Interpolation als Kernidee inner- und außermathematischer Anwendungen : mathematikdidaktische Analyse und Entwicklung von Unterrichtsmaterialien zum Kontext Computeranimation

• Interpolation as a basic idea of inner- and extra-mathematical applications : didactical analysis and design of teaching materials for computer animation

Peters, Agnes; Heitzer, Johanna Maria (Thesis advisor); Filler, Andreas (Thesis advisor)

Aachen (2016)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen, 2016

Abstract

Methods for approximating functions based on discrete data play an important role in extra-mathematical as well as inner-mathematical applications. Often, in extra-mathematical contexts only discrete data are given which represent a number of values of an unknown function. Inner-mathematically, such methods are used to describe complicated functional relations through simpler approximation functions. This doctoral thesis discusses one of the most important subthemes of approximation techniques for functional relations, namely interpolation. The basic idea is to compute a function which assumes the given values exactly. In this thesis only methods that use polynomial functions are discussed because they are commonly used and directly linked to mathematics teaching at school. These include the polynomial and Hermite interpolation, the interpolation with linear and cubic splines and with cubic Hermite splines. The topic links to the so-called "Steckbriefaufgaben" from analysis lessons. In addition to linear equation systems and polynomial functions, derivatives belong to the relevant mathematical tools. Beyond this, especially the applied perspective, which is always present in interpolation problems, makes the topic interesting for educational tasks. The look at applications supports an appropriate image of mathematics and is therefore required by the national standards of the KMK. However, there is a lack of authentic problems that can be solved by pupils. Additionally, in contrast to real situations the application-oriented exercises typically dealt with in analysis lessons start with concrete given functions whose model character is however rarely discussed. Therefore, it is the aim of this thesis to point out the potential of the mathematical idea of interpolation to convey an adequate image of mathematics and its applications. Furthermore, an animation technique called Keyframing is made accessible to educational tasks. The basic idea of this method is the interpolation of the numerous frames necessary for an animation based on a few keyframes. As an interdisciplinary context of mathematics, informatics, physics and arts and because of its direct reference to everyday experiences Keyframing is an authentic, illustrative and motivating subject. Naturally, curves and their parametric forms are used for the description of the movements of objects. Thus, the context motivates the examination of curves and their parametric representations and supports especially the dynamic perspective on curves. In summary, the concepts of functions and derivatives on the one hand and parametric representations and equation systems on the other hand as central themes of analysis, linear algebra and analytic geometry are linked. Regarding mathematical lessons, this thesis provides the relevant mathematical background for teachers. Based on experiences from workshops with pupils at RWTH Aachen University a learning environment for Keyframing is introduced. Possible approaches and focuses are pointed out alongside concrete tasks and examples. Furthermore, GeoGebra and Python files have been created for exploring the mathematical contents. The possibility to use the mathematical knowledge for generating an animation in a program like Synfig is particularly motivating for pupils.