My poodle used to do that (take one piece at a time to the rug) but Tatum just likes to make a mess! What she does is paw at the bowl until it tips over enough to dump some food out. Only then will she eat it - off of the floor. She's going to be really disappointed when we move into the new house. I already have new bowls there - they are ceramic and have the wider bottom that she can't tip over as easy. But they also aren't as deep - I think she tips the food out because she doesn't want to stick her nose down into the bottom. The other day she did that to her food bowl and it flipped completely over and only 3 little pieces of food escaped - the rest of it was trapped under the bowl. My husband said the look on her face was priceless! Like "what the heck do I do NOW??"!
And for those of you with new puppies.... yes, we had to throw out little pieces of food one at a time for her to chase just to get her to eat. It had to be a game for a while but she finally got to where she'd eat out of the bowl. All the experts say a dog WILL NOT starve themselves. They will eventually eat it if it's available. They just like to make us think they're helpless!