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The Geometry Standard of NCTM's Principles and Standards for School Mathematics (2000) describes the study of both two- and three-dimensional shapes. In the elementary grades, plane geometry often receives more emphasis than does three-dimensional geometry. In many classrooms, students actively explore two-dimensional shapes using a rich variety of materials, including tan-grams, geoboards, and pattern blocks. Materials and lessons for exploring three-dimensional shapes are not as prevalent.
This article describes how students in grades three to five can actively explore three-dimensional shapes using simple, inexpensive materials: paper and drinking straws. The article first discusses basic principles that underlie geometry instruction, then presents a variety of experiences that focus on the properties of prisms and pyramids and the relationships among different shapes.
The Van Hiele Model of Geometric Thinking
The van Hiele model of geometric thinking guided me in making instructional choices for my fourth-grade students (van Hiele 1999). Based on their experiences as teachers in the Dutch public schools, Pierre and Dina van Hiele developed the theory that students progress through the following distinct levels of development in their geometric thinking:
Visual level. Students judge figures by their appearances. A child might say, "It is a rectangle because it looks like a box."
Descriptive level. Students analyze figures based on properties and attributes. They are able to make generalizations about specific shapes, but they cannot make generalizations about how different properties of a shape relate to one another or how different types of shapes relate to one another.
Informal-deduction level. Students can generalize interrelationships of properties within the same shape and among different shapes. They can understand and develop abstract definitions, and they can meaningfully classify shapes in hierarchies.
The van Hiele theory asserts that the development of geometric thinking is more dependent on instruction than on age. The van Hieles proposed sequential phases of learning to help students move from one level to another. Instruction should include a sequence of activities that begins with "play" and exploration, gradually builds concepts and related language, and culminates in summary activities that help students integrate new ideas into their previous knowledge.
Background of My Students
Prior to studying three-dimensional geometry, my students had explored topics in plane geometry through a variety of hands-on activities. I recognized that...