Very first scale PATTERN for a beginner to learn

Both of your teachers are right and both are wrong.
They are both right in their teaching of 3-, 4- and 5-fret patterns. 3-fret patterns are easiest, but, in my experience, lead to a fixation on 3-fret patterns. 4 and 5 fret patterns get you out of that blinkered attitude and teach you different ways of looking at the same patterns. You also get a feel for the transition from one pattern to the next, which 3-fret patterns don't seem to give you.
I recently bought the Rosetta Stone of Guitar, which take a slightly different approach to patterns. I discovered that the RSoG approach moved me away from pattern-stagnation.

On the other hand, both of your teachers are wrong, in that they seem to be giving you the impression that patterns are the way to go. Patterns, whichever method you choose, are only a means to an end. Patterns are just that - patterns, they don't teach you about music - it's playing "by numbers".
Learning what the notes are and what they represent to your piece of music is really what you're trying to achieve. Patterns can't help you here.

The aforementioned RSoG and Kirk Lorange's Planetalk (I don't have Planetalk, so I'm going by what others have said) give you a bridge between the world of patterns and of music and allow you to see what relevance they have to one another.

You'll never escape the patterns, they are what makes up a piece of music, but it's the difference between "I'm playing the first pattern at the 5th fret" and "I'm playing in Am at the 5th position".

As for my first scale, it would have been C in the open position.

Ok, here's the stock responses with a bit more detail, and a secondhand anecdote to boot:

1. There is no wrong way

Scales are just sets of notes. How you finger that set of notes can vary from the easy (the second finger, sixth string root, which puts all the notes in a four-fret range) to the hard (the third finger root fingerings, which almost nobody uses because of the awkward stretches involved).

2. Keep working on it, you'll get it

No matter how difficult something seems, it'll seem easier after 6 months. A few years from now you may even wonder why you found it hard. Familiarity cures a lot of problems.

3. Do whatever works (with bonus anecdote)

Since # 1 is true, this one must also be true.

I've heard Jon Finn (guitar instructor at Berklee) talk about learning scale fingerings from William Leavitt (who wrote the Berklee guitar method books). Leavitt showed him many different fingerings, which Jon then went away and practiced. The next week he told Leavitt that some of them were very hard... and asked if Leavitt really used all of them when he played. Leavitt said no, he only used a handful. Jon asked why he'd shown the ones he didn't find useful. And Leavitt said:

I know what works for me. I'm not arrogant enough to know what works for you.

Anyway, what your current teacher says has a grain of truth in it. But that's like a single grain of salt - it won't do much for the soup.

The reason three-notes-per-string fingerings are "faster" is in the right hand: you can continually use economy picking. On the sixth string you'll go down, up, down... and now your pick keeps moving down through the first note on the fifth string. Because there's no need to change stroke direction when you change strings, you'll be able to play a scale in this fingering pattern marginally faster than any other pattern.

But marginal speed doesn't matter. It's not like you're going to be 30-50% faster - and that means you'll be NO faster in a practical setting! Unless you're playing all by yourself, it's the rest of the band that sets the tempo. So let's say you've reached a speed of 400 notes per minute in this scale fingering, but only 350 notes per minute in your second-best fingering. The band starts doing a tune at a tempo of 168.

No matter where you play, your fastest speed will be eighth notes. They'll happen 336 times per minute - within your speed for BOTH fingerings... but the difference isn't enough for you to play 8th note triplets (because that would need a speed of 504 npm). Unless a fingering is about 1/3 faster - and none are - having a "fast" fingering adds nothing to your speed most of the time.

Of course, that's "most" of the time, not "all" of the time. If you get lucky and the band plays a tune at 100 or 200bpm, you'll be able to keep up with a smaller beat division in the "best" scale fingering... but only if you stay in that one neck position!

3-on-a-string fingerings are favored by metal players, who like to whiz through a scale run to show off just how fast they can go. You'll also notice that many of these rapid-fire sections aren't in time with the beat. If that's your thing, hey - go for it. But if you're focusing on melodic construction and musical phrasing, you'll find that no one fingering is faster than the others from a practical standpoint.

Oh, and to answer the question, the first moveable scale fingering I learned is the 6th string 2nd finger root (two octaves within four frets). That's also the first one I teach. The 3-on-a-string fingering you show is the fifth one I teach - after the more compact ones have all been covered. I think students benefit more in the long run by knowing the fretboard first, and worrying about speed a bit later.

Note from another newbie, so take it for what it's worth.

When I first learned my first scales (pentatonic minors, but that isn't necessarily what I'd recommend. Beside the point, anyway.) I didn't learn a particular key, but just the intervals. If you start with the intervals, you can get used to the SOUND of the scale, whether by playing it in a three fret pattern, a four fret pattern, or even (just to start) up and down a single string. (When you get to where you hear that the intervals are right, map out the whole fretboard by ear. It's FUN! Then you can choose your own patterns.)

This has stayed with me, in that I don't spend a lot of time thinking about "A major scale" but rather just about the intervals that procede and follow an "A" to make it major. SO it's tone, tone, semitone, tone, tone, tone, semitone, starting anywhere there's an "A".

I don't know if this is right or wrong, NB- it seems to work for me so far, for two reasons: It focuses me on the notes that make the scale, wherever they fall, rather than the shapes that make the scale (Which would always exclude some other options). And it makes moving between "positions" sorta natural- if you follow the same intervals, you can go anywhere.

Best,
Ande

The 4 Scales You Should Know post off my website should help. It uses simple first patterns. Root on the 6th string-1st finger.

Hey! And here I thought that I was mercenary!!!

Cat

Thanks for letting me know what fingering patterns across the frets you all use.

I mentioned to my instructor how it was difficult to make the stretch and he offer to try something else, but I told him I wanted to work on it for a couple of more weeks before considering that.

I suspect that learning it the way he does it will make things easier for him to see in my playing as we go along.

f/b in beautiful wentzville, mo

It took me awhile to assimilate but Tom, that was some heavy (and well-instructed advice).

+1