My readers already know that I suffer little of Barton's half-baked ideas on teaching maths. I don't know that guy from Adam, and I really am not attacking him personally. He just happens to be the one who wrote the book.
But my criticisms are more technical than personal. the fact that they have political implications make a critical analysis even more urgent. Not out of hatred but out of the need for honesty and to better serve the discourse.
To wit: when he brings up the notion of "commutativity" he doesn't do it in a natural way. He says that the arithmetic operations of adding and multiplication are "commutative" but he doesn't really show why the operations of division and subtraction are "non-commutative"; when he talks about the number zero he speaks only of the strange, non-intuitive numerical value of nothingness, and says nothing really of the place-value system of hindu-arabic numbers (101 does not equal 11 does not equal 1001, say); when he talks about the "set" he doesn't get into the historical motives for constructing such a theory (namely, Cantor developed his theorems of transfinite arithmetic to try and put the definition of "irrational numbers" and infinitesmals on a more rigorous analytic foundations). Instead, to Barton and so many others, these are just empty words and not living, breathing, dynamical concepts.
The motives and reasons for such mathematical ways of thinking (ie, commutativity; theory of sets; rational/irrational numbers; etc. etc.) only come about naturally as one comes up against the limitations of what came before. The narrative informs the student, and they are able to follow the lines of reasoning much easier.
Barton talks about transmission teaching and closedness of subject. His confused discourse is a perfect example of that approach.