Let's not get started with that... those same people also say "viertel vier" to mean 3:15 (one quarter of the "fourth hour" has passed), which is really confusing to the uninitiated, so "viertel nach"/"viertel vor" is preferable IMHO...

It does make sense though (once you know where it comes from):

Before the ubiquity of watches, time was announced using church clocks and bell strikes. There's a big bell for hours (low pitch) and a smaller one for announcing quarters (higher pitch).

Signalling zero is not possible using "zero bell strikes", so 00:00 is signalled by 4 strikes of the quarters bell and 12 strikes of the hour bell.

Thus, the sequences go like:

11:15 1x quarter bell

11:30 2x quarter bell

11:45 3x quarter bell

12:00 4x quarter bell + 1x hour bell

Basically it makes sense then as all the quarters belong to the same hour.

I have used those time expressions for over 55 years. Never thought about the explanation.

Yes. The other explanation is that time is nothing special, it gets just counted like everything else. You wouldn't say it's a quarter to a full cake either.

If we follow your argument, the number 3.25 would be read "point two five to four" istead of "three point two five". Which is to say, the fact that the quarters are mentioned in connection to the next hour, not the previous one, is indeed unusual.

Quarters are named by the hour they are in, neither the one previous or after. The hour number four of the day starts at 3:01 (or 3:00:01 with precision in seconds) and is complete by 4:00, it's the same mistake that people make with centuries.

'dreiviertel Vier' is short for 'dreiviertel der vierten Stunde des Tages'.

We can agree that there is a big disagreement, but "viertel nach"/"viertel vor" just sounds plain wrong to me.

It's absolutely correct German. Native speaker here.

Me too. I know. Doesn't change that it sounds plain wrong to me. I guess probably how "dreiviertel X" sounds to you and others. Germany isn't uniform and only a nation state for some short time.

If I try to rationalize it, it is probably that a quarter (to me) is not a distance or difference, but a single thing. So my internal parser expects a Genitiv or another thing after it, not a preposition like 'vor' or 'nach'. 'Zehn Minuten vor/nach X' sounds fine to me, 'eine viertel Stunde vor/nach X' too, but 'viertel vor/nach X' just doesn't.

it's still correct and should be at least accepted as such, even if not explicitly taught

Let's not get started with that... those same people also say "viertel nach drei" to mean 3:15 (one quarter after the "third hour" has passed), which is really confusing to the uninitiated, so "viertel vier" is preferable IMHO...

Yeah? Well, you know, thats just like uh your opinion, man.

And, its dismissive and ignorant. IMHO.