Dynamic Euclidean Geometry: pseudo-Toulmin modeling transformations and instrumental learning trajectories

Stavroula Patsiomitou

doi:10.7176/jep/12-9-09

Abstract

The present paper attempts to bridge the world of DGS technology with the world Euclid bequeathed to us in his Elements. Competence in the DGS environment depends on the competence of the cognitive analysis as students seek to decode their ideas using the tools provided by the software. The dynamic notions (e.g., dynamic point, dynamic segment, instrumental decoding, hybrid-dynamic objects, active/ “alive” representations etc.), are taken as given and form the specific /particular theoretical basis for the constructive processes. Dynamic Euclidean constructions will be considered using pseudo-Toulmin’ diagrams. These considerations provide a theoretical basis for the idea that, in order to solve a mathematical construction problem in Dynamic Euclidean Geometry, we have to build up the interdependencies of tools in various sequential steps (based on theorems and definitions and the competence in using tools) which can be linked to the level of our conceptualization. The central idea is the following: Do the tools of Dynamic Euclidean Geometry determine a new kind of Geometry? Is Dynamic Euclidean Geometry a new kind of geometry? Does it have its own axiomatic system or its own undefined terms? In the paper, the notion of an instrumental learning path/trajectory is introduced as the interdependence/intra-dependence between dynamic tools, diagrams and mathematical objects during an instrumental decoding process. Keywords: Dynamic geometry , Euclid “Elements”, instrumental learning trajectories, Dynamic Euclidean Geometry DOI: 10.7176/JEP/12-9-09 Publication date: March 31 st 2021

Highlights

The present paper attempts to bridge the world of digital technology and the world Euclid bequeathed us in his "Elements"
In my previous research studies, I have introduced through my research studies among others the following notions: LVAR with active /or “alive” representations, “instrumental decoding”, “instrumental obstacles”, “hybrid-dynamic objects” which helped me to articulate my thoughts on what evoked through research with Dynamic Geometry System (DGS)
These notions did not exist until the moment that I had to find a way to discuss about the new kinds of objects that exist in the Dynamic Euclidean Geometry

Summary

Introduction

The present paper attempts to bridge the world of digital technology and the world Euclid bequeathed us in his "Elements". In my previous research studies, I have introduced through my research studies among others the following notions: LVAR with active /or “alive” representations, “instrumental decoding”, “instrumental obstacles”, “hybrid-dynamic objects” which helped me to articulate my thoughts on what evoked through research with DGS These notions did not exist until the moment that I had to find a way to discuss about the new kinds of objects that exist in the Dynamic Euclidean Geometry. A dynamic section contains meanings belonging to the same class that are united or joined into a whole, which in the concrete situation symbolically means they exist in one [“alive” book] section or they are dynamically linked It is not within the scope of this section to discuss these issues in any more detail, but like to Goldenberg, & Cuoco (1996) I would argue that “Dynamic Geometry needs its own axiomatic foundation to define the objects and postulates of its environment.” (cited in Jackiw, & Sinclair, 2009, p.415). The central idea is the following: Do the tools of Dynamic Euclidean Geometry determine a new kind of Geometry? Is Dynamic Euclidean Geometry a new kind of geometry? Does it have its own axiomatic system or --in other words-- its own undefined terms?

A cognitive analysis of DGS use

Euclidean Geometry and DGS

Proof and proving

Pseudo-Toulmin models for the analysis of Propositions in Euclid Elements