Geometry Teaching

tl;dr Version
Learn the basics of geometry by teaching it to an anthropomorphic robot. Aiming to be genuinely educational in teaching the player about the relationships between shapes. Also an exploration of a learning-by-teaching technique. Maybe not much of an actual game in the traditional sense, but you can't stop me! Hand Shake Right

Long Version
My original idea was to make a game that criticises the typical methods of teaching mathematics. Specifically bemoaning the tendency to have students memories formulae and techniques without really understanding how they work, so resulting in them only being able to apply them to situations they're specifically taught. I was going to link it in with Searle's Chinese Room thought experiment (if someone only knows how to find the area of a triangle by almost blind application of techniques, do they actually understand triangles at all?) Setting out ideas for how to do this I realised it might be more fun to try to make an alternative to these 'bad' methods rather than just criticise them. Also, bringing in philosophy of mind to an already dull game about maths? What was I thinking?

Game starts with a screen showing a rectangle with sides labelled a and b. Robot appears, and explains that all it knows about the world is that this is a rectangle, and that its area is a multiplied by b. It wants you to teach it about other shapes, but it can only learn by reference to what it already knows, which so far isn't much.

Game proceeds by a series of 'puzzles' where you are presented with a novel shape, and need to show the robot how to find its area. Your only way to interact is to perform various manipulations of the shape you're presented with (you're not allowed to write up the correct formula on the blackboard and have the robot copy it down.) Your aim is to make the shape into something that the robot recognises, so it can work out how to find the area by working back from your manipulations.

Example!
The game has just begun, so the robot only knows about rectangles. Our puzzle is this right-angled triangle:

We try duplicating it:

Then moving and rotating it:

The robot watches all this, looking for a shape it recognises. It suddenly pipes up, "I know that shape! It's a rectangle!"

It then looks back over how you arrived at the rectangle shape and sees that you duplicated the triangle once (the other transformations made didn't alter the area of the shape so it ignores them,) "So the area of this triangle must be half the area of the rectangle that you made. Now I know how to find the area of right-angled triangles! Hurray!"

The game will continue with fresh puzzles, but now the robot will be ready to recognise right-angled triangles as well as rectangles. For instance, when given a parallelogram you can show how it may be split into a rectangle with two right-angled triangles attached:

The robot will recognise the component shapes thanks you to drawing on those lines, and will work out how to find a parallelogram's area from its existing knowledge.

Actually making the game
This'll probably be more fun than playing it. Lots and lots of AI stuff - mainly shape recognition from vector data and manipulation of algebraic formulae. How to represent the shapes so that they're easily recognisable by the robot will likely turn out to be the biggest challenge. I could cheat and hard-code a lot of it (especially the correct area formulae) but that seems against the spirit of the exercise.

Oh, I'll make it in Flash, 'cus Flash is what I make things in.

Whimpering
There is a prize for "most dry game", right?
Could use a real name for this too, please!