When you are trying to get a horse's attention, the first thing you go for is his ears. Once an ear swivels in your direction, you've got his attention. It may take a little longer before he turns and faces you or before he walks up to you. But those bigger movements start with that little tiny movement of his ear.
The same principle applies to teaching your horse to back up. A lot of people make the mistake of going after the big moves before they've gotten the smaller ones. All it takes to back a horse is just the beginning of a movement backwards. That tiny bit of motion gets a little momentum started that eventually builds to the bigger movement you're after.
Backing is an illogical movement for horses. It is not a natural defense or play maneuver. The horse backs by moving its legs in diagonal pairs, just like it does at the trot. So it has to shift its weight from side to side in order to back. The horse that is backing properly does not squat on his hindquarters or get dragged backwards by the pressure of the halter or the bit. The rider must apply alternate side-to-side pressures to ask the horse to shift its weight from side to side in order to back.
The horse's front legs support its head and neck. The head acts like a weight at the end of a lever. When the weight of the head shifts sideways, that movement gets transferred to the front legs. As the head shifts just a little to one side, the horse shifts more weight to the front foot on that side to maintain his balance. If the head shifts a little farther to that side, the horse continues to shift his balance a little more and and you begin to get the feeling of wanting to move the foot that's not carrying very much weight anymore. Now you've got a tiny bit of momentum started that you can build on.
So the first step in getting a horse moving backwards is to move his head to one side or the other. Pulling straight ahead or pushing straight back doesn't create the need to shift his weight from one foot to the other, doesn't create any feeling of movement, and doesn't get any momentum going.
Use a corridor of aids to help the horse get a feel of the direction that's open for him to move. Heed the horse along a wall or fence up into a corner where it's horse logical to stop. Turn your primary line onto his secondary or neutral line to ask him to stop. Scratch and groom a little to show him he responded correctly. Now turn so you're standing at the horse's shoulder facing backwards.
Check to see which of the horse's front feet is a little farther forward and which one is a little farther behind. The foot that's a little farther forward is the first one you are going to ask him to move back because he already has less weight on it to start with.
Shift the horse's head toward the foot that's farther behind with a little tug. Don't hold the horse's head there, just ask him to shift a little more of his weight onto that foot. At the same time, reach over with the butt of your whip and put pressure on the shoulder above the foot that's a little farther forward. Hold the pressure until the horse moves back. Shift his head a little farther if you need to but don't increase the pressure on his shoulder. Just leave it there until the horse moves and then allow him to move away from the pressure. That front foot moving back puts pressure on the diagonal hind and the horse will eventually move it back, too. Now shift the horse's head to the other side and apply pressure to the other shoulder in the same sequence to move the other diagonal pair of legs.
In the beginning, it's not important how far the horse backs. Just ask for one step, one foot at a time. Remember, this is not a logical movement to the horse. So just ask for a small piece at first. If the horse is really having a problem with stepping backwards, just quietly return to heeding him forward and in circles and then go back to the fence or the wall and try another single step backwards. Eventually, the horse's understanding will increase.
Everything you do with a horse should start with a command or pressure that says "prepare" or get ready to do something followed by a pressure that says "execute" or now's the correct time to do it. Bringing the horse's head to the side is the command of preparation to back. Putting pressure on his shoulder is the command of execution to move a foot. Later, under saddle, you will use bit pressures to shift the horse's head and your weight and leg pressures will replace the shoulder pressures.
As the horse's understanding of what the pressure mean increases, then the speed with which he responds to them will increase. You will be able to keep the pressure up longer so he takes more steps and backs a greater distance. You will be able to switch the pressures from side to side more quickly to influence the cadence of his steps. And you will be able to influence the length of the horse's strides by how long you keep the pressure on before releasing it.