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FUNDAMENTALS OF TEACHING MATHEMATICS UNIT1:PLACE OF MATHEMATICS IN CURRICULUM A) Meaning and nature of mathematics: Mathematics is the study of topics such as quantity (numbers,structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Galileo Galilei (1564–1642) said, "The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth." 1|Page Carl Friedrich Gauss (1777–1855) referred to mathematics as "the Queen of the Sciences. "Albert Einstein (1879–1955) stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." The word mathematics comes the Greek μάθημα from (máthēma), which, in the ancient Greek language, means "that which is learnt",[24] "what one gets to know", hence also "study" and "science", and in modern Greek just "lesson". The word máthēma is derived from μανθάνω (manthano), while the modern Greek equivalent is μαθαίνω (mathaino), both of which mean "to learn". In Latin, term mathematics more and in commonly English meant until around "astrology" (or 1700, the sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. Aristotle defined mathematics as "the science of quantity Some just say, "Mathematics is what mathematicians do." Fields of mathematics Mathematics can, broadly speaking, be subdivided into the study of quantity, structure, space, and change (i.e. arithmetic, algebra,geometry, and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty. 2|Page Pure mathematics Quantity The study of quantity starts with numbers, first the familiar natural numbers and integers ("whole numbers") and arithmetical operations on them, which are characterized in arithmetic. Structure Many mathematical objects, such as sets of numbers and functions, exhibit internal structure as a consequence of operations or relations that are defined on the set. Mathematics then studies properties of those sets that can be expressed in terms of that structure; for instance number theory studies properties of the set of integers that can be expressed in terms of arithmetic operations. Space The study of space originates with geometry – in particular, Euclidean geometry. Trigonometry is the branch of mathematics that deals with relationships between the sides and the angles of triangles and with the trigonometric functions; it combines space and numbers, and encompasses the well-known Pythagorean theorem. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. . Applied mathematics Applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is amathematical science with specialized knowledge. The term applied mathematics also describes the professional specialty in which 3|Page mathematicians work on practical problems; as a profession focused on practical problems, applied mathematics focuses on the "formulation, study, and use of mathematical models" in science, engineering, and other areas of mathematical practice. Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, especially with probability theory Computational mathematics proposes and studies methods for solving mathematical problems that are typically too large for human numerical capacity. Meaning and Nature Of Mathematics "Mathematics is the Science of number and Space" "Mathematics is the Science of Measurement, quantity and Magnitude." These definitions clearly indicate that Mathematics is an accepted Science which deals with the quantitative aspect of our life and knowledge. Mathematics has been recognized as one of the Central Strings of human intellectual activity throughout the centuries. The Word "Mathematics" has been used in two distinct and different senses. i.e. One as a method used to solve the problems of quality, space, order etc. and the second as a set of laws or generalizations of truths that are discovered. Mathematics is a tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field. 4|Page In other words, as in mathematics we find results at the abstract level with the help of process of reasoning, therefore mathematics may be regarded as a science of abstract forms. As expressed by Hindi Mathematicians, "Mathematics(Ganita) means the Science of Calculation." From the above opinions we include that Mathematics is a Science of Quality and Space. It deals with the questions and problems involving size, portion, area, time interval, distance etc. It is also a science of calculation involving the use of number and symbols. It deals with the relationship between magnitudes. It deals with Numerical part of Man's life. In this way it is a Systematic, organized and exact branch of science, which deals with abstract concepts. NATURE OF MATHEMATICS 1. Mathematics, like everything else that man has created, exists to fulfil certain human needs and desires. 2. The dominant intellectual interest of mankind shows that, mathematics appeals very powerfully to mankind. 3. It involves man's high cognitive powers. 4. It has its own language-signs, symbols, terms and operations etc. 5. It has its own tools like intuition, logic, reasoning, analysis, construction, generality and individuality etc. 6. It helps in drawing conclusions and interpreting various ideas and themes. 7. Mathematics is the science of inductive, deductive reasoning. 5|Page Inductive-reasoning means when a particular property is true in a sufficient number of cases, then we conclude that it will true for all similar cases. Deductive-reasoning is based on axioms, postulates, self-evident truth, undefined terms and definitions. 1. "Mathematics is a way to settle in the mind as habit of reasoning." 2. Mathematics can be used in two distinct and different senses i.e. (i) The truth that are discovered and (ii) The methods used to learn truth. 3. Mathematics is of two types- Pure and Applied Mathematics Pure Mathematics deals with theories and principles without regard to their application to concrete things. Applied Mathematics is the practical side of pure mathematics. That is why each and every invention and discovery in the field of physical, biological or social sciences owe much to applied mathematics. "It is the science of number and space." "It is the science of measurements, quantity and magnitude." Rationality, accuracy, originality of thinking, certainty of results, transfer of learning in terms of similarity to the reasoning of day to day life and verification are the vital components of mathematics. Mathematics is an expression of the human mind reflects the active will, the contemplative reason and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. 6|Page BRANCHES OF MATHEMATICS 1)ARITHMATIC :ART OF COMPUTING—NUMBERS /CALCULATION 2)ALGEBRA:AL-JABAR-AL MUQABALAH---THE OPERATION OF TRANSFORMING A QUANTITY FROM ONE SIDE TO OTHER SIDE OF EQUATION---E.G---indices-graphs-directed numbers 3)GEOMETRY:measurement of earth-euclidian geometry ---------------------------------------------------------------------------------------------- 7|Page