It takes a great deal of thought to come up with an algorithm and to understand why it works, but actually performing one should be an almost entirely mechanical process. The whole point is to get the answer reliably and quickly with an absolute minimum of thinking.
This isn't a bug; it's a feature. There are situations where you want people operating on autopilot. Thought is slow, unpredictable and distracting. You probably don't want your tax preparer stopping to reflect on the subtleties of economic distortions while filling out your 1040 and if you're an administrator, you certainly don't want students thinking about the nature of numbers while doing long division on a standardized test that determines your next bonus. You could even argue that most of the progress of mathematics over the past three centuries is due to notation that makes much of the work thought-free thus allowing mathematicians and scientists to focus on more important matters.
If you're a teacher or just someone with an interest in math education, check it out and let me know what you think.