</a>Psychological recapitulation of historical development of mathematics

Dynamics of Symmetry Group Theorizing (Part #4)

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With respect to the above table, given the assumptions regarding the fundamental relationship between symmetry and the inner structure of the intellect, it might be assumed that the theory of recapitulation -- expressed as ontogeny recapitulates phylogeny -- also applies to progressive comprehension of such symmetry. In evolutionary biology the theory holds that embryonal development of an individual organism (its ontogeny) follows the same path as the evolutionary history of its species (its phylogeny). The theory has been variously refuted although it continues to be held to offer interesting insights as a first approximation. It may be equally suggestive in the case of progressive comprehension and learning.

Giorgio T. Bagni, et al. (History and Epistemology in Mathematics Education, 2003) indicate that this possibility of mathematics education "had its apotheosis" in a famous book by Benchara Branford (1921). Fulvia Furinghetti and Luis Radford (Historical Conceptual Developments and the Teaching of Mathematics: from phylogenesis and ontogenesis theory to classroom practice, 2000) review current renewed interest in this approach from the following perspective:

That is, in considering history not only as a window from where to draw a better knowledge of the nature of mathematics but as a means to transform the teaching itself. The specificity of this pedagogical use of history is that it interweaves our knowledge of past conceptual developments with the design of classroom activities, the goal of which is to enhance the students' development of mathematical thinking.... the question is how to relate the development of students' mathematical thinking to historical conceptual developments. Psychological recapitulation, which transposes the law of biological recapitulation, claims that in their intellectual development our students naturally traverse more or less the same stages as mankind once did; it has been taken as a guarantee (sometimes implicitly) to ensure the link between both domains. In its different variants, however, psychological recapitulation has been subject to deep revision recently, in part because of the emergence of new conceptions about the role of culture in the way we come to know and think.

A learner (whether child or adult) has to struggle ab initio -- at whatever rate -- through various inappropriate understandings of what can be comprehended, whatever the stimulus for learning. In this sense the pattern of learning may be understood as involving processes of experimentation with forms and stages of order somewhat analogous to the pattern in the biological case. In this respect the authors note the conclusion of Jean Piaget:

We mustn't exaggerate the parallel between history and the individual development, but in broad outline there certainly are stages that are the same.

And with respect to mathematics, they note the argument of physicist Rolando Garcia (Bringuier, 1980):

In modern mathematics, at the level of algebraic geometry, of quantum mechanics, although it's a much higher level of abstraction, you find the same mechanisms in action -- the processes of the development of knowledge or the cognitive system are constructed according to the same kinds of evolutionary laws.

Garcia argued (Bringuier, 1980, p. 103) that:

And it means that one can explain the development of knowledge by starting with biology; in other words, it is the developing biological being that becomes a thinking being, even a scientist, capable of making systems that explain nature -- not the system that explains nature but some systems that explain part of nature.

Of the argument of Garcia, Piaget notes (Bringuier, 1980, p. 95-6):

...take the history of geometry, for example... you find what I call "common mechanisms". In geometry, the common mechanisms are these: in the first stage all the geometric spatial relations the child constructs are strictly intrafigural, just as for Euclid.... the second stage is interfigural. It is the Cartesian coordinates... The third step is the algebraization of geometry, starting with Klein and the Erlangen program; all geometries are reduced to displacement groups or transformation groups. Now that's a mechanims common to the history of sciernce and psychogenesis.... You see how the elementary laws of formation appear, from simple to complex... It can't be done any other way. If you began with structures and ended with a description of the elements, you'd be reversing an order that is, as I call it, "natural" because it's required, so to speak, by the very nature of things.

With respect to what was then termed genetic development, Piaget and Garcia subsequently collaborated in a book Psychogenesis and the Hisotry of Science (1989) to counteract overly simplistic psychological interpretations of recapitulation. The concept was a feature of Piaget's theory of genetic epistemology. As noted in the major collective study by John Fauvel and Jan A. Maanen (History in Mathematics Education: the ICMI Study, 2000), they argued that we should try to understand the problem of knowledge in terms of the intellectual instruments and mechanisms allowing its acquisition. Citing the challenges to this approach by Lev Vygotsky (1997) from a cultural perspective, the study concludes (p. 147):

The examples of Piaget and Garcia and of Vygotsky, uncover the complexity of the problem of the relationship between phylogenesis and ontogenesis and the importance of working towards a clear theoretical framework.

And, more specifically, with the reservations (pp 168-170):

Indeed any attempt to put in relation the history of mathematics and the teaching or learning of mathematics necessarily induces an epistemological questioning both of individual cognitive development and of the interpretations of the historical development of mathematics.... an epistemological reflection on the development of ideas in the history of mathematics can enrich didactical analysis by providing essential clues which may specify the nature of the knowledge to be taught, and explore different ways of access to that knowledge. Nevertheless what appears to have happened in history does not cover all the possibilities..

In this light an individual struggles with understandings and hypotheses -- that may indeed be considered inadequate by others if they could be adequately communicated -- perhaps discarding them quickly, perhaps retaining them inappropriately for an undue amount of time. This may also be true of a community introduced to mathematical concepts for the first time (such as after some civilizational disaster).

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