"Given children's affinity toward, knowledge of, and
ability to gain geometric knowledge, it is important
that this domain of mathematics not be neglected.
Instruction in geometry needs to complement the
study of number and operation in pre-K to 8"
(National Research Council, 2001). Math Trailblazers
echoes this expectation by emphasizing the importance
of geometry in the mathematics curriculum.
This unit represents the focal point of geometry in
Second Grade. Students describe, analyze, and classify
two-dimensional shapes using their properties.
Students also discover relationships within and
among these properties as they advance their understanding
through stages, from basic intuition to
analysis and informal deduction.
Much of the approach to geometry found in the Math
Trailblazers curriculum is grounded in the insights of
Dutch educators Pierre van Hiele and Dina van
Hiele-Geldof. Ongoing research in mathematics education
continues to confirm the five levels of geometric
development first described by the van Hieles in
the 1950s (Burger and Shaughnessy, 1986). The five
levels are:
- Level 0: Visualization. Students judge geometric
objects by their appearance, but not by attributes.
For example, a student can identify a rectangle
because it "looks like a rectangle," but not
because it has opposite sides equal and four right
angles.
Level 1: Analysis. Students begin to describe the
properties of objects. A figure is no longer judged
because it "looks like one," but rather because it
has certain properties. For example, an equilateral
triangle has three equal sides, three equal angles,
and line symmetry.
Level 2: Informal Deduction. Students logically
order the properties of figures and are able to
deduce that one property precedes or follows
from another property. They see relationships
among figures. For example, a square has all the
properties of a rectangle; therefore, a square is a
rectangle. Students may also be able to define a
square based on its properties.
Level 3: Deduction. Students write formal proofs
based in an axiomatic system. A rigorous high
school geometry course is taught at Level 3.
Level 4: Rigor. Students can work with different
axiomatic systems. This level corresponds to college
work in geometry (Crowley, 1987; van Hiele,
1999).
The van Hieles found that each level, while not age
specific, builds on the previous level. Students proceed
from level to level sequentially and no level can
be omitted. Advancement depends on content and
method of instruction (Crowley, 1987; van Hiele,
1999). Moreover, a student's experiences with
lower-level reasoning at the elementary school level
are critical to success with geometry in later schooling.
Students who are at Level 0 or 1 when entering
high school geometry have a poor chance of success.
Students who begin high school geometry at Level 2
have at least a 50% chance at succeeding (Senk,
1989). Unfortunately, many upper elementary students
are still at Level 0. This is not surprising, as
researchers have found that most geometry questions
asked in standard elementary math textbooks were
answerable with Level 0 understanding (Fuys,
Geddes, and Tischler, 1988).
In this unit, we start with Level 0 ideas by asking students
to identify and draw various geometric figures.
Most of the work in this unit is at Level 1, where students
describe the properties of two-demensional
shapes. Level 2 ideas are introduced as students
explore ways to classify these figures.
