Jane- there IS a solution to your problem- buy a better board, dont just go to Joanns and buy one, get a creative mark artists cutting mat from jerrysmatarama.com, the are less expensive, 3 times as thick, i just bought one, but my mat from joanns was wearing out too, doesnt matter what size you get, the big one is only 34$. ill never buy another mat from joanns thats for sure, i got a 24x36 and it weighs 5 pounds!! and with a better board, you will have longer blade life too.
good luck,
katz