After tutoring someone about polynomial manipulation, my mind went back to this equivalence:
$(a+b)\times (c+d)=ac+ad+bc+bd$
I realized that, while it can be memorized as it is, it has a very simple geometrical interpretation:
After making a mental note of mentioning this in the future (and feeling a bit guilty for not thinking about it during the session), this other equivalence came to mind:
$(a+b)^3=a^3+3a^2b+3ab^2+b^3$
I realized this too had a similar geometrical interpretation:
So, I had the impulse to dig up old block toys and build the cube, with different sections differently colored.
At this point, I asked myself if bringing the toy cube to the next tutoring session would be a good idea.
I didn't have a similar thought with the first example: the rectangle can be easily drawn with pen and paper, it's an accessible illustration.
The cube, on the other hand, isn't as easy to draw, and not everyone has blocks lying around to mess with.
While I find the cube illustration more elegant and powerful than a plain equivalence to get accustomed to, is there a risk that presenting the cube would create or strengthen a distance between the learner and the subject?
If yes, how can the risk be addressed?
If no, why don't i recall seeing illustrations like these in my textbooks or on the blackboard?