1. Productive Mathematical Discussions: Working at the Confluence of Effective Mathematics Teaching Practices
Core Mathematics Partnership
Building Mathematical Knowledge and
High-Leverage Instruction for Student Success

2. Effective Mathematics Teaching Practices that Support Productive Mathematical Discussions
Tuesday, July 19, 2016
10:30 – 12:00

3. Effective Mathematics Teaching Practices
Establish mathematics goals to focus learning.
Implement tasks that promote reasoning and problem solving.
Use and connect mathematical representations.
Facilitate meaningful mathematical discourse.
Pose purposeful questions.
Build procedural fluency from conceptual understanding.
Support productive struggle in learning mathematics.
Elicit and use evidence of student thinking.
Use and connect mathematical representations (3) and facilitate meaningful mathematical discourse (4) are the most salient practices in this short clip and ones that will be examined further.
It could also be argued that there is evidence that the teacher 1) set a clear goal for student learning; 2) selected a task that promoted reasoning and problem solving; 5) asked purposeful questions; and 8) elicited and made use of student thinking. This highlights how the practices all work together.

4. Facilitate meaningful mathematical discourse
Establish math goals to focus learning
Implement tasks that promote reasoning and problem solving
Build procedural fluency from conceptual understanding
Facilitate meaningful mathematical discourse
Elicit and use evidence of student thinking
Pose purposeful questions
Use and connect mathematical representations
Support productive struggle in learning mathematics
Effective Mathematics Teaching Practices
“Building a Teaching Framework”

5. The Case of David Crane
Read The Case of David Crane in Smith & Stein (2011), pages 1-4.
As you read, consider:
What important mathematical ideas are evident in the work students produced in Mr. Crane’s class?

6. The Case of David Crane Discuss with your small groups:
What important mathematical ideas are evident in the work students produced in Mr. Crane’s class?
What effective mathematics teaching practices did you notice Mr. Crane engaged in that supported student learning?
What missed opportunities did you notice?

7. Facilitate meaningful mathematical discourse
Establish math goals to focus learning
Implement tasks that promote reasoning and problem solving
Build procedural fluency from conceptual understanding
Facilitate meaningful mathematical discourse
Elicit and use evidence of student thinking
Pose purposeful questions
Use and connect mathematical representations
Support productive struggle in learning mathematics
Effective Mathematics Teaching Practices
“Building a Teaching Framework”

8. The Case of David Crane
Read the analysis of the case (pp. 5-6). How do the points made reflect our discussion?

9. The Case of Peter Dubno: Counting Cubes
Building 1
Building 2
Building 3
Describe a pattern you see in the cube buildings.
Use your pattern to write an expression for the number of cubes in the nth building.
Use your expression to find the number of cubes in the 5th building. Check your results by constructing the 5th building and counting the cubes.
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 nth building.
HANDOUT -CountingCubesTask-MS-Dubno
Engaging teachers in solving the Counting Cubes Task will take approximately 45 minutes to an hour. The lesson guide (CountingCubes-LESSONGUIDE-MS-Dubno) contains suggestions for conducting the lesson and sample solutions that may be helpful to you. In addition, using the learning goals on slide 5 to guide your discussion of the task will help participants understand what the teacher featured in the video was trying to accomplish.
If time is limited you may want to tell teachers that the number of cubes in a building can be determined by using one of the equations shown below and ask teachers to define the variable and explain what each equation means in the context of the problem. [Note that the task asks students to write an expression not an equation. It is likely that students will do one or the other. If so it would be important to discuss the difference between an expression and an equation.]
5n – 4 = T where n is the building number (5 legs x the building number but subtract 4 cubes that overlap in the middle); and T is the total number of cubes
1 + 5 x (n-1) = T where n is the building number (the building number – 1 gives the number of cubes in each arm then multiply by 5 and add the 1 in the center); and T is the total number of cubes
4(n – 1) + n = T where n is the building number ( building number -1 gives the number of cubes in the 4 arms of the bottom layer + the building number which is the number of cubes in the center tower); and T is the total number of cubes
5n + 1 = T where n is the number of cubes in each of the 5 legs (5 legs x number of cubes in a leg + the 1 cube in the center); and T is the total number of cubes. This equation is uncommon and should be a point of discussion for the group. Defining the variable in a unique way is a good discussion for the group and for use in classrooms.
It is recommended that you share the 4 different equations (or additional equations if they arise) that represent different visualizations and/or representations, in an order from lower to higher complexity.
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

10. The Case of Peter Dubno: 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.
HANDOUT -Transcript--MS-Dubno

11. 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.
You may want to give teachers a copy of the Effective Teaching Practices to guide their discussion. (HANDOUT -Effective Mathematics Teaching Practices-MS-Dubno.)
Here are some of the things that teachers might notice:
T used a good task
T asked purposeful questions involving defining the variables and equivalent expressions and looking for similarities and differences in expressions
T asked students to make connections between the pictorial representation and their expression
T promoted discourse between/among the students in the class as they explained their solution and the solution of other groups that were similar or different than theirs
T elicited student thinking
You may want to chart these ideas so you can keep track of what has been said. If Using and Connecting Mathematical Representations and Facilitating Meaningful Mathematical Discourse (the primary focus of this module) are NOT mentioned, you might want to ask teachers if they noticed the way in which the picture was used (connections) or the nature of the talk in the classroom (discourse).

12. Establish math goals to focus learning
Implement tasks that promote reasoning and problem solving
Build procedural fluency from conceptual understanding
Facilitate meaningful mathematical discourse
Elicit and use evidence of student thinking
Pose purposeful questions
Use and connect mathematical representations
Support productive struggle in learning mathematics
Use and connect mathematical representations (3) and facilitate meaningful mathematical discourse (4) are the most salient practices in this short clip and ones that will be examined further.
It could also be argued that there is evidence that the teacher 1) set a clear goal for student learning; 2) selected a task that promoted reasoning and problem solving; 5) asked purposeful questions; and 8) elicited and made use of student thinking. This highlights how the practices all work together.
Effective Mathematics Teaching Practices
“Building a Teaching Framework”

13. Comparing the Two Cases
What are the similarities and what are the differences between The Case of David Crane and The Case of Peter Dubno? Do the differences matter?

14. Facilitate Meaningful Mathematics 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. Five Practices for Orchestrating Productive Discussions
anticipating likely student responses to challenging mathematical tasks;
monitoring students’ actual responses to the tasks (while students work on the tasks in pairs or small groups);
selecting particular students to present their mathematical work during the whole-class discussion;
sequencing the student responses that will be displayed in a specific order; and
connecting different students’ responses and connecting the responses to key mathematical ideas.

17. For your project… Briefly review Chapter 4 in Smith and Stein (2011).
Consider: as you learn more about the ways in which students think geometrically, what types of responses would you anticipate your students providing to the task you gave them?

18. Core Mathematics Partnership Project
Disclaimer
Core Mathematics Partnership Project
University of Wisconsin-Milwaukee,
This material was developed for the Core Mathematics Partnership project through the University of Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This material may be used by schools to support learning of teachers and staff provided appropriate attribution and acknowledgement of its source. Other use of this work without prior written permission is prohibited—including reproduction, modification, distribution, or re-publication and use by non-profit organizations and commercial vendors.
This project was supported through a grant from the Wisconsin ESEA Title II, Part B, Mathematics and Science Partnerships.