Module 9

Hey Jess! I hope you're having a great week. I really enjoyed this module, especially the case studies! They really opened my eyes to struggles that I never even realized students may face.

Quick Images Video
I did the activity along with the students, drawing what I saw, and I noticed that the ways they used to describe how they remembered them were similar to how I did. For example, with the first image, I thought of a crescent moon with a circle in the center. This is very similar to what the students said. However, some of the students thought of the most random things, like the letter C, a banana, or a jet ski. These are things that they can relate to in their own lives that they are reminded of when they see this image. Many others would have never connected this image with a jet ski. Some of the answers that followed these, though, were more geometrical in nature, like saying that it is a portion of a circle and that it has two points. You could tell if the students used real-life connections to remember this shape, or if they made geographic connections to remember this shape. One student, Eric, was very impressive because he was able to define this shape as a crescent and note that it has two curved sides. This definition seemed advanced for his age.

Shapes and Geometric Definitions

These case studies made me really think about children's views that I have never really considered. We as adults know what makes a triangle and we do not have a set idea in our mind that one type of triangle is correct, but these kids do. Below, I have listed my personal definitions for the geometric shapes discussed:

A triangle is a closed shape with three sides and three corners. 

A rectangle is a closed shape with four sides and four corners. It has to be made up of two sets of parallel lines of the same length, but not all four sides need to be the same length.

Creating my definitions seemed to be a little harder than I thought. I believe that I inherently know the definitions, but as I read further into the case study, I felt the need to add rules, such as a triangle's lines must be straight. Did you find it a bit difficult to come up with concrete definitions? 

This kind of leads into the next question, about definitions versus properties. I believe that a definition is an umbrella that all of the types of that shape must fall under, while the properties are certain aspects that can vary. The purpose of a definition is to see whether a shape applies to that title or not, as if it is the set rule. For example, a shape can have similar properties to a triangle, but if it does not have only three sides and three corners, it does not apply as a triangle.

The difference in definitions and properties is something that could help students understand why they think of things so specifically. Just because a shape does not have the properties they are comfortable with seeing does not mean that it is not that shape. If it fits the definition or rule, then it applies. The students go through a long process to discover why they feel that certain shapes are "real" while others are not. In doing this, they are building a greater understanding of the shape as a whole and the reasons they believe things to be true. Were you surprised by any of the specific comments the students made? In Andrea's case, Zachary's comment was very insightful, but I loved what Even said about if he were all stretched out and upside down, he would still be Evan. By learning to apply shapes they are not comfortable seeing to a specific definition, they are learning that just because something looks off to them, it does not make it any less real or true.

In Andrea's case, Susannah seemed very confused. She was confident in what she was saying, but she just could not comprehend that a triangle that did not look correct to her could still be a triangle. She wanted to help and give input, but the things she was saying just weren't true, such as shapes being too long or narrow to be triangles, or that it should be a rule that a triangle must have two "slanty sides." I wish her written conclusion would have been included so we could see where she ended in this debate. While reading through Andrea's case, I loved the comments Evan made. He always seemed to want to relate these triangles to himself. He kept assuring the class that, just as the triangles, if you turned him a certain way, he was still Evan. He clarified this thinking more in his written definition, where he said that you can turn a triangle any way and it still makes a triangle. Dolores's case is very similar to Andrea's, as they are both trying to figure out why their students are so set in their way about what makes a particular shape a shape. I have already shared some comments that surprised me from Andrea's students, but in Dolores's case I was shocked at the students, like Francie, who said that the triangle was not a shape at all! Not only did she not view it as a triangle because it did not have properties that she recognized, but she was not comfortable enough to label it as a shape whatsoever. 

In Natalie's case, the students are learning what sets aside squares and triangles from all figures with four sides and four corners. They seem to have a good start, but there are a lot of things to cover still. They need more specific definitions, like quadrilateral, that can help them separate these shapes. They also do not seem comfortable labeling a square as a type of rectangle. Their discussion after Roberto's comment shows that definitions have to be specific enough to differentiate shapes from other shapes, but broad enough to fit all types of that shape. 

Math Activity with Color Tiles

I am doing this week's blog from out of town, so I do not have my manipulative kit, but I completed this on the computer so you can see the picture below:

I missed one of the four tile shapes, and I did not find any ten or twelve sides until seeing it on the PowerPoint. I did remember most of the names, though. I could only find 4 shapes with 5 tiles.

For Further Discussion...

I made a point to try to think of things in a geometric way today for this portion of the blog. As I was driving, I was thinking about how the roads connect and what possible shapes they could make. I looked at my house and noticed that most of my rooms were square or rectangular, which is pretty normal, but some rooms had 5 sides and one of my spare bedrooms has six. I have a shelf that makes an elongated triangle and much of my décor is rectangular. While I was taking the time to purposefully think this way, I was thinking about how there are some people who's mind work this way every single day. They think of everything mathematically and this is the way they see the world. It was very interesting to reflect in this way. Every single thing is a shape of some sort. Tires on the car are cylinders, some fruit is spherical. There is geometry all around us if we take the time to look for it. Comparing shapes to common things like this can help us teach our students and allow them to make connections.