*By Joan A. Cotter, Ph.D.*

For years geometry was not considered a topic that primary students could or should study, even though they are surrounded by geometrical shapes. I had no geometry in school until 10th grade. Sometimes teacher education programs provide little guidance on how to teach geometry. I know an elementary teacher who graduated about ten years ago with no geometry in either high school or college.

**Geometry and Math
**Some second graders I worked with didn’t consider their geometry work to be math until they saw numbers. Yet, geometry is more than quantity; geometry is about measurements and relationships. To determine length, we need a linear unit. The same unit is used to determine area, or surface space, by finding the number of squares having sides 1 unit long that fill the area. Likewise, the same unit is used to determine volume, or space, by finding the number of 1-unit cubes that fill the space.

Because the lengths being measured aren’t always whole numbers, fractions and decimals were needed. Also, some relationships introduced other new numerical values. There is no unit that will produce whole numbers—not even fractions or decimals—for both a side and a diagonal of a square. The diagonal in a square equals the length of a side times the square root of 2, which is approximately 1.4142135.

Another example lies in a circle. The diameter fits around the circumference a little over 3 times. Since this ratio is not an exact fraction or decimal, it is given a special name, pi.

**Teaching Geometrical Names
**Today, most standards for the earliest grades recommend teaching the names of common geometrical figures and objects. This is not as simple as it sounds. Take for example a rectangle. Does the longer side need to be at the base, or bottom? No. Can the four sides be congruent (the geometrical word for equal)? Yes. However, the curriculum for a well-known preschool program refers to a rectangle as “long and low.” That’s one definition that will have to be unlearned. And yes, a rectangle can be a square—a special rectangle.

Triangles frequently suffer the same fate as rectangles. Children are often presented with an equilateral triangle as a model. Andy, 5, was adamant that a triangle must have all the sides equal. Also, triangles are often shown as having one edge parallel to the bottom of the page or table. Children need to see all types of triangles in all types of orientation. And 5-year-olds enjoy learning about equilateral, isosceles, scalene triangles, as well as acute, right, and obtuse triangles.

A more difficult problem for some children is distinguishing between two- and three-dimensional figures. I showed Jonathan, 4, a paper plate and reminded him it was a circle. Then I showed him a ball and told him it was called a sphere. He knew they were both round, but he really had to think about the sphere.

Until I was about 9 years old, because of an eye problem, I lived in a 2D world. I remember asking people how can bricks turn a corner; I didn’t understand that bricks had depth as well as width and height.

Babies are born into a 3D world. But, electronic screens and television along with workbooks flatten children’s perceptions into a 2D experience, especially when they start school. To understand cylinders, prism, and pyramids, children need to touch and explore these objects just like Jonathan did with the sphere.

**Drawing Board Geometry
**Children learn by manipulating and creating new objects. Since the time of Euclid, to make geometric constructions, traditional geometers use only a straightedge and compass. No measuring was permitted. Complying with these restrictions made it impossible for children to delve into the wonders of geometry.

As a freshman taking an engineering drafting course, I was introduced to the drawing board and tools: T-square, triangles, and compass. When my daughters were around 7 and 8, I located the equipment, substituted a simpler compass, and showed them how to make some constructions. To add some color to their work, they did the following. Before erasing the unwanted lines, they traced over their work with colored markers and then they erased all the pencil lines. Even the lines under the colored portions disappeared.

A drawback was that the size of the equipment was simply too big for children. The board and T-square were 2 feet wide. When daughter Connie took a drafting course, she was provided with an additional smaller portable drawing board and tools for doing homework on standard-size paper. When I saw this equipment, I asked her if I might borrow it and try it with my Montessori kindergartners.

First, I showed the 5- and 6-year-olds how to construct equilateral triangles. Next they divided them into halves several different ways and wrote the fraction on each half. The children continued by dividing equilateral triangles into thirds, fourths, sixths, eighths, ninths, and twelfths. Some children also did 18ths and 36ths. One girl, Stephanie, actually divided that triangle into 256 equal parts!

The children also constructed hexagons, stars, squares, and inscribed squares. At a number of conferences, I demonstrated these activities to hundreds of teachers.

**RightStart and Geometry
**These geometry activities are incorporated into RightStart Grade 2 and Level C. The student needs to be able to explain how they know where to draw every line. Lines are not drawn willy-nilly. This practice of justifying every step is consistent with the whole field of mathematics.

After a number of years, teachers asked if I could write more advanced lessons. About the same time, I realized that much of the math taught at the middle school level could be approached either algebraically with lots of equations or geometrically with visual representations. Since most preteens prefer visual information, I spent four years writing the RightStart middle school curriculum. These lessons are written to the student. The intent is to encourage them to learn how to read a math textbook and work through the mathematics more independently. Ω