For Goat on a Mountain, you have to first teach the horse to lower it's head. This is something you can do easily in a stall or alley. It has benefits way beyond the first stage of the Goat movement! IN fact, as it is a standing exercise, if the horse is comfortable and calm, you can pracitce this, too, in the stall or in the alley. Just make sure the horse has good footing. The further forward the hind feet come, the more they are likely to slip on a slippery surface.
In the Goat, the horse's balance will be on his front legs.
With the head down, and in a very calm state of mind, you can reach with a twig or a whip and GENTLY touch a hind foot. Usually the near hind first. See what reactions you get. For Tamarack, if I touched above the fetlock, he would lift the foot, then set it right back where it was. So I experimented with touching different places. His sweet spot was on the heel bulb. If I touched there, he would lift the foot and set it back down a bit more forward. If the horse is very calm about being touched on the hind feet or fetlocks in this way, you can even touch and hold the whip in place as the leg is lifted to help guide it a bit forward.
It can take a very long time for a horse to be able to move the back feet all the way to the front feet. So always reward for small tries, and don't get greedy
Do just a little bit every session.
This exercise is a very nice stretch for the back and haunches, as well as having a calming effect on the horse.
Some horses may lie down while trying this. If you go at it too long, the legs get tired.
Do not ask a horse to hold the extreme position (back feet touching the fronts) for too long, and if you feel your horse is struggling to hold the position, only ask it, then ask them to walk out of it right away.
Goat/brisk-walk transitions are a nice exercise too, and within this type of transistion you will tend to see a horse over step (reach further forward with the hind) more as he/she moves out of the goat.
As this is a static exercise with no forward movement, it should be taught in a very calm setting, with the horse in a very calm state of mind. If the horse is anxious at all, don't do it. Wait for a quieter moment.