MaT²SMC: materials for teaching together: science andmMathematics teachers collaborating for better results

Let us start with an important statement: Mathematics and Science teachers do a good, and often an outstanding, job in teaching young people the basic knowledge of their respective fields! It is not the intent of this book to criticize what they do or how they do it. Keeping that in mind, and noting the fact that the teaching content of these fields intersects and overlaps, we observed – and this took us by surprise – that there is hardly any collaboration or consultancy between mathematics and science teachers (or textbook authors). Mathematics teachers often use science contexts in tasks, and science teachers often use mathematics, however they are usually working independently. Science c…

Laboratorio di matematica in classe: due nuove macchine per problemi nel continuo e nel discreto

In this work we will introduce two mathematical machines (originally invented by the authors) that can be useful in class during laboratorial activities. We decided to introduce them together because they appeared us complementary about the subtended contents, in fact one of them solves a discrete problem (diophantine equations) involving concepts as integers, divisibility, and in general discrete math, while the other machine (solving differential equations in a grapho-mechanical way) involves the complementary concepts of continuous math: curves, tangents, derivatives. In this paper we will just present the machines, deepening the ideas and the subtended mathematical contents, but we will…

Movimento trazionale: dalle macchine matematiche ai computer

The Equiangular Compass

Tractional Motion Machines extend GPAC-generable functions

In late 17th century there appeared the Tractional Motion instruments, mechanical devices which plot the curves solving differential equations by the management of the tangent. In early 20th century Vannevar Bush’s Differential Analyzer got the same aim: in this paper we’ll compare the Differential Analyzer mathematical model (the Shannon’s General Purpose Analog Computer, or GPAC) with the Tractional Motion Machine potentials. Even if we will not arrive in defining the class of the functions generated by Tractional Motion Machines, we’ll see how this class will strictly extend the GPAC-generable functions.

Geometrical-mechanical artefacts for managing tangent concept

A CONSTRUCTIVE APPROACH TO THE INFINITESIMAL ANALYSIS: EPISTEMOLOGIC POTENTIALS AND LIMITS OF THE "TRACTIONAL MOTION"

Une quête d'exactitude : machines, algèbre et géométrie pour la construction traditionnelle des équations différentielles

In La Géométrie, Descartes proposed a “balance” between geometric constructions and symbolic manipulation with the introduction of suitable ideal machines. In particular, Cartesian tools were polynomial algebra (analysis) and a class of diagrammatic constructions (synthesis). This setting provided a classification of curves, according to which only the algebraic ones were considered “purely geometrical.” This limit was overcome with a general method by Newton and Leibniz introducing the infinity in the analytical part, whereas the synthetic perspective gradually lost importance with respect to the analytical one—geometry became a mean of visualization, no longer of construction. Descartes’s…

Tractional Motion Machines: Tangent-Managing Planar Mechanisms as Analog Computers and Educational Artifacts

Concrete and virtual machines play a central role in the both Unconventional Computing (machines as computers) and in Math Education (influence of artifacts on reaching/producing abstract thought). Here we will examine some fallouts in these fields for the Tractional Motion Machines, planar mechanisms based on some devices used to plot the solutions of differential equations by the management of the tangent since the late 17th century.

Geometrical-mechanical artefacts mediating tangent meaning: the Tangentograph

This work deals with the didactical use of geometrical-mechanical artefacts to acquire tangent concept in vygotskian perspective. We adopt Rabardel’s theory on instru-mental approach to distinguish artefacts and instruments specially to evince the history-to-education ontogenesis-phylogenesis process. From this point of view we trace a historical-epistemological pathway for the tangent up to set an ad hoc didactical counterpart. Specifically in this paper we deepen the kinematical properties of the tangent (introducing the XVII century so called tractional motion) designing a laboratorial didactic pathway for 12th grade students with the use of a particular geometrical-mechanical artefact f…

Un delicato equilibrio tra macchine, algebra e geometria: Descartes e una possibile estensione differenziale

A geometrical constructive approach to infinitesimal analysis: epistemological potential and boundaries of tractional motion

Recent foundational approaches to Infinitesimal Analysis are essentially algebraic or computational, whereas the first approaches to such problems were geometrical. From this perspective, we may recall the seventeenth-century investigations of the “inverse tangent problem.” Suggested solutions to this problem involved certain machines, intended as both theoretical and actual instruments, which could construct transcendental curves through so-called tractional motion. The main idea of this work is to further develop tractional motion to investigate if and how, at a very first analysis, these ideal machines (like the ancient straightedge and compass) can constitute the basis of a purely geome…