Learning to play a tune/piece in only one key is a very popular piano method. People learn in different ways but I find that I get a lot of understanding when I practice it in other keys as well. Transposing is very important to me. I analyse and feel the music when I transpose it. Standard piano lessons never really works for me. I want to understand the music I play.
Why is it that most piano students are taught to only play a piece in one key?
Like you, I have a hard time getting something to work properly without a deep knowledge of exactly how it works. My driving teacher spent hours repeatedly explaining the sequence of steps required to drive a manual transmission and I just couldn't get it. Later, when I looked up the internal mechanics of exactly how a manual transmission works, it immediately clicked and I could suddenly drive a manual without issue.
So I too have an insatiable need to understand exactly what (and why) I am playing in order to play it properly. When I decide to learn a new piece, I carefully analyze each chord and passage, trying to internalize the technical structure and decipher the composers original intent. I also take time to gain an understanding of why exactly the composer chose the key that they did; oftentimes this also involves some study of why they didn't chose certain other keys.
This often comes down, largely, to fingering and technique. As you get into more advanced pieces, certain keys just don't make sense and may be nearly impossible to play. At the end of the day, classical composers had a very deep understanding of music and their choice of key was not arbitrary. Figuring out why a piece is in a particular key may well give you more insight and understanding than figuring out how to play it in other keys.
Now, to actually answer your question: why are piano students taught to only play songs in one key?
The main point of piano lessons is to learn proper playing technique, not music theory. Naturally, some theory must be taught, but it is usually secondary and supportive. If additional knowledge of theory is desired, one must seek out other sources, like online tutorials, music theory dictionaries, college courses, or this site (obviously).
(Alternatively, you can do what I did and try to find a teacher that loves theory as much as you do so you can lure them into spending the entire hour session in a deep theory dive! That is, until your parents find out that you have been spending most of your lessons chatting and not actually playing anything, so they get mad at your teacher who explains to them that you are "advancing" into jazz improvisation for which you need a solid foundation of music theory, after which your sessions are split between discussing theory and screwing around on the piano...yup, he was by far my favorite teacher!)
The ability to transpose, either in advance or on the fly, is a practical application of music theory, not a physical skill. A steady progression of the student's skill and ability is the closest thing a piano teacher has to a deliverable "product", and is therefor the easiest way to demonstrate to the student (or their parents) that they are getting their moneys worth.
Lastly, as alluded to before, most pieces, especially more advanced ones, are intentionally written in a key that provides a stable structure for proper technique. For example, pieces that require the thumb to frequently pass under the other fingers are often written in keys with lots of sharps/flats to make this motion easier and more fluid. Practicing it in a different key can make it more awkward to play, and can even result in picking up bad habits, negatively impacting the advancement of technical skill, thereby undermining the very reason for taking lessons in the first place.
I think it depends on what you’re going to do with the piece. Personally, if I had time to learn the piece in more than one key then I think I should have chosen a more difficult piece.
It might take me 3 - 18 months to learn a piece. After that, I’d rather learn something else than learn to play the same piece again in a key different from the composers original intent.
But, I mainly play classical piano. Accompanists for musical theatre, jazz pianists, and some other genres quite often are asked to sight read songs and transpose on the fly. Those professionals aren’t exactly learning pieces in different keys, they are just playing them in a different key. If you would like to play like that then there’s nothing wrong with learning the same piece in multiple keys and/or learning to transpose on the fly.
In the classical world, the key in which a piece is written is an important, intentional choice. I’ve never heard of a classical work being professionally performed in a key other than the one it was written in, although I expect it may have happened from time to time.
And now I’m reminded of a slight contradiction to my first paragraph, which is that I’m currently learning the jazz standard “Autumn Leaves” in a different key from what it was written in to make it easier for me to play the melody on clarinet (for which I’m only a beginner).
So in summary: in the classical world, transposing pieces is rare, probably because that would make it a different piece (in the classical mind). In the musical theatre and jazz genres; transposing is very common, but often the pieces are not learned in different keys, merely transposed on the fly (an important skill). In pop and rock transposing happens for covers and less commonly originals. For example, as Ozzy Osbourne has aged, he has lost his ability to hit some high notes, so some of his most popular songs, like “Crazy Train” have been played in a lower key in live shows.
To my knowledge, most piano lessons (at least ones given to children) are oriented around playing classical music. For classical music, playing according to the written score is important, and deviations from the score are treated as mistakes.
As Todd Wilcox mentions in his answer, keys of classical music pieces are chosen intentionally. The notion that a classical piece of music must have an overarching key is so important that they are often labelled with their single key each in their name (e.g. "Minuet in G Major", "Symphony No. 5 in C Minor, Op. 67"). Entire cycles of preludes and etudes have one piece in each of the major and minor keys - changing the key of any one of these pieces is seen as sacrilege.
Sheet music isn't cheap. As illegal as all my piano books said the practice was, all my piano teachers photocopied my repertoire and gave the photocopies to me. I strongly suspect my piano teachers were not alone. I don't think they're about to bust sheet music transposing my repertoire.
One more note: piano repertoire for young children is often in keys with few accidentals such as C Major, F Major, and G Major, probably for readability's sake. Good luck convincing piano teachers to change the keys of those.
To give a counter-example, The Russian School of Piano Playing encourages teachers and students to introduce transposition right from the beginning:
„From the outset of musical training it is important that the pupil be taught to transpose. Such exercises promote the development of ear, memory and keyboard orientation.“
As a specific example, on piece number 20, a simple one-line melody piece, the instruction says:
„The pupil should play the exercise in different keys […].“
So I would conclude that there is at least one piano school which recognizes transposition as an important concept of piano lessons.
I don‘t think that this depends so much of the education level of teachers or the education system like Stretto says. But theres some truth in it, there are actually teachers - generally - who are good in just one subject and transport their knowledge without seeing behind the horizon.
There would be great benefit in the task to transpose a piece in other keys. (But this shouldn’t be done by just using the same fingersettings and moving the hand some intervals up or down.)
I’m convinced that a didactic of more experience in transposing would help a lot for better understanding a piece and playing music in general.
Why doesn’t this happen?
Because any other method wouldn‘t fit with the expections of the students, their parents who probably pay the lessons and also the teachers themselves.
The goal of most people is to learn as many pieces as possible in a short time, and not the goal of seeing through and internalizing fully a piece.
And not every teacher would be able to control whether the student is playing correctly.
Of course there are other points like the key characteristics, the respect of the authority of the composer, the availability of sheet music in other keys, the lack of fantasy or insight of the benefit of transposing, improvising and transfer learning!
But the main problem seems to me that people want to reach a goal direttissima and don’t see that there would be a short cut what looks like detour at first sight.
A good piano teacher should apply different approaches and use a large variety of methods: transposing, formal analysis, improvising, comparing pieces are just some. A lot of this happens unconsciously. What is unconscious should become aware.
Your question seems to telegraph the expected answer: the teaching is focused on recitals and advancing through graded methods. That kind of teaching isn't concerned with understanding why things happen in music.
EDIT
I wasn't going to add this initially, but there seems to be a lot of interest (disagreement) on this topic of transposing.
Recently I started working on Bach's Two Part Invention in C Minor and I simply did not want to memorize it. I've done that before with other music, and eventually I forget the stuff.
I decided to take most of the expository material and break it up into two beat units which I then practiced by sequencing the stuff up and down by step. I also inverted the parts between the hands. I made myself a sort of shorthand score of what I was doing, like this...
...the double notes at the end of phrases are supposed to show the continuation by sequence up/down. Fingerings above/below are RH/LH depending on how the hands invert. I don't expect anyone else to read this, but I think the basic idea should be clear enough.
Now I've moved on to playing each hands separately, but I play it in various keys. The music itself modulated up a fifth so you can just repeat the exposition and keep going around the circle of fifths. After working through a few keys and getting used to the changing "topographical" feel it seems I can play it in any key comfortably without practice.
I'm not done with the piece yet, and certainly this take more time than just memorizing it in C, but I definitely feel like I'm learn much, much more this way. My hands move more freely and I better understand the actual tonal content of the music.
At the very least, you aren't the only person that thinks transposing is a useful study method.
