I believe the ability to check your work is crucial.

So learn how to check your work. First, look at your answer and try to determine whether it makes sense, and then see if you made any silly algebra mistakes. Then if you're learning integration, for example, take the derivative and see if you get the original function back again. If you're learning differential equations, plug your purported solution in and see if it is actually a solution. In many situations, you have more than one method available to solve a problem, so try both and see if they produce the same thing.

In the real world you don't have a solution manual, so it's a valuable skill to be able to check your work without one. Furthermore, some students use solution manuals badly: if they don't get the right answer, they tinker with their work until their answer matches the right one, with no understanding of what they did wrong or what they did to correct it. It's a good idea to not have all of the answers available; for calculus, half seems about the right proportion.

This, of course, is precisely backwards of how math is taught. They try to teach the mathematic principles, and then from that you are supposed to deduce how to do the problems. This has never worked for me.

I'm not sure what you're talking about -- mathematics is taught lots of different ways: there is no single, monolithic, method for "how math is taught."