It always amazes me to see the differences between styles, and yes this is supposed to be all Tang Soo Do to boot! Thats not bad- just amazing!
Reading through the comments I felt like adding a couple things.
1) Even a bird learns to walk before they fly- at least in the nest. Without the basic motions and balance of doing ground kicks, its hard to believe there is any newbe learning jumpkicks first. Not that the kick is that hard, but its less basic. I know this isnt what the thread is actually about. (I'll get to that in a second) But I believe in progressive learning. Front, Round, Side are our white belt kicks. I learned from some of my early instructors, that when teaching mixed belts I have three options, I can have them repeat these three kicks as I move on for other students, I can have the beginner sit an watch, or I can add one or two orange belt kicks for the w.b. as a primer. Since class is short enough and I want to keep the students interst, I opt for the third option. For this, if the student is ready, I do add Jump front kick for a white belt. But ony if they are grasping the three basics first.
2) Anyone that says Jump Kicks are not effective or practical needs to practice their jumpkicks more. Ofcourse this is comming from someone who loves them. but I've used every kick I've ever been taught (and thats a lot of them) in free sparing. This includes Jump spin kicks as well as straight jump kicks, as well as both front and back leg kicks. I just read a Bruce lee quote that said- I dont fear the man who practices 10,000 kicks but someone who practices one kick 10,000 times. This sounds contradictory to the first part of my statement , but the point is to practice and you'll be good. That goes for any thing. So I propose take your 5 or 10 or 20 kicks and practice them each 10,000 times. I could tell you stories of how practice became instict which gave me the edge to instantly react to an opponent and sometimes that ment being able to throw an advanced kick with ease in the heat of the battle - sometimes that means hold back a kick and switching up mid stream. Practice of combinations and foot work allows for advanced kicks to be used. As I step down from one kick, I land in a prep postion for the next kick- which means knees bent. This isnt telegraphing because I am in motion from the previous technique and can throw any number of kicks from that position.
Now to the Kick-
1) We teach Front kick from both the front and back leg, we teach jump front kick from both the front back leg and we teach both of these with and without a pumping leg. So for front kick we have 5-6 variations. (not counting pi chagi (which is sometimes refered to as an inverted front))
2) as somone noted a proper fighting stance should have knees bent- however a little extra knee bend isnt the end of the world- espectially if you do the kick in combination and not off the line when sparing. So telling somone not to bend their knees is not help full. One of my teachers says, if your jump kick is not higher than your ground kick, then you are not jumping enough. Ofcourse the jump adds momentum and force to the attack- if done right- so height isnt the only posible advantage to the jump motion. I highly recomend a good knee bend so you can learn to spring off your feet. As you perfect the springing motion, try to work on eliminating any telegraphing (by concealing/eliminating the extra knee bend) First you need to break down the kick to do it right, then perfect and make more efficient.
To Practice - (First the Jump)- from fighting stance, bend knees a bit more, then work on springing up off your toes. So the full springing motion starts with both knees bent slightly and finishes where you are rolling up onto your toes (balls of feet) as your knees are extending. This motion should be a great base to sping from. It should be a continuous motion, all the way to the tucking of your knees. I like to teach to focus on bringing your knees as high as you can, then to tuck your feet. Concentration on the knees helps for height. When someone has trouble with this I have them just work on this jump- S0- First do this straight up and down, to practice the jump bringing the knees as high as possible. And land straight down.
After this is comfortable, add the hip/foot switch- w/o the kick so you end up with your opposite foot forward (jump, twist hip, land).
Now do the full kick. At the top of your jump, you are in the process of twisting the hip, then extend the kick. If you do the jump with a focus on bringing the knees high, then you can more easily kick (in my oppinion) while still in the air.
Many people dont focus on a proper jump and they end up landing with or before the kick. which make is an odd hop kick, since they are not in the air when the are executing the kick.
Good luck.