In answer to your question, I never used a divider in the crate, but when I had a WIRE crate I would cover it with a blanket to make it more like a cave. With a plastic airline type crate this isn't necessary. At 12 weeks with some consistency, as others have said, it shouldn't take long. Their bladders are pretty good at this point.
I have 2 dogs currently and one I got at about 6 weeks old, the other at 12 weeks. The difference in development and bladder control by 12 weeks is huge and this has to do with biology, not training. You will get there. Consistency is key.
To this day I send my dogs out to do their business very last thing before bed, whether they act like they need to go or not, and first thing in the morning they go straight out. They are now 5 and 12 years old. They know these are "business trips" and not play time and come straight back inside when they are done, no messing around, no playing, no running off to the neighbors. Be patient. Reward the good behaviors, it will pay off. The reward needs to come within about 30 seconds of the good behavior for them to make the connection, so other posters are right, you really need to catch them in the act of being good.
Other good acts to catch them in and reward include things like going to the crate on their own, laying down/sitting quietly (especially if you are doing something like cooking/eating), if they drop something or leave something they aren't supposed to have (a shoe, the cat, the chickens). It's easy to ignore a dog/puppy that's being good because well, they are just laying there, being good, but that's exactly when you need to fawn over them and pet them and praise them. Now when my dog really really wants my attention, instead of jumping in my face or climbing in my lap she goes to the middle of the room and flops down very emphatically and will deliberately lay down with her head on her paws and stare at me. This is not a natural behavior, but learned, she knows if she is calm and still and can quit making noise and quit wiggling that she will get rewarded, she will get petted, she will get to go out or play with her toys, eventually, and she will lay there staring me down for quite a long time too.