I agree that numerical methods are not a substitute for analytic tricks, you need analytic tricks to come up with the algorithms and to understand why they work! But learning how to use a particular calculator is not important (that is what I'd like to see replaced by learning the algorithms). And even for analytic tricks it's often much more enlightening to code the algorithm. For example I learned much more by coding an algorithm for factoring an arbitrary polynomial over Q than doing any number of factorizations by hand. I don't think that many people are even aware that there is an algorithm for factoring an arbitrary polynomial. They either don't even consider it a potential algorithmic problem or they think that it might be impossible to write an algorithm to solve it.