Transcript The van Hiele Model of Geometric Thought

SPATIAL SENSE
• What and why Spatial Sense?
• van Hiele Model Geometric
Thinking
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Dr. Lee Wai Heng & Dr. Ng Kok Fu
WHAT IS SPATIAL SENSE?

Spatial sense is an
intuitive feel for shape
and space. It involves the
concepts of traditional
geometry, including an
ability to recognize,
visualize, represent, and
transform geometric
shapes.
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It also involves other, less
formal ways of looking at
2- and 3-dimensional
space, such as paperfolding, transformations,
tessellations, and
projections.
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NCTM: GEOMETRY & SPATIAL SENSE
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Geometry is the area of
mathematics that involves
shape, size, space,
position, direction, and
movement, and describes
and classifies the
physical world in which
we live.
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Young children can learn
about angles, shapes,
and solids by looking at
the physical world.
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NCTM: GEOMETRY & SPATIAL SENSE
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Spatial sense gives
children an
awareness of
themselves in relation
to the people and
objects around them
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WHY CHILDREN SHOULD LEARN
GEOMETRY
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Spatial understandings
are necessary for
interpreting,
understanding, and
appreciating our
inherently geometric
world.
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Children who develop a
strong sense of spatial
relationships and who
master the concepts and
language of geometry are
better prepared to learn
number and
measurement ideas, as
well as other advanced
mathematical topics.
(NCTM, p. 48)
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WHY CHILDREN SHOULD LEARN
GEOMETRY

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The world is built of
shape and space, and
geometry is its
mathematics.
Experience with more
concrete materials and
activities prepare
students for abstract
ideas in mathematics

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Students solve problems
more easily when they
represent the problems
geometrically.
People think well visually.
Geometry can be a
doorway to success in
mathematics
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IMPORTANCE IN DAILY LIFE
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Spatial relationships
is connected to the
mathematics
curriculum and to real
life situations.
Geometric figures
give a sense of what
is aesthetically
pleasing.

Applications

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architectural use of
the golden ratio
tessellations to
produce some of the
world’s most
recognizable works
of art.
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IMPORTANCE IN DAILY LIFE

Well-constructed
diagrams allow us to
apply knowledge of
geometry, geometric
reasoning, and intuition to
arithmetic and algebra
problems.
Example:
Difference of 2 squares
a2 - b2 = (a-b) (a+b)
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Whether one is designing
an electronic circuit
board, a building, a dress,
an airport, a bookshelf, or
a newspaper page, an
understanding of
geometric principles is
required.
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van Hiele Model
of Geometric Thinking
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Background of van Hiele Model
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Husband-and-wife team of Dutch educators
(1950s): Pierre van Hiele and Dina van HieleGeldof noticed students had difficulties in
learning geometry
These led them to develop a theory involving
levels of thinking in geometry that students
pass through as they progress from merely
recognizing a figure to being able to write a
formal geometric proof.
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Levels of Thinking in Geometry
Level 1. Visual
 Level 2. Analysis
 Level 3. Abstract
 Level 4. Deduction
 Level 5. Rigor
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The development of geometric ideas progresses through a hierarchy of
levels. The research of Pierre van Hiele and his wife, Dina van HieleGeldof, clearly shows that students first learn to recognize whole shapes
then to analyze the properties of a shape. Later they see relationships
between the shapes and make simple deductions. Only after these
levels have been attained can they create deductive proofs.
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Levels of Thinking in Geometry
The levels progress sequentially.
 The levels are not age-dependent.
 The progress from one level to the next is
more dependent on quality experiences
and effective teaching.
 A learner’s level may vary from concept
to concept
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1 - Visual Level Characteristics
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The student
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identifies, compares and sorts shapes on the basis
of their appearance as a whole.
solves problems using general properties and
techniques (e.g., overlaying, measuring).
uses informal language.
does NOT analyze in terms of components.
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Visual Level Example
It is a flip!
It is a mirror image!
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2- Analysis Level Characteristics
The student
 recognizes and describes a shape (e.g.,
parallelogram) in terms of its properties.
 discovers properties experimentally by
observing, measuring, drawing and
modeling.
 uses formal language and symbols.
 does NOT use sufficient definitions. Lists
many properties.
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Analysis Level
It is a reflection!
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3 - Abstract Level Characteristics
The student can

define a figure using minimum
(sufficient) sets of properties.
 give informal arguments, and discover
new properties by deduction.
 follow and can supply parts of a
deductive argument.
http://www.mathopenref.com/kite.html
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Abstract Level
If I know how to find the area of the rectangle,
I can find the area of the triangle!
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Area of triangle = bh
2
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4 - Deductive Level Characteristics
The student

recognizes and flexibly uses the
components of an axiomatic system
(undefined terms, definitions, postulates,
theorems).
 creates, compares, contrasts different
proofs.
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Deductive Level Example
In ∆ABC, BM
is a median.
I can prove that
Area of ∆ABM = Area of ∆MBC.
A
M
∆ABM
∆MBC.
B
C
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5 - Rigor
The student

compares axiomatic systems (e.g.,
Euclidean and non-Euclidean
geometries).
 rigorously establishes theorems in
different axiomatic systems in the
absence of reference models.
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References
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Learning to Teach Shape and Space by Frobisher,
L., Frobisher, A., Orton, A., Orton, J.
Geometry Module
http://math.rice.edu/~rusmp/geometrymodule/index.htm
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Mind map of van Hiele model
http://agutie.homestead.com/FiLes/mindmap/van_hiele_geometry_level.html
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van Hiele model at Wikipedia
http://en.wikipedia.org/wiki/Van_Hiele_levels
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