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
Wednesday, July 20, 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.

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. 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.

6. The Hexagon Task
Trains 1, 2, 3, and 4 are the first 4 trains in the hexagon pattern. The first train in this pattern consists of one regular hexagon. For each subsequent train, one additional hexagon is added. For the hexagon pattern:
Compute the perimeter for the first 4 trains.
Determine the perimeter for the tenth train without constructing it.
Write a description that could be used to compute the perimeter of any train in the pattern. Explain how you know it will always work. (Use the edge length of any pattern block or the length of a side of a hexagon as your unit of measure.)
Consider the following: What are some correct and incorrect student approaches to this task?

7. Hexagon Task: Some possible solutions2 2
3 4 2
3 4 2
3 4 2
3 4 2
3 4 A
Tops & Bottoms, plus 2 ends
y = 4x + 2 B
Insides and Outsides
y = 4(x – 2) + 10
The two outside hexagons each contribute 5 sides
The inside hexagons each contribute 4 Number of inside hexagons is train number minus 2 C
Shared sides subtracted
y = 6x – 2(x – 1)
Each hexagon contributes 6 sides For each new hexagon past train 1, there is a pair of inside sides that have to be subtracted The number of shared pairs is the train number minus 1
Possible misconceptions: incorrect definition of perimeter, assuming the relationship is proportional (train 10 = 2 times train 5), incorrect formula based on the recursive (y=x+4)

8. 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 & Pasley, 2004)

9. Characteristics of Questions That Support Students’ Exploration
Assessing
Advancing
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
Walk away
Stay & listen

10. Hexagon Task Student Work
Consider each piece of the hexagon task student work.
What questions would you ask the student that assesses their thinking?
What questions would you ask to advance their thinking?

11. Hexagon Task: Some (more) possible solutions2 2
3 4 2
3 4 2
3 4 2
3 4 2
3 4 A
Tops & Bottoms, plus 2 ends
y = 4x + 2 B
Insides and Outsides
y = 4(x – 2) + 10
The two outside hexagons each contribute 5 sides
The inside hexagons each contribute 4 Number of inside hexagons is train number minus 2 C
Shared sides subtracted
y = 6x – 2(x – 1)
Each hexagon contributes 6 sides For each new hexagon past train 1, there is a pair of inside sides that have to be subtracted The number of shared pairs is the train number minus 1 D E
Table of values
y = 4x + 2
Slope-intercept form
Train Number
Perimeter 1 6 +4 2 10 3 14 4 18 5 22 26 7 30 8 34
Train Number
Perimeter  m
2 (b) 1 6 +4 2 10 3 14 4 18 5 22

12. Hexagon Task: Selecting and Sequencing
Imagine that you had determined the following learning goals for the lesson featuring the hexagon task:
Students will understand that:
An equation can be written that describes the relationship between 2 quantities, the independent (x) and dependent (y) variables;
Linear relationships have a constant rate of change between the quantities, are depicted graphically by a line, and can be written symbolically as y = mx + b, where m is the constant rate of change and the slope of the line, and b is the value of the y-quantity when x = 0 (i.e., the y-intercept);
Different but equivalent equations can be written that represent the same situation; and
Connections can be made between tables, graphs, equations and contexts.
Your students produced Solutions A-E. Consider:
Which of these five solution paths would you select for students to share in a whole-class discussion? (Select at least 3.)
In what order would you sequence these solutions, and why?
What teacher moves would you make to highlight connections between these solutions and your lesson goals?

13. Rules of Thumb for Selecting and Sequencing
Present strategies in sequence from concrete to abstract
First present strategies that afford broad student access, and then move to more unique or mathematically complex solutions
Sequence solutions so that the connections among solutions with common mathematical features can be highlighted

14. For your project… Review Chapter 5 in Smith and Stein (2011).
Consider: What specific moves do you want to focus on as a teacher to better connect student responses to move students towards your mathematical goals?

15. 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.