"I didn't say I teach purely intervallic reading. I said it doesn't matter what we call the notes, as long as you can see it on the page and translate it to the piano."
I just disagree slightly here. It was specifically about the idea of thinking step and jump etc. When in earlier stages, the different positions are so similar, that the student may only be feeling distances between fingers. After finding the first note, they may just think two fingers away etc. This may not develop any association to the geography of the piano. If they have to think of E being two notes away from C I believe they're much more likely to associate the symbol with the note on the piano- not just with a relative movement. They might be able to play it, but it doesn't mean they're building up any associations beyond the physical.
"You even admitted that the point is to associate the note on the page with the key on the piano. That's not intervallic reading at all. Purely intervallic reading would not necessarily produce the correct notes because if all you saw was "start, up a 2nd, down a 2nd, up a third," where would you play it on the piano?"
I don't "admit" this. I actively emphasise it. It's one of many important aspects. The problem is that once the position has been found, everything that comes AFTER this point can easily be purely done on intervals. It can mean a moments thought and then basically no more to follow. Keeping the letters involved helps keep many all important associations involved.
"Good question. I still think by intervals. I can quickly recognize 3 lines as a 5th, 4 lines as a 7th and 5 lines as a 9th. So if I see a note on 5 ledger lines above the treble staff with nothing else nearby, I think "9th from A" rather than "A C E G B = B." So, once again, it's not purely intervallic because I'm STARTING from a letter name."
Exactly. That's my point- there is constantly some form of recourse to absolute reference points. All it takes is one note to go wrong- and then you're lost, unless regularly checking. One thing I realised a few years ago is that, while I emphasised intervals in reading and though I depended almost entirely on it, I am still rubbish at reading the alto clef. I can go slowly and follow patterns, but the patterns aren't enough to give the fluency I have in regular clefs. Although I do not consciously think about letters while playing either treble or bass clef, it made me realise how much I am dependent on knowing them extremely well. Without that, intervallic reading is possible but rather hard work, in other clefs. This made me realise that concrete reference points play a huge role- including within that which I thought was scarcely related to anything but intervals.
The idea of thinking of the 9th is interesting. I can identify anything up to a 9th at one glance, when the notes are written. However, I'll have to think about the idea of trying to imagine it as a single entity, when one note is only imagined I find that thinking up in thirds has become pretty much instant, but split-seconds can make a difference. Also, I should stress that I see the letters and visualisation of the note as synomous. So, when I think through the letters I'm picturing the notes on the keyboard- not finding a letter first and then figuring out where it is on the piano as an extra step. When I say that students should continue to think in letters, I mean in the sense of relating each and every one to the note- not in the sense of letters as a separate thing. I think that's why the one finger thing is a great accompaniment to saying letters out loud.
"Just so you know, I like discussing these things and "debating" them with smart teachers. I hope you are not offended or think I'm picking a fight when I disagree. It's hard to tell what someone's sentiment is with just text on a page, but I'm saying all these things with a friendly smile

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Sure- very much intended in the same spirit here!
"Just find it
1. Find the first note: E
2. Up or down? down
3. up or down? down"
I'd just add that it pays to think down= playing D and be reminding yourself what D looks like on the piano etc. Logically, it's easiest to figure out what to play with up and down. However, if you also keep track of letters, many more valuable associations are necessarily created than if you just think up/down. Also, any mistake and the student is screwed- unless they keep associating with an absolute. Something has to trigger alarm bells. Again this is the difference between an accomplished reader using intervals and a student. Students who go by interval frequently end up playing a whole series of notes too high or too low. If they are only feeling distances, they have no clue as to how far off track they have got. Accomplished readers can spot a slip at once, and pull everything back.
"It seems to me like adding an extra step to translate the letter name and then find the letter name on the piano when all they really have to do is follow the ups and downs."
Yeah, this is the big thing- I don't encourage that extra step of read and then find on the piano. That's the classic mistake in the wrong kind of letter thinking. I encourage the student to do the practical things in the easiest way i.e up/down etc. However I ALSO encourage them to have complete awareness of what letter they should be associating with the note on the page and how both that letter and notated is symbol is associate with a specific note on the piano- not merely a shift between fingers. Deeper associations don't necessarily form, when the student just takes the most direct and obvious route.