Hey, welcome back to all the Not-Calculus-Dweebs who skipped the previous post. :-) Fair warning: I'm probably going to sneak some math in on you in this post anyway.
In working in a committee on a question for a calculus test, we ran into some difficulties in writing a question that would not be made completely pointless by the students' calculators. It led to an interesting argument between me and someone else over the use of calculators in math classes. Her position was that students should never use calculators in any math class. (Her words; I'm really not exaggerating here.)
I know this sentiment (or something similar) is not uncommon, but I've never understood it. I didn't see the point when I was a student of spending lots of time in long, tedious computations that the machine could do both faster and more accurately, and I don't see it now. I was slow on my times tables when I was in grade school, and hated math. Who wanted to do something that involved memorizing endless lists of facts? (I was particularly perplexed by the insistence that some people had on memorizing the times tables up to 12. If you knew up to 9, you could do all the rest by hand anyway; why bother?) Through grade school and junior high and even the first year of high school, I pretty much hated math. Math was boring. Math consisted of memorizing tables, finding common denominators, memorizing and practicing long division algorithms, and a bunch of other really boring things that amounted to spending 20 minutes of hard work with pencil and paper to tell you what a calculator could have told you in 20 seconds anyway.
The first time I realized I had some interest in math was in my geometry class in high school, where we started doing proofs. Proofs were interesting. This involved thinking and being creative, as well as being precise and logical.
Do I occasionally shake my head at my students inability to add, subtract, multiply, or divide without a calculator? Yes, I do indeed. (I had a student yesterday with the unreduced fraction 9/18.) Do I think this is a major problem? No, not really. If it's a common problem someone faces, they will learn how to deal with it through practice. If it's something they rarely have to deal with, I doubt it makes much difference.
As an aside, a certain amount of calculator dependency is different from a basic lack of understanding. I am concerned by students who cannot simplify things like (2)(7/2). A friend of mine had a student complain during an exam that "You said we wouldn't need a calculator!" The student was stuck because their answer had come out to the square root of 0^2. These are not just computational weakness, they are fundamental misunderstanding of how the operations work.
One point we debated over was whether students needed to memorize certain values of the trig functions, like that sin(pi/4) = 1/sqrt(2). I'm not sure that it matters much most of the time whether they say sin(pi/4) is 1/sqrt(2) or approximately 0.7071. She says that it's important to emphasize exactness. If that's true, I want to know what sin(1) is. Exactly. The point being that we like to talk about a very tiny subset of possible angles which we can reason out another expression for. I certainly feel like my students should at least be able to reason out these common angles, but don't see a point in memorizing them. (Of course, I admit I'm a total hypocrite sometimes; I usually feel fairly exasperated when they don't know what sin(0) or cos(0) are.)
So I'm basically not a big fan of lots of memorizing in mathematics, even when it's things I know, but I'm still conflicted sometimes. To what extent am I expecting my students to memorize things just because I know them?
I often think of a story my Math Ed professor in college told. He explained that when typewriters and pencils were becoming commonplace and available to all, the naysayers lamented loudly that there would be a vast decline in the quality of people's handwriting. People would no longer devote time and effort to perfecting their handwriting, and most people would end up with very poor script, perhaps all but illegible. And, he says, they were right. Handwriting did decline, and most people have horrendous penmanship when compared to people of 50 or 100 years ago. And...so? We are almost completely dependent on aids like typewriters (now computers) to communicate clearly with any speed. Sure, if absolutely necessary, we can still write by hand--slowly and laboriously, and probably still not as neatly as generations before. But the world didn't collapse; civilization didn't come to an end. People use crutches, but the crutches aren't going anywhere, so how much harm was really done? Now it seems silly to worry about this, and maybe in another 50 years it will seem silly to worry about whether someone uses a calculator for most of their basic math.