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AIMS AND OBJECTIVES OF TEACHING MATHEMATICS, AIMS AND OBJECTIVES OF TEACHING MATHEMATICS, Aims of Teaching Mathematics, 1. Practical Aim (Utilitarian Aim), One cannot do without the use of fundamental process of the subject mathematics in, daily life. Any person ignorant of mathematics will be at the mercy of others and will be, easily cheated. A person from labour class, a businessman, an industrialist, a banker to the, highest class of the society utilizes the knowledge of mathematics in one form or the other., Whoever earns and spends uses mathematics and there cannot be anybody who lives without, earning and spending., Counting, subtraction, multiplication, division, weighing, selling, buying etc., will, have got an immense practical value in life. The knowledge and skill in these processes can, be provided in an effective and systematic manner only by teaching mathematics in schools., Natural phenomena like rising and setting of the moon and the suns change of seasons, speed, of rotation of planets, etc., need time specification., Mathematics will continue to occupy a prominent place in man’s life. In all activities, of life like arranging a party, admitting a child to school, celebrating a marriage, purchasing, or selling a property, etc, mathematical considerations are uppermost in human mind. In order, to create system in life we have to fix timings, prices, rates, percentages, exchanges,, commissions, discounts, profit and loss, areas, volumes, etc., The following are the practical aims of teaching mathematics., 1. To enable the students to have clear ideas about number concept., 2. To give the individual an understanding of ideas and operations in number and quantity, needed in daily life., 3. To enable the individual to have clear comprehension of the way the number is applied to all, measures but most particularly to those frequently used concepts such as length, volume,, area, weight, temperature, speed etc., 4. To enable the individual to become proficient in the four fundamental operations of addition,, subtraction, multiplication and divisions., 5. To provide the basis of mathematical skills and processes which will be needed for, vocational purposes., 6. To enable the learner to acquire and develop mathematical skills and attitude to meet the, demands of (i) daily life (ii) future mathematical work and (iii) work in the related fields of, knowledge.

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7. To enable the learner to understand the concept of ratio and scale drawing, read and interpret, graphs, diagrams and tables., 8. To enable the individual to apply his mathematics to a wide range of problems that occur in, daily life., , 2., , Disciplinary Aim, , The principal value of mathematics arises from the fact that it exercises the reasoning, power more and climbs from the memory, less from any other school subjects. It disciplines, the mind and develops reasoning power. Locke is of the opinion that ‘mathematics is a way, to settle in the mind a habit of reasoning. A person who had studied mathematics is capable, of using his power of reasoning in an independent way’. The reasoning in the mathematical, world is of special kind processing characteristics that make it specially suited for training the, minds of the pupils. It can be studied under following heads –, Characteristics of Simplicity: In this subject teaching and learning advances by degrees, from simple to complex. It teaches that definite facts are always expressed in a simple, language and definite facts are always easily understandable., Characteristics of Accuracy: Accurate reasoning, thinking and judgment are essential for, the study of mathematics. The pupils learn the value and appreciation of accuracy and adopt, it as a principle of life. It is in the nature of the subject that it cannot be learn through, vagueness of thoughts and arguments., Originality of Thinking: Most work in mathematics demands original thinking., Reproduction and cramming of ideas of others is not very much of appreciated. The pupil can, safely depend on the memory in other subjects but in the mathematics without original, thinking and intelligent reasoning there cannot be satisfactory progress., Similarity to Reasoning of Life: Clear and exact thinking is as important in daily life as in, mathematical study. Before starting with the solution of a problem the pupil has to grasp the, whole meaning similarly in daily life while undertaking a task, one must have firm grip on, the situation. This habit of thinking will get transferred to the problem of daily life., Verification of Results: This gives a sense a achievement, confidence and pleasure. This, verification of results also likely to inculcate the habit of self-criticism and self-evaluation., , 1., , 2., , 3., , 5., , 6., , 1., 2., 3., 4., 5., 6., , The teaching of mathematics intends to realise the following disciplinary aimsTo provide opportunities that enable the learners to exercise and discipline mental faculties., To help the learner in the intelligent use of reasoning power., To develop constructive imagination and inventive faculties., To develop the character through systematic and orderly habits., To help the learner to be original and creative in thinking., To help the individual to become self-reliant and independent.

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3. Cultural Aim, Cultural aim helps the pupils to grow in cultured situation. The greatness of Indian, culture is once reflected through the glory of Indian mathematics of olden days. Similarly, having come to know the progress of Egyptians and Greeks in mathematics, one can be aware, of their progress in culture and civilization. Mathematics does not only acquaint us with the, culture and civilization but it also helps in its preservation, promotion and transmission to the, coming generation.The aim of teaching of mathematics is to develop cultured citizen who can, discharge their obligations to the society effectively and successfully. As the education, commission report (1964-66) was conscious of this need when it wrote ‘one of the, outstanding characteristics of scientific culture is quantification of mathematics. Therefore it, assures a prominent position in modern education…. Proper foundation in the knowledge of, the subject should be laid at the school’ That is why Hogben says, ‘mathematics is the mirror, of civilization’. Different laws of science and scientific instruments are based on the exact, mathematical concept. For example astronomy and physics are the most exact science and, their exactness is the outcome of the usefulness of mathematics., , The cultural aims can be summarized as follows,, 1. To enable the student to appreciate the part played by mathematics in the culture of the past, and that it continues to play in the present world., 2. To enable the student to appreciate the role played by mathematics in preserving and, transmitting our cultural traditions., 3. To enable him to appreciate various cultural arts like drawing, design making, painting,, poetry, music, sculpture and architecture., 4. To provide through mathematics ideas, aesthetic and intellectual enjoyment and satisfaction, and to give an opportunity for creative expression., 5. To help the student explore creative fields such as art and architecture., 6. To make the learner aware of the strength and virtues of the culture he has inherited., 7. To develop in the individual an aesthetic awareness of mathematical shapes and patterns in, nature as well as the products of our civilization., , 4. Recreational Value, Mathematics not only give pleasure through its application to various arts, it also, entertains through its own riddles, games and puzzles. While developing and subject, its, dedicated students have been playing with its numbers, figures, shapes and problems. We can, also have magic squares though which one can derive pleasure by getting an equal sum every, time after adding horizontally, vertically or diagonally., Entertainment in mathematics, 1), , ‘9’ is a wonder number. In the multiplication table of mathematics the sum of digits of every product is, 9:

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9X1=9, 9+0=9, 9X6=54, 5+4=9, 9X2=18, 1+8=9, 9X7=63, 6+3=9, 9X3=27, 2+7=9, 9X8=72, 7+2=9, 9X4=36, 3+6=9, 9X9=81, 8+1=9, 9X5=45, 4+5=9, 9X10=90, 9+0=9, , Similarly √999999999= 999999.999999 to six decimal places the characteristic is wonderful., , 9X9+7=88, 98X9+6=888, 987X9+5=8888, 9876X9+4=88888, etc., 2) 37 is a prime number, but it divides the following numbers completely,, 111,222, 333, 444, 555, 666, 777, 888, 999, etc,., 3) Using ‘8’ 8 times to get 1000?, 888+88+8+8+8=1000, 2, 4), (12) =144, 2, (144) =441, , 1., 2., 3., 4., 5., , 5) Statement: A door is half opened, that means that door is half closed!, i.e,. Door is half opened = ½ closed, X2, ½ X 2 opened = ½ X2 closed, --- 1 Door opened = 1 Door closed!!!, i.e., one door fully opened, it is fully closed!!!!, 7) 1 Re = 100 paise --- (1), 2 Re = 200 paise --- (2), ½ a Re = 50 paise ---- (3), Add (2) and (3), 2 ½ Re = 250 paise, 6) By using English alphabets except first four (ABCD),is it possible to write 100 words in 1, minute?, Ans: Yes!! Starting from ZERO TO NINETY NINE, We get 100 words without using ABCD!!!, 7) KEPREKAR CONSTANT, Born: 1905.Jan.17. Place: Mumbai, Steps:, Any 4 different numbers., Write the number in ascending and descending orders. (4321---1234), Take difference of 1 and 2., Again ascending and descending order and difference., Repeat maximum 7 times you will get constant 6174 (before also we can get this number)., Example: 8640 – 0468=8172, 8721 – 1278=3996, 9963- 3699= 6264, 6642 – 2466 =4176, 7641 – 1467=6164, * Note: For two digits we get 9 as constant., For three digits we get 495 as constant., For five digits we get 63954 as a constant.

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5. Social Aim:, The important social aims of teaching mathematics are as under,, 1., , To develop in the individual an awareness of the mathematical principles and operations, which will enable the individual to understand and participate in the general, social and, economic life of his community., , 2., , To enable the student to understand how the methods of mathematics such as scientific,, intuitive, deductive and inventive are used to investigate, interpret and to make decision in, human affairs., , 3., , To help the pupil acquire social and moral values to lead a fruitful life in the society., , 4., , To help the pupil in the formation of social laws and social order needed for social, harmony., , 5., , To provide the pupils scientific and technological knowledge necessary for adjusting to the, rapidly changing society and social life., , 6., , To help the learner appreciate how mathematics contributes to his understanding of natural, phenomena., , 7., , To help the pupil interpret social and economic phenomena., , Objectives of Teaching Mathematics – National Policy of Education (1986), At the end of high school stage, a pupil should be able to –, , , , , , , , , , , Acquire knowledge and understanding of the terms, concepts, principles, processes, symbols, and mastery of computational and other fundamental processes that are required in daily like, and for higher learning in mathematics., Develop skills of drawing, measuring, estimating and demonstrating., Apply mathematical knowledge and skills to solve problems that occur in daily life as well, as problems related to higher learning in mathematics or allied areas., Develop the ability to think, reason, analyze and articulate logically., Appreciate the power and beauty of mathematics., Show an interest in mathematics by participation in mathematical competitions, and, engaging in its learning, etc., Develop reverence and respect towards great mathematicians, particularly towards great, Indian mathematicians for their contributions to the field of mathematical knowledge., Develop necessary skills to work with modern technological devices such as calculations,, computers, etc., , Objectives of Teaching Mathematics – New Curriculum Document (2000), The learners, , Consolidate the mathematical knowledge and skills acquired at the upper primary stage.

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, , , , , , , , , , Acquire knowledge and understanding of the terms, symbols, concepts, principles, process,, proofs, etc., Develop mastery of basic algebraic skills., Develop drawing skills., Apply mathematical knowledge and skills to solve real mathematical problems by, developing abilities to analyze, to see interrelationship involved, to think and reason., Develop the ability to articulate logically., Develop necessary skills to work with modern technological devices such as calculators,, computers, etc., Develop interest in mathematics and participate in mathematical competitions and other, mathematical club activities in the school., Develop appreciation for mathematics as a problem-solving tool in various fields for its, beautiful structures and patterns, etc., Develop reverence and respect towards great mathematicians, particularly towards the Indian, mathematicians for their contributions to the field of mathematics., , Objectives of Teaching Mathematics, The objectives of teaching mathematics at the secondary state may be classified as, A., B., C., D., , under:, Knowledge and Understanding objectives, Skill objectives, Application objectives, Attitude objectives, , A. Knowledge and Understanding Objectives, 1., 2., 3., 4., 5., 6., , The student acquires knowledge and understanding of:, Language of mathematics i.e., the language of its technical terms, symbols, statements,, formulae, definitions, logic, etc., Various concepts i.e., concept of number, concept of direction, concept measurement., Mathematical Ideas, like facts, principles, processes and relationships., The development of the subject over the centuries and contributions mathematicians., Inter-relationship between different branches and topics of mathematics etc., The nature of the subject of mathematics., , B. Skill Objectives, The subject helps the student to develop the following skills:, 1. He acquires and develops skill in the use and understanding of mathematical language., 2. He acquires and develops speed, neatness, accuracy, brevity and precision in mathematical, calculations., 3. He learns and develops technique of problem-solving., 4. He develops and ability to estimate, check and verify results., 5. He develops and ability to perform calculations orally and mentally.

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6., 7., 8., 9., , He develops skills to use mathematical tools, and apparatus., He develops essential skill in drawing geometrical figures., He develops skill in drawing, reading, interpreting graphs and statistical tables., He develops skill in measuring, weighing and surveying., , C. Application Objectives:, The subject helps the student to apply the above-mentioned knowledge and skills in, the following way:, 1. He is able to solve mathematical problems independently., 2. He makes use of mathematical concepts and processes in everyday life., 3. He develops ability to analyze, to draw inferences, and to generalize from the collected data, and evidence., 4. He can think and express precisely, exactly, and systematically by making proper use of, mathematical language., 5. He develops the ability to use mathematical knowledge in the learning of other subjects, especially sciences., , D. Attitude Objectives:, The subject helps to develop the following attitudes:, 1. The student learns to analyze the problems., 2. Develops the habit of systematic thinking and objective reasoning., 3. He develops heuristic attitude and tries to discover solutions and proofs with his own, independent efforts., 4. He tries to collect enough evidence for drawing inferences, conclusions and generalizations., 5. He recognizes the adequacy or inadequacy of given data in relation to any problem., 6. He verifies his results., 7. He understands and appreciates logical, critical and independent thinking in others., 8. He expresses his opinions precisely, accurately, logically and objectively without any biases, and prejudices., 9. He develops self-confidence for solving mathematical problems., 10. He develops personal qualities namely, regularity, honesty, objectivity, neatness and, truthfulness., ., , Objectives of Teaching Arithmetic:, Arithmetic is the science of numbers and art of computation. It is the oldest branch of, the subject mathematics. Historically arithmetic developed out of a need for a system of, counting. It is considered to be essential for efficient and successful living. That is why

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arithmetic is divided as the science that deals with numbers with relations between numbers,, numbers in term, or abstraction arising from such concrete situations as counting measuring, and ordering the various quantities and objects that we encounter in everyday life., , The teaching of arithmetic has to fulfill two responsibilities., 1. The inculcation of an appreciative understanding of number system and an intelligent, proficiency in its fundamental process., 2. The socialization of number experiences., The following are the objectives of teaching arithmetic –, 1. To teach the learner mathematical type of thought, to understand the statement to analyze, them and to arrive at right conclusions., 2. To arose pupil’s interest in the quantitative side of the world around him and its use as a, simple tool in business., 3. To develop fundamental arithmetic concepts like the concept of number, order, units of, measurement, size and shape etc., 4. To give accuracy and facility in simple computation of the fundamental process., 5. To develop speed and accuracy in arithmetical calculation and computation., 6. To appreciate the use of arithmetic in daily life., 7. To help in the learning of other branches and higher studies in mathematics., , Objectives of Teaching Algebra, Algebra is called the science of letter. It refers to the methods of reasoning about, numbers by employing letters to represent their relationship. Algebra is concerned largely, with structure of number system, operations with numbers and statements involving numbers, as well as the solution of problems. Algebra is a language used to develop and express much, of the scientific data. Algebra comprehended a more general treatment of numbers and, number relation than thus arithmetic. It is concerned with the general statement about, numerical situation. Algebra refers to the operation of taking a quantity from one side of the, equation to another by changing its signs., The following are the objectives of teaching Algebra1., 2., 3., 4., 5., , To give compact formulae of generalization to be used in all cases., To provide an effective way for expressing complicated relations., To correct the weaknesses and supplement the deficiency of language as an instrument of, abstract investigation and exact statement., To inculcate the power of analysis., To verify the results in simpler and more satisfactory way.

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6., 7., 8., , To develop confidence among the pupils., To provide new and refined approach in the study of abstract mathematical relationship, though the use of new symbolism., To enable the pupils to use it for solving more difficult problem., , Objectives of Teaching Geometry:, The word geometry originally means measurement of earth. Geometry has two value:, a) the knowledge and, b) as a method of logical thinking., It is the science of lines and figures it is the science of space and extent. It deals with the, position, space and size of bodies but nothing to do with their material properties. Geometry, has two important aspects –, Demonstrative Geometry – It deals with the shape, size and position of figures by, pure reasoning based on definitions, self-evident truths and assumptions. Euclid, a great, Greek Mathematician was the father of demonstrative Geometry. His methods arte, intuitional, observational, intentional, constructive, informal, creative, experimental and so, on., Practical Geometry – It covers the constructional work of the subject. Most of the, work directly or indirectly based on demonstrative Geometry., The following are the objective of teaching Geometry, , Trigonometry Objectives, , , Measurement of Angles, Arcs, and Sectors, , o, , Using Radians, Degrees, or Grads to Measure Angles, , o, , Length of an Arc and Area of a Sector of a Circle, , o, , Circular Motion, , , , The Trigonometric Functions, , o, , Definition of the Six Trigonometric Functions, , o, , Values of the Trigonometric Functions at some multiples of 15 degrees., , o, , Trigonometric Functions for right triangles, , o, , Solving Right Triangles, , o, , Applications of Right Triangle Trigonometry, , o, , Circular Functions, , , , Graphs of Trigonometric Functions, , o, , Graphing Generic Sine and Cosine Functions, , o, , Shifting Generic Curves Right/Left or Up/Down, , o, , Using the Graphing Calculator to Graph Functions by Addition of Ordinates, , o, , Graphing the Tangent and Cotangent Functions, , o, , Graphing the Secant and Cosecant Functions, , o, , Qualitative Analysis of Trigonometric Functions, , , , Inverse Trigonometric Functions

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o, , Relations, Functions, and Their Inverses, , o, , Inverses of Trigonometric Functions, , o, , Finding Inverses of Trigonometric Functions Using a Calculator, , , , Basic Trigonometric Identities, , o, , Fundamental Identities, , o, , Opposite Angle Identities, , o, , Additional Techniques to Prove Identities, , , , Sum and Difference Identities, , o, , Sum and Difference Identities for Cosine, , o, , Some Identities Useful in Calculus, , o, , Sum and Difference Identity for Tangent, , o, , Identities Involving Sums and Differences of Pi and Pi/2., , , , Additional Identities, , o, , Double Angle Identities, , o, , Half Angle Identities, , o, , Identities to Rewrite Sums and Products, , , , Trigonometric Equations, , o, , Solving Basic Trigonometric Equations, , o, , Solving Trigonometric Equations Involving Factoring, , o, , Solving Trigonometric Equations Where the Argument is a Function, , o, , Using Identities to Solve Trigonometric Equations, , , , Law of Sines and Law of Cosines, , o, , Derivation of the Law of Sines, , o, , The Ambiguous Case, , o, , Applications of the Law of Sines, , o, , Derivation of the Law of Cosines, , o, , Applications of the Law of Cosines, , o, , Area of a Triangle, , , , Vectors, , o, , Addition of Vectors, , o, , Geometric Resolution of Vectors, , o, , Algebraic Resolution of Vectors, , o, , Work, Inclined Planes, and the Dot Product, , , , Complex Numbers, , o, , Algebraic Operations with Complex Numbers, , o, , Trigonometric and Polar Representation of Complex Numbers, , o, , DeMoivre's Formula, , Objectives of TeachingTrigonometry, 1., 2., , The students will be able,, To impart knowledge of trigonometric ratios and identities., To apply the knowledge of trigonometry to solve daily life problems.

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3., 4., 5., 6., 7., 8., 9., 10., 11., 12., 13., , To find heights and distances., To appreciate the use of trigonometry to solve problems., To develop creative thinking and reasoning., To understand that it is an essential tool., to know how structures are built., To realise that it very useful in technology and for engineers., To continue higher education., To understand the relationship between trigonometry and other branches of mathematics., Find the value of trigonometric ratios of some specific angles., Determine the trigonometric ratios of complementary angle., Apply the trigonometric identities in proving the given statement., , Objectives of TeachingCoordinate Geometry, 1., 2., 3., 4., 5., 6., , The students will be able to,, Draw a plan for the given situation., Appraise the Cartesian system., Identify the coordination of a point., Locate the quadrants in the Cartesian plane., Plot the points in the Cartesian plane., Write the abscissa and ordinate of a point., , ANDERSON’S, REVISED, BLOOM’S, INSTRUCTIONAL OBJECTIVES, , TAXONOMY, , OF, , Taxonomies of the Cognitive Domain, , Bloom’s Taxonomy 1956, , Anderson and Krathwohl’s Taxonomy, 2001, , 1. Knowledge: Remembering or retrieving, previously learned material. Examples of verbs, that relate to this function are:, , 1. Remembering:, Recognizing or recalling knowledge from, memory. Remembering is when memory, is used to produce or retrieve definitions,, facts, or lists, or to recite previously, learned information., , know, identify, relate list, , define, recall, memorize, repeat, , record, name, recognize, acquire, , 2. Comprehension: The ability to grasp or, construct meaning from material. Examples of, verbs that relate to this function are:, restate locate, , identify, , illustrate, , 2. Understanding:, Constructing meaning from different, types of functions be they written or, graphic messages or activities like, interpreting, exemplifying, classifying,

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report, recognize, explain, express, , discuss, describe, discuss, review infer, , interpret, draw, represent, differentiate, conclude, , 3. Application: The ability to use learned, material, or to implement material in new and, concrete situations. Examples of verbs that, relate to this function are:, apply relate, develop, translate use, operate, , organize, employ, restructure, interpret, demonstrate, illustrate, , practice, calculate, show, exhibit, dramatize, , 4. Analysis: The ability to break down or, distinguish the parts of material into its, components so that its organizational structure, may be better understood. Examples of verbs, that relate to this function are:, analyze, compare, probe, inquire, examine, contrast, categorize, , differentiate, contrast, investigate, detect, survey, classify, deduce, , experiment, scrutinize, discover, inspect, dissect, discriminate, separate, , 5. Synthesis: The ability to put parts together, to form a coherent or unique new whole., Examples of verbs that relate to this function, are:, compose, , plan invent, , propose, , summarizing, inferring, comparing, or, explaining., , 3. Applying:, Carrying out or using a procedure, through executing, or, implementing. Applying relates to or, refers to situations where learned, material is used through products like, models, presentations, interviews or, simulations., , 4. Analyzing:, Breaking materials or concepts into, parts, determining how the parts relate to, one another or how they interrelate, or, how the parts relate to an overall, structure or purpose. Mental actions, included in this function, are differentiating, organizing, and, attributing, as well as being able to, distinguish between the components or, parts. When one is analyzing, he/she can, illustrate this mental function by creating, spreadsheets, surveys, charts, or, diagrams, or graphic representations., , 5. Evaluating:, Making judgments based on criteria and, standards through checking and, critiquing. Critiques, recommendations,, and reports are some of the products that, can be created to demonstrate the

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produce, design, assemble, create, prepare, predict, modify tell, , formulate, collect set, up, generalize, document, combine, relate, , develop, arrange, construct, organize, originate, derive write, propose, , 6. Evaluation: The ability to judge, check, and, even critique the value of material for a given, purpose. Examples of verbs that relate to this, function are:, judge assess, compare, evaluate, conclude, measure, deduce, , argue decide, choose rate, select, estimate, , validate, consider, appraise, value, criticize, infer, , processes of evaluation. In the newer, taxonomy, evaluating comes before, creating as it is often a necessary part of, the precursory behavior before one, creates something., , 6. Creating:, Putting elements together to form a, coherent or functional whole;, reorganizing elements into a new pattern, or structure through generating,, planning, or producing. Creating, requires users to put parts together in a, new way, or synthesize parts into, something new and different creating a, new form or product. This process is the, most difficult mental function in the new, taxonomy.