Children begin learning about geometry as soon as they are able to experience the shapes and spaces in the world around them. Their early school experiences should build on the understandings they bring and help students begin to develop more formal geometric thinking.
Geometric reasoning involves thinking systematically about shape and space. Initially, students need to explore and discuss both two- and three-dimensional shapes so that they can identify and use these shapes according to conventional terminology. They should have many opportunities to explore a variety of examples and non-examples through drawings, pictures, blocks, paper folding, dot paper, geoboards, and everyday objects. They can begin to understand connections between shapes by decomposing (e.g. splitting a rectangle into two triangles) and composing (e.g. finding all the possible shapes they can make from two triangles).
Initially students learn to identify shapes they know by the overall way the shape looks, rather than by the properties of the shape. For example, students may be able name squares as such when they have horizontal and vertical sides, but when the square is presented with sides tilted diagonally, they may not recognize them as squares and instead call them diamonds. A key focus thus is to help students begin to focus on properties of the shapes they know and learn to identify shapes by their properties. Thus, as students learn that squares have four sides that are the same length and four right angles, they can recognize squares, even when they are oriented in unfamiliar ways. This focus on the properties of shapes, rather than solely on their overall appearance, sets the stage for later learning about how geometric properties are interrelated and the ability to reason about geometric shapes in abstract ways.
In addition to exploring shapes and their properties, students should have opportunities to investigate how shapes exist and move in space. As students learn about a space, they first begin by noticing and recognizing landmarks, then build knowledge of routes through that space, and finally learn to coordinate landmarks and routes into a mental map of the space. Students should have opportunities to practices these phases, for example, asking them to create a map of the classroom and mark different routes from the doorway to the reading station in the far corner. Learning about space and location is also a natural venue for connections to other subject areas, for example, ask students create maps based on books they are reading.