1. Transition to PA Common Core Mathematical Practices
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2. PA Common Core Local Curriculum Rigor Math Practices Dive Deeper
Assessment
Local Curriculum
This is one of five modules designed to assist educators in understanding and transitioning to PA Common Core. Today we will explore the standards for mathematical practice to build a basic understanding of the mathematical practices and the need for continued professional development to fully implement the Standards for Mathematical Practice for all students in every classroom.
Toolbox
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3. Please Do the Following
Connect to the Internet
Navigate to:
If a registered user, sign-in
If not a registered user, join now
Place your name and school district/organization on your name tent
Your School District/Organization
Your Name
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4. PA Common Core Introduction Essential Questions
• What are the Standards for Mathematical Practices and how do they relate to the PA Common Core?
• Can the characteristics of a student and classroom that exemplify mathematical practices be identified and implemented?
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5. Dan Meyer describes why we need to makeover math classrooms.
Math Class Makeover
Dan Meyer describes why we need to makeover math classrooms.
Watch and discuss the Dan Meyer video
Click on the icon to view the video.
NOTE: If you have time to extend the presentation:
A scale drawing of a very small object is larger than the object. The scale of the drawing is 2 cm:14 mm. Find the unknown measure (in cm).
width of object = 70 mm; width on drawing =  ? 
Use Dan Meyer’s problem example.
Take a drawing and scale it for a mural.
What content standard is addressed (comparing fractions, ratios, proportion, scale drawing)?
What practices standard(s) is addressed in your problem?
Take this traditional problem and change it to a problem that addresses both content and practice?
How?
Why would you want to?
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6. Explore Identify Expected Outcomes
the Standards for Mathematical Practice.
Identify
characteristics of a student and classroom that exemplify mathematical practice.
Review the expected outcomes for this session.
Explore the Standards for Mathematical Practice.
Identify characteristics of a student and classroom that exemplifies mathematical practice.
The purpose of this session is to build awareness and begin answering these questions:
What are the Practices?
What does a student who is using these practices look and sound like?
What learning opportunities are students experiencing to be proficient with all Practices? (Student-Content interaction, teacher-content interaction and teacher-student interactions).
How do these standards go hand in hand with the content standards to bring rigor and relevance to the study of mathematics?
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7. Looks Like/Sounds Like
When all students are engaged in learning mathematics, what does a classroom...
Look Like
Sound Like
A set of common standards is important; however, instruction and what students are doing in the classroom are key to success.
Before we look at the mathematical practices:
Please list the words and/or phrases that you think describe such a classroom.
Individuals or one member of a group can post their words and/or phrases on the chart paper posted for all participants to see. Give the participants a chance to read the completed list.
Are there any words and/or phrases you would like to add? Take a moment for whole group to think about the list and add anything that is missing. Additions to the list will be made periodically and reflection on the list will occur later when examining the practices.
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8. 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)
Read slide, 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.
- Common Core State Standards for Mathematics, page 6
The development of the standards for mathematical practice was based on ideas from two publications: NCTM’s Principals and Standards for School Mathematics and the National Research Council’s Adding It Up. 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.
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9. NCTM – Principles and Standards for School Mathematics
Problem solving
Reasoning and proof
Connections
Communication
Representation
“NCTM’s Principals and Standards for School Mathematics included a set of “process standards:” problem solving, reasoning and proof, communication, representation, and connections. Many of you may be familiar with these process standards.”
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10. Standards of Proficiency of Mathematical Practice
The National Research Council’s Adding It Up includes five strands of Mathematical Proficiency.
Conceptual Understanding: Comprehension of mathematical concepts, operations, and relations.
Procedural Fluency: Skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.
Strategic Competence: Ability to formulate, represent, and solve mathematical problems.
Adaptive Reasoning: Capacity for logical thought, reflection, explanation, and justification.
Productive Disposition: Habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.
The Mathematical Practices are intertwined with the Common Core Standards and neither should be taught separately from one another.”
Adding It Up: Helping Children Learn Mathematics
By Jeremy Kilpatrick,
Jane Swafford, & Bob Findell (Editors). (2001).
Washington, DC: National Academy Press
p. 117
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11. Standards for Mathematical Practice
1. Make sense of complex 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.
Using the process standards and the five strands of mathematical proficiency, the writers of the Common Core for Mathematics developed eight Standards for Mathematical Practice.
An important note about these standards is that they are for students. The purpose of the Mathematical Practices is to help the students know and understand the core standards at a deeper level. They engage the students in the content of your course. While they have implications on teachers and their instruction, in and of themselves, they are not goals for teachers.”
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12. Grouping the Standards for Mathematical Practice
Distribute Handout: Mathematical Practice Grouping Chart
Bill McCallum, one of the writers of the Common Core State Standards for Mathematics, described grouping the mathematical practices into four general categories:
Reasoning & Explaining (practice 2 & 3).
Modeling Using Tools (practice 4 & 5).
Seeing structure and generalizing (practice 7 & 8).
Read the first three words for each mathematical practice. What do you notice?
Allow several minutes for participants to work and then ask, What do you notice?
What are the phrases and words you listed earlier that describe these practices?
Lead discussion to help clarify what these practices look and sound like in the classroom as students become proficient in all of the practices. These areas would not be expected every day; however, as often as appropriate for the tasks and as needed for students to be proficient in all practices.
(McCallum, 2011)
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13. Standards for Mathematical Practice
1. Make sense of complex 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.
Looking at the Mathematical Practices, take a few minutes, individually, to read the highlighted words for each of the practices. What do you notice? Also, think about the implications the standards might have on your classroom.
Allow a few minutes for participants to think individually.
Now take a few minutes to work with a partner to discuss the highlighted words as well as the implications of the mathematical practices in the classroom.
Allow a few minutes for participants to discuss in partners.
Have the partners share their thoughts with the group.
What implications might the standards of mathematical practice have on your classroom?
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14. Mathematical Rigor Rigor Is Rigor Is Not
approaching mathematics with a disposition to accept challenge and apply effort;
engaging in mathematical work that promotes deep knowledge of content, analytical reasoning, and use of appropriate tools; and
emerging fluent in the language of mathematics, proficient with the tools of mathematics, and being empowered as mathematical thinkers.
Rigor Is Not
“difficult,” as in “AP calculus is rigorous.”;
enrichment activities for advanced students;
Problem Solving Friday
Adding two word problems to the end of a worksheet; and
adding more numbers to a problem.
The word rigor may have come up in your discussions about the mathematical practices. Mathematical rigor is an elusive term with multiple meanings. To a pure mathematician, rigor is a mark of excellence. To some K-12 educators and many parents, “rigorous” often means difficult as in “AP Calculus is rigorous”.
In your group, discuss any additional statements that could be placed in this chart.
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15. Buttons Task
Gita plays with her grandmother’s collection of black & 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.
Pass out copies of the Button Task to teachers.
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.
CTB/McGraw-Hill; Mathematics Assessment Resource Services, 2003
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16. 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.
National Council of Supervisors of Mathematics
CCSS Standards of Mathematical Practice: Reasoning and Explaining
CTB/McGraw-Hill; Mathematics Assessment Resource Services, 2003
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17. www.Inside Mathematics.org
A reengagement
lesson using the
Button 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 a instructional strategy whereby students are ask 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.
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, distribute the samples of student work from Learner A and Learner B provided with this package. 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.
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. 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. 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.
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18. 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.
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19. 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.
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20. 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 value did Mr. Dickinson generate by 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 Mathematical Practice for students?
What classroom implications related to implementation of CCSS resonate?
Preview the focus questions with teachers, then let them consider them individually before you facilitate a whole group discussion.
Handout: MOD 3 Button Task packet for review.
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21. Standards for Mathematical Practice
Standard 1: Make sense of problems and persevere in solving them. 4th Grade
For an elementary audience you may want to show video of a 4th grade example.
Click on the icon to view the video:
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22. Standards for Mathematical Practice
Standard 4: Model with mathematics. 9th Grade/10th Grade
For a secondary audience you may want to show video of a 9th/10th grade example. Click on the icon to view the video:
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23. Standards for Mathematical Practice in a Classroom
Traditional U.S. Problem
Which fraction is closer to 1: 4/5 or 5/4?
Same problem integrating content and practice standards
4/5 is closer to 1 than is 5/4.
Using a number line, explain why this is so.
(Daro, Feb 2011)
Lead the following discussion:
What is the difference between the traditional and the integrated problem?
What content standard is addressed (comparing fractions)?
What practices standard(s) is addressed (1,2,6, 4)?
Could you take a traditional problem and change to a problem that addresses both content and practice?
How?
Why would you want to change the problem?
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24. End of the 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.
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25. Reflection and Planning
Does our list of words/phrases describe a classroom where students are engaged in mathematical practice?
Use the reflection sheet to capture key thoughts about the practice standards.
Review the list created at the beginning of the session and make changes/additions to provide a reference for what a classroom engaging students in math practice may look and sound like.
Distribute MOD 3 Handout MP Overview Reflections form and have a few of the participants share responses to second part of Question 2:
What actions will you take to address and overcome these challenges?
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26. References National Council of Supervisors of Mathematics
Jean Howard
Mathematics Curriculum Specialist
(406) ;
Cynthia Green
ELA Curriculum Specialist
(406) ;
Judy Snow
State Assessment Director
(406) ;
National Council of Supervisors of Mathematics
CCSS Standards of Mathematical Practice: Reasoning and Explaining
Adding It Up: Helping Children Learn Mathematics
By Jeremy Kilpatrick,
Jane Swafford, & Bob Findell (Editors). (2001).
Washington, DC: National Academy Press
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