All to often with articles like this I can't help but think of grouchy old people with congnitive dissonance who are now conviced that (a) young people are stupid and (b) things were better in their day and (c) since they had to learn mental arithmetic the young one should learn it too.
In accordance with your statement about grouchy old people: I was amazed to find out that they allow calculators on the SAT. Now get off my lawn.
That's a very good question. This question has not been studied as rigorously as it should have been. Here are some suggestive observations. Professor W. Stephen Wilson surveyed many colleagues (other research mathematicians) once to ask if they thought advanced mathematics could be learned without a basic understanding of arithmetic. The responses he received
http://www.math.jhu.edu/~wsw/ED/list
included comments such as "I am shocked that there is any issue here" and "That it is even slightly in doubt is strong evidence of very distorted curriculum decisions" and "One of my favorite attacks is that we are _helping_ the students by insisting that they do things by hand because otherwise they can waste a lot of time when the calculator would fail them." One especially thoughtful comment, by a mathematician who has long thought deeply about teaching mathematics, was "It might be argued that we do not really require students to fiercely add, subtract, multiply and divide in our university courses - which is true. But we do require an automatic understanding of these operations and why they work because WE BUILD FROM THERE."
The longer story about understanding arithmetic--REALLY understanding it--and how that relates to learning beyond arithmetic can be found in the book Knowing and Teaching Elementary Mathematics by Liping Ma
http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic...
(well reviewed by two mathematicians who study mathematics education)
http://www.ams.org/notices/199908/rev-howe.pdf
http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf
A classic article on the subject is "Basic skills versus conceptual understanding: A bogus dichotomy in mathematics education"
http://www.aft.org/pdfs/americaneducator/fall1999/wu.pdf
A recent effort to embody strong conceptual understanding of basic skills into a mathematics textbook is Prealgebra by Richard Rusczyk, David Patrick, and Ravi Boppana,
https://www.artofproblemsolving.com/Store/viewitem.php?item=...
which points to what young students should be able to do WITHOUT a calculator if they really understand mathematics well.
One example I know, related to me by an economics professor, is teaching a lecture on economics in which the professor (the one who told me this story) said, as part of a calculation, "20 percent," and then was interrupted by a student who asked, "You just said '20 percent,' but you wrote '.2' on the blackboard. Why did you do that?"
Yes, you may never in life have to subtract 1/4 from 3/4, but even so, in the grand scheme of mathematics manipulating fractions is the easy part, and still useful if only as a really good place to practice and pick up useful manipulations on relatively concrete items, before you move on to more abstract versions of the same operations, which are used everywhere in mathematics.
(I'd also point out I phrased it as "at some point", not "at the beginning"; in general we can't solve the chicken-and-egg problem and just start Kindergartners on pure set and number theory. But at some point you need a decent foundation laid down or you will never build a strong structure.)
No one would argue that you should know the turn by turn direction to drive from the White House to the Liberty Bell and use that to generalize how to read a map. We don't force kids to memorize Harry Potter and then later expect them to generalize that our into literacy. History courses don't start with Supreme Court cases and wait until students generalize out the Constitution.
By starting with the abstraction and moving toward the concrete, we give the information context and allow the students a good mental framework to build upon as they learn the details. Starting with the concrete and moving towards the abstract is needlessly confusing and robs the students of the firm intellectual foundation that they need to understand the subject.
There's a lot more to programming than creating and calling functions, but I'm yet to see an interviewee struggle with how functions work in their putative "best" language but they're otherwise brilliant. You can't be progressing very far if you're burning that much mental effort on the very, very basics of the task at hand.
By starting with abstraction and moving towards the concrete, you're completely fighting everything about how we experience life and learn. Nothing else works that way. The only abstractions you can teach that way are series of words that students can mouth without understanding, but have no true comprehension of.... gee, that sounds a bit familiar.
As for the rest, I disagree that I'm fighting that standard learning pattern. In nearly every subject, we start with a generality and then move into specifics. We learn about the Roman Empire before we learn about Cicero. We learn about teeth before we learn about incisors. We learn about protons before we learn about quarks. We learn about function calls before we learn the standard library. We learn about China before we learn about Beijing. Almost all learning is performed through iteratively refining a central abstraction with increasing detail.
Now that I know you're not in favor of it, though, I'm perversely inclined to wonder if a sufficiently skilled teacher might not introduce young tykes to ZFC; and what sort of mathematicians that might produce in another few decades.
This is why it wouldn't surprise me if mental (or at least pencil-and-paper) arithmetic is correlated with understanding complicated mathematics. For one thing: to get to the complicated mathematics, you need algebra. A major part of algebra is arithmetic on polynomials. If I didn't understand arithmetic on integers, what hope have I on a more complicated entity?
Okay, forget polynomials. Maybe they're not important. How about logarithms? A good intuition of logarithms is that the log of a number tells you how many digits it has. Paired with a good intuition of multiplication (multiplying a 3-digit by a 2-digit results in at least a 4-digit) makes it feel obvious that log (MN) = log (M) + log (N).
Finally, how on earth can you understand calculus at all without realizing that division by a big number gives you a small number?
Using calculators certainly is one way to be "good at arithmetic" in the sense of always getting the correct answer without too much thought, but it does not enable arithmetic to be used as a building block for gaining intuition about more complicated things. And arithmetic is one of the best building blocks we have precisely because we're born with the beginnings of it.
At the heart of the modern emperical scientific revolution is the idea that you should seek confirmation from the real world for everything, because the world is much, much more stranger than our intuitions lead us to believe. After all, for a long time it would have suprised people to hear that the earth went around the Sun, that the mountain actually move, that there a tiny germs you can't see that can kill you, etc.
He was fond of working in generality rather than in specific examples. Once during a talk, he said "let p be a prime" and so on until a member of the audience asked him to pick a particular prime for the purposes of illustration.
"Okay, let p be 57."
Someone in the audience had to point out that 57 = 3 * 19.
http://www.math.rutgers.edu/~zeilberg/Opinion62.html
Gelfand was a mathematician who also cared deeply about mathematics pedagogy, and his textbooks are delightful.
http://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773
http://www.amazon.com/Functions-Graphs-Dover-Books-Mathemati...
http://www.amazon.com/Method-Coordinates-I-M-Gelfand/dp/0817...
http://www.amazon.com/Trigonometry-I-M-Gelfand/dp/0817639144...
Gelfand poses delightful problems that give students a workout in arithmetic (and CAN'T be done with a calculator) and that build conceptual understanding and interest in higher mathematics.
I am unsure how the traditional Calculus course is, but I took a slightly altered one at a technical institute, and we spent quite a bit of time learning the methods computers use to calculate integrals. I think it is safe to say that as long as I understand these methods and remember them, I am at no disability to use a computer, rather then solving integrals on paper.