This module was developed by Margaret Smith at the University of Pittsburgh. Video courtesy of Pittsburgh Public Schools and the Institute for Learning. These materials are part of the Principles to Actions Professional Learning Toolkit: Teaching and Learning created by the project team that includes: Margaret Smith (chair), Victoria Bill (co-chair), Melissa Boston, Fredrick Dillon, Amy Hillen, DeAnn Huinker, Stephen Miller, Lynn Raith, and Michael Steele. Principles to Actions Effective Mathematics Teaching Practices The Case of Elizabeth Brovey and the Calling Plans 2 Task Eighth Grade Principles to Actions Effective Mathematics Teaching Practices The Case of Elizabeth Brovey and the Calling Plans 2 Task Eighth Grade

Overview of the Session Solve and Discuss the Calling Plans Task Parts 1 and 2 Watch two video clips and discuss what the teacher does to support her students engagement in and understanding of mathematics Discuss the effective mathematics teaching practice of pose purposeful questions

The Calling Plans Task Long-distance company A charges a base rate of $5.00 per month plus 4 cents for each minute that you are on the phone. Long- distance company B charges a base rate of only $2.00 per month but charges you 10 cents for every minute used. Part 1:How much time per month would you have to talk on the phone before subscribing to company A would save you money? Part 2:Create a phone plane, Company C, that costs the same as Companies A and B at 50 minutes, but has a lower monthly fee than either Company A or B.

The Calling Plans – Part 2 Video Context School: Pittsburgh Classical Academy, Pittsburgh, PA Principal: Valerie Merlo Teacher: Mrs. Elizabeth Brovey, Math Coach Class: Pre-Algebra 8th Grade Class Curriculum:Connected Mathematics Project 2 Size:27 students At the time the video was filmed, Elizabeth Brovey was a coach at Classical Academy in the Pittsburgh Public School District. The students are mainstream eighth grade Pre-Algebra students. The lesson occurred in April. (Elizabeth Brovey is currently a coach at Propel Andrew Street High School, a charter school in Pittsburgh, PA.)

Mrs. Brovey’s Mathematics Learning Goals Students will understand that: 1.the point of intersection is a solution to each equation (Companies A, B, and C); 2.the rate of change (cost per minute) determines the steepness of the line; 3.if the y-intercept (monthly base rate) is lowered then the rate of change (cost per minute) must increase in order for the new equation to intersect the other two at the same point.

Connections to the CCSS Content Standards National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, DC: Authors. Define, Evaluate and Compare Functions8.F Uses Functions to Model Relationships Between Quantities 4.Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. 5.Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

Connections to the CCSS Standards for Mathematical Practice 1.Make sense of problems and persevere in solving them. 2.Reason abstractly and quantitatively. 3.Construct viable arguments and critique the reasoning of others. 4.Model with mathematics. 5.Use appropriate tools strategically. 6.Attend to precision. 7.Look for and make use of structure. 8.Look for and express regularity in repeated reasoning.

The Calling Plans Task – Part 2 The Context of Video Clip 1 Prior to the lesson: Students solved the Calling Plans Task – Part 1. The tables, graphs and equations they produced in response to that task were posted in the classroom. Video Clip 1 begins immediately after Mrs. Brovey explained that students would be working on the Calling Plans Task – Part 2 and read the problem to students. Students first worked individually and subsequently worked in small groups.

Lens for Watching the Video Clip 1 - Time 1 As you watch the video, make note of what the teacher does to support student learning and engagement as they work on the task. In particular, identify any of the Effective Mathematics Teaching Practices that you notice Mrs. Brovey using. Be prepared to give examples and to cite line numbers from the transcript to support your claims.

Effective Mathematics Teaching Practices 1.Establish mathematics goals to focus learning. 2.Implement tasks that promote reasoning and problem solving. 3.Use and connect mathematical representations. 4.Facilitate meaningful mathematical discourse. 5.Pose purposeful questions. 6.Build procedural fluency from conceptual understanding. 7.Support productive struggle in learning mathematics. 8.Elicit and use evidence of student thinking.

Pose Purposeful Questions Effective Questions should: Reveal students’ current understandings; Encourage students to explain, elaborate, or clarify their thinking; and Make the mathematics more visible and accessible for student examination and discussion. 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 deepen their understanding. (Weiss and Pasley, 2004)

Lens for Watching the Video Clip 1 - Time 2 As you watch the video this time, pay attention to the questions the teacher asks. Specifically: What do the questions reveal about students’ current understandings? To what extent do the questions encourage students to explain, elaborate, or clarify their thinking? To what extent to the questions make mathematics more visible and accessible for student examination and discussion?

The Calling Plans Task – Part 2 The Context of Video Clip 2 Following individual and small group work, Mrs. Brovey pulls the class together for a whole group discussion. Several different equations that satisfy the conditions of the problem are offered by students. Jake, a student in the class then proposed a theory that every time the rate increases by 1 cent the base rate decreases by 50 cents. Mrs. Brovey records the four possible phone plans for Company C (shown below) on the board and ask the class what patterns they see. C =.14m C =.13m + $.50 C =.12m + $1.00 C =.11m + $1.50 Video Clip 2 focuses on the discussion between teacher and students regarding the patterns they notice..

Lens for Watching the Video Clip 2 As you watch the second video this time, pay attention to the questions the teacher asks. Specifically: What do the questions reveal about students’ current understandings? To what extent do the questions encourage students to explain, elaborate, or clarify their thinking? To what extent to the questions make mathematics more visible and accessible for student examination and discussion? How are the questions in this clip similar to or different from the questions asked in video clip 1?

Pose Purposeful Questions: Teacher and Student Actions What are teachers doing? Advancing student understanding by asking questions that build on, but do not take over or funnel, student thinking. Making certain to ask questions that go beyond gathering information to probing thinking and requiring explanation and justification. Asking intentional questions that make the mathematics more visible and accessible for student examination and discussion. Allowing sufficient wait time so that more students can formulate and offer responses. What are students doing? Expecting to be asked to explain, clarify, and elaborate on their thinking. Thinking carefully about how to present their responses to questions clearly, without rushing to respond quickly. Reflecting on and justifying their reasoning, not simply providing answers. Listening to, commenting on, and questioning the contributions of their classmates.

Consider the Teacher and Student Actions Related to Posing Purposeful Questions What will you need to work on in order to pose purposeful questions in your own classroom? Where will you start?