Your Geometry Journey

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Your Geometry Journey

• Barb Maitski

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What are pentablocks?

• Pentablocks are an extension of traditional
  pattern blocks. A regular pentagon is the parent
  of this new family of blocks, which also includes
  two non-similar isosceles triangles, two
  non-similar rhombi, and a five-pointed star.
• Each of the pentablock shapes is derived from a
  regular pentagon. The angles of the blocks are
  multiples of 36, allowing all of the pieces to
  be combined naturally to form a wide variety of
  patterns.

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How are pentablocks used?

• Pentablocks are beneficial for exploring a wide
  range of concepts. They can be used to discover
  and explore mirror and rotational symmetry,
  congruence, tessellations, fractions, area
  equivalence, and angle measurement.
• Several articles were written by classroom
  teachers who used pentablocks and the van Hiele
  theory to analyze students ideas about geometry.
•  

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van Hiele Theory

• Dutch educators Pierre van Hiele and Dina van
  Hiele-Geldof developed a theory that students
  progress through distinct levels of development
  in their geometric thinking.
• It is just a theory, but a useful one for
  thinking about activities which are appropriate
  for your students and prepare them to move to the
  next level, and for designing activities for
  students who may be at different levels.

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Development of Geometric Thinking

• Geometric thinking is more dependent on
  instruction than on age.
• Sequential phases of learning help students move
  from one level to another.
• Level 0 Visualization
• Level 1 Analysis
• Level 2 Informal Deduction
• Level 3 Deduction
• Level 4 Rigor
• Instruction should include a sequence of
  activities that begins with play and
  exploration, gradually builds concepts and
  related language, and culminates in summary
  activities that help students integrate new
  ideas into their previous knowledge.

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Level 0 Visualization

• Students judge figures by their appearances.
• What does it look like?
• Early late elementary school

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Level 1 Analysis

• Students analyze figures based on properties and
  attributes. They begin to understand that if a
  shape belongs to a class, it has all the
  properties of that class.
• What are the defining characteristics?
• Late elementary school middle school

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Level 2 Informal Deduction

• Students can generalize interrelationships of
  properties within the same shape and among
  different shapes.
• Students begin if-then thinking.
• Middle school high school

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Level 3 Deduction Level 4 Rigor

• The objects of thought here are the relationships
  among properties of geometric objects.
• The structure of axioms, definitions, theorems,
  etc., begins to develop.
• 10th-grade geometry courses, but many students
  are not developmentally ready for it.
• The objects of thought are deductive axiomatic
  systems for geometry.

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Making Instructional Choices

• Students in pre-K-2 generally fall at level 0
  (visualization). This level describes students
  who reason about shapes primarily on the basis of
  visual considerations of the whole without
  explicit regard to the properties of the
  components.
• One goal of the schooling of these students is to
  move them to level 1 (analysis), where they can
  informally analyze component parts and
  attributes.

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Making Instructional Choices

• In the primary grades, students build the
  foundation for understanding shapes, both two-
  and three-dimensional. They learn what shapes
  look like, the features that distinguish shapes
  from one another, and ways to describe shapes.
• Most students in grades K-3 will be at Level 0
  (visualization) while students in grades 4-5 may
  be at Level 1 (analysis) and some possibly at
  Level 2 (informal deduction). It is important for
  elementary school teachers to provide their
  students with experiences that will help them
  move from Level 0 to Level 2 by the end of the
  eighth grade.