TITLE: Source for analysis of identification of structures in learner's mind and mathematical structures?
QUESTION [13 upvotes]: Concerning the structure of the learner's mind, psychologist Piaget claimed that 

There exists, as a function of the development of intelligence as a whole, a spontaneous and gradual construction of elementary logico-mathematical structures and that these 'natural' ('natural' the way that one speaks of the 'natural' numbers) structures are much closer to those being used in 'modern' mathematics than to those being used in traditional mathematics. (p. 79 in Piaget 1973). 

Piaget appears to postulate an affinity between, on the one hand, the structures of the mind and, on the other, the structures of modern mathematics (mainly following Bourbaki). The essay in question is 

Piaget, J. "Comments on Mathematical Education," in A. G. Howson, ed., Developments in Mathematical Education: Proceedings of the Second International Conference on Mathematical Education, 79--87, Cambridge: Cambridge University Press, 1973. 

Piaget's postulated affinity has apparently been challenged by some scholars in the context of the New Math controversy. 

Question. Is there a source that provides a detailed analysis of such a postulation of a connection between the structures in the learner's mind on the one hand, and Bourbaki-style structures in the foundations of mathematics, on the other? 

(Note that I am not looking for general sources on the New Math/Modern Math controversy, nor am I particularly interested in Piaget's work in general, but rather for an analysis of this particular identification of Piaget's psychological structures and Bourbaki's mathematical structures).
I just came across a book that might be relevant: 

All Positive Action Starts with Criticism: Hans Freudenthal and the Didactics of Mathematics. By Sacha la Bastide-van Gemert. Springer, 16 Jan 2015 

Here the author quotes Freudenthal as follows on page 211: 

It thus did not begin with the Sputnik shock. It had already begun in the early 1950s. They had even managed to convince Piaget, who did not understand anything of it except for the fact that the word "structure" appealed to him. With Piaget's name on the billboard they felt confident of the support of psychology. What now, psychology! Mathematics is ruled by a logical order and he who teaches mathematics is easily seduced to sacrifice the psychological, the educational order to the logical order. I have done my utmost to avoid this and in my 'fragment Rekendidactiek' of 1942, if not earlier, I wanted to warn others. But what was now happening before my very eyes? A logical order brought to ecstasy, a systematic of mathematics as a whole--that is how mathematics should be taught. 

It is clear from this that Freudenthal was sceptical of these developments but unfortunately he does not elaborate the details of his objections.

REPLY [6 votes]: You say that Freudenthal does not elaborate his objections, but in a sense he did just that in his book Mathematics as an educational task (1973), Appendix I, starting: “The somewhat summary criticism I administered on several occasions
to Piaget's work demands more detailed argument.” However, this addresses egregious mathematical misconceptions and methodological flaws (ill-chosen questions) more than the (“ontogeny parallels phylogeny”?) postulation you ask about. 
Against that, his argument seems much shorter:

(p. 46): Bourbaki. How convincing this organization of mathematics is! So
  convincing that Piaget could rediscover Bourbaki's system in 
  developmental psychology. Poor Piaget! He did not fare much better than Kant,
  who had barely consecrated Euclidean space as "a pure intuition" when
  non-Euclidean geometry was discovered! (...) Mathematics is
  never finished – anyone who worships a certain system of mathematics
  should take heed of this advice.
(p. 192): Impressed by Cantor's analysis he turned to studying the development of the number concept under this aspect. (...) Piaget believed that the concept of natural number could be entirely derived from potencies. (...) This may have seduced him to believe that it is
  also psychologically true; it was one of his ideas to trace in 
  developmental psychology the system of mathematics he happened to be 
  acquainted with.

Or, as reported by F. Goffree in The legacy of Hans Freudenthal (1993):

(p. 36): Piaget thought that the cognitive development in children took place from poor to rich structures whereas HF thought it was the other way round. Geometry showed something similar. In the Erlanger Program Klein had given a hierarchy of geometrical structures: topological, projective, affine and euclidean, in HF's terms from poor to rich. Children start by drawing irregular circles, anyone can see that. This was sufficient reason for Piaget to presume that their geometric development began
  with the topological structure. HF remarked condescendingly that those very same children were quite capable of distinguishing between correctly drawn circles and other figures.

(Note added: These last remarks, not sourced by Goffree, are in Revisiting mathematics education (1991), p. 27. So far as I can tell, the criticized “postulation” is most detailed in Épistémologie des mathématiques. Partie II. Chap. 8 & 11 (1961, transl. 1966).)