From anecdotal experience a dog who sits by their bowl and hits it with their paw until you fill it up can just be one of the most annoying things you can come by, especially when you are trying to limit their intake for health reasons. Canines often learn that annoying behaviors like that can get you to do what they want and therefore become so ingrained into their daily actions that it seems like something your dog may just have always been destined to do. This is not the case and you can correct these behaviors with a little understanding and some best practices around the home.
The Root of the Behavior
Most dogs will eat their whole bowl at mealtime. This is not because they were really hungry and needed to eat it all, but rather because as scavengers they do not understand that they do not need an unlimited supply of things to eat.
This behavior often leads them to asking for your food while you are eating, or constantly begging for you to give them treats or place more food in their bowl.
The problem is if you typically give them bits of food when they do this you are reinforcing the thought process and encouraging them to beg for more food from you in the future. It can be hard, but not giving your dog food when they beg is easily the best advice you could adhere to.
Once they have learned to do this, then each time you give them additional food you are reinforcing this behavior. It will not generally subside with time and can become quite an annoying behavior whenever you and your family sit down to dinner.
Children often fall victim to this approach and minding what your children give your dog can help prevent some headache for you down the road.
Just like them begging at the table, be sure that if they do something to get you to give them food then you do not react by giving them what they want. In my experience, this is tapping and kicking their bowl.
I made the mistake early in the dogs life of feeding her whenever she would kick her bowl, which to me signaled she was hungry. As she was at a healthy weight I thought nothing of it. As she has grown older and is not as healthy as she was in the past I have had to regulate her diet, and hear her protests in the constant clanging of metal bowl against tile.
Encouraging the Behavior
The only way to address it is really to bear through it. For awhile at least they are going to continue to beg, continue to annoy you to get what they want. They need to learn that whatever action they are doing will not get the result they are looking for. This can simply be learned with time and repetition.
Try to regulate their meal times and feed them at the same time each day. They will learn that at that time, typically in relation to something you do each day, that they will get fed. When you sit down to eat give them a toy or something they enjoy to distract them during that time. If your meals are at the same time each day consider feeding your dog at the same time you take your meal. That way they are eating there food while you enjoy yours and no begging ensues.
Try not to give them an alternative when they beg. If they beg for food and you give them a bone instead they will simply stop begging for food, but beg instead for a bone, as they have now learned that it what you give them when they beg. Ambivalence is what is going to cure these behaviors.
Other Solutions and Considerations
Begging for food can be incredibly annoying and the only way to stop this behavior is to work with your dog over a long period of time to teach them that it is not going to give them the results they are hoping for. A trainer or behavioral specialist can help you identify the best approach and the cause of the behaviors.
Quite simply, dogs beg because it works. As long as it keeps on working they will likely keep on doing it. To cure these behaviors you simply have to show your dog that begging will not yield the results they are looking for, whether it be to play fetch or get some food.
Regardless of how annoying these behaviors can be, correcting them can be quite simple. To a dog, begging is a form of communication and to you dog has no connotation in sadness. They beg because it tells you they want something. They look sad, not because they are, but because that is the face and actions that most commonly get them what they want.