Geometry and Spatial Sense

Spatial sense is necessary for understanding and appreciating the

many geometric aspects of our world. Insights and intuitions about

the characteristics of two-dimensional shapes and three-dimensional

figures, the interrelationships of shapes, and the effects of changes to

shapes are important aspects of spatial sense.

Geometry and Spatial Sense is a relevant area of mathematics for young children, who are naturally curious about their physical world. Learning experiences in the primary grades should provide students with opportunities to explore geometric and spatial concepts, and to relate them to real-life situations. By the time children enter school, they have already developed significant notions about shape and space – the result of everyday experiences in moving about in their environment and in interacting with the objects around them (Clements, Swaminathan, Hannibal, & Sarama, 1999). Classroom experiences can build on what students already understand about geometry and can help students:

• recognize and appreciate geometry in the world;

• develop reasoning and problem-solving skills related to geometric thinking;

• apply geometric ideas in other strands of mathematics (e.g., measuring lengths, perimeters, and areas of shapes; using concrete materials, such as square tiles, to represent numerical ideas; creating and extending geometric patterns);

• apply geometric ideas in other subjects (e.g., creating two- and three dimensional works in the arts, developing map skills in social studies, building structures in science and technology).

The three big ideas that form the basis for the curriculum expectations in Geometry and Spatial Sense for Kindergarten to Grade 3. An understanding of these big ideas assists teachers in providing instructional and assessment opportunities that promote student learning of important concepts

in Geometry and Spatial Sense. The big ideas in Geometry and Spatial Sense are the following:

• properties of two-dimensional shapes and three-dimensional figures

• geometric relationships

• location and movement

Teachers should recognize that these big ideas are conceptually related and interdependent, and that many instructional experiences reflect more than one of the big ideas. For example, an activity in which students construct models of three-dimensional figures provides opportunities for students to learn about the properties of three-dimensional figures and about the geometric relationships between three-dimensional figures and their two-dimensional faces. The discussion of each big idea in this section includes:

• an overview, which provides a general discussion of the big idea in the primary grades, an explanation of some of the key concepts inherent in the big idea, and in some instances additional background information on the concepts for the teacher;

• grade-specific descriptions of (1) characteristics of learning evident in students who have been introduced to the concepts addressed in the big idea, and (2) instructional strategies that will support those learning characteristics.

The following principles of instruction are relevant in teaching Geometry and

Spatial Sense in the primary grades:

• Student talk is important. Students need to talk about and talk through mathematical concepts, with one another and with the teacher.

• Representations of concepts promote understanding and communication. In Geometry and Spatial Sense, concepts can be represented in various ways (e.g., using manipulatives, familiar objects, illustrations, diagrams). As students investigate geometric ideas, it is important that they manipulate concrete materials and do not simply view pictures and diagrams of two-dimensional shapes and three-dimensional figures. As well, students should be encouraged to make their own representations of mathematical ideas using concrete materials, pictures, and diagrams.

• Students learn through problem solving. Problem-solving situations provide opportunities for students to reason about mathematical ideas and to apply concepts and skills in meaningful contexts.

• Students need frequent experiences using a variety of learning strategies (e.g., playing games, using movement, sorting, classifying, constructing) and resources (e.g., using models of two-dimensional shapes and three-dimensional figures, geoboards, pattern blocks, or tangram pieces). A variety of learning strategies should be used in instruction to address the learning styles of different children.

• Teachers can help students acquire mathematical language by using correct mathematical vocabulary themselves. Beginning in Kindergarten, teachers should model appropriate mathematical terminology as they discuss geometric ideas with their students. They should encourage students to use mathematical vocabulary that will allow them to express themselves clearly and precisely.

Students learn about geometric properties as they view, handle, and manipulate objects (Copley, 2000). At first, students describe objects using vocabulary related to observable attributes: colour, size (e.g., big, small, long, thin), texture (e.g., smooth, rough, bumpy), movement (e.g., slide, roll), material (e.g., wood, plastic). Instruction in the primary grades helps to focus students’ attention on geometric features of two-dimensional shapes and three-dimensional figures so that students begin to think about the properties that make a rectangle a rectangle, or a cylinder a cylinder. The emphasis in instruction, however, is on developing students’ ability to analyse and describe the geometric properties of shapes and figures, not on their ability to learn the definitions.