As this is the piano method which I use most frequently in my own piano lessons, and on which I was initially taught myself, I cannot promise an impartial review of this book: Me and My Piano parts I & II.
This particularly series of piano tutors (in two parts), the authors intended for the younger beginner. Waterman and Harewood had already produced another series when these books were published, which might best be described as more intense. I have certain strong views on how piano methods ought to progress and what they ought to teach a young pupil. Such tutors should, in my opinion, teach note reading and recognition first and foremost and at the earliest feasible stage; avoid unnecessary and monotonous exercises whose purpose is ambiguous and which serve principally to delay the student's progress in their piano studies; and, finally, it should progress with sufficient economy that the pupil who completes the method has already achieved a high degree of competence in understanding rhythm and pitch and in realizing them in performance.
Me and My Piano achieves these aims
Beginning simply with advice on posture at the piano, listed in simple and comprehensible bullet points, the method moves swiftly on to introduce elementary rhythms utilizing crotchets, minims, and semibreves. There is ample practice to allow the young pianist to become thoroughly acquainted with note-values before they start to read actual music, but not so much that an unimaginative teacher can hold students captive for weeks on end clapping and not playing.
It is perfectly possible, and I speak from experience, that an attentive piano student could leave their first lesson able to understand both simple rhythms and read notes on the treble clef stave. Me and My Piano is slower than some tuition books in that it teaches left and right hands separately, but this can be useful for the younger audience for which it was written and it is possible to start in the middle with the grand staff for slightly older pupils.
As for progress, it is sufficient. When a student at the piano has completed both parts of the book, they will be playing at a grade one level, approximately. It may be recommended for some to explore piano repertoire at a grade one standard before endeavouring to take an exam.
Comparison with other piano tuition books
It might be a little unfair, and not comparing like-with-like, to juxtapose Me and My Piano with intricately choreographed methods like the Alfred or Piano Adventures. My reason for thinking so is simply that life, and learning as part of life, are rarely so tidy. If that average pupil's brain worked like a computer to be programmed according to a predetermined and inflexible progression, such series might have greater value; Me and My Piano, in contrast to these, lays out the fundamental information that any student beginning their musical education will need to know, and various pieces at each stage that will help to practise these core skills. In my opinion, this makes it far more valuable to the piano teacher at the very least.
The most closely related piano tuition series would be John Thompson's Easiest Piano Course, which was published much earlier. The main advantage it might have over Me and My Piano is that it introduces the grand staff straight away, providing faster progression for the more able student; however, the early exercises in this work are also incredibly dull and repetitive. In theory, also, it might be a little better for teaching note-values, employing as it does constant rote learning from the start for letter names and their corresponding locations on the stave, but the trick would be getting any modern student to do it.