# Fun classroom exercise for mental rotation

I'm training to be a teacher and I am doing a maths lesson later next week. The topic is geometry, the students are 12-year-olds. More concretely, I've been given a selection of exercises that I may use if I wish so. For example:

Exercise text: This has been built according to the plan given. Draw all 4 side views.

Another example:

Exercise text: For a given cube building you find here two side views and the view from above. Which of the following plans fits all three views?

My guess is that the goal of this lesson is to train mental rotation.

I have come up with an exercise where first each student builds one of these cube houses, then draws the side views and then I pair up the students. One gives the views to the other who has to try and build it, then they switch roles.

Does anyone have any better more fun ideas for a lesson?

• I love the exercises. I'm sure there must be some other fun things (maybe very similar or more divergent) but I am not thinking of any right now. Nov 17, 2018 at 6:43
• My guess is that the goal of this lesson is to train mental rotation. This raises two questions that you don't seem to have considered. (1) Is this skill susceptible to training? (2) If so, what would be the purpose of training it? I happen to know a little bit about this because my mother did her PhD thesis in psychology on this sort of thing, and she used me as a guinea pig for some tasks. IIRC, there is a heavy genetic component to 3-d visualization, and there is also a big difference in average ability between the sexes (a rarity, since the evidence seems to be that in general, [...]
– user507
Nov 17, 2018 at 15:29
• If I had to do the two exercises in the question, I wouldn't attempt to visualize the results of mental rotations. I'd just work with the numbers. Be aware that some of your students might also work that way. Nov 17, 2018 at 16:15
• @BenCrowell Gender differences may be a valid point as such but any skill, also mental rotation, improves by training. Since I'm not doing a mental rotation competition but merely some not too difficult exercises this will hopefully not frustrate anyone. Or at least not just one gender. Nov 18, 2018 at 22:08
• @BenCrowell, I believe you are wrong. Yes, girls and women do worse on these tasks, but it is susceptible to training. Many boys have trained with legos, etc, and girls can train themselves to remove this gap. See research by Sheryl Sorby. For example: eed.osu.edu/news/2016/02/… Nov 19, 2018 at 6:46

I think these are great exercises as you outline them, especially the object-building aspect.

Maybe it also would be useful to ask if these two views show the same object (your 1st; they do), or not? One could delete one or more cubes so that the answer is No. Just for 3D-training.

Two views of the same object.
I'm thinking that going beyond side orthogonal views may be useful.

Also, one could arrange two polycube objects such that no two viewpoints could distinguish them, because one has a cavity and the other does not. The cavity would be invisible externally if the cubes are opaque.

This could lead into research questions exploring the shadows of polycube objects, including the famous cover of Gödel, Escher, Bach.

MSE question: Cover of “Gödel, Escher, Bach”
This is a fascinating topic that could serve to show these students that there is more beyond what is known, and even they can (dimly) glimpse the research frontier.

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