Zugl.: Köln, Univ., Diss., 2009

Zugl.: Diss. / Lizenzpflichtig

Zugl.: Karlsruhe, Universiẗat, Diss., 2007.

Den Mittelpunkt des folgenden Diskurses bildet ein Projekt des Neukantianers Hermann Cohen (1842-1918), das dieser unter dem Titel „Das Prinzip der Infinitesimal-Methode und seine Geschichte“ 1883 präsentiert hat. Sein Vorhaben, die Fruchtbarkeit der infinitesimalen Größe in der Mathematik und den Naturwissenschaften auch für die Philosophie, vor allem die Kantische Transzendentalphilosophie, nutzbar zu machen, erwies sich zu damaliger Zeit als wenig populär. Infolge von Schwierigkeiten mit der Interpretation seiner komplizierten Schrift und heftiger Kritik führender Mathematiker blieb sein Werk weitgehend unbeachtet. Anhand eines Blickes auf den Gang der Wissenschaft der Infinitesimal-Mathematik soll diese Kritik im Folgenden entkräftet und neu bewertet werden. Es zeigt sich hierbei, dass, anders als zu Lebzeiten Cohens, heute gezielt versucht wird, die infinitesimale Größe in die mathematische Lehre zu integrieren – auch wenn dies mit erheblichen, vor allem philosophischen Schwierigkeiten verbunden ist. Hierbei soll auch das wieder erstarkte Interesse an den Infinitesimalien in der Nonstandard-Analysis als Anreiz dienen, die Philosophie Cohens am heutigen Forschungsdiskurs teilhaben zu lassen. In jüngerer Zeit spielt zudem auch in der Smooth Infinitesimal Analysis die Position des Intuitionismus wieder eine Rolle, welche der um Hermann Cohen und Paul Natorp entstandenen „Marburger Schule“ nahesteht. Auf den folgenden Seiten soll anhand Cohens „Logik der reinen Erkenntnis“ (1902) eine Lesart für eine „Infinitesimal-Logik“ Cohens präsentiert werden, die die Gedanken Cohens zur Infinitesimal-Methode in ein philosophisches System eingliedert. Wie schon in Cohens "`Prinzip der Infinitesimal-Methode und seine Geschichte"' soll es auch hier als "`unmittelbar nützlich"' erscheinen, "`zugleich mit der Durchführung eines systematisch entscheidenden Gedankens seine geschichtliche Entwicklung zu verfolgen."' [Cohen 1883, Vorwort] Dieser Rückblick auf die bewegte Historie des Infinitesimal-Begriffs soll grob die Entwicklungen hin zur Schaffenszeit Cohens umreißen und sodann als Prüfstein für dessen Ideen gelten. / This study focuses on a project of the neo-Kantian Philosopher Hermann Cohen (1842-1918), entitled “Das Prinzip der Infinitesimal-Methode und seine Geschichte,” which was presented in 1883. In this work, Cohen attempted to make the fruitfulness of the infinisimal magnitude in mathematics and natural science beneficial to philosophy as well, especially to Kantian transcendental philosophy. This project has proved unpopular in contemporary philosophy; difficulties in interpreting his complex writing and harsh criticism by prestigious mathematicians have resulted in his work remaining largely unnoticed. This study provides a refutation and reappraisal of this criticism in the light of subsequent historical developments in [the pedagogy of] infinitesimal analysis. Nowadays, in contrast to Cohen’s lifetime, the infinitesimal magnitude is being reintegrated into mathematical teaching – even where it is associated with significant (especially philosophical) difficulties. In addition, the renewed intensity of interest in infinitesimals in Nonstandard Analysis should be a stimulus for reintegrating Cohen‘s philosophy into contemporary research. Recently, moreover, intuitionism, which was closely affiliated with the “Marburger Schule” established by Hermann Cohen and Paul Natorp, has once again come to play a role in Smooth Infinitesimal Analysis. Following Cohen‘s “Logik der reinen Erkenntnis” (1902), this study presents an interpretation of Cohen‘s “Infinitesimal-Logik” by incorporating Cohen’s thoughts about the infinitesimal method into a philosophical system. In the spirit of Cohen’s remarks in his “Prinzip der Infinitesimal-Methode und seine Geschichte,” it should be “immediately useful. . .simultaneously to apply a systematically decisive idea and to follow its historical development” [Cohen 1883, Foreword]. A review of the colorful history of the infinitesimal traces this development up to the point of Cohen’s productive period; from that point on it is used as a performance test for his ideas.

Zugl.: Dresden, Techn. University, Diss., 2007.

It is well known that mathematics students have to be able to understand and prove theorems. From our experience we know that engineering students should also be able to do the same, since a good theoretical knowledge of mathematics is essential for solving practical problems and constructing models. Proving theorems gives students a much better understanding of the subject, and helps them to develop mathematical thinking. The proof of a theorem consists of a logical chain of steps. Students should understand the need and the legitimacy of every step. Moreover, they have to comprehend the reasoning behind the order of the chain’s steps. For our research students were provided with proofs whose steps were either written in a random order or had missing parts. Students were asked to solve the \"puzzle\" – find the correct logical chain or complete the proof. These \"puzzles\" were meant to discourage students from simply memorizing the proof of a theorem. By using our examples students were encouraged to think independently and came to improve their understanding of the subject.

Several years of my experiences in the use of real-time experiments are now upgraded in order to enhance also the teaching of mathematics. The motion sensor device enables us to get real time x(t) and v(t) graphs of a moving object or person. We can productively use these graphs to introduce differentiation on visual level as well as to show the integration procedure. The students are fully involved in the teaching as they are invited to walk in front of the sensor. This approach motivates them by the realistic aspects of mathematical structures. The method could help to fulfill the credo of teaching: comprehension before computation. The steps of such an approach are explained and discussed in further detail below.

Certainly one of the most powerful and important modeling languages of our time is the Calculus. But research consistently shows that students do not understand how the variables in calculus-based mathematical models relate to aspects of the systems that those models are supposed to represent. Because of this, students never access the true power of calculus: its suitability to model a wide variety of real-world systems across domains. In this paper, we describe the motivation and theoretical foundations for the DeltaTick and HotLink Replay applications, an effort to address these difficulties by a) enabling students to model a wide variety of systems in the world that change over time by defining the behaviors of that system, and b) making explicit how a system\'s behavior relates to the mathematical trends that behavior creates. These applications employ the visualization and codification of behavior rules within the NetLogo agent-based modeling environment (Wilensky, 1999), rather than mathematical symbols, as their primary building blocks. As such, they provide an alternative to traditional mathematical techniques for exploring and solving advanced modeling problems, as well as exploring the major underlying concepts of calculus.

In this paper I present a theoretical analysis (genetic decomposition) of the cognitive constructions for the concept of infinite Riemann sums following Piaget\'s model of epistemology. This genetic decomposition is primarily based on my own mathematical knowledge as well as on my continual observations of students in the process of learning. Based on this analysis I plan to suggest instructional procedures that motivate the mental activities described in the proposed genetic decomposition. In a later study, I plan to present empirical data in the form of informal interviews with students at different stages of learning. The analysis of those interviews may suggest a review of my initial genetic decomposition.

It is well known that mathematics students have to be able to understand and prove theorems. From our experience we know that engineering students should also be able to do the same, since a good theoretical knowledge of mathematics is essential for solving practical problems and constructing models. Proving theorems gives students a much better understanding of the subject, and helps them to develop mathematical thinking. The proof of a theorem consists of a logical chain of steps. Students should understand the need and the legitimacy of every step. Moreover, they have to comprehend the reasoning behind the order of the chain’s steps. For our research students were provided with proofs whose steps were either written in a random order or had missing parts. Students were asked to solve the \"puzzle\" – find the correct logical chain or complete the proof. These \"puzzles\" were meant to discourage students from simply memorizing the proof of a theorem. By using our examples students were encouraged to think independently and came to improve their understanding of the subject.