In choir our conductor gave us the advice that your lips should be tingling when you hum, which worked well for lower notes. However, when me and the first sopranos got up to the higher notes, we couldn't hum in the same way and it just felt tight in the throat and not a very nice sound. Is there a good technique to do this or is there no nice way to hum high notes?!
Just talking from personal experience, but humming usually needs you you to have you lips closed. This means that the air flow will need to go through the nose. If the air flow through the nose is blocked to some extend it will be harder to do high notes because it is harder to get the air flow required. In such a case it might help to slightly open your mouth.
Also you need the keep resonance in mind. Singing technique is about creating as much sound with as little effort as possible. This means you need to adjust the resonance of your mouth in such a way that you get the most from the note you are producing to strain your voice as little as possible.
The resonance frequency of a cavity like the mouth will increase if we make the cavity smaller or the opening larger (this is similar to how closing the bottom of a wind instrument will make the note lower).
So when singing we would usually adjust this resonance by opening our mouth wider when singing higher notes. This is not something that can be done when humming (unless again if you open your mouth slightly on high notes). What you can do on the other hand is to reduce the space of the mouth cavity by raising your tongue up and close off between palate and tongue, and further by raising the back of your tongue into the palate as you go higher. Optimally you’d find at each pitch the point where you get the optimal resonance.
Generally, to get a good sound, you'll be inclined to open your mouth a little more the higher you go—mm for low notes, nn for middle ones, then oo, then eventually ah; done right, it should be gradual and not too noticeable. This is the easy way to experience what good resonance feels like at every part of your range, which is a stepping stone to maintaining that resonance with any mouth shape. (Spoiler: the resonance moves to different areas, notably including your nose; it won't feel the same at every pitch.)
My musical theatre teacher knew this well and had us sopranos do some funny things with vowels when we went extremely high in chorus numbers, arguing that resonance was more important to the overall texture than diction up there, and generally she was right.
Do what your teacher asks, but she won't mind if you cross into nn territory, maybe further. Don't forget to open up air inside your mouth as you mm, and don't get stuck trying to resonate in a specific way, just let it happen.