My son is 13, and just moved up in violin size, from 1/2 to 3/4. How do you determine where to place your fingers on the fingerboard?
He’s not in class right now, but starts a music camp in a week. He needs to practice on the new violin to get used to it, but I think it would help if he didn’t develop any bad habits on the new violin.
I know how to play harmonics on a guitar, and I could try that on his violin, if that’s the right approach.
I started in the sixth grade with a full size. There’s a general area on the neck for each position. You find the exact correct position by ear.
here’s an online chart. you can click the notes for the pitch. I wish we had these 30 years ago. 
Some of these positions are rarely used. I don’t recall using any beyond third.
shifting positions
Really it’s just done by ear. You can make it a bit easier by using other strings as a reference. Third finger on any string, for example, should be exactly one octave above the string below. Just play, say, third finger E string (a high A) and the A string at the same time. Mark where the finger is with a pencil and then put the tape down right there. Similarly, first finger on any string will be a perfect fourth below the next string, which is another relatively easy interval to tune.
You could also use a piano or something else as a source of absolute pitch. Play both together to find out where the finger should be placed on your violin. Don’t worry too much about precise placement; tapes are a crude sort of marker anyways.
Unless he’s very small for his age, your son should be playing a full-size violin by now. How is it that he’s been playing a 1/2 until recently? I started playing at the age of 10, and always had a full-size instrument. It really wasn’t a problem.
As for finger placement, it’s more of an “ear” thing than a “finger” thing. He should be able to correct when he hears that he’s playing flat, and if he has any sense of intonation it shouldn’t take him long to make the transition. Exercises like scales should help, assuming he’s very mindful of every note.
A lot of people frown on putting tape on the fingerboard. But If you decide to do this I suggest just one piece of tape for the third finger. Somehow, the other fingers will find their place. Ideally, you shouldn’t be looking at your fingers because, among other things, it interferes with reading written music.
Another approach is to slowly play scales as you watch a guitar tuner to see whether it’s in tune.