How to switch light on by standing on a mat
You have a few options for this, either a step switch or touch sensitive switch only if the mat can conduct. 

Here is normal step switch which is pretty rugged, inexpensive and can hold up to students bouncing back and forth on the mat. The only issue is that it stays on per step and has to be stepped on again to toggle it off. You will most likely have to do what the previous poster mentioned and swap the internal switch out for a compression based switch shouldn't be too hard, but at least you'll have the housing and wiring already manufactured if you buy this.