Touching the transcendentals by tractional constructions: historical and foundational research, educational and museal applications by new emerging technologies
Project Calculus marked an epochal change in the evolution of scientific thought. However, this subject poses several difficulties to our understanding because it involves the manipulation of infinitary objects. Research in mathematics education highlighted obstacles and proposed different approaches in this field of mathematics.
In contrast to today's mainstream conception, which relegates geometry to an ancillary tool of mere visualization, we shall investigate new ways of presenting calculus centered around its geometric and constructive components. The core idea of our project is the utilization of "tractional motion," which appeared during Leibniz's stay in Paris, to construct transcendental curves. In this project, we shall advance three objectives.
The first objective concerns the research of new elements in the historical development of tractional motion and some missing elements in the related mathematical contents. Special attention is devoted to Giovanni Poleni (1683-1761), the first scholar who related the late-17th-century approach to inverse tangent problems by the dragged weight (Huygens, Leibniz) with the early-18th-century implementation by the direction of a wheel. Concerning the undisclosed mathematical potential of tractional motion, we also aim to provide a general method to solve complex differential equations with these tools. We hope that such a method could provide new visual and dynamic insights.
The second objective concerns material and digital machines in education laboratory activities. We propose experimental activities not only with material tractional machines but also with Virtual-Reality tools. After defining a methodology to explore the potential of VR machines in mathematics education (starting from activities involving machines already studied in the literature by material explorations), we propose investigating VR tractional machines.
As the third objective, we propose investigating the adoption of emerging technology (digital fabrication and the already introduced Virtual Reality) to allow the scientific diffusion of tractional motion even without the presence of craftsman-made artifacts. Virtual Reality is also studied to provide a museum vulgarization tool for mathematics.
As a dissemination strategy, we plan to write papers, attend national and international conferences, and organize a final symposium on the project. From the perspective of material artifacts, we plan to construct machines for exhibitions in some Festivals of Science and prepare exhibition pathways for museum and educational activities. About the digital counterpart, we aim to realize and freely share two Virtual Reality applications (one on the Scheiner's pantograph and the other on a tractional machine) and the digital design of several mathematical machines (so that anyone can easily download the files and construct the machines by digital fabrication tools).