I too have found that, after spraining my ankle, if I can keep moving it will heal much quicker---the key is to NOT twist it again; at least not until it's recovered.
I've wondered if keeping active on the ankle keeps the swelling down and allows for "healing juices" to flow to the injury, Or, if after you've had numerous sprains, it just doesn't matter what you do. In any case, I know the medical pepole would say keep off it and maybe even suggest crutches.
I can attest (with a single data point..) that keeping off a sprained ankle because you're not allowed to put any weight on the knee above it is an excellent way to screw up the ankle for long periods...
Get an Active Ankle brace to prevent reinjury and just keep training. Those braces are work very well.