In calculus III we spend the majority of our time studying functions in 3-dimensional space. I told the students early on that making a real 3-D model would, at times, be the only way they would really be able to “see” it. Most of the semester our 3-D grapher has been sufficient because we only needed general information.

Today we started triple integrals. To totally understand the six ways to write the triple integral you must totally understand and “see” the solid that makes up the domain. I did a couple of problems and then set them loose on a new problem. I supplied them with scissors, tape, coffee stirrers, and paper so they could make a model.

I walked around the room and watched their progress. About half of them started by trying to make a model with the supplies and the other half tried to use the 3-D grapher and paper and pencil. After a short time the majority were working with scissors and tape! It was so much fun to see the AHA moment when their brain really saw the solid they were after and they understood the misconception that had occurred when just trying to decipher the information with just pencil and paper.

It is always interesting to me that when I am really learning something new for the first time I am back at that concrete learner level again