BrailleOperatic said:
Is painting art?
Is music art?
Is writing art?
Is story telling art?
Is performance art?
Does the application of code truly counteract any of these things? Games are art through simple Gestalt, if nothing else (and given the complexities a nuances of computer code, I'd go so far as to assert that it too is art) and every game is art. Not every game is GOOD art mind you, but judging video games by Mortal Kombat is a lot like judging books by Twilight, and no one is questioning the legitimacy of literature as a medium.
Coding is definitely not an art and while many feats of genius are performed by coders each day I'm sure most of them would describe what they do as a mix of engineering and science. Sure there's an artistry involved of sorts but coding and the resultant programs aren't art. You might as well say mathematics is an art and I'm pretty sure you'd annoy a fair few professional mathematicians in doing so.
The answer to all your questions is "Yes, they can be". Games CAN be art, music CAN be art. Thing is a lot of it isn't art. Even visual media can not hit the "art" mark, concept art for instance is mostly not art as it exists to serve a functional purpose (it's concept art, it does what the name says it does). (However this all comes down to your definition of art).
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Playing devil's advocate here I'd say your art teacher would err on the conservative side of art insofar as it comes to its definition. She most likely will not buy "art is subjective, anything can be art", on the most cynical level saying such a thing denegrates a large part of her chosen subject though I doubt that would be the only reason she'd oppose such a view.
If you want to get her behind this (and you need your teacher's support for any project, her entire role is to help you succeed in producing not only something full of artistic merit but also full of the stuff markers like), you'll want to be extremely cautious in what you show her. Don't show her mere concept art, as that's not a game. Show her something like Krystian Majewski's "Trauma", something indie and groundbreaking. Show her that interactivity is a powerful means of conveying emotion/meaning and not a gimmick, not something just for "fun". I'd rule out anything like bioshock, not only because it's primary purpose is the $$$ but also because the reality is your project will be nothing like a AAA title or even like "Trauma" (which took years to create).
As for your own game (and the following qualities would be great in any game you show her) you'll want something short, with a simple and engaging premise, striking visual style (even though you'll be lacking graphical fidelity) and great music (going to be tough but there's some good non-copyright work out there). Honestly making a game is a huge amount of work, however I would not discourage you from making an electronic interactive work that borrows heavily from videogames.
If you want to continue making a videogame, you're going to be working against skepticism from your markers and technical limitations. However it is possible and if you have a clear idea of what you want to do and a plan to implement then it don't let anyone get in your way.