1. The National Council of Supervisors of Mathematics
The Common Core State Standards
Illustrating the Standards for Mathematical Practice:
Reasoning about Problems and Unpacking Others’ Reasoning
Facilitators will need to have the following resources for each participant:
A copy of the PPT (formatted with two slides per page)
A copy of the participant handouts
- Common Core Content Standards for Mathematics
- Bulleted version of the Common Core State Standards for Mathematical Practice
(If you can, it is nice to have these Standards printed on card stock and laminated!)
- The handout for this session
Download the video files (4 clips). Refer to Facilitator’s notes on Slide 12.
4. Materials needed: colored counters and/or square tiles and graph paper available for participants to use when they work on the math task
Ask each participant to bring with them a copy of a lesson they have recently taught.

2. We hope you will help us grow and improve our NCSM resources!
Module Evaluation
Facilitator: At the end of this Powerpoint, you will find a link to an anonymous brief e-survey that will help us understand how the module is being used and how well it worked in your setting.
We hope you will help us grow and improve our NCSM resources!

3. Common Core State Standards
Mathematics
Standards for Content
Standards for Practice
Share both sets of Standards with Participants.
The first thing to understand about the Common Core State Standards is that there are two types of mathematics standards: content standards and practice standards. Neither is intended to stand alone; both support and enhance the development of the other. Together they weave a new picture of what mathematics education might look like with implications for teaching, assessing and learning. Both types of standards are equally important. Both types need to be implemented, and both will be assessed.

4. Today’s Goals
To explore the mathematical standards for Content and Practice
To consider how the Common Core State Standards (CCSS) are likely to impact your mathematics program and plan next steps
In particular, participants will
Examine opportunities to develop skill in reasoning about problems and unpacking others’reasoning
Discuss the goals.

5. Standards for Mathematical Practice
“The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important ‘processes and proficiencies’ with longstanding importance in mathematics education.” (CCSS, 2010)
The new Standards for Mathematical Practice were based on the NCTM Process Standards and the National Research Council’s Stands of Mathematical Proficiency.
Remind participants of the vital nature of the Standards for Mathematical Practice with respect to students developing a powerful set of core mathematical competencies. These practices do not stand alone and are not intended to be taught as stand alone lessons. They are an integral part of learning and doing mathematics and need to be taught with the same intention and attention as mathematical content.

6. Standards for Mathematical Practice
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Refer participants to the handout with the description of the CCSS Standards for Mathematical Practices.

7. Structuring the Practices
This organization of the Standards for Mathematical Proficiency was developed by one of the principal authors of the Common Core State Standards for Mathematics, Dr. William McCallum, University of Arizona. His rationale for this organization is as follows:
In the progressions project, we’ve been discussing how best to represent the standards for mathematical practice. The practices are signposted throughout the documents, but we’ve also been thinking about how to provide some structure for the practice standards that will help people avoid fruitless tagging exercises in their efforts to integrate the practice standards into the content standards. If you think about it long enough you can associate just about any practice standard with any content standard, but this sort of matrix thinking can lead to a dilution of the force of the practice standards—if you try to do everything all the time, you end up doing nothing. This diagram is an attempt to provide some higher order structure to the practice standards, just as the clusters and domains provide higher order structure to the content standards.

8. Standards for Mathematical Practice
Individually review the Standards for Mathematical Practice.
Choose a partner at your table and discuss a new insight you had into the Standards for Mathematical Practice.
Then discuss the following question.
This opening reflection and discussion are designed to be used in one of two ways depending on your group’s familiarity with the Standards for Mathematical Practice.
Step 1. Select the appropriate option for your group; then move to step 2 below.
Option 1.
If most of the participants in your group have already begun to think about the practices, the facilitator can choose to have them focus on one or two practices. In this situation, have participants read and think individually (no more than 3- 5 min.) about the 4th and 5th Standards for Mathematical Practice; then ask them to share their thinking about the standard with a partner. See questions 1 and 2 on the slide. Jump to step 2 below.
Option 2.
If most participants have not already begun to think about the practices, the facilitator can choose to have the group explore the entire set of 8 standards. In this situation, have participants select a partner and then ask them to divide all 8 of the Standards for Mathematical Practice with their partner. Give participants time to read and think individually (no more than 5 min.) about their portion of the 8 Standards for Mathematical Practice. Let them know you will ask them to summarize each of their four standards for their partner before they discuss question 2 on the slide above. Jump to step 2 below.
Step 2 Once partners have had a chance to discuss the Standard of Practice they were assigned, move them on to a discussion of the focus question: What implications might this standard for mathematical practice have on your classroom? Ask participants make notes as they discuss their Standard(s) of Practice and the focus question.
As you move about the room listening in on the discussions, you may find opportunities to push the discussion with questions like: What specific types of mathematical experiences will students need to become proficient with the practices?
As a whole group, discuss the focus question and chart the list of classroom implications related to the practices. This does not need to be a long discussion, just enough to encourage participants to begin to connect these practices to what they already know about the practices and to get a list of related implications for use later on.
Let participants know we will return to these questions at the end of the session.
What implications might the Standards for Mathematical Practice have on your classroom?

9. Buttons Task
Gita plays with her grandmother’s collection of black and white buttons. She arranges them in patterns.
Her first 3 patterns are shown below.
Pattern #1 Pattern # Pattern # Pattern #4
Draw pattern 4 next to pattern 3.
How many white buttons does Gita need for Pattern 5 and Pattern 6? Explain how you figured this out.
How many buttons in all does Gita need to make Pattern 11? Explain how you figured this out.
Gita thinks she needs 69 buttons in all to make Pattern 24. How do you know that she is not correct?
How many buttons does she need to make Pattern 24?
Let participants know that we will use this task to explore the CCSS Content and Practice Standards. First we will do the task and discuss it, then we will look to see how a teacher used this task in his fifth grade.
A copy of the Button Task is in the handout. Distribute.
Consider modeling the pattern on a document camera with counters as you preview the four questions, then move to the next slide for directions on how to proceed with the task. Make graph paper, colored counters or square tiles available to teachers to use as they work on the task.

10. Buttons Task Individually complete parts 1 - 3.
Then work with a partner to compare your work and complete part 4. Look for as many ways to solve parts 3 and 4 as possible.
Consider each of the following questions and be prepared to share your thinking with the group:
What mathematics content is needed to complete the task?
Which mathematical practices are needed to complete the task?
Have participants follow the directions on the slide, working first individually, then in pairs on the Button Task.
Whole Group Discussion: Depending on available time, consider selecting two or three papers with interesting solution strategies and or representations to share with participants before you begin the discussion of question 3. Don’t linger on the discussion of the task solution strategies, remember the focus of today’s session is the Standards for Mathematical Practice so save time for that part of the discussion.
Chart the comments from participants regarding both the mathematical content and practices needed to successfully complete the task. You may also what to push the conversation by asking which elements of the task and/or the way the task was facilitated, triggered students to use specific practices.

11. The Nature of Tasks Used in the Classroom …
Will Impact Student Learning!
Tasks as
they appear
in curricular
materials
Student
learning
First, as we have discussed in the overview session, we need to pay attention to what tasks we select because the nature of the tasks will impact student achievement, but . . .

12. But, WHAT TEACHERS DO with the tasks matters too!
The Mathematical Tasks Framework
Tasks as
they appear
in curricular
materials
Student
learning
Tasks as
they appear
in curricular
materials
Student
learning
Tasks as
set up by
teachers
enacted by
teachers and
students
PAUSE FOR THE SLIDE’S ANIMATION TO BE COMPLETED.
. . . having good tasks is only part of the story. How a teacher uses the task can significantly impact students’ learning opportunity. Tasks are important, but teacher decisions also matter!
Teacher actions and reactions, that is their instructional decisions in the classroom, …
influence the nature and extent of student engagement with challenging tasks,
and
affect students’ opportunities to learn from and through task engagement.
In particular, the way teachers choose to use tasks can significantly influence the opportunities students have to develop skills associated with the mathematical practices.”
Stein, Grover & Henningsen (1996)
Smith & Stein (1998)
Stein, Smith, Henningsen & Silver (2000)

13. www.InsideMathematics.org A re-engagement lesson using the
Buttons Task
Francis Dickinson
San Carlos Elementary
Grade 5
Source of the video is the website Inside Mathematics. Development of the website was funded by the Noyce Foundation with a grant to the Silicon Valley Mathematics Initiative in California. The Noyce Foundation and the Silicon Valley Mathematics Initiative have graciously granted NCSM permission to utilize the website for this resource.
The phrase ‘reengagement’ is intended to suggest an instructional strategy whereby students are asked to reconsider some aspect of a task in which they have already been engaged for the purpose of deepening or broadening their understanding. It is an excellent strategy for a variety of instructional goals, and this particular video was chosen in part to highlight a powerful type of lesson with which teachers may be unfamiliar.
Norms for Watching Video
Video clips are examples to allow for discussion of teaching and learning, not for criticism or evaluation of the teacher. What is in the video is a very limited piece of the lesson, so be wary of assumptions we may make about what came before or after.
All comments should be made respectfully.
Always assume another’s comments are not intended to offend.
1. VIDEO 1: First select and show approximately 2 minutes segment from the Lesson Planning video clip beginning at 3:00 and ending at 4:49. This segment shows the teacher describing the lesson he has designed and his mathematical goals are for his students. Look for notes about the teacher, his classroom, and school on the website.
2. Next, direct participants’ attention to the samples of student work from Learner A and Learner B provided in the handout. Allow teachers to look over the two samples of student work (Learner A and Learner B) and consider the nature of mathematics content and the mathematical practices students might be engaged in as they complete this task.
VIDEO 2: Next, show approximately 2 minutes segment from the Problem 2 video clip beginning at 0:00 minutes and ending at 1:50 minutes. In this segment participants will see the teacher launch the task.
VIDEO 3: Next share a segment or two of students trying to make sense of the sample student work. The video segments from 3:04 to 4:40 showing two girls working on the task and another segment from 5:40 to 6:07 with two boys are nice samples. Again, the question for participants to consider as they watch the video is the nature of mathematics content and the mathematical practices students are engaged in as they complete this task.
4. VIDEO 4: Finally, show approximately 2 minutes segment from the Closure video clip beginning at 0:00 minutes and ending at 2:00 minutes. If you have a bit more time in the session you may want to play this video all the way to the end for a total of approximately 5:10 minutes. Then proceed to the focus questions on the next slide.

14. Learner A Pictorial Representation
What does Learner A see staying the same? What does Learner A see changing? Draw a picture to show how Learner A sees this pattern growing through the first 3 stages. Color coding and modeling with square tiles may come in handy.
Verbal Representation
Describe in your own words how Learner A sees this pattern growing. Be sure to mention what is staying the same and what is changing.

15. Learner B Pictorial Representation
What does Learner B see staying the same? What does Learner B see changing? Draw a picture to show how Learner B sees this pattern growing through the first 3 stages. Color coding and modeling with square tiles may come in handy.
Verbal Representation
Describe in your own words how Learner B sees this pattern growing. Be sure to mention what is staying the same and what is changing.

16. Buttons Task Revisited
Which of the Standards of Mathematical Practice did you see the students working with? Cite explicit examples to support your thinking.
What did Mr. Dickinson gain from using the same math task two days in a row, rather than switching to a different task(s)?
How did the way the lesson was facilitated support the development of the standards of practice for students?
What implications for teachers implementing CCSS does this activity suggest to you?
Preview the focus questions with teachers, then let them consider them individually before you facilitate a whole group discussion.

17. Again, WHAT TEACHERS DO with the tasks matters too!
The Mathematical Tasks Framework
Tasks as
they appear
in curricular
materials
Student
learning
Tasks as
they appear
in curricular
materials
Student
learning
Tasks as
set up by
teachers
enacted by
teachers and
students
(This slide duplicates slide 11.)
Having good tasks is only part of the story. How a teacher uses the task can significantly impact students’ learning opportunity. Looking for good tasks that will engage students in use of the 8 mathematical practices is an important start, but we have just seen how the ways in which the teacher uses the tasks also contributes significantly to students’ opportunities to learn.
Stein, Grover & Henningsen (1996)
Smith & Stein (1998)
Stein, Smith, Henningsen & Silver (2000)

18. Planning to Support Students’ Opportunity to Learn
Select a typical task (or a related set of problems) from your instructional materials and design a lesson so that it offers more opportunities for students to develop both the content and practice standards.
Participants work together to create a modified task that gives students the opportunity to develop mathematical practices with the content.

19. Next Steps and Resources
Review the implications you listed earlier and discuss with your table group one or two next steps you might take as a district, school, and classroom teacher.
Refer to list of implications generated from slide 7.
Hopefully some of these ideas will come up:
Think about how posing the task can affect the learning generated as students engage in the task and try not to pose tasks that are too limiting. Provide opportunities for students to work more extensively with a single problem. Provide opportunities for students to talk over mathematical ideas.
Think about whether my students know when and how to use tools, including representational tools, to make sense of and solve problems. Can they make decisions about what tools will help them? Can they use representations to apply mathematics to a real-world or contextual situation? Look for ways to support them in this area.

20. Today’s Goals
To explore the mathematical standards for Content and Practice
To consider how the Common Core State Standards (CCSS) are likely to impact your mathematics program and plan next steps
In particular, participants will
Examine opportunities to develop skill in reasoning about problems and unpacking others’reasoning
Recap the day’s goals and ask if there are any final questions or thoughts participants would like the group to consider.

21. End of Day Reflections
Are there any aspects of your own thinking and/or practice that our work today has caused you to consider or reconsider? Explain.
2. Are there any aspects of your students’ mathematical learning that our work today has caused you to consider or reconsider? Explain.
Ask participants to take the final 10 minutes to respond to these questions in writing. This activity is intended to offer participants a moment to synthesize the thinking they did during this session and to leave facilitators with some evidence of where participants’ current thinking is following the session.

22. Join us in thanking the Noyce Foundation for their generous grant to NCSM that made this series possible!
This series was made possible by a generous grant from the Noyce Foundation to NCSM.
In addition, video for some of the modules comes from the website Inside Mathematics. Development of the website was also funded by the Noyce Foundation with a grant to the Silicon Valley Mathematics Initiative in California. The Noyce Foundation and the Silicon Valley Mathematics Initiative have graciously granted NCSM permission to utilize the website for this resource.

23. Project Contributors Geraldine Devine, Oakland Schools, Waterford, MI
Aimee L. Evans, Arch Ford ESC, Plumerville, AR
David Foster, Silicon Valley Mathematics Initiative, San José State University, San José, California
Dana L. Gosen, Ph.D., Oakland Schools, Waterford, MI
Linda K. Griffith, Ph.D., University of Central Arkansas
Cynthia A. Miller, Ph.D., Arkansas State University
Valerie L. Mills, Oakland Schools, Waterford, MI
Susan Jo Russell, Ed.D., TERC, Cambridge, MA
Deborah Schifter, Ph.D., Education Development Center, Waltham, MA
Nanette Seago, WestEd, San Francisco, California
Hope Bjerke, Editing Consultant, Redding, CA
Thank you to the project contributors.

24. Help Us Grow!
The link below will connect you to a anonymous brief e-survey that will help us understand how the module is being used and how well it worked in your setting.
Please help us improve the module by completing a short ten question survey at: