by Glenda Lappan, NCTM President 1998-2000
NCTM News Bulletin, December 1999
The foundations for geometric thinking are laid early. Basic shapes, structures, and locations, along with colors and sounds, are some of the first things infants learn to recognize. From day one, infants begin to struggle to make sense of objects--faces, hands, feet, toys, bottles, and so forth--and relationships among them.
As children grow, they encounter two-dimensional representations--"pictures"--of familiar three-dimensional objects. With experience they learn to "read" these pictures, understand what they represent, and how those representations are related to the objects they represent. Children also learn to make their own such representations to capture and show their ideas to others. This is an early stage of learning how to interpret and communicate spatial information.
These early experiences vary from child to child. Some learn how to think with and reason about spatial information with little effort. Many do not, yet research shows that we can improve students' knowledge and ability to visualize and reason about the spatial aspects of the world in which they live. But the troubling question is, do we?
Both the Third International Mathematics and Science Study (TIMSS) data from 41 nations and our own National Assessment of Educational Progress (NAEP) data show that geometry is an area of dismal performance for our students at all levels. For example, the TIMSS results for geometry at grade 8 show that 24 nations scored significantly higher than U.S. students and only four nations scored significantly lower. The international average on geometry at grade 8 was 56 percent correct. U.S. students scored 48 percent correct, on average. Contrast this with fractions and number sense where the United States was slightly above the international average of 58 percent correct with only 13 nations scoring significantly higher. Although this is not good enough, it is a great deal better than we do in geometry.
So what do we do about this? We must build a geometry strand that engages our students in this interesting and important area of mathematics throughout their school experience, from pre-K through grade 12. NCTM's updated Standards document, Principles and Standards for School Mathematics,* gives us the tools to do this.
The Geometry Standard outlines the foundation of substantive geometry learning across the grades, from pre-K through grade 12. At each of the four grade bands of Principles and Standards, the Geometry Standard includes specific expectations. These expectations allow teachers and others to examine their own frameworks and curricula to see whether they provide the experiences students need to achieve the overall goals of the Standard. These expectations are more specific than those implied in the NCTM 1989 Standards documents and should help us to build an excellent geometry strand for our students.
The Geometry Standard covers the skills and concepts of visualization, spatial reasoning and representation, and analyzing characteristics and properties of two- and three-dimensional shapes and their relationships. It also includes locating objects in space and describing changes in objects under such transformations as translation, rotation, reflection, and dilation.
All this wonderful mix of identifying and describing, moving and transforming geometric objects is a part of geometry right along with argument and proof. As recommended in the strand, activities such as examining enlargements and other kinds of transformations of geometric objects will help students explore important geometric ideas. For instance, are two geometric objects congruent? Are they similar? If so, why? If not, why not? If we know certain properties of a geometric object, what else can we infer? What happens if we take a line or curve and rotate it in space? What sort of object does it sweep out as it rotates? Can we describe the curve and the object it sweeps out in mathematical terms? With questions like these, not only are we exploring geometry, we are connecting to other areas and levels of mathematics.
But we cannot get to this level unless we give students the time throughout their school mathematics to explore geometry to its fullest--to play, observe, analyze, conjecture, imagine, represent, transform, create arguments, and simply experience the beauty and joy of geometry.
We have a beautiful, important strand of mathematics that has too often not been given appropriate attention in the curriculum. Many of our students who have difficulty with number shine in geometry. We know that the ability to look at situations geometrically, spatially, and analytically enhances our understanding and our problem-solving success. We owe it to our students to integrate geometry into all grade levels of our programs. Let's make geometry integral to our curriculum so that all students are given the tools to make sense of--and be successful in--the three-dimensional world in which they live.?
* Principles and Standards for School Mathematics outlines the foundations of excellent pre-K–12 mathematics education programs and extends the vision of NCTM's three Standards volumes: Curriculum and Evaluation Standards for School Mathematics, Professional Standards for Teaching Mathematics, and Assessment Standards for School Mathematics. It will be released on 12 April 2000 at the opening session of NCTM's 78th Annual Meeting in Chicago.