He is 16 weeks. We have a large pen in the kitchen. At one end is his sleeping area and at the other end is a large cookie tray for pooing and peeing. Sometimes he pees on the lg. tray and sometimes poos on it. Most of the time he does his business anywhere in the pen including on his blankets, etc. It is too cold to take him outside.

Does anyone have a suggestion on how to train him to do his business always on the tray?

What breed of dog? Any reason you aren't trying to train your puppy to go outside?

What are your current training methods?

He is a Dachsund. It is too cold to go outside at the moment. We hope to start the "outside" process soon. In anycase, is it possible to train him to use the tray if we aren't around to take him out?

Our current method is to take his nose right up to the poo or pea and say NO and then put him on the tray and say "good boy". We are truly not sure what we're doing.

First off, you should be able to train him to go outside. Get him a coat. Training him first to go inside and then getting him to go outside will be a hard transition.

NEVER rub your dogs face in what it's done. If you don't catch him RIGHT in the act, his mind will not be able to make the connection as to why he is being punished.
Even punishing him this way when you catch him IN the act has negative consequences, as he may be afraid to urinate or defecate in your presence - which will make housebreaking outside very hard.

I don't know a good way to train dogs to go inside as I don't do that myself (our Chihuahuas had to learn to go outside in the winter and we live in a pretty cold climate).
You need to learn a routine with your dog. He should be going out every few hours. Make SURE to take him out after every time he plays, eats, drinks, wakes up, etc etc. If you come home and he has had an accident - ignore it. As I mentioned earlier, he will not be able to make the connection as to why you are mad at him.
Just take him out. Praise him to no END when he pees and poos outside. Ask him if he wants to go outside before you take him out. I've had people get their dog to ring a bell with their nose before taking them out, so they eventually make the connection to ring the bell when they need to go.

I'm sure someone will come along with better ideas, but I'm super tired. I hope this helped a tiny bit.

Thank you for your time and suggestions. We will begin the outdoor process, but must figure out the indoor training, as well.

Don't even bother with the indoor training. Any time the dog needs to go potty, then take him outside. What I did with my little puppy when I got her in Jan, was any time she needed to go out, I went out with her. Not only does that reassure you that the puppy is going potty, but the puppy will actually go. If you just shuv the pup outside and peek through the window, the pup will more than likely sit by the door shivering. Pottying outside is always a big deal. You need to be there every time and praise.

As for showing the puppy its mistakes... Puppies don't understand that. Their moms always clean up after them. They're not going to get why they are being scolded for pottying in the nest. What I do with all my puppies during house training is I never let them see me upset with them when they go potty... It may sound gross, but it's a good thing when they go potty in front of you. You can correct it. You can tell them to go outside. IF you are going to scold... you have to catch them doing it... Even if its 10 seconds later, the puppy will not understand why you are upset. It has to be caught in the act. And don't scare them. If you scare them, they will sneak off and go potty somewhere else, and the problem will never be resolved. When I catch my pups, I usually say a stern "NO" then an upbeat OUTSIDE.

Seems like the puppies living place is too large. If your able to, get a crate and crate the puppy. I always advise putting the kennels in rooms so you can get up when its time for puppy to go outside. Remember, your not just training the puppies mind, your also training their bladders to hold it for a certain period.