I was also taught to sew darts from the wide end to the point. However many, many years ago I learnt from a book (Short Cuts to Sewing Skills by Thelma Lane) to pull up the bobbin thread and reverse thread this through the needle, then pull sufficient thread through and tie on to the top thread (it is trial and error as to how much to pull through as it all depends on the length of the dart), wind all this surplus thread on to the cotton spool/reel and then start stitching at the point, you place the needle right on the fold of fabric and then stitch the dart with a few stitches on the fold and then slowly tapering outwards from the point. It does take a little practice, but the beauty of this method is that you do not have ends to sew/knot off which can create a little bulk, also they cannot unravel due to insecure sewing/tie off). Just remember when all the darts have been stitched to unwind all the knotted part off the top spool – there is usually quite a bit, can always be saved for any hand sewing.
So ever since I have sewn my darts this way, perhaps it is a bit of a fiddle pulling up fresh thread from the bobbin for each dart you sew, but to me it is worth it.
Incidentally I actually use a small needle threader to thread the needle the reverse way or if you are nimble fingered try knotting the thread, just a single knot and gradually ease this through the needle.
Incidentally I found this book full of ideas on better machine stitching, for a quick fix for seam allowances marked on the machine bed, use masking tape – especially useful for those with machines that do not have the throat plate marked in this way.