Hi all,
SungWon Hwang and I just had 2 companion articles published in J Math
Beh that deal with the relationship of the abstract and the concrete.
Anyone interested will find them through your library, or, when you
don't have access, I can provide you with a copy. (Because of
copyright issues, I cannot put it on the server.)
Cheers,
Michael
Roth, W.-M., & Hwang, S.-W. (2006). Does mathematical learning occur
in going from concrete to abstract or in going from abstract to
concrete? Journal of Mathematical Behavior, 25, 334–344.
Roth, W.-M., & Hwang, S.-W. (2006). On the relation of abstract and
concrete in scientists’ graph interpretations: A case study. Journal
of Mathematical Behavior, 25, 318–333.
TITLE: Does Mathematical Learning Occur in Going from Concrete to
Abstract or in Going from Abstract to Concrete?
Abstract: The notions of abstract and concrete are central to the
conceptualization of mathematical knowing and learning. It is
generally accepted that development goes from concrete toward the
abstract; but dialectical theorists maintain just the opposite:
development consists of an ascension from the abstract to the
concrete. In this article, we reformulate the relationship of
abstract and concrete consistent with a dialectical materialist
approach to conscious human activity, as it was developed in the line
of cultural-historical psychology. Our reformulation of development
in and through interpretation shows that rather than being a movement
from concrete to abstract or from abstract to concrete, development
occurs in a double ascension that simultaneously moves in both
direction: it is a passage of one in the other. In the proposed
approach, the theoretical contradictions of earlier approaches to the
issue of abstract have been eliminated.
Keywords: Abstract; Concrete; Generalization; Dialectical Logic;
Double Ascension.
TITLE: On the Relation of Abstract and Concrete in Scientists’ Graph
Interpretations: A Case Study
Abstract: The notions of abstract and concrete are central to the
conceptualization of mathematical knowing and learning. Much of the
literature takes a dualist approach, leading to the privileging of
the former term at the expense of the latter. In this article, we
provide a concrete analysis of a scientist interpreting an unfamiliar
graph to show how engagement with some object leads to the working
out of existing, concrete practical understanding and the
articulation of categorical statements (“generalizations”); because
the scientist knew something at the end of his interpretive work that
he did not prior to it, the event is understood to constitute an
episode of learning. The analysis shows that rather than being a
movement from concrete to abstract or from abstract to concrete,
development occurs in a movement that appears to be simultaneously
from concrete to abstract and from abstract to concrete.
Keywords: Abstract; Concrete; Generalization; Dialectical Logic;
Consciousness._______________________________________________
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