Hey! You got pics? What kind of substrate are these birds on? Do you see any injury? Joint swelling? Are his legs splayed?
If he's been doing this for a week, he needs help soon. It could be something else, and you might need a vet.
Ducklings and goslings are very susceptible to niacin (a B vitamin) deficiency, which causes (and is usually the first suspect with) leg problems. What are you feeding the duckies? Just plain chick starter or poultry ration isn't really good enough, especially if the bird is young or laying. Some people supplement with brewer's/nutritional yeast, while I like to grind some B vitamin (make sure it's the right kind, niacin, B3 complex) into the food. How much you put in depends on the dosage of your tablets and how much food you're giving at one time. If you grind the tablets, be sure to wet the food a little, because you want it to stick to the food...otherwise, it'll just settle at the bottom. Not sure if the protocol is the same with yeast.
Also, if the legs are splayed, you may need to make a hobble. There will probably be some wiggling and complaining from the duckling, but given that it's done right, it can save the ducky's life...Splayed legs can be so uncomfortable/painful that the duckling will stop eating and drinking, so it's definitely a life-or-death situation.
edit: Please keep us updated! Well wishes for the little one.
edit 2: Also, if the duckling can swim, that's generally good PT! Get the little one swimming a couple times a day.
