Asymptotic Stability and Smooth Lyapunov Functions

Dragging in Dynamical Geometry Software (DGS) is described by introducing a hierarchy of its functions. This is suitable for classifying different attitudes and aims of students who investigate a geometric problem, such as exploring, conjecturing, validating and justifying. Moreover the hierarchy has cognitive features and can be used to describe the twofold modalities, namely ascending and descending in which students interact with external representations (e.g. Cabri drawings). Switching from one modality to the other through dragging often allows them to produce fruitful conjectures and to pass from the empirical to the theoretical side of the question. The genesis of such different functions in students does not happen automatically but is the consequence of specific didactical interventions of the teacher in the pupils' apprenticeship of Cabri practises. A worked-out example illustrates the theoretical concepts introduced in the paper. Kurzreferat: Der Text fuhrt eine Funktionshierarchie fur den Gebrauch des Zugmodus in Dynamischer Geometrie Software (DGS) ein. Diese ist zur Klassifikation von Verhaltensweisen und Zielen der Lernenden geeignet, wenn diese geometrische Problem explorieren, Vermutungen aufstellen, bestatigen und rechtfertigen und dabei theoretische und empirische Aussagen in Cabri-Zeichnungen in Beziehung setzen (wechselweise „auf„ und „absteigen“). Der Wechsel zwischen diesen Sichtweisen unter Nutzung des Zugmodus erlaubt es ihnen oft, gehaltvolle Vermutungen zu formulieren und von einer empirischen zu einer theoretischen Sicht auf das Problem zu wechseln. Der Sichtwechsel stellt sich allerdings nicht automatisch ein, sondern ist die Folge besonderer didaktischer Masnahmen des Lehrenden. Ein ausgearbeitetes Beispiel illustriert die neu eingefuhrten theoretischen Begriffe.

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A cognitive analysis of dragging practises in Cabri environments

Control of underactuated spacecraft with bounded inputs

Despite widespread agreement that the activity of reasoning-and-proving should be central to all students' mathematical experiences, many students face serious difficulties with this activity. Mathematics textbooks can play an important role in students' opportunities to engage in reasoning-and-proving: research suggests that many decisions that teachers make about what tasks to implement in their classrooms and when and how to implement them are mediated by the textbooks they use. Yet, little is known about how reasoning-and-proving is promoted in school mathematics textbooks. In this article, I present an analytic/methodological approach for the examination of the opportunities designed in mathematics textbooks for students to engage in reasoning-and-proving. In addition, I exemplify the utility of the approach in an examination of a strategically selected American mathematics textbook series. I use the findings from this examination as a context to discuss issues of textbook design in the domain of rea...

Reasoning-and-Proving in School Mathematics Textbooks

Spin-axis stabilization of symmetric spacecraft with two control torques

Global feedback stabilization of the angular velocity of a symmetric rigid body

Global stability and external stability of dynamical systems

The logic of discovering and that of justifying have been a permanent source of debate in mathematics, because of their different and apparently contradictory features within the processes of production of mathematical sentences. In fact, a fundamental unity appears as soon as one investigates deeply the phenomenology of conjecturing and proving using concrete examples. In this paper it is shown that abduction, in the sense of Peirce, is an essential unifying activity, ruling such phenomena. Abduction is the major ingredient in a theoretical model suitable for describing the transition from the conjecturing to the proving phase. In the paper such a model is introduced and worked out to test Lakatos' machinery of proofs and refutations from a new point of view. Abduction and its categorical counterpart, adjunction, allow to explain within a. unifying framework most of the phenomenology of conjectures and proofs, encompassing also the method of Greek analysis-synthesis.

Valeria Andriano

Papers

Global feedback stabilization of the angular velocity of a symmetric rigid body

Global stability and external stability of dynamical systems

Abduction and Conjecturing in Mathematics