The problem with all the methods outlined by Rogers is English. Mastery of English should never be a requirement for learning Math. Use of English should be minimized and eliminated if at all possible.

Arithmetic is a separate language with separate symbols and precise definitions and a few key manipulation rules. And it is abstract. Really, it can be thought of (and in fact is) just a symbol manipulation game. There really are about a handful of symbols and simple rules that kids need know.

The question then, for me, is how to present these simple rules and then show how these rules are used to say meaningful things about some small portion of the real world.

The application of arithmetic takes some perhaps simplified scenario from the real world, and simplifies it further so that the arithmetic can be used to model some narrow aspect of the real world.

The problem with the Core Standards is that, from my reading, it pushes "authentic" assessment and learning. Thus the verbal diarrhea that the examples illustrate.

There is also a problem with the number line. It too is just one way to visualize or apply the abstract game of arithmetic. It itself is also an abstraction. Depending on prior knowledge, the number line can be useful or too abstract for kids to understand. Rogers misses that, and fails to understand that the number line is not that obvious.