How can I show the admission board that I'm better than my academic record says I'm a last year student of a degree in mathematics ( and another in physics) in Spain.

I'm starting graduate education next year and I need some advices on how to improve my chances to get into graduate programmes.

Here in Spain the grading system is based almost entirely in exams in which I'm not very good at. I will explain this, since I don't want to sound like I'm complaining.

I have some concentration issues while working under pressure so I loose stupid points that matter a lot in the academic world. Somehow it is very hard to me to detect mistakes in these situations and fix them.

I'm writing this because working on my own I managed to get quite a high level in some topics that I'm interested in. This could be confirmed by some  teachers that spent some time talking with me.

I think I could function in academia for a number of reasons:


My supervisor thinks my thesis is promising and advanced.
I participated in summer schools (research oriented) having really good results. Some speakers are willing to write a recommendation letter for me.
The few hand out problems that I get to do here usually are marked as A.
Classmates rely on me when the need to understand some subject related to my interests. They appreciate my way of thinking.


I'd love to learn how to show this in my applications. How can I tell the universities that maybe I'm better than what my academic record says?
Abellán,

I grew up and attended college in Spain myself, then jumped to the US for a Ph.D. and been here for 20 years already. So I know the Spanish system well and have first-hard experience with the transition.

Your best bet is the recommendation letters. You really have to make sure that they come from people that know you well and that can speak at length and in depth about your qualities and potential. Letters from Spain don't always go to this level of depth; it would be best if your letter writers have had prior experience with this (of course not at the expense of their knowledge of you).

Letters are very important--the most consistently underrated component of an application by the students in my experience.

@lafemmecosmique's advice is very good. The best way to make your letters look outstanding is to do outstanding work with the people that will write them.

Good luck to you.