A Synergy between History of Mathematics and Mathematics Education: A Possible Path from Geometry to Symbolic Algebra

Emilia Florio

doi:10.3390/educsci10090243

Abstract

This paper proposes an experimental path aimed at guiding upper secondary school students to overcome that discontinuity, often perceived by them, between learning geometry and learning algebra. This path contributes to making students aware of how the algebraic language, formalized in the most powerful form by Descartes, grafts itself onto the geometric language. This is realized by introducing a problem included in a text written by Abū Kāmil before the year 870. This awareness acquired by the students, when accompanied by some semiotic considerations, allows the translation of the problem from “spoken” algebra to “symbolic” algebra, and it represents the background for a possible use of the same problem within the framework of analytic geometry. This proposition manifests a didactic and popular efficacy that supports and favors the recognition of the object it is talking about in different contexts, helping to create a unitary vision of mathematics.

Highlights

This work proposes a didactic path, designed for an upper secondary school class, aimed at showing a connection between geometric knowledge and algebraic knowledge starting from the use of the first problem of the chapter on the regular pentagon and decagon of the Kitāb fıal-jabr wa al-muqābala (Book on algebra and science of reduction and cancellation) of Abū Kāmil [1].The choice of a proposition taken from the history of mathematics is encouraged by the fact that knowledge of the latter and its use in teaching constitute a significant part of the cultural background of future mathematics teachers [2]
Useful feedback on the training of teachers are found in Reference [5] with the practice-based theory of mathematical knowledge for teaching, in Reference [6,7,8] where the authors specify the positive elements of the teacher education through the history of mathematics, in Reference [9] where the importance of the historical and cultural dimensions in mathematics education is underlined, in Reference [10], concerning the support of history in changing individual’s epistemic beliefs about the nature of mathematical knowledge, and in Reference [11] for the implementation of teachers’ skills in the cultural analysis of content
All this helps to recognize, in different contexts, results obtained in a specific context, making possible an intellectual experience of a continuum of meaning instead of a discontinuity, which is often found in mathematics teaching, between learning geometry and learning algebra

Summary

Introduction

The choice of a proposition taken from the history of mathematics is encouraged by the fact that knowledge of the latter and its use in teaching constitute a significant part of the cultural background of future mathematics teachers [2]. The knowledge of how a mathematical concept was born and evolved can contribute to a better understanding of that concept itself [3], and the study of historical sources contains a potential suitable to increase awareness of possible misconceptions, obstacles, and impediments related to various mathematical concepts and ideas [4]. The same choice of Abū Kāmil’s proposition is motivated by the fact that the teaching and learning of mathematics are favorably affected by a programming that takes into account the social context in which they develop [12,13]. A further motivation is given from the cultural elements [14,15,16]

Methods

Results

Discussion

Conclusion