I graduated from an undergraduate maths course from a prestigious university. However, I was rejected from attending it's master's program due to poor showing in the final exams. I'd like to study at said university again sometime within the next few years, applying for the same course. In applying though, I will almost definitely have to use the same final exam results. What can I do to show that I have improved, if I indeed have? I am not currently enrolled in any graduate level program.
At many schools, particularly those rated highly, the most important parts of your PhD application are your letters of recommendation. What many applicants don't grasp is that a PhD program is so different from undergrad that past performance in the latter is only a poor indicator of future performance in the former. For most students, the hard part of a PhD is doing original research. So, as an applicant, you must convince the committee that you will excel at doing research.
One way to make your case is to have already successfully done research (or some "research-like" project). In the absence of this, you must somehow convince your letter writers that you will succeed in research (and then they convince the admissions committee). Meet with your letter writers and ask them what they suggest. You might even ask them if you could work on some research with them (prior to being admitted to the PhD program).
Have you considered directly contacting the admissions committee or a former teacher and asking for advice? They'll probably be wary of saying anything that sounds like "if you do x,y, and z, you'll be accepted", but even if they're unwilling to share specific advice they might be willing to tell you what other experience (beyond the exams) they consider during the admissions process. A bonus of contacting the admissions committee in a respectful and professional manner is that if you do indeed follow through on any recommendations they give you, you can mention this in you application as evidence of your commitment to improving. You might also ask if you can retake the test--either officially (i.e. if they'll replace your old results with your new ones) or unofficially (i.e. your old results will still be the official ones, but at least you have some metric showing your improvement). Another option is to talk with a professor in the department with whom you have rapport and ask them about how to frame your rejection in your application when you reapply. For example, is it considered good form in your field to discuss what led you to do poorly on the exams (whether this was some unrelated life experience, or a lack of preparation)?