The humming-top is an exercise in which the horse turns around his own axis, while the human stands in one point. Here is a video example: Titum doing humming-tops
One way to start it is by using a tiger (target stick). There are other ways to teach it, but the nice thing about using the tiger is that it is a pressureless way that works without sending the horse away from you.
Before you start working on humming-tops with a target, your horse should have learned to touch the target with his nose. Now you want him to make a turn while following the target. This requires you to hold the target over the horse's back and move it away from his head, so that he has to turn away from you to touch it. Be careful if you have a horse that tends to be afraid of plastic bags, especially when they move over his head or back. In this case, some preparatory work might be necessary.
Once you have established a turn with the target, the next step is to fade out the target so that you will be able to do it only on voice cue or with body language later. With Titum, this worked best by becoming sloppy with the tiger. Whereas in the beginning I had made a complete turn with the tiger, later on I started to begin the turn with the tiger, but I did not complete it, but just moved the tiger upwards once the horse has started to move. When he was used to it, it was enough to hold up the tiger so that the bag hangs somewhere above the horse, and he would turn under it. Then you can just place the tiger next to you with the bag in the air etc. Make the tiger less important step by step.
Here is a video of our second day, when we began to fade out the tiger: Intermediate steps
The next bigger step is to do it completely without a tiger. This was quite difficult for Titum and me in the beginning. Before, I had not really used the tiger anymore for targetting during the turn, but the mere fact that I had it in my hand had profoundly changed my body language. Without a tiger, Titum just did not understand what I wanted. What we did to overcome this problem was using a "disabled tiger", a short and thick branch that had no real use and didn't act as a target for Titum, but I had something to hold in my hand and to pretend that I was holding a tiger. While you work with the disabled tiger, watch your body language. What movement do you make? Then you can try it with no tiger at all.
The body language cue you will end up with directly after you have eliminated all kinds of tigers will probably include standing next to the horse and pointing over his back with your hand. It might be helpful in the beginning not to stand in front of the horse, but at the side where he will turn away from. This is because when your horse doesn't know what to do, you can just reach over his back and initiate the turn by letting him follow your hand.
Once your horse knows the basic humming-tops, there are lots of possibilities for varying them. For example you can increase their speed so that they become a spin in trot, or perhaps even a chain of jumps. You can incorporate them into moving forwards, so that your hhorse is spinning while you are walking or running with him.