Concerning the structure of the learner's mind, psychologist Piaget claimed that 

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> 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). 
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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 

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

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> ***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? 
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(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: 

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> All Positive Action Starts with Criticism: Hans Freudenthal and the Didactics of Mathematics. By Sacha la Bastide-van Gemert. Springer, 16 Jan 2015 
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Here the author quotes Freudenthal as follows on page 211: 

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