Geometry is an essential component of mathematics instruction. "Geometry
helps us represent and describe in an orderly manner the world in which
we live" (NCTM 1989, 48).
Research in the field of early childhood mathematics education (Fuys and
Liebov 1993; Del Grande 1985;Fruedenthal 1973) confirms that children
are naturally intrigued by, and motivated to learn more about, the geometry
that defines their worlds. Although it is important to provide a rich
geometry program in the primary grades, research reveals that the little
attention given to geometry is typically for exposure purposes (Bruni
and Seidenstein 1990;Porter 1989). Therefore, any classroom time devoted
to geometry is precious.
As
practitioners in early childhood mathematics, we have observed that many
young children have numerous misconceptions about geometry, perhaps as
a result of a lack of exposure to vocabulary and too few authentic experiences
in the primary grades. It is also possible that misinformation conveyed
by adults during formal and informal instruction minimizes and contradicts
the budding geometric constructs of young children. To search for the
source of misconceptions and to improve the geometry program, we must
consider the current practices used to teach geometry.
UNCOVERING
THE ROOTS OF MISCONCEPTIONS SQUARES AND RECTANGLES A common
activity involving geometry is for young children to recognize and name
various shapes. Many children are taught the names of shapes through experiences
with adults, peers, television, books, and computer games. Although these
experiences may be rich, they may lack depth or be inaccurate. For example,
children are often taught to categorize rectangles and squares separately.
Typically, a polygon with four equal sides and four equal angles is referred
to as a square, whereas the polygon with equal length, parallel sides,
but unequal perpendicular sides is referred to as a rectangle. We hear
children orally refer to rectangles as being "long" or "tall." Their system
for differentiating between squares and rectangles is based on narrow
experiences with a few specific examples.
These
constructions may cause confusion later as educators clarify for children
that squares also fit the description of rectangles. The new information
does not connect logically to what they have already learned. Although
the first classification of rectangles versus squares is essentially correct,
it does not allow for growth in understanding that a square is a more
specific classification of a rectangle, just as a rectangle is a more
specific classification of a parallelogram, and that a parallelogram is
a specific classification of a quadrilateral (see fig.1). The relationships
go unexplored, and each shape is viewed exclusively. It seems that providing
incorrect or incomplete information at the onset, in hopes of re-teaching
and altering the paradigms of thought later in students' educational careers,
has become an acceptable means of addressing a rather complex classification
system.
We
do not expect young children to understand fully the intricacies of quadrilaterals
and their classification system, but we believe that children would fare
better by learning about quadrilaterals as a whole versus focusing on
a few specific examples and attempting to expand their understandings
later. To aid understanding, define a quadrilaterals a four-sided figure
and give students various-sized sticks with which to build a variety of
quadrilaterals. After the formations are made, students can observe the
creations of their classmates, and the teacher can initiate a discussion
about the similarities and differences among the shapes. The discussion
should include the terminology of corners and sides and lead students
to further sort the quadrilaterals into several different categories.
Students have the opportunity to discover and express the attributes of
the quadrilaterals that they have explored. Within these categories, the
names of particular quadrilaterals, such as square, rectangle, and parallelogram,
could be introduced. This discovery-oriented approach involves looking
at the similarities and differences between and among shapes instead of
memorizing specific descriptors of individual shapes. This activity will
set the stage for students to understand that many types of quadrilaterals
exist and that these shapes have some elements in common.
TRAPEZOIDS
AND TRIANGLES A staff member was asked to gather several tables
for a special school event. As he returned with a table in the shape of
a trapezoid, he asked, "Shall we use this triangular table?" Many adults
use similar terms and incorrectly label shapes and solids in the presence
of students. As role models, we must pay particular attention to the language
of mathematics and must consider how our words are interpreted. Perhaps
we should examine our own content knowledge of geometry. In many situations,
the misuse of terminology is not intentional; rather, it is the result
of gaps in learning and possibly the superficial way in which we learned
the geometry lexicon. As young learners, we may have identified shapes
by memorizing specific attributes, which we may not have fully understood.
When memorization occurs without an attachment to well-developed concepts,
learners use or hear erroneous terminology that can lead to misconceptions.
Likewise, when young learners are offered only regular or common examples
of shapes, they connect one label to one shape, which limits applicability
and understanding.
A recent experience with first grader Grace highlights the issue. Grace
was given the pattern in figure 2 and asked to continue it. She studied
the pattern and began to read aloud. "Triangle, triangle, wrong triangle,
triangle, triangle, wrong triangle, triangle.... The next shape is a right
triangle!" Clearly, Grace's construct of a proper triangle included only
equilateral triangles. Ironically, right triangles were deemed "wrong
triangles" by Grace. It is clear that Grace has learned only about one
type of triangle in her mathematical experience, so this situation presented
a perfect teachable moment. Because Grace called the unfamiliar shape
a triangle, even though it was a "wrong" triangle, she was ready to learn
about different types of triangles, various angles, and the labels equilateral
triangle and right triangle.
One
may argue that young children are not developmentally ready to process
the extensive vocabulary and abstractions associated with geometry. However,
later re constructions may require much more intellectual sophistication.
Does it make sense to set subsequent hurdles for children simply because
we are afraid to give them academic challenges early on?
THREE-DIMENSIONAL SHAPES Children are exposed to misconceptions
about geometry from a variety of sources, including books. The Silly Story
of Goldie Locks and the Three Squares (MacCarone 1996) was written to
help teach mathematics and includes an introductory note to parents, as
well as suggestions for follow-up activities. To an educator, this book's
familiar story line and instructional component look appealing, and parents
and teachers may think that they are contributing to an understanding
of geometry. However, the book's illustration of Goldie Locks's three
beds is problematic. (See fig. 3.) The first bed "was shaped like a circle";
the second, "like a triangle"; and the last, "like a rectangle." In reality,
the shapes look like a circle, a triangle, and a rectangle but are actually
a cylinder, a triangular prism, and a rectangular prism. Although the
text does state that the beds are shaped "like" a circle, triangle, and
rectangle, few children would distinguish between "like" and actual without
emphasis and further discussion, and parents and teachers may be misled
as well.
Similarly,
many young children are taught to label a cube as a square and a sphere
as a circle. One popular preschool television show's search for squares
resulted in a collection of cubes, and a search for circles resulted in
spheres. Although the faces of a cube are squares and a sphere depicted
in two dimensions would be called a circle, the terminology and characteristics
of two-dimensional and three-dimensional representations were not explored.
Asking
students to search for particular shapes around the room or at home is
a popular activity, but we must be careful about the questions that we
ask and the responses that we give as students communicate their findings.
This point is illustrated in the dialogue that occurred in a primary classroom
between a teacher (T) and different students (S1 to S5).
T:
What shapes did you find around the room?
S1: I found a rectangle.
T:
Would you like to show us the rectangle you found?
S1:
Right here [pointing to the front of the classroom door]. The door of
our classroom.
T: So, the front part of the door looks like a rectangle?
S1:
Yes.
T: Why did you call it a rectangle?
S1:
I know it's a rectangle because these two sides are the same [pointing
to opposite sides] and these two sides are the same [pointing to the other
pair of opposite sides].
S2:
And there are four square corners!
T:
So, everyone agrees that we can call this shape a rectangle? Could we
call it anything else?
S3:
A quadrilateral!
T: Does anyone see any other quadrilaterals on this door?
S4:
I do. I see another rectangle on the side of the door. [Student4 outlines
the four edges making up the sides of the door.] It is really tall and
thin.
T: Are there any other rectangles that make up the door? [Discussion continues,
and students eventually name all six surfaces of the door as rectangles.]
T:
We can see six different rectangles on the door. When you put those rectangles
together to form a solid, like this door, it is no longer a rectangle.
It is made up of six rectangles on its surface. It is called a rectangular
prism. Let's use these materials to build our own rectangular prisms [students
receive cardboard attribute blocks and tape].
Even
though the lesson's focus was on two-dimensional shapes, the identification
of the door as a rectangle warranted a discussion of three-dimensional
solids, as well. It is better to introduce new terminology to students
rather than subsequently attempt to distinguish between two- and three-dimensional
shapes. Although all students may not remember the term rectangular prism,
they will have been given the opportunity to discuss and explore the concept,
which will be a building block for future geometric understanding.
AVOIDING
MISCONCEPTIONS ABOUT GEOMETRY IN THE CLASSROOM Educators should
keep in mind several key elements as they introduce and teach geometry
concepts to young children. The following list, although not exhaustive,
is intended to inspire thoughts about program evaluation. |