Or practice scale patterns with one hand while doing something else with the other?
Once the basic fingering of a scale is in place, the real challenge is to master passages that use them in snakey, winding ways. In my opinion the most complicated/challenging key is C major, because there are SO many alternate fingers. By comparison, in B major both hands will put thumb on B and E, and where the other fingers go is logical and soon automatic.
There is a difference between a scale - in theoretical form, starting on the tonic and running up and down multiple octaves - and passages that keep reversing directions and that somtimes skip notes in the scale (more of what Hanon does).
I prefer to teach pieces that keep one hand fairly stationary, leaving the other to sort of "explore patterns", in a practical and musical way. You can find lots of examples in Mozart, to name just one composer, where one hand is doing almost nothing while the other is racing around.
The biggest challenge is finding enough materials that balance that out with scale patterns in the LH while the RH is fairly stationary. This why I teach Bach later than most teachers, because so often BOTH hands are doing complicated scale-like movements at the same time.
I tell my students. There are three levels of Bach:
2) Very hard
As a student, I like exercises. Let me see if I can explain why.
If we take NeilOS' example of 2 measures from K. 284, while I was working on that as an exercise I would most of the time be feeling like these 2 stupid measures were keeping me from being able to play that section of the piece. Plus, unless the exercise included to transpose those two measures into all keys, or diatonically up and down the scale, I'd only be learning how to play those two measures, and not something that might apply in lots of places. I'd rather have an exercise in advance that works me through a particular skill, and then when I meet it in a piece, there's a big payoff because it makes that section easier and more familiar than it would be otherwise.
Sometimes that is a great idea, other times it just doesn't work. Remember, playing the same pattern in different keys may demand totally different fingerings. Not long ago I fumbled through the famous part of the "Minute Waltz" in C major. Of course PART of the problem was trying to transpose it on they fly, which almost totally negated any finger memory, and I could no longer just feel the pattersn. But in addition, it felt abnormally awkward in C. I think my student probably thought I was incompetent. I sounded like a bumbling student. But then I played it again in the right key, and it went like greased-lightning. On the other hand, doing some of the patterns in C major - in the K. 545 first movement, Mozart - feels TOTALLY different in Db, and much, MUCH harder.
I spent about three months last year working every day on a variety of exercises (new exercise every week or two) involving chord inversions. The result is that now when I meet an inverted triad in closed position, BOOM, my fingers know how to fall into place. No effort. Compare that to a time slightly before that three month period, when I was trying to learn a method book piece with chord inversions. No luck at all. What the piece presented me to practice was too narrow for me to get comfortable with the inversions, plus I would never have had the patience to practice that one piece for three months. It would have felt like I was making no progress at all.
That is a perfect example of a 12 key skill that IS practical to have. We continually encounter common chords in all inversions, in all keys. Though I would add that "inversion" eventually means bass note, because in the long run most chords end up being in open-voicing form. Looking at any chord, in any voicing, and being able to instantly get back to the simple, root position chord, is what I call "unscrambling". Those of us who have played for many years do this on autopilot, so we only realize who hard it is when we run into a very unusual chord, spaced widely, inverted, and have to think a bit to get to what the chord really is.
And in my experience this "unscrambling" is done much more quickly and extensively by jazz pianists, or artists who are known for "crossing over", at home in both worlds.
I've been working on scales on and off for the past year, and the big payoff for me is that now when I meet a scale in my music, BANG, I can play it. So that's one part of the piece done and dusted, instead of, say, encountering G major for the first time in a sonatina and being held up by those few measures.
I had to absorb and use scales so early in my life that I honestly don't even remember when and how I first learned them. I THINK I had them nailed on brass before piano, or maybe it happened on both instruments. Are you aware that all major and conventional minor scales are always alternating patterns of 3 and 4 finger groups?
I'm currently working through exercises that, along with whatever else they're asking me to practice (generally either chromatically or diatonically up and down an octave), they ask for variations in dynamics. This is constituting a months-long study in varying dynamics between my two hands, and when I emerge from it, the next time a piece has some dynamic challenge, I'll be halfway or more to being able to play it already, instead of having a brand-new skill stopping me.
There are enough complicated things that happen in every new piece, that if I can have pre-learned some of them through exercises in advance, I find that a great gain.
You can see how two extremes can lead to different problems. If you have no skills developed apart from music, when you run into a specific technical "demand" that is foreign, you have to stop everything until that skill is mastered.
But if you try to prepare for all eventualities, in advance, there will always be new things you could never predict that you have to more or less master and then use - there is no way to prepare for them because they do not exist until you run into them.
I would say reasonable technical preparation, apart from playing music, is reasonable and good. Now much you absorb from music you are learning, on the fly, will be personal. For instance, if you know scales are alternating 3s and 4s (groups), when you hit scale patterns, alterned, that have more or fewer notes, you can adapt really fast. Here is a very common example, descending:
F E Eb D C Bb A G F (etc.)
There you will use 4 on E and Bb (RH) to zoom down the piano. No major or minor scale will totally prepare you for such an alteration, and this is a common one.
I've been looking longingly at the Chopin C# minor Nocturne, and the fast runs at the end are way beyond my skill right now. But for me, what that suggests is that I find someone who can teach me how to do multi-octave scales very fast, and practice that. Then I can pick up the Nocturne and not have one part in particular standing out and stopping me dead in my tracks.
Which one? Always give an Opus number.
I wonder what goes into people's makeup that makes them prefer one approach or the other? And also wonder if teachers are sensitive to their students' proclivities in this matter, and ever adjust how they teach in response?
Absolutely. Each composer makes slightly different technical demands (understatement), so what works for Brahms and what works for Bach fugues is going to be VERY different. For every obvious reasons, becoming very good at playing the music of both composers, eventually, will stretch you.
As for scales, what about this?
C Bb G F#/Gb F Eb C. Blues scale. How will you finger it? How will you finger it in all 12 keys?
You are now in a different world. Octatonic scales, really fast? Another different world. Whole tone scales? Yet another.
And so on...