Many of the Systema folks believe that training on a hard floor is better (e.g., more honest training partner) but the Toronto school has really nice mats
Having done it on mats from thick to thin and all the way to basketball floors and even concrete, I firmly believe that you START on good mats, and use mats (or soft ground) for the majority of your falling practice, but that you need to take it to progressively harder surfaces.
I personally learned to fall mostly on Aikido mats, but have taken safe falls on the steel of oil rigs and parking lot concrete (non of this involved 'self-defense' but it might well have saved me serious injury, even death.)
Make learning safe. Don't exact so much punishment from practice that people will be unwilling to pay the price. Do SOME practice in realistic conditions.
Falling is likely a more important skills than martial arts per se -- you may never be criminally attacked or need your empty hand skills, but if you live long enough you will almost certainly fall and may be severely injured if you cannot find the ground safely.
This is a common (many times indirect) cause of death in the elderly. And fear of falling frequently limits their social activity and isolates older people who avoid physical activities.
Learn to fall safely. Practice safely.
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HerbM