# The National Council of Supervisors of Mathematics

## Presentation on theme: "The National Council of Supervisors of Mathematics"— Presentation transcript:

The National Council of Supervisors of Mathematics
The Common Core State Standards Illustrating the Standards for Mathematical Practice: Mathematical Modeling and Constructing Viable Arguments 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!) Gym Task Gym Task Summary of Student Understandings and Misunderstandings DVD Task Student H – (consider printing on green paper as in the video) Student A – (consider printing on blue paper as in the video) Download the video files (8 clips) A downloadable video in provided with clips for Slides 16, 19 and 21.  If the downloadable video is not available, it can be accessed in the full version at Video clips: Comparing Linear Functions (7th Grade) Lesson Video Slide # 16 Clips 1a and 1b - Introduction a. 0-2:30 and b. 6:45-7:50 Slide # 19 Clips 2a, 2b, and 2c - Problem 1: Part A 1:39-4:04, Part B 1:20-2:45, Part C 0-1:33 Slide # 21 Clip 3a, and 3b - Problem 2: Part A 2:40-3:40 and Part B 1:07-3:11

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!

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.

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 explaining and mathematical modeling Each of the modules in this series is designed to address the first two goals listed above. When participants leave a session it is our hope that they will have grown in their understanding of both the content and practice standards from CCSS though the exploration and discussion of rich mathematics tasks. In addition, each of the modules is designed to help participants think about these new standards in relation to their own practice and begin to identify productive starting points for implementing the CCSS.

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 NRC’s Strands of Mathematical Proficiency.

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.

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.

Standards for Mathematical Practice

Common Core State Standards for Mathematical Practice
In our work today, we will examine the Standards for Mathematical Practice through a classroom vignette. What is the nature of mathematical tasks in these classrooms? What do you hear or see in a mathematics classroom that is building the mathematical practices? Video clips from the insidemathematics.org website will be shown in later slides. Advise participants to focus on these questions as they watch the video clips. 9

Standards of Mathematical Practice through Reengagement

Using Formative Assessment to Plan Instruction
Carlo thinks he will go to the gym about 20 times a month. Calculate how much each of these options would cost Carlo for one month. Which of these options is the least expensive for Carlo? Explain. Pay as you go Pay only \$6 each time you work out Regular deal Pay \$50 a month and \$2 each time you work out All-in-one price! Pay just \$100 per month for unlimited use of our great facilities Refer participants to the “Gym” handout. Give participants a few minutes to examine the task. (Participants need only enough time with the task to be able to engage in a subsequent conversation about a summary of student work in the next slide. Modeling interaction of this task is not necessary. Consider giving time to work individually, jot notes and/or debrief at small tables.) 11

Using Formative Assessment to Plan Instruction

Opportunities for Content and Practice in Tasks

Opportunities for Content and Practice in Tasks
Individually complete the task. Then work with a partner to compare your work and discuss the following questions: What mathematics content is needed to compete the task? When using tables to model and compare the DVD plans, what information and processes will students need to use strategically? What aspects of the explaining and modeling Standards of Practice might students need to use to complete the task? Have participants follow the directions on the slide, working first individually, then in pairs on the DVD Task. Highlight partner conversations in a whole group debrief. Whole Group Discussion: Mathematics content Interpret verbal representations. Use a unit rate to create a table of values. Compare linear functions. The intent of this question is to consider skills and processes that students who are not yet fluent in using tabular representations will need to develop. (This leads into the subsequent analysis of student work.) For example, students will need to . . . extend their tables far enough to examine when the costs are equal. use common comparisons (cost per month vs. cost per movie). Possible bullets for question c: SP3 Understand and use stated assumptions. SP4 Identify important quantities in a practical situation Map relationships using such tools as tables. Analyze those relationships mathematically to draw conclusion. Interpret their mathematical results in the context of the situation. Reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

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 which tasks we select because the nature of the tasks will impact student achievement. We listed several aspects of explaining and modeling that students have opportunities to engage with during the DVD task.

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 – so animation can occur! “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)

Standards for Mathematical Practice: Opportunities in Task Enactment
In what ways did the teacher’s launch increase students’ opportunities to begin explaining and modeling? What evidence do you see that students are building these standards of practice? Video Clips 1a and 1b: Part A, Introduction 0-2:30 and 6:45-7:50 (time stamps refer to video as posted on the InsideMathematics.org website) Teacher moves: Made connections to the real world Asked students to restate Asked students to talk with their shoulder partner Encouraged/required multiple representations Asked students to discuss and then note how they started creating tables Student responses: Able to identify what the problem is asking Know a strategy to engage in the task (make a table) and can describe some details to attend to in making the tables Because of Movie Flix’s constant cost of \$18, Charles reasoned that the tables for each of the other plans also needed to reach \$18. He communicated his additive reasoning strategy. 17

Using Student Work to Develop Standards for Mathematical Practice
Use the questions posed by Mr. Dimas to analyze the tables generated by Student H on the next slide. Do the tables make mathematical sense, and do they match the plans? Would these tables help us tell if the three plans will ever cost the same? Questions in italics are the same questions that the teacher asks his students to think about as they look at student work. 18

Using Student Work to Develop Standards for Mathematical Practice
Student H Movies 1 2 3 4 5 6 7 \$3 \$6 \$9 \$12 \$15 \$18 \$21 Money \$13 \$14 \$16 \$17 \$19 \$36 \$54 \$72 \$90 \$108 \$126 Movie Buster Online Flix Summary of student errors shown on this slide: The placement of “movies” and “money” labels is confusing. (The labels on this slide intentionally mimic the student work. This student did not label all three of the tables, thereby allowing room for confusion and an inability to accurately respond to the original prompt.) The Mail Fix table looks like \$18/movie instead of \$18/month (following the labels given with Movie Buster). The students may have either neglected the cost per movie or neglected the change in unit rate. The notion that mathematical comparisons in this situation can only be made with like units is a big mathematical idea in this particular case or context. Likewise, the notion that just because there is a correct mathematical pattern doesn’t mean that the table is correct for this context. The tables all make mathematical sense as the patterns continue logically, but Mail Fix does not match the plan. While the reader can infer the tables refer to movies and money, there is not consistent labeling for the second and third table. Conclusions related to the cost analysis aren’t noted other than the work on the table. If student H made a conclusion, it might be that for 6 rentals the Movie Buster and Online Flix were the same. Possible Discussion: What instructional strategies do you see the teacher using? (agree/disagree) He asks, “Why are we creating these three tables?” He asks students to consider both mathematical sense and correct modeling (matching the plan). What might asking about the “mathematical sense” of a table afford students? Mail Flix 19

Using Student Work to Develop Standards for Mathematical Practice

Using Student Work to Develop Standards for Mathematical Practice
Use the questions posed by Mr. Dimas to analyze the table generated by Student A. Does the table make mathematical sense, and does it match the plans? Would this table help us tell if the three plans will ever cost the same? # of Movies MB OF MF 12 18 1 3 25 2 6 38 9 51 4 64 5 15 77 90 Give participants a few minutes to examine the work of Student A. What additional concepts might this student’s work add to a conversation? The inclusion of 0 and whether or not it makes sense to include 0 Student A used an increase of \$13 per movie in the OF Plan, not distinguishing between the flat fee and the cost per movie. When graphical representations are added to the student’s repertoire, this flat fee will be the y-intercept. 21

Using Student Work to Develop Standards for Mathematical Practice
As you watch the video, consider these questions. What evidence do you see that suggests students are developing competency with explaining and modeling? In what ways did interactions between students support their ability to develop competency with explaining and modeling? Note: This slide is a duplicate of slide 19, but now refers to Student A. Be aware of the language of each standard and be prepared to support some key points of the practices. Highlight both the addition of the 0 and discussions that ensued and the error with OF. Clips 3a and 3b Problem 2 (time stamps refer to video as posted on the InsideMathematics.org website) Part A 2:40-3:40 Listen to or read the arguments of others, decide whether they make sense, and ask useful questions – students are asked to both analyze the tables given to them and also participate in a classroom discussion of the work. Students learn to determine domains to which an argument applies-Victoria doesn’t like including 0 but Melanie thinks it makes sense. Part B 1:07-3:11 Students learn to determine domains to which an argument applies – Katie says that 0 makes sense in the table and articulates what the values in the table represent in this context. (Jessica still thinks that including these values creates confusion.) Reflect on whether the results make sense, possibly improving the model if it has not served its purpose. Kyle explains what changes need to be made to the Online Flix plan. The teacher gives students the opportunity to fix their tables. This gives students the opportunity to compare the correct and incorrect tables. 22

Considering Next Steps
Create verbal, graphical, and tabular representations of these carnival ticket plans. Will any of the three plans ever cost the same? Explain. Bracelet Unlimited tickets with a \$12.00 bracelet Dollar Deal No Entrance Fee \$1.00 per ticket Discounted Plan \$4.00 Entrance Fee, with discounted tickets (\$0.50/ticket) Teachers in this vignette used the Carnival task after the DVD task. Ask: How might you use this task to extend or assess student knowledge of the content and competence in the practices? Possible responses: This task adds the graphical model. Students need to have opportunities to make connections between representations. A teacher might look for how well students are able to project their knowledge of tabular representations onto graphical representations. 23

Planning and Teaching to Develop Standards for Mathematical Practice
How does the process of reengagement seen in this vignette support students in developing Standards for Mathematical Practice? What instructional decisions did Mr. Dimas make that seemed to support the development of Standards for Mathematical Practice for students? Discuss the concept of “reengagement” as used in the process to deepen and clarify student understanding. One way to think about Reengagement is in a sequence of assessment activities: 1. Give the Assessment Task 2. Analyze student work for aspects to reengage around (e.g., common misconceptions, incomplete understandings, particular representations or innovative thinking) 3. Reengage: Select and Share Representative Solutions 4. Summarize 5. Independent Practice First question: Because the teachers first used formative assessment to guide the development of the activity, they could form activities that followed based on the results. Students first engage in the task by trying to solve the problem and make mathematical sense of the situations. Teachers strategically choose solutions that illustrate common misconceptions. Reengaging with these solutions allow students to reconsider skills related to tabular representations. Students then begin to work on analyzing other students’ thinking and the process they used in solving and deepening their understanding of using tables to model situations. Second question: The teacher chooses tasks that asked students not only to represent plans with a table but to do cost analysis comparing the plans. He chooses to spend another day on a task, giving students an opportunity to refine both their reasoning and mathematical models. He uses talk moves that encourage student engagement and sense making with modeling the DVD scenarios. (Chapin, S., O’Connor, C, Anderson, N. (2003). Classroom Discussions: Using Math Talk to Help Students Learn. Sausalito, CA: Math Solutions Publications) 24

Real-World Situations

Next Steps and Resources

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 explaining and mathematical modeling Revisit the goals.

Source of the mathematics tasks, video, sample student work, teacher reflections 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.

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.]

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. You could ask participants to write these reflections on a postcard that you will collect and review. You could mail the reflections back to participants at a designated time so they can self-evaluate based on their own reflections.

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.

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: