Why do this problem?
This might be a good starting exercise, if, as a teacher, you've never done any similar imaginative thinking (visualising) in Maths. You may find it best to introduce these kinds of ideas in small groups to start with, before having the whole class embark on such an adventure.
It can be very hard to prevent yourself from saying too much when the pupils are explaining what is going on in their minds. Questions such as "Tell me about what you are imagining" and "Tell me about how you're counting" are probably useful "open" questions to use. Try to avoid halting a pupil's thinking by saying "Yes I know exactly what you mean" too soon! A natural progression from the
problem is to look at the other "kinds" of small cubes which make up the larger cube, for example those with just one face painted, three faces painted, four faces painted etc. and of course no faces painted.
When going on to the larger 4 by 4 by 4 cube, it would be worth asking pupils to predict the numbers before thinking too much.
If you are keen to focus on generalisations for still larger cubes, you could look at Painted Cube