Download Levels of Geometric Thinking

Introduction to Geometric Thinking
Historical Note Kay, D. College Geometry: A Discovery Approach. p. 63
About all we know concerning the man Euclid is that he was a professor of mathematics around 300 B.C., taught at
the University of Alexandria in Greece, and was the author of the celebrated Elements. The stories told of Euclid
show that he was a serious, uncompromising scholar. When asked by King Ptolemy whether there were an easier
way to learn geometry (other than studying the Elements), Euclid is supposed to have responded, “There is no royal
road to geometry!” Euclid’s great contribution to mathematics was the elegant, logical arrangement of the Elements.
This monumental work consists of 13 books containing 465 propositions, with detailed arguments for each, all based
on only 10 basic axioms. The work was an immediate success, so much so that all other works of mathematics soon
disappeared-only the Bible has been printed in more editions. The Elements have had a great and lasting impact on
science, its influence even being felt up to modern times. While the propositions found in the Elements were
themselves not new, their logical arrangement and ingenious arguments were. Euclid’s arguments were so effective
that many of them have survived to this day as the simplest method of proof, in spite of major revisions of his work
to make it more rigorous.
Building Blocks of Geometry:
Euclid based his development of the geometry of the plane on a set of undefined terms and a set of 5 postulates (or
axioms), and some common notions. If you are interested in gaining a more historical understanding of Euclid’s
geometry, then check out the following web site: http://www.perseus.tufts.edu/
From this page, Search Perseus: Euclid, then click on Euclid, Elements.
Postulates
1.
To draw a straight line from any point to any point.
2.
To produce a finite straight line continuously in a straight line.
3.
To describe a circle with any centre and distance.
4.
That all right angles are equal to one another.
5.
That, if a straight line falling on two straight lines make the interior angles on the same side less than two
right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less
than the two right angles.
Common Notions
1.
Things which are equal to the same thing are also equal to one another.
2.
If equals be added to equals, the wholes are equal.
3.
If equals be subtracted from equals, the remainders are equal.
4.
Things which coincide with one another are equal to one another.
5.
The whole is greater than the part.
From these 10 statements, 465 propositions (theorems) were developed. As a historical note Euclid’s work is very
interesting. As a way to teach geometry to young students, it is not very practical. It is a difficult and time
consuming to develop the early propositions, and many students would be lost in the development, and they would
not appreciate why they were doing such a geometrical exercise.
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.
Level 0 – Visualization
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 able to understand that all shapes in a group such as parallelograms have the same properties,
and they can describe those properties.
Level 2 – Informal deduction (relationships)
Students deduce properties of figures and express interrelationships both within and between figures.
Students at level 2 are able to notice relationships between properties and to understand informal deductive
discussions about shapes and their properties.
Level 3 – Formal deduction
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).
Level 4 – Rigor
Students rigorously compare different axiomatic systems.
Students at this level think about deductive axiomatic systems of geometry.
mathematics majors think about Geometry.
This is the level that college
In general, most elementary school students are at levels 0 or 1; some middle school students are at level 2. The CA
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 7th 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.”