Thanks Michael, this is very helpful.
Ed Wall
>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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