1. Developing Procedural Fluency from Conceptual Understanding The Road to Proportional Reasoning
Core Mathematics Partnership
Building Mathematical Knowledge and
High-Leverage Instruction for Student Success
January 7, 2016

2. Learning Intentions & Success Criteria
We are deepening our understanding of the highly effective mathematics teaching practice Building Procedural Fluency from Conceptual Understanding.
Success Criteria:
We will be successful when we can articulate how procedural fluency was built from conceptual understanding in three different situations.
Number Talk – procedural fluency with number talk
Grade level posters
Eureka problems

3. A Look at the
Effective Mathematics
Teaching Practices

4. 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.
Handout: MathTeachingPracticesList-ES-Casey.pdf
Facilitation Suggestions
The purpose of this slide is to shift the participants’ attention to the eight Effective Mathematics Teaching Practices.
Distribute the handout listing the eight Effective Mathematics Teaching Practices.
National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.

5. Jot down your responses.
Teaching Practice Focus: Build Procedural Fluency from Conceptual Understanding
Jot down your responses.
What does it mean to be fluent with computational procedures?
Why is it important to build procedures from conceptual understanding?
Facilitation Suggestions
Give participants 1-2 minutes to jot down their initial thoughts in response to the two questions on the slide.
Then the participants can read the excerpts from the book on the next two slides.
Participants should be looking for answers to the two questions and compare it to their own initial responses:
What does it mean to be fluent with computational procedures?
Why is it important to build procedures from conceptual understanding?

6. Build Procedural Fluency from Conceptual Understanding
Effective teaching of mathematics
builds on a foundation of conceptual understanding;
results in generalized methods for solving problems; and
enables students to flexibly choose among methods to solve contextual and mathematical problems.
Recap of reading
National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. (p. 42)

7. Procedural Fluency
Being fluent means that students are able to choose flexibly among methods and strategies to solve contextual and mathematical problems, they understand and are able to explain their approaches, and they are able to produce accurate answers efficiently.
Fluency builds from initial exploration and discussion of number concepts to using informal reasoning strategies based on meanings and properties of the operations to the eventual use of general methods as tools in solving problems. This sequence is beneficial whether students are building toward fluency with single- and multi-digit computation with whole numbers or fluency with, for example, fraction operations, proportional relationships, measurement formulas, or algebraic procedures.
Facilitation Suggestions
Discuss the first question: What does it mean to be fluent with computational procedures?
National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. (p. 42)

8. Procedural Fluency
Being fluent means that students are able to choose flexibly among methods and strategies to solve contextual and mathematical problems, they understand and are able to explain their approaches, and they are able to produce accurate answers efficiently.
Fluency builds from initial exploration and discussion of number concepts to using informal reasoning strategies based on meanings and properties of the operations to the eventual use of general methods as tools in solving problems. This sequence is beneficial whether students are building toward fluency with single- and multi-digit computation with whole numbers or fluency with, for example, fraction operations, proportional relationships, measurement formulas, or algebraic procedures.
Facilitation Suggestions
Discuss the first question: What does it mean to be fluent with computational procedures?
Now add to your notes about the 1st question.
National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. (p. 42)

9. Conceptual Understanding & Procedural Fluency
When procedures are connected with the underlying concepts, students have better retention of the procedures and are more able to apply them in new situations (Fuson, Kalchman, and Bransford 2005).
Martin (2009, p. 165) describes some of the reasons that fluency depends on and extends from conceptual understanding:
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.
Facilitation Suggestions
Discuss the second question: Why is it important to build procedures from conceptual understanding?
Now add to your notes
National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. (p. 42)

10. “Multiplication Strings”
Watch the Video
“Multiplication Strings”
8 minutes
Watch the video up to right before the final student talks. How is the teacher helping students think multiplicatively? Up to this point, students are doing a lot of additive thinking. Ask the table groups to turn and talk to identify instructional moves that could be used to help students move to multiplicative thinking.
Then finish the video.

11. Discussing the Video
Discuss the teacher actions that Build procedural fluency from conceptual understanding. Be prepared to give examples and to cite line numbers from the transcript to support your observations.
Handouts: Transcript-ES-Casey.pd
Materials: VideoClip-ES-Casey.mp4
Facilitator Suggestions
At this point, participants have established some common understanding of this mathematics teaching practice

12. Creating a Trajectory of Learning for Proportional Reasoning from 1st Grade to 7th Grade
Part 2 of the evening

13. Building Proportional Reasoning Across Grades
Silently review your homework reflection of your grade level’s work and its impact on the development of proportional reasoning.
Beginning with the youngest grade, share your reflection with the teachers at your table.
Create a poster that shows the development of proportional reasoning across the grades.
35 miniutes
3 minutes for each teacher to share.
10 minutes to create poster
10 minutes for gallery walk and final discussion.

14. How does the work in all the grades help students have procedural fluency for proportional reasoning based on conceptual understanding?
What instructional moves do you need to make to ensure that you are developing proportional reasoning in your students?
Stand and find someone to talk with. What are you understanding about building procedural fluency from conceptual understanding?

15. Selected Problems from Eureka Grade 6 Module 1
(packet from last session)
Part 2 of the evening

16. Using Ratio Reasoning to Solve Problems
Work in pairs to complete Exercise 2 from Eureka MathTM Grade 6, Module 1, Lesson 7.
How does the conceptual understanding of ratios lead to the understanding of fractions and ratios?

17. Using Ratio Reasoning to Solve Problems
Work in pairs to complete Exercise 1 from Eureka MathTM Grade 6, Module 1, Lesson 11
How did the use of ratio tables help to surface the concept of equivalent ratios?

18. Building Procedural Fluency in Grade 6
As a table group…
Review the 4 problems in the packet.
Review the student learning goals for each lesson.
Discuss…
How do these lessons build procedural fluency from conceptual understanding?
Lesson 2, 5, 7, 11
Putting into a 4 square

19. Learning Intentions & Success Criteria
We are deepening our understanding of the highly effective mathematics teaching practice Building Procedural Fluency from Conceptual Understanding.
Success Criteria:
We will be successful when we can articulate how procedural fluency was built from conceptual understanding in three different situations.

20. Homework: January 28 Read NCTM’s Principles to Action
Page 42 to the top of page 46
The table of teacher and student actions on pp. 47 and 48.
How do the general ideas in this section relate to the development of proportional reasoning at your grade level?

21. 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.