# Classroom Case Studies, 6-8

## This is the final session of the Geometry course! In this session, we will examine how geometry as a problem-solving process might look when applied to situations in your own classroom. This session is customized for three grade levels. Watch Videos 11 and 12 in the 10th session for grade 6–8 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 6–8 teachers (former course participants) who have adapted their new knowledge to their classrooms.

Part A: Geometry and Reasoning
Part B: Developing Geometric Reasoning
Part C: Problems That Illustrate Geometric Reasoning
Homework

In the previous sessions, we explored geometry as a problem-solving process. You put yourself in the position of a mathematics learner, both to analyze your individual approach to solving problems and to get some insights into your own conception of geometric reasoning. It may have been difficult to separate your thinking as a mathematics learner from your thinking as a mathematics teacher. Not surprisingly, this is often the case! In this session, however, we will shift the focus to your own classroom and to the approaches your students might take to mathematical tasks involving geometry.

As in other sessions, you will be prompted to view short video segments throughout the session; you may also choose to watch the full-length video for this session. Note 1

### Learning Objectives

In this session, you will do the following:

• Explore the development of geometric reasoning at your grade level, including the van Hiele model of geometric learning
• Review mathematical tasks and their connection to the mathematical themes in the course
• Examine children’s understanding of geometric concepts

### Key Terms

Previously Introduced

Congruent: Two figures are congruent if all corresponding lengths are the same, and if all corresponding angles have the same measure. Colloquially, we say they “are the same size and shape,” though they may have different orientation. (One might be rotated or flipped compared to the other.)

Convex Polygon: A convex polygon is any polygon that is not concave.

Platonic Solid: A Platonic solid is a solid such that all of its faces are congruent regular polygons and the same number of regular polygons meet at each vertex.

Rectangle: A rectangle is a quadrilateral with four right angles.

Square: A square is a regular quadrilateral.

Triangle Inequality: The triangle inequality says that for three lengths to make a triangle, the sum of the lengths of any two sides must be greater than the third length.

New in This Session

Van Hiele Levels: Van Hiele levels make up a theory of five levels of geometric thought developed by Dutch educators Pierre van Hiele and Dina van Hiele-Geldof. The levels are (0) visualization, (1) analysis, (2) informal deduction, (3) deduction, and (4) rigor. The theory is useful for thinking about what activities are appropriate for students, what activities prepare them to move to the next level, and how to design activities for students who may be at different levels.

### Notes

Note 1

This session uses classroom case studies to examine how children in grades 6-8 think about and work with geometry. If possible, work on this session with another teacher or group of teachers. A group discussion will allow you to use your own classroom and the classrooms of fellow teachers as case studies to make additional observations.