Peanut butter- just like for hair. Work it into the gum, just as if it were in your hair, gently pulling and breaking it up. The oils in the PB will gradually break up the gum, and prevent it from re-sticking. I'd be loathe to use goo-gone or a petroleum distillate based product- the carpet is made of a petro-chemical and I'd think might dissolve!
It would be easy to wash out the PB once the gum is gone.
Of course, if the mats are old, H and A have them at reasonable prices, and being 'ruined' by gum is a good excuse to replace them... Just imagine the conversation with your significant other: "I swear they're ruined, Honey, and I need new ones. You don't want me to ruin my floors, do you?"