Geometric Definitions Case Studies

Susannah knows that triangles have 3 sides and 3 vertices. I am thinking though that in her head she visualizes what the standard triangle looks like. If a triangle fits or is close to her vision she considers it a “real” triangle. If is too different than she places it in the triangle family, but it is not a “real” triangle. She makes clear that her definition of “real” triangles are similar to triangle B. Near the end of the discussion she is proposing the idea of turning the triangle to see the sides in different ways, leading another student to understand that no matter which way you turn a triangle it is still a triangle.

In Mrs. Rivera’s class the students are working on developing definitions of a square and a rectangle. They are using square as part of their rectangle definition. They state “two squares make a rectangle.” They have not yet figured out the distinguishing quality between a rectangle and a square. At the end Brett is becoming close because he is turning an orange square pattern block around and around in his hand. He is thinking through what makes that shape different from a rectangle. Roberto’s definition makes it clear that he is at a level one on the Van Hiele Model. He is using properties to describe a rectangle. The students are describing the properties that are visual and describe a rectangle, but they are not considering what makes it different from a square. They are engaging in productive talk though that is leading them closer and closer to discovering the difference between the two shapes. Once they can see that I think it will be easier for them to come to a clear definition of both shapes.

I was especially intrigued by Zachary’s comments as it seems to be very insightful to me. He seems to really understand the properties of a triangle and is articulating this thoughts very well. I think that his point is very valid and reflects the thinking of many of the students in that class and many other classes as well. I like how he is trying to a conscience effort to make sure his thoughts coincide when he says “we just have to make our eyes and our heads meet.” In Dolores’s study the students are making it clear that they understand what a triangle is, but because they are not use to seeing it that way it is much harder to classify irregular triangles, as triangles.

The purpose of a definition is to give us something to go by when trying to classify shapes correctly. It is gives us an understanding of if a shape meets the classifications to be called by a certain name.