Step 3 (Age 5)
Objects and Spatial Relations

Kindergartners move to more sophisticated understanding at the thinking about parts level by measuring via the repeated use of a unit. However, they initially may not be precise in such iterations. Beginning to develop aspects of thinking at the level of relating parts and wholes, they can explore the concept of the inverse relationship between the size of the unit of length and the number of units required to cover a specific length or distance, recognizing it at least at an intuitive level. However, they may not appreciate the need for identical units. Work with manipulative units of standard measure (e.g., 1 inch or 1 cm), along with related use of rulers and consistent discussion, will help children learn both the concepts and procedures of linear measurement.

Kindergartners also can learn to fill containers with cubes, filling one layer at a time, intentionally, all of which involves relationships at the thinking about parts level of thinking. In a similar vein, they can learn to accurately count the number of squares in a rectangular array, using increasingly systematic strategies, including counting in rows or columns. They represent a complete covering of a rectangle’s area (although initially there may be some inaccuracies, such as in the alignment of drawn shapes).

Compositions and Decompositions

Kindergartners understand length composition explicitly. For example, they can add to lengths to obtain the length of the whole. They can use a simple ruler (or put a length of connecting cubes together) to measure one plastic snake and measure the length of another snake to find the total of their lengths. Or, more practically, they can measure all sides of a table with unmarked (foot) rulers to measure how much ribbon they would need to decorate the perimeter of the table. Their use of rows or columns in covering a rectangular area also implies at least an implicit composition of units into a composite unit.

Instruction to Support the Teaching-Learning Path

To move children through the teaching-learning path, teachers of the youngest children should observe children in their play, because they encounter and discuss measurable quantities frequently (Ginsburg, Inoue, and Seo, 1999). Using such words as “bigger/larger/smaller,” and, as soon as possible, “longer/shorter” and “taller/shorter” directs children’s attention

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