I have cut both ways and still do from time to time..Makes no difference as long as your cut is accurate...

Are you by any chance left-handed? I found out I had my blade on the wrong side of the cutter after about 3 years of trying to improve my cutting.

I work in a local quilt shop and we teach students to use rulers..they are more accurate..and always be consistent whatever method you choose..use the same ruler for an entire project. It really does not matter if you choose to square up the left or right side of your fabric..personal preference..

DebraK, I smile every time I see your avatar with your cute little feet in your cute little slippers.

JeanneS,
As long as you're not cutting toward yourself , what ever works for you is fine.
Sharon W.

The main problem I had when cutting it your way is if my cutter went crooked and I was cutting, for example, a three in strip of fabric, I had to pretty much put aside that piece for scrap or try to salvage part of it. But, if the three inch strip is under my ruler, it is completely protected. Then if I cut a little wonky my 3 in. strip is fine and I can square up that little wonky part and only lose about an inch or less of fabric. While both ways may be totally accurate as far as cutting. I waste less fabric with the strip I want to use under my ruler.

Thanks so much for all your comments! You have given me some great advice and I am going to practice some of your techniques. I don't quilt with a lot of really tiny pieces so haven't had any issues, but can see the benefits of cutting with the strip under the ruler. Maybe like some of you I will get comfortable using different methods.