1 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical...

1National Council of Supervisors of MathematicsIllustrating the Standards for Mathematical Practice: Seeing Structure and Generalizing

The National Council of Supervisors of Mathematics

The Common Core State Standards

Illustrating the Standards for Mathematical Practice:

Seeing Structure and Generalizing

In Grades 9-12

www.mathedleadership.org


Module EvaluationFacilitator: 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


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 to plan next steps

In particular, participants will

• Examine opportunities to develop skill in seeing structure and generalizing


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.


Sidewalk Patterns

1. Draw the next pattern in this series.

In Prague some sidewalks are made of small square blocks of stone.

The blocks are in different shades to make patterns that are in various sizes.


Sidewalk Patterns

3. What do you notice about the number of white blocks and the number of gray blocks?

2. Complete the table below.

Pattern number, n 1 2 3 4

Number of white blocks 12 40

Number of gray blocks 13

Total number of blocks 25


Sidewalk Patterns4. The total number of blocks can be found by squaring

the number of blocks along one side of the pattern.

a. Fill in the blank spaces in this list. 25 = 52 81 = ___ 169 =___ 289 = 172

b. How many blocks will pattern number 5 need?

c. How many blocks will pattern n need?

5. a. If you know the total number of blocks in a pattern you can work out the number of white blocks in it. Explain how you can do this.

b. Pattern number 6 has a total of 625 blocks. How many white blocks are needed for pattern number 6? Show how you figured this out.


Sidewalk Patterns

1. Individually complete parts 1-5.

2. Compare your work with a partner’s work.

3. Consider each of the following questions and be prepared to share your thinking with the group:

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

b) How are the mathematical practices related to seeing structure and generalizing relevant to 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


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


www.InsideMathematics.org


Student A


Student C


Student F


Student D


Comparing Students A, C, F and D

• What evidence supports the conclusion that these students are using structure?

• What evidence supports the conclusion that these students are using regularity in repeated reasoning?


Watch the animation on the following slide and consider:

What aspects of seeing structure and generalizing are evident in this work?

What are some possible next steps for this student?


4 sets of 1 by 3 rectangles

RED =4•1•3


RED =4•2•5


RED =4•3•7

4 sets of 1 by 1 and a 3 by 3BLUE =4•1•1+3•3

4 sets of 2 by 2 and a 5 by 5

BLUE =4•2•2+5•5

4 sets of 3 by 3 and a 7 by 7

BLUE =4•3•3+7•7


Sidewalk Patterns

A student showed the calculations at the right for finding the number of gray squares.

•What structure was he attending to?

•What questions might help him use his thinking to generalize the number of gray tiles?


Student A


Student B


Student C


Student E


Comparing Students A, B, C and E

Compare the explanations given by these four students.

•How is each using regularity in repeated reasoning?

•What evidence might indicate that some or all of these students are making a generalization?


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 to plan next steps

In particular, participants will

• Examine opportunities to develop skill in seeing structure and generalizing


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.


Join us in thanking the

Noyce Foundation

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

http://www.noycefdn.org/


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


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:

http://tinyurl.com/samplesurvey1

1 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical...

Documents

Transcript of 1 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical...