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. In many occupations like accountancy, banking, tailoring, carpentry, taxation, insurance etc., which fulfills the needs of man can be carried out by the use of mathematics. These agencies depend on mathematics for their successful functioning. It has become the basis for the world’s entire business and commercial system. Ignorance of mathematics in the masses is a formidable obstacle in the way of country’s progress. Individual resources add up to form national resources. An individual who is ignorant of calculation is often ruins himself and also causes for the national loss by wasting his time, energy and money. There are family budget, school budget, factory budget, national budget, etc., which owe the fundamentals of mathematics. 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. In the absence of these fixations life in the present complex society revert back into confusion and chaos. The number imparts system through our life. In this complex world passing through scientific and technological age the practical value of mathematics is going to be increasingly felt and recognized.

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.

7.      To enable the students to make appropriate approximations.

8.      To enable the learner to understand the concept of ratio and scale drawing, read and interpret graphs, diagrams and tables.

9.      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’. Its truths are definite and exact. The learner has to argue the correctness or incorrectness of a statement. 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 –

1.            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.

2.            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. Accuracy, exactness and precision compose the beauty of mathematics. He learns to influence and command others by accuracy.

3.            Certainly of Results: There is no possibility of difference of opinion between the teacher and the pupil. It is possible for the learner to remove his difficulties by self-effort and to be sure of the removal. He develops faith in self-effort, which is the secret of success in life.

4.            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. This practice in originality enables the situation with confidence in his future life.

5.              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.

6.                  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.

      7.    Concentration of Mind: Every problem of education and life demandsconcentration.

Mathematics cannot be learn without whole-hearted concentration.  Hamilton says, ‘the study of mathematics cures the vice of mental distraction and cultivates the habit of continuous attention’.

In every field new ideas and new methods are being introduced rapidly. In this every advancing society the important thinking is not only to learn facts but also to know how to learn facts. This is the discipline of the mind. Mathematics has a vast scope of application. Mathematics has the ability to apply knowledge to new situation and acquire the power to think effectively so that intellectual power of the learner is strengthened.

     The teaching of mathematics intends to realise the following disciplinary aims,

1.      To provide opportunities that enable the learners to exercise and discipline mental faculties.

2.      To help the learner in the intelligent use of reasoning power.

3.      To develop constructive imagination and inventive faculties.

4.      To develop the character through systematic and orderly habits.

5.      To help the learner to be original and creative in thinking.

6.      To help the individual to become self-reliant and independent.

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. Further after knowing what our ancestors have done we as the students of mathematics bring new idea in the body of mathematical knowledge. And thus increase our cultural heritage. Culture is not possible unless there is a proper development of the power of reasoning and judgment. The aim of teaching of mathematics is to develop cultured citizen who can discharge their obligations to the society effectively and successfully. Our entire present civilization depends on intellectual penetration and utilization of nature has its real foundation in the mathematical science. 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’. Mathematics played a major role in bringing man to the advanced stage of development. The prosperity and the cultural advancement of man depend on the advancement in mathematics. 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.

Mathematics is the backbone of our civilization. What we have in our modern culture and civilization owes its depth to science and technology, which in turn, depends upon the progress in mathematics. Moreover in various cultural arts like poetry, drawing, painting, music, architecture and design making, mathematics is playing a vital role and therefore it can be safely said that mathematics is intimately linked with the culture and civilization.

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:

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

4)        (12)²=144    

  (144)²=441

5)   1X8+1=9

                      12X8+2=98

                       123X8+3=987

                       1234X8+4=9876

                       12345X8+5=98765

                       123456X8+6=987654

                       1234567X8+7=9876543

                       12345678X8+8=98765432

                       123456789X8+9=987654321

             6) Statement: A door is half opened, that means that door is half closed!

i.e,. Door is half opened = ½ closed

                         X 2

                   ½ X 2 opened = ½ X2 closed

          ---à  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

              (2) – (3) -à 1 ½ Re = 150 paise

            (2) X (3)  à 2 ½ Re = 200 X 50

                                      1 Re = 10000 paise!!!!

Divide (2 ) by (3) -à 2/1/2 Re = 200/50 paise

4 Re= 4 paise!!!

         8) 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!!!

        9) KEPREKAR CONSTANT

              Born: 1905.Jan.17. Place: Mumbai

Steps:

1.       Any 4 different numbers.

2.       Write the number in ascending and descending orders. (4321---1234)

3.       Take difference of   1 and 2.

4.       Again ascending and descending order and difference.

5.       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.

10) Easy multiplication, the number end with 5 (two digits only)

25X25   -à * 5x5=25àkeep as it is.

          625             * add 1 to second digit, (1+2=3).

                * 3x2=6   

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.

·         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 awareness of the need for national unity, national integration, protection of the environment, observance of small family, norms, removal of social barriers, and elimination of sex biases.

·         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 under:

A.    Knowledge and Understanding objectives

B.     Skill objectives

C.     Application objectives

D.    Attitude objectives

E.     Appreciation and Interest objectives

A. Knowledge and Understanding Objectives

                        The student acquires knowledge and understanding of:

1.      Language of mathematics i.e., the language of its technical terms, symbols, statements, formulae, definitions, logic, etc.

2.      Various concepts i.e., concept of number, concept of direction, concept measurement.

3.      Mathematical Ideas, like facts, principles, processes and relationships.

4.      The development of the subject over the centuries and contributions mathematicians.

5.      Inter-relationship between different branches and topics of mathematics etc.

6.      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.

6.      He develops and ability to think correctly, to draw conclusions, generalizations and inferences.

7.      He develops skills to use mathematical tools, and apparatus.

8.      He develops essential skill in drawing geometrical figures.

9.      He develops skill in drawing, reading, interpreting graphs and statistical tables.

10.  He develops skill in measuring, weighing and surveying.

11.  He develops skill in the use of mathematical tables and ready references.

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.

6.      He develops the students’ ability to apply mathematical in his future vocational life.

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.

11.  He develops mathematical perspective and outlook for observing the realm of nature and society.

E. Appreciation and Interest Objectives:

            The student is helped in the acquisition of appreciation and interest in the following way:

   1. He appreciates the role of mathematics in everyday life.

2. He appreciates the role of mathematics in understanding his environment.

3. He appreciates mathematics as the science of all sciences and art of all arts.

4. He appreciates the contribution made by mathematics in the development of civilization and culture.

5.  He appreciates the contribution of mathematics with field and other branches.

6.  He develops the interest in the learning of the subject.

7.  He feels enter by mathematical recreations.

8.  He develops act interest in the activities of mathematics clues.

9.  He develops act interest inactive library reading, mathematical projector.

10. He appreciates the aesthetic nature of mathematics by observing symmetry,

      Similarity, order and arrangement in mathematical facts, principles and processes.

11. He appreciates the contribution of mathematics in the development of other

branches of knowledge.

12.  He appreciates the recreational values of the subject and learn to utilize it in his

leisure time.

13. He appreciates the vocational value of mathematics.

14. He appreciates the role of mathematical language, graphs and tables in giving

       Precision and accuracy to his expression.

15. He appreciates the power of computation developed through the subject.

16. He appreciates the role of mathematics in developing his power of acquiring

       Knowledge.

17. He appreciates mathematical problems, their intricacies and difficulties.

18. He develops interest in the learning of the subject.

19. He feels entertained by mathematical recreations.

20. He takes an active interest in the activities of mathematics club.

21. He takes an active interest inactive library reading, mathematical projects, and     doing practical work in mathematics laboratory.

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 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 need of a good command of arithmetic by a housewife, by a farmer, by a successful merchant, by a skilled worker is too obvious to need any discussion. Also its utilitarian, cultural and disciplinary values are too obvious to need any argument at this stage.

            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 impart a working knowledge of practical arithmetical applications which are useful in life.

7.      To appreciate the use of arithmetic in daily life.

8.      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. It presents a radically new and different approach to the study of quantitative relationships characterized by a new symbolism, new concepts, and a new language much higher degree of generalization and abstraction than has been encountered in arithmetic. But it is primarily taught for manipulative skill. Solutions of problems by equations give a power of generalization and use of formulae and idea of functionality.

The following are the objectives of teaching Algebra-

1.            To give compact formulae of generalization to be used in all cases.

2.            To provide an effective way for expressing complicated relations.

3.            To correct the weaknesses and supplement the deficiency of language as an instrument of abstract investigation and exact statement.

4.            To inculcate the power of analysis.

5.            To verify the results in simpler and more satisfactory way.

6.            To develop confidence among the pupils.

7.            To provide new and refined approach in the study of abstract mathematical relationship though the use of new symbolism.

8.            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

1.                  To enable the learner to acquire a mass of geometrical facts.

2.                  To implement geometrical principles like equality, symmetry similarity in every nature of things.

3.                  To develop the ability to draw accurate figures.

4.                  To demonstrate the nature and the power of pure reason.

5.                  To systematize the information received by the pupils in the pre-school stage.

6.                  To aid the pupils in becoming familiar with the basic geometrical concepts and space perceptions and in understanding the fundamental techniques such as the use of set square, protractor, compass, etc.

7.                  To acquaint with the pupil the good geometrical notation.

Instructional Objectives:

I.                   Remembering: It is cognitive level.

The pupil –

         1. Recalls the mathematical laws, principle, rule formulae, etc.

         2. Recognizes the mathematical laws, principle, formulae, etc.

II. Understanding: Goes deep into the content

                        The pupil –

1.      Cites or gives examples

2.      Gives reasons

3.      Identifies

4.      Compares

5.      Finds relationship

6.      On the basis of observation draws conclusion

7.      Draws inference or  the result

8.      Converts verbal form to symbolic form or vice versa

9.      Classifies mathematical data

III. Applying:

                        The Pupil –

1.      Analyses the problem into its components

2.      Judges the adequacy of the given data

3.      Suggests the alternate methods

4.      Suggest the most appropriate method

5.      Generalizes

 IV. Skill:

                        The Pupil –

1.      Reads mathematical figures, statements, problems, charts, tables, etc.

2.      Labels the geometrical figure.

3.      Draws the most appropriate, neat and proportionate geometrical figures

4.      Solves oral problems quickly and accurately

5.      Solves written problems quickly and accurately.

Trigonometry Objectives

• Measurement of Angles, Arcs, and Sectors
• Using Radians, Degrees, or Grads to Measure Angles
• Length of an Arc and Area of a Sector of a Circle
• Circular Motion
• The Trigonometric Functions
• Definition of the Six Trigonometric Functions
• Values of the Trigonometric Functions at some multiples of 15 degrees.
• Trigonometric Functions for right triangles
• Solving Right Triangles
• Applications of Right Triangle Trigonometry
• Circular Functions
• Graphs of Trigonometric Functions
• Graphing Generic Sine and Cosine Functions
• Shifting Generic Curves Right/Left or Up/Down
• Using the Graphing Calculator to Graph Functions by Addition of Ordinates
• Graphing the Tangent and Cotangent Functions
• Graphing the Secant and Cosecant Functions
• Qualitative Analysis of Trigonometric Functions
• Inverse Trigonometric Functions
• Relations, Functions, and Their Inverses
• Inverses of Trigonometric Functions
• Finding Inverses of Trigonometric Functions Using a Calculator
• Basic Trigonometric Identities
• Fundamental Identities
• Opposite Angle Identities
• Additional Techniques to Prove Identities
• Sum and Difference Identities
• Sum and Difference Identities for Cosine
• Some Identities Useful in Calculus
• Sum and Difference Identity for Tangent
• Identities Involving Sums and Differences of Pi and Pi/2.
• Additional Identities
• Double Angle Identities
• Half Angle Identities
• Identities to Rewrite Sums and Products
• Trigonometric Equations
• Solving Basic Trigonometric Equations
• Solving Trigonometric Equations Involving Factoring
• Solving Trigonometric Equations Where the Argument is a Function
• Using Identities to Solve Trigonometric Equations
• Law of Sines and Law of Cosines
• Derivation of the Law of Sines
• The Ambiguous Case
• Applications of the Law of Sines
• Derivation of the Law of Cosines
• Applications of the Law of Cosines
• Area of a Triangle
• Vectors
• Addition of Vectors
• Geometric Resolution of Vectors
• Algebraic Resolution of Vectors
• Work, Inclined Planes, and the Dot Product
• Complex Numbers
• Algebraic Operations with Complex Numbers
• Trigonometric and Polar Representation of Complex Numbers
• DeMoivre's Formula
• Polar Coordinates
• The Polar Coordinate System
• Parametric Equations
• Other Graphs in Polar Coordinates

Objectives of TeachingTrigonometry

The students will be able,

1.       To impart knowledge of trigonometric ratios and identities.

2.       To apply the knowledge of trigonometry to solve daily life problems.

3.       To find heights and distances.

4.       To appreciate the use of trigonometry to solve problems.

5.       To develop creative thinking and reasoning.

6.       To understand that it is an essential tool.

7.       to know how structures are built.

8.       To realise that it very useful in technology and for engineers.

9.       To continue higher education.

10.   To understand the relationship between trigonometry and other branches of mathematics.

11.   Find the value of trigonometric ratios of some specific angles.

12.   Determine the trigonometric ratios of complementary angle.

13.   Apply the trigonometric identities in proving the given statement.

 Objectives of TeachingCoordinate Geometry

The students will be able to,

1.       Draw a plan for the given situation.

2.       Appraise the Cartesian system.

3.       Identify the coordination of a point.

4.       Locate the quadrants in the Cartesian plane.

5.       Plot the points in the Cartesian plane.

6.       Write the abscissa and ordinate of a point.

Specific Objectives:

1. To find areas of plane rectilinear figures in the coordinate plane.

 2. To find the angle between two straight lines.

3. To understand the normal form of a straight line equation and to apply the knowledge in finding distances.

4. To find the equation of a circle and the points of intersection of a straight line and a circle.

5. To find the equations of tangents to a circle.

6. To find the equations of families of straight lines and circles.

ANDERSON’S REVISED BLOOM’S TAXONOMY OF INSTRUCTIONAL OBJECTIVES

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: know identify relate list define recall memorize repeat record name recognize acquire	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.
2. Comprehension: The ability to grasp or construct meaning from material. Examples of verbs that relate to this function are:   restate locate report recognize explain express identify discuss describe discuss review infer illustrate interpret draw represent differentiate conclude	2. Understanding:  Constructing meaning from different types of functions be they written or graphic messages or activities like interpreting, exemplifying, classifying, summarizing, inferring, comparing, or explaining.
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	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. 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	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. 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 produce design assemble create prepare predict modify tell plan invent formulate collect set up generalize document combine relate propose develop arrange construct organize originate derive write propose	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 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. 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	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.

Instructional Objectives:

1. Remembering: The pupil acquires knowledge of mathematics.

Learning outcomes: The pupil,

·         Recalls mathematical terms, facts, processes, principles, formulae definitions, signs and symbols, relationships, generalizations etc.,

·         Recognizes terms, instruments, process, formulae, signs and symbols, relationships, generalisations etc.,

2. Understanding: The pupil develops understanding in mathematics.

Learning outcomes: The pupil,

·         Explains mathematical terms, concepts, principles, relationships etc., in his own words.

·         Defines mathematical terms and concepts.

·         States mathematical principles, relationships etc.

·         Gives illustrations for mathematical concepts, principles, etc.,

·         Identifies mathematical terms, concepts, relationships, figures, processes etc.

·         Finds similarities between mathematical terms, concepts, relationships, figures etc.

·         Differentiates between mathematical terms, concepts, relationships, figures etc.

·         Classifies mathematical terms, concepts, figures etc.,

·         Verbalises symbolic relationships and vice versa.

·         Frames mathematical formulae, generalisations on the basis of data.

·         Uses the formula to solve problems.

·         Substitute relevant numbers, symbols and signs in the mathematical formulae and operations.

·         Calculate the answers for given problems.

·         Uses appropriate units to write answers.

·         Finds solutions for given problems.

3. Applying:The pupils applies knowledge of mathematics to novel situations.

Learning outcomes:The pupil,

·         Analyses a problem or data into component parts.

·         Judges the adequacy, inadequacy or superfluity of data.

·         Establishes relationships among data.

·         Gives a number of methods of solving a problem.

·         Select the most appropriate formulae or principles or methods or process to solve problem,

·         Reasons deductively.

·         Reasons inductively.

·         Makes a generalization.

·         Draws inferences.

·         Predicts results on the basis of data.

4. Skill:  a) The pupil acquires skills in handling mathematical instruments with ease.

Learning outcomes:The pupil,

·         Draws freely satisfactory free-hand figures.

·         Selects the most appropriate mathematical instruments.

·         Takes necessary precautions in taking measurements while constructing geometrical figures.

·         Takes measurements correctly.

b)Drawing geometrical figures and Graphs

Learning outcomes: The pupil,

·         Draws figures to given specifications.

·         Draws figures quickly.

·         Uses appropriate marking to denote different parts of a figure.

c) Computation:

Learning outcomes: The pupil,

·         Does oral calculation correctly.

·         Does oral calculation quickly.

·         Does a written calculation correctly.

·         Does a written calculation quickly.

·         Uses correct notations and symbols.

·         Avoids unnecessary steps in the solution of a problem.

·         Is systematic in working of a problem.

           d) Reading of Tables, charts, Graphics etc.

Learning outcomes:The pupil,

·         Selects appropriate mathematical tables.

·         Uses mathematical tables, charts, ready reckoners etc., correctly.

·         Co-ordinates the different sections of the graphs correctly.

·         Reads graphs correctly.

TASK ANALYSIS:

      It is a set of questions set on the learnt materials in the class room should not be left in the school itself, but be stored for use in future life. To help the learner to register the information and for its longer retainment, teacher should make students involved in certain activities related to the class room learning.

Example (Problem): A rectangle is having 18 metre length and area 180 sq. metre. What will be breadth?

Knowledge:

1.       Recall   What is given?                      5. What is the unit of Area?

2.       What is length given?                        6. What is the formula to calculate area of

3.       What is the area of rectangle?             rectangle?

4.       What is the unit of length?