1. Based on the work of Dr. M.S. Smith, University of Pgh. Key Ingredients to Developing Mathematical Understanding: Anticipating, Monitoring, Selecting, Sequencing, and Connecting Student Thinking Thinking Through a Lesson Protocol

2. Based on the work of Dr. M.S. Smith, University of Pgh. Background on the Thinking Through the Lesson Protocol (TTLP) Framework for developing lessons Dr. Margaret Smith- University of Pittsburgh “As a teacher, how will you determine what your students learn and the effectiveness of your instruction?”  Developed as a result of nearly a decade of work on QUASAR at the Learning Research and Development Center at the University of Pittsburgh  “The effectiveness of a lesson depends significantly on the care with which the lesson plan is prepared.”- Brahier, 2000

3. Based on the work of Dr. M.S. Smith, University of Pgh. “The cumulative experiences of (numerous elementary and secondary teachers with varying levels of teaching experience) suggests that the Thinking Through a Lesson Protocol can be a useful tool in planning, teaching, and reflecting on lessons and can lead to improved teaching.” Smith, Bill, Hughes (MTMS, October 2008, p.137)

4. Based on the work of Dr. M.S. Smith, University of Pgh. Part III Sharing & Discussing the Task: Orchestrating Whole- Group Discussions

5. Based on the work of Dr. M.S. Smith, University of Pgh. The Role of the Teacher  Worthwhile tasks alone are not sufficient for effective teaching. Teachers must also decide what aspects of a task to highlight, how to organize and orchestrate the work of students, what questions to ask to challenge those with varied levels of expertise, and how to support students without taking over the process of thinking for them and thus eliminating the challenge. NCTM, 2000, p.19

6. Based on the work of Dr. M.S. Smith, University of Pgh. Communication  Communication is an essential part of mathematics and mathematics education. It is a way of sharing ideas and clarifying understanding. Through communication, ideas become objects of reflection, refinement, discussion, and amendment. The communication process also helps build meaning and permanence for ideas and makes them public. NCTM, 2000, p. 60

7. Based on the work of Dr. M.S. Smith, University of Pgh. Sharing and Discussing the Task With your group, analyze all student work samples Based on the math goals for this task, select four pieces of work. Consider the following in your selections:  Identify the solution paths that you would want to have shared during the whole-group discussion.  Specify the order in which they would be shared.  Explain your selection and ordering.

8. Based on the work of Dr. M.S. Smith, University of Pgh. Sharing and Discussing the Task  Review the chart of responses to be shared and identify any patterns you observe in this data.

9. Based on the work of Dr. M.S. Smith, University of Pgh. Sharing and Discussing the Task  Can we begin to generate any general “Rules of Thumb” for determining which responses to share and discuss?

10. Based on the work of Dr. M.S. Smith, University of Pgh. Part II: Supporting Students’ Exploration Imagine that you are walking around the room as your students work on the task, observing what they are doing. Consider what you would say to the students who produced the following responses:  Emma & Emma  Harry & Ese  Nellie & Nate  Suzanne & Rose, in order to assess and advance their understanding. Specifically for each response, indicate what questions you would ask:  To assess the students’ understanding of key mathematical ideas, problem-solving strategies or representations.  To advance the students’ understanding toward the target goals.

11. Based on the work of Dr. M.S. Smith, University of Pgh. Supporting Students’ Exploration With your group, discuss:  What are the characteristics of assess questions?  What are the characteristics of advance questions?

12. Based on the work of Dr. M.S. Smith, University of Pgh. 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 they have done 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

13. Based on the work of Dr. M.S. Smith, University of Pgh. Reflecting on our work  In what ways might assessing and advancing questions help you support students’ engagement with a high-level task?  Does the distinction between assess and advance questions really matter?

14. Based on the work of Dr. M.S. Smith, University of Pgh. The importance of Questions  Teachers provoke students’ reasoning about mathematics through the tasks they provide and the questions they ask. (NCTM,1991)  Asking questions that reveal students’ knowledge about mathematics allows teachers to design instruction that responds to and builds on this knowledge. (NCTM, 2000)

15. Based on the work of Dr. M.S. Smith, University of Pgh. The Importance of Questions  Teachers’ questions are crucial in helping students make connections and learn important mathematics and science concepts. Teachers need to know how students typically think about particular concepts, how to determine what a particular student or group of students thinks about those ideas, and how to help students think about those ideas, and how to help students deepen their understanding. Weiss & Paskey, 2004

16. Based on the work of Dr. M.S. Smith, University of Pgh. Purpose of TTLP  To prompt teachers to think deeply about a specific lesson in order to consider how to advance students’ mathematical understanding  To focus on students’ mathematical thinking  Requires anticipating a range of student solutions or solution strategies  Prompts the development of questions that will support students’ engagement and learning  Address ways to facilitate the learning of all students  To move beyond structural components of lesson planning

17. Based on the work of Dr. M.S. Smith, University of Pgh. Connecting the TTLP to Practice 1. What questions will you ask to focus student thinking? 2. What will you see or hear that lets you know how students are thinking about the mathematical ideas of the lesson? 3. Which solution paths do you want to have shared during the class discussion? In what order will the solutions be presented? Why? 4. What will you see or hear that lets you know that students in the class understand the mathematical ideas that you intended for them to learn?

18. Based on the work of Dr. M.S. Smith, University of Pgh. Teacher reflection: “I may not have it sitting on my desk, going point to point with it, but I think: What are the misconceptions? How am I going to organize work? What are my questions? Those are the three big things I think about when planning a lesson.” Smith, Bill, Hughes- (MTMS, October 2008, p. 137)

19. Based on the work of Dr. M.S. Smith, University of Pgh. This material is based on work supported by the National Science Foundation under Grant No. EHR-0314914. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the granting agency.