Cutting Mat Has Grooves
The idea of using the back of the mat for cutting batting and fleece is a good idea too.
Cutting mat has grooves. The other posts about using a sharp blade are true as well a dull blade cuts a wider groove allowing the fabric to be pushed into the mat. When the moisture is gone the mat dries out and will have cut marks and dull your blade. A v grove provides an accent line around the outside of a mat opening giving an effect similar to the ruled lines on a french mat. Soak the mat in room temp water for about and hour let dry.
Cut the artwork opening in the normal fashion with the mat face in. I have one old mat soaking now. When you use a rotary blade on a hard plastic cutting mat you leave a small groove in. I just gently bend the cutting mat over the edge of the table or over a coffee can.
To clean the fabric whiskers from my rotary cutting mat i dry scrub it with a greenie which is really designed for scrubbing dishes. The mat will be soft and be like new again. I would think if you have to push so hard that you leave grooves in your mat you either need to replace your blades more often or definitly switch mats my 24 x36 mat took almost 10 years before it was faded kind of needing to be changed out you really should not have to apply much pressure when cutting with a rotory cutter and you should never use a knife or other tool on a cutting mat they are made for rotory cutters anything else will damage the mat once there is signs of damage it. Reset the mat borders for the desired spacing of the v groove from the mat opening.
If you have a self healing cutting mat it needs to be re hydrated to stay self healing. It reminds me of a c filter material there is a trick to removing all of the fabric particles from the cuts in the mat.