Brian Gleason, EMAT 8990, Sp 09
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Undoing the Genetic Principle
The “Genetic Principle,” sometimes articulated as “Ontogeny recapitulates Phylogeny,”
was originally proposed as a theory of biological evolution. The crux of this theory states that an
organism, in its growth to maturity, passes through stages approximating the adult stages of
lower life forms. As popularly adopted by mathematics educators (though often only privately),
the genetic principle is taken to mean that a student’s understanding of a topic will necessarily
parallel the historical development of that topic. As stated by Herbert Spencer, “The genesis of
knowledge in the individual, must follow the same course as the genesis of knowledge in the race”
(Howson et al., 1981, p. 33). For instance, the genetic principle would claim that a geometry
student should (or perhaps must) learn Euclidean geometry before non-Euclidean geometry
before topology, as this sequence roughly parallels the historical development of geometry (at
least the documented development).
While not often explicitly given as justification for the ordering of topics within a
proposed geometry curriculum, it is nevertheless the case that a majority of American students
receive an education in geometry (if any) that is consistent with this principle. There is an
alternative view of learning however, that takes a point of view very nearly opposing the genetic
principle, this view is similar to Dienes’ Stages (Howson et. al, 1981). This alternative view
claims that a student’s learning of a subject should proceed from the less constrained to the more
constrained. Specifically, this view argues that topology is more “natural” to young children than
is Euclidean geometry because topological spaces are much less constrained than standard
Euclidean space, by not requiring things such as metrics or continuity. Indeed, one may note that
American children’s cartoons seem to be oriented to a topological view of the world1, one in
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I owe this insight to my fellow students in the Fall 2008 section of EMAT 8010.
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Brian Gleason, EMAT 8990, Sp 09
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which a cat may be flattened by a steam roller and then pop back into full form again, a
transformation quite impossible working from Euclid’s axioms.
The question I propose for research is simply this: Which of these two views of a child’s
geometric development is the most accurate, or are they both wrong together? There exist several
possible means by which this question could conceivably be investigated. One could attempt to
observe understanding of basic topological principles in very young children. For example,
activities could be designed incorporating the notions of open sets, compactness, connectedness,
and continuous deformation. These activities could be used to test for children’s rudimentary
grasp of these concepts. Another way in which the answer to this question could be pursued
would be to develop and implement a geometry curriculum where Euclid and his axioms serve as
the summit, rather than the base of the curricular progression. Thus, the students would study
more general topological spaces (albeit informally), gradually adding such concepts as metrics,
smoothness, and orientability. The students who pursued this curriculum could then be studied to
see if they were able to construct these concepts in ways the genetic principle deems impossible.
These studies would have difficulty, of course, not the least of which is the fact that a
curriculum such as the one described here is, to my knowledge, not anywhere in practice. This
curriculum would not only have to be designed then, but would also have to be implemented in
the midst of a society that rather strongly embraces the genetic principle. This mismatch could
lead to eventual corruption of the curriculum, which would render the question unanswered.
References
Howson, G., Keitel, C., & Kilpatrick, J. (1981). Curriculum development in mathematics.
Cambridge, England: Cambridge University Press.
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