There's a lot of helpful information about trying to explain to you how all of this works and giving your mum the benefit of the doubt that she will see things your way, which is hopefully what's going to happen - but this smells of controlling, and I'm getting warning lights in my head that your mum isn't going to simply change her mind just by presenting her with facts.
So, here's something not a lot of people are aware of - but structured debate is actually a stupidly bad way of trying to convince someone to agree with you. In fact, arguing in GENERAL is a bad way of doing things. This is because of a psychological phenomena known as "cognitive dissonance", which is:
the state of having inconsistent thoughts, beliefs, or attitudes, especially as relating to behavioural decisions and attitude change.
For example,
look at this hilarious video about one man coming to term with the facts that steel isn't always heavier than feathers. Sure, this is a skit designed to be funny, but remember that comedy is typically an embellishment of what happens in real life. This man is facing physical distress because something that he took for gospel is all of a sudden proven wrong to him. The way that people typically react to cognitive dissonance? They actually dig in their heels, and believe their original opinion
even harder, even with given physical evidence of their thought process being wrong.
For the curious, here's a great excerpt from The New Yorker about it.This makes everything seem a little hopeless, doesn't it? And tbh, it can make things very sticky. However, there is someone who should know your mother better than anyone - your father, the man who has lived with her at this point for probably half of her life. You said your dad was honestly very supportive of you doing what you want. This is the best way to combat cognitive dissonance - by knowing how someone thinks, and being able to use their own logic against them - make them believe for themselves, and have them come to the conclusion on their own terms, that your opinion is the right one. (the other way is just recognising it exists and doing your best daily to combat it, but that's a self thing, not an other people thing)
Also a fan of Heidi's talking to career councillors/teachers at school to see if they can help.