 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 6-8 classrooms, students are expected to do the following:

 • Precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties • Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects • Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship • Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling • Examine the congruence, similarity, and line or rotational symmetry of objects using transformations • Draw geometric objects with specified properties, such as side lengths or angle measures • Use two-dimensional representations of three-dimensional objects to visualize and solve problems such as those involving surface area and volume • Use geometric models to represent and explain numerical and algebraic relationships • Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life

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 that 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 minimal 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. 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.

 Van de Walle, John A.. Geometric Thinking and Geometric Concepts (2001). In Elementary and Middle School Mathematics: Teaching Developmentally, 4th ed. (pp. 342-349). Boston: Allyn and Bacon. Reproduced with permission from the publisher. Copyright © 2001 by Pearson Education. All rights reserved. Principles and Standards for School Mathematics (Reston, VA: National Council of Teachers of Mathematics, 2000). Standards on Geometry: Grades 6-8, 41, 232. Reproduced with permission from the publisher. Copyright © 2000 by the National Council of Teachers of Mathematics. All rights reserved.   Session 10, Grades 6-8: Index | Notes | Solutions | Video