Home Chapter 21 Summary

# Summary

#### MAIN IDEAS

There is no difficulty identifying the main idea that I want to get across to teachers in this chapter: Geometry can be one of the most fun, exciting, engaging, profitable, and enticing areas of the curriculum if we only let it. (You may want to add your own adjectives to this list.) It is, however, non-trivial to make this point within the limited time allotted by a methods class. As described on page 408 and briefly listed on Transparency T-118, there are two types of goals that all teachers should be aware of: Growth in geometric thought and growth in understanding of geometric content. The issue of geometric thought is reflective of the van Hiele theory. Geometric content has always been a more difficult concern since there is very little consensus about what content should be taught and at what grade level. To help your teachers with a perspective on content development, I have turned to the Principles and Standards document. There the authors describe four goals for geometric content: 1) shapes and properties, 2) location (essentially coordinate geometry), 3) transformation, and 4) visualization.

#### The van Hiele Theory

It is quite valuable to help teachers try to understand something of the van Hiele theory, the characteristics of the levels, and implications for instruction. Without this theoretical framework, they have very little to guide their interaction with students and to help them progress in their thinking.

One aspect of the van Hiele theory that I have come to see as important in terms of daily instruction is the notion that the product of thought at one level becomes the object of thought at the next. To the extent that teachers can grasp that idea, it sends a message about working where students are while simultaneously prodding them forward. Level 0 children need not do trivial activities. Transparency T-117 includes a copy of Figure 21.3 illustrating this progression of objects and products is included here.

#### Geometric Content and Organization of the Chapter

I understand that you are not teaching a content course, but it is almost certain that your teachers have a very weak background in geometric content. Until NCTM's Principles and Standards was published, I was not sure what content to explore in my class. The Standards authors organized geometry in four content areas which I think are quite useful. Therefore, following the discussion of the van Hiele theory are four sections containing activities for each of the four areas. Within each, the first three Van Hiele levels further organize the activities as illustrated in the following table.

 Shapes and Properties Location Transformations Visualization Level 0 Level 1 Level 2

This organization allows your teachers to easily find a content area and explore it across the van Hiele levels, or simply look at activities for a level of their interest. You of course can do the same as you make choices for what activities to do in class. [My apologies to those who just got familiar with the fifth edition in which the organization was horizontal across this table. I hope you find this to be an improvement.]

#### Materials, Activities, and Resources

To the extent that you are able in a short period of time, you want to expose your students to as many materials and activities as possible. The best thing you can do is select activities that are easily reproducible, modifiable to a wide grade range, use a popular set of materials, and are minimally threatening to teachers who are insecure about their own content base. How you can share a lot of things and still demonstrate a spirit of inquiry is a true dilemma. You may only do one or two small activities with geoboards, for example, but a real activity is better than holding up a geoboard but never using it. It is probably also an error to spend a lot of time with materials to which teachers are not very likely to have access.

By all means, explain to your students that this rather long chapter is about 80% a resource. I do not require my students to "know" any particular activity found there. I believe they should have some speaking knowledge of the four areas of geometry and probably know at least several representative activities in each. They will have the book as a reference and as a resource for both teaching and learning. Encourage teachers to grow over time with respect to geometry. The organization of the chapter should allow them to select activities that best suit their needs.