Curriculum Units – Level 4

Getting into Shapes

In Getting Into Shapes, students explore 2- and 3-dimensional shapes with a focus on their properties, relationships among them and spatial visualization. The reasoning skills that they build upon in this unit help them to develop an understanding of more complex geometric concepts. They learn new, more specialized vocabulary and learn how to describe properties of shapes with this terminology allowing for greater clarity and precision in their explanations. They move from describing properties to comparing and contrasting properties of 2- and 3-dimensional shapes by classifying them into different groups based on their properties.

To better understand how students grow in their understanding of shapes—and thus understand the development of the activities in this unit—it helps to examine a model of geometric thinking created by Pierre van Hiele and Dina van Hiele-Geldof, contemporaries and colleagues of Jean Piaget. Their model represents a five-level hierarchy of ways of understanding spatial ideas (van Hiele, 1999). The levels describe how one thinks and what types of geometric ideas one thinks about rather than the amount of knowledge someone has. At the lowest of the levels, visualization, figures are judged by their appearance alone. For example, a student might claim, “I know it is a rectangle because it looks like a box.” At this level, if a student saw a tilted square they would not identify it as a square because it does not look like one. At the next level, the descriptive level, students focus on the properties of shapes. They recognize that an equilateral triangle has three congruent angles, three congruent sides and symmetry. However, the properties are not necessarily in any order. Ordering occurs at the next level, the informal deduction level. At this level students can put properties in order and use them to formulate definitions. They are able to see the relationships among shapes and group them into classifications. For example, they would recognize that a square is also a rectangle and both of these shapes can be classified as parallelograms.

The last two levels, deduction and rigor, are much more advanced. Students at the deduction level are generally studying high school geometry. They are able to work with abstract statements about geometric properties and begin to appreciate the structure of a system complete with definitions, postulates and theorems. Students who study college-level geometry as a branch of mathematical science and focus on the axiomatic system itself and its relationship to other systems are at the level of rigor.

The focus in this unit is on moving students from their initial level of geometric thinking to the third level, informal deduction. Students will examine shapes, describe them using their definitions and properties and then find relationships among shapes based on these properties. They will classify and reclassify shapes according to different properties. These kinds of experiences help students develop the more sophisticated reasoning skills used in informal deduction.

In addition to studying physical models, students develop and use mental images. Beginning at a very early age, students display the ability to use spatial logic through simple tasks such as manipulating building blocks. Early student opportunities shape their ability to use spatial visualization as a problem solving tool. Activities such as building puzzles and hidden pictures provide explorations using spatial orientation, perceptual constancy and eye-hand coordination skills. There are many Internet sites where students can participate in spatial reasoning and logic exercises. For example, the National Council of Teachers of Mathematics hosts the Illuminations site where students can manipulate 2- and 3-dimensional shapes. They determine the faces, edges and vertices of shapes and paint surfaces while rotating the shapes in space.

In Chapter 2, students study transformations of 2-dimensional shapes and learn how to mentally transform a shape by changing its position or orientation. They use the coordinate grid to connect location to transformations. These activities are aimed at developing students’ sense of spatial visualization. This, in turn, will help students make better sense of the world around them.