Purpose:

To determine students' depth of understanding of the concept of volume [Note: This is one of several subtasks involving different uses and configurations of unit cubes.]

Materials:

16 wooden cubes

Empty container with a 96-cube capacity

Task:

What is the maximum number of cubes that would fit in the container?

Not enough cubes were supplied to enable the students to determine the answer by actually filling the container and counting. They had to use the cubes, along with some other approaches to solving the problem. Scorers were not only looking for the results obtained, but also the students' solution strategies.

Multiple-Choice Question:

The large block below is made up of 4 layers of small cubes. Each layer has 3 rows with 3 cubes in each row. How many small cubes are there?

Results:

Just over half of fourth graders and 70 percent of eighth graders tested were successful on the performance subtask. The fourth graders did significantly worse on the multiple-choice question.

Conclusion:

It appears that with less exposure to instruction in volume, the fourth graders benefitted from the use of the unit cube manipulatives. However, it seems that some older students, having routinized volume-finding on the prototypic, non-authentic context (multiply the three given dimensions of the rectangular solid) were somewhat baffled by the more authentic, real-world problem.

In fact, it was apparent that those students, and even many who succeeded on the performance subtask, did not see that v=lwh could easily be applied by first using the given cubes to determine the three dimensions. Many put all 16 cubes in the box and then visualized filling the rest of the space.