The roles of Geometry and Arithmetic in the Development of Elementary Algebra: Historical Remarks from a Didactic Perspective (original) (raw)
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Approaches to Algebra, 1996
ln order to provide a brief overview of some of the historical affiliations between geometry and arithmetic in the emergence of algebra, we discuss some hypotheses on the origins of Diophantus' algebraic ideas, based on recent historical data. The first part deals with the concept of unknown and its links to two different currents of Babylonian mathematics (one arithmetical and the other geometric). The second part deals with the concepts offormula and variable. Our study suggests that the historical conceptual structure of our main modern elementary algebraic concepts, that of unknown and that ofvariable, are quite different. The historical discussion allows us to raise some questions concerning the raie geometry and arithmetic could play in the teaching of basic concepts of algebra in junior high school.
Through a broad structural analysis and a close reading of Old Babylonian mathematical "procedure texts" dealing mainly with problems of the second degree it is shown that Old Babylonian "algehra" was neither a "rhetorical algebra" dealing with numbers and arithmetical relations between numbers nor built on a set of fixed algorithmic procedures. Instead, the texts must be read as "naive" prescriptions for geometric analysis-naive in the sense that the results are seen by immediate intuition to be correct, but the question of correctness never raised-dealing with measured or measurable but unknown line segments, and making use of a set of operations and techniques diff8rent in structure from that of arithmetical algebra.
1987
Through a broad structural analysis and a close reading of Old Babylonian mathematica] "procedure texts" dealing mainly with problems of the second degree it is shown that Old Babylonian "algebra" was neither a "rhetorical algebra" dealing with numbers and arithmetical relations between numbers nor built on a set of fixed algorithmic procedures. Instead, the texts must be read as "naive" prescriptions for geometric analysis-naive in the sense that the results are seen by immediate intuition to be correct, but the question of correctness never raised-dealing with measured or measurable but unknown line segments, and making use of a set of operations and techniques different in structure from that of arithmetical algebra. The investigation involves a thorough discussion and re-interpretation of the technical terminology of Old Babylonian mathematics, elucidates many terms and procedures which have up to now been enigmatic, and makes many features stand out which had not been noticed before. The second-last chapter discusses the metamathematical problem, whether and to which extent we are then entitled to speak of an Old Babylonian algebra ; it also takes up the overall implications of the investigation for the understanding of Old Babylonian patterns of thought. It is argued that these are not mythopoeic in the sense of H. and H. A. Frankfort, nor savage or cold in a Lévi-Straussian sense, nor however as abstract and modern as current interpretations of the mathematical texts would have them to be. The last chapter investigates briefly the further development of Babylonian "algebra" through the Seleucid era, demonstrating a clear arithmetization of the patterns of mathematical thought, the possible role of Babylonian geometrical analysis as inspiration for early Greek geometry, and the legacy of Babylonian "algebraic" thought to Medieval Islamic algebra. Jens Heyrup presented the progress of the project in the four Workshops on Concept Development in Babylonian Mathematics held at the Seminar für Vorderasiatische Altertumskunde der Freien Universität Berlin in 1983, 1984, 1985 and 1988, and included summaries of some of my results-without the detailed arguments-in various contexts where they were relevant. This article is then meant to cover my results coherently and to give the details of the argument, without renouncing completely on readability. Admittedly, the article contains many discussions of philological details which will hardly be understandable to historians of mathematics without special assyriological training, but which were necessary if philological specialists should be able to evaluate my results ; I hope the non-specialist will not be too disturbed by these stumbling-stones. On the other hand many points which are trivial to the assyriologist are included in order to make it clear to the non-specialist why current interpretations and translations are only reliable up to a certain point, and why the complex discussions of terminological structure and philological details are at all necessary. I apologize to whoever will find them boring and superfluous. It is a most pleasant duty to express my gratitude to all those who have assisted me over the years,-especially Dr.
Diophantus and premodern algebra: New light on an old image
As a theme of historical research Diophantus’ work raises two main issues that have been intensely debated among researchers of the period: (i) The proper understanding of Diophantus’ practice; (ii) the recognition of the mathematical tradition to which this practice belongs. The traditional answer to this range of questions – since medieval Islam, through the Renaissance and the Early Modern period, up to the leading historians of mathematics of the 20th century – was that Diophantus’ book is a book on algebra. This traditional approach has been criticized recently by some historians of mathematics who point out the anachronistic methodology that historians in the past often were using in analyzing ancient texts. But, criticizing the methodology by which one defends a historical claim does not mean necessarily that the claim itself is wrong. The paper discusses some crucial issues involved in Diophantus’ problem-solving, thus, giving support to the traditional image about the algebraic character of Diophantus’ work, but put in a totally new framework of ideas.
Altorientalische Forschungen, 2000
Through a broad structural analysis and a close reading of Old Babylonian mathematical "procedure texts" dealing mainly with problems of the second degree it is shown that Old Babylonian "algehra" was neither a "rhetorical algebra" dealing with numbers and arithmetical relations between numbers nor built on a set of fixed algorithmic procedures. Instead, the texts must be read as "naive" prescriptions for geometric analysis-naive in the sense that the results are seen by immediate intuition to be correct, but the question of correctness never raised-dealing with measured or measurable but unknown line segments, and making use of a set of operations and techniques diff8rent in structure from that of arithmetical algebra.
1990
Through a broad structural analysis and a close reading of Old Babylonian mathematical "procedure texts" dealing mainly with problems of the second degree it is shown that Old Babylonian "algehra" was neither a "rhetorical algebra" dealing with numbers and arithmetical relations between numbers nor built on a set of fixed algorithmic procedures. Instead, the texts must be read as "naive" prescriptions for geometric analysis-naive in the sense that the results are seen by immediate intuition to be correct, but the question of correctness never raised-dealing with measured or measurable but unknown line segments, and making use of a set of operations and techniques diff8rent in structure from that of arithmetical algebra.
The aim of this paper is to employ the newly contextualised historiographical category of ''premodern algebra'' in order to revisit the arguably most controversial topic of the last decades in the field of Greek mathematics, namely the debate on ''geometrical algebra''. Within this framework, we shift focus from the discrepancy among the views expressed in the debate to some of the historiographical assumptions and methodological approaches that the opposing sides shared. Moreover, by using a series of propositions related to Elem. II.5 as a case study, we discuss Euclid’s geometrical proofs, the so-called ''semi-algebraic'' alternative demonstrations attributed to Heron of Alexandria, as well as the solutions given by Diophantus, al-Sulamī, and al-Khwārizmī to the corresponding numerical problem. This comparative analysis offers a new reading of Heron’s practice, highlights the significance of contextualizing ''premodern algebra'', and indicates that the origins of algebraic reasoning should be sought in the problem-solving practice, rather than in the theorem-proving tradition.
THE FOUR SIDES AND THE AREA Oblique Light on the Prehistory of Algebra
The present essay traces the career of a particular mathematical problem-to find the side of a square from the sum of its four sides and the area-from its first appearance in an Old Babylonian text until it surfaces for the last time in the same unmistakeable form during the Renaissance in Luca Pacioli's and Pedro Nunez' works. The problem turns out to belong to a non-scholarly tradition carried by practical geometers, together with other simple quasi-algebraic "recreational" problems dealing with the sides, diagonals and areas of squares and rectangles. This "mensuration algebra" (as I shall call it) was absorbed into and interacted with a sequence of literate mathematical cultures: the Old Babylonian scribal tradition, early Greek so-called metric geometry, and Islamic al-jabr. The article explores how these interactions inform us about the early history of algebraic thinking.
DIOPHANTHO'S ARITHMETIC: A CONTRIBUTION OF GREEK MATHEMATICS AS A STRATEGY FOR TEACHING EQUATIONS IN BASIC EDUCATION (Atena Editora), 2022
In this article, we investigated and discuss the didactic potential of historical problems in mathematics, aiming to locate classical problems and their possible formalizations, so that we could understand their elements and compare them. The investigation and study of equations from the work Arithmetic of Diophantus of Alexandria (3rd century), allowed us to select problems of a historical nature in an integration process, aiming to offer Basic Education teachers, notes and suggestions for the exploration of this type. of problems as a means of overcoming learning difficulties in the classroom. Since the use of the History of Mathematics promotes an integration of the mathematics of the past with the mathematics of the present day and provides a way of treating the contents and contextualized mathematical knowledge.
Notes for a History of the Teaching of Algebra
Handbook on the History of Mathematics Education, 2013
Abundant literature is available on the history of algebra. However, the history of the teaching of algebra is largely unwritten, and as such, this chapter essentially constitutes some notes that are intended to be useful for future research on this subject. As well as the scarcity of the works published on the topic, there is the added difficulty of drawing the line between the teaching of algebra and the teaching of arithmetic-two branches of knowledge whose borders have varied over time (today one can consider the arithmetic with the four operations and their algorithms and properties taught in schools as nothing more than a small chapter of algebra). As such, we will be very brief in talking about the more distant epochs, from which we have some mathematics documents but little information on how they were used in teaching. We aim to be more explicit as we travel forwards into the different epochs until modern times. We finish, naturally, with some reflections on the present-day and future situation regarding the teaching of algebra.