History in Mathematics Education. A Hermeneutic Approach
The paper discusses the possibility of bringing history in the mathematics classroom by studying historical sources with students. A manuscript by Johann Bernoulli about the differential calculus which was brought to a grade 11 classroom serves as an exam
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History in Mathematics Education. A Hermeneutic Approach Hans Niels Jahnke
Abstract The paper discusses the possibility of bringing history in the mathematics classroom by studying historical sources with students. A manuscript by Johann Bernoulli about the differential calculus which was brought to a grade 11 classroom serves as an example. Reading a source is fundamentally a hermeneutic activity and can be conceptualised by the term ‘horizon merging’. In the so-called hermeneutic circle the horizons of the reader and the author of a text are supposed to merge by a repeated reading. In contrast to common ideas about the genetic principle the hermeneutic approach described in the present paper assumes that students have already some experience with and knowledge of the modern counter-part of the concepts treated in the source. Reading a source is an activity of applying mathematics in a way completely new to students. It provides opportunities for reflecting deeply about their images of the respective mathematical concepts. Keywords Johann Bernoulli · Concept image · Differential · Genetic principle · Hermeneutics · Hermeneutic circle · Horizon merging · Historical source · Infinitely small quantity
Preliminary Remark Before I entered the field which usually is called “History and pedagogy of mathematics” (HPM) I had met two rather different notions of what this could mean. One was the idea to consider history of mathematics as a collection of interesting mathematical problems some of which were suitable to be treated at school. This idea was mainly supported by teachers at schools respectively math educators who saw themselves predominantly as teachers. To be sure, some work in this direction is impressive but I asked myself where there was any substantial relation to the history of mathematics. All these problems were meaningful by themselves and could be treated without any reference to history.
H.N. Jahnke (B) Faculty of Mathematics, University of Duisburg-Essen, 45117 Essen, Germany e-mail: [email protected] M.N. Fried, T. Dreyfus (eds.), Mathematics & Mathematics Education: Searching for Common Ground, Advances in Mathematics Education, DOI 10.1007/978-94-007-7473-5_6, © Springer Science+Business Media Dordrecht 2014
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The other idea was in a vague sense related to what could be called the “genetic principle”. Prominent mathematicians like Felix Klein and Otto Toeplitz believed that history of mathematics could contribute to the learner’s understanding by making visible great lines of development and thereby connecting seemingly unconnected subjects. Klein had exemplified this by his magnificent book “Lectures on the development of mathematics in the 19th century” in which he had reconstructed the immediate pre-history of the mathematics of his time. One can imagine that students who attended these lectures got a sound idea of what was going on in mathematics. But these lectures were definitely intended for an advanced audience. Otto Toeplitz, on the other hand, was more involved in t
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