The
van Hiele Levels of Geometric Thinking
Cathcart, et al. Learning Mathematics in Elementary and Middle Schools. p.282-3
Dina and Pierre van Hiele
are two Dutch educators who were concerned about the difficulties that their
students were having in geometry. This
concern motivated their research aimed at understanding students’ levels of
geometric thinking to determine the kinds of instruction that can best help
students.
The five levels that are
described below are not age-dependent, but, instead, are related more to the
experiences students have had. The
levels are sequential; that is, students must pass through the levels in order
as their understanding increases. The
descriptions of the levels are in terms of “students” – and remember that we
are all students in some sense.
Students recognize shapes by their global, holistic
appearance.
Students at level 0
think about shapes in terms of what they resemble and are able to sort shapes
into groups that “seem to be alike.” For
example, a student at this level might describe a triangle as a “clown’s
hat.” The student, however, might not
recognize the same triangle if it is rotated so that it “stands on its point.”
Level 1 – Analysis
Students observe the component parts of
figures (e.g., a parallelogram has opposite sides that are parallel) but are
unable to explain the relationships between properties within a shape or among
shapes.
Student at level 1
are capable of describing the properties of shapes. Thus, they
are able to understand that all shapes in
a group such as parallelograms have the same properties, and they can describe
those properties.
• Students can separate shapes into
groups.
• Students can describe the
properties of given groups.
ACTIVITIES
Level 2 –
Informal deduction (relationships)
Students at level 2 are able to notice
relationships between properties and to understand informal deductive discussions
about shapes and their properties.
This is when the students begin to understand the relationships
between shapes and their different characteristics.
• Students can see relationships
between properties.
• Students can understand informal
deductive discussion concerning shapes and their properties.
ACTIVITIES
Students can create formal deductive
proofs.
Students at level 3 think about relationships
between properties of shapes and also understand relationships between axioms,
definitions, theorems, corollaries, and postulates. At this level, students are able to “work
with abstract statements about geometric properties and make conclusions based
more on logic than intuition”
(Van de Walle).
Students
can create formal deductive proofs. This
is when the students understand axioms to solve problems.
• Students can think about
relationships between properties of shapes
• Students can distinguish between
axioms, definitions, theorems, corollaries, and postulates.
• Students can work with geometric
abstracts and make conclusions based on logic.
ACTIVITIES
Students rigorously compare different
axiomatic systems.
During
this level the students are using different premises while developing different
shapes. Students at this level think about deductive
axiomatic systems of geometry. This is
the level that college mathematics majors think about Geometry.
• Students think about deductive
axiomatic systems of geometry.
ACTIVITIES
In general, most elementary school students are
at levels 0 or 1; some middle school students are at level 2. State standards are written to begin the
transition from levels 0 and 1 to level 2 as early as 5^{th} grade
“Students identify, describe, draw and classify properties of, and
relationships between, plane and solid geometric figures.” (5^{th} grade, standard 2 under
Geometry and Measurement) This emphasis
on relationships is magnified in the 6^{th} and 7^{th} grade
standards.
Interestingly, the sixth National Assessment of Educational
Progress report (1997) reported that “most of the students at all three grade
levels (fourth, eight, and twelfth) appear to be performing at the ‘holistic’
level (level 0) of the van Heile levels of geometric
thought.”
Bibliography
Fuys, David, Dorothy Geddes,
and Rosamond Tischler. "The
van Hiele Model of Thinking in Geometry among
Adolescents." Journal for Research in Mathematics
Education Monograph No. 3.
Malloy‚Carol
E., and Susan Friel. “Perimeter and Area through the van Heile Model.”
Math Teaching in the Middle School.
Volume 5, Issue 2, October,1999.
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van Hiele, P.M.(1986). Structure
and Insight, Academic Press
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(1983). The van Hiele levels of geometric thought in
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