Transcript Developing Geometric Thinking: The Van Hiele Levels

Van Hiele’s Learning
Theory
Mara Alagic
Levels of Geometric Thinking
Precognition
 Level 0: Visualization/Recognition
 Level 1: Analysis/Descriptive
 Level 2: Informal Deduction
 Level 3: Deduction
 Level 4: Rigor
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Learning Theory Phases
Information
 Guided Orientation
 Explication
 Free Orientation
 Integration
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Outline
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Learning theory phases
Information
 Guided Orientation
 Explication
 Free Orientation
 Integration
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Example
Theme: Properties of two-dimensional shapes
 Tools: Seven tangram shapes
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Information
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Students should be encouraged to freely explore
the materials; they will discover some properties
& structures
While students are playing, the teacher can
observe and informally assess students’ thinking
and language
Give each student a tangram set and ask 'What can
you do with these pieces?'
 Encourage students to share and talk about the
shapes and pictures they have made
 Students begin to see how pieces fit together; they
begin to discover the properties and relationships
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Guided Orientation
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Activities are presented so that students’ attention
is focused on particular characteristics of the
shapes; Questions might be generated from the
first phase:
Choose two shapes. How many different shapes can
you make with them?
 Use three shapes to make a new shape. How many
ways can you make this shape?
 Which pieces can be made from three other pieces?
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Have students record what they find by tracing
around the shapes, and share with others
One activity will often lead to another, e.g. …
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Explication
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Tasks and games that deliberately develop the
vocabulary associated with the ideas that have
been encountered so far
The teacher clarifies terms the children are
already using and introduces new terms
Questions provide opportunities for the terms to
be used:
Which shapes have a right angle?
 How many lines of symmetry does each shape
have?
 What is the same about all the triangles? Which
shapes have parallel sides? …
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Free Orientation
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Students engage in more challenging tasks that
draw on the knowledge and skills previously
developed
Activities and problem-solving tasks are openended or can be completed in different ways
The aim is to apply what they have learned and
become more skillful
How many ways can you make a square from some
or all of the pieces? Complete classic tangram
puzzles of outlines of animals
 Draw a tangram square (like the one illustrated
above) on an 8x8 square grid, examine the pieces
carefully in relation to the grid, then work out a way
to enlarge all the pieces
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Integration
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Opportunities are given for the students to pull
together their new knowledge and reflect on it
as a whole
They should be able to express or summarize
what they have learned in some way:
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For example, small groups of children can design charts,
class books and displays that present what they have
learned about the tangram shapes
Small groups, pairs of students and individuals can design
presentations (posters, class books, technology aided),
games and lists of questions about what they have learned
about the tangram shapes
This will help students to logically organize their
knowledge of properties of the shapes
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References
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Van Hiele, P. M. (1959). Development and learning
process. Acta Paedogogica Ultrajectina (pp. 1-31).
Groningen: J. B. Wolters.
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A method of initiation into geometry at secondary
schools. In H. Freudenthal (Ed.). Report on methods
of initiation into geometry (pp.67-80). Groningen: J.
B. Wolters.
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Fuys, D., Geddes, D., & Tischler, R. (1988). The van
Hiele model of Thinking in Geometry Among
Adolescents. JRME Monograph Number 3.
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