Constructing Knowledge by Transformation, Diagrammatic Reasoning in Practice



The aim of this article is to present a certain view on the reading of mathematical texts. Using an example it will be shown that readings of mathematical texts may be codetermined by different readers’ transformations. As an example of such a kind of reading, a theorem from a nineteenth-century-geometry textbook is introduced. To comprehend the meaning of this geometry and to be able to follow all the arguments offered, the text and some proofs are transformed into a more up-to-date kind of geometry. This transformation can be characterized by a transfer of the author’s use of diagrams to the reader’s use of diagrams. In addition, this transfer of use can be supplemented occasionally by the use of an alternative set of diagrams. This means that the use of diagrams that are well known to the reader of a mathematical text may considerably support his/her construction of knowledge.


Semiotics Diagrammatic reasoning Transformation Geometry History of mathematics Construction of knowledge Writing Reading Problem solving 


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© Springer Science+Business Media, LLC 2014

Authors and Affiliations

  1. 1.Institute of MathematicsAlpen-Adria University of KlagenfurtKlagenfurtAustria

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