NF.B4 and 5.NF.B5) • 20% of the students need additional work on fraction standards previously addressed (4.NF standards). These students also need opportunities to struggle with and make sense of the problem. (MP1) • 5% of the students still do not recognize the importance of knowing what the “whole” is when talking about fractions. (Part of 4.NF.A2) • 5% of the students struggle to pay attention and their understanding of mathematics is two grade levels below fifth grade.

11. The CCSS for Mathematics: Grade 5 Common Core State Standards, 2010, p. 36 - 37, NGA Center/CCSSO

12. Mathematical Practice Standards Related to the Task 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.

13. Identify Goals: Solving the Task(Small Group Discussion) Solve the task. Discuss the possible solution paths to the task.

14. Bobby’s Hike Task Bobby said that he wanted to go for a four-mile hike. Bobby stops every mile for a sip of water from his water bottle. How many times does Bobby stop? Be sure to show how you found your answer with both diagrams and an explanation in words. What equations involving fractions match your diagram? Extension: It takes Bobby hour to travel one mile. How often does Bobby stop for water? How do you know your answer is correct? Show with words, diagrams, and a fractional equation. Is there another equation that matches your diagram?

15. Identify Goals Related to the Task(Whole Group Discussion) Does the task provide opportunities for students to access the Mathematical Content Standards and Practice Standards that we have identified for student learning?

16. Identify Goals: Essential Understandings (Whole Group Discussion) Study the essential understandings associated with the Number and Operations – Fractions Common Core Standards. Which of the essential understandings are the goals of Bobby’s Hike Task?

17. The CCSS for Mathematics: Grade 5 Common Core State Standards, 2010, p. 36 - 37, NGA Center/CCSSO

18. Essential Understandings (Small Group Discussion)

19. Selecting and Sequencing Student Work for the Share, Discuss, and Analyze Phase of the Lesson

20. Analyzing Student Work(Private Think Time) Analyze the student work. Identify what each group knows related to the essential understandings. Consider the questions that you have about each group’s work as it relates to the essential understandings.

21. Prepare for the Share, Discuss, and Analyze Phase: Selecting and Sequencing Student Work (Small Group Discussion) Assume that you have circulated and asked students assessing and advancing questions. Study the student work samples. • Which pieces of student work will allow you to address the essential understanding? • How will you sequence the student’s work that you have selected? Be prepared to share your rationale.

22. The Share, Discuss, and Analyze Phase: Selecting and Sequencing Student Work(Small Group Discussion) In your small group, come to consensus on the work that you select, and share your rationale. Be prepared to justify your selection and sequence of student work.

23. The Share, Discuss, and Analyze Phase: Selecting and Sequencing Student Work(Whole Group Discussion) What order did you identify for the EUs and student work? What is your rationale for each selection?

24. Group A

25. Group B

26. Group C

27. Group D

28. Group E

29. Group F

30. Group G

31. The Share, Discuss, and Analyze Phase: Selecting and Sequencing Student Work(Whole Group Discussion) What order did you identify for the EUs and student work? What is your rationale for each selection?

32. Academic Rigor in a Thinking CurriculumThe Share, Discuss, and Analyze Phase of the Lesson

33. Academic Rigor In a Thinking Curriculum A teacher must always be assessing and advancing student learning. A lesson is academically rigorous if student learning related to the essential understanding is advanced in the lesson. Accountable Talk discussion is the means by which teachers can find out what students know or do not know and advance them to the goals of the lesson.

34. Accountable Talk Discussions Recall what you know about the Accountable Talk features and indicators. In order to recall what you know: • Study the chart with the Accountable Talk moves. You are already familiar with the Accountable Talk moves that can be used to Ensure Purposeful, Coherent, and Productive Group Discussion. • Study the Accountable Talk moves associated with creating accountability to: • the learning community; • knowledge; and • rigorous thinking.

35. Accountable Talk Features and Indicators Accountability to the Learning Community • Active participation in classroom talk. • Listen attentively. • Elaborate and build on each others’ ideas. • Work to clarify or expand a proposition. Accountability to Knowledge • Specific and accurate knowledge. • Appropriate evidence for claims and arguments. • Commitment to getting it right. Accountability to Rigorous Thinking • Synthesize several sources of information. • Construct explanations and test understanding of concepts. • Formulate conjectures and hypotheses. • Employ generally accepted standards of reasoning. • Challenge the quality of evidence and reasoning.

36. Accountable Talk Moves

37. Accountable Talk Moves (continued)

38. The Share, Discuss, and Analyze Phase of the Lesson: Planning a Discussion (Small Group Discussion) From the list of potential EUs and its related student work, each group will select an essential understanding to focus their discussion. Identify a teacher in the group who will be in charge of leading a discussion with the group after the Accountable Talk moves related to the EU have been written. Write a set of Accountable Talk moves on chart paper so it is public to your group for the next stage in the process.

39. An Example: Accountable Talk Discussion The Focus Essential Understanding Continuous and Discrete Figures A fraction can be continuous (linear model), or a measureable quantity (area model), or a group of discrete/countable things (set model) but, regardless of the model, what remains true about all of the models is that they represent equal parts of a whole. Group F Group G • Explain how your model shows the problem. • Who understood what he said about the number line? (Community) • Can you say back what he said how the model shows the hike? (Community) • Who can add on and talk about the section of the number line? (Community) • The denominator tells the number of equal parts in the whole. (Marking) • Do we see in both models? (Rigor) • Tell us how you found in your picture (Group G). (Rigor)

40. Problematize the Accountable Talk Discussion(Whole Group Discussion) Using the list of essential understandings identified earlier, write Accountable Talk discussion questions to elicit from students a discussion of the mathematics. Begin the discussion with a “hook” to get student attention focused on an aspect of the mathematics.

41. An Example: Accountable Talk Discussion The Focus Essential Understanding Continuous and Discrete Figures A fraction can be continuous (linear model), or a measurable quantity (area model), or a group of discrete/countable things (set model) but, regardless of the model, what remains true about all of the models is that they represent equal parts of a whole. Group F Group G • One group used a number line and one group used an area model. How can this be? Can they both model the problem? (Hook) • Can Group F explain where the whole and where the stops are? • Who understood what they said about the divisions of the line? (Community) • Can you say back what they said about the meaning of the numerator and denominator for Bobby’s hike? (Community) • Each group made statements about the model being accurate to the context. Where do we see division in each of the models? (Rigor)

42. Revisiting Your Accountable Talk Prompts with an Eye Toward Problematizing Revisit your Accountable Talk prompts. Have you problematized the mathematics so as to draw students’ attention to the mathematical goal of the lesson? If you have already problematized the work, then underline the prompt in red. If you have not problematized the lesson, do so now. Write your problematizing prompt in red at the bottom and indicate where you would insert it in the set of prompts. We will be doing a Gallery Walk after we role play.

43. Role Play Our Accountable Talk Discussion • You will have 15 minutes to role play the discussion of one essential understanding. • Identify one observer in the group. The observer will keep track of the discussion moves used in the lesson. • The teacher will engage you in a discussion. (Note: You are well-behaved students.) The goals for the lesson are: • to engage all students in the group in developing an understanding of the EU; and • to gather evidence of student understanding based on what the student shares during the discussion.

44. Reflecting on the Role-Play: The Accountable Talk Discussion The observer has 2 minutes to share observations related to the lessons. The observations should be shared as “noticings.” Others in the group have 1 minute to share their “noticings.”

45. Reflecting on the Role Play: The Accountable Talk Discussion(Whole Group Discussion) Now that you have engaged in role playing, what are you now thinking about regarding Accountable Talk discussions?

46. Zooming In on Problematizing(Whole Group Discussion) Do a Gallery Walk. Read each others’ problematizing “hook.” What do you notice about the use of hooks? What role do “hooks” play in the lesson?

47. Step Back and Application to Our Work What have you learned today that you will apply when planning or teaching in your classroom?

48. Summary of Our Planning Process Participants: • identify goals for instruction; • Align Content Standards and Mathematical Practice Standards with a task. • Select essential understandings that relate to the Content Standards and Mathematical Practice Standards. • prepare for the Share, Discuss, and Analyze phase of the lesson. • Analyze and select student work that can be used to discuss essential understandings of mathematics. • Learn methods of problematizing the mathematics in the lesson.