MTH 232
An Introduction to Chapter 9:
Geometric Figures
Philosophy of this Course
• To study shapes in geometry in an informal
manner while at the same time being
mathematically careful with the material.
Historical Perspective
• Before 600 B.C., geometry was both informal
and practical.
• From 600 B.C. to 300 B.C., geometry was
transformed into a theoretical science.
• Euclid’s Elements introduced theorems
(statements requiring proof) that were
deduced from axioms (statements accepted
without proof).
Intuition and Experience
1. Undefined terms are identified and careful
definitions of other concepts are made from
those terms.
2. Geometric facts are then discovered by
explorations of pictorial representations and
physical models.
Activity-based Learning: the Van Hieles
• The van Hieles were Danish mathematics
teachers in the late 1950s.
• Their research concluded that the knowledge
construct for themselves through hands-on
activities is essential to learning geometry.
• Learning geometry progresses through five
levels.
van Hiele Levels
Level 0: Recognition of shape. Children
recognize shapes holistically. Only the overall
appearance of the shape is observed, with no
attention given to the parts of the figure.
Level 1: Analysis of single shapes. Children are
aware of the parts of certain figures (e.g., a
rectangle has four straight sides that meet at
“square” corners).
Continued
Level 2: Relationships among shapes. Children
understand how common properties create
abstract relationships among figures, and can
make simple deductions about figures.
Level 3: Deductive reasoning. The student at this
level views geometry as a formal mathematical
system and can write deductive proofs.
Finally
Level 4. Geometry as an axiomatic system. This
level is highly abstract and is reached only in
high-level university courses. No dependence is
placed on concrete or pictorial models.
Your Challenge
• As an elementary school teacher, you want
your students to progress through the first
three van Hiele levels.
• To do this, you must create interesting
activities that support each child’s
progression.
NCTM Principles and Standards
• Students first learn to recognize a shape by its
appearance.
• In grades Pre-K through 2, geometry begins with
describing and naming shapes.
• Formal terminology is gradually introduced in
order to focus attention and clarify ideas.
• Teachers must provide materials (geoboards,
tangrams, pattern blocks and geometric solids).
• The learning environment must encourage
students to explore shapes and their attributes.
Continued
• Students need to see both examples and nonexamples of shapes that correspond to a
particular geometric concept.
• Interactive computer programs provide a rich
environment for activities in which students
compose (put together) and decompose (take
apart) shapes.