Use of leg aids on young throughbred

Question
I am currently leasing a 4 year old TB at  lesson / show barn that I have been taking hunt seat lessons at for approx. 6 months. The horse is very quiet to work around and likes to be handled. I am now starting to ride him on my own time outside of lessons and find him to be a bit lazy and not very forward when he is asked to trot. Of course when I am handed a crop by the instructor he behaves a bit better even without using it but I would like to work with the horse and get him to be more forward on his own with out a "scare tactic" so to speak. My last time out I worked him asking him to trot by squeezing with my legs & strongly asking him ,verbally, to trot, I would then halt him and ask him to pick up the trot again . By the end of 45 min he seemed to be a bit more responsive . Am I on the right track?
Also, why is it that some horse are so sensitive to leg aids only requiring the slightest sqeeze and others more stubbborn? Can the lazy ones become more forward when asked to work?  

Answer
Hi Anne,
I did not get the original question, only this reminder.
It sounds as if overall you had a good experience for the last 6 months riding hunt seat. Here are some factors to consider when you ride lessons horses:
-- how many other students are there in the class
-- how many hours /day or days/week the horse is ridden
  Consider this not because the horse is tired -- he is not tired physically, strictly speaking.  He is more "tired mentally" if you will, since he is your teacher, and he also teaches others about riding.  Yes, that is stressful to a horse -- no matter how good the instructor and the lesson program is.

about "lazy" horses:
A school horse has to be somewhat less sensitive.  Most riders are at first intimidated by a horse that moves off very fast. And I am sure you have seen the riders who ask a horse to go faster, but get uneasy when he does, and pull on the reins , unintentionally punishing the horse because it responded to the riders leg.

"My last time out I worked him asking him to trot
                 by squeezing with my legs & strongly asking him ,verbally, to trot, I would then halt him and ask him to pick up the trot again . By the end of 45 min he
                 seemed to be a bit more responsive . Am I on the right track? "
If you see an improvement, yes you are on the right track. Great!  Try to soften you aids each time tho, so you eventually only need to use subtler signals.  (Assuming you ride this same horse )  I think it works partially because stopping is a reward, and partially because you are getting the horse to pay attention to you  by varying your signals.  It's boring to go once around the ring -- but by stopping and starting, you are giving the horse a job to do, and wake him up, so to speak.
Just be aware that the ultimate goal is to use subtle aids.

my suggestions:
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think in terms of asking-waiting to see a change - rewarding for the slightest change
asking to trot / or trot with more life and energy:
if my seat + smallest squezze has no effect -
follow with many tiny kicks.  I "annoy " the horse, rather that constantl squeeze.  My legs would get very tired way vefore the horse responds.  If I kick-kick-kick (very sligtly-an onlooker might not notice) if the horse responds by moving on, I stop the annoying signal, I stop -- praise verbally, and allow the horse to trot on on a long rein.
If there is no response withing 10 sec or so, I increase the power of the kicks...again, stop and keep troting if the horse morves on.
In other words, I increase the rythmic signal in my legs rapidly if there is no response.  Stop the kicking and praise if the horse responds.

Now, if you are already in a trot but it's way to slow:  I'll move in rythm with the hind legs of the horse.  I'll ask with my left leg when the horses's left hind leaves the ground, and with my righ leg when the horses's righ t leg leaves the ground.
(it's like pushing a swing at the proper time to make it swing wider).

about "stubborn" horses:
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Also, keep in mind  that every rider "trains" the horse:  if the rider before you is strong, and has a "strong" leg, and you have a gentle signal, the horse will ignore you at first.  Imagine the horse is like an employee who is used to his boss loudly speaking with a booming voice all the time.  The boss is not mad, he just has a loud booming voice.  If you are to replace that boss for a day, and you whisper an instruction to the employee, he will simply not hear you. He is used to someone shouting at him.  He will not listen for a whisper.  You then label that employee "stubborn" or "lazy".   He is neither-- he is simply used to having  a "loud" boss.

Moreover, all schoolhorses, and all horses ridden by more than one person are going to be less sensitive.  If you rode my personal horse.....and you squeezed your legs like you do when you ask your horse to trot -- welll, my horse might be jumping over a 5 ft corral fence, since I am using very slight leg signals with her..   So when I teach beginners, I cannot let them ride the super-sensitive and responsive horse -- it would be WAY to fast and responsive.  Think about  how you might a teenage boy how to drive -- you might not let him drive the Ferrari race car with a stick shift the first time around.

So the horses who move at the slightest squeeze generally do so because no one kicks them in an inapropriate fashinon on a regular basis.  And there is no such thing as a stubborn horse.  What we call stubborn is simply a miscommunication.  He simply does not understand what you are asking.  

He knows that human students don't know "horse language" yet... and he responds to the rider the best he can.  As a rider, you are doing the righ thing: practice, learn, ask questions...:)  And we are never done learning.