Supporting Teachers’ Development of NCTM’s Effective Mathematics Teaching Practices: An Exploration of New Resources Teachers’ Development Group 2016 Leadership Seminar on Mathematics Professional Development March 16, 2016 Portland, OR The materials included in this presentation 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. A This talk is based on a presentation given in February 2016 at the 20 th Annual AMTE Meeting in Irvine, CA by members of the project team highlighted in blue. Peg Smith University of Pittsburgh

Session Overview Introduce Principles to Action Toolkit Engage Participants in Selected Toolkit Modules – Narrative Module: “The Case of Vanessa Culver and the Pay it Forward Task” – Video Module: “The Case of Kelly Polosky and the Triangle Task” – Student Work Module: “Using Questions to Assess and Advance Students Thinking” Consider Ways to Use Materials in the Professional Education of Teachers

Principles to Actions Professional Learning Toolkit: Teaching and Learning

Principles to Actions Professional Learning Toolkit: Teaching and Learning Purpose - Develop materials to support teacher learning of the Eight Effective Mathematics Teaching Practices. Each grade-band module engages teachers in analyzing instructional artifacts (e.g., mathematical tasks, narrative and video cases, student work samples) in order to help them refine or develop practices that support the learning of all students.

Project Team Peg Smith (chair), University of Pittsburgh Victoria Bill (co-chair) IFL, University of Pittsburgh Melissa Boston, Duquesne University Fredrick Dillon, Math Instructional Coach, Ideastream Amy Hillen, Kennesaw State University DeAnn Huinker, University of Wisconsin–Milwaukee Stephen Miller, IFL, University of Pittsburgh Lynn Raith, Mathematics Curriculum Specialist K-12 Michael Steele, University of Wisconsin–Milwaukee

Project Team Peg Smith (chair), University of Pittsburgh Victoria Bill (co-chair) IFL, University of Pittsburgh Melissa Boston, Duquesne University Fredrick Dillon, Math Instructional Coach, Ideastream Amy Hillen, Kennesaw State University DeAnn Huinker, University of Wisconsin–Milwaukee Stephen Miller, IFL, University of Pittsburgh Lynn Raith, Mathematics Curriculum Specialist K-12 Michael Steele, University of Wisconsin–Milwaukee

Components of a Toolkit Module PowerPoint Slides (w/facilitation notes) Discussion of Focal Effective Mathematics Teaching Practice(s) Math Task Featured Artifacts of Practice to Be Analyzed (i.e., video clips, narrative cases, student work samples) Additional Materials (lesson guides, sample solutions)

Narrative Module “The Case of Vanessa Culver and the Pay it Forward Task” Narrative Module “The Case of Vanessa Culver and the Pay it Forward Task”

2 hour Concurrent Session

Overview of Module Focus:Introduction to the effective teaching practices at the high school level Goal:To engage high school teachers in considering the effective teaching practices and how they play out during instruction Activities: – Read and analyze a narrative case that illustrates the effective teaching practices. – Consider each practice and how it is exemplified in the case.

The Pay It Forward Task In the movie “Pay it Forward”, a student, Trevor, comes up with an idea that he thought could change the world. He decides to do a good deed for three people and then each of the three people would do a good deed for three more people and so on. He believed that before long there would be good things happening to billions of people. At stage 1 of the process, Trevor completes three good deeds. How does the number of good deeds grow from stage to stage? How many good deeds would be completed at stage 5? Describe a function that would model the Pay It Forward process at any stage.

The Case of Ms. Culver Read the Case of Ms. Culver and study the strategies used by her students. Make note of what Ms. Culver did before or during instruction to support her students’ learning and understanding of exponential relationships. Talk with a neighbor about the actions and interactions that you identified as supporting student learning.

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.

Effective Mathematics Teaching Practices Establish mathematics goals to focus learning (lines 3-6) - used goals to guide instructional decision making during the lesson. Implement tasks that promote reasoning and problem solving (Lines 10-16) – the task had multiply entry points and could be solved in different ways (see work of groups). Use and connect mathematical representations – task evoked different representations (see work of groups) which were presented and connected (lines 81-82). Facilitate meaningful mathematical discourse – students presented work, explained, and asked questions (lines 30-36); students debated whether the function was y=3x or y=3 x (lines 42-75). Pose purposeful questions - teacher asked questions to assess what students knew (lines ),to help them make progress (line 23-26), and to challenge them (lines) 43-43). Build procedural fluency from conceptual understanding – developed an understanding of what an exponential function is and how it can be expressed algebraically. This provided a basis for ultimately using the general equation in more procedural ways. Support productive struggle in learning mathematics – provided suggestions when they were at at impasse (lines 23-26), facilitated a discussion about a misconception (lines 42-75). Elicit and use evidence of student thinking – the task required thinking, the work produced by students form the basis for the classroom discussion, and the exit ticket at the conclusion of the lesson provided insight on what students understood (lines 81-84).

Setting clear goals and selecting appropriately challenging tasks sets the stage for everything else. Implementing challenging tasks in ways that maintain their demands requires posing purposeful questions, using and connecting representations, eliciting and using evidence of student thinking, and facilitating meaningful discourse. Supporting students’ productive struggle is essential to learning with understanding. Conceptual understanding must come before, and serve as a basis on which to build, procedural fluency. Four Take-Aways from the Module

Video Module “The Case of Kelly Polosky and the Triangle Task” Video Module “The Case of Kelly Polosky and the Triangle Task”

Overview of Module Focus:Build procedural fluency from conceptual understanding Goal:To engage middle school teachers in how considering how tasks that promote reasoning and problem solving can provide the foundation for developing conceptual understanding Activities: – Solve and discuss the triangle task – Watch and discuss a video clip – Discuss the effective mathematics teaching practice of procedural fluency from conceptual understanding – Analyze a set of tasks and consider which ones might help students in realize that the generalization they found for right triangles works for all triangles.

Overview of Module Focus:Build procedural fluency from conceptual understanding Goal:To engage middle school teachers in how considering how tasks that promote reasoning and problem solving can provide the foundation for developing conceptual understanding Activities: – Solve and discuss the triangle task – Watch and discuss a video clip – Discuss the effective mathematics teaching practice of procedural fluency from conceptual understanding – Analyze a set of tasks and consider which ones might advance students from a generalization for right triangles to a generalization for all triangles

The Triangle Task Using the triangles shown above and the grid paper, construct a formula for the area of a right triangle. You may cut your triangles out and manipulate them in any way that might help you make your argument. After you explore with your particular triangle, make a general mathematical argument using words, symbols, and diagrams about the formula for the area of any right triangle. Task adapted from Everyday Mathematics Grade 5, Copyright © 2007 by Wright Group/McGraw-Hill, Chicago.

The Triangle Task Video School District: Wilkinsburg School:Kelly Elementary School Coach: Darin Cole Teacher: Ms. Kelly Polosky Class: 5 th grade Curriculum:Everyday Mathematics This small district is adjacent to the City of Pittsburgh and features a diverse student body. A coach, Mr. Cole, is also present in the video interacting with students. The students have just begun the geometry unit. Prior to the day’s lesson, the students sorted polygons and non-polygons and they identified the characteristics of polygons. They have also found the formulas for the area of rectangles and squares.

Ms. Polosky’s Mathematics Learning Goals Students will understand that: 1.The area of a triangle is ½ of its length times its width. 2.The relationship between area, length and width of a triangle can be generalized to a formula. 3.There are several equivalent ways of writing the formula for the area of a triangle and each can be related to a physical model.

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. Geometry ★ 6.G Solve real-world and mathematical problems involving area, surface area, and volume. 1.Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

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 Triangle Task: Video Context Prior to this clip: Ms. Polosky launched the task, ensuring that students understood what they were being asked to do Ms. Polosky provided students with cardboard triangles and other tools such as scissors and grid paper In this clip, we see Ms. Polosky facilitating a whole-class discussion that focused on comparing two possible formulas for the area of a right triangle generated by students.

Lens for Watching the Video As you watch the video, make note of what the teacher does to support student learning and engagement as they work on the task. Time 1 Identify any of the Effective Mathematics Teaching Practices that you notice Ms. Polosky using. Time 2 Make note of how you see Ms. Polosky supporting students in developing conceptual understanding as a basis for procedural fluency.

Procedural Fluency from Conceptual Understanding Mathematics classrooms should: Provide students opportunities to use their own reasoning strategies and methods to solve problems; Press students to explain and discuss why the procedures they are using work for particular problems; Use visual models to support students’ understandings of general methods To use mathematics effectively, students must be able to do much more than carry out mathematical procedures. They must know which procedure is appropriate and most productive in a given situation, what a procedure accomplishes, and what kind of results to expect. Mechanical execution of procedures without understanding their mathematical basis often leads to bizarre results. (Martin, 2009, p. 165)

Ms. Polosky Building Conceptual Understanding Provided students with tools (triangles on grids, scissors) and the challenged students to create a formula for finding the area of a right triangle. Pressed students to explain their formulas using the representations they had created (lines 8, 13, 17, 27, 26, etc.) Challenged students to compare different versions of the formula, relate each to the picture, and explain why they are equivalent (lines 44; 79-81)

Tasks that promote reasoning and problem solving should provide the opportunity for students to explore and make sense of ideas. Wrestling with ideas helps students build a solid foundation of conceptual understanding. Developing meaning for rules is critical to being able to use them appropriately and fluently. Three Take-Aways from the Module

Student Work Module “Using Questions to Assess and Advance Student Thinking” Student Work Module “Using Questions to Assess and Advance Student Thinking”

Overview of Module Focus:Posing Purposeful Questions Goal:To develop teachers capacity to ask questions about a student’s work that assess what they know about mathematics and advance them towards the goal of the lesson. Activities: – Read a portion of a lesson transcript and consider what the teacher is trying to accomplish with the questions she asks. – Examine students’ responses to a task and consider how to support their work through questioning

Teacher Questioning Example Ms. Rhee’s math class was studying statistics. She brought in three bags containing red and blue marbles. The three bags were labeled as shown below: Ms. Rhee shook each bag. She asked the class, “If you close your eyes, reach into a bag, and remove 1 marble, which bag would give you the best change of picking a blue marble? Which bag would you choose? ____________ Explain why this bag gives you the best choice of picking a blue marble. You may use the diagrams above in your explanation. The QUASAR assessment team, under the direction of Dr. Suzanne Lane, created this task. It appeared on the QCAI (QUASAR Cognitive Assessment Instrument) at grades red 25 blue Bag X Total = 100 marbles Bag Y Total = 60 marbles 40 red 20 blue 100 red 25 blue Bag Z Total = 125 marbles

Teacher Questioning Example T:Can you tell me what you did here? S:I compared the red marbles to the blue marbles in each bag. So I got 75 to 25 for bag X, 40 to 20 for bag Y, and 100 to 25 for bag Z. Then I reduced them down to 3 to 1, 2 to 1 and 4 to 1 that is just the same as 3, 2 and 4. T:What do you mean “reduced them down”? S:Well I found a number that you could divide both the top and bottom by that would give you 1 on the bottom. So for X it was 25, for Y it was 20, and for Z it was 25. T:Okay. So how did your work help you decide that “your chances would be higher” with bag Z? S:So I looked at the 3, 2 and 4. Bag Z had the biggest number (4) and the most marbles. So I decided that bag Z would be the best. T:Okay, so let’s go back to the ratios you made. Can you tell me what 3 to 1, 2 to 1 and 4 to 1 mean? S:Well it means that there are 3 red marbles for every 1 blue marble in bag X, 2 red marbles for every 1 blue marble in bag Y, and 4 red marbles for every 1 blue marble in bag Z. T:So what if each of the bags only had one blue marble. How many marbles would be in bag X, Y and Z? S: Bag X would have 4 marbles, Y would have 3 and Z would have 5. T: So I would like you to draw a picture or build a model of each of the smaller bags and think about which one of these smaller bags would give you the best chance of getting a blue marble. I will be back. What do you notice about the questions the teacher asks?

Teacher Questioning Example T:Can you tell me what you did here? S:I compared the red marbles to the blue marbles in each bag. So I got 75 to 25 for bag X, 40 to 20 for bag Y, and 100 to 25 for bag Z. Then I reduced them down to 3 to 1, 2 to 1 and 4 to 1 that is just the same as 3, 2 and 4. T:What do you mean “reduced them down”? S:Well I found a number that you could divide both the top and bottom by that would give you 1 on the bottom. So for X it was 25, for Y it was 20, and for Z it was 25. T:Okay. So how did your work help you decide that “your chances would be higher” with bag Z? S:So I looked at the 3, 2 and 4. Bag Z had the biggest number (4) and the most marbles. So I decided that bag Z would be the best. T:Okay, so let’s go back to the ratios you made. Can you tell me what 3 to 1, 2 to 1 and 4 to 1 mean? S:Well it means that there are 3 red marbles for every 1 blue marble in bag X, 2 red marbles for every 1 blue marble in bag Y, and 4 red marbles for every 1 blue marble in bag Z. T:So what if each of the bags only had one blue marble. How many marbles would be in bag X, Y and Z? S: Bag X would have 4 marbles, Y would have 3 and Z would have 5. T: So I would like you to draw a picture or build a model of each of the smaller bags and think about which one of these smaller bags would give you the best chance of getting a blue marble. I will be back.

Teacher Questioning Example T:Can you tell me what you did here? S:I compared the red marbles to the blue marbles in each bag. So I got 75 to 25 for bag X, 40 to 20 for bag Y, and 100 to 25 for bag Z. Then I reduced them down to 3 to 1, 2 to 1 and 4 to 1 that is just the same as 3, 2 and 4. T:What do you mean “reduced them down”? S:Well I found a number that you could divide both the top and bottom by that would give you 1 on the bottom. So for X it was 25, for Y it was 20, and for Z it was 25. T:Okay. So how did your work help you decide that “your chances would be higher” with bag Z? S:So I looked at the 3, 2 and 4. Bag Z had the biggest number (4) and the most marbles. So I decided that bag Z would be the best. T:Okay, so let’s go back to the ratios you made. Can you tell me what 3 to 1, 2 to 1 and 4 to 1 mean? S:Well it means that there are 3 red marbles for every 1 blue marble in bag X, 2 red marbles for every 1 blue marble in bag Y, and 4 red marbles for every 1 blue marble in bag Z. T:So what if each of the bags only had one blue marble. How many marbles would be in bag X, Y and Z? S: Bag X would have 4 marbles, Y would have 3 and Z would have 5. T: So I would like you to draw a picture or build a model of each of the smaller bags and consider -- Which one of these smaller bags would give you the best chance of getting a blue marble? Be ready to explain why. I will be back.

Characteristics of Questions That Support Students’ Exploration 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 Provide information to the teacher about what the student understands 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 AssessingAdvancing Stay & listen Walk away

Assessing and Advancing Learning What does each student appear to know and/or understand? Be prepared to share and justify your conclusions. For each response, write questions that you could ask… – to assess students’ thinking? – to advance students’ thinking?

Student A What questions could you ask to assess their understanding? What questions could you ask to advance their understanding?

Student A What questions could you ask to assess their understanding? How did you get 1/3, ½ and ¼? What does each number in the comparison mean? What questions could you ask to advance their understanding? One of the other students said that ¼ of the marbles in Bag X were blue. Can both answers be right?

Student B What questions could you ask to assess their understanding? What questions could you ask to advance their understanding?

Student B What questions could you ask to assess their understanding? Can you tell me what you did? What questions could you ask to advance their understanding? Does your method always work? Try these: Jar W has 3B and 9 R; Jar V has 200 B and 400 R.

Student C What questions could you ask to assess their understanding? What questions could you ask to advance their understanding?

Student C What questions could you ask to assess their understanding? How did you know y was better than x? What were you comparing? How did you know that y was better than z? What were you comparing? Why does y have a better chance? What questions could you ask to advance their understanding? How are your comparisons of x and y and y and z the same and how are they different?

Four Take-Aways from the Module Asking students to explain is the only way you can find out they understand. All students should be asked to explain whether they are correct or incorrect. Advancing students beyond incorrect approaches requires providing them with a challenge that helps them confront their error Assessing and advancing questions must be tailored for the individual. One size does not fit all and one question will not take students from where they are to the lesson goal. Planning good questions in advance of a lesson is essential. They don’t usually happen by chance.

Coming in 2017 from NCTM Three Book Series: Taking Action: Implementing Effective Mathematics Teaching Practices in Grades PreK-5 (Huinker, Bill, & Hillen) Taking Action: Implementing Effective Mathematics Teaching Practices in Grades 6-8 (Smith, Steele, & Raith) Taking Action: Implementing Effective Mathematics Teaching Practices in Grades 9-12 (Boston, Dillon, Miller, & Smith) Peg Smith, Series Editor

Questions or Comments

Reflecting on the various Toolkit modules, are you or could you use these modules in your work with teachers? Video CasesStudent Work AnalysisNarrative Cases

Students’ mathematical understandings Target mathematical goal

Students’ mathematical understandings Assess Target mathematical goal

Students’ mathematical understandings Assess Target mathematical goal

Student’s current understanding Mathematical trajectory

Students’ mathematical understandings Target mathematical goal