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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 here.

Piaget's postulated affinity has apparently been challenged by some scholars in the context of the New Math controversy. Is there a source that provides a detailed analysis of such a postulation?

(Note that I am not looking for general sources on the New Math/Modern Math controversy, but rather for an analysis of this particular identification).

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.

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    $\begingroup$ Is the essay in question online? I'd like to read more context to understand what, exactly, is the claim Piaget is making. $\endgroup$ – mweiss Jun 25 '17 at 17:03
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    $\begingroup$ Crossposted to Mathoverflow: mathoverflow.net/questions/272840/… $\endgroup$ – Tommi Jun 26 '17 at 5:27
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This is not on general New Math controversy but more on Piaget. I suggest reading a few articles in general about issues with his assertions.

Here is a simple overview, but several more are available by Googling Piaget and criticism. Some of the comments may indirectly or directly relate to your questions on development of the mind.

See here:

https://www.psychologicalscience.org/observer/jean-piaget#.WVAHDGfrvIU

In particular:

"The second criticism of Piaget concerns the nature of the stages themselves. Stage theories of development have fallen out of favor in developmental research. Although stages are seen as useful heuristics for describing the trajectory of human behavior, several problems have pushed stage theories aside. One problem is that stages often fail to capture the complexities of intraindividual and interindividual variation in development. Furthermore, in the case of Piaget’s theory, intelligence is now viewed more as a modular system than as a unified system of general intelligence. In Piaget’s system, once an individual has mastered an overall intellectual operation, it should apply across all domains of the mind, as a “structured whole” (or “structure d’ensemble,” to use Piaget’s phrase). Contemporary developmental approaches to intelligence consider modules of the mind to be largely independent, and do not assume a structured whole."

I would suggest that consideration will show that it is common that learners or even yourself does not immediately apply an intellectual structure (say in EE) to a problem in a domain that seems different (say econ). Of course we do have the ability to do this at times. But it is an aspect of general intelligence and not infallible. This is one reason why word problems are trickier than non-word problems.

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    $\begingroup$ +1, but this answer would be improved if it did not also tell the questioner to google the question, and have meta-commentary on the vote score of the post itself, I think. I hope you will make an account at some point so you can receive and discuss comments like this. Thanks for the answer! $\endgroup$ – Chris Cunningham Jun 26 '17 at 3:29
  • $\begingroup$ Hi guest, what I find particularly interesting about this critic's comments on Piaget is the evidence he provides that Bourbaki-inspired concepts like sets (not that "emsemble" is "set" in French) may not only not be helpful but may actually be detrimental in attempts to analyze the learner's mind and acquisition of knowledge. I was hoping to find some even more specific criticism of the connection implied by Piaget between structures in the learner's mind and "structures" a la Bourbaki. $\endgroup$ – Mikhail Katz Jun 26 '17 at 12:21
  • $\begingroup$ I notice that the author of the article you linked writes "Piaget's theory has been influential in curriculum design, with scores of books and papers addressing the application of his theory to education (e.g., Elkind, 1976; Furth, 1970; Furth & Wachs, 1975; Jardine, 2006). " Do any of these challenge Piaget's assumption concerning "structures" that I outlined? $\endgroup$ – Mikhail Katz Jun 26 '17 at 17:37
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Many decades ago, c. 1965 and subsequently, I read Piaget's stuff with interest, as it was (seemingly) a fresh viewpoint. From my naive viewpoint at the time, various of his assertions seemed plausible. With some decades' hindsight, his "insights" with regard to "mathematics" were of an unfortunately superficial and cliched type, much as a very intelligent but essentially ignorant person might present today.

Still, yes, an ill-informed person (or an idiot) can say (apparently profound) things which are correct. There is no rule that prevents... Yet, this is not an argument in favor of pronouncements of intelligent people about things they actually are ignorant of.

And, for the sake of not wasting time, we are not obliged to debunk every ill-informed assertion of otherwise-very-intelligent people.

In particular, I'd suspect (based only on general considerations, not on specific information) that people've not been very concerned on rebutting what turned out to be somewhat-fanciful speculations. (This is considerably in the vein of the more recent non-necessity of rebutting every wrong thing that is asserted on the internet. Woof.)

So, at best, this may be a meta-answer to the meta-issue of whether anyone currently cares about Piaget's opinions. They were certainly interesting (to a naive kid like me, anyway, and apparently others), but their eventual naivete may not have been construed as needing explicit refutation in scholarly literature. I don't know, but that would be my guess.

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  • $\begingroup$ Freudenthal fought this tooth-and-nail at the time. He was convinced that the resulting damage was real. See the quote I added at the parallel question at MO. $\endgroup$ – Mikhail Katz Jun 27 '17 at 11:51
  • $\begingroup$ Hm! Though, on statistical grounds, the chances might be good that someone (a mathematician) might react. Maybe there was less noise then, unlike the internet now... $\endgroup$ – paul garrett Jun 27 '17 at 17:08
  • $\begingroup$ Certainly Piaget's ideas were, and are, discussed at length in the math education literature. Running a google scholar search on "Piaget" "mathematics" since 2013 appears to return 15,700 results. I don't quite get what the OP is asking RE: "New Math" but ... plenty of refutation and explication as pertains to Piaget's work in, or related to, math ed. $\endgroup$ – Benjamin Dickman Jun 28 '17 at 6:37
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    $\begingroup$ I am asking for analysis of the specific assumption I mentioned, namely, that structures of modern mathematics shed light on child learning because they somehow contain information about the structure of the human mind. @BenjaminDickman $\endgroup$ – Mikhail Katz Jun 28 '17 at 6:49
  • $\begingroup$ P.S. In other words, I am interested in an analysis of this specific issue rather than "Piaget" in general. @BenjaminDickman $\endgroup$ – Mikhail Katz Jun 28 '17 at 6:56
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For Piaget, 'the key tool of the constructivist is the group' (yes, the mathematical notion of a group!) and a child is understood as part of a feedback loop embedded in their environment consistent with much of the other cybernetic discourse, Norbert Weiner in particular.

Nothing in these books targets the actual teaching of mathematics.

Read and enjoy:

  • Genetic Epistemology by Piaget

  • Structuralism by Piaget

To begin his work on Genetic Epistemology, Piaget attempts to problematize the contemporary visions for scientific knowledge. Piaget believes that:

"Scientific thought, then, is not momentary; it is not a static instance; it is a process. More specifically it is a process of continual construction and reorganization." - GE, 2.

He offers two examples to support this. First, is Louis de Broglie who shifted from adhering to Niels Bohr's view of indeterminism. de Broglie later changed his mind, hence his knowledge is constantly in the process of being constructed.

Second example is Bourbaki. The mother structures weaknesses have been exposed, and Bourbaki's architectures were giving way to Eilenberg and MacLane's use of categories.

"As a result, today part of the Bourbaki group is no longer orthodox but is taking into account the more recent notion of categories."- GE, 3.

So, knowledge is not static, it is actively in construction.

Then, logical developments in children are subdivided into figurative and operative aspects. Figurative refers to imitative behavior whereas operative represents a transformation from one state to another. These operative modes are what matters if knowledge is in fact an active construction, and the actions on things are what matter.

"it includes actions themselves, which transform objects or states and it also includes the intellectual operations, which are essentially systems of transformation...to know is to assimilate reality into systems of transformations...knowing an object does not mean copying it-it means acting upon it. It means constructing systems of transformations that ccan be carried out on or with this object...knowledge, then, is a system of transformations that become progressively adequate." - GE, 15.

Piaget believes then that he has argued "the roots of logical and mathematical structures are to be found in the coordination of actions, even before the development of language..."(GE, 21)

Operating on something for Piaget means that:

  • An operation is an action that can be internalized

  • An operation is reversible

  • An operation always supposes some conservation

  • No operation exists alone

Thus, for Piaget, "every operation is related to a system of operations, or to a total structure as we call it."- GE 22.

The Bourbaki's had three 'mother structures'--algebraic, order, and topological. An example of an algebraic structure would be a group, ring, or field.

The group is Piaget's favorite tool. In mathematics, a group is a structure that consists of a collection of things and some action(s) on these things. Similarly, a group has its four axioms of closure, associativity, identity, and existence of inverses. These map closely to Piaget's description of operative knowledge, and it was deliberate.

Piaget described an interaction with the Bourbaki member Dieudonne at a meeting on 'Mathematical Structures and Mental Structures' to emphasize the connection:

Dieudonne gave a talk in which he described the three mother structures. Then I gave a talk in which I described the structures that I had found in children's thinking, and to the great astonishment of us both we was that there was a very direct relationship between these three mathematical structures and the three structures of children's operational thinking. We were, of course, impressed with each other, and Dieudonne went so far as to say to me: "This is the first time that I have taken psychology seriously. It may also be the last, but at any rate it's the first."-GE, 26

For Piaget, algebraic structures are readily observable through the logic of classes, i.e. classification. If you give children some different shapes, according to Piaget, around age 7 or 8 they will be able to classify them operationally as described above. Otherwise, Piaget noticed a range of pre-operational thinking in younger children.

For example, the lowest level was figurative behavior where children would make shapes of similar shapes, i.e. a triangle out of triangles, circle of circles. Later, children dismiss the figurative aspect and piles the shapes. Piaget claims that this child does not yet understand class inclusion however.

he cannot deduce, for instance, that the total class must necessarily be as big as, or bigger than , one of its constituent subclasses. A child of this age will agree that all ducks are birds and that not all birds are ducks. But then, if he is asked whether out in the woods there are more birds or more ducks, he will say, "I don't know; I've never counted them.""- GE, 27.

So, the child is still not evincing all the characteristics of operational thinking. This relationship of class inclusion is what gives rise to THE OPERATIONAL STRUCTURE OF CLASSIFICATION. This structure is something like an algebraic structure, though not exactly as distributivity is not a feature. (birds + birds = birds, hence (bird + bird) - bird = 0 but bird + (bird - bird) = 0)

Note the similarity to how he is describing the changing characteristic of a child's knowledge based on group like behavior. Only when the child engages with classification tasks in such a way as to demonstrate all four axioms, they are demonstrating 'pre-operational' thinking, and these different forms are what he bases his stages of development on.

The other structures are there too. Piaget goes on to describe the example of an ordering structure through the problem of seriation, and that of topological through children's use of either topological, euclidean, or projective frames in geometric contexts. Piaget believed children think like topologist's first, and you can find more on this in his geometry book, The Child's Conception of Geometry.

I think you get a sense for how he uses structures as models for cognition here though, and if you're interested I suggest reading Genetic Epistemology and Structuralism. They're not long and very interesting to see how involved Piaget was with using mathematics and physics to build his system.

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  • $\begingroup$ Hi @jf and thanks for your input. I emphasized in my question that what I am looking for is not an analysis of Piaget's oeuvre but rather an analysis of the specific issue of the putative connection between the structures in the learner's mind and the mathematical structures like sets, commutativity and other laws for arithmetic operations, definition of number is equivalence classes of equipotent collections a la Frege, etc. Only the next-to-last paragraph of your answer indirectly touches upon this issue. $\endgroup$ – Mikhail Katz Jun 29 '17 at 8:12
  • $\begingroup$ If you get a chance try to focus your answer on the question being asked rather than the question you feel should be asked. $\endgroup$ – Mikhail Katz Jun 29 '17 at 8:12
  • $\begingroup$ @MikhailKatz word. edited above and tried to change things a bit and give an example. Hope this is better. $\endgroup$ – jfkoehler Jun 29 '17 at 15:38
  • $\begingroup$ Thanks! Could you elaborate if possible on what Dieudonne's three structures were, and what Piaget's three structures were, and what their perceived similarity was exactly? $\endgroup$ – Mikhail Katz Jun 29 '17 at 16:20
  • $\begingroup$ @MikhailKatz tried. I don't know how familiar you are with structures in mathematics, but a group is the important one for Piaget. His rules for operational knowledge should be compared to the group axioms and the example of classification for children to the mathematician's description of a group. Structuralism and GE are must reads if you're interested in this. Hope I've helped some... $\endgroup$ – jfkoehler Jun 30 '17 at 2:04

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