From Vecten’s Theorem to Gamow’s Problem: Building an Empirical Classification Model for Sequential Instructional Problems in Geometry

Stavroula Patsiomitou

doi:10.7176/jep/10-5-01

Abstract

In the current study, I will be presenting a literature review regarding the importance of students building a problem’s representation and the role modeling a real-world problem plays in students’ progressive mathematization. I shall introduce five types of geometrical problems applying the meaning of Linking Visual Active Representations (LVARs). Concrete examples will be presented in the next sections (i.e., Euclid’s proof of the Pythagorean Theorem, Vecten’s theorem, Gamow’s problem). I shall also introduce the meanings of hybrid object and diagram, as well as the meaning of dynamic section in a dynamic geometry environment, through examples. To summarize, I created an empirical classification model of sequential instructional problems in geometry. Its contribution to our knowledge in the area of the didactics of mathematics lies in the fact that this sequence of problems is regarded as a process whereby students develop a sequentially deeper understanding and increasingly more coherent reasoning that raises their van Hiele level. Keywords: dynamic section, hybrid object, Euclid “Elements”, Pythagorean Theorem, Vecten’s Theorem, Gamow’s problem, problem-solving. DOI : 10.7176/JEP/10-5-01

Highlights

Defining problem and problem solving in mathematics education The word “problem” is derived from the Greek word “provlema” with etymology from the verb “provalein”, whose meaning covers “projecting, showing, revealing, displaying, presenting”: i.e. ‘provalein’ refers to a goal presented in a question. (See https://etymonline.com)
Can we identify different levels of investigation in problem-solving in order to enhance the abstract thinking of our students?
Building on earlier works (e.g., Patsiomitou, 2008a, b) I shall review the building of sequential dynamic diagrams of the problem in “The Geometer’s Sketchpad” ( Sketchpad) (Jackiw, 1988) dynamic geometry software, applying the meaning of Linking Visual Active Representations (LVARs) (e.g., Patsiomitou, 2008 a, b, c, d, 2009 a, b, 2010, 2011, 2012a, b, 2013, 2014, 2018a, b), to ‘dynamically’ scaffold students thinking

Summary

Introduction

Defining problem and problem solving in mathematics education The word “problem” is derived from the Greek word “provlema” with etymology from the verb “provalein”, whose meaning covers “projecting, showing, revealing, displaying, presenting”: i.e. ‘provalein’ refers to a goal presented in a question. (See https://etymonline.com). Building on earlier works (e.g., Patsiomitou, 2008a, b) I shall review the building of sequential dynamic diagrams of the problem in “The Geometer’s Sketchpad” ( Sketchpad) (Jackiw, 1988) dynamic geometry software, applying the meaning of Linking Visual Active Representations (LVARs) (e.g., Patsiomitou, 2008 a, b, c, d, 2009 a, b, 2010, 2011, 2012a, b, 2013, 2014, 2018a, b), to ‘dynamically’ scaffold students thinking. I shall investigate the importance of the modeling process for students’ understanding and for the development of their progressive mathematization abilities; modeling is especially important for students who struggle with real-world geometric problem.

Results

Conclusion