1 National Council of Supervisors of Mathematics Illustrating the Practices - Mathematical Modeling...

1National Council of Supervisors of MathematicsIllustrating the Practices - Mathematical Modeling and Constructing Viable Arguments

The National Council of Supervisors of Mathematics

The Common Core State Standards Illustrating the Standards for

Mathematical Practice:Mathematical Modeling and

Constructing Viable Argumentswww.mathedleadership.org

http://www.mathedleadership.org/


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Common Core State Standards

Mathematics

• Standards for Content

• Standards for Practice


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


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)



1. Make sense of 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.


Structuring the Practices



What implications might the Standards for Mathematical Practice have on your classroom?

1. Individually review the Standards for Mathematical Practice.

2. Choose a partner at your table and discuss a new insight you had into the Standards for Mathematical Practice.

3. Then discuss the following question.


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?


Standards of Mathematical Practice through Reengagement

Teacher Goals:– Students will make mathematical

sense of tables and discover mathematical reasoning.

– Students will go beyond the notion that the table is either right or wrong to evaluate both mathematical sense and correct modeling.

Outline:1. Gym Task (Formative Assessment)2. DVD Rental Task (Reengagement Vignette)3. Carnival Task (Culminating Activity)


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


Using Formative Assessment to Plan Instruction

Consider the summary of 6th grade results of the Gym task.

What are some possible next steps to deal with misconceptions or misunderstandings documented in the table?


Opportunities for Content and Practice in Tasks

Create verbal and tabular representations of these 3 DVD rental plans .

Do the three plans ever cost the same? Explain.

Movie Buster

$3 per movie rented

Online Flix

$12 per month plus $1 per

movie rented

Mail Flix

$18 per month regardless of the number of movies rented


Opportunities for Content and Practice in Tasks

1. Individually complete the task.

2. Then work with a partner to compare your work and discuss the following questions:

a) What mathematics content is needed to compete the task?

b) When using tables to model and compare the DVD plans, what information and processes will students need to use strategically?

c) What aspects of the explaining and modeling Standards of Practice might students need to use to complete the task?


The Nature of Tasks Used in the Classroom …

Tasks as they appear in curricular

materials Studentlearning

Will Impact Student Learning!


But, WHAT TEACHERS DO with the tasks matters too!

Stein, Grover & Henningsen (1996)Smith & Stein (1998)Stein, Smith, Henningsen & Silver (2000)

The Mathematical Tasks Framework

Tasks as they

appear in

curricularmaterials

Studentlearning

Tasks as they

appear in

curricular materials

Student learning

Tasks as set up by teachers

Tasks as enacted

byteachers

andstudents


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?


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?



Movies

1 2 3 4 5 6 7

$3 $6 $9 $12 $15 $18 $21

Money

1 2 3 4 5 6 7

$13 $14 $15 $16 $17 $18 $19

1 2 3 4 5 6 7

$18 $36 $54 $72 $90 $108 $126

MovieBuster

OnlineFlix

Mail Flix

Student H



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?



# of Movies

MB OF MF

0 0 12 18

1 3 25 18

2 6 38 18

3 9 51 18

4 12 64 18

5 15 77 18

6 18 90 18

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?



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?


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.

Dollar Deal

No Entrance Fee

$1.00 per ticket

Bracelet

Unlimited tickets with a

$12.00 bracelet

Discounted Plan

$4.00 Entrance Fee, with

discounted tickets

($0.50/ticket)


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?


Adapted from Lesh, R., Post, T., & Behr, M. (1987). Representations and Translations among Representations in Mathematics Learning and Problem Solving. In C. Janvier, (Ed.), Problems of Representations in the Teaching and Learning of Mathematics (pp. 33-40). Hillsdale, NJ: Lawrence Erlbaum.

Geometric/Graphical

Verbal - Written and Oral

Tabular

Contextual

Symbolic

Pictures

Oral Language

ManipulativeModels

Real-World Situations

WrittenSymbols

Representation Stars

Elementary Secondary


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.


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


www.InsideMathematics.org


Join us in thanking the

Noyce Foundation

for their generous grant to NCSM that made this series possible!

http://www.noycefdn.org/

http://www.noycefdn.org/


End of Day Reflections

1. 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.


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


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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:

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