California Preschool Learning Foundations - ECEZero2Three ...

MATHEMATICS
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directly to determine equality or
inequality, and as they order three or
more objects by size. Older preschool
children begin to make progress in
reasoning about measuring quantities
with less dependence on perceptual
cues (Clements 2004a, Clements and
Stephen 2004). Children start to compare
the length of objects, indirectly,
using transitive reasoning, and to
measure the length of objects often by
using nonstandard units. They develop
the ability to think of the length of a
small unit (i.e., a block) as part of the
length of the object being measured
and to place the smaller unit repeatedly
along the length of the larger
object.
Geometry
Geometry is the study of space and
shape (Clements 1999). Geometry
and spatial reasoning offer a way to
describe, interpret, and imagine the
world. They also provide an important
tool for the study of mathematics and
science. The research literature shows
that young children bring to kindergarten
a great deal of knowledge about
shapes. This finding is important
because teachers and curriculum
writers seem to underestimate the
knowledge about geometric figures that
students bring to school. This underestimation
and teachers’ lack of confidence
in their own geometry knowledge
usually result in teachers’ minimizing
the time dedicated to teaching geometry
concepts to children (Clements
2004a; Lehrer, Jenkins, and Osana
1998).
The literature recommends that
young children be given the opportunity
to work with many varied examples
of a particular shape and many
“nonexamples” of a particular shape
(Clements 2004a). For example,
children need to experience examples
of triangles that are not just isosceles
triangles. They need to experience
triangles that are skewed—that is, a
triangle where the “top” is not “in the
middle,” as in an isosceles triangle.
They need also to experience triangles
with a varying aspect ratio—the ratio
of height to base. Without the opportunity
to experience a wide range of
triangles, children may come to
“expect” triangles to have an aspect
ratio that is close to 1 and, consequently,
often reject appropriate examples
of triangles because they are too
“pointy” or too “flat.” In addition, children
need to experience nonexamples
of triangles so that they can develop a
robust and explicit sense of the properties
of a triangle.
In 1959, Van Hiele developed a hierarchy
of ways of understanding spatial
ideas (Van Hiele 1986). The hierarchy
consists of the following
levels: 1—Visualization, 2—Analysis,
3—Abstraction, 4—Deduction, and
5—Rigor. Van Hiele’s theory has
become the most influential factor
in geometry curricula. Recently,
researchers have suggested a level of
geometric thinking that exists before
the visual level: a “precognition” level
in which children cannot yet reliably
identify circles, triangles, and squares
(Clements 2004b; Clements and others
1999).
Shape knowledge involves not only
recognition and naming but also an
understanding of shape characteristics
and properties. One way in which children
demonstrate this understanding
is through their ability to put together
shapes into new shapes (Clements
Preschool Learning Foundations, Volume 1 • California Department of Education