Exploring Mathematical Connnections for ALL Students

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Exploring Mathematical Connnections for ALL
Students

• Dr. David S. Allen
• Melisa J. Hancock
• Kansas Staff Development Council
• Wichita, Kansas 2006

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• The limits of my language are the limits of my
  mind. All I know is what I have words for.
• (Ludwig Wittgenstein)

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Three Views on Teaching Mathematics

• View 1- Using symbols that are used across
  cultures, is universal and therefore, easily
  accessible to language learners.
• View 2- The best thing to do for ELL is Just
  Good Teaching, in other words embracing the
  recommendations outlined in Principles and
  Standards (2002).
• View 3- Integrate standards-based practices with
  intentional language instruction.
• (Jenny Bay-Williams, KSU)

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EQUITY PRINCIPLE

• Excellence in mathematics education requires
  equityhigh expectations and strong support for
  ALL students.

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Mathematics for ALL Children

• STOP AND THINK FOR A MINUTE
• Do you personally believe the Every Child
  statement? Children with disabilities, children
  from impoverished homes, minority childrencan
  ALL of these children learn to think
  mathematically?

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What is Sheltered Instruction?

• A means for making grade-level academic content
  more accessible for English language learners
  while at the same time promoting their English
  language development.
• The practice of highlighting key language
  features and incorporating strategies that make
  the content comprehensible to students.
• An approach that can extend the time students
  have for getting language support services while
  giving them a jump start on the content subject
  they need for graduation.

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Over the Decade . . .

• The National Council of Teachers of Mathematics
  (NCTM) has advocated for changes in school
  mathematics programs so that ALL students have
  the opportunities to engage in high-quality
  mathematics that will prepare them for today and
  a world tomorrow they can only imagine.

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SIOP Strategies

• Preparation
• Building Background
• Comprehensible Input
• Learning Strategies
• Interaction
• Practice/Application
• Lesson Delivery
• Review/Assessment

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EQUITY PRINCIPLE

• Excellence in mathematics education requires
  equityhigh expectations and strong support for
  all students.

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Process Standards

• Problem Solving
• Communication
• Connections
• Representation
• Reasoning and Proof
• Back

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Model SIOP Lessons

• Tangrams 3rd-6th Grade
• Graphic Organizer to develop understanding of
  Geometric Vocabulary
• Discovering Pi 5th 8th
• Graphic Organizer to develop understanding of
  Mathematical Relationships

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Tangrams

• The Tangram Lesson
• Available on Website (Written for fourth grade)
  http//www.educ.ksu.edu/allen/Math/Presentations/P
  rofDev.html
• Uses the 5E-Inquiry Model
• Applicable to all grades (you must determine
  where your students are and what types of
  experiences they are ready for)
• Goals and Objectives
• Given a set of tan pieces TLW identify linear
  congruence between the pieces using correct
  geometric vocabulary with no errors.
• Given a set of tangram puzzles TLW identify the
  correct transformations acted upon each tan piece
  as the pieces are manipulated during the puzzle
  completion with no errors.
• Given a sheet of graph paper with a shape drawn
  upon it TLW draw the shape in a second location
  after the shape has been acted upon by two
  geometric transformations.

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Task 1

• Using the three small pieces (two small triangles
  and the medium size triangle) create these five
  geometric shapes.
• Square
• Trapezoid
• Parallelogram
• Rectangle
• Triangle

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Triangle
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Rectangle
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Trapezoid
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Parallelogram
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Square
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Explanation Task 1

• Linear Congruent Relationships
• The hypotenuse of the small triangle is congruent
  to the leg of the medium size triangle.
• The hypotenuse of the medium sized triangle is
  congruent to twice the length of the leg of the
  small triangle.
• The two small triangles are congruent because
• The legs of both triangles are congruent.
• The hypotenuse of both triangles are congruent.
• The angles of both triangles are congruent.

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Explanation Task 1

• 2. Transformations
• Flips or Reflections
• Slides or Translations
• Turns or Rotations
• Triads
• Three Little Pigs
• The Trinity
• The Three Tan Pieces

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Task 2

• Using the five small pieces (two small triangles,
  medium size triangle, rhombus, parallelogram)
  create these five basic geometric shapes.
• Square
• Trapezoid
• Parallelogram
• Rectangle
• Triangle

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Rectangle
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Trapezoid
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Parallelogram
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Triangle
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Square
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Explanation Task 2

• Linear Congruent Relationships
• The leg of the small triangle is congruent to the
  side of the square.
• The leg of the small triangle is congruent to the
  small side of the parallelogram.
• Therefore the side of the square is congruent to
  the small side of the parallelogram.
• The hypotenuse of the small triangle is congruent
  to the long side of the parallelogram.
• The leg of the medium triangle is congruent to
  the long side of the parallelogram.

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Task 3

• Using all seven tan pieces create these five
  basic geometric shapes.
• Square
• Trapezoid
• Parallelogram
• Rectangle
• Triangle

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Rectangle
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Parallelogram
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Trapezoid
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Triangle
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Square
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Connecting the Tasks

• Working with Three Small Pieces
• Identifying Linear Congruent Relationships
• Examining Transformations
• Working with Five Small Pieces
• Application of Linear Congruent Relationship
  Identification
• Strengthening Language Descriptions of
  Transformations
• Working with Seven Pieces
• Similar Task to Three Small Pieces
• Introduce concept of Ratio and Proportion
• 4. Examining Area is Another Lesson

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Conclusion

• Create a school climate built on the expectation
  of high achievement by ALL students which means
  the application of standards based pedagogical
  perspectives. (NCTM, TSOL, NSTA)
• Energize teachers and students in ways that
  challenge current expectations.
• Evaluate the process for placing students in
  mathematics classes to ensure that groups of
  students are not being excluded from a
  challenging mathematics program.

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Exploring Mathematical Connnections for ALL
Students

• Dr. David S. Allen
• Melisa J. Hancock
• Kansas Staff Development Council
• Wichita, Kansas 2006