Knowing what a test is going to be like is pretty useful if that test matters at all. Otherwise there is a higher chance that the people who fare well on the test will be those who happened to work more on the topics or skills which come up. I'm happy my lecturer for group theory in university let me know which of the 5+ page proofs of theorems from the course wouldn't need to be known for our exam - it wouldn't have been very good use of my time to fully internalise all of them. However they were interesting and I think it added value that they were in the course.
I think you just have an idealised picture of how tests work. In the best maths exams I have taken, you know all you could within reason about what you need to know and what will be the content of the test, and yet the questions still surprise you and cause you to think in new ways.
Knowing what a test is going to be like is pretty useful if that test matters at all.
Knowing and understanding the subject matter is likely to be pretty useful if the test matters at all. This is why you study and understand the entire course, which is the point of the course. However, being told explicitly which portions you will be tested on goes against this goal.
which of the 5+ page proofs of theorems from the course wouldn't need to be known for our exam - it wouldn't have been very good use of my time to fully internalise all of them
What? If you're doing mathematics correctly, you are ably to derive the proofs on the fly. You don't have to memorize "five pages of proofs". You understand the concept, you understand the mechanics, and you derive the proof as necessary.
I think you just have an idealised picture of how tests work.
I'll refrain from saying what I think about your "picture of how tests work".
Maybe I worded my post badly; I was trying to say that your idea of a good test isn't in line with how tests are designed in my experience of education. If you agree with the sentiment that children should be punished for trying to optimise for exams, then yes I think your picture of how tests work (at the moment, in school especially and somewhat college) is wrong. I see nothing wrong with a well designed test, made to cover a whole course and to be fruitless to optimise for. At least in the UK in school, all STEM type exams are not like that as far as I remember. Humanities exams are much closer in style to this.
I do think GCSE and A-level science exams could be much better designed. At college level though I'm not sure the way you describe is the only good way to do things.
I'll refrain from saying what I think about your idea of how to "do mathematics correctly".
I think you just have an idealised picture of how tests work. In the best maths exams I have taken, you know all you could within reason about what you need to know and what will be the content of the test, and yet the questions still surprise you and cause you to think in new ways.