Hans-Joachim Vollrath’s research while affiliated with University of Wuerzburg and other places

Gegen Ende des 19. Jahrhunderts hatte die Philosophische Fakultät der Universität Würzburg eine beachtliche Sammlung mathematischer und naturwissenschaftlicher Instrumente als Unterrichtshilfen zusammengetragen. Die meisten dieser inzwischen „unmodern“ gewordenen Instrumente wurden 1877 an das Bayerische Nationalmuseum München verkauft. Eins dieser Instrumente ist ein Satz von 6 Messingstäben in einem mit Leder bezogenen Holzköcher. Die Stäbe sind bekannt als Architekturstäbe von Nikolaus Goldmann (1611–1665), einem deutschen Mathematiker und Architekturtheoretiker, der in Leiden lebte. Er beschrieb die Stäbe, die zur Konstruktion der 5 klassischen Säulenordnungen dienen sollten, in seinem Tractatus de stylometris, Leiden 1662. Im Folgenden werden diese Architekturstäbe didaktisch analysiert. Abstract By the end of the 19th century the Philosophical Faculty of Würzburg University had compiled a notable collection of mathematical and scientific instruments as teaching aids. Most of these “old-fashioned” instruments were sold to the Bayerisches Nationalmuseum München in 1877. One of the items is a set of six brass rods in a leather-covered case known as “Architectural rods” designed by Nikolaus Goldmann (1611–1665), a German mathematician and architect who lived in Leiden. He described the rods that should be used for the construction of the five classical orders of columns in his Tractatus de stylometris, Leiden 1662. In the following paper the rods are analvsed didactically.

This paper is a historical investigation about the relationship between concept formation and problem development in mathematics. A survey of the history of infinite series reveals different roles of concepts for generating and solving problems. Further historical considerations lead to didactical implications.

Aus Anlass des 25-jährigen Bestehens des JMD berichtet das erste Herausgeberteam über die Gründung der Zeitschrift, die damit verbundenen Ziele und Erwartungen der GDM, die Überlegungen und Diskussionen zur Konzeption, die Entwicklung der Verfahrensweise und über Erfahrungen der Anfangsphase. Abstract On the occasion of the 25th anniversary of JMD’s foundation the first team of editors reports about the start of the journal, the underlying goals and expectations of the GDM, the discussions about the character of the journal, the development of the procedure and about experiences in the beginning.

This paper reflects the influence of MARTIN WAGENSCHEIN (1896-1988) on the didactics of mathematics. From his point of view certain deficits and false orientations are pointed out. Some hints for possible stimulations by his ideas are given.

“Didactical phenomenology” has become one of FREUDENTHALs key-concepts in mathematics education. It indicates a research program which he made explicit in his book Didactical Phenomenology of Mathematical Structures (1983). There he analysed the central concepts of mathematics instruction. It was his intention to show that children do not learn them primarily as concepts but “grasp them as mental objects and carry them out as mental activities”. This paper points out some methodological roots of his approach and refers to possible applications in empirical research about the development of mathematical thinking in children.

Die Frage, welche Vorstellungen über Geraden und Strecken Lernende im Geometrieunterricht erwerben, war Gegenstand einer Untersuchung an Schülerinnen und Schülern der 7. und 8. Jahrgangsstufe zweier Gymnasien in Bayern. Sie war angeregt durch die These von Struve (1990), in der Elementargeometrie könnten die Lernenden keine angemessenen Vorstellungen über die Unbegrenztheit der Geraden gewinnen. Die Untersuchung zeigte, daß die Begrenztheit der Strecken und die Unbegrenztheit der Geraden als die entscheidenden Merkmale gesehen wurden. Abstract Straight lines and segments are taught from grade 5 on in German schools. Concerning the straight lines Struve (1990) doubts that children gain adequate images of the unboundedness of straight lines from elementary geometry. This paper reports about an investigation of children’s images of these concepts. Students from grade 7 and grade 8 of Bavarian Gymnasiums were tested with respect to their images of straight lines and segments. The boundedness of the segments and the unboundedness of the straight lines were seen as the crucial differences by most of the students.

Understanding is a fundamental category of mathematics education. There are several characteristics of mathematics which can support or rather restrain understanding. This leads to paradoxes caused by complementarieties, e.g. the general and the special, the whole and the detail, intuition and rigour, variety and essence, process and result. This paper elaborates such paradoxes of understanding and offers councels for teaching. © 1993, GDM - Gesellschaft für Didaktik der Mathematik. All rights reserved.

The concept of “functional thinking” has a long tradition in mathematics education. It was a fundamental goal in the Meran proposals for mathematics teaching (1905). After a period of enthusiastic acceptance it became out of fashion during the “new math” era. But it has been rediscovered in the last decade. This paper is a report about this development. It reviews contributions to this concept, and it tries to make evident that functional thinking is a fruitful didactical concept. © 1989, GDM - Gesellschaft für Didaktik der Mathematik. All rights reserved.

The ability to reason is an important goal in mathematics education. Reasoned arguments are often used to generate understanding. But it is also necessary to discuss the role of argument as a subject of interest. Therefore a suggestion is made for a teaching sequence on reasoning in arithmetic for lower ability groups at the pre-college level. It is based on an investigation in some German schools (grade 7 and 9) about students’ ability to prove elementary statements such as 1/2 < 3/4. It was found that many different arguments were used on different levels, and that students were conscious of different possible levels of reasoning. The great variety of types used may lead to discussions about the different arguments.

Bei den Vorbereitungen einer Ausstellung der Universität Würzburg zum 400. Geburtstag des Universalgelehrten Athanasius Kircher (1602–1680) habe ich mich eingehend mit einem von ihm entwickelten Lehrmittel befasst. In seiner barocken Sprache hatte er es Organum mathematicum (Mathematische Orgel) genannt. Sein Schüler Kaspar Schott (1608–1666) schrieb dazu ein ausführliches Handbuch mit dem Titel Organum mathematicum, das 1668 postum erschien. Exemplare dieses Lehrmittels finden sich im Bayerischen Nationalmuseum in München und im Istituto e Museo di Storia della Scienza in Florenz. Im Folgenden soll die Mathematische Orgel didaktisch analysiert werden. Dabei stütze ich mich auf Schotts Buch und eine gründliche Untersuchung des Münchner Exemplars. Abstract As part of the preparations for an exhibition about the famous universal scholar Athanasius Kircher (1602–1680) I had the opportunity to study an apparatus for teaching that he had developed. He had called it Organum mathematicum (Mathematical Organ). His disciple Kaspar Schott (1608–1666) had written a handbook for this apparatus with the title Organum mathematicum (1668). This book is an outstanding source for understanding the Organum. Models of the Organum can be found in the “Bayerisches Nationalmuseum München” and in the “Istituto e Museo di Storia della Scienza Firenze”. On the basis of Schott’s book and the study of the Munich model the Mathematical Organ will be analysed didactically.