Today I put my Math 30-3 students in random groups of 3, setup my whiteboards against walls and posts, gave them interlocking cubes, and had them work on the following problem:

Think of a 2x2x2 cube composed of 8 smaller cubes:

If this cube was dipped in paint, then each of the 8 smaller cubes would have paint on 3 faces:

Now think of a 3x3x3 cube. If this cube is dipped in paint, each cube will have paint on either 0, 1, 2, or 3 faces. How many cubes will have 0, 1, 2, or 3 faces covered with paint.

**First Class**

In my first class, most of the students were engaged in this activity (there was only one eye-roll that I noticed 🙂 ). However, except for one group, each group only had one member who was actively engaged the entire time.

The majority used interlocking cubes to physically count the number of cubes in each category, but one group drew a 3D cube on their board--not something I would expect.

I tried to point students to other groups when they needed help, but it's really hard not to just jump in and say, "here's how you can do it". There were only a couple of times that I had success with two groups discussing strategies with each other.

When groups were confident that they had the correct solutions, I asked them to work on 4x4x4, then 5x5x5, and (for one group), 6x6x6.

None of the groups changed their strategies for the larger cubes, however when the last group finished 6x6x6 I challenged them with 7x7x7 by identifying patterns, rather than physically making a 7x7x7 cube. There were only a few minutes left in class, but the "lead" student identified the patterns for 2 faces and 0 faces. For the others, she was only looking at additive patterns.

Memorable quote of this class: "Can you take a picture of my work? I want to show it to my boyfriend."

**Second Class**

My second class had a significantly different feel to it. There was a lot of trouble with terminology--particularly face, edge, cube, and square. I took a minute to show hands-on the difference between these terms. There were also a handful of students sitting around doing nothing, even when I encouraged them to work with their group members.

A number of students were telling me they knew "the answer" and gave me a single number. It took me a while to figure out, but I think they were adding up the total number of cubes. I had a lot of difficulty convincing students that, given the question asked, there couldn't be just one answer. Weird, given how straight forward the question was (at least to me 🙂 ).

As with my first class, one group chose to draw 3D diagrams and were quite successful with this approach:

As with my other class, none of the groups changed their strategies when asked to think about larger cubes (4x4x4, etc).

Memorable quote of this class: "When are we going to start doing real math?" 🙂

**Conclusions**

- My whiteboard alternatives work very well. Erasing was no problem (I only used black and blue, since I've noticed red and green don't erase well on "regular" whiteboards), and the X on the back provided enough stability.
- I was quite pleased with the engagement overall, and I'm assuming the level of engagement will increase as students realize that this will be the default mode of operation in my classes
- I came up with a couple of ways to address groups that had either correct or mostly correct work, without acting as a typical teacher--i.e. the holder of all mathematical knowledge
- I referred almost-correct groups to correct groups and said something like, "You have very similar answers. Now it's your job to discuss with each other to decide what the correct answer is."
- I referred almost-correct groups to other almost-correct groups and asked the same question as above
- Each class had a group working faster than the rest of the class. At each stage, I would say, "It looks like you're convinced that answer is correct. How about you move on to the next question?"

- For each class, I took a picture of work that represented the understanding of the class as a whole and posted it on our moodle site.

**Next Steps**

I'm planning to revisit this problem in each class tomorrow. I'll start by having groups try to describe the characteristics of each type of cube (for example, the cubes with 3 faces covered are in the corners). Depending on how well that goes, I might challenge them with with an nxnxn cube, or give them another problem to work on.