>There probably is much that could be argued in support of Liping Ma's and Mark Saul's respective arguments (and no doubt many have done exactly that, as you are doing in support of Liping Ma's thesis).
I have confessed sympathy with her viewpoint but haven't really tried supporting her arguments, other than it seems self evident to me that arithmetic is a coherent subject. For Mark Saul to take that on as "problematic" is not a good place to go. As I've just pointed out, he pretty much flubs it from an argumentation standpoint: 2 strawmen followed by a change of subject.
But going a bit deeper, I also confess I'm not sure I've pegged his 2nd "strawman" accurately. What he's really questioning is whether the standard algorithms are useful to teach, even *if* they are internally coherent. That's an oft-debated question, so calling it a strawman is inaccurate. The way he presents it is flimsy though, so its like a strawman. This sets up his shift of attention -- he wants his readers to reject, or at least question the notion that arithmetic (in his view, standard algorithms) is central to elementary math education, because arithmetic is just a delivery system for teaching logic (like a cigarette for delivering nicotine.)
Perhaps we could debate that single point first, if there are any takers for Saul's view.
My view is pretty much that arithmetic is not just standard algorithms, but I wouldn't neglect those. It is in fact a "coherent subject" par excellence. Here's a couple questions I might ask an elementary school teacher who professes a solid understanding of elementary arithmetic:
* How would you characterize those (rational) fractions that can be represented by a non-repeating decimal number?
* As a function of n (whole number) what percentage of all fractions with denominator < 10**n do such numbers comprise?
Is knowledge of standard algorithms used? No. Would most elementary math teachers be able to answer these simple question easily? I wonder. Would be being able to discuss such question easily and naturally be indicative of at least some familiarity with arithmetic? I think so.
Arithmetic is a very old subject, and arguably along with geometry is one of the oldest branches of mathematics. Its coherence embraces much more than standard algorithms. Its usefulness goes beyond answers that can be found with a calculator.