I like what several of you are saying, i do disagree with some as well. I think my advice here will fall somewhere in-between everyone elses. After i write what i would do to try and adjust turn radius, i will respond to several other people on this thread.
I would move back to garlands, this task allows to board to seek the fall line (point downhill), then turn back across the hill staying on the same edge the entire time. If you do several of these in a row it makes the shape of Christmas garlands in the snow. So yes you will only be going one direction over and over again, then you change edges and go the other direction.
Alright, so how to do garlands properly (or how i do them):
-Keep your shoulder in line with your board for all of this-
Toeside: First start off in a sideslip to find a great toeside stance, do this by bending your knees, allowing your ankles to flex (aka collapse, avoid pushing down on your toes as this will extend your ankles), relax into the shin of your boots, now notice how if you push your hips over your toe edge (like your standing at a urinal) you have plenty of weight on your edge, and as a result have great control, cool, maintain this position. To allow the board to seek the fall line (aka point downhill) roll your knee out towards the nose of your board, and gently lower your heel, (keep your knee bent, or even better bend it more during this part) this will flatten out the board under your front foot and you will pretty quickly point down the fall line. Then when you have reached a desired speed, drive your front knee toward your pinky toe until the board is coming across the hill, then drive both knees into the shins of your boots to slow down. then repeat, til your extremely comfortable with it
Heelside: Start from a stop, or barely moving on your heelside with your board across the hill. To get the board to seek the fall line, relax your front ankle, and add weight to your front foot by flexing your front knee this should also keep your shin in contact with your boot here. wait for the board to point directly down the hill. Then roll your front knee out (towards the nose) and pull up on your front toes, then both toes til your across the hill almost stopped. then repeat, til your extremely comfortable with it
Linking turns: Now that i am sure you understand all of that. To make shorter and shorter radius turns. Exaggerate the main points from above, and speed up the timing.
So, from toeside to heelside, exaggerate rolling your front knee out (while flexing it to add weight to it) and start pulling up on your front foot toes, as soon as you feel the board start to steer across the fall line pull up on your back toes as well. Make sure to look across the hill here.
Now from heel to toe, relax (aka flatten out) your front foot, while bending your front knee to add more weight to it, immediately start driving your front knee toward your pinky toe, as the board steers across the hill do the same with your back knee make sure to be looking across (NOT down) hill here
If that works, great have fun. If its still not fast enough, think about making every movement a little more powerful, and make it happen a little faster (imagine the difference between stomping on the gas pedal and slowly depressing it to the floor, it ends at the same place, but it got there faster), now make the next move sooner as well (speeding up the timing). If you know you are doing all of that and you still want it to be faster, add a slight steering with the hip (like theres a light on your hip facing your nose, and you want to steer with it), but do NOT let your shoulders follow (keep them in line), and be sure to only use this with the proper ankle and knee movement i talked about above...
Hope this helps!
Responses to others
MGD81: I like alot of what you said, especially about knee steering on toeside. However i do not like to turn the shoulders to get pointed down the fall line. Will it work? Yes, but it will not be as efficient, as fast, or as balanced as it could be, and will cause other problems later on in the turn, and later on in snowboarding
Snowolf: Your knowledge here is also your downfall in my opinion. Everything you said is right on. However i don't think cootcraig on day 5 needs the techie definitions, or big words, or dynamic turns. Your explanations of movements and performances and blending them is amazing, impressive really. I'm just not so sure he will be able to put this mass of knowledge into action. Save the AASI terms for clinics, keep it simpler for lessons.
Example: I could say to utilize plantarflexion with your rear foot and dorsiflexion with your front foot to fully use twist to initiate a heelside turn, then flex both of your anterior tibialus muscles through the completion of the turn making sure to use more angulation than inclination. But who would that help?