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Adding + It Up: Helping Children Learn Mathematics
The basic idea of measuring area is that of coverings region by units that “just fît” (an idea that is sometimes called tiling). In many ways the development of area measure parallels that of length, but it lags behind. First and second graders often treat length measure as a surrogate for area measure. For example, some children will measure the area of a square by measuring the length of one side, moving the ruler parallel to itself a bit and measuring the length again, and so on, treating length as a space-filling attribute.87 When provided with geometric manipulatives (squares, right triangles, circles, and rectangles) for use in finding the area measure of a variety of shapes, most students in grades 1 to 3 freely mix units and then report the total count of those units.
As they progress through the elementary grades, students usually begin to differentiate area measure from length measure, and the space-filling (tiling) requirement of the unit becomes more apparent to most of them. Other aspects of area measure, however, remain problematic. Students find it very difficult to decompose and then recompose shapes or even to see one shape as a composition of others, an idea that is fundamental to conservation.88 For example, students in grades 1 to 3 often cannot think of a rectangle as an array of units.89
By the end of the elementary grades, students typically understand core concepts like using identical units and covering the object for length measure but not for area measure. Younger children often employ resemblance as the prime criterion for selecting a unit of area measure, suggesting the need for attention to the qualities of a unit that make it suitable for measuring area. The common instructional practice of declaring that the square is the unit of area measure may lead to procedural competence but may violate students’ preconceptions about what makes a unit suitable.
Teaching experiments with area measure have revealed that second graders could develop a comprehensive understanding of area measure when they began by solving problems involving partitioning and redistributing areas without measuring.90 It is worth emphasizing that this approach makes conservation of area a fundamental construct rather than an afterthought. Later, when the children explored the suitability of different units (e.g., beans) for finding the areas of irregular shapes like handprints, they found that units like squares had desirable properties of space filling and identity. By the end of the school year, these children had little difficulty creating two-dimensional arrays of units for rectangles and even for irregular (nonpolygonal) shapes.