How can one check-up to make sure that a lesson has been truly studied and it is time to move on to the next lesson without a teacher?
This is a good question, Kristina, and is important, I suspect, for a lot of those self-studying with Alfred's and other methods.
The first step is to make a recording. You can't concentrate enough while you're playing to pick out all the errors and as a current thread will show, I can't even concentrate enough on a recording
If you're not consistent enough to get an error free recording in a reasonable time that should point to a flaw.
One of the first things to listen for is wrong notes. If your own ear isn't strong enough to tell (and I'm finding out that mine is in such a category), get someone else to listen honestly and objectively. At the Alfred's level it doesn't have to be a professional but if you can't get anyone, posting a recording in the forums (or a PM to a willing volunteer) with a request to have them pointed out is an option. I am currently undergoing this kind of therapy.
Next up is dynamics. Are you obeying all the P's and F's, all the crescendos, decrescendos and hairpins (< and >)?
Are you getting all the accents (typically the first note of each measure but some are individually marked) and touches, legato, staccato, non-legato?
Are your phrases complete and unbroken and is there clear separation between phrases? Are they correctly inflected, with dynamics as well as pitch? This might be a bit advanced but it's never too soon to start improving in this area.
Is the melody clear enough above the accompaniment or, in canon, individually brought out? Listen to a good recording of a song and hear how much more prominent is the voice. This is difficult stuff to pull off on one instrument and it isn't worth holding yourself back trying to achieve it but it is still worth considering regarding your playing in general.
Is the tempo reasonable for the piece and your present ability? This is subjective but for me I need to be playing fast enough that I'm not having to concentrate in order to get the right notes.
In pieces that are beyond our sight-reading ability we go through what I consider to be distinct stages. This is specific to my (current) way of learning but it has universal features. The first, for me, is mechanical accuracy with a complete disregard for rhythm or tempo. Just being able to play the right notes with the right dynamics in the right order. This may entail difficult stretches, leaps, changing fingers on one key, a change of the whole hand position and other mechanical difficulties.
The second stage is when I'm familiar with the mechanical actions and can begin to think of the piece as music and start play rhythmically but slowly and thinking carefully about which finger and which key for each note in each phrase.
The third stage is when I can play each phrase and concern myself only with the first notes of each. The rest of the phrase is a semi automatic response to either the printed notes or the memorised sound or a mixture of both. There will be a much greater fluency in the playing and it is usually accompanied by a fairly noticeable rise in tempo without my having to bother about it. There will still be hesitations and pauses at the start of some phrases or patches of difficulty that I need to isolate and work on slowly.
Finally, there is a stage where I don't hesitate or stumble at any phrase or section and this is again signalled by a further automatic rise in tempo. At this stage the piece is memorised and fluent (for me) but not everybody memorises and I don't know how well others judge their arrival at this stage.
It's at this stage that a piece can
be considered finished - for me. Others get to this stage before they add dynamics, phrasing and "polish". It's an individual thing - we all have our own methods, objectives, agendas and criteria.
Whether I'm at recital speed is another issue and, for me, is not an issue. There is a limit to my speed and my tempo usually increases over time from familiarity. I don't have to work at it. I'm not concerned with winning races. My own feeling is that tempo is personal and dynamic. If you can keep time with a metronome then you can keep time. If you can keep time with a metronome at recital speed then you can play faster without - if you choose to (it takes extra thinking time to synchronise with a metronome that slows down the paying) - and you are then free to 'interpret' the tempo and vary it in the piece.
If I can increase my speed in scales or in one of my faster pieces, it affects my playing across the board so I don't need to bring each piece up to a speed beyond comfortable.
Our progress is not measured by the difficulty or level of the pieces we play but by how well we play them. If you're progressing through Alfred's ask yourself honestly if the last piece you've learned is representative of your current ability. If it is, move on. If it isn't linger a while longer.
And as has been noted, there's no harm moving on to other pieces and returning to unfinished material after a spell away. We don't learn in gradual curves but by sudden jumps followed by longer plateaus. We need variety of material to see us through the plateaus and the Achievement of the Week thread to celebrate the jumps.