I don't think there is a general method for solving problems that nobody has solved before, but if you happen to know of one I'd love to hear it. :D
Speaking as an undergraduate physics student, the issue of already established results being the main focus of the curriculum is mainly solved through the inclusion of practical labs. These undergraduate labs are supposed to demonstrate that what we're learning works in practice and to illustrate specific phenomena that are counterintuitive. In practice, my freshman and sophomore labs didn't really do that; underclassmen physics labs are sometimes notorious for not working when inexperienced undergrads are working with old equipment.
However, in my junior lab, something that really helped was that we were working with better equipment, the experiments (for the most part) worked, and we were seeing not just that established results were being confirmed, but that the experiments (photoelectric effect, Ramsauer-Townsend, electron diffraction) were well explained by quantum models and (in some cases very) poorly explained by classical models. We were able to see why scientists had shifted from classical theory to quantum theory, which I gave major rationalist points.
The curriculum could overall be improved by something like the gallery of failed atomic models, but this would take a lot of time and would probably be best offered as a supplemental, elective course that gave in-major credit.
A Wall Street Journal article by Harvard professor of government Harvey Mansfield claims that the social sciences and humanities are inferior to the sciences. The article implicitly urges undergraduates to major in science. From the article:
Do you agree with this? As a game theorist I probably have a rather biased view of the situation. It's certainly true that the ideal of the scientific method is vastly better than the practice of economists, but I think that majoring in economics provides better training for a rationalist than majoring in any of the sciences does.
Economics explicitly considers what it means to be rational. Although it infrequently considers ways in which humans are irrational, I'm under the impression that the hard sciences never do this. Furthermore, because economists can almost never perform replicable experiments we have to rely on what everyone in the profession recognizes as messy data; therefore we’re far more equipped than hard scientists to understand the limits of using statistical inference to draw conclusions from real world situations. Although I have seen no data on this, I bet that a claim by nutritionists that they have found a strong causal link between some X and heart disease would be treated with far more skepticism by the average economist than the average hard scientist.