Go to any engineering school...you'll see the same thing, with very good reason. It's quite easy when doing complex calculations to miss a decimal point or transcribe a 1 into a 7 (depending on your personal printing). The point is that as long as the process is correct, experience will tell if the answer makes sense or not. If the process is correct and the same every time, the answers will generally be correct, if an answer doesn't make sense, commonly off by a factor of 2 or 10, you can go back through your steps and see where the error was made.
Now the specific methods, I learned multiplication the correct way, long division, etc. I whole-heartedly endorse those methods and on many of my class notes you will find long division or multiplication calculations scribbled into the margins when I'm too lazy to pull out a calculator. These algorithms are meant as shortcuts for experienced mathematicians (or students) not as a way of teaching the fundamentals.
The teaching of basic traditional mathematics are true tested methods. All the great mathematical minds have been taught through proven traditional teaching methods. In India they teach traditional math and the esults are astounding.
Applying advanced teaching methods to young minds who do not have a set foundation is leading them astray. My nephiew was so utterly frustrated that he would simply tune out. I showed him a boring 'long division' like textbook and showed him how the old school of math worked and since he 'got it' right away, he felt liberated and would proudly display his homework when he accomplished it. I spoke to his teacher and she went on that Michael should stick to the teaching method that they were teaching and I said to her: Weren't you schooled in the trad method? Isn't the most important thing that a 10 yr old boy can make sense and is enthusiastic of the math he asked to do?
Michael never again did that 'new math' until he applied new techniques as a carpenter. (A good one at that!)
cheers!