# The Shape of Space

Children can develop fairly sophisticated ideas about spatial visualization and spatial patterns during the primary grades. Many people think that skills so essential to success in a number of careers - carpentry, architecture, statistics, mechanics, engineering, map making, surveying, and design - are simply inherited. We have learned that children can develop and improve these skills when given the opportunity to do so.

Patterns consist of repeated elements. One of the most common spatial patterns that we experience and use are shapes. For example, a simple shape like a square contains repeated elements - a side and a right angle, repeated 4 times. Children's reasoning about shape generally follows a predictable sequence. Awareness of this typical sequence helps adults understand how a child's growth is progressing and what to expect next. However, the sequence is not lock-step; children often reason in all of these ways at the same time, but most of their reasoning will be at one of these levels.

#### 1. Resemblance.

Children begin to reason about shape by resemblance. In other words, unfamiliar shapes are identified by their resemblance to familiar shapes. For example, children often describe a rectangle as a "pushed-out square," a triangle as a steeple, and so on.

#### 2. Attributes.

As children develop a vocabulary about shape, they begin to view shapes as having characteristics or attributes. Often, children will notice that some shapes have "square corners" (like those of a square), and others, "pointy tips" (like those of triangles containing less than 60 degrees angles). When children first reason about attributes, they often don't make connections among them. For example, seeing a six-sided shape, children may notice that there are six sides ("edges"), but must count the number of angles ("corners") to determine their number. As children solve problems, they begin to coordinate attributes, so that if a closed shape has four sides, then they also know that it has four angles.

#### 3. Properties.

When children reason about shapes via resemblance or attributes, their thinking is guided primarily by what they see. With experience talking about and drawing shapes, they can think about shapes as collections of properties. They can then discover rules about what makes something a shape. For example, they reason that a square is shape with properties of closure, equal sides, and 4 right angles, so that any shape that they can imagine (even if they don't actually see it) with these properties is a square. So, they can reason that the two figures drawn below are both squares, even though one is "tilted." (Children often think of the tilted square as a "diamond.")

It is very important for children to discover such properties and be able to talk about them. One of our goals is to help children develop rules about shapes, like "triangles have 3 sides and 3 corners (angles)". By developing these rules, children learn to make mathematical arguments and to apply their knowledge to shapes they have never seen before. Geometry is a form of reasoning that people have developed to describe and discuss space systematically. When children understand shapes as collections of properties (rules) they are on their way to higher level thinking and reasoning.

#### Building Knowledge About Shape

Children's experiences and their conversations about shapes help them understand patterns and relationships. One way to develop knowledge about shapes is to talk about how shapes are alike and different. This is the way that we focus on in the homework accompanying this newsletter.

Another way for children to develop knowledge about shape is to think about shapes as paths - as a series of moves and turns.

For example, you can walk the path of a square by moving forward 10 paces, then turning right 1/4 of a turn, then moving forward 10 paces, turning right 1/4 of a turn, and so on.

Remember that in one of the previous newsletters, you worked with your child to write directions for a path between two landmarks (e.g., your home and a friend's home). Your child will use a computer language, LOGO, that helps children develop ideas about shapes as paths.

Here's a small sample of LOGO's commands:

• F 5 - Moves the turtle FORWARD five turtle lengths.
• B 5 - Moves the turtle BACK five turtle lengths.
• TR 1 / 2 - Turns the turtle 1/2 of a whole turn to the right.
• TL 1 / 2 - Turns the turtle 1/2 of a whole turn to the left.
• FD 10 - Moves the turtle FORWARD 10 steps (a step is about 1/12 of the turtle's length)
• BK 10 - Moves the turtle BACK 10 steps.
• RT 180 - Turns the turtle 180 degrees to the right. This is the same as 1/2 of whole turn because each degree is 1/360 of a circle.
• LT 180 - Turns the turtle 180 degrees to the left.

After writing directions in the last newsletter, children have been using LOGO to make a model of their path on the computer. For example, if you walked 20 paces, turned right 1/4 and walked 10 paces, you might model this walk with LOGO as FD 20 TR 1/4 FD 10. If you have a computer at home, the Verona Schools have a site license for LogoWriter, a form of LOGO that combines word processing and turtle graphics. The site license will allow you to use LOGO at home. Please contact Betty Wottreng for details.