The webinar will begin at 3:30. While you are waiting, please mute your sound. During the webinar please type all questions in the question/chat box in the go-to task pane on the right of your screen.

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WELCOME

• The webinar will begin at 330. While you are
  waiting, please mute your sound. During the
  webinar please type all questions in the
  question/chat box in the go-to task pane on the
  right of your screen.
• As always, this webinar and the supporting
  materials will be available on our wikispace.
• www.ncdpi.wikispaces.net

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Making Mathematics Accessible

• Department of Public Instruction
• Mathematics Consultants

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Mathematics Section Contact Information
Kitty Rutherford Elementary Mathematics Consultant 919-807-3934 kitty.rutherford_at_dpi.nc.gov Amy Scrinzi Elementary Mathematics Consultant 919-807-3839 amy.scrinzie_at_dpi.nc.gov
Robin Barbour Middle Grades Mathematics Consultant 919-807-3841 robin.barbour_at_dpi.nc.gov Johannah Maynor High School Mathematics Consultant 919-807-3842 johannah.maynor_at_dpi.nc.gov
Barbara Bissell K-12 Mathematics Section Chief 919-807-3838 barbara.bissell_at_dpi.nc.gov Susan Hart Program Assistant 919-807-3846 susan.hart_at_dpi.nc.gov
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• Our teachers come to class,
• And they talk and they talk,
• Til their faces are like peaches,
• We dont
• We just sit like cornstalks.
• Cazden, 1976, p. 64

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• ALL students can
• generate
• mathematical understandings!

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• Whenever a teacher reaches out to an individual
  or small group to vary his or her teaching in
  order to create the best learning experience
  possible, that teacher is differentiating
  instruction.
• -- Carol Tomlinson

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Instructional Strategy

• Concrete-Representational-Abstract (CRA)
• Concrete doing stage
• Representational seeing stage
• Abstract symbolic stage

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Division of Fractions

• 5 ? ?
• 5 3 15
• Why?

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Division of Fractions

• 5 ? ?
• 5 3 15
• Why?

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Division of Fractions

• 5 ? ?
• 5 3 15
• Why?

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Algorithms

• Algorithms without understanding
• Errors practiced and hard to break
• Extensive practice time
• Limited retention

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Algorithms

• Algorithms with understanding
• Conceptual development
• Reduction in practice time
• Extended retention and application

Deborah Ball, Secretarys summit on Mathematics,
Washington, D.C., 2003, http//www.ed.gov/rschstat
/progs/mathscience/ball.html (accessed November
12, 2006
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• Can all students explain the WHY-TOs not just the
  HOW-TOs?

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• When planning,
• What task can I give that will build student
  understanding?
• rather than
• How can I explain clearly so they will
  understand?
• 
  Grayson
  Wheatley, NCCTM, 2002

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The Border Problem

• Sue is tiling a 10 by 10 patio. She wants darker
  tiles around the border. How many tiles will she
  need for the border? Show the arithmetic you
  used to get your solution. Describe your method
  and explain why it makes sense. Use algebraic
  expressions to write a rule for each method you
  found.

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• Sue is tiling a 10 by 10 patio. She wants darker
  tiles around the border.
• How many tiles will she need for the border?
• Show the arithmetic you used to solve the
  problem.
• Describe your method.
• Explain why your method makes sense.
• Use algebraic expressions to write a rule for
  each method you found.

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• Sue is tiling a 10 by 10 patio. She wants darker
  tiles around the border.
• How many tiles will she need for the border?
• Show the arithmetic you used to solve the
  problem.
• Describe your method.
• Explain why your method makes sense.
• Use algebraic expressions to write a rule for
  each method you found.

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The Border Problem

• Several Possibilities
• 10 9 9 8 36
• 9 x 4 36
• 100 - 64 36

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The Border Problem
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Standards for Mathematical Practices

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.

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What is the area and perimeter of this
shape? How do you know?
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Make sense of problems and persevere in solving
them.
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(No Transcript)
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Instructional Task

• With a partner, using color tiles solve the task
  below
• What rectangles can be made with a perimeter of
  30 units? Which rectangle gives you the greatest
  area? How do you know?
• What do you notice about the relationship between
  area and perimeter?

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Compared to.





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What is the area of this rectangle? What is the
perimeter of this rectangle?
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• When thinking about the concept of area and
  perimeter what mathematical terms come to mind?
• In two minutes list all terms you can think of
  in the center box.

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• When thinking about the concept of area and
  perimeter what mathematical terms come to mind?
• In two minutes list all terms you can think of
  in the center box.

area perimeter multiply array
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(No Transcript)
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Strategies for Developing Mathematical
Understanding

• Allow mathematics to be problematic for students.
• All students need to struggle with challenging
  problems
• Teacher must refrain from doing too much of the
  mathematics
• Problem solving leads to understanding!

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Strategies for Developing Mathematical
Understanding

• Focus classroom activity on the methods used to
  solve problems.
• Opportunity for students to share ones own
  method
• Hear alternative methods of solving a problem
• Examine the advantages and disadvantages of these
  different methods (efficiency)
• Class discussions should revolve around sharing,
  analyzing, and improving methods. Mistakes
  become opportunities for learning.

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Strategies for Developing Mathematical
Understanding

• Determine what mathematical information should be
  presented and when this information should be
  presented.
• Presenting too much information too soon removed
  the problematic nature of problem
• Presenting too little information can leave the
  students floundering

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ENCOURAGING MATHEMATICAL DISCOURSE

• Teachers
• Use effective questioning
• Be nonjudgmental about a response or comment
• Let students clarify their own thinking
• Require several responses for the same question
• Require students to ask a question when they need
  help.
• Never carry a pencil
• Mathematical
  Teaching in the Middle School, April 2000, Never
  Say Anything a Kid Can Say.

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But Most Importantly

• Never Say Anything a Kid can Say!!

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Please dont let student sit like Cornstalks!

• Our teachers come to class,
• And they talk and they talk,
• Til their faces are like peaches,
• We dont
• We just sit like cornstalks.
• Cazden, 1976, p. 64

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http//www.ncdpi.wikispaces.net
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Mathematics Section Contact Information
Kitty Rutherford Elementary Mathematics Consultant 919-807-3934 kitty.rutherford_at_dpi.nc.gov Amy Scrinzi Elementary Mathematics Consultant 919-807-3839 amy.scrinzie_at_dpi.nc.gov
Robin Barbour Middle Grades Mathematics Consultant 919-807-3841 robin.barbour_at_dpi.nc.gov Johannah Maynor High School Mathematics Consultant 919-807-3842 johannah.maynor_at_dpi.nc.gov
Barbara Bissell K-12 Mathematics Section Chief 919-807-3838 barbara.bissell_at_dpi.nc.gov Susan Hart Program Assistant 919-807-3846 susan.hart_at_dpi.nc.gov
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