Let me start picking apart Mark Saul's critique one point at a time. He sees the notion of coherence talked about my Ma as problematic. He seems at a loss to know really what she's talking about.
Can it be he does not see arithmetic as a coherent subject? (It seems wild, but that could be the case.) After posing and knocking down a couple strawmen in quick succession, he puts up his "real" argument: some kids may not get a point in arithmetic because of differing underlyingreasons, arguing that in fact distinctly different cognitive abilities may be to blame for different sorts of misconceptions (pg. 505).
Now, how does that support his claim that Ma's "coherence" is in fact illusory or problematic? I don't think it does in the least. A coherent subject does not imply that a single cognitive ability underlies or supports it.
In fact, Saul quickly abandons his claim of "coherence" being problematic, and instead jumps to the conclusion that one was really teaching arithmetic in order to strengthen these underlying cognitive skills. We teach arithmetic in order to teach logic, to paraphrase him. People say things like this all the time: we teach sports in school because it teaches other socially desirable qualities, yadda yadda. But why make this claim about arithmetic? We all use arithmetic everyday in our lives. Do we need to justify it on other grounds? Why?
So, is arithemtic a coherent subject or not? Why do we teach it? Is it really just a delivery system for logic?