One of the issues raised by parents is how to look at change in children's understanding of mathematics. In short, as a parent, how do I know if my child is really learning anything worthwhile about mathematics? Although there is no simple answer to this question, we can offer some guidelines for tracking change. In this newsletter, we will focus on spatial reasoning: reasoning about shape, measurement, depiction and navigation. Although children experience space all the time by virtue of living in the world, we intend to help children develop a mathematical language to describe their experiences.
Taking a Long View
Most programs of assessment try to measure change in short periods of time - a few minutes, a lesson, a few days. CGI takes a different view - changes in children's understanding of mathematics occur gradually over a long period of time - years not days. Generally, we expect children's spatial reasoning to develop in two ways:
(l) First, over time, children's reasoning about space becomes more representational - they are able to represent space by symbolizing, and talking. What was first known perceptually drawing, becomes known again mathematically. Children's representational skills grow.
(2) Second, over time, as children's reasoning about space becomes more representational, children begin to integrate representations - drawings are integrated with language, diagrams are labeled with numbers, and so on.
We look for growth in children's reasoning about the structure of space. This includes ideas like dimension (2-D, 3-D), transformation (motions in a plane), and visualization (how to take a space "apart" and put it back together mentally). We look for the development of language to describe and think about shapes, including describing relationships among different shapes and their properties (ideas like sides, faces, angles, and systems of classification). Early in the school year, children generally do not have very elaborate ways of talking about or representing spatial structure. Your child's portfolio provides snapshots of his or her development in this area.
We look forgrowth in children's reasoning about measure of length, area, and volume. This includes ideas like appropriate units of measure, the importance of measuring with like (identical) units, and ways to iterate units to obtain a measure. Your child's portfolio should show the development of his or her thinking about length, area, and volume.
Look for growth in children's abilities to draw maps or scale models, use nets to describe 3-D space, and to make and interpret diagrams.
Look for growth in children's reasoning about large- scale space. This includes ideas like po-sition and direction, using a compass, making maps, using Logo, writing directions, and describing routes.