Been playing for probably ten years or so and I learned oringinally to play with all of my appendages. Why is the normal tendency to teach from the beginning to reach over the snare with the right hand instead of teaching to learn high hat and fills with both. I remember seeing some drummer do this just a couple of weeks into learning to play and it made sense to me that this should be a focus, develops both hands equally from the beginning. Of course I had to have someone move my hand for me while I focused on the rest of my body but it only took a good day of practice and I had it. I'm new to this forum and I'm not really an "in the box book taught drummer" so work with me here. I'm just wondering why we automatically refer to any hand as playing strong hand, seems if we teach confidence and teach to expect independence from the beginning it would help up and comers greatly.

I started out as a "wood-be" "ear taught" drummer and I really missed out on the musicianship of playing. Grooves are much easier to figure out when you can just read them huh? Anyway, don't want to sound like I think I'm the man when it comes to independence, I still have problems seperating my hand from feet at higher speeds. Full independence is still definitely an unreached goal for myself.

On my first drum lesson, my teacher had me playing both right handed and open handed and see which was more comfortable. He then got me to stick with the one I was most comfortable with (right hand lead).
After that however, he never talked about open handed playing again, and he probably only got me playing both at first because I told him I was kind of amidextrous, in that I do some things better left-handed and others right-handed.

Well, I'd say it has a lot to do with kit setup & configuration.  Since the "normal" config places ride cymbal on right, it makes perfect senese to teach/learn the ride pattern using the right hand.  Once you've learned that, you can ride on hats using the same skill.

A beginner student has a lot to learn.  I'd place the importance of being able to perform the standard "ride" pattern with left hand way down the priority list.  I'd consider that an intermediate-level skill.  Developing independence is a good thing, but a new student has much more to learn the first couple of years (or even longer).

I'll take a stab at this.

If you think back to the evolution of the "trap set," the ride was usually set up to the right ( of a right handed drummer). There was no hi-hat originally, the hi-hat evolved from the sock cymbal (although I've heard some old timers still refer to it as the sock cymbal). The sock cymbal looked like a hi hat but was only about 12 inches high and usually had much smaller cymbals.  

As the trap set evolved and the hi-hat came into play as a surface on which to keep time, the same ride patterns were being played on them so it was only natural to use the same hand for most players. Given that traditional grip was usually employed with the left hand it was easier to cross over than to try to play the hi hat  with the left hand. In the early days due to recording techniques and the way in which the songs were presented the drums were kind of relegated to the background so there usually were not any intricate things going on with the ride hand.

I'm assuming that this (the corss over) was the natural thing to do. I don't know if there was actually a formally accepted method of drum set instruction back then.  
Keep in mind, that back in the 20s and 30s playing of a drum set was still considered "non-legit." The "real" drummers were still considered to be the orchestral players. Drum set players had a reputation of being undiscplined upstarts that only played popular music. It wasn't until the early swing years when drummers like Davey Tough, Chick Webb and later Krupa started to legitimize the drum set in the musician's world.

As players and techniques have evolved over the years it kind of seems unatural to learn to cross over, but if I were a teacher I would tech both ways. I find that playing both ways opens up a lot of interesting possibilities. (Well actually Phillips, and Cobham found that out way before me :-)