Supporting Rigorous Mathematics Teaching and Learning

1. Supporting Rigorous Mathematics Teaching and Learning Illuminating Student Thinking: Assessing and Advancing Questions Tennessee Department of Education High School Mathematics Algebra 1

2. Rationale Effective teaching requires being able to support students as they work on challenging tasks without taking over the process of thinking for them (NCTM, 2000). Asking questions that assess student understanding of mathematical ideas, strategies or representations provides teachers with insights into what students know and can do. The insights gained from these questions prepare teachers to then ask questions that advance student understanding of mathematical ideas, strategies or connections to representations. By analyzing students’ written responses, teachers will have the opportunity to develop questions that assess and advance students’ current mathematical understanding and to begin to develop an understanding of the characteristics of such questions.

3. Session Goals Participants will: learn to ask assessing and advancing questions based on what is learned about student thinking from student responses to a mathematical task; and develop characteristics of assessing and advancing questions and be able to distinguish the purpose of each type.

4. Overview of Activities Participants will: • analyze student work to determine what the students know and what they can do; • develop questions to be asked during the Explore Phase of the lesson; • identify characteristics of questions that assess and advance student learning; • consider ways the questions differ; and • discuss the benefits of engaging in this process.

5. The Structures 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 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: By engaging 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 3. Focus the Discussion on Key Mathematical Ideas 4. Engage in a Quick Write

6. Bike and Truck Task Distance from start of road (in feet) Time (in seconds) A bicycle traveling at a steady rate and a truck are moving along a road in the same direction. The graph below shows their positions as a function of time. Let B(t) represent the bicycle’s distance and K(t) represent the truck’s distance.

7. Bike and Truck Task • Label the graphs appropriately with B(t) and K(t). Explain how you made your decision. • Describe the movement of the truck. Explain how you used the values of B(t) and K(t) to make decisions about your description. • Which vehicle was first to reach 300 feet from the start of the road? How can you use the domain and/or range to determine which vehicle was the first to reach 300 feet? Explain your reasoning in words. • Jack claims that the average rate of change for both the bicycle and the truck was the same in the first 17 seconds of travel. Explain why you agree or disagree with Jack.

8. The Common Core State Standards (CCSS) for Mathematical Content : The Bike and Truck Task Which of CCSS for Mathematical Content did we address when solving and discussing the task?

9. The CCSS for Mathematical ContentCCSS Conceptual Category – Algebra Common Core State Standards, 2010, p. 65, NGA Center/CCSSO

10. The CCSS for Mathematical ContentCCSS Conceptual Category – Algebra Common Core State Standards, 2010, p. 65, NGA Center/CCSSO

11. The CCSS for Mathematical ContentCCSS Conceptual Category – Algebra Common Core State Standards, 2010, p. 65, NGA Center/CCSSO

12. The CCSS for Mathematical ContentCCSS Conceptual Category – Functions Common Core State Standards, 2010, p. 69, NGA Center/CCSSO

13. What Does Each Student Know? Now we will focus on three pieces of student work. Individually examine the three pieces of student work A, B, and C for the Bike and Truck Task in your participant handout. What does each student know? Be prepared to share and justify your conclusions.

14. Response A 14

15. Response B 15

16. Response C 16

17. What Does Each Student Know? Why is it important to make evidence-based comments and not to make inferences when identifying what students know and what they can do?

18. Supporting Students’ Exploration of a Task through Questioning Imagine that you are walking around the room, observing your students as they work on theBike and Truck Task. Consider what you would say to the students who produced responses A, B, C, and Din order to assessandadvancetheir thinking about key mathematical ideas, problem solving strategies, or use of and connection between representations. Specifically, for each response, indicate what questions you would ask: • to determine what the student knows and understands(ASSESSING QUESTIONS) • to move the student towards the target mathematical goals (ADVANCING QUESTIONS).

19. Cannot Get Started Imagine that you are walking around the room, observing your students as they work on theBike and Truck Task. Group D has little or nothing on their papers. Write an assessing question and an advancing question for Group D. Be prepared to share and justify your conclusions. Reminder: You cannot TELL Group D how to start. What questions can you ask them?

20. Discussing Assessing Questions Listen as several assessing questions are read aloud. Consider how the assessing questions are similar to or different from each other. Are there any questions that you believe do not belong in this category and why? What are some general characteristics of the assessing questions?

21. Discussing Advancing Questions Listen as several advancing questions are read aloud. Consider how the advancing questions are similar to or different from each other. Are there any questions that you believe do not belong in this category and why? What are some general characteristics of the advancing questions?

22. Looking for Patterns Look across the different assessing and advancing questions written for the different students. Do you notice any patterns? Why are some students’ assessing questions other students’ advancing questions? Do we ask more content-focused questions or questions related to the mathematical practice standards?

23. 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.

24. Reflection • Why is it important to ask students both assessing and advancing questions? What message do you send to students if you ask ONLY assessing questions? • Look across the set of both assessing and advancing questions. Do we ask more questions related to content or to mathematical practice standards?

25. Reflection Not all tasks are created equal. Assessing and advancing questions can be asked of some tasks but not others. What are the characteristics of tasks in which it is worthwhile to ask assessing and advancing questions?

26. Preparing to Ask Assessing and Advancing Questions How does a teacher prepare to ask assessing and advancing questions?

27. Supporting Student Thinking and Learning In planning a lesson, what do you think can be gained by considering how students are likely to respond to a task and by developing questions in advance that can assess and advance their learningin a way that depends on the solution path they’ve chosen?

28. Reflection What have you learned about assessing and advancing questions that you can use in your classroom next school year? Turn and Talk

29. Bridge to Practice • Select a task that is cognitively demanding, based on the TAG. (Be prepared to explain to others why the task is a high-level task. Refer to the TAG and specific characteristics of your task when justifying why the task is a Doing Mathematics Task or a Procedures With Connections Task.) • Plan a lesson with colleagues. • Anticipate student responses, errors, and misconceptions. • Write assessing and advancing questions related to the student responses. Keep copies of your planning notes. • Teach the lesson. When you are in the Explore Phase of the lesson, tape your questions and the student responses, or ask a colleague to scribe them. • Following the lesson, reflect on the kinds of assessing and advancing questions you asked and how they supported students to learn the mathematics.