1. This module was developed by Lynn Raith, Mathematics Curriculum Specialist K-12. Video courtesy of NYC District 2 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 Peter Dubno and the Counting Cubes Task Eighth Grade Principles to Actions Effective Mathematics Teaching Practices The Case of Peter Dubno and the Counting Cubes Task Eighth Grade

2. Overview of the Session Solve and Discuss the Counting Cubes Task Watch the video clip and discuss what the teacher does to support his students’ engagement in and understanding of mathematics Discuss the effective mathematics teaching practices of use and connect mathematical representations and facilitate meaningful mathematical discourse.

3. The Counting Cubes Task Adapted from “Counting Cubes”, Lappan, Fey, Fitzgerald, Friel, & Phillips (2004). Connected MathematicsTM, Say it with symbols: Algebraic reasoning [Teacher’s Edition]. Glenview, IL: Pearson Prentice Hall. © Michigan State University Building 1 Building 2 Building 3 1. Describe a pattern you see in the cube buildings. 2. Use your pattern to write an expression for the number of cubes in the n th building. 3. Use your expression to find the number of cubes in the 5 th building. Check your results by constructing the 5 th building and counting the cubes. 4. Look for a different pattern in the buildings. Describe the pattern and use it to write a different expression for the number of cubes in the n th building.

4. The Counting Cubes Lesson Video Context School: The Lab School, New York City Community School District 2 Directors: Sheila Breslaw and Robert Menken Teacher: Peter Dubno Class: Grade 8 Curriculum:Connected Mathematics Size:17 students

5. Peter Dubno’s Mathematics Learning Goals Students will understand that: An equation can be written that describes the relationship between 2 quantities (i.e., the number of cubes and the building number); Different but equivalent equations can be written that represent the same situation; Connections can be made between pictorial and symbolic representations; and Variables must be clearly defined.

6. 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. Functions 8.F 8.F.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.

7. 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.

8. The Counting Cubes Task The Context of Video Earlier in the class: Students solved the Counting Cubes Task. The tables and equations students produced in response to the task were posted in the classroom. The Video Clip begins as pairs of students explain the thought processes they used to connect the volume or number of cubes in each picture with their equation. The students then point out differences and similarities in the equations generated.

9. Lens for Watching the Video Clip As you watch the video, make note of what the teacher does to support student learning and engagement with the mathematics as they explain their thinking. In particular, identify any of the Effective Mathematics Teaching Practices that you notice. Be prepared to give examples and to cite line numbers from the transcript to support your claims.

10. 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.

11. Use and Connect Mathematical Representations Effective teaching of mathematics engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving (NCTM, 2014, p.24). Teachers should: Allocate instructional time for students to use, discuss, and make connections among representations; Encourage students to explain, elaborate, or clarify their thinking; and Ask students to use the pictures to explain and justify their reasoning.

12. Use and Connect Mathematical Representations Representations embody critical features of mathematical constructs and actions, such as drawing diagrams and using words to show and explain the meaning of fractions, ratios, or the operation of multiplication. When students learn to represent, discuss, and make connections among mathematical ideas in multiple forms, they demonstrate deeper mathematical understanding and enhanced problem-solving abilities (Fuson, Kalchman, & Bransford, 2005; Lesh, Post, & Behr, 1987). NCTM, 2014, p. 24

13. Use and Connect Mathematical Representations Visual SymbolicPhysical ContextualVerbal

14. Facilitate Meaningful Mathematical Discourse In effective discourse teachers: Engage students in purposeful sharing of mathematical ideas, reasoning, and approaches, using varied representations; Select and sequence student approaches and solution strategies for whole-class analysis and discussion; Facilitate discourse among students by positioning them as authors of ideas, who explain and defund their approaches; and Ensuring progress toward mathematical goals by making explicit connections to student approaches and reasoning.

15. Facilitate Meaningful Mathematical Discourse Mathematical discourse includes the purposeful exchange of ideas through classroom discussion, as well as through other forms of verbal, visual, and written communication. The discourse in the mathematics classroom gives students opportunities to share ideas and clarify understandings, construct convincing arguments regarding why and how things work, develop a language for expressing mathematical ideas, and learn to see things from other perspectives (NCTM 1991, 2000). NCTM, 2014, p. 29

16. Lens for Watching the Video Clip - Time 2 As you watch the video this time, pay attention to the students discourse and the connections they make between representations. Specifically: 1.What does the discourse reveal about students’ understandings of the connections between the pictorial and algebraic representations? 2.To what extent does the discourse facilitate students explanations, or clarifications of their thinking? 3.To what extent does the discourse make mathematics more visible and accessible for student examination and discussion?

17. How might your apply what you have learned about the effective mathematics teaching practices to your own classroom instruction?