Childrens Conception of Space and Geometry

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Childrens Conception of Space and Geometry
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Geometry For Young Children

• The NCTM (National Council of Teachers of
  Mathematics) Curriculum Standards (1989) In
  grades k-4, the mathematics curriculum should
  include two- and three-dimensional geometry so
  that students can
• describe, model, draw, and classify shapes
• investigate and predict the results of combining,
  subdividing, and changing shapes,
• develop spatial sense

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• relate geometric ideas to number and measurement
  ideas
• recognize and appreciate geometry in their world

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Role of Geometry

• Helping children develop mathematical thinking
  skills
• by exposing to a variety of shapes and
  orientations of these shapes, as well as time to
  discuss and make figures
• vocabulary words can be introduced and modeled
  their use when children are working with geometry

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Materials for Involving Children in Geometry

• Magnetic strips
• geoboards
• models
• pattern blocks,
• mirrors
• tangrams

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Checklist for geometric work

• Sorts geometric shapes into two or more piles
• matches three-dimensional shapes
• matches two-dimensional shapes
• constructs simple three-dimensional shapes
• constructs simple two-dimensional shapes
• describes shapes

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• Extends patterns with three- and two-dimensional
  shapes
• describes patterns with shapes
• names shapes in spontaneous conversation
• recognizes shapes in the classroom and
  environment
• makes geoboard designs
• duplicates geoboard designs

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• Transfers geoboard designs to paper
• seems to enjoy work with geometry

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The Van Hieles ModelDina Van Hiele -Geldof,
Pierre Van Hiele

• Functions of experience and education rather than
  development
• Level 0 -- Visualization
• recognize and name figures according to general,
  holistic impression
• can match alike figures and can copy figures
• can classify and sort figures according to their
  overall appearance.
• Can reproduce figures on a geoboard but cannot
  see a square is a rhombus,..

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• Level 1 -- Analysis
• focus on attributes of figures, sees a rectangle
  has 4 sides, etc..
• not able to work very successfully with classes
  of figures and their relationships, not able to
  recognize a rectangle as parallelgram.
• Level 2 -- Informal Deduction
• construct relationships between classes of
  figures
• make logical conclusion and follow simple,
  logical proofs.

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• Level 3 -- Formal Deduction
• relationships and definitions are beginning to be
  clarified but only with guidance, square seems as
  rectangle which is also a parallelogram.
• high school geometry
• use axioms, theorems, abstract definitions, make
  formal arguments about relationships
• Level 4 -- Rigor
• Mathematicians
• elaborating on and compare axioms

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• People may be on different Van Hiele levels for
  different topics
• Wirszup (1976) if a childs introduction to
  geometry is with measurement and other concepts
  of Level II and III withoug a sound grounding in
  the visual geometry of Level I then he is doomed
  to failure.

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Level 0 Activities

• Let a child feels and manipulates shapes behind a
  screen, he can always point to the correct one
  from the 4 figures shown on the screen.
• Drawing the order was circle, square, triangle
  and rhombus 20 of 6 1/2 years can correctly
  reproduce the three straight-line shapes.

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• Construction
• given 6 sticks to construct
• 3 1/2 years manages a square
• 2/3 of 6 1/2 years succeeded with the rhombus
• reasons why better than drawing ?
• Naming(10 years old)
• square 95
• rectangle 91
• rhombus 74
• triangle 76
• pentagon 18
• why?

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Level I Activities

• Fitting shapes together important for the
  developing of the right angle notion flatness
• work with 2- and 3- dimensiona shapes is
  essential groundwork for developing an
  understanding of the notions of area and volume
  and the measurement of these as well as providing
  a foundation for the study of geometric
  transformations.

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• Relations between two right angles
• sets of parallel lines

Forming different shapes by moving the right
angles
By changing angles of intersection, distances
between lines, different figures can be formed
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How Do Children Form Misconceptions About Space

• Misperception
• misgeneralization
• p. 29 to p. 37

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Figures are normally presented as
When asked to identify the following figures
(a)
(b)
(c)
(d)
Age recognising (c) as a square 5 years
54 6 56 7 80
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Age recognising (b) as a triangle 5
38 6 47 7 24 8 65 9 50 10 67
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Ten year olds to recognize the following as right
angle
83
93
63
56
63
60
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Ten year olds to recognize the following as
parallel lines (Kerslake)
73
71
43
32
38
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Identification of the following as
rectangles (Kerslake 1979)
5
85
89
22
87
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Spatial Perception

• The ability to recognize and discriminate stimuli
  from space and interpret those stimuli by
  associating them with previous experiences
• Seven spatial abilities that seem to have strong
  relationships to academic development and to the
  development og geometric ideas

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Spatial Abilities

• Eye-motor coordination ability to synchronize
  vision and body movements.
• Figure-ground perception recognizing figures
  embedded in a background
• Perception Constancy ability to recognize that a
  figure has invariant properties such as size and
  shape in spite of the variability of its
  impression as seen from different view points

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• Position-in-space relationship of one object to
  another and to the observer
• rotation, reversals, change of position, mirror
  patters
• differentiate b,d,p,q
• Perception of spatial relationship
• seeing two or more objects
• perceive effects of that motions or
  transformation such as slide and flips have on
  shapes

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• Visual Discrimination noting similarities and
  difference between objects and figures
• Visual Memory recalling objects or design that
  are no longer in view.