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Especially for young hitters, a tee is key to realizing the purpose and effect of proper 'contact points' in relation to where a pitch crosses the plate and grooving the feel for the approach to each.
Re "swinging through the baseball", the tee also enables you to set up your mostly-deflated volleyballs and basketballs to reinforce the swinging through concept and feel.
My 13 yo loves the game that can be made of this...
When hitting (righty) off the tee in a tunnel we move an L-screen over to the right, about 50 ft away, such that it's top bar would be in line with the oppo alley. Then it's a game for points, 1 each for clanking that top rail.
No doubt the tee can be one of the more valuable tools in developing a good swing. I can also be detrimental to hitter if he is not shown how to approach his practice on the tee. If he kid is approaching the batting tee like approach the driving range with my driver than nothing good will ever come from hitting off of it.
I don't allow kids in our organization (www.cbcbaseball.net) to pull balls off the tee. We have them work the barrel to the inside half of the ball and stay up the middle or into the opposite field gap. I'm a believe that the tee can help a hitter develop a good hand path inside the ball. It goes back to the old adage that practice doesn't make it perfect...perfect practice makes it perfect.
Obviously all the physical factors involved in developing a good swing has to take place when working on the tee. What I have found is that focusing on staying inside the ball and driving it up the middle or to the opposite gap brings the other important physical factors into the swing naturally. Not to say a player doesn't need to work on other parts of his swing, which you as instructor will determine, but this approach is a good start to teaching a kid the value of hitting off the tee.
Maybe it sends some mysterious kinetic message (like Reactive Neuromuscular Training (RNT)).
But those demos by coach were pretty bad. I wouldn't want anyone's muscles memorizing that mess of tentative, unexplosive, unstable, backside-spinning, poor dynamic balance mechanics.