I don't think the one who started this thread is being arrogant. Realistically, learning independently is one of the most important things for graduate school in the first place, since one's goals as a graduate student are not the same as those of an undergraduate. The one thing you should do is...
For engineering, I'd certainly aim to have at least a 3.5, and for physics, even higher than that. Beyond a certain point, however, letters of recommendation plus context of undergraduate institution will be taken into account.
I would agree with the above that a mathematics master's sounds a better option at this point. A minor often covers core fundamentals, while a major involves more in depth study with electives and such, and realistically many of these can be undertaken in a master's program.
It may be tough to speak for such a thing, but does anyone know what the profile of students who get the chance to attend the likes of great math schools like Chicago, Michigan, Princeton, Harvard, and so on actually are like on average? Is a significant factor the fact they were able to learn...