I have the following two options available on how to utilize the next year. First some background:
I'm an Economics and Mathematics major at a college in Pakistan. I'm, however, interested in going to graduate school in mathematics and working with mathematical physicists. Starting from the end of my second year, I started taking physics course. I'll have a dozen or so physics courses by the time I graduate; I have the basic, and a few upper division math courses. I continue, however, to study mathematics outside of class.
I can choose to take an additional year before going to graduate school. I doubt I'll be able to fulfill the requirements of a major in mathematics (and my school's policies are a bit ambiguous as well); however, at the very least, I could take as much mathematics courses as I can. If I were to do so, I'll probably be able to complete the department's requirements of math courses, so one could assume that I have completed all the requirements necessary to fulfill the department's choice. I may be offered a RA position in the physics department; so I could fund my additional year with that money, along with all my savings.
Just take the year off. Continue to study mathematics informally with the instructors. That is, cover material outside of class/college, follow up with them regualrly so they can write good letters. Depending on whether I get to work on some other problem, I could still take the RA position and save money to apply for graduate schools etc.
Here are some more details that'll help you advise.
Mathematics courses I have taken:
I have taken single variable calculus, multivariable calculus, linear algebra, real analysis (two semesters), probability and point-set topology.
I have also been studying material outside of class. I finished linear algebra II on my own, and I am working on multivariable analysis right now. Regardless of which option I choose, I will continue studying more mathematics, revising old mathematics etc.
Here's my assessment of the options thus far.
Frankly, I don't think I'll want to work in this position for an extended period of time. I wrote a physics paper with a professor on (open) quantum dynamics; since he wishes to work on similar problems, he offered me a position so I can continue working on similar problems.
I don't think it'll be (extremely) helpful in my eventual application because the path I wish to take is somewhat different.
I, however, can get paid and use this money for either option.
I get to cover my cores -- ODEs, Complex Variables etc.
I'm not too sure how much it'll help, but I'll have graded coursework.
I'll still, however, be covering some of the same material outside of class on my own.
It'll be hectic; I'll have to complete 2 semesters worth 35-40 credit hours. I will also have the responsibilities of the RA position; on top of that, add applying to graduate schools in the mix.
Flexible schedule designed at my own will and pace.
If both the above options would have the same effect, then I can just study material on my own. For instance, assuming I take the RA position, I'll be on campus the next year. I can audit courses which my advisers will be teaching next semester -- for instance, on of my advisers will be teaching a graduate level course in algebra. I can sit through classes, work on problem sets, and hopefully then the instructor can compare my work when she writes the letter; of course, I won't be graded relative to my peers, though.
Another local university provides short, intensive courses in mathematics. It's a graduate school, so most courses are advances courses -- for example, in advanced algebra, differential geometry. Since I'll have more time on me, I can perhaps take 1 or 2 of these courses, and cover the more advanced material. You can see the this link for a description of such short courses that have been offered in the past.
Also, with the first option, I'd have to go through the boring, plug and chug courses -- ODEs etc. Very important to train yourself, though.
It'd be great if someone could weigh out both these options, especially considering the graduate admissions process at math departments.