The Concept of Variable in the Passage from the Arithmetical
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COMENIUS UNIVERSITY BRATISLAVA FACULTY OF MATHEMATICS AND PHYSICS DEPARTMENT OF DIDACTIC MATHEMATICS THE CONCEPT OF VARIABLE IN THE PASSAGE FROM THE ARITHMETICAL LANGUAGE TO THE ALGEBRAIC LANGUAGE IN DIFFERENT SEMIOTIC CONTEXTS Doctoral Thesis by ELSA DEL PILAR MALISANI SUPERVISOR: PROF. FILIPPO SPAGNOLO To Carolina, Andrea and Andrés ACKNOWLEDGMENTS My most sincere thanks go to my supervisor, Professor Filippo Spagnolo, for his great availability, for his unconditioned support and for having encouraged and solicited me to write this thesis. I want also to thank Professor Ivan Trenþanský for his attention and the precious support he gave me during the realization of the present work. My gratitude goes to the Professors Luis Radford, Vladislav Rosa, Jozef Fulier and Michaela Regecová for having accepted the task of “reporters”. I thank to Nanette Sortino and Giulia Sarullo for having helped me with the translation into English and for having understood the complex problem of my research. All of my gratitude goes to my husband Andrés, to my son Andrea and to my daughter Carolina for their incomparable patience, and for having given me the strength to reach the end of this important journey. A particular thanks to my family of origin, I owe what I am to them. To all of you and many people that I do not mention here for space constraints, I just want to say… Thanks, and God Bless You! 1 INDEX Introduction ……………………………………………………….….. 6 Story of the preceding works .………………………………………………………6 Purpose of the research ……………………………………………………………8 Applications ……………………………………………………………………….. 9 Structure of the thesis………………………………………………………………. 9 Bibliography ……………………………………………………………………….10 Chapter 1. Historical evolution of the algebraic language 12 1.1 Introduction …………………………………………………………………….12 1.2 The symbolism ………………………………………………………………….13 1.3 Methods of resolution of equations …………………………………………….18 1.3.1. Methods of resolution of the equations of first degree ……………...…...18 1.3.1.1. The geometric procedure of Euclid ……………………………18 1.3.1.2. The methods of the false position ……………………………….18 1.3.1.2.1. The methods of the simple false position ……………. 19 1.3.1.2.2. The methods of the double false position ……………. 20 1.3.1.3. The regula infusa ………………………………………………. .22 1.3.2. Methods of resolution of the equations of second degree ….………....... 24 1.3.2.1. The geometric procedure of Euclid .…………………………. 24 1.3.2.2. The procedure of al-Khowârizmî ……………………………… 25 1.3.2.3. The “cut-and-paste geometry” …………………………………. 26 1.3.3. Methods of resolution of the equations of third degree ………………… 27 1.3.3.1. The procedure of al-Khayyam ………………………………… 27 1.3.3.2. The procedure of al-Tusi ………………………………………. 29 1.3.4 Methods of resolution of the indeterminate equations ………………….. 30 1.3.4.1 The procedure of Diophantus ………………………………….. 30 1.3.4.2 The method of pulverization ………………………….………... 30 1.3.4.3 The procedure of Abu Kamil …………………………………… 32 1.3.5. European methods up to 1500 ………………………………………….. 32 1.3.6 Conclusions on the methods of resolutions ………………………………34 1.4 The negative numbers as obstacle. The incomplete numerical field ………….. 36 1.5 Generalization of the problems ……………………………………………….. 38 2 1.6 The variable as “thing that varies” ……………………………………………..39 1.7 Conclusions …………………………………………………………………..... 41 Notes ………………………………………………………………………………. 43 Bibliography ………………………………………………………………………. 45 Chapter 2. The magic square. An experience on the transition between the arithmetical language and the algebraic language 48 2.1. Introduction …………………………………………………………………… 48 2.2. Hypothesis …………………………………………………………………….. 50 2.3. Experimental reports for scholastic level …………………………………….. 50 2.3.1 Middle School ………………………………………………………….... 50 2.3.1.1 The a-didactic situation and its phases ………………………….. 50 2.3.1.2 The a-priori analysis …………………………………………….. 51 2.3.1.3 Quantitative analysis of the data …………………………………52 2.3.1.4 Qualitative analysis ……………………………………………... 54 2.3.1.5 Discussion of the results ………………………………………… 56 2.3.2 High School ……………………………………………………………… 57 2.3.2.1 The a-didactic situation and its phases ………………………….. 57 2.3.2.2 Description of the phase of validation ………………………….. 59 2.3.2.3 The a-priori analysis …………………………………………….. 59 2.3.2.4 Quantitative analysis of the data …………………………………61 2.3.2.5 Qualitative analysis ………………………………………………63 2.3.2.6 Discussion of the results …………………………………………64 2.4 Conclusions ……………………………………………………………………. 65 Bibliography ………………………………………………………………………..66 Appendix N° 1 …………………………………………………………………….. 68 Appendix N° 2 …………………………………………………………………….. 69 Appendix N° 3 …………………………………………………………………….. 70 Appendix N° 4 ……………………………………………………………………. 72 Appendix N° 5 ……………………………………………………………………. 74 Appendix N° 6 ………………………….…………………………………………. 75 3 Chapter 3. The notion of variable in different semiotic contexts 84 3.1. Introduction …………...……………………………………………………… 84 3.2 Methodology of the research……………………………………………………85 3.3 A- priori analysis ……………………………………………………………… 87 3.4 The hypotheses and the a-priori table ………………………………………… 90 3.5 Quantitative analysis ………………………………………………………….. 92 3.5.1 First implicative analysis and comments of the first problem ………….. 93 3.5.2 Falsification of H1 ………………………………………………………. 95 3.5.3 Profile of the pupils ………………………………………………………95 3.5.4 The hierarchical tree …………………………………………………….. 96 3.5.5 The factorial analysis by S.P.S.S. ……………………………………….. 97 3.5.6 Second implicative analysis and comments of the first problem ……….. 98 3.5.7 Falsification of H2 ……………………………………………………… 99 3.5.8 Third implicative analysis and comments of the second problem ……… 100 3.5.9 Falsification of H3 ………………………………………………………. 101 3.5.10 Fourth implicative analysis and comments of the fourth problem ……. 101 3.5.11 Fifth implicative analysis and comparison between the first and fourth problem ……………………………………………………………….. 103 3.5.12 Falsification of H4 ………………………………………………………104 3.6 Conclusions ……………………………………………………………………. 104 Notes ………………………………………………………………………………. 106 Bibliography ……………………………………………………………………….107 Appendix N° 1 …………………………………………………………………….. 110 Appendix N° 2 ……………………………………………………………………...111 Appendix N° 3 …………………………………………………………………….. 114 Appendix N° 4 …………………………………………………………………….. 116 Appendix N° 5 ……………………………………………………………………..116 Chapter 4. The variable between unknown and “thing that varies”. Some aspects of the symbolic language 117 4.1. Introduction ………………………………………………….……………….. 117 4.2. Methodology of the research ……... …………………………………………. 118 4 4.3 Analysis of the protocols of the first problem …………………………………118 4.3.1 First pair: Serena and Graziela …………………………………..…….. 118 4.3.1.1 Types of language ……………………………………………… 118 4.3.1.2 Resolutive procedure ………………………………………….. 119 4.3.1.3 Comments ……………………………………………………… 119 4.3.2 Second pair: Vita and Alessandra ………………………………………. 120 4.3.2.1 Types of language ……………………………………………… 120 4.3.2.2 Resolutive procedure ……………………………………………120 4.3.2.3 Comments ………………………………………………………122 4.4 Analysis of the protocols of the second problem ………………………………124 4.4.1 First pair: Serena and Graziela ………………………………………… 124 4.4.1.1 Resolutive procedure ………………………………………….. 124 4.4.1.2 Comments ………………………………………………………125 4.4.2 Second pair: Vita and Alessandra ………………………………………. 125 4.4.2.1 Resolutive procedure ………………………………………….. 125 4.4.2.2 Comments ……………………………………………………... 127 4.5 Final conclusions …………………………………………………………….. 129 Notes ………………………………………………………………………………132 Bibliography ………………………………………………………………………132 Appendix N° 1 ……………………………………………………………………..134 Appendix N° 2 ……………………………………………………………………. 135 Appendix N° 3 …………………………………………………………………… 139 Chapter 5. Final conclusions 160 Bibliography ……………………………………………………………………… 167 5 INTRODUCTION STORY OF THE PRECEDING WORKS The aim of the experimental research effected in Malisani (1990, 1992) was to study the cognitive performance of the students between the ages of 14-15 in the assignment of resolution of algebraic and geometric problems. We wanted to know more specifically how the different kind of logical structure of a problem affect the resolving performance (types of solutions, steps of the resolving algorithm and errors); and if it is verified that the isomorphism of logical structures in the algebraic and geometric contexts does not implicate isomorphism in the performance of the students. The problems belonging to the algebraic context are concerned with the resolution of equations of first degree with one unknown, of the type: y = k1 . (x - k2 ) [1] for a determined value of x or y, being k1 and k2 positive constants and such that k2 < x. In this type of equation the variables x and y and the constants k1 and k2 can represent any elements, therefore, they have only a formal significance. The geometric problems consider, instead, the application of the “theorem of the sum of the interior angles of a convex polygon” that has equation: s = 180°. (n - 2) [2], in which every variable and every constant represent determined geometric objects or relations among these objects. For example: s is the sum of the interior angles of a convex polygon, 180° is the sum of the interior angles of a triangle, n is the number of sides of a polygon, n ± 2 is the number of triangles that are determined in the polygon tracing the diagonals from a vertex to the others. In this case the variables and the constants have a geometric significance. We observe that the equations [1] and [2] are isomorphic with regard to their logical structure, because if we fix a variable (for example: y and s) they require the same steps for their resolution. These