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Say I am using MIT OCW or self-studied using college books. How can I prove my mastery of the subjects if I didn't sit for any exams, i.e. There are no grades?
Are there any online colleges/traditional colleges that let students just sit for the college exams without taking their classes?
I mean advanced undergraduate courses such as Linear Algebra, Real Analysis and Complex Analysis... Everyone knows about AP, SAT II and CLEP. But, thanks anyway!
My son self studied some higher math. I just put it on his transcript. There is a lot of honesty and trust in the application process. I think they look for patterns - like you took BC in 10th or 11th and did well, so if you say you did self study in later grades, it just makes sense.
There are some other options, too. You can get the teacher's manual or test booklet or the online teacher's portal and give yourself the final exam. In some classes, you could do a research paper. Ultimately, though, what anotherparent says is correct. If you did it, include it, even if you don't have an objective assessment.
UIUC offers distance Linear Algebra, Differential Equations and Probability and stats. Stanford EPGY may offer some higher-level courses too. But these cost money.
You might be able to have a professor at a local university administer a test and then get a letter indicating that he proctored the exam and feels that the student knows the material. Another approach could be to take the actuarial examinations.
I think you can put down those courses and there will be trust there IF your SAT2 and other test scores are high. If you have a 500 on the Math2, for instance or a 3 on the Calc AB, and are looking for them to be impressed with your mastery of those upper level courses, you are going to run into problems.
Do you have contact with any mathematicians? A letter of recommendation from someone who can attest to your high level of math would be one good way to show your achievement.