Both horses and riders ultimately learn from and cement behaviors through repetition. This is why you need to make sure that you are repeating things CORRECTLY.

Maintain your good score for your trot or canter lengthenings by performing a clear, precise, and active transition back to the working gait at the end of the lengthening.

Many times riders ride each stride to a jump, but then sort of "freeze" right when they get there. Usually this is because of nerves, or because the rider is worried about trying to make some preconceived takeoff spot happen.

Make sure you ride each stride, including the takeoff, and even over and across the jump if necessary. Don't "jump ship" and leave your horse feeling alone right when he needs you most.

If you want to develop a nicely shaped crest (the top of the neck muscling) on your horse, you have to make sure your horse is truly seeking the connection - reaching into the bit, rather than backing away from it. 

Riders who can accept and learn from criticism will generally improve the most.

I once heard Ingrid Klimke call a galloping jump a "rhythm fence." And that is a perfect way to look at it!

"Don't ride to other people's ability – always ride to your ability and your horse's ability." ~ Andrew Hoy

Good lengthening and extended work requires straightness in the form of even loading of the hind legs.

"Since the criteria of a correct seat are the same as the criteria of good posture in general, being constantly attentive to one’s bearing when standing or walking is excellent training. A correct vertical posture of the head and the trunk on horseback is not a special posture applicable only to riding." ~ Kurt Albrecht

"If I want my horse to be steady and up to the bridle then I have to provide a steady target [quiet hands] and use leg to keep him up there."

If you do all of your training on the exact same kind of footing, (i.e. either all firm ground, or all sand), you might be putting your horse at an increased risk of possible injury. Training on a variety of different surfaces will develop both the strength and the elasticity of tendons and ligaments, which will help to prevent injury.

"It’s important that the rider doesn’t disturb the horse – leaning this way or that – and that is the same with this pulling and pushing. You give a half halt, but half halt is not just pull back and then let go. First of all you have to push the horse into your contact, and while you do a half halt, the horse should not get tighter in the neck and not get slower in the hind legs. Actually we want to engage the hind legs. It’s something you have to work on all the time, and get to feel it. When you tell the rider, now this, now that, you are already too late. You have to practice this, so that the riders get to feel it themselves." ~ Monica Theodorescu

The half pass helps to develop the brilliance of the extended trot, by increasing hip and stifle flexibility and range of motion, and freeing up the horse's shoulders.

Horses learn by a process of trial and error based on trying to make themselves more comfortable.

True lightness does not come from playing with the bit until the horse feels lighter in your hand. Only real hind leg engagement and lowering of the quarters will bring about the true lightness that you are looking for.

"You are connected with your seat to the horse’s back, connected with your legs to the horse’s body, and connection can only work if it goes through the whole horse and into the rider’s body, and from the rider’s body back to the whole horse." ~ Susanne Miesner

"I discovered that the horse is life itself, a metaphor; but also an example of life's mystery and unpredictability, of life's generosity and beauty, a worthy object if repeated and ever changing contemplation." ~ Jane Smiley

"Dressage is the art of teaching the horse to carry you. Riding is the art of learning to be a good load to carry." ~ Richard Weiss

Both horses and riders ultimately learn from and cement behaviors through repetition. Make sure you are repeating things correctly to avoid cementing incorrect responses. 

You have to have your core muscles engaged when riding, so that the horse's movement doesn't displace you.