UNIT 1 INTRODUCTION OF MATHEMATICS Structure 1.1 Introduction 1.2 Objective 1.3 What is mathematics? 1.4 Characteristics of Mathematics 1.4.1 Objectivity 1.4.2 Logical 1.4.3 Structure 1.4.4 Symbolism 1.4.5 Constructiveness 1.4.6 Brevity 1.5 History of mathematics 1.6 Contribution of Indian Mathematician 1.7 Correlation with Physics, Chemistry, Biology and Geography 1.8 Let Us Sum Up 1.9 Answers to Check Your Progress 1.10 References 1.1 Introduction Children are naturally attracted to the science of number. Mathematics, like language, is the product of the human intellect. It is therefore part of the nature of a human being. Mathematics arises from the human mind as it comes into contact with the world and as it contemplates the universe and the factors of time and space. It under girds the effort of the human to understand the world in which he lives. All humans exhibit this mathematical propensity, even little children. It can therefore be said that human kind has a mathematical mind. Montessori took this idea that the human has a mathematical mind from the French philosopher Pascal. Maria Montessori said that a mathematical mind was ―a sort of mind which is built up with exactity‖. The mathematical mind tends to estimate, needs to quantify, to see identity, similarity, difference, and patterns, to make order and sequence and to control error. The present unit presents the general ideas about the introduction of mathematics which are the needful for students to understand the characteristics of mathematics, history of mathematics, contribution of Indian mathematicians and the correlation of mathematics with other subjects as like as physics, chemistry, biology and geography. 1.2 Objectives On completion of this unit students will be able to …. 1. understand the nature and history of mathematics; 2. understand the characteristics of mathematics; 3. recognize the symbolism, constructiveness, logical, structural, brevity characters of mathematics; 4. know the contribution of Indian mathematician; 5. correlation between mathematics and other subjects likewise physics, chemistry, biology and geography. 1.3 What is Mathematics? All will agree that mathematics is very fascinating subject. According to the renowned Indian mathematician Shakuntala Devi (the super computer), ―mathematics is only a systematic effort to solving puzzles posed by nature‖. While solving of the mathematical puzzles and riddles may provide pleasant relaxation to some, undoubtedly these items have a way of holdings of students‘ interest as little else can. What is mathematics? The question can sound very silly. It is very difficult to give a correct answer for this question. But it is difficult only to give answer, not impossible. A very famous answer was given by the great German Mathematician of the 20th century, David Hilbert, who said that ‗Mathematics is what competent people understand world to mean‘. It is said that Mathematics is the gate and key of the Science. According to the famous Philosopher Kant, ―A Science is exact only in so far as it employs Mathematics‖. So, all scientific education which does not commence with Mathematics is said to be defective at its foundation. Neglect of mathematics works injury to all knowledge. One who is ignorant of mathematics cannot know other things of the World. Again, what is worse, who are thus ignorant are unable to perceive their own ignorance and do not seek any remedy. So Kant says, ―A natural Science is a Science in so far as it is mathematical‖. And Mathematics has played a very important role in building up modern Civilization by perfecting all Science. In this modern age of Science and Technology, emphasis is given on Science such as Physics, Chemistry, Biology, Medicine and Engineering. Mathematics, which is a Science by any criterion, also is an efficient and necessary tool being employed by all these Sciences. As a matter of fact, all these Sciences progress only with the aid of Mathematics. So it is aptly remarked, ―Mathematics is a Science of all Sciences and art of all arts‖. Mathematics is a creation of human mind concerned chiefly with ideas, processes and reasoning. It is much more than Arithmetic, more than Algebra more than Geometry. Also it is much more than Trigonometry, Statistics, and Calculus. Mathematics includes all of them. Primarily mathematics is a way of thinking, a way of organizing a logical proof. As a way reasoning, it gives an insight into the power of human mind, so this forms a very valuable discipline of teaching-learning programmes of school subjects everywhere in the world of curious children. So the pedagogy of Mathematics should very carefully be built in different levels of school education. In the pedagogical study of mathematics we mainly concern ourselves with two things; the manner in which the subject matter is arranged or the method the way in which it is presented to the pupils or the mode of presentation. Mathematics is intimately connected with everyday life and necessary to successful conduct of affairs. It is an instrument of education found to be in conformity with the needs of human mind. Teaching of mathematics has it s aims and objectives to be incorporated in the school curricula. If and when Mathematics is removed, the back -bone of our material civilization would collapse. So is the importance of Mathematics and its pedagogic. 1.4 Characteristics of Mathematics Mathematics is not just a study of numbers, nor is it simply about calculations. It is not about applying formulas, either. It can perhaps be better described as ―a field of creation through accurate and logical thinking‖. Mathematics has a long, rich history and continues to grow rapidly. New findings are regularly presented at conferences in and outside of Japan and articles based on such findings are published in mathematics journals in countries around the world. Mathematics is a very diverse filed. The appendix shows a list of subcategories classified under mathematics by the American Mathematical Society (the list is an excerpt from the AMS‘s Mathematical Reviews journal). Many of you must be surprised to see so many subfields of mathematics. ―Algebra‖ and ―Geometry‖, for instance, may be familiar subjects of mathematics in high school, but they are divided further into more specified categories in the list, which also includes combinations of subfields, such as ―algebraic geometry‖. Mathematics is an academic discipline of great depth, with a number of unsolved problems. It has progressed on cumulative contributions from countless mathematicians in the world who have tackled those problems while creating new areas of inquiry. Some of the subfields in the appendix, such as fluid mechanics, quantum theory, information and telecommunication, and biology, may seem irrelevant to mathematics at first glance. These fields, however, take mathematical approaches to describing and analyzing phenomena under study. They illustrate how a number of subfields of mathematics have benefited from and evolved through interactions with other disciplines. Mathematics, on the other hand, has made considerable contributions to the advancement of other academic fields. In fact, mathematics is often described as the foundation of scientific studies. One of the major characteristics of mathematics is its general applicability. One equation, for instance, can represent a particular phenomenon in physics as well as certain logic in economics. This general nature of mathematical equations enables unified treatment of diverse phenomena in various academic fields. Furthermore, mathematical theorems are no respecter of age or seniority any theorem has to be proven through appropriate mathematical procedures whether you are a novice student researcher or an eminent professor of mathematics and once proven true, mathematical theorems will never be reversed. This ―universality‖ of mathematics is another important feature that allows the discipline to transcend time and space. 1.4.1 Objectivity Objectivity is a central philosophical concept which has been variously defined by sources. A proposition is generally considered to be objectively true when its truth conditions are met and are ―mind-independent‖—that is, not met by the judgment of a conscious entity or subject. Objectivity in mathematics is traditionally thought of as one of the more desirable necessities for its own credibility. The basic assumptions are, on the one hand, that any mathematical description of nature must be independent of the wishes, moods, and/or needs of an individual mathematician (or group of mathematicians), and on a more fundamental basis that nature itself is objective, because it is ruled by laws and not by the concerns or intentions of its inhabitants. The latter is highly critical in that the laws of nature are deemed to be inviolable, quantitative, inexorable, and general; i.e., they should not bend for a purpose. By objective I mean there is ―universal agreement‖ (or at least near universal agreement). Everyone agrees that ―1 + 1 = 2‖. In fact, mathematics is possibly the only discourse where there is a real sense of universal agreement. This is the reason mathematics is so successful in facilitating the study of so many diverse fields. One can argue that mathematics is the most successful language ever invented. Their existence is an objective fact, independent of our knowledge of them. Our mathematical knowledge is objective and unchanging because it‘s knowledge of objects external to us, independent of us, which are indeed changeless. 1.4.2 Logical The qualities that most people tend to associate with mathematics are things like precision and logic and therefore mathematics is closely linked with clear, ‗correct‘ thinking. Because it teaches students to think, the study of mathematics is beneficial even in the absence of plausible practical uses of the mathematics topics selected. This notion is based on the transfer of learning theory which for centuries was used to justify teaching Latin.