The trick is to not cut the root of the onion off. Cut the whole onion in half going through the root so that each half has a part of the root on it. Then lay one half flat and cut off the end without the root and skin the onion. When chopping make cuts away from the root then make cuts perpendicular to these starting at the end with out the root. Your very last cut should sever the root from the onion and hey presto you'll have chopped onion on the right and a piece of root on the left - that is if I have explained it properly ; -)
Don't laugh but I always suck on a spoon and I have yet to cry ..... well only from the tears of laughter when I see what an idiot I look in the mirror!!!!
This is a very late answer but if your eyes water when cutting onions, run the cold tap and stick your wrist under it, this takes the sting away really quickly! Don't ask me why!!