For years geometry was not considered a topic that primary students could or should study, even though they are surrounded by geometrical shapes. Some second graders I worked with didn’t consider their geometry work to be math, until they saw numbers included in their work.

Sometimes teacher education programs provide little guidance on how to teach geometry. I know an elementary teacher who graduated recently with no geometry in either high school or college!

**Geometry and Math**

Geometry is more than quantity; geometry is about measurement and relationships between those measurements.

- To determine
**length**, we need a linear unit. - To determine
**area**, or surface space, we need to find the number of squares having sides 1 unit long that fill the area. - Then, to determine
**volume**, or space, we need to find the number of 1-unit cubes that fill the space.

Because the lengths being measured aren’t always whole numbers, fractions and decimals are needed.

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.

If the sides are 1 unit long, the diagonal is the square root of 2. If the sides are 10 times longer, the diagonal is 10 times longer than the square root of 2, which is approximately 14.142135.

Another example of these relationships 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.

For example, consider a rectangle.

First of all, what is the definition of a rectangle? It is *a quadrilateral *(four sides)* in which all four angles are right angles *(90º) *and opposite sides are parallel*.

Does the longer side of a rectangle need to be at the base, or bottom? No. Curriculum for a well-known preschool program refers to a rectangle as “long and low.” Sadly, that’s one definition that will have to be unlearned.

Can the four sides of a rectangle be congruent? In other words, can the four sides be equal? Yes.

And yes, a rectangle can be a square—albeit a special rectangle. Did you notice the definition of a rectangle says nothing about the length of the sides? It only says is that the opposite sides must be parallel.

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.

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.

Five-year-olds enjoy learning about equilateral, isosceles, and 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 two-dimensional 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.

To understand cylinders, prism, and pyramids, children need to touch and explore these objects just like Jonathan did with the sphere.

Babies are born into a 3D world. Yet electronic screens and television screens, along with workbooks, flatten children’s perceptions into a 2D experience, especially once they start school.

**Drawing Board Geometry**

Children learn by manipulating and creating new objects. Since the time of Euclid, to make geometric constructions, traditional geometers, persons skilled in geometry, use only a straightedge and compass. No measuring was permitted. Complying with these restrictions made it very difficult 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: they traced over their work with colored markers and erased all the pencil lines. Even the lines under the colored portions disappeared.

A drawback to the traditional drafting tools was that the size of the equipment was simply too big for children. The board and T-square were 2 feet wide (although is nice for a student with some dexterity issues). When daughter Connie took a drafting course in college, she was provided with a 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. I have demonstrated these activities to hundreds of teachers at conferences all across the United States.

**RightStart Math and Geometry Lessons**

These geometry activities are incorporated into RightStart Level C and Second Grade. Lines are not drawn willy-nilly. The student needs to be able to explain how they know where to draw every line. 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.

We must give children the opportunity to explore the world of geometry. This introduces them to another branch of mathematics. A child who may not excel in arithmetic may well flourish in geometry, fostering and strengthening their love for math.

Rebecca Ray says

My son, age 11, has used Right Start Mathematics since first grade. We love it! The games and manipulatives make learning easier and often fun. The short worksheets do not overwhelm and discourage him with what seem like and endless list of problems to work. I love your approach to starting Geometry early. However, this has been the most difficult thing for my son. He struggles to “see” things when we work with 2D and 3D shapes, such as different ways to divide the shapes, being able to mirror-draw, etc. The T-square and triangle frustrate him because he struggles with some fine motor skills. (He was born at 25 weeks, and has tested within normal limits of Early Intervention standards.) He is very bright, but does not enjoy the physical act of writing, coloring, cutting, etc mostly because (I believe) he lacks the same dexterity that other children his age have with fine motor skills. I normally take the geometry lessons slow, and don’t force him to use the T-square and triangle. But now I’m thinking that is a mistake! What tips can you give to help me teach him and help him learn? How can I keep him from getting so frustrated because this is a weaker area for him?

Rachel Anderson says

Hi, Rebecca.

My name is Rachel and I wanted to let you know that I sent you an email giving you some steps on how to help your son learn to work with the drawing tools. If you did not receive the email, let me know at [email protected].

Have a great day!

Rachel

Hi Rachel,

Maybe Right Start would consider doing a blog post on the email you sent to Rebecca giving steps on how to learn to work with the drawing tools? I love Right Start and it sounds like it would be beneficial for all of us to read! =)

Thanks!

Hi, Karol.

Here is the reply that I sent Rebecca. 😉

_________________

First of all, thank you for your comments. I, too, have enjoyed teaching RightStart Math to my children, including two struggling learners. 😉

Next, I want to encourage you that you are doing a great job. As a mom of struggling learners, I have learned that sometimes the lessons need to be adapted to adjust for my children’s learning abilities. As your son has some fine motor skill struggles, you have also adapted by slowing down the lessons and adjusted the lessons to fit his learning abilities.

Now that he is a little older, perhaps it is time to help him build some of his motor skills. I would recommend that you first just have him draw horizontal lines using the t-square. If you have RS2 Level C, take a look at Lesson 61. You do not even need to have the worksheet to do this type of activity. Simply have him draw horizontal lines on a blank sheet of paper. The struggle will be to keep the t-square from shifting as he draws the line. When my children worked through this activity, I would sometimes even help hold the t-square in place until they were comfortable with the practice. Feel free to do the same to help your son with this skill.

If your son does well with the horizontal lines, then move on to vertical lines. You can take a look at Lesson 62 in Level C for instructions on that skill. Again, don’t worry about having the worksheet. Simply use a blank sheet of paper. That way, if your son is a perfectionist, he will not be given ‘extra stress’ by trying to match the lines up.

If all goes well in those two lessons, then continue to work on diagonal lines as shown in Lesson 63.

If your son continues to do well in these activities, continue through the Geometry lessons in Level C until your son demonstrates frustration. Then pull off for a while until you feel he may be willing to try again. This may be a couple months to a couple of years, depending on the level of his motor skills struggle. 😉

I hope that gives you a place to start. Of course, if you have any further questions, please do not hesitate to email me back.

_____________________

If you have a specific situation you would like assistance with, please do not hesitate to email us at [email protected]. We are here to help you!

Rachel