Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

Session 10, Part B:
Developing Geometric Reasoning (40 minutes)

In This Part: Introducing van Hiele Levels | Analyzing with van Hiele Levels

The National Council of Teachers of Mathematics (NCTM, 2000) identifies geometry as a strand in its Principles and Standards for School Mathematics. In grades pre-K through 12, instructional programs should enable all students to do the following:

 • Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships • Specify locations and describe spatial relationships using coordinate geometry and other representational systems • Apply transformations and use symmetry to analyze mathematical situations • Use visualization, spatial reasoning, and geometric modeling to solve problems

In grades pre-K-2 classrooms, students are expected to do the following:

 • Recognize, name, build, draw, compare, and sort two- and three-dimensional shapes • Describe attributes and parts of two- and three-dimensional shapes • Investigate and predict the results of putting together and taking apart two- and three-dimensional shapes • Recognize and apply slides, flips, and turns • Recognize and create shapes that have symmetry • Create mental images of geometric shapes using spatial memory and spatial visualization • Recognize and represent shapes from different perspectives

Dutch educators Pierre van Hiele and Dina van Hiele-Geldof developed a theory of five levels of geometric thought. It is just a theory, but a useful one for thinking about activities which are appropriate for your students and prepare them to move to the next level, and for designing activities for students who may be at different levels.

Level 0: Visualization. The objects of thought at level 0 are shapes and what they look like. Students have an overall impression of the visual characteristics of a shape, but are not explicit in their thinking. The appearance of the shape is what's important. Students may think that a rotated square is a "diamond" and not a "square" because it looks different from their visual image of square. (Early elementary school and, for some, late elementary school)

Level 1: Analysis. The objects of thought here are "classes" of shapes rather than individual shapes. Students are able to think about, for example, what makes a rectangle a rectangle. What are the defining characteristics? They can separate that from irrelevant information like the size and the orientation. They begin to understand that if a shape belongs to a class like "square," it has all the properties of that class (perpendicular diagonals, congruent sides, right angles, lines of symmetry, etc.). (Late elementary school and, for some, middle school)

Level 2: Informal Deduction. The objects of thought here are the properties of shapes. Students begin "if-then" thinking; for example, "If it's a rectangle, then it has all right angles." Students can begin to think about the minimum information necessary to define figures; for example, a quadrilateral with four congruent sides and one right angle must be a square. Observations go beyond the properties into mathematical arguments about the properties. Students can engage in an intuitive level of "proof." (Middle school and, for some, high school)

Level 3: Deduction. The objects of thought here are the relationships among properties of geometric objects. Students can explore relationships, produce conjectures, and start to decide if the conjectures are true. The structure of axioms, definitions, theorems, etc., begins to develop. The students are able to work with abstract statements and draw conclusions based more on logic than intuition. (This is the goal of most 10th-grade geometry courses, but many students are not developmentally ready for it.)

Level 4: Rigor. The objects of thought are deductive axiomatic systems for geometry. For example, students may compare and contrast different axiomatic systems in geometry that produce our familiar Euclidean plane geometry, finite geometries, the geometry on the surface of a sphere, etc. Note 4

 For more information on the van Hiele levels and how to work with students within each level, read the article "Geometric Thinking and Geometric Concepts" by John A. Van de Walle from Elementary and Middle School Mathematics.