Syllogistic Analysis and Cunning of Reason in Mathematics Education Yang Liu Faculty of Health, Science and Technology Mathematics DISSERTATION | Karlstad University Studies | 2013:28 Syllogistic Analysis and Cunning of Reason in Mathematics Education Yang Liu DISSERTATION | Karlstad University Studies | 2013:28 Syllogistic Analysis and Cunning of Reason in Mathematics Education Yang Liu DISSERTATION Karlstad University Studies | 2013:28 ISSN 1403-8099 ISBN 978-91-7063-505-2 © The author Distribution: Karlstad University Faculty of Health, Science and Technology Department of Mathematics and Computer Science SE-651 88 Karlstad, Sweden +46 54 700 10 00 Print: Universitetstryckeriet, Karlstad 2013 WWW.KAU.SE Abstract This essay explores the issue of organizing mathematics education by means of syllogism. Two aspects turn out to be particularly significant. One is the syllogistic analysis while the other is the cunning of reason. Thus the exploration is directed towards gathering evidence of their existence and showing by examples their usefulness within mathematics education. The syllogistic analysis and the cunning of reason shed also new light on Chevallard's theory of didactic transposition. According to the latter, each piece of mathematical knowledge used inside school is a didactic transposi- tion of some other knowledge produced outside school, but the theory itself does not indicate any way of transposing, and this empty space can be filled with the former. A weak prototype of syllogism considered here is Freudenthal's change of perspective. Some of the major difficulties in mathematics learning are connected with the inability of performing change of perspective. Conse- quently, to ease the difficulties becomes a significant issue in mathematics teaching. The syllogistic analysis and the cunning of reason developed in this essay are the contributions to the said issue. Contents Introduction 1 1 Some Cases 9 2 The Problem and the Approach 29 3 Three Examples 35 3.1 Contingency and Necessity . 35 3.2 On 5 + 7 = 12 . 38 3.2.1 The Concept of the Number Five . 38 3.2.2 Adding and Subtracting . 45 3.2.3 The Additive Group of Integers . 49 3.3 On 1=2 + 1=3........................... 49 3.3.1 Numerical Values of Fractions . 50 3.3.2 Fractions . 51 3.3.3 The Field of Rationals . 55 3.4 On the Equation ax2 + bx + c =0 ............... 55 3.4.1 Constants . 56 3.4.2 Variables . 56 3.4.3 Equations . 56 4 The Syllogistic Analysis 65 4.1 Structural Analysis . 65 4.2 Nicholas Bourbaki . 69 4.3 Syllogism . 75 4.4 Sides . 80 5 The Cunning of Reason 85 5.1 A Philosophical Discussion . 87 5.1.1 The World in Motion and the World in Form . 87 5.1.2 Body and Mind . 93 5.1.3 Reason . 96 5.2 The Cunning of Reason in Three Contexts . 104 5.2.1 Lev Vygotsky . 104 i ii CONTENTS 5.2.2 Jean Piaget . 108 5.2.3 Gottlob Frege . 111 5.3 Axiomatization of the Cunning of Reason . 117 6 Numbers 123 6.1 Natural Numbers . 124 6.2 Counting . 128 6.3 Negative Integers and Integers . 131 6.4 Fractions . 142 6.5 Numbers in Triangular Structure . 153 6.6 Numbers in Minimum Structure . 164 6.7 Numbers in Peano Structure . 176 7 Probability and Statistics 183 7.1 Probability in Itself . 183 7.2 Probability for Others . 185 7.3 Probability for Itself . 186 7.4 Statistics . 189 7.5 Independent Identical Distributions . 191 7.6 Point Estimation . 191 7.6.1 Schema . 191 7.6.2 Characteristic Examples . 192 7.7 Interval Estimation . 193 7.7.1 Schema . 193 7.7.2 Characteristic Examples . 193 7.8 Hypothesis Testing . 195 7.8.1 Schema . 195 7.8.2 Characteristic Examples . 196 Afterword and Acknowledgment 199 Bibliography 203 Introduction HALFWAY upon the road of our life, I came to myself amid a dark wood where the straight path was confused. And as it is a hard thing to tell of what sort was this wood, savage and rough and strong, which in the thought renews my fear, even so is it bitter; so that death is not much more; but to treat of the good which I there found, I will tell of the other things which there I marked. Alighieri Dante, Divine Comedy, translated by Arthur John Butler, 1894. The change of perspective, as a didactical principle, is one of the most important contributions made by Freudenthal (1978, pp. 242{252) in the domain of mathematics education. The importance of this principle is very clear as Freudenthal said: Yet I believe that the observation I reported here is the most important I have been confronted with for quite a while, and I am sure that the questions it gives rise to are the most urgent we are expected to answer. Some of learning difficulties, even for those who are excellent in arithmetic, is caused by being unable to perform change of perspective. (ibid., pp. 245-249) The situation that the student gazes at a task and says of I do not know where to begin is certainly recognizable for many teachers. The student for some mysterious reason is locked in one perspective or what amounts to the same as trapped in one context. The change of perspective does not mean to overthrow the old perspective, on the contrary, this old perspective is equally valuable as the new one. The change of perspective consists in a synthesis of two different perspectives. The equation x + 7 = 11 was used by Freudenthal in his enunciation on the change of perspective. For my purpose I shall use the equation 2x+4 = 6 instead. In general, an equation is a synthesis A = B of two objects A and B, as a result, an equation is a syllogism. To solve the equation 2x+4 = 6 is to relate it to another equation x = 1 through some intermediate equations. I shall exhibit the procedure of relating two equations now, it is 2x + 4 = 6 2x = 2 x = 1: 1 2 INTRODUCTION Altogether there are three equations (syllogisms), and solving equation be- comes finding interrelationships among the equations. Mathematics, according to Encyclopedia Britannica (2012), is the science of structure, order and relation (see the quotation at the beginning of the chapter four). The simplest order is that between two objects, say, 3 < 5; and the same holds for the relation. In the sense of the present essay, an order is a syllogism and a relation is a such as well. One of the most important mathematical structures, the group structure, is also a syllogism, for it concerns itself the multiplication of two elements. Mathematics, according to Encyclopedia Britannica (2012) again, deals with logical reasoning. The logical reasoning in mathematics is either induc- tive or deductive. The deductive reasoning, before Frege, was completely identified with the Aristotelean syllogistic reasoning. An Aristotelean syllo- gism has the form of ( A ! B =) A ! C B ! C where the sign ! means the word is. The curricula in Sweden after 1962 seem, according to Tomas Englund (oral communication), to evolve in the following fashion: centralization −! de-centralization −! re-centralization with Lgr 62, Lpo 94 and Lgr 11 as the characteristic documents. The formula is understood in the sense of descriptions on aims and knowledge in school- ing. This attitude is in fact confirmed by a recent thesis of Nordin (2012). Those five paragraphs above exhibit the change of perspective, the equa- tion solving, the group structure, the Aristotelean deductive reasoning, and the development of curricula in Sweden since 1962. A close examination of these examples reveals a common part that they share. They all have some- thing to do with syllogisms and their interrelationships, and these words are in fact the principal words of this essay. As a principle, the exact meanings of the principal concepts used shall not be given here but in the main body of the text. In a broadest sense, a syllogism is thesis −! antithesis −! synthesis: In order to be able to talk about interrelationships among syllogisms, we shall treat a syllogism as a structure, just like a water molecule is made up by an oxygen atom and two hydrogen atoms in which case the said interrelationships are on the level of water. Syllogism and interrelationship being the principal issues, the present essay is constructed with the center at the following questions: 3 • how can teaching and learning of mathematics be organized by syllo- gisms? • what makes the syllogistic organization possible? The aim here is therefore to inquire into these two fundamental questions and to indicate answers. Of course, the inquiry starts with a fundamental belief that teaching and learning of mathematics can to a great extent be organized by the syllogism. This belief does not grow from the middle of nowhere, it is an outcome from a huge accumulation of mathematics and mathematics teaching practices. And it should not be taken as an answer to the fundamental questions, not even a partial one, for what concerns mathematics education is not a simple positive answer but more importantly the how-issue in the questions. It is this issue that the essay is principally addressed to. In the process of the inquiry it is revealed that two aspects of syllogism are particularly significant. The one, which lies outside syllogism, is the syllogistic analysis; and the other lying inside is the cunning of reason. In fact it is these two aspects that are really the underpinnings of the present essay. Thus, teaching and learning of mathematics can be greatly organized by syllogistic analysis and skillful use of the cunning of reason; and the syllogistic organization is possible because of our deductive faculty and the cunning of reason.