If you have not yet got one, then the purchase of a Turbo or Titan Peeler. http://www.asseenontvvideo.com/511654/Titan-Peeler.html is a good and worth while investment.
You'll need some newspaper (Sunday Telegraph is a good choice) several sheets of newspaper or if no newspaper then some supermarket plastic shopping bags will do.
Notice the set up. I have arranged the Titan Peeler on the chopping board, the peelings of the carrots already done to the right of the board and the carrots to be done on the empty bag.
I start by topping and tailing the carrots, using a knife. Then I'll peel four carrots (This is just my choice, as I like small numbers). Then using a knife I'll chop two carrots together (This is entirely possible with a suitable knife).
I then empty the carrots into a colander, rinse and repeat until all the carrots in the bag are peeled and chopped. Once they are all done empty the carrots into the pan and cover with water.
If you'd like the full report, then you can purchase it for the grand total of ninety nine pence from the Lulu button on this blog.