Problem Centered Mathematics

Mathematics is More than Number

Children develop ideas about space at the same time as they develop ideas about number. Unfortunately, children's ideas about space are often ignored in the mathematics classroom. This is a problem for two reasons. First, reasoning about space is basic to many forms of mathematics that children encounter in their schooling. Second, geometry skills are very useful and practical especially for professions involving design.

Before children begin school, they have many experiences that can be used to teach them about space and spatial reasoning. Common experiences such as finding their way about the neighborhood, building with blocks and legos, drawing and looking at pictures, and noticing common shapes in nature lead children to ideas about:

CGI builds upon these common experiences to help children learn to describe and act upon space much in the way in which CGI attempts to help children learn to use numbers. Children's reasoning about direction, pattern (shape), depiction, and measurement unfolds in typical ways. CGI geometry works with these characteristic ways of thinking to make geometry more accessible to young children.

Goals of the CGI approach to geometry and spatial reasoning are similar to those for learning about number:

  1. Learning by solving problems: Although children can learn much about space simply through observation, they learn more when they solve problems that help them represent and think about what they perceive.
  2. Learning to solve spatial problems in multiple ways: Children learn the most about space when they can take different approaches to the same problem. It is better to draw or build a pattern than just to look at it.
  3. Learning to communicate about space: Communication is an important part of mathematics. Children learn by describing space in many ways - talking, drawing, and constructing - and all of these forms of description help children communicate about what they understand.
  4. Learning to argue mathematically: Geometry offers children unique opportunities to learn about mathematical arguments. For example, is it true that because all squares have 4 equal sides, then if a shape has 4 equal sides, it is a square?

Way Finding

As we noted above, one of the ways that children learn about space is by finding their way in the neighborhood. Some interesting ideas in geometry can grow out of these experiences - ideas about distance, direction, making and reading maps. However, these ideas don't just grow naturally - children need some help in thinking about them. In this newsletter we will discuss some ways to help children begin to build these ideas.

Giving Directions

For homework this week, you can try this activity in your neighborhood, at a park, or anywhere else that is convenient.

Have your child think about a landmark you can walk to (a special place like a tree or a building) and ask them to write directions for you to get to that place. (It might be fun to make this a "treasure hunt" i.e. pick the library and check out a special book when you get there or the ice cream store for a cone). Notice how your child handled length and direction in the written instructions.

One way people have handled length (distance) is with the idea of a pace. A pace is two steps. If you start with your left foot (holding your right foot on the ground), one pace is the distance covered when your right foot again strikes the ground. For most of us, it is about five feet. One helpful way to count paces is to have a "number" leg and an "and" leg. Students count by saying "and (left foot) -one (right foot), and (left foot) -two (right foot).

Sometimes children have difficulty thinking about paces, so they can also use their feet or make something else up. The important thing is that they think about how to measure length. We are trying to build to the concept of a ruler so we want children to discover the benefits of standard units of measure. It would be best if children did not start out with rulers right away.

All paths are not straight lines. We often need to turn to get from one place to another. One of the first ways of describing the direction of a turn is to mark one's right or left. The concepts of left and right are difficult to learn. With experience primary grade children obtain a firm grasp of these basic concepts. A further development is to consider whether one turns to the right or left. If children think of one whole turn of their bodies, then they can think of parts of this whole turn. (We use degrees to measure parts of turns but this is not a good place to start with children). For children it best to start with simple fractions like 1/4 or 1/2, so we can describe a 90 degree angle as "turn right 1/4".

Children's directions often consist entirely of familiar landmarks like "Go to a neighbor's house, then go to the corner store." We would like them to be able to go beyond this so that anyone could follow the directions even if they don't know the landmarks.

So then a child's directions might be something like this: Go out the front door and walk 10 paces. Turn right one-fourth of a turn. Walk another 20 paces. You are there.