Aspects of negative numbers in the early 17th century |
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| Authors: | Yannis Thomaidis |
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| Institution: | (1) Pesonton Iroon 1, GR 56430 Stavroupolis, Thessaloniki, Greece |
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| Abstract: | This paper argues that the questions, posed by researchers in the field of didactics of mathematics, require new historical research which mainly concerns the problems related to the emergence and evolution of concepts. Motivated by recent historico-didactical studies on negative numbers, the author explores two different types of problems through which these numbers started being used systematically in mathematics. The first problem deals with the correspondence between the terms of an arithmetical and a geometrical progression, which constitutes the theoretical basis of logarithms; the second deals with the application of algebraic syntactical ruies in the theory of equations. In the specific context of these problems, concepts, such as  negative logarithm or negative root, were established in the early 17th century, long before the appearance of a general concept of negative quantity in mathematical textbooks. The analysis of these problems reveals the conventional character of negative numbers and poses certain questions about the meaning of the various concrete models, traditionally employed in their teaching (via temperature, debits and credits, etc.). Recent, large-scale empirical research has shown a major percentage of failure in understanding negative numbers and their operations; this fact is related to the meanings attributed to negative numbers during their introduction at school. The matter of revising traditional teaching models is considered in connection with a constructive learning hypothesis; there is a need for new problem-situations, which entirely justify the meaning of the concept that must be used and constructed by the pupil and allow a fruitful interaction with it. The case of negative numbers provides an illuminating example of the role historical problems can play in the creation of situations like these. |
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