How can I dice vegetables on a small cutting board?
If you can't get a bigger cutting board, here are a couple pointers to getting the most done  CLEANLY in a small space (mostly tricks from my kitchen manager):


Use a sharp knife.  It's easier to control, and when you chop, pieces don't move as much.  Hone it at least daily to keep the edge aligned.
Organize your work on the board.  If you're right-handed, keep raw ingredients on the left side, and finished, cut product on the other.  
Keep the spot you're using to cut clear.  It may only be 4-6 square inches, but it shouldn't have anything to get in the way.  I like to use the bottom middle of the board for cutting and the corners &amp; top to organize stuff in various stages of cutting.
Remove trash (trimmings, ends, unusable vegetable scraps) from the board.  Either keep a trashcan next to the counter, or use a bowl to collect garbage.  This frees up space on your cutting board.  
You can use additional bowls to store uncut material, and finished results.  This frees up more cutting board area. 


All of these tips help, but I'll tell you what my chef told me:  the bigger your cutting board, the more room you have to work, and the faster you can go.  As your knife skills improve, you will be able to work in a smaller space without problems.  But first, start off with plenty of room.  As you you gain experience, your cuts become more precise, you organize your space, and work faster and cleaner.

With this advice in mind, you should move to a larger cutting board.  The extra space to work make it much easier to develop proper knife skills, because you're not trying to cramp yourself in.  I suggest the largest board that you can comfortably fit in your sink for cleanup.  With  professional knife skills, I can handle prepping multiple items on an 8.5" x 9.5" cutting board; however, I find that my 11" x 14" board provides a lot more room to organize my work, and is more comfortable to use.  At work, I use the biggest board I can fit on my station.