Hans Niels Jahnke

Proof and application

This paper outlines an epistemological conception which attempts to relate the formal aspects of mathematical proof to its pragmatic dimensions. In addition to the key concept of application, the paper makes use of several concepts from the domain of analytical philosophy, to present a view of proof that might best be categorized as a dialectical one. A number of implications for teaching are discussed.

The use of original sources in the mathematics classroom

The study of original sources is the most ambitious of ways in which history might be integrated into the teaching of mathematics, but also one of the most rewarding for students both at school and at teacher training institutions.

A history of analysis

Antiquity by R. Thiele Precursors of differentiation and integration by J. van Maanen Newtons method and Leibnizs calculus by N. Guicciardini Algebraic analysis in the 18th century by H. N. Jahnke The origins of analytic mechanics in the 18th century by M. Panza The foundation of analysis in the 19th century by J. Lutzen Analysis and physics in the nineteenth century: The case of boundary-value problems by T. Archibald Complex function theory, 1780-1900 by U. Bottazzini Theory of measure and integration from Riemann to Lebesgue by T. Hochkirchen The end of the science of quantity: Foundations of analysis, 1860-1910 by M. Epple Differential equations: A historical overview to circa 1900 by T. Archibald The calculus of variations: A historical survey by C. Fraser The origins of functional analysis by R. Siegmund-Schultze Index of names Subject index.

The Conjoint Origin of Proof and Theoretical Physics

This paper examines the historical fact that the Greeks invented not only the idea of mathematical proof but also and simultaneously “theoretical physics.” This simultaneity was not accidental; rather, the two events were connected and influenced each other. The link between them was an idea in the Greek philosophy of science called “saving the phenomena.” This paper establishes a connection between this idea and the pre-Euclidean meaning of the term “axiom.” It then demonstrates how this idea continued into modern mathematics as well as maintaining its “traditional” centrality in the sciences. The last part of the paper applies these ideas to the teaching of proof, explaining why and how the relationship between hypotheses and consequences should be made a focus in the teaching of proof.

Another Approach to Proof: Arguments from Physics

In the first part of the paper we will explore the use of arguments from physics in mathematical proof and give some reasons why this approach might be worthwhile. In the second part we will relate this idea to Freudenthals concept of local organization. The third part of the paper will present the results of an empirical study conducted in Canada on the classroom use of arguments from physics in mathematical proof.

Conceptions of Proof – In Research and Teaching

This chapter first analyses and compares mathematicians’ and mathematics educators’ different conceptualisations of proof and shows how these are formed by different professional backgrounds and research interests. This diversity of views makes it difficult to precisely explain what a proof is, especially to a novice at proving. In the second section, we examine teachers’, student teachers’ and pupils’ proof conceptions and beliefs as revealed by empirical research. We find that the teachers’ beliefs clearly revolve around the questions of what counts as proof in the classroom and whether the teaching of proof should focus on the product or on the process. The third section discusses which type of metaknowledge about proof educators should provide to teachers and thus to students, how they can do this and what the intrinsic difficulties of developing adequate metaknowledge are.

Proving and Modelling

This paper discusses the complementary roles of modelling and proof. The two are inseparably linked, and the authors argue that this should be reflected in teaching. Two examples are discussed. The first describes a teaching unit using arguments from statics to prove geometrical theorems, the second discusses the role of thought experiments in general and a specific thought experiment for deriving Pick’s formula.

Die Bestimmung des Umfangs der Erde als Thema einer mathematikhistorischen Unterrichtsreihe

ZusammenfassungDer vorliegende Aufsatz dokumentiert und analysiert eine mathematikhistorische Unterrichtsreihe, die der erstgenannte Autor im regulären Unterricht einer 9. Klasse eines Gymnasiums im Frühjahr 2002 gehalten hat. Thema dieser Reihe war die Auseinandersetzung mit Texten, die der griechische Mathematiker und Astronom Kleomedes (ca. 1. Jh. n. Chr. (?)) hinterlassen hat. Durch die Arbeit mit diesen Texten lernten Schüler die Mathematik aus einem ungewohnten Blickwinkel kennen, der ihnen neue Einsichten über ihr eigenes Verhältnis zum Fach vermitteln konnte. Für den Lehrer ergaben sich darüber hinaus Möglichkeiten, typische Verste-hensprobleme von Schülern unter hermeneutischen Gesichtspunkten zu reflektieren.AbstractThe paper discusses a historico-mathematical teaching unit that was held by the first-mentioned author in spring 2002 in a class of grade 9 of a German Gymnasium. In this unit the pupils studied a historical source by the Greek mathematician and astronomer Cleomedes (1st cent. A.D. (?)). By doing so they have obtained a fresh and unusual view on mathematics that has given them new insights into their attitudes to the subject. The teacher on the other hand has got several opportunities to reflect upon typical problems of comprehension from a hermeneutic point of view.

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 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.

Historical Sources in the Mathematics Classroom: Ideas and Experiences

Reading historical sources in the mathematics classroom should lead to authentic mathematical experiences and introduce students to the cultural context of mathematics.