I do believe that beginner students should simply play as much as they can in the first year. This is a tricky situation to describe in words, I do believe the beginner is among the hardest student to teach because what works for the individual is always very different. One might argue that an advanced student is the same but this is not so, they understand how to learn and play the piano thus directions you give can be general but understood. Where with beginners if you are general they will have no idea what you are on about. You must get into the beginners head and hands and nurture small change in them.
I have never taught pieces out of a single book, I don't believe that one book can be written in such a way that it is relevant for all students. If we do a book we do whatever I think will help the student learn the piano AND what interests the student. If these two factors are not present then usually we do not do the piece. There is a point in slowly increasing difficulty of a students playing, like I mentioned in my previous post, making the LH more complicated. These address basic coordination issues.
It is important to ask your student to know what they like to learn. What pieces interest them as opposed to what doesn't. Some have no idea, they just play whatever you give them, more often than not, these students do not learn the piano for very long. Those that have a taste and are eagre to choose pieces they like, these students usually last a little longer because they have a personal interest in the music they learn. This is why I don't like teaching out of a single book. It takes away our ability to choose pieces and know what we like. Most teachers teach from books because they do not know enough pieces, as a teacher it is our responsbility to get all that material, be able to weigh which pieces are useful and what each piece aims to develop. Then we have to consider our beginner student and question whether a piece is suitable or not. This process is quite complicated and difficult to explain in words, it of course is dependant on the student.
Difficulty in the piano is subjective. A preliminary piece played with one hand might be hard for an early beginner! Putting one note in the Lh together with the Rh might be an astronomically large jump in difficulty! I am exaggerating of course, but this is the situation with most beginners. Any time you give them something they haven't experienced before it can result in a slow learning curve. Thus an axiom that teachers can use to deal with improving beginners is, baby steps. Make small changes and work with the technique that the student currently displays. Do not try to recreate or even create technique in your beginner student.
You must always work in terms of your students hands and ability level. That is what you make small changes to, not wipe it clean and start from your own starting point. This often means playing a lot of pieces which are not much difficult than each other, but merely demonstrate basic tools of playing piano. The Hook method (Lh support Rh melody) mentioned in this thread is a good starting point for early beginners. In fact I am yet to meet an early beginner that needs to learn single handed pieces because they have no ability to use 2 hands at once, get them using 2 hands straight away.
There is also no point in obsessing about good technique if the student does not practice by themselves. If the student does not display a discipline to their work then there is no point in being specific in teaching them. If they cannot get the basic things how can we expect them to get the fine touches? We simply teach them the core basic tools of piano playing, they will play 2nd rate, but so what? We as teachers have to allow for students who do not practice a lot (and this will be the majority of your students if you teach the general public). Those that do practice and are organized and disciplined in their work, it is often a lot easier to know how to improve them since we can be more detailed and be confident that the student can practice their difficulties alone and not only with our supervision. You get those students that can only learn when the teacher is there with them, it is important to get them to work on their own.