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I am an early beginner (about 1.5 months of self study) who is struggling with moving my hands without looking at the keys. I realize this is a skill that will take years to fully develop, but I feel like I'm at a loss regarding how best to approach practicing this skill.

Perhaps my most fundamental question is this: should I be thinking of jumps in terms of how far I need to move my hands, or in terms of the absolute location of the key I will strike? That is, say I'm moving from C4 to C3 - should I be be trying to mentally encode "this is what it feels like to move my hand down an octave" (this is how I've been thinking about it), or should I be thinking "this is where the C3 key is on the keyboard"?

I have also been surprised that I've found very few exercises designed to hone this skill. I was thinking of just moving up and down the keyboard with different patterns, but thought that instead of reinventing the wheel, surely there is already an established approach that I just haven't come across.

Don't worry about it and just look down. Seriously, getting the feel of big intervals is something that develops naturally over time as you play lots and lots of music with jumps.

Green check
You are combining two different issues;
Find a single note: it is very important that you work on ‘seeing a note on a score and being able to instantly play the right key’

Jumps from one chord or note to the next:
First. You need to instantly know where you want to go. That means you are good at seeing the note on the score and then playing the note

Then, Q’s advice, as above, not to worry about it. Just play. Your body will develop the proprioception to go from one note/chord to the next.

How do you get there? Play a lot of music at your level

It's quite straightforward: don't jump before you can walk.

Get used to intervals first. (Look it up if you don't know what an interval is.) Get very familiar with spaces between your fingers, with your hand in the same position. Your fingers just move to the right keys - after you've gotten very familiar with 5-finger positions. Remember, you didn't run before you could walk, and you certainly didn't jump before you could stand.

But now - after years of walking, running etc - you can jump over a cat that's just run in front of you while you're walking, without having to stop and think: how far, how high?.....am I going to end up stepping on it, because I can't react instantly??

Then, get very familiar with the keyboard. Can you instantly strike all the C's on the keyboard - with any finger - without having to 'think'/'count'? How about A's? How about all the other white notes (stay with the white ones for now)?

There is a pedagogical reason why blind jumps are not taught to beginners. Because first of all a beginner needs to develop a confident touch. You won't feel confident in your blind jumps for many years to come. Besides proprioception and spatial feeling it requires arms being flexible and deft like tentacles, and now your arms are like crab's claws. Everything that brings uncertainty to playing is considered undesirable at the beginner stage.

Answering the question, yes, you need to think about keys, like C3 and D4, not about abstract distances.

Thank you all for the feedback, it has been really helpful. I think the consensus is that the proprioception will come with time, and that I should make sure I am mastering the basics before trying to jump (ha) to blind hand movements.

I'd start with some Count Basie in C. Why would your arms be like crab claws? Take it as slowly as you need to.

Well certainly don't overdo it, but attempting to play without looking at the keys for a few minutes might be helpful. If you're curious about blind jumps, it's about remembering the layout of the keyboard, and judging distances/hand positions. For example, you can think of an octave in terms of playing it with fingers 1 and 5. An exercise which helps is to try to visualize the keyboard in your head, and then visualize your hands on the keys. You may eventually be able to play entire pieces in your mind's eye like that. And I think that's a large part of developing proprioception and feeling a sense of oneness with the instrument.

Start with trying to imagine yourself playing a small section of a piece in your head. At least that's what I did. Now I can play a lot of things comfortably without looking at the keys and it's very useful in some ways.

Oscar Peterson was famous for never missing a jump. On a show once though he was missing right, left and centre. Someone fortunately noticed the flap had been left down on his Bosondorfer ! All was well once it was flipped back up.

I have been practicing to play ragtime and stride for a few months now. Everyone has always said go slow at first until you build muscle memory. Go so slow that it is impossible to make a mistake. Speed and ability to play without seeing the keys comes automatically over time. That is good advice. I can now play a C bass note and go up an octave and play the chord in most inversions most of the time, but,not always I am still learning. Just look at the keys and play a C bass. Then move up an octave and play a C chord. Go slow over and over and you will notice your improvement. Do not do but a few minutes a day so as not to make your hand or arm sore.

Like you said, it takes years, don't bee to harsh on yourself. I have 4 years under the belt and I still fail occasionally even simple jumps.

As for you practical question: relative, by all means, in time you will have a strong feeling of how distant an octave is, how distant a tenth etc. and you will be able to reuse that knowledge in all instances.

If one can only do a few minutes before one's arm is sore, one should try to figure out how to change one's technique to avoid soreness so quickly.

This is something that will come to you over time. I remember when I first started out I tried to play Satie's Gymnopedies because it was supposed to be easy, but the jumps seemed impossible because I was still having to look at what my right hand was playing. It's a part of developing familiarity with the geography of the keyboard and your relation to it. There are a lot of pressing foundational skills for a beginner, and if you're receiving instruction you probably won't have too much material with lots of jumps to it at this stage. If I were practicing it I'd probably stick to within an octave's range at first, jumping to different notes of an arpeggio and using different fingers (1 and 5 the most since these will be used the most in jumps, 4 and 2 to a lesser extent, 3 rarely and mostly for marcato playing in my somewhat limited experience). Do it pretty slowly--you don't want to tense up your forearms, nor do you want to lock your wrist, and I would aim keep a pretty relaxed hand. I'd also try to center my sounding finger over the key so you can have a controlled attack. The benefit to sticking to arpeggios (or notes of a chord) is that a high percentage of jumps are to notes of a common chord, and as you gain more experience and comfort you can gradually extend the range of your jumps. This is something you should probably talk about with your teacher, because a lot is going on when you jump, and you don't want to habituate something you will later have to unlearn in order to progress.

Just a few comments:
First, I don't think your arm should be getting sore at all even as a beginner. This would mean that you are tensing up your hand, which isn't good.

Second, I think one of the most effective way to practice jumps is shadow practice. Play the first note, then move to the new position very quickly but do not press those notes. In my experience, you get a feeling for the new position quite quickly. That said, it improves with time and keyboard familiarity.

1.5 months of study... there is nothing wrong with being aware of the things you're thinking of - but if that is your most fundamental question, then you're overthinking it, and you're misguided.

At this point you should be following a syllabus of some sort which will put your focus where it should be, which is playing actual beginner pieces (which do not involve jumps), rhythms, scales and maybe for you some Czerny. Avoid Hanon for now; all you'll get there are bad habits.
And if you're very proactive lots and lots of theory.

If you're doing jumps, then I'll venture a guess that you're already turning into a Beethoven / Chopin drone. Playing nonsense like Fur Elize, the Moonlight Sontate and Etude's. It's a disease. Spend enough time on forums like this and you'll find lots of talk in the beginner forum by muppets flexing their muscles. It doesn't matter why, there are idiots everywhere.
In fact, the minute you read anything to do with Mozart, Bach, Beethoven, Rachmaninoff, Chopin etc.... just ignore the post, log off and go back to what you should be doing. The stuff I've already mentioned.
It's is extremely frustrating, yes I know.
It will get much much more frustrating and unless you start slow you will get nowhere, and give up.
Whether you say you can, or say you can't either way you're right. Ford.

Why not play the note also? I think practicing what will actually be done makes more sense that practicing something close to/related to what will be done. Also, moving reasonably quickly may make more sense than moving very quickly.

There are two things that go into a jump. One is the arc you go in while you move from one position to another and getting your hand shape correct corresponding to the target figuration. The second part is getting the articulation of the second group of notes. It's very useful to practice both separately. Moving very quickly makes you freer because you can't move quickly if you waste energy. You need to reach the target some time before you play it so that you have enough time to articulate it properly.

You say it's useful to practice the two things separately but without giving a reason. Just because one can describe something as being composed of two parts doesn't mean it makes sense to practice them separately. One could break the jumping movement into more than two parts, but it would likewise make little sense to practice each part separately.

I get the impression you are an intermediate player, but think you have all kinds of secrets to technique. Am I correct about your level?

Likewise, I don't think your reason for moving quickly makes much sense. I see no reason why moving reasonably (fast enough for the passage in question)as opposed to "very quickly" would cause one to waste energy. I don't think wasting energy is even a consideration.

You can judge all you want. Yes, you can say I am an intermediate player if you wish, I am certainly not very advanced. But what I'm talking about here is standard fare and a very common practice strategy. Just because you may not have been exposed to it doesn't mean it doesn't exist. I have consistently got the impression from your posts that your approach to the piano is quite dogmatic, and you appeal to authority way too much. If you want demonstrations of what I'm talking about here, just Google/YouTube shadow practice, and how to learn jumps on the piano. I think I have seen at least a dozen excellent pianists talk about it. Enough said.

Moving fast, well I'm talking about the transitions during shifts in hand position. If you're playing scales for example, the best strategy may be different.

There is nothing in my comments on this post that is dogmatic or an appeal to authority. It seems to the contrary that you are the one appealing to authority by referencing YT videos.