Dear Pianoscholar,
Congratulations and best wishes for embarking on the study of Music. Its world will merge with yours, and every time you are together, you will be "elsewhere" in the wonderful non-verbal language, walking on clouds.
Ok. Für Elise is a mandatory milestone for every pianist. Even a piece defined as easy is not easy... and "Für Elise" is not. First of all, the good balance between right and left hand, the great cantability of the central episode, and the phrasing! Correct rhythmic division and... atmosphere!!!
As far as the A minor arpeggio episode you are studying is concerned, it must be kept in mind that the tempo is the same!!!! It's true... many pianists perform it faster, almost to identify the passage as a "connection" before the "reprise," but it is not necessary, and perhaps not quite appropriate. One should try to play it in tempo.
If, however, we want to talk about speed, then you can try studying it faster... but not continuously. How? With a short pause on every beat (waiting for an empty metronome click) in two different ways; it is important!!!
1) stop at every beat and restart by repeating the notes (of the right and left hand together) from and to the arrival point. And so on. Try to play a bit fast within the pauses.
2) stop at every beat and restart going forward until the next beat, but without repeating the notes of the arrival point, instead starting FROM the arrival point. Try to play a bit fast within the pauses.
What is the advantage? One can observe which note and position errors the hand makes within the two pauses without sacrificing speed. I will give a lesson on this concept. I want to clarify and affirm here, however, that studying slowly does not always prepare one for speed and clarity at speed. Let's take an example. If we walk and cover 20 meters, our feet, our knees, our whole body perform certain movements. If we force ourselves to cover those 20 meters running, naturally our movements will be, besides being faster, also different. If we have to run, we cannot always practice and "train" by walking. In reality, the movements in speed, even for the hand, the arm, the wrist, the forearm, etc., are much more complex, but the advantage is that they are smaller and shorter in duration. Another world!
So for now and in the future: Prepare, yes, the piece by studying slowly and assimilating the notes with the correct fingering... but then observe speed... by studying with pauses.
Fast study with pauses does not "distort" what will happen in performance, but rather "dissects" the continuity of the performance and allows us to observe and correct the complexity of the movements.
Furthermore, later you will discover that the fingering chosen for a piece when playing it slowly is not necessarily appropriate or effective for the same piece played at the required speed.
I hope, even if very briefly, to have been of help to you and above all to have been clear. Let me know
Paolo