## Teaching Geometry

Recently I have read two different articles that have challenged my approach to teaching geometry. “Young Children’s Idea about Geometric Shapes” and “The Earliest Geometry” were both written by Douglas H. Clements and Julie Sarama. When I am not away at college, I work at IXL Learning Center in my hometown. I have worked there since high school and I can honestly say that I have been teaching geometry all wrong.

Believe me when I say that nearly all of the kids at the center can recite the name of any shape I show them, but after reading these articles I wonder if they really understand the difference between shapes. Do they only know the shapes because we as teachers show them the same example of a triangle each day? Do they truly know what a trapezoid is, or do they only memorize the name that goes with the cute, purple shape with a smiley face on it? Of course they are given geometric manipulatives to use during their daily centers, but still I find myself questioning if they understand the characteristics of each shape. I’ll ask them how they know that a triangle is in front of them and they typically answer the same way, “It has three sides and three points.” This is where I make another mistake, I assume that they know what a side or angle really is. Instead of questioning them to deepen their understanding, I congratulate them on their answer that they have memorized. Have they ever been taught what an angle even is? The answer is most likely no.

Since these kids were two years old, we as teachers encouraged straight memorization. I mean, a two year old is not going to understand that a trapezoid has two parallel sides. So, we show them the same pictures of shapes that they will see every day all the way up until they graduate from Pre-k. By the time they are five years old they are expected to perfectly recite the name of every shape, and all of the teachers conclude that they have mastered pre-kindergarten geometry. Is it fair to say that they have mastered shapes when they fail to identify a different example of a triangle?

What is really unfair is that as they move into kindergarten, they are already expected to know the different shapes. According to Clements, “By the time they enter kindergarten, most children have many ideas about shapes. Yet teachers often do not ask children to extend their ideas.” As you can see, these articles really had me questioning how shapes are taught. Young students have yet to be taught what parallel lines and congruent sides are. Clements also explains that many of our perceptions of shapes solidify by the age of six. However, at six years old we have not learned what characteristics classify each shape. After it is determined that a child can recall the name of each shape, it is expected that they know this in later grades and the idea is not really revisited until more complex ideas are introduced. I recall learning angles and congruence around sixth grade. At that time, I remember seeing different examples of each shapes. It wasn’t until years after learning shapes that I learned the difference between isosceles, scalene, and equilateral triangles. I had never known that they existed in my lower elementary education. I thought an equilateral triangle was they only way a triangle could look like. Perhaps this is why it can be sometimes be difficult for us as college students to identify different quadrilaterals.

It all starts before our elementary education. So, this brings me back to teaching at the learning center. What we assume to be enhancing their knowledge is actually contributing to the problem. The lessons I have been teaching and help teach are actually setting up students for confusion later on.

One of the best parts about Clements’ writing is that he provides various methods and activities to further students’ thinking. He talks about ways students can picture being in a shape or identifying a shape by feeling it instead of looking at it. Many of his activities that I would consider using and seem to be very beneficial for students.

In class, we created multiple representations of the eleven different types of quadrilaterals. Originally, we created some of geoboards; eventually, we drew them on dot paper and used them for games such as matching and go fish. Some of the examples are included below.

Instead of geoboards, I found a way to use popsicle sticks to create shapes on ABeeCpreschool.blogspot.com. Students can create other representations of triangles and quadrilaterals. Teachers should encourage creativity to create shapes that fit the qualities of a shape but appear differently than the standard representation.

Clements mentions providing examples and non-examples. In another education course of mine, I learned how to create and implement concept diagrams. I have included an example below that compares and contrasts several types of shapes.

As I mentioned before, the few weeks of this class and the readings we have done so far have already changed my perception of teaching geometry and mathematics in general. When I head back to the learning center I will have a new understanding of what tools children need to learn shapes. Instead of asking a child what shape I am showing them and being satisfied with a memorized answer, I will question them and ask them to explain their thinking. I plan to introduce new visual representations of each shape so that when I hold up a triangle with the point facing down I will not receive the answer, “It’s an upside down triangle.”