# Geometry & Spatial Reasoning

The Development of Volume at the middle levels

## Longitudinal study of Volume

The study started with a concern about a group of sixth graders inability to use a visualization strategy to solve volume problems.

I was interested in seeing if the development of visual spatial abilities of fourth graders, with relationship to volume, could be developed so that those mental representations could be learned in fourth grade, remembered, and used to conceptualize density in sixth grade. Previous sixth grade students did not have a good understanding of volume, which is necessary for understanding density.

So when I was given an opportunity to teach a fourth grade class, that would in two years be in my sixth grade class volume, the study was born.

I started by presenting a challenge to this group of fourth graders.

I asked them to find the number of white (cm^{3}) Cuisenaire cubes in a
red cube (2 cm^{3}) , green cube (3 cm^{3}), purple cube ... orange cube. The sizes of these cubes increased by one cm. From
1x1x1, 2x2x2, 3x3x3 ... 10x10x10.

All the cubes were available for students and all decided to solve the challenge by constructing each cube with white cubes and then count the number of white cubes. Students worked individually and in pairs to construct each cube. The total time spent for construction, charting, and discussion was five days with an average of 50 minutes a day. I did not anticipate that it would have taken so long. However, I believed it was time well spent as it would improve their visual spatial abilities as well as construct a concept of volume. So I could hardly wait until next year.

The next year, now being fifth graders, I gave them the same challenge. They proceeded to construct the cubes with the white cubes. However, most stopped their construction with the 4 x 4 x 4 cube or the 5 x 5 x 5 cube, with a few making all ten. However, the activity this time took about two and one-half days.

Next year as sixth graders they were given the same challenge. All started with the white cube. A few constructed a 2 x 2 x 2 cube, counted eight cubes, were satisfied that 2 * 2 * 2 was eight and multiplied for the rest of the cubes. A few other students constructed cubes beyond the 2 x 2 x 2 but none constructed any larger than 5 x 5 x 5 and used an algorithm (formula) to calculate the rest. The total time was within 50 minutes.

The construction of volume for all students was pretty good and most were able to connect the concepts of volume and mass and construct a concrete understanding of density. Their understanding of a proportional relationship of volume and mass was fragile but that seemed to be related to their limited understanding of ratio and proportion.