Geometric Thought in Elementary School

Geometry is a part of our curriculum in Kindergarten through Fifth Grade. In Kindergarten, one of the standards reads, The student describes shapes and space, and uses basic shapes, spatial reasoning, and manipulatives to model objects in the environment and to construct more complex shapes and in Fifth Grade two of the standards read, The student describes draws, identifies, and analyzes two- and three-dimensional shapes and The student uses appropriate geometric vocabulary to describe properties and attributes of two- and three- dimensional figures (Example: obtuse & acute angles, equilateral, scalene, & isosceles triangles).

As children's' understanding of Geometry develops in elementary school, they move through three of van Hiele's five levels of Geometric Thought (Van de Walle, Teaching Student-Centered Mathematics). Visualization, Level 0, is when students recognize a shape based on its appearance. At this stage, students recognize a rectangle because it looks like a rectangle. As a student's knowledge develops and they move into Level 1, Analysis, they are able to understand classes of shapes. They don't just recognize a rectangle on its appearance, but are able to say it is a rectangle because it has four sides, four right angles, and opposite sides are parallel. Knowing the properties of the shapes gives the shape its name. When students move into Informal Deduction, Level 2, they are able to see the relationships and connections among the properties of a class of shapes. They would be able to conclude that a square is a type of rhombus because rhombi are polygons with four equal sides and opposite sides are parallel and congruent. A square fits this definition, therefore a square is a type of rhombus. Students in elementary school work through these stages, and then move into Level 3, Deduction, and Level 4, Rigor, later in their school career.

For our March Standard Snapshot, Kindergarten and Fifth Grade teachers used work from their Geometry unit to share with parents. The Kindergarten students were in the beginning of their unit and the Fifth Grade students were in the middle of their unit.

Kindergarten Student Sample

You will notice in this Kindergarten sample that the student has mastered Level 0, Visualization, because they are able to name the shape, a rhombus, based on its appearance. They have not proven, through this assignment, that they have mastered the Kindergarten standard The student describes shapes, and uses basic shapes, spatial reasoning, and manipulatives to model objects in the environment and to construct more complex shapes. The sample shows the naming of the basic shape and the model object, "a braslit," but the student has not described why the shape is a rhombus. To see if the student has developed understanding of Level 1, Analysis, the student would have to know that the shape is a rhombus because it is a four sided shape (a quadrilateral) with 2 pairs of parallel sides and four equal sides. This level of understanding is not expected of a Florida student until Grade 3.

By Fifth Grade, students are expected to move into Level 2, Deduction, where they identify the shapes within a class, but also can see the relationships between and among the properties.

(Click on the work to make it larger.)

Fifth Grade Student Work

Questions on the Student Sheet:

1. A square is a kind of rhombus. How can this be?

2. Name all the shapes above that are parallelograms. How can they be parallelograms and have other names as well?

3. An equilateral triangle is isosceles. How can this be?

4. Some obtuse triangles are scalene. Some obtuse triangles are isosceles. Sketch one or two examples of each.

5. Obtuse triangles cannot be equilateral. Explain why this is true.

This Fifth Grade assignment, Some Shapes Fit Many Categories, asks questions to assess students' geometric thought. Students who master this assignment are able to identify, draw, and analyze shapes, and are able to make connections about the relationships among their properties. They have moved into Level 2 of Geometric Thought. Students who have not yet mastered the assignment are likely at Level 1 of Geometric Thought, Analysis. After the teacher analyzes the student work he/she will be able to assess which students need more exploration with the concept through small group instruction.

When the Standard Snapshots were turned in, I was intrigued to see that both Kindergarten and Fifth Grade were sending home work in Geometry. It got me thinking about where students begin their geometry journey and where we expect them to be when they leave elementary school. It also got me thinking about the process that occurs for that learning, and the key role that each teacher plays with instructional delivery along the way. In addition, it made me go back into the standards to explore the vertical alignment from K-5. I came across a document that aligns the comparisons and is an easy to read resource for teachers. They are able, through this one document, to see what students learned previously and what students must be able to master during instruction in their classroom. It reaffirmed for me the importance of each piece of the puzzle in their journey, so students master the necessary skills and concepts before they emerge as middle schoolers.