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. Learning Latin was presumed to make it easier to learn other languages. The same argument can be used to justify the inclusion of chess in school curricula. There is no objective data to support the notion, and certainly only a small minority of mathematics teachers believes it to have merit. In any event, the type of logic which is encouraged in school mathematics is characterized by being prescriptive, convergent and linear as opposed to descriptive and divergent and creative. Hardly the sort of thing we would normally admit to encouraging. 1.4.3 Structure Mathematics looks at nature and classifies objects according to their shapes, sizes and spaces. One- faced solid – the moebus strip or belt is used as belts in garri-processing machines. Also two-faced solids are used as belts in automobiles. Mathematics differentiates plane figures from solid shapes. Among the plane shapes are Δ for triangle, □ for square and for pentagonal shapes. Among the solid shapes are Δ the tetrahedron,  the cube and  the cuboids. Sizes, shapes and space are very much utilized in housing schemes, in building bridges, in Engineering construction, in surveying and in works of art. Space perception is necessary for students for students to form proper sense of land utilization for agricultural and other purpose. It looks for order, common properties and constant occurrences in objects, numbers or shapes. Example (i) certain number can be arranged to form squares such as       1 4 9 16 etc These are Square numbers. Example (ii) other numbers form triangular numbers such as        1 3 6 10 etc Example (iii) similarly, we have pentagonal and hexagonal numbers. Some other number patterns can be formed in the form of operation thus 10-1=9 100-1=99 1000-1=999 10000-1=9999 etc The geometric patterns are very much used in tessellations of surfaces, tiling and carpeting of floors and walls, and in fine and applied and in envelopes of surfaces and curves. 1.4.4 Symbolism Mathematics makes use of symbols to represent concrete objects, words or expressions, other symbols, abstract ideas or concept. For example, the concept of twoness is represented by the symbols for addition and Δ represents a triangle. The beauty of mathematics as a symbolic language lies in the simplicity and brevity of the symbolic forms the impersonal character of the symbols devoid of emotions. Examples i) The beauty of ―Mr. Bex, the magistrate is on trial for armed robbery‖ we can write ―p is on trial for armed robbery? ii) Instead of ―Mrs. May, the first lady is the defense witness number one‖, we can write, ―y is DW1‖. iii) For the following long sentences ―A games master bought 5 rackets and 8 tennis balls from one shop for a total sum of ₨ 132. In the second shop he spent ₨ 156 to buy 7 rackets and 4 tennis balls. Find the unit prices of a racket and a tennis ball. We can simply write. Solve simultaneously 5r + 8b and 7r + 4b = 156 where r is the price per racket and b is the price per tennis ball. iv) From these examples, we see the mathematical symbols are freely used in law courts. Other examples are ―PH/2‖ which stands for ―prosecution witness number 2‖. Of course in keeping records to cases, of prisoners, fines terms of imprisonment, dates and case references, mathematics concept are utilized. v) In fact, in all science subjects; physics, chemistry, engineering, agriculture, biology, integrated

science, geography and others mathematical symbols are freely used. For example H2O in chemistry is the chemical symbol for water, while NaCl is the formula for common salt. This mathematically shows that one molecule of sodium chemically combines with one molecule of chlorine. The mastery of the use of this symbolic language by students will inculcate in them the habit of brevity, clarity and precision of expression and will bring them nearer to unity with other human beings in the world. 1.4.5 Constructiveness The utility of academic and school mathematics in the modern world is greatly overestimated, and the utilitarian argument provides a poor justification for the universal teaching of the subject throughout the years of compulsory schooling. Thus although it is widely assumed that academic mathematics drives the social applications of mathematics in such areas as education, government, commerce and industry, this is an inversion of history. Five thousand years ago in ancient Mesopotamia it was the rulers‘ need for scribes to tax and regulate commerce that led to the setting up of scribal schools in which mathematical methods and problems were systematized. This led to the founding of the academic discipline of mathematics. Even today the highly mathematical studies of accountancy, actuarial studies, management science and information technology applications are mostly undertaken within professional or commercial institutions outside of the academy and with little immediate input from academic mathematics. The mathematization of modern society and modern life has been growing exponentially, so that by now virtually the whole range of human activities and institutions are conceptualized and regulated numerically, including sport, popular media, health, education, government, politics, business, commercial production, and science. Many aspects of modern society are regulated by deeply embedded complex numerical and algebraic systems, such as supermarket checkout tills with auto mated bill production, stock control; tax systems; welfare benefit systems; industrial, agricultural and educational subsidy systems; voting systems; stock market systems. 1.4.6 Brevity Brevity is a recurring characteristic of mathematics mathematical truth turns out to be relevant in awfully distinct ways of brevity in phenomena from across the universe to across the street. Why is this? What is it about mathematics and the concepts that it captures that causes this? As Mathematics has progressively advanced and abstracted its natural concepts, it has increased the host of subjects to which these concepts can be fruitfully applied. It is hard to believe that brevity is a characteristic of mathematics. Yet, for the practitioner of mathematics, brevity is a strong part of the culture. The mathematician desires the brevity possible concise exposition. Through greater abstraction, a concise exposition is possible at the price of additional terminology and machinery to allow all of the various particularities to be subsumed into the exposition at the higher level. Thus, it is believed that the mathematician‘s desire for a concise exposition that leads to the attendant brevity of mathematics, especially in contemporary mathematics. 1.5 History of Mathematics What came before 250,000 years ago? Archaeological timelines date the Cenozoic epoch of earth‘s history, when the earth‘s climate was stable, at 25 million years ago. Primates were already living in trees by 13 million years ago, and by the time of the first ice age 700,000 years ago (Early Pleistocene), the early ancestors of man (Australopithecus) had already descended from tree living and were using crude stone tools. If we broaden our inquiry beyond the evidence for mathematical understanding and use, and into the possession of the cognitive precursors for mathematics, we find that both number sense (but not measurement or counting per se) and the perception of shape and change (though perhaps not their description or communication) are not at all unique to the human species. Investigations have found evidence of number sense in animals (birds, dogs, monkeys, dolphins). Perception of the passage of time, the ability to distinguish one from many (in particular, quantities other than two), and the ability to distinguish shapes from each other, have all been documented in various animals. Dating the capability for mathematical cognition then becomes a question of the timeline of intelligent, perceptive life itself. From the evidence at hand of the number sense in animals and primates, it is clear that the first crude-tooled ancestors of man (Australopithecus) of 700,000 years ago and the intelligent tree-dwelling primates of 13 million years ago, both had the mental capacity for the cognition of the precursors of mathematics. In India, Mathematics has its roots in Vedic literature which is nearly 4000 years old. Between 1000 B.C. and 1000 A.D. various treatises on Mathematics were authored by Indian mathematicians in which were set forth for the first time, the concept of zero, the techniques of Algebra and algorithm, square root and cube root. This method of graduated calculation was documented in the Pancha- Siddhantika (Five Principles) in the 5th Century. But the technique is said to be dating from Vedic times circa 2000 B.C. In India around the 5th century A.D. a system of Mathematics that made astronomical calculations easy was developed. In those times its application was limited to Astronomy as its pioneers were astronomers. As astronomical calculations are complex and involve many variables that go into the derivation of unknown quantities. Algebra is a short-hand method of calculation and by this feature it scores over conventional Arithmetic. In ancient India conventional Mathematics termed Ganitam was known before the development of Algebra. This is borne out by the name - Bijaganitam, which was given to the algebraic form of computation. Bijaganitam means ‗the other Mathematics‘ (Bija means ‗another‘ or ‗second‘ and Ganitam means Mathematics). The fact that this name was chosen for this system of computation implies that it was recognized as a parallel system of computation, different from the conventional one which was used since the past and was till then the only one. Some have interpreted the term Bija to mean seed, symbolizing origin or beginning. And the inference that Bijaganitam was the original form of computation is derived. Credence is lent to this view by the existence of Mathematics in the Vedic literature which was also shorthand method of computation. But whatever the origin of Algebra, it is certain that this technique of computation originated in India and was current around 1500 years back. Aryabhatta an Indian mathematican who lived in the 5th century A.D. has referred to Bijaganitam in his treatise on Mathematics, Aryabhattiya. An Indian mathematician - astronomer, Bhaskaracharya has also authored a treatise on this subject. the treatise which is dated around the 12th century the treatise which is dated around the 12th century A.D. is entitled ‗Siddhanta-Shiromani‘ of which one section is entitled Bijaganitam. The ancient India astronomer Brahmagupta is credited with having put forth the concept of zero for the first time Brahmagupta is said to have been born the year 598 A.D. at Bhillamala (today‘s Bhinmal ) in Gujarat, Western India. Not much is known about Brahmagupta‘s early life. We are told that his name as a mathematician was well established when K Vyaghramukha of the Chapa dyansty made him the court astronomer. Of his two treatises, Brahma-sputa siddhanta and Karanakhandakhadyaka, first is more famous. It was a corrected version of the old astronomical text, Brahma siddhanta. It was in his Brahma-sputa siddhanta, for the first time ever had been formulated the rules of the operation zero, foreshadowing the decimal system numeration. With the integration of zero into the numerals it became possible to note higher numerals with limited characters. 1.6 Contribution of Indian Mathematicians The most fundamental contribution of ancient India in mathematics is the invention of decimal system of enumeration, including the invention of zero. The decimal system uses nine digits (1 to 9) and the symbol zero (for nothing) to denote all natural numbers by assigning a place value to the digits. The Arabs carried this system to Africa and Europe. 1.6.1 Aryabhatta I (475 A.D. -550 A.D.) Aryabhatta I is the first well known Indian mathematician. Born in Kerala, he completed his studies at the University of Nalanda. Some conjecture that Aryabhatta I was born in the region lying between Narmada and Godavari, which was known as Ashmaka, and is now identified with Maharashtra, though early Buddhist texts describe Ashmaka as being further south, dakshinapath or the Deccan, while other texts describe the Ashmakas as having fought Alexander, which would put them further north. Other traditions in India claim that he was from Kerala and that he travelled to the North, or that he was a Maga Brahmin from Gujarat. His first name ―Arya‖ is hardly a south Indian name while ―Bhatt‖ (or Bhatta) is a typical north Indian name even found today specially among the ―Bania‖ (or trader) community. In the section Ganita (calculations) of his astronomical treatise Aryabhatiya (499 A.D.), he made the fundamental advance in finding the lengths of chords of circles, by using the half chord rather than the full chord method used by Greeks. He gave the value of as 3.1416, claiming, for the first time, which was close to the actual value π = 3.14159265 correct to 8 places. He gave it in the form that the approximate circumference of a circle of diameter 20000 is 62832. He also gave methods for extracting square roots, summing arithmetic series, solving indeterminate equations of the type ax -by = c, and also gave what later came to be known as the table of Sines. He also wrote a text book for astronomical calculations, Aryabhattasiddhanta. Aryabhatta I gives a systematic treatment of the position of the planets in space. He gave the circumference of the earth as 4967 yojanas and its 1 diameter as 1581 /24 yojanas. Since 1 yojana = 5 miles this gives the circumference as 24835 miles, which is an excellent approximation to the currently accepted value of 24902 miles. Aryabhatta I gives the radius of the planetary orbits in terms of the radius of the Earth/Sun orbit as essentially their periods of rotation around the Sun. He believes that the Moon and planets shine by reflected sunlight; incredibly he believes that the orbits of the planets are ellipses. He correctly explains the causes of eclipses of the Sun and the Moon. Even today, this data is used in preparing Hindu calendars (Panchangs). In recognition to his contributions to astronomy and mathematics, India‘s first satellite was named Aryabhatta I. This remarkable man was a genius and continues to baffle many mathematicians of today. His works was then later adopted by the Greeks and then the Arabs. 1.6.2 Varahamihira (505 A.D. – 587 A.D.) Not much is known of Varahamihira‘s life. According to one of his works, he was educated in Kapitthaka. He is considered to be one of the nine jewels (Navaratnas) of the court of legendary ruler Vikramaditya (thought to be the Gupta emperor Chandragupta II Vikramaditya). However, far from settling the question this only gives rise to discussions of possible interpretations of where this place was. Dhavale discussed in his problem that we do not know whether he was born in Kapitthaka, wherever that may be, although we have given this as the most likely guess. We do know, however, that he worked at Ujjain which had been an important centre for mathematics since around 400 AD. The school of mathematics at Ujjain was increased in importance due to Varahamihira working there and it continued for a long period to be one of the two leading mathematical centers in India, in particular having Brahmagupta as its next major figure. He was the first one to mention in his work Pancha Siddhantika that the ayanamsa, or the shifting of the equinox is 50.32 seconds. The most famous work by Varahamihira is the Pancasiddhantika (The Five Astronomical Canons) dated 575 AD. This work is important in itself and also in giving us information about older Indian texts which are now lost. The work is a treatise on mathematical astronomy and it summarizes five earlier astronomical treatises, namely the Surya, Romaka, Paulisa, Vasistha and Paitamaha siddhantas. Varahamihira made some important mathematical discoveries. Among these are certain trigonometric formulae which translated into our present day notation correspond to sin x = cos(π/2 - x), sin2x + cos2x = 1, and (1 - cos 2x)/2 = sin2x. Varahamihira improved the accuracy of the sine tables of Aryabhata I. He calculated the binomial coefficients, known in the European civilization as Pascal‘s triangle and also he worked on magic squares. Among Varahamihira‘s contribution to physics is his statement that reflection is caused by the back-scattering of particles and refraction (the change of direction of a light ray as it moves from one medium into another) by the ability of the particles to penetrate inner spaces of the material, much like fluids that move through porous objects. Another important contribution to trigonometry was his sine tables where he improved those of Aryabhatta I giving more accurate values. It should be emphasized that accuracy was very important for these Indian mathematicians since they were computing sine tables for applications to astronomy and astrology. This motivated much of the improved accuracy they achieved by developing new interpolation methods. Varahamihira‘s other most important contribution is the encyclopedic Brihat-Samhita. It covers wide ranging subjects of human interest, including astrology, planetary movements, eclipses, rainfall, clouds, architecture, growth of crops, manufacture of perfume, matrimony, domestic relations, gems, pearls, and rituals. The volume expounds on gemstone evaluation criterion found in the Garuda Purana, and elaborates on the sacred Nine Pearls from the same text. It contains 106 chapters and is known as the ―great compilation‖. 1.6.3 Brahmagupta (598 A.D. - 665 A.D.) Brahmagupta is renowned for introduction of negative numbers and operations on zero into arithmetic. His main work was Brahmasphutasiddhanta (The Opening of the Universe) in 628, which was a corrected version of old astronomical treatise Brahmasiddhanta. This work was later translated into Arabic as Sind Hind. He formulated the rule of three and proposed rules for the solution of quadratic and simultaneous equations. He gave the formula for the area of a cyclic quadrilateral as where s is the semi perimeter. He was the first mathematician to treat algebra and arithmetic as two different branches of mathematics. He gave the solution of the indeterminate equation Nx²+1 = y². He is also the founder of the branch of higher mathematics known as ―Numerical Analysis‖. He gave the following formula, used in G.P series a + ar + ar2 + ar3 +……….. + arn-1 = (arn-1) ÷ (r – 1) Brahmagupta‘s understanding of the number systems went far beyond that of others of the period. In the Brahmasphutasiddhanta he defined zero as the result of subtracting a number from itself. He gave some properties as follows - When zero is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by zero becomes zero. Really Brahmagupta is saying very little when he suggests that n divided by zero is n/0. He is certainly wrong when he then claims that zero divided by zero is zero. However it is a brilliant attempt to extend arithmetic to negative numbers and zero. Brahmagupta developed some algebraic notation and presents methods to solve quardatic equations. He presents methods to solve indeterminate equations of the form ax + c = by. Majumdar in writes - Brahmagupta perhaps used the method of continued fractions to find the integral solution of an indeterminate equation of the type ax + c = by. Brahmagupta also solves quadratic indeterminate equations of the type ax2 + c = y2 and ax2 - c = y2. For example he solves 8x2 + 1 = y2 obtaining the solutions (x, y) = (1, 3), (6, 17), (35, 99), (204, 577), (1189, 3363), ... For the equation 11x2 + 1 = y2 Brahmagupta obtained the solutions (x, y) = (3, 10), (161/5, 534/5), ... He also solves 61x2 + 1 = y2 which is particularly elegant having x = 226153980, y = 1766319049 as its smallest solution. Rules for summing series are also given. Brahmagupta gives the sum of the squares of the first n natural numbers as n (n+1) (2n+1) / 6 and the sum of the cubes of the first n natural numbers as (n (n+1) / 2)2. No proofs are given so we do not know how Brahmagupta discovered these formulae. In his renowned book Brahmasphutasiddhanta it was briefly discussed about solar and lunar eclipses, planetary conjunctions and positions of the planets. Brahmagupta believed in a static Earth and he gave the length of the year as 365 days 6 hours 5 minutes 19 seconds in the first work, changing the value to 365 days 6 hours 12 minutes 36 seconds in the second book the Khandakhadyaka. After Brahmagupta, the mathematician of some consequence was Sridhara, who wrote Patiganita Sara, a book on algebra, in 750 A.D. Even Bhaskara refers to his works. After Sridhara, the most celebrated mathematician was Mahaviracharaya or Mahavira. He wrote Ganita Sara Sangraha in 850 A.D., which is the first text book on arithmetic in present day form. He is the only Indian mathematician who has briefly referred to the ellipse (which he called Ayatvrit). The Greeks, by contrast, had studied conic sections in great detail. 1.6.4 Bhaskara I (600 A.D. – 680 A.D.) Bhaskara I was a 7th century Indian mathematician, who was apparently the first to write numbers in the Hindu-Arabic decimal system with a circle for the zero, and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhatta‘s work. This commentary, Aryabhatiyabhasya, written in 629 CE, is the oldest known prose work in Sanskrit on mathematics and astronomy. Little is known about the life of Bhaskara I is appended to his name to distinguish him from a 12th-century Indian astronomer of the same name. In his writings there are clues to possible locations for his life, such as Valabhi, the capital of the Maitrika dynasty, and Ashmaka, a town in Andhra Pradesh and the location of a school of followers of Aryabhata. His fame rests on three treatises he composed on the works of Aryabhata. Two of these treatises, known today as Mahabhaskariya (―Great Book of Bhaskara‖) and Laghubhaskariya (―Small Book of Bhaskara‖), are astronomical works in verse. Bhaskara‘s works were particularly popular in South India. Planetary longitudes, heliacal rising and setting of the planets, conjunctions among the planets and stars, solar and lunar eclipses, and the phases of the Moon are among the topics Bhaskara discusses in his astronomical treatises. He also includes a remarkably accurate approximation for the sine function sin x = 16x (π – x)/[5π2 – 4x(π – x)], where 0≤x≥ π/2. In his commentary on the Aryabhatiya, Bhaskara explains in detail Aryabhata‘s method of solving linear equations and provides a number of illustrative astronomical examples. Bhaskara particularly stressed the importance of proving mathematical rules rather than just relying on tradition or expediency. One of the approximations used for π for many centuries was √10. Bhaskara I criticized this approximation. He regretted that an exact measure of the circumference of a circle in terms of diameter was not available and he clearly believed that π was not rational. 1.6.5 Mahavira (800 A.D. – 870 A.D.) All that is known about Mahavira‘s life is that he was a Jain (he perhaps took his name to honour the great Jainism reformer Mahavira [c. 599–527 BCE] and that he was the author of Ganitasarasangraha (―Compendium of the Essence of Mathematics‖) during the reign of Amoghavarsha (c. 814–878) of the Rashtrakuta dynasty. The work comprises more than 1,130 versified rules and examples divided in nine chapters the first chapter for ―terminology‖ and the rest for ―mathematical procedures‖ such as basic operations, reductions of fractions, miscellaneous problems involving a linear or quadratic equation with one unknown, the rule of three (involving proportionality), mixture problems, geometric computations with plane figures, ditches (solids), and shadows (similar right-angled triangles). Ganitasarasangraha was translated into Telugu language by Pavuluri Mallana as Saar Sangraha Ganitam. Everyone knows you can‘t divide by 0. It‘s undefined. But in 830 A.D. the Indian mathematician Mahavira said you could do it. The short version is a ÷ 0 = a. In other words, dividing by 0 is identical to dividing by 1. It has no effect, just like adding or subtracting 0 has no effect. At the beginning of his work, Mahavira stresses the importance of mathematics in both secular and religious life and in all kinds of disciplines, including love and cooking. While giving rules for zero and negative quantities, he explicitly states that a negative number has no square root because it is not a square (of any ―real number‖). Besides mixture problems (interest and proportions), he treats various types of linear and quadratic equations (where he admits two positive solutions) and improves on the methods of Aryabhata I (b. 476). He also treats various arithmetic and geometric, as well as complex series. For rough computations, Mahavira used 3 as an approximation for π, while for more exact computations he used the traditional Jain value of √10. He also included rules for permutations and combinations and for the area of a conch like plane figure (two unequal semicircles stuck together along their diameters), all traditional Jain topics.

1.6.6 Sridhara (870 A.D. – 930 A.D.) Sridhara is now believed to have lived in the ninth and tenth centuries. However, there has been much dispute over his date and in different works the dates of the life of Sridhara have been placed from the seventh century to the eleventh century. The best present estimate is that he wrote around 900 AD, a date which is deduced from seeing which other pieces of mathematics he was familiar with and also seeing which later mathematicians were familiar with his work. We do know that Sridhara was a Hindu but little else is known. Two theories exist concerning his birthplace which is far apart. Some historians give Bengal as the place of his birth while other historians believe that Sridhara was born in southern India. Sridhara is known as the author of two mathematical treatises, namely the Trisatika (sometimes called the Patiganitasara ) and the Patiganita. However at least three other works have been attributed to him, namely the Bijaganita, Navasati, and Brhatpati. Information about these books was given the works of Bhaskara II (writing around 1150), Makkibhatta (writing in 1377), and Raghavabhatta (writing in 1493). We give details below of Sridhara‘s rule for solving quadratic equations as given by Bhaskara II. There is another mathematical treatise Ganitapancavimsi which some historians believe was written by Sridhara. The Patiganita is written in verse form. The book begins by giving tables of monetary and metrological units. Following this algorithms are given for carrying out the elementary arithmetical operations, squaring, cubing, and square and cube root extraction, carried out with natural numbers. Through the whole book Sridhara gives methods to solve problems in terse rules in verse form which was the typical style of Indian texts at this time. All the algorithms to carry out arithmetical operations are presented in this way and no proofs are given. Indeed there is no suggestion that Sridhara realized that proofs are in any way necessary. Often after stating a rule Sridhara gives one or more numerical examples, but he does not give solutions to these examples nor does he even give answers in this work. After giving the rules for computing with natural numbers, Sridhara gives rules for operating with rational fractions. He gives a wide variety of applications including problems involving ratios, barter, simple interest, mixtures, purchase and sale, rates of travel, wages, and filling of cisterns. Some of the examples are decidedly non-trivial and one has to consider this as a really advanced work. Other topics covered by the author include the rule for calculating the number of combinations of n things taken m at a time. There are sections of the book devoted to arithmetic and geometric progressions, including progressions with a fractional numbers of terms, and formulae for the sum of certain finite series are given. Of all the Hindu Acharyas the exposition of Sridharacharya on zero is the most explicit. He has written, ―If 0 (zero) is added to any number, the sum is the same number; If 0 (zero) is subtracted from any number, the number remains unchanged; If 0 (zero) is multiplied by any number, the product is 0 (zero)‖. He has said nothing about division of any number by 0 (zero). He gave the first correct formulas in India for the volume of a sphere and of a truncated cone. He used two approximations for 22 π, the traditional Jain value of √10 as well as /7. Bhaskara II cites Shridhara‘s rule for quadratic equations that allows two solutions of a single equation, in so far as they are positive, probably from Shridhara‘s lost work on bija-ganita. The book ends by giving rules, some of which are only approximate, for the areas of some plane polygons. In fact the text breaks off at this point but it certainly was not the end of the book which is missing in the only copy of the work which has survived. We do know something of the missing part, however, for the Patiganitasara is a summary of the Patiganita including the missing portion. Shukla examines Sridhara‘s method for finding rational solutions of Nx2 ± 1 = y2, 1 – Nx2 = y2, Nx2 ± C = y2, and C – Nx2 = y2 which Sridhara gives in the Patiganita. 1.6.7 Aryabhata II (920 A.D. – 1000 A.D.) Aryabhata II was a mathematician and astronomer of India, and wrote the well-known book the Mahasiddhanta. He is believed to be born in India during c.920 and died at c.1000. These dates are approximately suggested by the modern historians; however there are historians like G.R.Kaye believed that Aryabhata II lived before al-Biruni, whereas Datta in 1926 proved that these dates were too early. Though Pingree considers that Aryabhatta‘s main publications was published between 950 and 1100, but R.Billiard has proposed a date in the sixteenth century. Aryabhatta II‘s most eminent work was Mahasiddhanta. The treatise consists of eighteen chapters and was written in the form of verse in Sanskrit. The initial twelve chapters deal with topics related to mathematical astronomy and covers the topics that Indian mathematicians of that period had already worked on. The various topics that have been included in these twelve chapters are the longitudes of the planets, lunar and solar eclipses, the estimation of eclipses, the lunar crescent, the rising and setting of the planets, association of the planets with each other and with the stars. The next six chapters of the book include topics such as geometry, geography and algebra, which were applied to calculate the longitudes of the planets. In about twenty verses in the treatise, he gives elaborate rules to solve the indeterminate equation by = ax + c. These rules have been applied to a number of different cases such as when c has a positive value, when c has a negative value, when the number of the quotients is a even number, when this number of quotients is an odd number, etc. Aryabhata II also gave a method to calculate the cube root of a number, but his method was not new, being based on that given many years earlier by Aryabhata I. Aryabhata II constructed a sine table correct up to five decimal places when measured in decimal parts of the radius. Indian mathematicians were very interested in giving accurate sine tables since they were used to calculate the planetary positions as accurately as possible. 1.6.8 Bhaskara II (1114 A.D. -1185 A.D.) or Bhaskaracharya Bhaskaracharya is the most well known ancient Indian mathematician. He was born in 1114 A.D. at Bijjada Bida (Bijapur, Karnataka) in the Sahyadari Hills. He writes about his year of birth as follows, ‗I was born in Shake 1036 (1114 AD) and I wrote Siddhanta Shiromani when I was 36 years old.‘ Bhaskaracharya has also written about his education. Looking at the knowledge, which he acquired in a span of 36 years, it seems impossible for any modern student to achieve that feat in his entire life. See what Bhaskaracharya writes about his education, ‗I have studied eight books of grammar, six texts of medicine, six books on logic, five books of mathematics, four Vedas, five books on Bharat Shastras, and two Mimansas‘. Bhaskaracharya calls himself a poet and most probably he was Vedanti, since he has mentioned ‗Parambrahman‘ in that verse. He became head of the Ujjain school of mathematical astronomy (Varahamihira and Brahmagupta had helped to found this school or at least ‗build it up‘). He was the first to declare that any number divided by zero is infinity and that the sum of any number and infinity is also infinity. He gave the value of Π = 22/7. He is famous for his book Siddhanta Siromani (1150 A.D.). It is divided into four sections - Leelavati (a book on arithmetic), Bijaganita (algebra), Goladhayaya (chapter on sphere -celestial globe), and Grahaganita (mathematics of the planets). Leelavati contains many interesting problems and was a very popular text book. Bhaskara introduced chakrawal, or the cyclic method, to solve algebraic equations. Six centuries later, European mathematicians like Galois, Euler and Lagrange rediscovered this method and called it ―inverse cyclic‖. Bhaskara can also be called the founder of differential calculus. He gave an example of what is now called ―differential coefficient‖ and the basic idea of what is now called ―Rolle‘s theorem‖. Unfortunately, later Indian mathematicians did not take any notice of this. Bhaskara‘s work on calculus predates Newton and Leibniz by half a millennium. He is particularly known in the discovery of the principles of differential calculus and its application to astronomical problems and computations. While Newton and Leibniz have been credited with differential and integral calculus, there is strong evidence to suggest that Bhaskara was a pioneer in some of the principles of differential calculus. He was perhaps the first to conceive the differential coefficient and differential calculus. As an astronomer, Bhaskara is renowned for his concept of Tatkalikagati (instantaneous motion). In his book ―Surya Siddhant‖ he wrote on the gravitational force, that helps to keep the planets, the Sun and the moon in their respective orbits much before the world could even waken and realize to these findings. ―Kuttaka‖ the Quadratic Indeterminate equations was given by him in 12th Century well before the European mathematicians got it in the 17th Century. Brahmagupta in the 7th Century developed an ―Astronomical Model‖ using which Bhaskara was able to define ―Astronomical Quantities‖. He accurately calculated the time that earth took to revolve around the SUN as 365.2588 days that is a difference of 3.5Minutes of modern acceptance of 365.2563 days. 1.6.9 Narayana Pandita (1340 A.D. – 1400 A.D.) Narayana Pandita was the son of Narasimha. Narayana Pandita was a renowned mathematician of India. Plofker writes that his texts were the most significant Sanskrit mathematics treatises after those of Bhaskara II, other than the Kerala School. He wrote the Ganita Kaumudi in 1356 about mathematical operations. One of the unusual features of Narayana‘s work Karmapradipika is that he gave seven methods of squaring numbers which are not found in the work of other Indian mathematicians. Narayana also gave a rule to calculate approximate values of a square root. He did this by using an indeterminate equation of the second order, Nx2 + 1 = y2, where N is the number whose square root is to be calculated. If x and y are a pair of roots of this equation with x < y then √N is approximately equal to y/x. To illustrate this method Narayana takes N = 10. He then finds the solutions x = 6, y = 19 which give the approximation 19/6 = 3.1666666666666666667, which is correct to 2 decimal places. Narayana then gives the solutions x = 228, y = 721 which give the approximation 721/228 = 3.1622807017543859649, correct to four places. Finally Narayana gives the pair of solutions x = 8658, y = 227379 which give the approximation 227379/8658 = 3.1622776622776622777, correct to eight decimal places. Note for comparison that √10 is, correct to 20 places, 3.1622776601683793320. Narayana made minor contributions to the ideas of differential calculus found in Bhaskara II‘s work. Narayana has also made contributions to the topic of cyclic quadrilaterals. Narayana is also credited with developing a method for systemic generation of all permutations of a given sequence. Naryana‘s Ganita Kaumudi contains a detailed discussion of magic squares and similar figures. Narayana gave the rules for the formation of doubly even, even and odd perfect magic squares along with magic triangles, rectangles and circles. He used formulae and rules for the relations between magic squares and arithmetic series. He gave methods for finding ―the horizontal difference‖ and the first term of a magic square whose square‘s constant and the number of terms are given and he also gave rules for finding ―the vertical difference‖ in the case where this information is given. 1.6.10 Srinivasa Aiyangar Ramanujan (1887 A.D. – 1920 A.D.) Srinivasa Ramanujan was one of India‘s mathematical geniuses. He made wonderful contributions to the field of advanced mathematics. Srinivasa Aiyangar Ramanujan is undoubtedly the most celebrated Indian Mathematical genius. He was born at Erode in Tamil Nadu on December 22, 1887. Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quartic. The following year, not knowing that the quintic could not be solved by radicals, he tried (and of course failed) to solve the quintic. By 1904 Ramanujan had begun to undertake deep research. He investigated the series ∑(1/n) and calculated Euler‗s constant to 15 decimal places. He began to study the Bernoulli numbers, although this was entirely his own independent discovery. Continuing his mathematical work Ramanujan studied continued fractions and divergent series in 1908. At this stage he became seriously ill again and underwent an operation in April 1909 after which he took him some considerable time to recover. On 14 July 1909, he married a nine-year-old girl his mother arranged for him. However, Ramanujan did not live with his wife until she was 12-years-old. Ramanujan continued to develop his mathematical ideas and began to pose problems and solve problems in the Journal of the Indian Mathematical Society. He developed relations between elliptic modular equations in 1910. After publication of a brilliant research paper on Bernoulli numbers in 1911 in the Journal of the Indian Mathematical Society he gained recognition for his work. Despite his lack of a university education, he was becoming well known in the Madras area as a mathematical genius. During this period, he published many papers and was becoming well known in Chennai as a mathematical genius. In 1913, while he worked as a clerk in the Indian Mathematical Society, Ramanujan wrote to Cambridge mathematician, GH Hardy, and told him about his work. He had read Hardy‘s 1910 book Orders of Infinity. Ramanujan‘s name will always be linked to Godfrey Harold Hardy, a British mathematician. It is not because Ramanujan worked with Hardy at Cambridge but it was Hardy who made it possible for Ramanujan to go to Cambridge. In 1914, Professor G. H. Hardy brought Ramanujan to Trinity College, Cambridge, to begin an extraordinary collaboration. Ramanujan sailed from India on 17 March 1914. It was a calm voyage except for three days on which Ramanujan was seasick. He arrived in London on 14 April 1914. On 16 March 1916 Ramanujan graduated from Cambridge with a Bachelor of Science by Research (the degree was called a Ph.D. from 1920). On 18 February 1918 Ramanujan was elected a fellow of the Cambridge Philosophical Society and then three days later, the greatest honour that he would receive; his name appeared on the list for election as a fellow of the Royal Society of London. He had been proposed by an impressive list of mathematicians, namely Hardy, McMahon, Grace, Larmor, Bromwich, Hobson, Baker, Littlewood, Nicholson, Young, Whittaker, Forsyth, and Whitehead. His election as a fellow of the Royal Society was confirmed on 2 May 1918, and then on 10 October 1918 he was elected a Fellow of Trinity College Cambridge, the fellowship to run for six years. During an illness in England, Hardy visited Ramanujan in the hospital. When Hardy remarked that he had taken taxi number 1729, a singularly unexceptional number, Ramanujan immediately responded that this number was actually quite remarkable it is the smallest integer that can be represented in two ways by the sum of two cubes 1729=1³+12³=9³+10³. J. E. Littlewood was so much impressed on the work of Ramanujan. In his words ―Every positive integer is one of Ramanujan‘s personal friends‖. Bertrannd Arthur William Russell (1872-1970) wrote to Lady Ottoline Morell. ―I found Hardy and Littlewood in a state of wild excitement because they believe; they have discovered a second Newton, a Hindu Clerk in Madras… He wrote to Hardy telling of some results he has got, which Hardy thinks quite wonderful‖. Ramanujan was a mathematical genius in his own right on the basis of his work alone. He worked hard like any other great mathematician. He had no special, unexplained power. As Hardy wrote, ―I have often been asked whether Ramanujan had any special secret; whether his methods differed in kind from those of other mathematician; whether there was anything really abnormal in his mode of thought. I cannot answer these questions with any confidence of conviction; but I do not believe it. My belief that all mathematician think at bottom, in the same kind of way and that Ramanujan was no exception.‖ The Norwegian mathematician Atle Selberg, one of the great number theorists of this century wrote; ―Ramanujan‘s recognition of the multiplicative properties of the coefficients of modular forms that we now refer to as cusp forms and his conjectures formulated in this connection and their later generalization, have come to play a more central role in mathematics of today, serving as a kind of focus for attention of quite a large group of the best mathematicians of our time. Other discoveries like the mock-theta functions are only in the very early stages of being understudy and no one can yet assess their real importance. So the final verdict is certainly not in and it may not be in for a long time, but the estimates of Ramanujan‘s nature in mathematics certainly have been growing over the years. There is no doubt about that.‖ Often people tend to speculate what Ramanujan would have achieved if he had not died or if his exceptional qualities were recognized at the very beginning. There are many instances of such untimely death of gifted persons, or rejection of gifted persons by the society or rigid educational system. In mathematics we may cite the cases of Niels Henrik Abel (1809-1829) and Evarista Galois (1811-1832). Abel solved one of the great mathematical problems of his day – finding a general solution for class equations called quintiles. Abel solved the problem by proving that such a solution was impossible. Galois pioneered the branch of modern mathematics known as group theory. What is important is that we should recognize the greatness of such people and take inspiration from their work. 1.6.11 D.R. Kaprekar (1905 A.D. – 1988 A.D.) Dattaraya Ramchandra Kaprekar was an Indian mathematician who discovered several results in number theory, including a class of numbers and a constant named after him. Despite having no formal postgraduate training and working as a schoolteacher, he published extensively and became well known in recreational mathematics circles. D. R. Kaprekar was born in Dahanu, a town on the west coast of India about 100 km north of Mumbai. He was brought up by his father after his mother died when he was eight years old. His father was a clerk who was fascinated by astrology. Although astrology requires no deep mathematics, it does require a considerable ability to calculate with numbers, and Kaprekar‘s father certainly gave his son a love of calculating. Kaprekar received his secondary school education in Thane and studied at Fergusson College in Pune. In 1927 he won the Wrangler R. P. Paranjpe Mathematical Prize for an original piece of work in mathematics. He attended the University of Mumbai, receiving his bachelor‘s degree in 1929. Having never received any formal postgraduate training, for his entire career (1930–1962) he was a schoolteacher at Nashik in Maharashtra, India. He published extensively, writing about such topics as recurring decimals, magic squares, and integers with special properties. Kaprekar had the fascination for numbers as a child continued throughout his life. He was a good school teacher, using his own love of numbers to motivate his pupils, and was often invited to speak at local colleges about his unique methods. He realized that he was addicted to number theory and he would say of himself – ―A drunkard wants to go on drinking wine to remain in that pleasurable state. The same is the case with me in so far as numbers are concerned”. Many Indian mathematicians laughed at Kaprekar‘s number theoretic ideas thinking them to be trivial and unimportant. Working largely alone, Kaprekar discovered a number of results in number theory and described various properties of numbers. In addition to the Kaprekar constant and the Kaprekar numbers which were named after him, he also described self numbers or Devlali numbers, the Harshad numbers and Demlo numbers. He is well known for ―Kaprekar Constant 6174‖ which was discovered in 1949. Take any four digit number in which all digits are not alike. Arrange its digits in descending order and subtract from it the number formed by arranging the digits in ascending order. If this process is repeated with reminders, ultimately number 6174 is obtained, which then generates itself. Kaprekar described another class of numbers that are known as the Kaprekar numbers. A Kaprekar number is a positive integer with the property that if it is squared, then its representation can be partitioned into two positive integer parts whose sum is equal to the original number (e.g. 45, since 452=2025, and 20+25=45, also 9, 55, 99 etc.) However, note the restriction that the two numbers are positive; for example, 100 is not a Kaprekar number even though 1002=10000, and 100+00 = 100. He also constructed certain types of magic squares related to the Copernicus magic square. Initially his ideas were not taken seriously by Indian mathematicians, and his results were published largely in low-level mathematics journals or privately published, but international fame arrived when Martin Gardner wrote about Kaprekar in his March 1975 column of Mathematical Games for Scientific American. Today his name is well-known and many other mathematicians have pursued the study of the properties he discovered. 1.7 Correlation between Mathematics and Other Subjects Mathematics is the queen of all sciences‘ – those are the words of Carl Friedrich Gauss the greatest mathematician of all time. Mathematics is an important tool for science. Math is most widely used in other sciences. Physics, Chemistry, astronomy, engineering rely most heavily upon mathematical ideas. Students who consider studying Physics or Chemistry will need a relatively strong Math background. 1.7.1 Mathematics and Physics Physics and mathematics have always enjoyed a close relationship, beginning in the Renaissance with Johannes Kepler‘s (1571-1630) in 1609 discovery of the three laws of planetary orbits. In 1687 Isaac Newton (1642-1727) introduced the theory of gravity. James Clerk Maxwell (1831-1879) was able to unify the forces of electricity and magnetism in 1865 with the theory of electromagnetism. In the twentieth century mathematical theories from the fields of geometry were instrumental in constructing Albert Einstein‘s (1879-1955) theory of general relativity as well as in the later development of superstring theory. All of these theories have been predicated upon the prior development of mathematical techniques that had been invented for pure and applied purposes. In the late seventeenth century Isaac Newton could not have developed the theory of gravity without calculus, a set of mathematical techniques he had developed for studying rates of change. Physics is the fundamental study of Nature, in other words, in physics we want to find out how stuff works. Mathematics, on the other hand, is harder to pin down since it exists only in the human mind. Mathematics epitomizes the word ―abstract‖. But still this can‘t be the complete story because many mathematical structures can be used to describe how physical objects interact and the nature of the relationships among them. In physics we use mathematics as a tool to understand Nature. In mathematics, the pure notions of numbers and other structures do not need physics to exist or explain or even justify them. But the surprising thing is that often some newly discovered abstract formulation in mathematics turns out, years later, to describe physical phenomena which we hadn‘t known about earlier. When, as a student of physics, you see this for the first time, it is truly overwhelming. Since mathematics is a product of our imagination, then somehow the structure of the universe itself seems to be imprinted on the human mind. And if that is the case, the relationship between mathematics and physics does indeed boil down to the chicken-and-the-egg question. Mathematics and Physics have always been closely interwoven, in the sense of a two-ways process i) Mathematical methods are used in Physics. That is, Mathematics is not only the language of Physics (i.e. the tool for expressing, handling and developing logically physical concepts and theories), but also, it often determines to a large extent the content and meaning of physical concepts and theories themselves. ii) Physical concepts, arguments and modes of thinking are used in Mathematics. That is, Physics is, not only a domain of application of Mathematics, providing it with problems ―ready-to-be- solved‖ mathematically by already existing mathematical tools. It also provides ideas, methods and concepts that are crucial for the creation and development of new mathematical concepts, methods, theories, or even whole mathematical domains. Any distinction between Mathematics and Physics, seen as general attitudes towards the description and understanding of an (empirical, or mental) object, is related more to the point of view adopted while studying particular aspects of this object, than to the object itself. 1.7.2 Mathematics and Chemistry Chemistry is the natural science which explores the composition and properties of substances. Math is essential for chemistry. The necessary mathematical background for the study of chemistry includes basic algebra, some trigonometry, and calculus. The following are some examples i) Being able to balance chemical equations is a very important skill for chemistry students. It‘s a simple mathematical exercise. Balancing a chemical equation refers to establishing the mathematical relationship between the amounts of reactants and products involved in the chemical reaction. Let‘s go more in detail. A chemical equation is a statement that describes what happens in a chemical reaction. In a chemical equation, we place the reactants (substances undergoing chemical reaction) on the left side of the equation and the products (substances produced in a chemical reaction) on the right side of the equation. We have reactants and products separated by an arrow and the arrow always points in the direction of the products. Consider the reaction of carbon with oxygen gas to produce carbon-dioxide.

C + O2 ---> CO2 (2 is subscript) The above equation is already balanced, because, it has an equal number of atoms of each element in the reactants and the product. One carbon atom (C) and two oxygen atoms (O) on the left side of the equation and it‘s the same on the right side too. Let‘s look at one more example Sodium chloride is the common salt. Sodium and chlorine form sodium chloride.

Na + Cl2 ---> NaCl (2 is subscript) The above equation is NOT balanced. It has two chlorine atoms on the left side, but, only one on the right side of the equation. Let‘s balance this chemical equation.

2Na + Cl2 ---> 2NaCl (2 is subscript only in Cl2) It works! Notice that now there are equal number of atoms of each element in the reactants and the product. Chemical equations can be balanced conveniently using matrices or simultaneous equations. A number of fields of chemistry use a significant amount of Mathematics. ii) Electrochemistry is a branch of chemistry that studies the chemical action of electricity and the production of electricity by chemical reactions. Diffusion in electrochemistry is completely based on differential equations. iii) Biochemistry is the study of the chemical processes in living organisms. Even biochemistry has important topics which depend heavily on binding theory and kinetics. Chemistry involves many calculations, and calculations require math. That‘s pretty much the relationship between the two. You can‘t do chemistry without math, but you can do math without chemistry. Relation of mathematics with chemistry is just as cake without cream. It starts with atomic number, mass and ends with nuclear energy. It is useful in rate of chemical equations, precision, speed of atoms, there different type energies. It is dominantly useful in physical chemistry e.g. calculating molarity, molality, normality, ppm, osmosis. How can we forget periodic table based on atomic number? Thus mathematics is necessary for chemistry to exist. 1.7.3 Mathematics and Biology The benefits of interdisciplinary collaborations and the need for researchers to read and think broadly have become truisms of today‘s scientific world. In this special issue of Science, as well as in Science‘s online sites, our keen-sighted contributors report on gleanings from their travels through the lands of mathematics and biology. The primary purpose for encouraging biologists and mathematicians to work together is to investigate fundamental problems that cannot be approached only by biologists or only by mathematicians. If this effort is successful, future years may produce individuals with biological skills and mathematical insight and facility. At this time such individuals are rare; it is clear, however, that a greater percentage of the training of future biologists must be mathematically oriented. Both disciplines can expect to gain by this effort. Mathematics is the ―lens thro ugh which to view the universe‖ and serves to identify the important details of the biological data and suggest the next series of experiments. Mathematicians, on the other hand, can be challenged to develop new mathematics in order to perform this function. The interaction between biology and mathematics has been a rich area of research for more than a century. Statistics and stochastic processes have their origins in biological questions. Galton invented the method of correlation, motivated by questions in evolutionary biology. Fisher‘s work in agriculture led to the analysis of variance. The attempt to model the success (survival) over many generations of a family name led to the development of the subject of branching processes; more recently, the compilation of DNA sequence data led to Kingman‘s coalescence model and Ewens‘ sampling formula. In the area of classical applied mathematics, biological applications have stimulated the study of ordinary and partial differential equations fundamentally, especially regarding problems in chaos, pattern formation, and bifurcation theory. In molecular biology, mathematical and algorithmic developments have allowed important insights, for example, recognition of the unexpected homology between a non co gene product and a growth factor that forms the basis of the molecular theory of carcinogenesis. Statistical link age analysis helped locate the cystic fibrosis gene. An understanding of the topology of DNA has been enhanced greatly by the close cooperation of biologists and mathematicians. Classical analysis has played a central role in image reconstruction. Radon‘s techniques, first developed in 1917, formed the centerpiece of computerized axial tomography that led to a Nobel Prize in 1979. 1.7.4 Mathematics and Geography Geography (from Greek – geografia) is the study of the earth and its features, inhabitants, and phenomena. A literal translation would be ―to describe or write about the Earth‖. The first person to use the word ―geography‖ was Eratosthenes (276-194 B.C.). Geography is the study of man and his environment. The environment refers to our surroundings and things found in them both physical and cultural which may differ from place to place. It is important to study our environment so that we understand and maintain it. Mathematics and Geography can be linked each other in terms of locating the axis of certain geographical places. Geography makes use of various mathematical concepts such as trigonometry, theories related to vectors, x-axis, y-axis coordinates etc. to facilitate in obtaining useful information and for the further analysis of data. The knowledge of mathematics allow the geographers to study the surface of earth, its mass, analysis of population, analysis of earth, characteristics and identification of different patterns in the same geographical regions etc. Sometimes mathematics and geography are taught as single subject to the students of geography to learn the mathematical tools which facilitate learning important dimensions of geography. Mathematics helps in calculating the various aspects of geography like frequencies of repetition of patterns in surface and soil. It also helps in geography prediction. For geographers, there must be an answer to the What, How, Why and the How Much. To coordinate all of the desired answers in its entirety, a geographer has to be a systematic and an intuitive thinker. This mixture is definitely not in the way at the application of mathematics in systematic thinking. So many values are passed over in geographical research because of a lack of emphasis in a more quantitative study. The earth and its simple mathematical relations are not sufficiently well understood by geographers. Literature about the subject is scarce and far too dispersed in subject matter of non- geographic fields of study. Generally it is accepted, that mathematics is the most difficult division of geography, but it is becoming a very indispensable discipline. Qualitative analysis alone cannot answer all the questions posed by a modern society with a modern industry and a dynamic activism. The divisions or portions of geography that need a mathematical background to insure an orderly comprehension are (a.) Form and Shape of the Earth (b.) Movements of the Earth and its immediate gravitational and electromagnetic relations. (c.) Elements of longitude and variables of time determination. (d.) Cartography and Map interpretation (e.) Climatology (f.) Physiography. Mathematics is the study of numbers while in Geography you use mathematics to calculate time, distance and so on. A passive knowledge of mathematics is not enough; it must be activated and brought up to date and in accordance with the modern contemporary scientific concepts of the physical world. Let it be clear that this is not an attempt to make a mathematical science out of geography, for it cannot be done as it is obvious to any geographer. 1.8 Let Us Sum Up This unit provides an opportunity to the students to understand the characteristics of mathematics. What is mathematics? Students may able to give the proper answer about it. Mathematics is a creation of human mind concerned chiefly with ideas, processes and reasoning. Mathematics is much more than Arithmetic, more than Algebra more than Geometry, much more than Trigonometry, Statistics, and Calculus. Students can understand the contribution of Indian mathematicians in the field of mathematics. Aryabhatta, Varahamihira, Brahmagupta, Bhaskara I, Mahavira, Sridhara, Aryabhata II, Bhaskara II or Bhaskaracharya, Narayana Pandita, Srinivasa Aiyangar Ramanujan and D.R. Kaprekar are the most eminent Indian mathematicians whose contributions are incredible in the field of mathematics. Students are also capable to have knowledge of correlations of mathematics with Physics, Chemistry, Biology and Geography. 1.9 Answers to Check Your Progress i) What is mathematics? ii) Explain the characteristics of mathematics. iii) Write an essay on history of mathematics. iv) Write the contribution of two Indian mathematicians. v) What is the correlation of mathematics with physics and chemistry? vi) What is the correlation of mathematics with biology and geography? 1.10 References D. G. Dhavale, (1974). The Kapitthaka of Varahamihira, in Proceedings of the Symposium on Copernicus and Astronomy, New Delhi, 1973, Indian J. History Sci. 9 (1), 77-78; 141. P. K. Majumdar, (1981). A rationale of Brahmagupta‘s method of solving ax + c = by, Indian J. Hist. Sci. 16 (2), 111-117. P. Jha, (1982). Aryabhata I and the value of π, Math. Ed. (Siwan) 16(3), 54-59. K S Shukla, (1977). The Panca-siddhantika of Varahamihira. II, Ganita 28, 99-116. K S Shukla, (1973). The Pancasiddhantika of Varahamihira. I (errata insert), Ganita 24 (1), 59-73. Alladi, Krishnaswami (1998). Analytic and Elementary Number Theory A Tribute to Mathematical Legend Paul Erdos. Norwell, Massachusetts Kluwer Academic Publishers. p. 6. ISBN 0-7923-8273-0. Vagiswari, A. (2007). ―Aryabhata II‖. In Thomas Hockey et al (eds.). The Biographical Encyclopedia of Astronomers. New York Springer, p. 64, ISBN 978-0-387-310022-0. K Shankar Shukla, On Sridhara‘s rational solution of Nx2+1=y2, Ganita 1 (1950), 1-12. R C Gupta, Narayana‘s method for evaluating quadratic surds, Math. Education 7 (1973), B93-B96. Kim Plofker (2009), Mathematics in India 500 BCE–1800 CE, Princeton, NJ Princeton University Press, ISBN 0-691-12067-6. Dilip M. Salwi (2005-01-24). ‖Dattaraya Ramchandra Kaprekar‖. Retrieved 2007-11-30.

UNIT 2 AIMS, OBJECTIVES AND VALUES OF TEACHING MATHEMATICS Structure 2.1 Introduction 2.2 Objective 2.3 Aims of Teaching Mathematics 2.4 Objectives of Teaching Mathematics 2.5 Values of Teaching Mathematics 2.6 Need and importance of objective based teaching of Mathematics 2.6.1 Importance of objective based teaching of Mathematics 2.6.2 Need of objective based teaching of Mathematics 2.7 Specification of Objectives 2.8 Let Us Sum Up 2.9 Answers to Check Your Progress 2.10 References 2.1 Introduction Mathematics is crucial not only for success in school, but in being an informed citizen, being productive in one‘s chosen career, and in personal fulfillment. In today‘s technology driven society, greater demands have been placed on individuals to interpret and use mathematics to make sense of information and complex situations. What are the aims, objectives and values of teaching of mathematics? This unit gives the clear-cut ideas about the aims, objectives and values of teaching of mathematics. 2.2 Objectives On completion of this unit students will be able to …. 6. understand the aims and objectives of the teaching of mathematics 7. realize the practical, disciplinary, cultural and social value of teaching mathematics 8. understand the needs and importance of objective based teaching of mathematics 9. know the specification of objectives. 2.3 Aims of Teaching Mathematics The meaning of mathematics teaching at regular school cannot be just the mediation of mathematics. The main task rather is the development and education of young men. Thus general aims have to be in the fore. Normally you find such general aims in the guidelines or in the preface of the curriculum- guide. In the concrete plan of the lessons under the heading ―objectives‖ you mostly find only subject orientated aims. Thus the general aims are often out of the mind of the teacher. In addition the assessment of general aims is difficult. The aims of teaching and learning mathematics are to encourage and enable students to: i) recognize that mathematics permeates the world around us ii) appreciate the usefulness, power and beauty of mathematics iii) enjoy mathematics and develop patience and persistence when solving problems iv) understand and be able to use the language, symbols and notation of mathematics v) develop mathematical curiosity and use inductive and deductive reasoning when solving problems vi) become confident in using mathematics to analyze and solve problems both in school and in real- life situations vii) develop the knowledge, skills and attitudes necessary to pursue further studies in mathematics viii) develop abstract, logical and critical thinking and the ability to reflect critically upon their work and the work of others ix) develop a critical appreciation of the use of information and communication technology in mathematics x) appreciate the international dimension of mathematics and its multicultural and historical perspectives. xi) promote enjoyment and enthusiasm for learning through practical activity, exploration and discussion; xii) promote confidence and competence with numbers and the number system; xiii) to develop the ability to solve problems through decision-making and reasoning in a range of contexts; xiv) develop a practical understanding of the ways in which information is gathered and presented; xv) explore features of shape and space, and develop measuring skills in a range of contexts; xvi) understand the importance of mathematics in everyday life. 2.4 Objectives of Teaching Mathematics Mathematics teaches us how to make sense of the world around us through developing a child‘s ability to calculate, to reason and to solve problems. Mathematics has got many objectives of progressive life which determine the need of teaching the subject in schools. The objectives of teaching mathematics can be studied under the following heads: A. Knowledge and understanding B. Investigating patterns C. Communication in mathematics D. Reflection in mathematics A. Knowledge and Understanding Knowledge and understanding are fundamental to studying mathematics and form the base from which to explore concepts and develop problem solving skills. Through knowledge and understanding students develop mathematical reasoning to make deductions and solve problems. Students should be able to: i) know and demonstrate understanding of the concepts from the five branches of mathematics (number, algebra, geometry and trigonometry, statistics and probability, and discrete mathematics) ii) use appropriate mathematical concepts and skills to solve problems in both familiar and unfamiliar situations including those in real-life contexts iii) select and apply general rules correctly to solve problems including those in real- life contexts. B. Investigating Patterns Investigating patterns allows students to experience the excitement and satisfaction of mathematical discovery. Mathematical inquiry encourages students to become risk- takers, inquirers and critical thinkers. The ability to inquire is invaluable and contributes to lifelong learning. Through the use of mathematical investigations, students are given the opportunity to apply mathematical knowledge and problem- solving techniques to investigate a problem, generate and/ or analyze information, find relationships and patterns, describe these mathematically as general rules, and justify or prove them. At the end of the course, when investigating problems, in both theoretical and real- life contexts, student should be able to: i) select and apply appropriate inquiry and mathematical problem solving techniques ii) recognize patterns iii) describe patterns as relationships or general rules iv) draw conclusions consistent with findings v) justify or prove mathematical relationships and general rules. C. Communication in Mathematics Mathematics provides a powerful and universal language. Students are expected to use mathematical language appropriately when communicating mathematical ideas, reasoning and findings — both orally and in writing. At the end of the course, students should be able to communicate mathematical ideas, reasoning and findings by being able to: i) use appropriate mathematical language (notation, symbols, terminology) in both oral and written explanations ii) use different forms of mathematical representation (formulae, diagrams, tables, charts, graphs and models) iii) move between different forms of representation. Students are encouraged to choose and use ICT tools as appropriate and, where available, to enhance communication of their mathematical ideas. ICT tools can include graphic display calculators, screenshots, graphing, spread sheets, data bases, and drawing and word- processing software. D. Reflection in Mathematics Mathematics encourages students to reflect upon their findings and problem- solving processes. Students are encouraged to share their thinking with teachers and peers and to examine different problem- solving strategies. Critical reflection in mathematics helps students gain insight into their strengths and weaknesses as learners and to appreciate the value of errors as powerful motivators to enhance learning and understanding. On the basis of the above discussion we conclude that the objective of teaching mathematics which can be given in the following form: i) develop a positive attitude towards learning mathematics ii) perform mathematical operations and manipulations with confidence, speed and accuracy iii) think and reason precisely, logically and critically in any given situation iv) develop investigative skills in mathematics v) identify, concretize, symbolize and use mathematical relationships in everyday life vi) comprehend, analyze, synthesize, evaluate, and make generalizations so as to solve mathematical problems vii) Collect, organize, represent, analyze, interpret data and make conclusions and predictions from its results viii) apply mathematical knowledge and skills to familiar and unfamiliar situations ix) appreciate the role, value and use of mathematics in society x) develop willingness to work collaboratively xi) acquire knowledge and skills for further education and training xii) communicate mathematical ideas 2.5 Values of Teaching Mathematics At present there is little knowledge about what values teachers are teaching in mathematics classes, about how aware teachers are of their own value positions, about how these affect their teaching, and about how their teaching there by develops certain values in their students. Values are rarely considered in any discussions about mathematics teaching, and a casual question to teachers about the values they are teaching in mathematics lessons often produces an answer to the effect that they don‘t believe they are teaching any values. Values in mathematics education are the deep affective qualities which education aims to foster through the school subject of mathematics and are a crucial component of the classroom affective environment. As a result of demands that students become more economically oriented and globally conscious, mathematics educators are being challenged about which values should be developed through mathematics education. Our concern is that, although values teaching and learning inevitably happen in all mathematics classrooms, they appear to be mostly implicit. Mathematics has got many educational values which determine the need of teaching the subject in schools. These values can be studied under the following heads: A. Practical Value B. Disciplinary Value C. Cultural Value D. Social Value A. Practical Value Mathematics has great practical value. Everyone uses some mathematics in every form of life. A common man sometimes can do without reading or writing but he cannot do without counting and calculating. Any person who is ignorant of mathematics can be easily cheated. He will always be the mercy of others. We have to make purchase s daily. We buy cloth, food items, fruit, vegetables, grocery etc. We have to calculate how much we have to pay for everything. A house-wife also needs mathematics for looking after her house, preparing family budgets and estimates, writing various expenses and noting down various household transactions. Mathematics is needed by all of us whether rich or poor, high or low. Not to speak of engineers, bankers, accountants, businessmen, planners etc., even petty shopkeepers, humble coolies, carpenter and labourers need mathematics not only for earning their livelihood but also to spend wisely and save for future. Whoever earns and spends uses mathematics. We are living in a world of measurements. We have to measure lengths, are as, volumes and weights. We have to fix timings, prices, wages, rates, percentages, targets, exchanges etc. In the absence of these fixations, the life in the present complex society will come to a standstill. There will be utter confusion and chaos. Just think if a fairy descends on earth and removes all mathematics. There will be no calendar, no maps, no accounts, no fixations or measurements, no industrial activity, no plans or projects. Thus we see that mathematics has tremendous value or application in our daily life. It is essential for leading a successful social life. B. Disciplinary Value Mathematics trains or disciplines the mind also. It develops thinking and reasoning power. According to Locke, ―Mathematics is a way to settle in the mind a habit of reasoning.‖ Mathematics is ‗an exact and definite science‘. Every student of mathematics has to reason properly without any prejudices or unnecessary biases. Reasoning in mathematics has the characteristics of simplicity, accuracy, objectivity, originality etc. Besides reasoning, mathematics has the following disciplinary values also. i) Development of the power of concentration. The faculty to concentrate one‘s mind can only be learnt by the study of mathematics. ii) Development of inventive faculty. The study of mathematics develops inventive faculty of the students. The solving of a difficult problem in mathematics is just like making a discovery. iii) Will power. Mathematics develops patience and perseverance in the students. It strengths their will power In addition to practical, cultural and disciplinary values, mathematics has so many other values. Mathematics teaches the art of economical living. It teaches economy in time, speech, thought and money. Thus we see that mathematics has many educational values which show the increasing importance of the subject in schools and in social life. C. Cultural Value Mathematics has got a great cultural value which is steadily increasing day by day. Mathematics has made a major contribution to our cultural advancement. The progress of our civilization has been mainly due to the progress of various occupations such as agriculture, engineering, industry, medicine, navigation, rail road building etc. These occupations build up culture. Mathematics makes direct or indirect contribution to the development of all occupations. Hogben says, ―Mathematics is the mirror of civilization‖. The history of mathematics shows how mathematics has influenced civilization and culture at a particular time. Progress in mathematics, of Greeks and Egyptians in the past led to their cultural advancement and the progress of their civilization. Mathematics is a pivot for cultural arts such as music, fine arts, poetry and painting. Perhaps that is why the Greeks, who were the greatest geometers of their times, were quite a dept in fine arts. All knowledge is related to our experience in the social and cultural worlds that we inhabit, and all knowledge comes to us as it passes through social and cultural systems and institutions through the socializing of norms, values, conventions, and practices. Some have even argued that the dichotomy between ―every day‖ and ―school‖ mathematics is false (Moschkovich, 2007). It is also true that knowledge is not neutral with respect to power— some types of knowledge are more aligned with communities of practice that hold more power, whereas other types of knowledge are more aligned with communities of practice that have less power. When viewed through this lens, any discussion of boundaries between mathematical knowledge and cultural knowledge must respect that these issues of power are implicated in our definitions, issues of concern, and the very conversation in which we are engaged through our scholarship. Furthermore, knowledge is fundamentally tied to the kinds of people we (and others) view us to be and the trajectory we (and others) view ourselves to be on. In other words, issues of identity are critical to understanding both the development of mathematical knowledge for individuals and communities but also to considering how we draw lines between cultural and domain knowledge. Mathematics is an intimate historical part of most cultures. Further mathematical systems and concepts, like all knowledge, have an intrinsic beauty of their own which makes them worthy of study for their own sake. No further justification is required and no liberal education can be considered complete without it. D. Social Value Social values of teaching mathematics are taken as ways of understanding that are embedded in rational logic - focusing on universal knowledge statements. They are seen by society in general as essential components of schooling and the social interactions. Hence, teachers face substantial political and social pressures from outside the school (e.g. system-wide assessments of student performance, purposes for teaching seen as being directly related to technological development, etc.). In teaching of mathematics, the social values are clearly seen in the involvement of teaching learning process. Yet mathematics plays a much more prominent role as a gatekeeper in society than does science. For example, it is often used as a selection device for entry to higher education or employment, even when the skills being tested are unrelated to the ultimate purpose. In broad terms (e.g. modelling or simulations which reduce costs and/or danger), mathematics is considered to be publicly important; at the very same time as it is considered to be personally irrelevant (Niss, 1994), apart from the obvious examples of cooking, shopping and home maintenance. Politically, mathematics has been ascribed a formatting role in society (Skovsmose, 1994). Values which favour or support the social group or society and which concern the individual‘s duty to society as related to mathematics education. Examples from this category are co-operation, justice and appreciation of the beauty of mathematics. 2.6 Need and Importance of Objective Based Teaching of Mathematics Mathematics is the queen of science and the language of nature. Its importance should be clear to any reasonable person. It is easy however to diminish the value of certain areas of research because they‘re currently thought as having little practical use. Evolutionary needs brought our mind to prefer knowledge that can be employed for the solution of specific problems in the real world, rather than deeply abstract ones. 2.6.1 Importance of Objective Based Teaching of Mathematics On the net I found an incredible lecture by the brilliant mathematician Timothy Gowers, entitled ―The Importance of Mathematics‖. In this keynote, Prof. Gowers makes a very strong case in favor of the value of math, of financing its relatively cheap research and its deep implication on human progress. Mathematics is a magnificent subject. The importance of Mathematics which will be quality math help for students are as follows: i) Mathematics is a tool for the subjects like Physics and chemistry in higher secondary and above. ii) Nothing ca n be done in Architecture and Designing without the knowledge of Mathematics. iii) It enables students to interact with numbers. iv) Business is all about making money. v) If one I well versed in basic Mathematics there‘s no need for him to be having a personal auditor for keeping accounts of his income and outcome. vi) Auditors must be avoided as they can cheat you very easily if you are a duffer in Mathematics. vii) All the constructions on earth require mathematics. 2.6.2 Need of Objective Based Teaching of Mathematics Mathematics is an understandable and even excusable fallacy that there are useful field s of math and useless ones, based on the perception of their applied or theoretical nature. But it‘s still a misconception. Each theorem and discovery is a little piece of a larger puzzle that we conveniently categorize into aptly labeled macro - areas. Discoveries and mathematical ideas that are perceived as ―useful‖ to day because they‘re applicable to engineering, for example, were at a certain point in time considered absolutely abstract and useless, or at least derived or intrinsically connected to some that were. Mathematics matters all of it. Mathematics applications in our day to day life are very important. The basic needs of Mathematics education for students are as follows: A. For Life Enhancement The everyday use of arithmetic and the display of information by means of graphs are an everyday commonplace. These are the elementary aspects of mathematics. Advanced mathematics is widely used, but often in an unseen and unadvertised way. The mathematics of error-correcting codes is applied to CD players and to computers. The stunning pictures of far away planets sent by Voyager II could not have had their crispness and quality without such mathematics. Voyager‘s journey to the planets could not have been calculated without the mathematics of differential equations. Whenever it is said that advances are made with supercomputers, there has to be a mathematical theory which instructs the computer what is to be done, so allowing it to apply its capacity for speed and accuracy. The development of computers was initiated in this country by mathematicians and logicians, who continue to make import ant contributions to the theory of computer science. The next generation of soft ware requires the latest methods from what is called category theory, a theory of mathematical structures which has given new perspectives on the foundations of mathematics and on logic. The physical sciences (chemistry, physics, oceanography, astronomy) require mathematics for the development of their theories. In ecology, mathematics is used when studying the laws of population change. Statistics provides the theory and methodology for the analysis of wide varieties of data. Statistics is also essential in medicine, for analyzing data on the causes of illness and on the utility of new drugs. Travel by aero- plane would not be possible without the mathematics of airflow and of control systems. Body scanners are the expression of subtle mathematics, discovered in the 19th century, which makes it possible to construct an image of the inside of an object from information on a number of single X-ray views of it. Thus mathematics is often involved in matters of life and death. These applications have often developed from the study of general ideas for their own sake: numbers, symmetry, area and volume, rate of change, shape, dimension, randomness and many others. Mathematics makes an especial contribution to the study of these ideas, namely the methods of precise definitions; careful and rigorous argument; representation of ideas by many methods, including symbols and formulae, pictures and graphics; means of calculation; and the obtaining of precise solutions to clearly stated problems, or clear statements of the limits of knowledge. B. For Career Development Students who choose to ignore mathematics, or not take it seriously in high school, forfeit many future career opportunities that they could have. They essentially turn their backs on more than half the job market (see the areas listed below). The vast majority of university degrees require mathematics. The importance of mathematics for potential future careers cannot be more emphasized. For example, degrees in the following areas require good knowledge of Mathematics and Statistics: the physical sciences (like Chemistry, Physics, Engineering), the life and health sciences (like Biology, Psychology, Pharmacy, Nursing, Optometry), the social sciences (including Anthropology, Communications, Economics, Linguistics, Education, Geography) the tech sciences (like Computer Science, Networking, Software development), Business and Commerce, Actuarial science (used by insurance companies), Medicine.

2.7 Specification of Objectives Aims of teaching mathematics gives answer to the question ―why is mathematics taught?‖ when mathematics is taught and learnt by the pupils it is expected that evidence of learning should be available in their behavior. They are called terminal behavior. When aims are required to this level of specificity, they become objectives. Objectives signify what the subject of mathematics is trying to produce in pupils when it is taught. The term ‗aim‘ is used in a general sense and is a very broad and wide term whereas the term ‗objective‘ is used is used in a more explicit sense. Aims are general and long term goals while objectives are specific immediate and attainable goals. Aims may be common to more than one subject whereas objectives are specific to one subject. Aims are general but objectives are precise and clearly defined. In short, an objective is an end view of the possible achievement in terms of what a student is able to do when the whole educational system is directed towards educational aims. It is point of something towards which action is desired. In short, objectives are reliable goals. Values can be called ‗outcomes‘. Objectives should terminate outcomes. Instructional objectives are brief, clear statements that describe instructional intent in terms of the desired learning outcomes. These statements describe the terminal behavior of students and are to demonstrate in terms of changes in student behavior. The term ‗specification‘, ‗specific objective; and ‗behavioral objective‘ are used synonymously. A specification is a stated desirable outcome before the student undergoes the learning experience. It is too specific as to clear and therefore meaningful. Is should be tangible, observable and in a way measurable also. General instructional objectives are stated using terms as to know, to understand, to apply, to appreciable. More meaningful specific instructional objectives are stated as to compare, to discriminate, to identify, to illustrate, to verify, to classify, to differentiate. These words describe behavior that demonstrates the level of mastery reached by the learner. It is not enough if we are able to state objectives. We must be able to spell out specifically what the learner does in a given situation and how he does it when he was achieved a given objective. Dr. Benjamin S. Bloom who headed the taxonomy group has done a lot in analyzing the objectives into three domains, namely cognitive, affective and psychomotor. The cognitive domain includes those educational objectives which are related to the recall of knowledge and to the development of intellectual abilities and skills. The affective domain includes those objectives which involve changes in interests, attitude, values and appreciations. The psychomotor includes objectives related to the motor skill area. Bloom‘s Taxonomy (1976) guides in identifying and defining instructional objectives. Bloom‘s method of classification of objectives attempts to arrange the different classes of behaviors in hierarchical order from the simple to the complex. Behavior in class makes use of and builds on behavior in the preceding class. The Bloom‘s Taxonomy is in three parts: A) Cognitive domain emphasizes intellectual outcomes. B) Affective domain emphasizes feeling and emotion. C) Psychomotor (conative) domain emphasizes motor skills. A) Cognitive Domain Cognitive domain with ‗knowing‘ and includes activities such as remembering and recalling knowledge, thinking, problem solving, creating etc. it is in the cognitive domain that much work in curriculum development has been done. It is in this domain that the clearest definition of objective are it be found in terms of pupils‘ behavior. Cognitive taxonomy contains six major classes of objectives arranged in an hierarchical order on the basis of complexity of task and arranged from simple to complex behavior and from concrete to abstract behavior. Each of these six classes is sub divided further. The condensed version of the cognitive domain taxonomy is presented in the following. i) Knowledge: Knowledge as defined here includes those behaviors and test situations which emphasize the remembering, either by recognition or recall of ideas, materials of phenomena. The behavior expected of the student in a recall situation is very similar to the behavior he was expected to have during the original learning situation. In the learning situation the student is expected to store in his mind certain information and the behavior expected later is the remembering of this information. Knowledge is broken down into six types such as the following: 1. Knowledge is specifies the recall of specific and isolate bits of information. 2. Knowledge is terminology of reference for specific verbal and non verbal symbols. 3. Knowledge is specific facts of dates, events, persons, places sources of information, etc. 4. Knowledge is trends and sequences of the process, direction and movements of phenomena with respect to time. 5. Knowledge of criteria of the criteria by which facts, principles, techniques and procedures are employed in a particular subject. 6. Knowledge of theories and structures of the body of principles, generalization, together with their interrelations which present a clear, rounded and systematic view of a complex phenomenon, or field. ii) Comprehension: It is defined as the ability to grasp the meaning of material. The learning outcomes go one step beyond the simple understanding of materials. It represents the lowest level of understanding and includes translation, interpretation and extrapolation. iii) Application: It is the ability to use learnt material in new and concrete situation. Learning outcomes in this area requires a higher level of understanding than those under comprehension. iv) Analysis: It refers to the ability to breakdown material into its component parts so that the organizational structure may be understood. Learning outcomes here represents a higher intellectual level than comprehension and application because they require an understanding of both of content and structural form of the materials. This includes an analysis of elements, relationships and organizational principles. v) Synthesis: Synthesis refers to the ability to put together elements and parts of material so as to form a new whole. Learning outcomes in the area stress creative behaviors with major emphasis on the formulation of new pattern or structure. This includes the production of a unique communication of a plan or proposed sets of operations and derivation of a set of abstract relations. vi) Evaluation: Evaluation is concerned with the ability to judge the value of material (statement, novel, poem, research, report) for a given purpose. The judgments are to be based on different criteria. This includes estimating and verifying answers, criticizing proofs and judging the significance of a problem. B) Affective Domain Affective objectives emphasize a feeling or emotion or a degree of acceptance or rejection. It includes those objectives which deal with interests, attitudes, appreciations, values and emotional sets or biases to teach. Affective taxonomy is divided into five major classes arranged in a hierarchical order on the basis of the level of involvement. A condensed version of affective domain taxonomy in presented in the following. i) Receiving (attending): This means that the learner should be sensitized to the existence of certain phenomena and stimuli. This includes awareness, willingness to receive and controlled or selected affection. ii) Responding: This is concerned with responses that go beyond merely attending to phenomena. A person is actively involved in attending to them. This includes acquiescence in responding, willingness to respond and satisfaction in response. iii) Valuating: This includes acceptance of a value preference for a value and commitment to or a conviction in regret to certain point of view. iv) Organization: For situation where more than one value is relevant, the need arises to (a) the organization of the value into a system (b) the determination of the relationship among them, and (c) the establishment of the dominant and pervasive value. It includes conceptualization of a value and organization of a value system. C) Psychomotor Domain ―The psychomotor domain includes those objectives which deal with manual or motor skills.‖ A comprehensive taxonomy of objectives in the psychomotor domain is based on the concept of coordination between various muscular activities. Behavior which includes muscular action and requires neuromuscular coordination is grouped under the domain. The teacher‘s job is to provide such activities as may help develop neuromuscular coordination. The educational objectives tentatively proposed under this domain are: i) Imitation: Performance. ii) Manipulation of Acts: This includes differentiating among various movements and selecting proper one. iii) Precision in Reproducing given Acts: This includes accuracy, proportion and exactness in performance. iv) Articulation among Different Acts: This includes coordination sequence and harmony among acts. v) Naturalization: Here a pupil‘s skill attains its highest level of proficiency in performing an act with the least expenditure of psychic energy. Thus the behavior of a pupil is governed by his development in the three domains cognitive, affective and psychomotor. The cognitive domain is mainly concerned with the ability of a student to do a task, while the affective domain is concerned with the will or desire or attitude to do that task. Thus the cognitive and affective domains are the theoretical aspects of human behavior and the psychomotor domain is the practical aspect. The Department of Curriculum and Evaluation of the N.C.E.R.T. launched a programme of examination reform in 1969. Many boards of Secondary Education and a few Universities in India is participated this programme. They brought out a list of instructional objectives. The classification of instructional objectives in mathematics drawn by this team offers a detailed inventory of terminal behavior. Writing Instructional Objectives Gronlund has identified a number of verbs to state the ―Mathematical behavior‖. Behavioral objectives are selected beginning with a verb. Mager has added two more dimensions for stating behavioral objective, viz. the criteria of performance and the conditions under which the behavior is to occur. Here are some of the verbs that describe mathematical behavior.

Add Derive Group Prove

Bisect Estimate Integrate Reduce

Calculate Extrapolate Interpolate Solve

Check Extract Measure Tabulate

Computer Graph Plot Verify

Gronlund has also procedure for writing instructional objectives. Stating General Instructional Objectives Use the following suggestions as a guide to facilitate this procedure. i) Begin each general objective with a verb (knows, understand, appreciate, etc.). ii) State each objective in terms of student performance (other than teacher performance). iii) State each objective as a learning product (rather than in terms of the learning process). iv) State each objective as that it indicates terminal behavior (rather than the subject matter to be covered during instruction). v) State each objective so that it includes only one general learning outcome (rather than a combination of several outcomes). Stating Specific Learning Outcomes i) State general instructional objective as expected learning outcomes. ii) Place under each general instructional objective a list of specific learning outcomes that describe the terminal behavior of students are to demonstrate when they have achieved the objective. a) Begin each specific learning outcome with an action verb that specifies definite observable behavior. b) List a sufficient number of specific learning outcomes under each objective to describe adequately the behavior of students who have achieved the objective. iii) When defining the general instructional objectives in terms of specific learning outcomes revise and refine the original list of objectives as needed. iv) Be careful not to omit complex objectives (e.g. critical thinking appreciation) simply because they are difficult to define in specific behavioral terms. Objective: 1 The pupil acquires knowledge of terms, concepts, symbols, definitions, principles, processes and formulae of mathematics at the secondary stage. Specifications The pupil a) recalls, reproduces. b) recognizes

Objective: 2 The pupil develops understanding of terms, concepts, symbols, definitions, principles, processes and formulae of mathematics at the secondary stage. Specifications The pupil a) gives illustrations (b) detects error and corrects them (c) compares (d) discriminates between closely related concepts (e) classifies as per criteria (f) identifies relationship among the given data (g) translate verbal statement into symbolical statements and vice verse (h) estimates the results (i) interprets (j) verifies. Objective: 3 The pupil applies his knowledge and understanding of mathematics to unfamiliar situations (new problems). Specifications The pupil a) analyses and finds out what is given and what is required (b) finds out the adequacy superfluity or relevancy of data (c) establishes relationship among the data (d) selects the most appropriate method for solution of problems (e) suggests alternative methods (f) generalizes (i.e. reasons inductively) (g) infers (i.e. reasons deductively). Objective: 4 The pupil acquires skills in a) computation (b) drawing geometrical figures and graphs (c) reading tables, charts, graphs, etc. Specifications The pupil a) carries out oral calculations with ease and speed (b) carries out written calculations with ease and speed (c) draws geometrical figures and graphs (d) handles geometrical instruments with ease (e) measures accurately (f) draws free hand figure with ease (g) draws figure to specialization or to scales (h) draws figures accurately (i) reads tables, charts, graphs, etc. with speed and accuracy (j) interprets graphs. Objective: 5 The pupil appreciates the role of mathematics in day to day life. Specifications The pupil a) appreciate the role of mathematics in solving problems in other branches of science (b) appreciate the symmetry of figures, designs and pattern (c) appreciate the development of qualities like brevity and exactness through the study of mathematics. Objective: 6 The pupil develops interest in mathematics. Specifications a) reads literature on mathematics (b) writes popular articles on mathematical topics for school journal (c) solves mathematical puzzles (d) participates in the activities of mathematics club (e) gives short cuts for solving problems (f) does additional study in mathematics (g) brings to the teacher to the additional problems not related to syllabus. Objective: 7 The pupil acquires positive attitude towards mathematics. Specifications The pupil a) likes his teachers of mathematics (b) appreciates the symmetry of figures, designs and patterns (c) participates and promotes the activities of the mathematics club in the school (d) likes to be in the company of students of mathematics (e) helps students who are weak in mathematics. Objective: 8 The pupil develops scientific attitude through the study of mathematics. Specifications The pupil a) accepts the proposition only when logically proved (b) examines all the aspects of problem (c) points out errors boldly if convinced (d) accepts error boldly (e) respects the opinion of others (f) keeps an open mind and does not regard arguments as finals (g) develops habits of logical thinking. Objective: 9 The pupil acquires good personality traits through the study of mathematics.

Specifications The pupil develops traits like (a) punctuality (b) regularity and orderliness (c) concentration (d) accuracy (e) neatness. We have tried to state objectives in the teaching of mathematics and spell out specifications for each objective. The teacher‘s job is to provide practical assistance in the form of instructional methods which will enable the pupils achieve these objectives. The teacher can provide learning experience through demonstration, explanation, questioning, comparison, contrast, analysis, synthesis, the use of audio-visual materials, etc. what is more important is the correct choice of learning experiences which alone will bring about the expected learning outcomes in pupils. This will depend upon factors like the standard of the pupils, motivational level, nature of learning unit, availability of materials, etc. 2.8 Let Us Sum Up This unit ‗Aims, Objectives and Values of Teaching Mathematics‘ brings out the aims and values of teaching mathematics at secondary school level. Teachers should determine the aims of teaching mathematics for on them depends the content of mathematics. The methods of teaching and the evaluation procedure, mathematics has both intrinsic value and utilitarian or practical value. Of course any value is utilitarian but for the purpose of discussion, the division may be taken to refer to the non – material and material sides respectively. Objectives form the basis for the planning of instruction. They are the means of achieving desired goals of instruction. The objectives show the expected learning outcomes in pupils. On the basis of objectives only, a teacher can plan the learning experiences to be provided to his pupils. The objectives are generally stated in terms of knowledge, understanding, application and skills that a pupil develops as a result of classroom teaching. While we assess the learning outcome, we keep in mind the original objectives with which we started teaching a lesson. If the learning outcomes are in accordance with the objectives with which we started a lesson, there the purpose of teaching can be said to be realized completely. 2.9 Answers to Check Your Progress i) Write a short note on aims and objectives of the teaching of mathematics. ii) What are the values of the teaching of mathematics? iii) What are the needs and importance of objective based teaching of mathematics? iv) What do you mean by specification of objectives? 2.10 References Niss, M. (1994). Mathematics in society. In R. Biehler, R. W. Scholz, R. Strasser, & B. Winkelmann (eds.), Didactics of Mathematics as a Scientific Discipline (pp. 367-378). Dordrecht: Kluwer Academic Publishers. Skovsmose, O. (1994). Towards a Philosophy of Critical Mathematics Education. Dordrecht: Kluwer Academic Publishers. Allan C. Ornstein and Daniel U. Levine (1977). An Introduction to the Foundation of Education, ISBN 10: 0395358043 / ISBN 13: 9780395358047. N.C.E.R.T. – Booklet No. 6, Evaluation in Mathematics – Improving Instruction in Mathematics.

UNIT 3 INSTRUCTIONAL METHODS AND STRATEGIES OF TEACHING MATHEMATICS

Structure 3.1 Introduction 3.2 Objective 3.3 Instructional Methods of Teaching Mathematics 3.4 Inductive and Deductive Methods 3.4.1 Comparison between Inductive and Deductive Methods 3.5 Analytic and Synthetic Methods 3.5.1 Analytic Method 3.5.2 Synthetic Method 3.6 Lecture 3.6.1 Procedure of Lecture Method 3.6.2 Advantages of the Lecture Method 3.6.3 Disadvantages of the Lecture Method 3.7 Lecture cum Demonstration 3.7.1 Requirements of Good Demonstration 3.8 Heuristic 3.8.1 Definition 3.8.2 Procedure 3.8.3 Advantages of the Heuristic Method 3.8.4 Disadvantages of the Heuristic Method 3.8.5 Conclusion 3.9 Laboratory 3.9.1 Procedure of Laboratory Method 3.9.2 Advantages of the Laboratory Method 3.9.3 Disadvantages of the Laboratory Method 3.10 Project 3.10.1 Advantages of the Project Method 3.10.2 Disadvantages of the Project Method 3.11 Craft Centered Methods 3.12 Problem Solving Methods 3.13 Instruction Strategies of Teaching Mathematics 3.13.1 Drill 3.13.2 Team Teaching 3.14 Grading Problems 3.14.1 Characteristics of a Good Grading System 3.14.2 Problems Associated with Grading System 3.14.3 Advantages of Grading System 3.14.4 Disadvantage of Grading System 3.15 Mathematics Teacher: Characteristics and Roles 3.16 Year Plan 3.17 Unit Plan 3.18 Lesson Plan 3.18.1 Elements of a Good Lesson Plan 3.18.2 Objectives of the Lesson 3.18.3 Division of a Lesson 3.19 Speed and Accuracy with Mathematics Work 3.20 An Understanding of Components of Any Five Skills in Microteaching 3.20.1 Steps of Microteaching 3.20.2 Microteaching Cycle 3.20.3 Five Major Teaching Skills in Microteaching 3.21 Link Practice 3.22 Individual Instruction Method 3.23 Advantages of the Individualized Instructional Method 3.24 Disadvantages of the Individualized Instructional Method 3.25 Let Us Sum Up 3.26 Answers to Check Your Progress 3.27 References 3.1 Introduction Instructional strategies used by teachers stem from particular learning theories and in turn produce certain kinds of outcomes. For most of the twentieth century, arguments persisted about which learning theories and which instructional strategies were the most accurate and most effective in affecting student learning. Each method has certain advantages and disadvantages; some are more suited for certain kinds of instruction than others. Each of the different methods requires greater or lesser participation by students. Instructional or learning strategies determine the approach a teacher may take to achieve learning objectives. The instructional strategies teachers use help shape learning environments and represent professional conceptions of learning and of the learner. This unit discusses the different instructional methods and instructional strategies. It also presents some ideas on the year plan, unit plan, lesson plan and microteaching. 3.2 Objectives On completion of this unit students will be able to: i) understand and use different types of teaching methods in mathematics ii) select and apply different strategies of teaching mathematics iii) aware the characteristics and roles of mathematics teacher iv) understand and make a difference between year plan, unit plan and lesson plan v) understand the different components of microteaching 3.3 Instructional Methods of Teaching Mathematics Instructional methods are only as good as they contribute to the achievement of a learning objective. In fact, it is often helpful to think of methods as roads which lead to cities (objectives) and of training materials (visuals aids, case study, write-ups, role play, descriptions) as the materials with which the roads are constructed. Instructional methods are used by teachers to create learning environments and to specify the nature of the activity in which the teacher and learner will be involved during the lesson. While particular methods are often associated with certain strategies, some methods may be found within a variety of strategies. Students have different intellectual capacities and learning styles that favor or hinder knowledge accumulation. As a result, teachers are interested in ways to effectively cause students to understand better and learn. Teachers want to bring about better understanding of the material he /she wants to communicate. It is the responsibility of the educational institutions and teachers to seek more effective ways of teaching in order to meet individual‘s and society‘s expectations from education. Improving teaching methods may help an institution meet its goal of achieving improved learning outcomes. The following are main instructional methods of teaching mathematics which are discussed in this unit. 3.4 Inductive and Deductive Methods Teaching methods can either be inductive or deductive or some combination of the two. The inductive teaching method or process goes from the specific to the general and may be based on specific experiments or experimental learning exercises. Deductive teaching method progresses from general concept to the specific use or application. These methods are used particularly in reasoning i.e. logic and problem solving. To reason is to draw inferences appropriate to the situation. Inferences are classified as either deductive or inductive. For example, ―Ram must be in either the museum or in the cafeteria.‖ He is not in the cafeteria; therefore he is must be in the museum. This is deductive reasoning. As an example of inductive reasoning, we have, ―Previous accidents of this sort were caused by instrument failure, and therefore, this accident was caused by instrument failure. The most significant difference between these forms of reasoning is that in the deductive case the truth of the premises (conditions) guarantees the truth of the conclusion, where as in the inductive case, the truth of the premises lends support to the conclusion without giving absolute assurance. Inductive arguments intend to support their conclusions only to some degree; the premises do not necessitate the conclusion. Inductive reasoning is common in science, where data is collected and tentative models are developed to describe and predict future behaviour, until the appearance of the anomalous data forces the model to be revised. Deductive reasoning is common in mathematics and logic, where elaborate structures of irrefutable theorems are built up from a small set of basic axioms and rules. However examples exist where teaching by inductive method bears fruit. EXAMPLES: (INDUCTIVE METHOD): i) Ask students to draw a few sets of parallel lines with two lines in each set. Let them construct and measure the corresponding and alternate angles in each case. They will find them equal in all cases. This conclusion in a good number of cases will enable them to generalize that ―corresponding angles are equal; alternate angles are equal.‖ This is a case where equality of corresponding and alternate angles in a certain sets of parallel lines (specific) helps us to generalize the conclusion. Thus this is an example of inductive method. ii) Ask students to construct a few triangles. Let them measure and sum up the interior angles in each case. The sum will be same (=180°) in each case. Thus they can conclude that ―the sum of the interior angles of a triangle = 180°). This is a case where equality of sum of interior angles of a triangle (=180°) in certain number of triangles leads us to generalize the conclusion. Thus this is an example of inductive method. iii) Let the mathematical statement be, S(n): 1 + 2 + …… + n =. It can be proved that if the result holds for n = 1, and it is assumed to be true for n = k, then it is true for n = k +1 and thus for all natural numbers n. Here, the given result is true for a specific value of n = 1 and we prove it to be true for a general value of n which leads to the generalization of the conclusion. Thus it is an example of inductive method. EXAMPLES: (DEDUCTIVE METHOD): i) We have an axiom that ―two distinct lines in a plane are either parallel or intersecting‖ (general). Based on this axiom, the corresponding theorem is: ―Two distinct lines in a plane cannot have more than one point in common.‖ (Specific). Thus this is an example of deductive method. ii) We have a formula for the solution of the linear simultaneous equations as and (general). The students find the solutions of some problems like based on this formula (specific). Thus this is an example of deductive method. 3.4.1 Comparison between Inductive and Deductive Methods

S. Inductive Method Deductive Method No.

1. It gives new knowledge It does not give any new knowledge.

2. It is a method of discovery. It is a method of verification.

3. It is a method of teaching. It is the method of instruction.

4. Child acquires firsthand knowledge and Child gets readymade information and information by actual observation. makes use of it.

5. It is a slow process. It is quick process.

6. It trains the mind and gives self It encourages dependence on other confidence and initiative. sources.

7. It is full of activity. There is less scope of activity in it.

8. It is an upward process of thought and It is a downward process of thought and leads to principles. leads to useful results.

To conclude, we can say that inductive method is a predecessor of deductive method. Any loss of time due to slowness of this method is made up through the quick and time saving process of deduction. Deduction is a process particularly suitable for a final statement and induction is most suitable for exploration of new fields. Probability in induction is raised to certainty in deduction. The happy combination of the two is most appropriate and desirable. There are two major parts of the process of learning of a topic: establishment of formula or principles and application of that formula or those principles. The former is the work of induction and the latter is the work of deduction. Therefore, friends, ―Always understand inductively and apply deductively‖ and a good and effective teacher is he who understands this delicate balance between the two. Thus: ―his teaching should begin with induction and end in deduction.‖ 3.5 Analytic and Synthetic Methods 3.5.1 Analytic Method The word analysis comes from analyein, which is Greek for ―to break up‖. It is often helpful to break a problem or a phenomenon in to small pieces: if one studied each piece, independently of the other pieces, one might have a better chance of understanding the pieces. From that, one might understand the whole. According to the Webster Comprehensive Dictionary (1982), the ―Analysis‖ means, the resolution of a whole unit into its parts or elements or the process of resolving a problem into its first element (inductive reasoning). According to Trowbridge (1986), Analysis is the ability to break down material to its fundamental elements for better understanding of the organization. Analysis may include identifying parts, clarifying relationships among parts and recognizing organizational principles of scientific system. The analytical method proceeds from unknown to known facts. In this method the problem is analysed to find out the relations. A statement is analysed into simpler statements and then truth is discovered. It is based on inductive reasoning and critical thinking. All the related facts are analysed to seek help in proceeding to the known conclusion. It is a logical method which leaves no doubt in the minds of students in understanding the core concept and discourages cramming and rote memory of the learner. It facilitates the understanding of the students and motivates them to discover facts by him (Rehman, 2000). It is a psychological method based on the principle of interest, which inculcates the spirit of inquiry and investigation in the students (Katozai, 2002). There are some demerits of the approach as well, stated by Rehman (2000) and Katozai (2002), that it‘s a time taking approach because the teaching learning process through analysis take more time to impart knowledge from teacher to students. In solving a problem if known facts are not proper sequence or in a logical order, the students feel it boring and laborious. Example If a/b=c/d, prove that (ac-2b2)/b= (c2-2bd)/d The unknown part is (ac-2b2)/b= (c2-2bd)/d is true, If a c d – 2 b2 d = b c2 – 2 b2 d is true, If a c d = b c2 is true, If a d = b c is true That is, if a/b = c/d is true, Which is known. 3.5.2 Synthetic Method The word synthesis comes from syntithenai, which is Greek for ―to put together‖. Knowledge is created, said John Locke in his Essay Concerning Human Understanding, by combining perceptions, ideas, and other bits of knowledge. Immanuel Kant, in his Critique of Pure Reason imagined that there were two operations: i) Analysis. One understands something by taking it apart and looking at the pieces. ii) Synthesis. One understands something by combining it or comparing it with other things, or by looking at interrelations between its constituent parts. According to the Webster Comprehensive Dictionary, the ―Synthesis‖ means the assembling of separate or subordinate parts into a new form. It is a process of reasoning from whole to a part and from general to the particular (deductive reasoning). According to Trowbridge (1986), the synthesis requires the formulation of new understanding of scientific systems. If analysis stresses the parts, synthesis stresses the whole components of scientific systems may be recognized into new patterns. Unlike analysis, synthesis asks your students to put parts together, to make patterns that one, new to them. Synthetic approach is just apposite to the analytical method. In this method we proceed from known to unknown as synthesis means combing together various parts. In mathematics various facts are collected and combined to find out the result which is unknown (Rehman, 2000). According to Katozai (2002), it is the process of putting together known bits of information to reach the point where unknown formation because obvious and true. According to Rehman (2000) and Katozai (2002) there are certain merits and demerits of the synthetic method. It is a short method and save time in teaching learning process. It is suitable both for intelligent and weak students. But at the other hand, it encourages the memory work and does not develop any reasoning power and students are unable to discover new idea. Procedure The known part is a/b = c/d Subtract 2b/c on both sides (But why and how the child should remember to subtract 2b/c and not any other quantity) a/b – 2b/c = c/d – 2b/c or, (ac – 2 b2)/b c = (c2 -2 b d ) / c d or, (ac – 2 b2)/b = (c2 -2 b d ) / d which is unknown. Thus, we conclude that both the methods go together. Analysis help in understanding and synthetic helps in retaining knowledge. The teacher should realise that he may offer help for the analytic form of the solution and that the synthetic work should be left to the pupils (Marwaha, 2009). The objective of this is to analyse the comparative effectiveness of the analytical and synthetic methods of teaching mathematics at secondary level.

3.6 Lecture The ―Lecture Method‖ in the math classroom is a means by which ―the expert‖ presents the material of the course in an organized way to ―the learners‖, going from theory to examples and back again. The learners typically take notes and try to pick up as many ideas and insights as they can from the expert during those class hours when the method is in use. In math classes extensive use of a blackboard during a lecture is common. To be effective the method requires that the expert and the learners possess both background and skill sets. Among other things, the expert should possess mastery of his or her subject way beyond the details of the course, as well as skills of exposition and delivery-with-style. Sensitivity to how the lecture is being perceived is vital for a lecturer, and the ability to adapt a presentation ―on-the-fly‖ to address unexpected problems is important too. The learners must come to the class with a solid foundation in course prerequisites and have the ability to concentrate for many minutes in a row. They must have the skill of effective note taking and be well organized. 3.6.1 Procedure of Lecture Method The teacher prepares his talk at home and pours it out in the class. The students sit silently, listen attentively and try to catch the point. He may not even write anything on the black board simultaneously or may not even argue a point with the listeners by cross questioning. During the lecturing there are three major points to keep in mind that are following: A) Effective Communication with all students i) Choose a simpler word when lecturing, while offering a more complex term in hand-outs. ii) Learn and use students‘ names. iii) Eliminate slang and informal expressions. For example, ―That is not necessary‖ is easier to understand than ―You don‘t have to do that.‖ iv) Limit the use of two and three-word verbs (run into, get across, etc.). For example, ―I will organize that‖ is easier to understand than ―I‘ll set it up.‖ v) Use Latin-based root words in place of more casual choices. Latin-rooted words in English generally indicate a more formal or academic speaking style. The non-native speaker is more likely to have studied a more formalized, generic form of English in his or her home country or intensive ESL program. vi) Refer students with difficulty in oral or written expression to tutorial or training programs for extra help. vii) When using fictitious names, include ones such as Nguyen (―new-win‖) or Durai (―do-rye‖) in addition to the ―traditional‖ Smith, Jones and Brown. B) Don’t just teach the material, teach how to learn i) Ask questions in class and wait to get answers (at least five seconds) — allow silence. ii) Show students how to perform important course skills — model the process of analyzing a research report, rather than assuming that students know how to do this. iii) Have extra sessions on note-taking and effective study practices. iv) Encourage students to use campus tutorial, study skills, and writing services. v) Provide extra material or exercises for students who lack essential background knowledge or skills. vi) Recognize that not all students seek advice and guidance when needed; be prepared to reach out to those who might not otherwise seek help. C) Remember it’s all about people i) Personalize the course for students. ii) Find out about students‘ learning styles, interests, and backgrounds at the start of the course. (See ―Getting to Know Your Students,‖ earlier in this chapter, for specific suggestions.) iii) In large classes, find ways for students to get to know one another, and encourage students to form study groups. iv) Incorporate the contributions of foreign-born scholars in citations of scholarly accomplishments. v) Include real-life examples of documents prepared for and by internationally-based clients rather than focusing on just North American and western European samples. vi) Use topics such as the impact of international trade on local economies to solicit and encourage the opinions and participation of students who have lived abroad. vii) Encourage international students to undertake research projects which will provide experience relevant to their future environment, since many will return to their countries of origin after graduation. viii) Pair undergraduate students with mature students already in the workforce so that shared experience contributes to realistic research and writing. 3.6.2 Advantages of the Lecture Method i) The teacher controls the topic, aims, content, organization, sequence, and rate. Emphasis can be placed where the teacher desires. Gives the instructor the chance to expose students to unpublished or not readily available material. ii) Students can interrupt for clarification or more detail. iii) The lecture can be taped, filmed, or printed for future use. iv) Can be used to arouse interest in a subject. v) Can complement and clarify text material. vi) The teacher can serve as a model in showing how to deal with issues and problems. vii) Complements certain individual learning preferences. Some students depend upon the structure provided by highly teacher-centered methods. viii) Facilitates large-class communication. ix) Other media and demonstrations can be easily combined with the lecture. x) The lecture can be used to motivate and increase interest, to clarify and explain, to expand and bring in information not available to the students, and to review. 3.6.3 Disadvantages of the Lecture Method i) Places students in a passive rather than an active role, which hinders learning. ii) Most students have not learned to take good notes. iii) Lecture information is forgotten quickly, during and after the lecture. iv) Students are not very active when only listening. v) The communication is mostly one-way communication from the teacher to her students. Encourages one-way communication. Usually there is little student participation. The students who do participate are few in number and tend to be the same students each class. vi) Lectures are not effective when teaching thinking objectives. vii) There is no immediate and direct check of whether learning has taken place. viii) The lecture method encourages student dependence on the teacher. ix) It is difficult to maintain student interest and attention for a full hour of lecture. x) Requires a considerable amount of unguided student time outside of the classroom to enable understanding and long-term retention of content. In contrast, interactive methods (discussion, problem-solving sessions) allow the instructor to influence students when they are actively working with the material. xi) Requires the instructor to have or to learn effective writing and speaking skills. xii) Some of the students may already know the content of the lecture while some may not be ready for the lecture. 3.6.4 Conclusion The Method neither suits the subject nor the learner. It goes against the independent and original thinking of the learner. There is no student participation in learning process. Most of the time his face is towards the class and the back is towards the blackboard. This is defective. He should mostly face the black board. 3.7 Lecture cum Demonstration The dictionary meaning of the word ―demonstration‖ is the outward showing of a feeling etc.; a description and explanation by experiment; so also logically to prove the truth; or a practical display of a piece of equipment to snow its display of a piece of equipment to show its capabilities. In short it is a proof provided by logic, argument etc. To define it is a physical display of the form, outline or a substance of object or events for the purpose of increasing knowledge of such objects or events. Demonstration involves ―showing what or showing how‖. Demonstration is relatively uncomplicated process in that it does not require extensive verbal elaboration. Now it will be easy to define what is lecture cum demonstration method. To begin with, this method includes the merits of lecture method and demonstration method. The teacher performs the experiment in the class and goes on explaining what she does. It takes into account the active participation of the student and is thus not a lopsided process like the lecture method. The students see the actual apparatus and operations and help the teacher in demonstrating experiments and thereby they feel interested in learning. So also this method follows maxims from concrete to abstract. Wherein the students observe the demonstration critically and try to draw inferences. Thus with help of lecture cum demonstration method their power of observation and reasoning are also exercised. So the important principle on which this method works is ―Truth is that works‖. 3.7.1 Requirements of Good Demonstration The success of any demonstration following points should be kept in mind. i) It should be planned and rehearsed by the teacher beforehand. ii) The apparatus used for demonstration should be big enough to be seen by the whole class. If the class may be disciplined she may allow them to sit on the benches to enable them a better view. iii) Adequate lighting arrangements be made on demonstration table and a proper background table need to be provided. iv) All the pieces of apparatus be placed in order before starting the demonstration. The apparatus likely to be used should be placed in the left hand side of the table and it should be arranged in the same order in which it is likely to be used. v) Before actually starting the demonstration a clear statement about the purpose of demonstration be made to the students. vi) The teacher makes sure that the demonstration lecture method leads to active participation of the students in the process of teaching. vii) The demonstration should be quick and slick and should not appear to linger on unnecessarily. viii) The demonstration should be interesting so that it captures the attention of the students. ix) It would be better if the teacher demonstrates with materials or things the children handle in everyday life. x) For active participation of students the teacher may call individual student in turn to help him in demonstration. xi) The teacher should write the summary of the principles arrived at because of demonstration on the blackboard. The black board can be also used for drawing the necessary diagrams. 3.8 Heuristic Method 3.8.1 Definitions i) According to H.E.Armstrong, ―This is the method of teaching which places the pupils as far as possible in the attitude of a discoverer.‖ ii) According to Westaway, ―the heuristic method is intended to provide training in method. Knowledge is a secondary consideration altogether. Discovery method is a teaching strategy which enables students to find the answers themselves. It is a learner centred approach hence it is called a heuristic method. It is of two types, notably, the guided discovery and the unguided discovery. In the guided discovery, the teacher (you) guides the students to discover for themselves solutions to given problems by providing them with general principles, but not the solution to the scientific problem. The unguided discovery type involves the students discovering for themselves both the general principles and solution to a scientific problem. It is sometimes called the pure discovery. 3.8.2 Example The following heuristic worked-out example is intended to provide an overview of the most important aspects of this type of example. This specific example is not only meant to prove that the interior angles in any triangle add up to 180° but also helps demonstrate to students various aspects of proving in general. The problem is given below: Alex and Chris have drawn different triangles and respectively measured the sum of their three angles. Both were surprised to discover that this sum was 180° for all of the triangles. Alex and Chris were sure that this was not an accidental result. Their conjecture was: In every triangle, the sum of its interior angles is 180°. In the following, we will look closely at their problem-solving work. You are encouraged not only to read this solution but to repeat all the problem solving steps that Alex and Chris followed. 1) Exploration of a problem situation: Equipment: a pair of scissors, a protractor, paper. (a) Draw a triangle ABC, mark its angles α, β, and γ. Measure the size of these angles. What is the sum of α, β, and γ? Note this size: ……………………………… Repeat the experiment three or four times. Note the size of all the angles you get: ……………………………… (b) Draw a triangle ABC, mark its angles α, β, and γ. Take your scissors, cut it out, tear off the corners of the triangle and put them together to form a new angle. What size does this angle probably have? Note this size: ……………………………… Repeat the experiment three or four times. Note the size of all the angles you get: ……………………………… (c) Draw a triangle ABC, mark its angles α, β, and γ. Take your scissors, cut it out. Using ABC as an outline draw, get more triangles. Cut them out and combine them so that you get a straight line at the bottom.

c b a c b

a b a b a

There is probably a straight line on top. This would provide evidence that congruent triangles may be used to completely inlay a plane. What does this mean for α, β, and γ? Accordingly, all these experiments suggest that the angles of an arbitrary triangle add up to 180°.

(2) Conjecture: Let ABC be a triangle, and α, β, and γ its angles. Then α + β + γ = 180°. Mathematical conjectures need to be proved. In order to perform this mathematical proof it is necessary to i) clarify what one knows about angles and respectively about triangles, ii) identify arguments which might be important for the proof and to, iii) organize correct arguments in a logical sequence. (3) What information is available on angles? There are some possible prerequisites for the proof, which are well known about angles. In particular you may recall the following facts: i) A straight line is regarded to cover an angle of 180°. ii) Vertical angles are congruent. iii) When parallel lines are cut by a transversal then the corresponding angles are congruent. iv) When parallel lines are cut by a transversal then alternate interior angles are congruent. Comparing this information and the experimental data may lead to an idea of the proof. (4) The idea of a proof: A straight line is regarded to cover an angle of 180°. Accordingly, one may argue that the angles of an arbitrary triangle are congruent to angles which add up to a straight line. (5) The proof of the conjecture: There is a triangle ABC, and α, β, and γ are its angles. Draw the line d, which is the parallel line to AB and which touches C. Mark the angles α‘ and β‘ as shown in the drawing C a‘ c b‘

a b A B We know that α and α‘ and respectively β and β‘ are alternate interior angles, and that AB and d are parallel lines. Accordingly, α = α‘ and β = β‘. As d is a straight line, it is obvious that α + β‘ + γ‘ = 180° and, therefore, you can conclude, that α + β + γ = 180° as well. (6) Looking back: As a result of the problem solving process we know now for sure that the interior angles in every triangle add up to 180°. In the language of mathematics one would say that we found a proof for this conjecture. The solution: Alex and Chris have found a proof for their conjecture. They proved: In every triangle, the sum of its interior angles is 180°. This heuristic worked-out example reveals the most important aspects of this type of presentation. It provides particularly information on the use of heuristics in the problem-solving process. Firstly, the exploration of the situation gives evidence for the conjecture. Secondly, the drawing suggests a specific problem-solving context. It does not necessarily lead to correct (or useful) mathematical arguments. For example, this proof does not use the (correct) argument that vertical angles are congruent. It is important to accept that mathematical work means making an adequate choice out of possible arguments. It is usually not at all clear in the beginning which (significant) arguments will be integrated in the proof. However, if the correct arguments are combined in a correct way, we come to a conclusion for all possible triangles. There are many ways to use worked-out examples in the mathematics classroom. However, as outlined above, most students process examples in a superficial or passive way so that they need some guidance. In addition, young students are hardly able to fully concentrate on elaborated examples for a longer period of time. With respect to the heuristic example detailed above, the students‘ work should be supervised and summarized at least a few points during the problem-solving process: after the exploration of the problem, after the identification of suitable arguments, and after their combination in a proof. Furthermore, a teacher could present the problem to the students and let them explore it. Thereafter, the students would be encouraged to reflect on their actions by reading and discussing these parts of the worked-out example. Heuristic worked-out examples cannot be used as isolated tools for teaching, but are meant to be integrated frequently in the mathematics classroom. Students need to learn how to extract important information and to concentrate on a series of problem solving steps while viewing an example. Accordingly, students will have to learn how to work independently with heuristic worked out examples.

3.8.3 Advantages of the Heuristic Method Discovery method, whether it is the directed type or undirected, makes the student an active participant rather than a mere passive recipient. i) Since the method poses a challenge for the student to discover the information or knowledge for himself, retention of any information or knowledge so discovered will be increased. This is a psychological method as the student learns by self-practice. ii) It creates clear understanding and a meaningful learning. iii) The student learns by doing so there is a little scope of forgetting. iv) It develops self-confidence, self-discipline and a sense of achievement in the students. The students acquire command of the subject. He has clear understand and notions of the subject. v) The training acquired in finding out things for one-self independently can be applied to new learning and problem solving. vi) The joy in discovering something provides the students with intrinsic motivation. It inculcates in the student the interest for the subject and also develops willingness in them. vii) Discovery method brings home to pupils their notions of the nature of scientific evidence; students learnt that answers to questions can often be obtained from investigations they can carry out for themselves. viii) Discovery method helps students develop manipulative skills and attitudes which constitute one of the fundamental objectives of science teaching. ix) Since discovery operates at the highest levels of the cognitive domain, it encourages analytical and synthetic thought as well as intuitive thinking. 3.8.4 Disadvantages of the Heuristic Method i) Discovery method is time consuming and progress is comparatively slow. Apparatus have to be set up and result(s) of the investigation awaited. ii) The method leaves open the possibility of not discovering anything. Students may end up discovering things other than what was intended to be discovered. This could be highly demoralizing to them particularly if great effort has been expended. iii) It does not suit larger classes. iv) The teacher may find it difficult to finish the syllabus in time. v) It suits only hard working and original thinking teachers. vi) The method is expensive considering the equipment and materials needed. vii) Discovery method is not suitable for lower classes as they are not independent thinkers. Discovery of a thing needs hard work, patience, concentration, reasoning and thinking powers and creative abilities. 3.8.5 Conclusion Heuristic method is not quite suitable for primary classes. However, this method can be given a trail in high and higher secondary classes. 3.9 Laboratory Method Mathematics is a subject which has to be learnt by doing rather than by reading. The doing of mathematics, gives rise to the need of a suitable method and a suitable place. Laboratory method and mathematical laboratory are the proper answers to it. This activity method leads the pupil to discover mathematical facts. It is based on the principles of learning by doing, learning by observation and proceeding from concrete to abstract. In one sense it is only an extension of the inductive method. It is a more elaborated and practical form of the Inductive method. Pupils do not only listen for information, but do something practical also. This is an activity method designed to be carried out by an individual student or a group of students for the purpose of making personal observations from experiments in which students can get conclusions by themselves. Laboratory method is based on the principles of ―learning by doing‖ and ―learning by observation‖ and proceeding from concrete to abstract. Students do not just listen to the information given but do something practically also. Principles have to be discovered, generalized and established by the students in this method. Students learn through hands on experience. This method leads the student to discover mathematical facts. After discovering something by his own efforts, the student starts taking pride in his achievement, it gives him happiness, mental satisfaction and encourages him towards further achievement. The method, if properly used, should help in the removal of the abstract nature of mathematics. It makes the subject interesting as it combines play and activity. Example: Making and observing models, paper folding, paper cutting, and construction work in geometry.

3.9.1 Procedure of Laboratory Method The construction work in geometry is on the whole a laboratory work. The drawing of a line, construction of an angle, construction of a triangle or a quadrilateral or parallelogram etc all involve the use of some equipment and therefore their nature is that of practical or laboratory work. There can be many more illustrations to explain the procedure. For calculating the area of a triangle, cut out a triangle of cardboard. Find the weight of a unit area of the cardboard and then find the weight of the triangle. His weight divided by the weight of the unit area of the same cardboard will give you the approximate area of the triangle. 3.9.2 Advantages of the Laboratory Method i) Learning through this method extends and reinforces theoretical learning through reality. ii) Laboratory method offers students the opportunity to develop scientific attitudes such as objectivity, critical thinking, carefulness, open mindedness etc. iii) Because the method implies learning by doing students tend to be more interested because of active involvement. iv) Students become familiar with how scientific knowledge is acquired by performing experiments, recording observations and results, summarizing data and drawing conclusions. v) Through laboratory method, the student learns how to handle apparatus and other instruments, thereby developing manipulative skills. vi) Laboratory method promotes problem solving and self reliance. vii) Getting involved in laboratory activities can also enable students to learn much about the inter- relationship between science and technology. 3.9.3 Disadvantages of the Laboratory method i) It can be expensive if separate equipment and materials have to be provided. ii) It is time-consuming because of the careful planning and preparation required. iii) Acquisition of skills which results from exposure to the laboratory method, is of questionable value as objectives for some of the students who will have little use of them later. iv) It is an inefficient practice of teaching where ordinary telling method or simple demonstration is perfectly adequate. 3.10 Project Method Project method is a natural hearted, problem solving and purposeful activity carried to completion in a social environment this is the most concrete of all types of activity methods. It is the revolt against the traditional, bookish and passive environment of school. The project method of teaching is the practical outcome of the John Dewey‘s philosophy of pragmatism. Pragmatism has made a unique construction in the shape of project method enunciated by the Kilpatrick the follower of Dewey. The Project Method of teaching centers on an assignment of interest undertaken by an individual student or a group or a whole class. In this method, the students are guided when necessary. In project method, study through workshop and source methods are also studied, concrete activity rather than academic work take the dominant place in the project method. The project method also transcends the subject barrier which is not done by other methods. In project method the teacher instead of following the lecture method substitutes ―the subject‖ with few outstanding problems and proceeds to solve the same by experiment method with the active co-operation of the students. The purpose of this method is to learn pupils into the trained investigators and prepare them for learning by living. Definitions of Project Method According to Ballard ―A project is a bit of real life that has been imported into the School‖. According to J. A. Stevenson ―A project is a problematic act carried to the completion in its Natural setting‖. According to Kilpatrick ―A project is a whole hearted purposeful activity proceeding in a social environment‖. According to W.W. Charters ―In the topical organization principle are learned first while in the projects the problem are proposed which demands in the solution the development of principles by the learner as needed ―. 3.10.1 Advantages of the Project Method i) Since emphasis is on doing by the student, opportunity is provided to develop his initiative as well as greater understanding of how to learn. ii) Motivation to work is high since it is based on the natural interests of students. It thus offers opportunity for creative ability particularly for specially talented students. iii) It gives students specific areas to work on sometimes with acquisition of some new skills and attitudes. iv) Group project afford opportunity for developing leadership and organizing abilities. 3.10.2 Disadvantages of the Project Method i) Projects are very time-consuming and what is ultimately learned may not justify the expense, efforts and time put in to complete the project. ii) Students often get sidetracked particularly if they lack good grasp of facts necessary in carrying out the projects. iii) It may be difficult to determine the extent to which the individual has participated. iv) It is difficult to choose a project that will interest all the students in the class at one time. v) It favours the independent students, those without independent study skills may suffer. vi) Some students may not participate in the project work at all. 3.11 Craft Centered Methods Under this method, an attempt is made to replace the isolated nature of the various projects by a single activities centered. Basic craft like spinning, weaving, pottery, carpentry, or agriculture. The choice of the craft depends on local conditions. The craft if the chief concern and learning is only incidental. As in the case of the project method, here also it is difficult to teach the body of mathematics around a single craft. This is not possible to give integrated instruction. At best these two plans can provide the necessary motivation and a real life situation for practice. Many teachers have used the results of educational research, as well as vast amounts of anecdotal evidence, to craft teaching methods that are innovative, interactive, student-driven, and responsive to a variety of learning styles. These craft – centered approaches require active participation from teachers and students. They put much of the responsibility for learning on the student and focus on creating vibrant communities of people united for the common purpose of learning. 3.12 Problem Solving Methods Life is full of problems and we term one as successful, who is able to use the knowledge acquired and reasoning power to find solutions to these problems. Problem– solving may be a purely mental difficulty or it may be physical and involve manipulation of data. Problem-solving method aims at presenting the knowledge to be learnt in the form of a problem. It begins with a problematic situation and consists of continuous, meaningful, well-integrated activity. The problems are test to the students in a natural way and it is ensured that the students are genuinely interested to solve them. This method aims at presenting the knowledge to be learnt in the form of a problem. It begins with a problematic situation and consists of continuous meaningful well-integrated activity. Choose a problem that uses the knowledge that students already have i.e. you as a teacher should be able to give them the problem and engage them without spending time in going over the things that you think they should know. After students have struggled with the problem to get solution, have them share their solutions. This method will help them in developing divergent thinking. Example: Put a problem of finding the amount of water in a given container instead of deriving the formula of volume (cylinder filled with water). Definition According to Gagne ―Problem solving is a set of events in which human beings was rules to achieve some goals‖. According to Ausubel ―Problem solving involves concept formation and discovery learning‖. According to Risk, T.M. ―Problem solving is a planned attacks upon a difficulty or perplexity for the purpose of findings a satisfactory solution‖. Goals of Mathematical Problem-Solving The specific goals of problem solving in Mathematics are to: i) Improve pupils‘ willingness to try problems and improve their perseverance when solving problems. ii) Improve pupils‘ self-concepts with respect to the abilities to solve problems. iii) Make pupils aware of the problem-solving strategies. iv) Make pupils aware of the value of approaching problems in a systematic manner. v) Make pupils aware that many problems can be solved in more than one way. vi) Improve pupils' abilities to select appropriate solution strategies. vii) Improve pupils' abilities to implement solution strategies accurately. viii) Improve pupils' abilities to get more correct answers to problems. Steps of Problem Solving Method i) Recognizing the problem or sensing the problem. ii) Interpreting, defining and delimiting the problem. iii) Gathering data in a systematic manner. iv) Organizing and evaluating the data. v) Formulating tentative solutions. vi) Arriving at the true or correct solution. vii) Verifying the results. Merits of Problem Solving Method The merits or advantages of problem solving method are as follows: i) Method is scientific in nature. ii) Develops good study habits and reasoning power. iii) Helps to improve and apply knowledge and experiences. iv) Stimulates thinking of the child. v) Students learn virtues such as patience, cooperation, and self-confidence. vi) Learning becomes more interesting and purposeful. vii) Develops qualities of initiative and self-dependence in the students, as they have to face similar problematic situations in real life too. viii) Develops desirable study habits in the students. Limitations of Problem Solving Method i) The limitations are mainly due to ineffective use of the problem solving method. When a classroom is completely teacher dominated then in such a classroom the problem solving method will fail. ii) Difficult to organize e contents of syllabus according to this method. iii) Time consuming method. iv) All topics and areas cannot be covered by this method. v) There is a lack of suitable books and references for the students. vi) Method does not suit students of lower classes. vii) Mental activity dominates this method. Hence there is neglect of physical and practical experiences. 3.13 Instruction Strategies of Teaching Mathematics Instructional strategies determine the approach a teacher may take to achieve learning objectives. Learning or instructional strategies determine the approach for achieving the learning objectives and are included in the pre-instructional activities, information presentation, learner activities, testing, and follow-through. The strategies are usually tied to the needs and interests of students to enhance learning. Contemporary conceptions of instructional strategies acknowledge that the goals of schooling are complex and multifaceted, and that teachers need many approaches to meet varied learner outcomes for diverse populations of students. The following are main instructional strategies of teaching mathematics which are discussed below. 3.13.1 Drill Drill is repeated exercise. The meaning of drill is to perform a training exercise. In another words drill is a disciplined, repetitious exercise as a means of teaching mathematics and perfecting a skill or procedure. In mathematics, only the same concept or the same principle is repeated with different numbers, different context and different words. There is no other way to fix concepts, learn principles and to master processes. Drill refers to any exercise, physical or mental, enforced with regularity and by constant repetition. Drill activities help learner‘s master materials at their own pace. Drills are usually repetitive and are used as a reinforcement tool. Effective use of drill depends on the recognition of the type of skill being developed, and the use of appropriate strategies to develop these competencies. There is a place for drill mainly for the beginning learner or for students who are experiencing learning problems. Its use, however, should be kept to situations where the teacher is certain that it is the most appropriate form of instruction. As an instructional strategy, drill is familiar to all educators. It promotes the acquisition of knowledge or skill through repetitive practice. It refers to small tasks such as the memorization of spelling or vocabulary words, or the practicing of arithmetic facts and may also be found in more advanced learning tasks or physical education games and sports. Drill likes memorization, involves repetition of specific skills, such as addition and subtraction, or spelling. To be meaningful to learners, the skills built through drill should become the building blocks for more meaningful learning. The method of conducting drill should also be varied in competitions, pupils questioning, games and novel methods designed by the imagination of the teacher make instruction and pleasant and enjoyable. Advantages of Drill i) It develops power and skills in teaching learning process. ii) It shows pupils various situations where the same powers are usable. iii) It makes responses automatic, saving energy and time for the higher functions. iv) Without drill in any essential skills, there is no mathematical growth. v) Can be used with large numbers of students making it a time-efficient way of conveying knowledge. vi) Good drill and practice provides feedback to students, explains how to get the correct answer, and contains a management system to keep track of student progress. vii) They can be used to build confidence as more answers are correctly provided. Disadvantages of Drill i) It should be confined only to the basic minimum skills in the case of the majority of pupils, otherwise it dose the backward pupils great harm. ii) For most students, it's boring, uninteresting, and unappealing. iii) Lacks the 'deeper-understanding' that ―hands-on learning‖ provides, and without perpetual reinforcement, the 'knowledge' obtained through drill and practice is quickly lost (most students only learn for the amount of time required to do well on the examination). iv) It has no application in the student's real-world, i.e. rarely will the student make the connection with why these things are important. v) Can encourage competition, which many Western Educators frown upon. 3.13.2 Team Teaching Definitions of Team Teaching According to the freedictionary.com i) A method of classroom instruction in which several teachers combine their individual subjects into one course which they teach as a team to a single group of students. ii) (Social Science / Education) a system whereby two or more teachers pool their skills, knowledge, etc., to teach combined classes. iii) a method of coordinated classroom teaching involving a team of teachers working together with a single group of students. According to Shaplin and Olds (1964) ―team teaching is a type of instructional organization, involving teaching personnel and the students assigned to them, in which two or more teachers are given responsibility, working together, for all or a significant part of the instruction of the same group of the students‖. Olson (1967) defines team teaching as ―an instructional situation where two or more teacher possessing complimentary teaching skills, cooperative plan and implement the instruction for a single group of students using flexible scheduling and grouping techniques to meet the particular instructional of the students‖. According to Trump (1968) ―the term ‗team teaching‘ applies to an arrangement in which two or more teachers and their assistants, taking advantages of their respective competencies, plan, instruct and evaluate in one or more subject areas a group of elementary or secondary students equivalent in size to two or more conventional classes, using a variety of technical aids to teaching and learning through large group instruction, small group discussion and independent study‖. Team teaching involves a group of instructors working purposefully, regularly, and cooperatively to help a group of students of any age learn. Teachers together set goals for a course, design a syllabus, prepare individual lesson plans, teach students, and evaluate the results. They share insights, argue with one another, and perhaps even challenge students to decide which approach is better. Teams can be single-discipline, interdisciplinary, or school-within-a-school teams that meet with a common set of students over an extended period of time. New teachers may be paired with veteran teachers. Innovations are encouraged, and modifications in class size, location, and time are permitted. Different personalities, voices, values, and approaches spark interest, keep attention, and prevent boredom. The team-teaching approach allows for more interaction between teachers and students. Faculty evaluates students on their achievement of the learning goals; students evaluate faculty members on their teaching proficiency. Emphasis is on student and faculty growth, balancing initiative and shared responsibility, specialization and broadening horizons, the clear and interesting presentation of content and student development, democratic participation and common expectations, and cognitive, affective, and behavioral outcomes. This combination of analysis, synthesis, critical thinking, and practical applications can be done on all levels of education, from kindergarten through graduate school. Working as a team, teachers model respect for differences, interdependence, and conflict-resolution skills. Team members together set the course goals and content, select common materials such as texts and films, and develop tests and final examinations for all students. They set the sequence of topics and supplemental materials. They also give their own interpretations of the materials and use their own teaching styles. The greater the agreement on common objectives and interests, the more likely that teaching will be interdependent and coordinated. Teaching periods can be scheduled side by side or consecutively. For example, teachers of two similar classes may team up during the same or adjacent periods so that each teacher may focus on that phase of the course that he or she can best handle. Students can sometimes meet all together, sometimes in small groups supervised by individual teachers or teaching assistants, or they can work singly or together on projects in the library, laboratory, or fieldwork. Teachers can be at different sites, linked by video-conferencing, satellites, or the Internet. Breaking out of the taken-for-granted single-subject, single-course, single-teacher pattern encourages other innovations and experiments. For example, students can be split along or across lines of sex, age, culture, or other interests and then recombined to stimulate reflection. Remedial programs and honors sections provide other attractive opportunities to make available appropriate and effective curricula for students with special needs or interests. They can address different study skills and learning techniques. Team teaching can also offset the danger of imposing ideas, values, and mindsets on minorities or less powerful ethnic groups. Teachers of different backgrounds can culturally enrich one another and students.

Advantages of Team Teaching All students do not learn at the same rate. Periods of equal length are not appropriate for all learning situations. Educators are no longer dealing primarily with top-down transmission of the tried and true by the mature and experienced teacher to the young, immature, and inexperienced pupil in the single- subject classroom. Schools are moving toward the inclusion of another whole dimension of learning: the lateral transmission to every sentient member of society of what has just been discovered, invented, created, manufactured, or marketed. For this, team members with different areas of expertise are invaluable. Of course, team teaching is not the only answer to all problems plaguing teachers, students, and administrators. It requires planning, skilled management, willingness to risk change and even failure, humility, open-mindedness, imagination, and creativity. But the results are worth it. Teamwork improves the quality of teaching as various experts approach the same topic from different angles: theory and practice, past and present, different genders or ethnic backgrounds. Teacher strengths are combined and weaknesses are remedied. Poor teachers can be observed, critiqued, and improved by the other team members in a nonthreatening, supportive context. The evaluation done by a team of teachers will be more insightful and balanced than the introspection and self-evaluation of an individual teacher. Working in teams spreads responsibility, encourages creativity, deepens friendships, and builds community among teachers. Teachers complement one another. They share insights, propose new approaches, and challenge assumptions. They learn new perspectives and insights, techniques and values from watching one another. Students enter into conversations between them as they debate, disagree with premises or conclusions, raise new questions, and point out consequences. Contrasting viewpoints encourage more active class participation and independent thinking from students, especially if there is team balance for gender, race, culture, and age. Team teaching is particularly effective with older and underprepared students when it moves beyond communicating facts to tap into their life experience. The team cuts teaching burdens and boosts morale. The presence of another teacher reduces student- teacher personality problems. In an emergency one team member can attend to the problem while the class goes on. Sharing in decision-making bolsters self-confidence. As teachers see the quality of teaching and learning improve, their self-esteem and happiness grow. This aids in recruiting and keeping faculty.

Disadvantages of Team Teaching Team teaching is not always successful. Some teachers are rigid personality types or may be wedded to a single method. Some simply dislike the other teachers on the team. Some do not want to risk humiliation and discouragement at possible failures. Some fear they will be expected to do more work for the same salary. Others are unwilling to share the spotlight or their pet ideas or to lose total control. Team teaching makes more demands on time and energy. Members must arrange mutually agreeable times for planning and evaluation. Discussions can be draining and group decisions take longer. Rethinking the courses to accommodate the team-teaching method is often inconvenient. Opposition may also come from students, parents, and administrators who may resist change of any sort. Some students flourish in a highly structured environment that favors repetition. Some are confused by conflicting opinions. Too much variety may hinder habit formation. Salaries may have to reflect the additional responsibilities undertaken by team members. Team leaders may need some form of bonus. Such costs could be met by enlarging some class sizes. Nonprofessional staff members could take over some responsibilities. 3.14 Grading Problems The two most common types of grading systems used are norm-referenced and criterion-referenced. Many teachers combine elements of each of these systems for determining student grades by using a system of anchoring or by presetting grading criteria which are later adjusted based on actual student performance. i) Norm-Referenced Systems In norm-referenced systems, students are evaluated in relationship to one another (e.g., the top 10% of students receive an A, the next 30% a B). This grading system rests on the assumption that the level of student performance will not vary much from class to class. In this system, the instructor usually determines the percentage of students assigned each grade, although it may be determined (or at least influenced) by departmental policy. ii) Criterion-Referenced Systems Norm-referenced tests measure students relative to each other. Criterion-referenced tests measure how well individual students do relative to pre-determined performance levels. Teachers use criterion- referenced tests when they want to determine how well each student has learned specific knowledge or skills. In criterion-referenced systems, students are evaluated against an absolute scale, normally a set number of points or a percentage of the total (e.g., 95-100 = A, 88-94 = B). Since the standard in this grading system is absolute, it is possible that all students could get As or all students could get Bs. Commonly in the Indian universities for the Undergraduate and Graduate courses having the following grading system: Quality of Performance Grade

Excellent Exceptional Achievement A

A-

Good Extensive Achievement B+

B

B-

C+

Satisfactory Acceptable Achievement C

D

Failure Inadequate Achievement F

(to secure credit, course must be repeated.)

3.14.1 Characteristics of a Good Grading System

i) Grades should be relevant to major course objectives Although this may seem obvious, students often complain that there is no connection between the stated course objectives and the way they are evaluated. For example, one frequent lament goes something like this: ―Professor X said the most important thing he wanted us to get out of this class is to be able to think critically about the material, but our entire grade was based on two multiple choice exams which tested our memory of names, dates, and definitions!‖ When preparing your grading system for a course, begin with a list of your objectives for the course. Assign relative weights to the objectives in terms of their importance. Be sure the items you are including as part of the grade (e.g. exams, papers, projects) reflect the objectives and are weighted to reflect the importance of the objectives they are measuring. ii) Grades should have recognized meaning among potential users Because the purpose of grades is to communicate the extent to which students have learned the course materials, grades should be based primarily on the students' performance on exams, quizzes, papers, and other measures of learning specified at the beginning of the course. Items such as effort, attendance, or frequency of participation, although contributing factors to student learning do not actually reflect the extent to which students have learned the course materials. iii) The grading process should be impartial and compare each student to the same criteria If you are willing to offer extra credit or opportunities to retake exams or rewrite assignments, the offer should be made to the whole class rather than only to individuals who request these opportunities. iv) Grades should be based on sufficient data to permit you to make valid evaluations of student achievement It is rarely justifiable to base students' grades solely on their performance on one or two exams. Unless the exams are extremely comprehensive, one or two exams would provide an inadequate sampling of course content and objectives. There is also likelihood that an off-day could lower a student's grade considerably and be an inaccurate reflection of how much he/she has learned. Generally speaking, the greater the number and variety of items used to determine grades, the more valid and reliable the grades will be.

3.14.2 Problems Associated with Grading System

i) Grading system makes it more difficult to identify the truly exceptional students, as more students come to get the highest possible grade. ii) Grading system is not uniform between schools/educational institutes. This places students in more stringently graded schools/educational institutes and departments at an inequitable disadvantage. iii) Grade inflation is not uniform among disciplines. iv) Grading system may motivate less productive students to keep studying whereas countries with no grade inflation may discourage students from studying by demoralizing them. v) Higher grades at some schools may reflect better performance than others (although with no national standard, there can be no way to compare one school to another by grades). 3.14.3 Advantages of Grading System There are many advantages of the grading system. A few of them are- i) The pressure is very less on students. ii) The grading system also focuses on sports with extra curricular‘s. iii) The spirit of competition is less among the students, which decreases the pressure. iv) Helps in overall development of the child v) More students take rigorous courses. vi) More challenging courses can be offered. vii) It increases a student‘s GPA. viii) Higher class rankings for those who take more demanding courses. ix) Students are more competitive with peers from other schools with weighted grading for first- choice and more elite college acceptance. x) Better chance for students to receive more in scholarship monies. xi) More likelihood for students to have higher self-esteem 3.14.4 Disadvantages of Grading System i) Lack of consistency from school to school as to what courses are weighted and how much they are weighted. ii) Not all courses, even honors and AP, are equally demanding. iii) It may send a message to those who are taking regular courses, that their work is not as highly valued as weighted classes, which may lower self-esteem and attempts to strive for high grades. iv) College admissions offices tend to look at the overall GPA and not if the grades had been weighted. v) If a student is afraid of getting a low grade in a more rigorous course, he/she may opt to take a less demanding course in order to earn a higher grade. vi) Tracking of students could become more common. vii) Students at the lower academic end of the spectrum would not have equal opportunities to take a more engaging academic program. viii) Litigation by parents may occur if they believe the system is hindering their children from equal access to the curriculum. ix) Smaller schools have fewer opportunities to offer a wide array of weighted and non-weighted courses. x) Fine arts courses may not be taken because it is possible that a non-weighted grade will lower a student‘s GPA. 3.15 Mathematics Teacher – Characteristics and Role The following characteristics and the role of the teacher describe in an exemplary mathematics teacher. The characteristics are based on the effectiveness of the teaching learning environment in the mathematics classrooms. The role of the mathematics teacher is essential for highly effective mathematics instruction in the development of the learners‘ mathematical knowledge and mathematical ability. The characteristics and the role of the good mathematics teacher are given following: i) Teacher creates learning environments where students are active participants as individuals and as members of collaborative groups. ii) The teacher models literacy and numeracy strategies to enable students to communicate and analyze mathematical problems/tasks and solutions. iii) Teacher motivates students and nurtures their desire to learn in a safe, healthy and supportive environment which develops compassion and mutual respect. iv) Teacher cultivates cross cultural understandings and the value of diversity. v) Teacher encourages students to accept responsibility for their own learning and accommodates the diverse learning needs of all students. vi) Teacher displays effective and efficient classroom management that includes classroom routines that promote comfort, order and appropriate student behaviors. vii) Teacher provides students equitable access to technology, space, tools and time. viii) The teacher provides access to the common core curriculum by utilizing differentiated teaching strategies, interventions, manipulatives, calculators, information technology, etc. ix) Teacher effectively allocates time for students to engage in hands-on experiences, discuss and process content and make meaningful connections x) Teacher designs lessons that allow students to participate in empowering activities in which they understand that learning is a process and mistakes are a natural part of learning. xi) Teacher creates an environment where student work is valued, appreciated and used as a learning tool. xii) The teacher provides opportunities for students to share mathematical ideas with others and to problem solve. xiii) A good math teacher should convey the beauty of the subject. xiv) Great teachers provide positive encouragement and credit where it's due. xv) Teacher uses multiple methods and systematically gathers data about student understanding and ability. xvi) B – Teacher uses student work/data, observations of instruction, assignments and interactions with colleagues to reflect on and improve teaching practice. xvii) Teacher guides students to apply rubrics to assess their performance and identify improvement strategies. xviii) Teacher instructs the complex processes, concepts and principles contained in state and national standards using differentiated strategies that make instruction accessible to all students. xix) Teacher structures and facilitates ongoing formal and informal discussions based on a shared understanding of rules and discourse. xx) Teacher designs learning opportunities that allow students to participate in empowering activities in which they understand that learning is a process and mistakes are a natural part of the learning. xxi) Teacher makes lesson connections to community, society, and current events. The teacher poses real-world problems involving community, society and current events for students to solve by applying mathematical reasoning. xxii) Teacher demonstrates an understanding and in-depth knowledge of content and maintains an ability to convey this content to students. xxiii) Teacher maintains on-going knowledge and awareness of current content developments. xxiv) Teacher designs and implements standards-based courses/lessons/units using state and national standards. 3.16 Year Plan A year plan is usually prepared for a number of units to cover a particular topic or set of related ideas for the year. Its main purpose is to put the relevant units into broader perspective and at the same time force a deeper analysis of the content to be taught. Year plan includes concepts, ideas, assessments tasks that you want to provide for your students. This can be in list form, or done as a chart, diagram, flow chart or outline - whatever helps you capture your vision of expectations for your students for the year. It sketches an overall plan for the year and also create a rough year plan with a month by month calendar that provides a plan for how your will incorporate your overall plan into your school year. For the preparation of year plan it is necessary to develop your generative topics/questions/ideas/concepts that will guide your curriculum throughout the year. These may be called themes or big ideas by some curriculum planners. Examples: For standard X: How to solve a quadratic equation? What are sets and functions? What are the properties of matrix addition and multiplication? What are the trigonometric identities and basic proportionality and angle bisector theorems? Finalize your year plan for your portfolio. Select one unit to develop fully.

3.17 Unit Plan A unit plan is usually prepared for a number of lessons to cover a particular topic or set of related ideas. Its main purpose is to put the daily lessons into broader perspective and at the same time force a deeper analysis of the content to be taught. It allows the teacher to ask for special teaching aids. Such aids may require special funds which may take some time to get from the bursary department. The following questions are considered when writing unit plans: i) Why is this unit important? What are the objectives of the unit? What does it have that will appeal to the student? Which are the keys to future progress? ii) What are the central ideas and unifying concepts around which activities may be organized? What should be stressed most? How should the class time allocated to this unit be subdivided? iii) What teaching strategies are appropriate? Is this the first time students are meeting these ideas or is it to give wider perspective to ideas previously introduced? Can students develop the ideas themselves? What materials are available to provide a varied attack on the unit? iv) What previous concept, skills and experiences are needed for this unit? How can the content be modified for students of varying ability? What extra practice can be provided for weak students? What special teaching techniques may be used with them? What can other students do whenever students receive special attention? What enrichment topics should be included for all students? What activity should be assigned to bright students? v) What teaching techniques will best suit this class? What are the difficult spots that require special attention? How did I teach this material previously? Should I change my approach or techniques? What lessons require practical work? vi) What teaching aids will this unit require? What supplementary books or pamphlets would be helpful for students? What models are appropriate? What should the bulleting board display? Are any excursions suitable? Who might be a suitable outside speaker or class participant? vii) What kind of evaluation should I use? What ways are best suited to the content of this unit and this class? viii) What kinds of assignments should the students prepare? Are long-term assignments appropriate? Can the students learn part of the material independently? These questions help the teacher to escape from the textbook. They provide the basis for developing a careful plan of attack that may be put into operation through the daily lesson plans. A unit plan should contain these elements: i) Statement of the objectives; ii) Description of the skills and the previous knowledge of the students; iii) Content outline to be taught and the basic skills and important ideas selected for mastery; iv) Selection of possible learning activities and stating briefly the teaching procedures and techniques; v) A list of materials to be used; vi) Description of the assignments and evaluation instruments. 3.18 Lesson Plan A lesson plan is the preparatory notes on the subjects to be taught on daily basis. It is the layout of how the teacher intends to handle a lesson from the beginning to the end. A lesson plan is the instrument with which a good teacher can effectively perform his daily classroom teaching. A good teacher is expected to plan his lesson on daily basis stating the steps or procedures to follow to achieve the stated objectives. 3.18.1 Elements of a Good Lesson Plan There are many opinions about the different ways of setting down notes, and all have certain advantages and disadvantages. Again you would not need to write notes in the same way for a History lesson as for a lesson in Mathematics. i) However it is clear, that every lesson should have a beginning, a middle and an end. Lesson notes are necessary because they limit teachers to particular bounds in their delivery. ii) Lesson notes make teachers select the facts they teach. iii) Lesson notes are details of the work to be done in one lesson. iv) They are also meant to guide the teacher‘s methods of delivery and activities in the lesson.

3.18.2 Objectives of the Lesson i) A good lesson plan will have specific aims and objectives to be achieved. ii) The objectives must be simple. iii) The objectives must be capable of being achieved within the stated period. 3.18.3 Division of a Lesson Most lessons are made of four parts which are the 1) Introduction 2) Development 3) Conclusion 4) Summary and evaluation. Time of these parts may vary from lesson to lesson but every part must have its own fair share of time. 1) Introduction A good lesson should begin with good introduction, which is interesting and could arouse students‘ interest and attention. A good lesson should have the following qualities. i) Should present the problem to be answered during the lesson. ii) Should be able to tell what the lesson is about. iii) Should make use of teaching aids like pictures and diagram. iv) Should be able to have questions that can be used to revise the work covered in the previous lesson. There are different ways of introducing lesson thus every lesson should not be introduced in the same way. 2) Development Development is the main body of the lesson. This is a period of exposition, when the teacher teaches new materials and this could come in different stages or steps. 3) Conclusion This is towards the end of the lesson. This should be made within a reasonable time and should have enough room for evaluation. Conclusion can take the following forms i) Brief revision of what has been covered in the lesson. ii) Students reporting on what they have gained. iii) Linking the conclusion with the next topic. iv) Giving homework which could serve as a logical outcome of the lesson. 4) Summary and Evaluation This is a brief review of the whole lesson where the teacher goes over the lesson again informing the students the ground covered. After all the steps have been properly covered the teacher gives the students assignments on area covered. When an assignment is given, date of submission must be indicated. Areas not clear in the lesson will be made clearer while attempting the assignment. Assignments must be marked, recorded and scripts returned to the students to enable them correct their mistakes. This is the last stage of the lesson. It deals with finding out the extent to which the teacher has succeeded in imparting the knowledge. This is the stage where the success of the lesson is determined. The teacher will know if the objective of the lesson has been achieved or not. Evaluation involves asking questions from the students based on the topic treated. Students also will know whether they have followed or not. If the students respond to questions very well it means the teacher has succeeded but if otherwise he has failed. A teacher who fails to evaluate his lesson is not a good teacher. 3.19 Speed and Accuracy with Mathematics Work Mathematics Education systems focused on helping students to solve a number of different types of problems raised around with us by using some combination of mental and written knowledge and skills. It takes a typical students hundreds of hours of study and practice to develop a reasonable level of speed and accuracy in performing addition, subtraction, multiplication, and division on integers, decimal fractions, and fractions. Even this amount of instructional time and practice spread out over years of schooling tends to produce modest results. Speed and accuracy decline relatively rapidly without continued practice of the skills. Both speed and accuracy matter when it comes to mathematics facts, but start with accuracy. Learner should memorize the facts even if he is not quick. Of course, if it takes a great deal of time to tell an answer of what is more important speed or accuracy? But in learner should more emphasis on accuracy instead of speed. Practice, and make the facts into games, flash cards and practice sheets. Over time, learner will build speed. Think of riding a bike. You have to learn how to balance and ride the thing before you can race a friend. Help him learn the skill set first and then help him build speed with those skills. 3.20 An understanding of the Components of any 5 skills in Microteaching Microteaching is a unique model of practice teaching. It is a viable instrument for the desired change in the teaching behavior or the behavior potential which, in specified types of real classroom situation, tends to facilitate the achievement of specified types of objectives. The pupil-teachers trained using the microteaching instrument are expected to have a greater range of technical teaching skills to choose from for overcoming day to day classroom teaching problems. Allen (l966) defined micro-teaching as ―Scaled-down teaching encounter in class size and class time.‖ Bush (1966) defined it as ―A teacher-education technique which allows teachers to apply clearly defined teaching skills to carefully prepared lessons in a planned series of five to ten minutes encounters with a small group of real students, often with an opportunity to observe the results on Video-tape‖. Cooper and Stround (1966) defined it as ―A scaled-down encounter in which the intern teaches for a short period of time, to a group of four students on some topic in his teaching subjects.‖ Allen and Ryan (1969) described microteaching as ―A teacher instructs four or five students for a short time and then talks it over with another adult. An experienced observer would emphasize the fact that the teacher concentrated on a specific training skill or technique and utilized several sources of feedback, such a supervisor, the students, the teacher‘s own reflections and the play-back of Video- tapes. The experienced observer would also note that the teacher has an opportunity to repeat the entire process by re-teaching the lesson and again having his performance critiqued and that in the second and subsequent cycles he teaches different.‖ Passi (1976) defined microteaching as ―A training technique which requires student teachers to teach a single concept using specified teaching skill to a small number of pupils in a short duration time. The most important point in microteaching is that teaching is practiced in terms of definable, observable, measurable and controllable teaching skills.‖ According to G.A. Brown (1978) microteaching may be described as ―A scaled down teaching situation in which a teacher teaches a brief lesson to small group of pupils or fellow trainees. The lesson may last from three or four minutes to twenty minutes. In most cases about 10 minutes is preferred. The small group may consist of three or four pupils or peers or up to fifteen pupils or peers.‖ Sharma (1981) defined microteaching as ―A specific teacher training technique through which trainee practices the various teaching skills in a specific situation with the help of feedback with a view to increase the student‘s involvement. Specific situation means small time to practice (5 – 7 minutes), small number of pupils (5-7) and small length of practicing material.‖ 3.20.1 Steps of Microteaching The Micro-teaching programme involves the following steps: Step I Particular skill to be practiced is explained to the teacher trainees in terms of the purpose and components of the skill with suitable examples. Step II The teacher trainer gives the demonstration of the skill in Micro-teaching in simulated conditions to the teacher trainees. Step III The teacher trainee plans a short lesson plan on the basis of the demonstrated skill for his/her practice. Step IV The teacher trainee teaches the lesson to a small group of pupils. His lesson is supervised by the supervisor and peers. Step V On the basis of the observation of a lesson, the supervisor gives feedback to the teacher trainee. The supervisor reinforces the instances of effective use of the skill and draws attention of the teacher trainee to the points where he could not do well. Step VI In the light of the feed-back given by the supervisor, the teacher trainee re-plans the lesson plan in order to use the skill in more effective manner in the second trial. Step VII The revised lesson is taught to another comparable group of pupils. Step VIII The supervisor observes the re-teach lesson and gives re-feed back to the teacher trainee with convincing arguments and reasons. Step IX The ‗teach – re-teach‘ cycle may be repeated several times till adequate mastery level is achieved. 3.20.2 Microteaching Cycle The six steps generally involved in micro-teaching cycle are Plan Teach Feedback Re-plan Re-teach Re-feedback. There can be variations as per requirement of the objective of practice session. These steps are diagrammatically represented in the following figure: Plan Teach

Re-feedback Feedback

Re-teach Re-plan

Diagrammatic representation of a Micro-teaching Cycle 3.20.3 Five Major Teaching Skills in Microteaching It is not possible to train all the pupil teachers in all these skills in any training programme because of the constraints of time and funds. Therefore a set of teaching skills which cuts across the subject areas has been identified. They have been found very useful for every teacher. The set of these skills are: A. Skill of probing questions, B. Skill of explaining and illustrating with examples, C. Skill of reinforcement, D. Skill of stimulus variation, E. Skill of using blackboard A. Skill of Probing Questions Sometimes you ask an open-ended question to get more information and you only get part of what you need. Now it‘s time for a probing question. A probing question is another open-ended question, but it‘s a follow-up. It‘s narrower. It asks about one area. Here‘s an example: ―What topic areas are you interested in?‖ This question would be better than reading off 50 topics to the caller. It‘s a probing question. A few other examples are: ―Are you able to tell me more about the form you received?‖ ―What did you like best about Paris?‖ Probing questions are valuable in getting to the heart of the matter. Components of the Skill of Probing Questions Lower Order Questions: These questions lay emphasis on important facts and information required for the learning task. They do not ensure the understanding of the knowledge. Middle Order Questions: these questions stimulate three mental process – translation, interpretation and application. Translation is the expression of a fact, concept or generalization through different media or feelings. The process of interpretation involves comparison and explaining relationships between ideas, concepts, facts, generalizations, definitions, values and skills. Comparison refers to finding out similarities or dissimilarities between two or more ideas or concepts. Application refers to the use of the knowledge acquired in one area or situation to the solution of some specific problem in another situation. Higher Order Questions: These questions stimulate the highest level of mental processes in pupil. They require analysis, synthesis and evaluation of facts, concepts, generalizations, values and skills. They are very important in classroom teaching because these questions intend to develop the creative and reasoning abilities of the pupils. Analysis refers to examining materials, situations, environments with a view to separate them into components involving inductive and deductive reasoning. Synthesis refers to unique and original interpretation of facts, concepts and generalizations. It also implies interpreting facts in imaginary situations stimulating the pupils to reorganize their present knowledge in hypothetical questions. Evaluation means to judge, justify or defend a position in the perspective of appropriate standards and values. B. Skill of Explaining and Illustrating with Examples Skill of Explaining Explanation is a key skill. Generally, the skill of explanation is complex Explanation is to explain or to give understanding to another person. It leads from the known to the unknown, it bridges the gap between a person‘s knowledge or experience and new phenomena, and it may also aim to show the interdependence of phenomena in a general sable manner. It assists the learner to assimilate and accommodate new data or experience. In a classroom, an explanation is a set of interrelated statements made by the teacher related to a phenomenon, an idea:, etc. in order to bring about or increase understanding in the pupils about it. The teacher should practice more and more of desirable behaviors like using explaining links using beginning and concluding statements and testing pupil understands behaviors like making irrelevant statements, lacking in continuity, using inappropriate vocabulary, lacking in fluency, and using vague words and phrases as far as possible. The explanation serves two purposes: (1) to introduce the subject by giving some background about its usefulness and application; and (2) to describe the subject in a simple, complete, and tantalizing way. The explanation should create a desire to become proficient in the subject under study The Components of Skill of Explaining Involved i) Clarity ii) Continuity iii) Relevance to content using beginning and concluding statements iv) Covering essential points v) Simple vi) Relevant and interesting examples appropriate media vii) Use of inducts, deductive approach, it can be functional, causal or sequential. Skill of Illustrating with Examples The teacher should illustrate the concepts with suitable examples to make the pupils understand difficult, complicated and new concepts. The examples should be concrete, pin-pointed, relevant, simple, understandable and interesting. These examples must lead the students from simple to complex. Examples are necessary to clarify, verify, or substantiate concepts. Both inductive and deductive uses of examples can be used effectively by the teacher. Effective use of examples includes: i) Starting with simple examples and progressing to more complex ones. ii) Starting with examples relevant to students. iii) Relating the examples to the principles or ideas being taught. iv) Checking to see if the objectives of the lesson have been achieved by asking students to give examples which illustrate the main points. C. Skill of Reinforcement Reinforcement is a term that belongs to the stimulus response (S-R) theoretical paradigms. Reinforcement is a theoretical construct. It was first used by Pavlov in connection with his classic experiments with dogs. According to Dictionary of Education by Good, reinforcement is defined as: ―Strengthening of a conditioned response by reintroducing the original unconditioned stimulus‖. ―Increase in response strengthens when the response, leads to the reduction of a drive‖. Reinforcing desired pupil-behavior through the use of positive reinforcing behavior is an integral part of learning process. This skill involves teacher encouraging pupils‘ responses or any desirable behavior using verbal statements like good, continue, etc. or non-verbal cues like a smile, nodding the hand, etc. D. Skill of Stimulus Variation The skillful change in the stimuli is known as the skill of stimulus variation. Just to avoid boredom, it is the teacher‘s skills to stimulate the students, increase their active participation, enthusiasm and spirit of study. The chief aim of the teacher‘s teaching in the class room is to make the lesson impressive and interesting. For this, use of various types of methods and techniques are an essentiality. In order to attract the students, teacher may present various types of stimuli and can function as stimulus. He/she presents various stimuli, and she / he can function as stimulus. He/she presents various stimuli such as movement of the body, gesture, changes in speech, focusing of the feeling, change in the interaction style in the students, pause and change in the order of audio visual aids. He/she can attract the learners by changing all these aspects which function as stimuli. A resourceful teacher, have to develop the skill in him /her to attract and hold the attention of his / her students throughout the teaching. He/she should deliberately change his/ her attention-drawing behavior in class. For the success of any lesson, it is essential to secure and sustain the attention of the pupils learning is optimum when the pupils are fully attentive to the teaching-learning process. How to secure and sustain the attention is main theme of this skill. It is known that attention of the individual tends to shift from one stimulus toothier very quickly. It is very difficult for an individual to attend to the same stimulus for more than a few seconds. Component Skills of Stimulus Variation The behaviors associated with the skill of stimulus variation are; Teacher movement: Varying movements by continuously changing location within the classroom, means making movements from one place to another with some purpose. As for writing on the black board, to conduct experiment, to explain the chart or model, to pay attention to the pupil who is responding to some question etc. This captures pupils‘ attention and every position they are sitting they do not feel the distance. Pupils feel that the teacher is with them .The movements are in a way of reducing the distance from students. Teacher Gestures: Gestures are the movements of the parts of the body the teacher‘s body movements – which communicate certain meaningful ideas to the students. These include movements of head, hand and body parts to arrest attention, to express emotions or to indicate shapes, sizes and movements. All these acts are performed to become more expressive. Changes in Speech Pattern Silence / Pause: The use of pitch in voice projection as a stimulus, should indicate, relative important of information. That‘s it should indicate happiness or sorrow. In other words slowing speech pattern is a way of stimulating pupils‘ interests. Variation can be in volume or accent of voice When the teacher wants to show emotions or to put emphasis on a particular point, sudden or radical changes in tone, volume or speed of the verbal presentation are brought out. The change in the speech pattern makes the pupils attentive and creates interest in the lesson. Do not be monotonous. Speak attractively and energetically. Speak loudly enough for the students at the back to hear what the teacher is saying, but not too loud so that the students at the front get shocked. Pausing: Pausing means ―stop talking‖ by the teacher for a moment. When the teacher becomes silent during teaching, it at once draws the attention of the students with curiosity towards the teacher. The message given at this point is easily received by the pupils. Silence can indicate that what has just said is important or it can indicate that what happened is unacceptable. Silence can also indicate the opportunity allowance for pupils to think and respond to the asked question. Change in Audio-Visual Sequence: A continuous change in the sequence of using audio visual aids concentrates the attention of the pupil upon the teacher. He should use sometime visual and sometimes audio-aids. Vary Methodology: For example use of demonstrations, group work, songs group work projects or individual projects, like in Art and craft, music social studies etc. Introducing variation is because each and every pupil would have something which is interested in. Dramatization, storytelling and jokes are part of stimulating learners at every angle. Give children current reading publications, oral reports, outdoor activities, guest speakers, tests in small groups and individual tests. Different pupils prefer to learn in different modes so a teacher has to vary methodology Variation in Questioning: Questions is also a stimulus as it provides pupils with the opportunity to express. Questions themselves should be varied, from low order and high order questions. .When asked, do not directly answer the question; let the student finish his question first so as to give time to you to prepare for a suitable and right answer; do not cut off the student‘s question or explanation Focusing: It is used to concentrate the attentions of the pupils on some specific point or event. It includes verbal focusing, gesture focusing and verbal or oral gesture focusing. The teacher draws the attention of the pupils to the particular point in the lesson either by using verbal or gesture focusing. In verbal focusing the teacher makes statements like, ―look here‖ listen to me‖ ―note it carefully‖. In gestural focusing pointing towards some object with fingers or underlining the important words on the black board. Varying of Non – verbal Gestures: In the lesson delivery the teacher should be able to use hands, eyes, body or even clap, stamp feet as a way of maintaining pupils interest in the lesson. Gestural focusing: With the use of gestures only i.e. the movements of head and hand the teacher can attract the attention of the students to a particular point. Even if he/she underline the point on the blackboard, the students will be attracted more towards it when appropriate gestures are used for this purpose. Verbal and gestural focusing: In the verbal focusing, the words are repeated again and again to concentrate the attention such as ―Look here, attend to me, don‘t see outside because I am going to announce something important‖. In the gesture focusing, the attention of the pupils is concentrated with the help of gestures towards some desirable direction or an object When both verbal and gestures focusing devices are used to focus the attention of students, it has more impact on them. Student Movement: The teacher can change the focus of attention of the students by involving them in physically doing something. He/she can involve them in experiments, handling apparatus, or dramatization. By doing so, he/she can sustain their interest in the teaching-learning process. Variation in Interaction Style: Interaction between the teacher and the pupils is very essential in the class room teaching otherwise it becomes monotonous. Therefore the style of interaction in the class room should go on changing. Interaction style can be used as stimulus, for example teacher/student or student/student or student /teacher .Variation in interaction so as to stimulate students‘ interests. Some variation interaction styles are used in group work or individual work. E. Skill of using Blackboard Blackboard is the main teaching tool in the Indian schools. It is used to provide clarity, variety, sequences and acceptability in teaching. Operation Blackboard‘ aspires to equip every single school in the country with blackboards which implies that every potential teacher should be proficient in the use of blackboard. Components of the Skill of using Blackboard i) Legibility in handwriting ii) Neatness in writing iii) Brevity and simplicity in writing iv) Topic matter or diagram should be relevance v) Orderliness in writing vi) Bold letters, capital letters, bracketing, punctuation marks, putting trick marks should have variation vii) The Blackboard should be cleaned before and after the class viii) Chalk should be held properly ix) Pointer should be used to focus thinks x) Blackboard should not be rubbed quickly. Students must be given time to copy 3.21 Link Practice When mastery has been attained in various skills, the teacher trainee is allowed to teach the skills together. This separate training programme to integrate various isolated skills is known as link practice. It helps the trainee to transfer effectively all the skills learn in the micro teaching sessions. It helps to bridge the gap between training in isolated teaching skills and the real teaching situation faced by a student teacher. Desirable number of pupils: 15 – 20. Preferable duration: 20 minutes. Desirable number of skills: 3 - 4 skills. Link practice or integration of skills can be done in two ways. i) Integration in Parts: 3 or 4 teaching skills are integrated and transferred them into a lesson of 15 – 20 minutes duration and again 3 or 4 skills are integrated and are transferred all the skills to one lesson. ii) Integration as a Whole Student teacher integrates all the individual teaching skills by taking them as a whole and transferred them into a real teaching situation. Advantages of Microteaching i) It helps to develop and master important teaching skills. ii) It helps to accomplish specific teacher competencies. iii) It caters the need of individual differences in the teacher training. iv) It is more effective in modifying teacher behavior. v) It is an individualized training technique. vi) It employs real teaching situation for developing skills. vii) It reduces the complexity of teaching process as it is a scaled down teaching. viii) It helps to get deeper knowledge regarding the art of teaching. Disadvantages of Microteaching i) It is skill oriented; content not emphasized. ii) A large number of trainees cannot be given the opportunity for re-teaching and re-planning. iii) It is very time consuming technique. iv) It requires special classroom setting. v) It covers only a few specific skills. vi) It deviates from normal classroom teaching. vii) It may raise administrative problems while arranging micro lessons. 3.22 Individual Instruction Method This method is a programmed instruction in which the learning programmes are presented in carefully structured steps and the steps depend on the individual student and the nature of materials to be learned. For example the pace of learning depends on individual students. 3.22.1 Advantages of the Individualized Instructional Method i) It allows the student to go at his own pace. ii) It makes the student to participate. iii) It gives the teacher quick knowledge of individual student i.e. whether the lesson is understood or not, since test is usually given at the end of every lesson. iv) It can be used effectively to make up for lack of background by particular member of the class. v) It reduces a student‘s anxiety as he depends on himself. 3.22.2 Disadvantages of the Individualized Instructional Method i) It is time consuming. ii) It is highly demanding of equipments and materials. iii) It requires very little or no interaction among the students. 3.23 Let Us Sum Up This unit provides several instructional methods and strategies of teaching mathematics which are more beneficial in compare to the present traditional system of instruction of instruction. Inductive and deductive, analytic and synthetic, lecture, lecture cum demonstration, Heuristic, laboratory, project, craft centered methods and problem solving methods are popular methods of teaching mathematics. Teacher can use the drill and team teaching techniques during the teaching period. Teacher explains to their students about the year plan, unit plan and lesson plan. This unit also provides the knowledge about the skills in microteaching. 3.24 Answers to Check Your Progress i) Illustrate the inductive and deductive teaching methods in mathematics. ii) Give a short note on analytic and synthetic teaching method in mathematics. iii) Discuss the problem solving method and its limitations. iv) What is team teaching? Describe its advantage and disadvantage in teaching mathematics. v) What do you mean by instructional strategy of teaching mathematics? Explain drill. vi) What are the characteristics and roles of mathematics teacher? vii) What do you mean by year plan and unit plan? viii) Write an essay on lesson plan. ix) Describe the five skills of microteaching. x) What is individual instructional method? 3.25 References Allen, Dwight, and Ryan, Kevin. (1969). Microteaching. Reading, MA: Addison-Wesley. Allen, Dwight, and Wang, Weiping. (1996). Microteaching. Beijing, China: Xinhua Press. Allen, Mary E., and Belzer, John A. (1997). ―The Use of Microteaching to Facilitate Teaching Skills of Practitioners Who Work with Older Adults.‖ Gerontology and Geriatrics Education 18 (2):77. Bransford, J., A. Brown, and R. Cocking, Eds. (1999). How People Learn: Brain, Mind, Experience, and School. National Research Council. National Academy Press, Washington, DC. Available online at: http://www.nap.edu/openbook/0309065577/html/index.html Fakude, R. A. (1981): Teaching Arithmetic and Mathematics in the Primary School, Ibadan: University Press, Ibadan. N. R. Gamsky, (1970). Team Teaching, Students Achievement and Attitude. Journal of Experimental Education, 39 (1), pp. 42-45. Odili, G. A. (1986): Teaching Mathematics in the Secondary Schools, Anachnna Educational Books. Passi, B. K. and Lalitha, M. S. (1977). Microteaching in Indian Context. Deptt. of Education, Indore university, Indore Passi, B. K. and Shah, M. M. (1976). Microteaching in Teacher Education, CASE Monograph No. 3, Baroda Passi, B.K. (1976). Becoming Better Teachers. Baroda : Centre for Advanced Study in Education, M. S. University of Baroda. Sharma, N.L. (1984). Micro-teaching: Integration of Teaching Skills in Ssahitya Paricharya, Vinod Pustak Mandir, Agra. Sharma, R. A.: Technology of Teaching, International Publishing House, Meerut. Singh, L. C. and Shama, R. D. (1987).: Microteaching Theory and Practice, Deptt. of Teacher Education, NCERT, New Delhi The Webster Comprehensive Dictionary. (1982). Encyclopedic ed. Vol. I & II. Ferguson Publishing Company. Chicago. Trowbridge. (1986). Becoming a Secondary School Science Teacher, 4th ed. Merrill Publishing Company. Columbus. Vaidya, N. (1970). Micro – teaching: An Experiment in Teacher Training. The Polytechnic Teacher, Technical Teacher, Technical Training Institute, Chandigarh. UNIT 4 LEARNING STRATEGIES IN MATHEMATICS Structure 4.1 Introduction 4.2 Objective 4.3 Group Learning inside the Classroom 4.4 CAI 4.4.1 How Is CAI Implemented? 4.5 Computer Games 4.5.1 The Advantages & Disadvantages of Computer Gaming 4.6 Problem Solving 4.6.1 Problem – Solving Techniques 4.7 Group Learning outside the Classroom 4.7.1 Clubs and Societies 4.7.2 Social Events 4.7.3 Academic and Learning Skills 4.7.4 Welfare and Advice 4.7.5 Tutoring 4.7.6 Summary of Teaching outside the Classroom 4.8 Field trips 4.8.1 How to Plan an Educational Field Trip? 4.8.2 Importance of Field Trips in Education 4.9 Surveys 4.9.1 Advantages of Surveys 4.9.2 Disadvantages of Surveys 4.10 Projects 4.11 Individual Learning outside the Classroom 4.11.1 Office hours learning 4.12 Assignments 4.13 Instructional materials – Need and Importance 4.14 Web Based Learning 4.14.1 Advantages of Web Based Learning 4.14.2 Disadvantages of Web Based Learning 4.15 Let Us Sum Up 4.16 Answers to Check Your Progress 4.17 References 4.1 Introduction The present traditional system of instruction is highly group oriented. Educators and administrations have always stressed the need for an instructional system suited to the needs and abilities of the individual. This system gives scope to the learner to work at his own paces. An instructional system is individualized when the characteristics of each learner play a major role in the selection of objectives materials, procedures and time. 4.2 Objectives On completion of this unit students will be able to …. 10. understand the CAI; 11. know the nature of group learning inside the classroom; 12. get educational benefit of computer games; 13. aware the problem solving strategy; 14. comprehend the group learning outside the classroom; 15. realize the importance of field trip, survey and project; 16. know the nature of individual learning inside the classroom; 17. make assignment and understand the need and importance of instructional material. 4.3 Group Learning inside the Classroom Group learning inside the classroom is more than merely having students sit together, helping the others do their work. Directing students who finish their work early to assist others isn‘t a form of cooperative learning either. Neither is assigning a group of students to ―work together‖ unless you assure that all will contribute their fair share to the product. Steps for setting up group learning experiences: i) Develop a positive classroom environment. Devise ways for students to become acquainted early in the year. Have them work on a mural, news letter, play or other project. Mode land encourages polite, respectful behavior to ward others. Reward students for such social skills as helping others, giving and accepting praise, compromise, etc. ii) Previous to organizing collaborative groups and assigning academic tasks, develop a cooperative climate. This can be accomplished by engaging students in fun team-building activities in which they support each other in a team effort to achieve non-academic or easily achieved academic goals. These activities might take the form of noncompetitive, active games such as those described in the books like the one titled. iii) Consider upcoming academic tasks and determine the number of students who will be assigned to each group. The size of the group will depend on the students‘ ability to interact well with others. Two to six students usually comprise a group. iv) Decide how long the groups will work together. It may range from one task, to one curriculum unit, to one semester, to a whole year. Most often the teacher will vary the composition of groups every month or two so that each student has a chance to work with a large number of class mates during the term or year. v) Determine the academic and behavioral/interpersonal objectives for the task. vi) Plan the arrangement of the room for the upcoming group – oriented tasks. Arrange group seating so that students will be close enough to each other to share materials and ideas. Be sure to leave yourself a clear access lane to each group. vii) Prepare materials for distribution to the group. Indicate on the materials that students are to work together. Avoid work activities that don‘t really encourage (or require) students to actively collaborate in a group. When student are working on independent tasks, simply clustered at tables, a revision is necessary. viii) Determine roles for group members. In addition to cooperating and ―brainstorming‖ with others, each group member should be assigned a duty to perform during the project. 4.4 CAI – Computer Assisted Instruction Acronym for computer-based training (CBT), a type of education in which the student learns by executing special training programs on a computer. CBT is especially effective for training people to use computer applications because the CBT program can be integrated with the applications so that students can practice using the application as they learn. Historically, CBTs growth has been hampered by the enormous resources required: human resources to create a CBT program and hardware resources needed to run it. However, the increase in PC computing power, and especially the growing prevalence of computers equipped with CD – ROMs, is making CBT a more viable option for corporations and individuals alike. Many PC applications now come with some modest form of CBT, often called a tutorial. Computer – assisted instruction (CAI) refers to instruction or remediation presented on a computer. Many educational computer programs are available online and from computer stores and textbook companies. They enhance teacher instruction in several ways. Computer programs are interactive and can illustrate a concept through attractive animation, sound, and demonstration. They allow students to progress at their own pace and work individually or problem solve in a group. Computers provide immediate feedback, letting students know whether their answer is correct. If the answer is not correct, the program shows students how to correctly answer the question. Computers offer a different type of activity and a change of pace from teacher – led or group instruction. Computer – assisted instruction improves instruction for students with disabilities because students receive immediate feedback and do not continue to practice the wrong skills. Many computer programs can move through instruction at the student‘s pace and keep track of the student‘s errors and progress. Computers capture the students‘ attention because the programs are interactive and engage the students‘ spirit of competitiveness to increase their scores. Also, computer – assisted instruction moves at the students‘ pace and usually does not move ahead until they have mastered the skill. Programs provide differentiated lessons to challenge students who are at risk, average, or gifted. CAI package was an interaction media play an important role in computer context act as a teacher. CAI package can provide a more stability presentation. Audience can obtain the message and information in dynamic form through CAI package. CAI package will be present in various computer interfaces either in two or three dimension. CAI package also present in good visual looking with nice graphic and animation. CAI package also can included hundred of note in a small right package. The main objective of this study is to introduce a fundamental of this digital image and a few common methods to process a digital image which have a very wide range of application in our modern world. Personal Digital Assistants (PDAs) such as Palm Pilots and Pocket PCs are handheld computers that serve as an organizer of personal and professional information. PDAs are now being broadly accepted in a variety of educational settings. PDAs come with software that allows educators and students to perform a range of tasks, including synchronizing data with desktop or laptop computers, accessing e- mail, managing appointments and course assignments. PDAs have been used to augment and supplant computers in classrooms because they are readily available, inexpensive, and easy for educators to use (Ray, 2002). PDAs are effective classroom organizational tools for educators. Student can use the PDA as an education tool for learning with this developed CAI package. Computer-assisted instruction (CAI) refers to instruction or remediation presented on a computer. Many educational computer programs are available online and from computer stores and textbook companies. They enhance teacher instruction in several ways. Computer programs are interactive and can illustrate a concept through attractive animation, sound, and demonstration. They allow students to progress at their own pace and work individually or problem solve in a group. Computers provide immediate feedback, letting students know whether their answer is correct. If the answer is not correct, the program shows students how to correctly answer the question. Computers offer a different type of activity and a change of pace from teacher – led or group instruction. Computer-assisted instruction improves instruction for students with disabilities because students receive immediate feedback and do not continue to practice the wrong skills. Computers capture the students‘ attention because the programs are interactive and engage the students‘ spirit of competitiveness to increase their scores. Also, computer assisted instruction moves at the students‘ pace and usually does not move ahead until they have mastered the skill. Programs provide differentiated lessons to challenge students who are at risk, average, or gifted. 4.4.1 How Is CAI Implemented? Teachers should review the computer program or the online activity or game to understand the context of the lessons and determine which ones fit the needs of their students and how they may enhance instruction. 1. Can this program supplement the lesson, give basic skills practice, or be used as an educational reward for students? 2. Is the material presented so that students will remain interested yet not lose valuable instruction time trying to figure out how to operate the program? Does the program waste time with too much animation? 3. Is the program at the correct level for the class or the individual student? 4. Does this program do what the teacher wants it to do (help students organize the writing, speed up the writing process, or allow students to hear what they wrote for editing purposes)? Teachers should also review all Web sites and links immediately before directing students to them. Web addresses and links frequently change and become inactive. Students might become frustrated when links are no longer available. Writing programs are beneficial to writing instruction because they allow students to learn in a variety of ways and can speed up the writing process. With proper training, students can learn to focus on the message instead of the mechanics. Computer – assisted instruction (CAI), a program of instructional material presented by means of a computer or computer systems. The use of computers in education started in the 1960s. With the advent of convenient micro computers in the 1970s, computer use in schools has become widespread from primary education through the university level and even in some preschool programs. Instructional computers are basically used in one of two ways: either they provide a straightforward presentation of data or they fill a tutorial role in which the student is tested on comprehension. 4.5 Computer Games It is true that the most striking feature of the present century is the process of science. Many great inventions have been born and one of the greatest advances in modern technology has been the invention of computer and computer games. Computer games are being used to provide numerous helpful benefits. They help us to treat a variety of disorders and disabilities, to make us have fun and entertainment, and serve a range of educational functions. When the computer was first invented, it was only supposed to become a really advanced calculator. For about thirty years ago (now it is 1999), when you mentioned the word game, most people would think of board games, like chess, backgammon and monopoly. As the technique evolved, more and more people used computers. An idea of using the computer for entertainment popped up. The first computer games were very simple and were in black and white. The very first computer game was named Colossal Cave, and was built up only by text. The text told you where you were, and on your command you could move or pick up something. The most important aspect of playing computer games is treating some diseases. Researchers are finding that computer games can be actually tweaked to treat people with phobias. The belief behind this treatment method is that exposing people to the source of their fear with in a controlled environment may actually lead to a cure. Two of the most common phobias that have been treated with computer games are a fear of confined spaces and heights. Teachers have found that computer games specifically prepared for language disabilities provide a unique way for children to overcome their disabilities. Another important aspect is that computer games providing us to have fun. When we are very bored tired or stressed, computer games are good source of enjoyment. Moreover, we have fun playing computer games because they give us time to be with our friends. It is a great opportunity to socialize. We can talk with our friends about other things at the same time. In addition, when we spend time with our friends and relax with friends, we have more energy for other work. Last but not least, computer games serve a range of educational functions. Computer games encourage different ways of learning, imagination, creativity and exploration. For example, Simulation games could be used as a means of preparing learners for the world of work. Computer games also help pupils to develop key learning skills such as cognitive process, logical thinking and independent decision making. To sum up, computer games have a main role in our lives. We can use of them for therapy disease, having fun and educating better.

4.5.1 The Advantages & Disadvantages of Computer Gaming Computer gaming is much maligned at times for the level of violence in some games and for the amount of time that it claims that could be put to more productive uses. Some advantages of gaming are that it has educational applications, increases visual processing of information and teaches problem solving skills. Advantages i) Educational Benefits Games have educational uses, for example, teaching economic skills like negotiating and purchasing strategies, and are used in college level economics courses to do so. For example, many games allow players to accumulate points or money and then purchase things that will advance their interests in the game. These features simulate markets, and in fact can become commodities outside the game. ii) Information Handling Gaming teaches people to process visual information more quickly than non-gamers. In a society where visual media is pushing more and more information at the audience all the time, this can give gamers an edge in processing it. iii) Behavioral Problem Solving Gaming teaches problem solving skills. Players are given a set of rules to follow and must figure out the best strategy for achieving the goals set in the gaming context. If the game has multiple players or teams, the problem solving is done in a social context, which teaches people to communicate and cooperate. Disadvantages i) Violence Created Violence in games has an unquestionable negative effect on players, making them more likely to be violent themselves, according to Craig Anderson of the American Psychological Association. He points out that experiments in the lab and in the field, and longitudinal and cross-sectional studies have all shown this to be true. The interactive nature of the games makes them more influential in teaching violence than more passive media like television or books. ii) Time Wasted Time spent in front of the computer has increased for kids ages 8 to 18, and media use now stands at an average of 7.5 hours per day, taking into account computers, television and other media, as reported in a study by the Kaiser Foundation. Much of the time is spent multitasking or combining TV and video games, for example. Half surveyed said they had a computer game console of their own in their room. In addition time spent reading has decreased. 4.6 Problem Solving A problem is any situation where you have an opportunity to make a difference, to make things better; and problem solving is converting an actual current situation into a desired future situation. Whenever you are thinking creatively and critically about ways to increase the quality of life (or avoid a decrease in quality) you are actively involved in problem solving. Problem solving is related to other terms such as thinking, reasoning, decision making, critical thinking, and creative thinking. Problem solving is an essential, if sometimes neglected, skill that demands attention from the earliest grades. Students must learn to question and apply mathematical concepts to problem – solving situations on a regular basis. These strategies are different problem solving techniques students use when faced with a challenging problem. Here are explanations of some problem solving strategies used in elementary problem solving. 4.6.1 Problem – Solving Techniques These techniques are usually called problem solving strategies. Abstraction: solving the problem in a model of the system before applying it to the real system. Analogy: using a solution that solves an analogous problem Brainstorming: (especially among groups of people) suggesting a large number of solutions or ideas and combining and developing them until an optimum is found. Divide and Conquer: breaking down a large, complex problem into smaller, solvable problems. Hypothesis Testing: assuming a possible explanation to the problem and trying to prove (or, in some contexts, disprove) the assumption. Lateral Thinking: approaching solutions indirectly and creatively. Means – ends Analysis: choosing an action at each step to move closer to the goal. Method of Focal Objects: synthesizing seemingly non- matching characteristics of different objects into something new. Morphological Analysis: assessing the output and interactions of an entire system. Proof: try to prove that the problem cannot be solved. The point where the proof fails will be the starting point for solving it. Reduction: transforming the problem into another problem for which solutions exist. Research: employing existing ideas or adapting existing solutions to similar problems. Root cause Analysis: eliminating the cause of the problem Trial – and – Error: testing possible solutions until the right one is found. 4.7 Group Learning outside the Classroom Learning outside the classroom will help students to develop personally. A successful travel experience will positively change their attitudes. Learning outside the classroom is an enriching experience that will make a lasting impact. Real life learning outside the classroom It‘s often difficult to comprehend information without firsthand experience. In particular young people find it hard to make a connection with something they‘ve read about as it often feels like it has no relevance to them. Learning outside the classroom provides a personal link that makes information relevant and memorable. Learning outside the classroom develops knowledge as well as skills. Travel inspires learning. It introduces students to new people and places, to spectacular scenery, unusual wildlife and makes them think about how the world is changing. Learning outside the classroom also teaches important practical skills such as team work, money management and leadership, all of which are important for further education and the workplace. 4.7.1 Clubs and Societies At GIC and the University of Glasgow there are clubs and societies to cater for all interests. This is a great way to make friends, socialize with students at the College and the University, and be part of a team. Some of the clubs and activities we‘ve organized at the College include: Tea club singing and music Foot ball Chess Debating You can take out membership at the University sports centre and join any of their sports clubs. You can also join cultural or special interest groups at the University. These are run by fellow students and are a great way to find people with similar interests and hobbies to you. 4.7.2 Social Events We organize day trips and events throughout the year. There are some popular social events mentioned belowe. i) Edinburgh Castle and City Tour ii) Stirling Castle and Loch Lomond Tour iii) Edinburgh Military Tattoo iv) Highland Games v) Visit to local farmers market vi) Walking tours around Glasgow vii) Ceilidh dancing 4.7.3 Academic and Learning Skills If you think that you need some extra support with your mathematics or want to know the meaning of the science vocabulary in your seminars, we can help! Our academic staffs offer workshops and sessions to support all areas of your learning. These include: Pronunciation Presentations Informal & formal English, Mathematics support Science & Engineering vocabulary Exam revision 4.7.4 Welfare and Advice It can be quite difficult to work out how to apply for a new visa or get a job in the UK, so we run many different drops – in and information sessions to answer your queries. For example: How to renew my visa volunteering Private accommodation working in the India. Our Student Services staffs are also available to answer your questions and are happy to help. 4.7.5 Tutoring Tutoring may be needed and expected, especially in introductory courses. It should be provided before difficulties become overwhelming. In light of the varied backgrounds and expectations of students in most classrooms, it is essential that you know how to refer students to academic and non-academic resources they are likely to need. Tutoring may be needed and expected, especially in introductory courses. It should be provided before difficulties become overwhelming. Accordingly, you can be most helpful by providing students with opportunities for obtaining prompt feedback, comments, and assessment (short papers, quizzes, lab reports, etc.) early in the term. You also may have to help students revise their expectations of tutoring. Students from different backgrounds might view tutoring in very different ways. Some students come to tutoring for clarification, some expect to be shown how to get the answers, and others come to be shown the answers. It is important to explain what tutoring and problem sessions can do; what topics, questions, and problems will be addressed; and what students should do before, during, and after such sessions. Scheduling tutoring sessions before or after assignments are due emphasizes the function of the sessions. A stigma can be attached to seeking tutoring services because needs or other deficiencies in preparation are viewed as signs of innate inability. However, the students who do best are usually those who take advantage of every learning situation. Tutoring and problem – solving sessions should be portrayed positively. These sessions are frequently the best opportunities for students to get to know the teachers and to see how they think. Methods and answers are important, but personal contact can be crucial to a student‘s success. 4.7.6 Summary of Teaching outside the Classroom  Assist students in learning problem – solving methods.  Mentor students and encourage their interest in your field.  Make office hours accessible to students by accommodating their personal concerns (e.g., night safety) and schedule conflicts.  Encourage students to seek tutoring as needed. 4.8 Field Trips A field trip is a type of learning experience provided for the purpose of experiencing something that cannot be encountered within the classrooms. When specific things are to be learned or gained the best method of learning is making a trip to the place concerned itself. If an educational field trip is organized and objectively planned it results not only in giving first hand information but also broadens and changes the attitudes of pupils in the right direction. A field trip or excursion, known as school trip in the UK and New Zealand and school tour in Ireland, is a journey by a group of people to a place away from their normal environment. The purpose of the trip is usually observation for education, non- experimental research or to provide students with experiences outside their everyday activities, such as going camping with teachers and their classmates. The aim of this research is to observe the subject in its natural state and possibly collect samples. In western culture people first come across this method during school years when classes are taken on school trips to visit a geological or geographical feature of the landscape, for example. Much of the early research into the natural sciences was of this form. Charles Darwin is an important example of someone who has contributed to science through the use of field trips. It is generally for students in grades K – 12 and college. Field trips are generally domestic, but for high school students, it becomes international. 4.8.1 How to Plan an Educational Field Trip? Educational field trips can be a wonderful experience for both teacher and student, but a poorly planned trip creates chaos. When planning a trip it is important to remember that most children if left to their own devices could end up in trouble. i) Decide on an objective for the field trip. What do you want students to learn or accomplish while they are on the field trip? Select a destination. Contact the location to ensure field trips are welcome. ii) Determine what students will do once they are there, how many chaperons you will need, how you will get to your destination, how much the trip will cost and which students will be allowed to participate. iii) Submit paperwork to school/ district administration outlining why you feel a field trip is necessary. Be persuasive as possible. Fill out and submit all field trip request forms. iv) Find parent chaperons. Inform the place of your destination: arrival time and coordinate activities and meals. v) Arrange transportation to and from the location. Plan activities to occupy children‘s time and attention. vi) Pass out permission slips that your school or district provides. Collect permission slips at least two days before the field trip. vii) Hand out a list of required items, and remind children of what is acceptable behavior for the field trip. Go over the outline of the field trip with students. Allow the students to ask questions. 4.8.2 Importance of Field Trips in Education Field trips require significant planning and coordination for teachers and administrators, but students often see a field trip as a free day out of the classroom. However, students will likely have an educational experience that they never could have had in the classroom. i) Interactive Learning Field trips help students interact with what they are learning. The experience goes beyond reading about a concept; students are able to see it, manipulate it or participate in it physically. Students are able to see elements with their eyes rather than reading about it and believing what they are told because it‘s in print. Visiting a farm and milking a real cow is much more powerful than reading about milking a cow. ii) Entertainment Field trips provide entertainment for students. They often serve as a powerful motivator for students, stirring up excitement as the trip nears. Breaking away from the routine provides kids with a refresher that might make them more focused back in the classroom. Learning and fun make a great combination. Field trips are considered fun, but the children learn as well, whether they realize it or not.

iii) Extension of Classroom Study Field trips take the book learning from the classroom and extend it to life. Students often question the importance of topics they study in class. Field trips, particularly for older students, can answer the question of how learning can be applied in life. For example, a field trip to a bakery proves that measurement and chemistry apply beyond the science classroom. There are also plenty of opportunities to incorporate the field trip experience back into classroom activity after returning to school. Through presentations, slide shows and answering questions, the kids can instill the lessons garnered on the field trip. iv) Social Interaction Leaving the classroom for a field trip places the kids in a different social environment. They encounter a new set of adults and possibly other children during the course of the average field trip. These interactions teach them how to behave in different settings. They employ more self-control because it is a less contained environment than the classroom. It fosters a sense of teamwork and community among the students as they experience a field trip together. v) New Experiences Many children don‘t get to experience the typical field trip locations with their families. A school trip gives students the opportunity to experience new venues. Because of money constraints or lack of resources, not all parents are able to take their kids to zoos, museums and other field trip destinations. While field trips take a great deal of work and energy, broadening the horizons of the students is worth it. 4.9 Surveys In statistics, survey methodology is the field that studies the sampling of individuals from a population with a view towards making statistical inferences about the population using the sample. Polls about public opinion, such as political beliefs, are reported in the news media in democracies. Other types of survey are used for scientific purposes. Surveys provide important information for all kinds of research fields, e. g., marketing research, psychology, health professionals and sociology. A survey may focus on different topics such as preferences (e. g., for a presidential candidate), behavior (smoking and drinking behavior), or factual information (e. g., income), depending on its purpose. Since survey research is always based on a sample of the population, the success of the research is dependent on the representativeness of the population of concern (see also sampling (statistics) and survey sampling). Survey methodology seeks to identify principles about the design, collection, processing, and analysis of surveys in connection to the cost and quality of survey estimates. It focuses on improving quality within cost constraints, or alternatively, reducing costs for a fixed level of quality. Survey methodology is both a scientific field and a profession. Part of the task of a survey methodologist is making a large set of decisions about thousands of individual features of a survey in order to improve it. 4.9.1 Advantages of Surveys They are relatively easy to administer. Can be developed in less time compared with other data- collection methods? Can be cost- effective? Few ‗experts‘ are required to develop a survey, which may will increase the reliability of the survey data. If conducted remotely, can reduce or obviate geographical dependence. Useful in describing the characteristics of a large population assuming the sampling is valid. Can be administered remotely via the Web, mobile devices, mail, e-mail, telephone, etc? Efficient at collecting information from a large number of respondents. Statistical techniques can be applied to the survey data to determine validity, reliability, and statistical significance even when analyzing multiple variables. Many questions can be asked about a given topic giving considerable flexibility to the analysis. Support both between and within- subjects study designs. A wide range of information can be collected (e. g., attitudes, values, beliefs, and behavior). Because they are standardized, they are relatively free from several types of errors. 4.9.2 Disadvantages of Surveys The reliability of survey data may depend on the following: Respondents‘ motivation, honesty, memory, and ability to respond: Respondents may not be motivated to give accurate answers. Respondents may be motivated to give answers that present themselves in a favorable light. Respondents may not be fully aware of their reasons for any given action. Structured surveys, particularly those with closed ended questions, may have low validity when researching affective variables. Self-selection is biased. Although the individuals chosen to participate in surveys are often randomly sampled, errors due to non- response may. That is, people who choose to respond on the survey may be different from those who do not respond, thus biasing the estimates. For example, polls or surveys that are conducted by calling a random sample of publicly available telephone numbers will not include the responses of people with unlisted telephone numbers, mobile (cell) phone numbers, people who are unable to answer the phone (e. g., because they normally sleep during the time of day the survey is conducted, because they are at work, etc.), people who do not answer calls from unknown or unfamiliar telephone numbers. Likewise, such a survey will include a disproportionate number of respondents who have traditional, land - line telephone service with listed phone numbers, and people who stay home much of the day and are much more likely to be available to participate in the survey (e. g., people who are unemployed, disabled, elderly, etc.). Question design and survey question answer- choices could lead to vague data sets because at times they are relative only to a personal abstract notion concerning ―strength of choice‖. For instance the choice ―moderately agree‖ may mean different things to different subjects, and to anyone interpreting the data for correlation. Even ‗yes‘ or ‗no‘ answers are problematic because subjects may for instance put ―no‖ if the choice ―only once‖ is not available. 4.10 Projects The project method of education is the assignment of a purposeful activity that is intended to stimulate the student‘s wholehearted interest. The project method originated in vocational schools around the turn of the twentieth century as a supplement to regular methods of instruction. Its purpose was to provide real life application of principles already learned in the classroom. The theory behind the project method was articulated by philosopher John Dewey and one of his foremost interpreters William Heard Kilpatrick. Dewey argues that the purpose of education is to provide students with experiences that sustain and enhance their growth. (Dewey does not use the term ―project.‖) Experience, for Dewey, is a process, both active and passive. ―When we experience something,‖ Dewey states, ―we act upon it, we do something with it; then we suffer or undergo the consequences. We do something to the thing and then it does something to us in return‖. The result of this trying and undergoing is learning – learning from experience. In the early 20th Century, William Heard Kilpatrick expanded the project method into a philosophy of education. His device is child - centered and based in education, proponents of the project method attempt to allow the student to solve problems with as little teacher direction as possible. The teacher is seen more as a facilitator than a deliver of knowledge and information. Students in a project method environment should be allowed to explore and experience their environment through their senses and, in a sense, direct their own learning by their individual interests. Very little is taught from textbooks and the emphasis is on experiential learning, rather than rote and memorization. A project method classroom focuses on democracy and collaboration to solve ―purposeful‖ problems. Kilpatrick devised four classes of projects for his method: construction (such as writing a play), enjoyment (such as experiencing a concert), problem (for instance, discussing a complex social problem like poverty), and specific learning (learning of skills such as swimming). 4.11 Individual Learning outside the Classroom 4.11.1 Office Hours Learning Because students often are reluctant to visit a teacher‘s office to discuss their concerns, some teachers have held their office hours in more public places such as bars, which they thought would provide a more relaxing and informal atmosphere. Although these teachers reported that more students came to see them as a result of holding their office hours in these places, some students avoided meeting their teachers in this situation. Consider, for example, a female student whose male teacher holds office hours in a bar. The teacher has put the student in a situation which may make her feel that she is the object of the teacher‘s personal, rather than professional, attention (which undermines the intellectual climate goals of the university). Therefore, while you might consider offering some office hours in non-traditional places, be careful that you choose places that are neutral and non-threatening to students. Coffee shops and student unions are some possible settings which are less intimidating. Also consider the time that you hold your office hours. Vary the time when you meet with students, so that students who are busy or employed may find a time which works with their schedules. Offer appointments to students whose schedules do not match yours. If you hold office hours late in the afternoon or in the evening, when there are few people in the building, you may make students feel uncomfortable. Female students may be concerned for their personal safety if they have to walk to your building after dark, or if they have to enter a darkened building. When meeting with students, keep your door open or slightly open unless there is a third person in the room. By keeping the door open, you create a less personally threatening atmosphere in your office. One way to make students more comfortable when they come to your office is to offer both group and individual office hours. Students who typically avoid one-on-one office meetings with their professors might be more likely to come if they know that all the attention in the meeting will not be focused on them. For example, if you find that several students make similar mistakes in their homework sets, suggest that they come to your office together, if possible, for a mini-tutorial in a workshop format. The students will realize that they are not alone in their difficulties and can learn from each other’s mistakes. 4.12 Assignments The meaning of assignment is 1. the act of assigning. 2. something, such as at ask, that is assigned. See Synonyms at task. 3. a position or post of duty to which one is assigned. The Assignment is the most common method of teaching especially in teaching of Science. It is a technique which can be usually used in teaching and learning process. It is an instructional technique comprises the guided information, self learning, writing skills and report preparation among the learners. The Assignment method is an important step in teaching and learning process (Douglas). Assignments helps to create ability to gather and understand credible sources, the ability to document sources, and the ability to present your information in a way that ordinary people can understand will be extremely beneficial to your future. Ability to write good essays and course work‘s enable students to receive higher marks and get good jobs after the graduation. To Write an effective assignments read assignment carefully as soon as you receive it. Do not put this task off – reading the assignment at the beginning will save you time, stress, and problems later. An assignment can look pretty straightforward at first, particularly if the instructor has provided lots of information. That does not mean it will not take time and effort to complete. You may even have to learn a new skill to complete the assignment. Do not feel compelled to answer every question. Pay attention to the order of the questions. Be concise write effectively and furiously. Once you start writing assignments its goes on, you need to prepare a specific schedule to overcome it. If you have problem in understanding the question, rewrite the question in your own words. Translate the whole question into your own words. Try to avoid using any of the same wordings as in the question. You will probably find that your translation will be double the length of the original question. No matter! – It will help you to fully understand the point or emphasis of the question. If you have any difficulty in translating the question into your own words the dictionary will help. This translation and redefinition should really help you in the next stages of assignment writing. Advantages of Assignments i) Provides opportunity in self learning for the students ii) Better learning experiences will be gained when combined with other science teaching methods. iii) Assignment provides sufficient flexibility in learning pace of the students. The slow learners too adapt with this method. iv) Teachers‘ interruption is very much reduced and the students‘ active participation is encouraged. v) Teacher acts as a role of guide only. vi) The students received a better training in the learning by doing method in this method. vii) The information seeking and retrieval behaviour is developed among the students. viii) It gives better understanding in scientific method and projects. Disadvantages of Assignments There are some demerits and limitations in this assignment method for both teachers and students. i) It is time consuming and burden process. ii) Teacher has to collect the information from various sources before assigning the work to the students. iii) Work burden extends in holidays too. There will be no encouragement for his work. iv) There are no source books and guide books are available in the market. Teacher has to prepare the same at his own risk of time and money. v) There are divergent group of students in a class, it poses problems for teacher assigning a unique or uniform topic for assignment. vi) Time consuming. Need to spend more time in seeking information and its retrieval. vii) The time limit given threatens the students which makes the substandard work. viii) The slow learners stay behind. They tend to copy others works. ix) It is found hard for the students having little scientific attitude. x) The report writing is little bit costly. 4.13 Instructional Materials – Need and Importance ―Instructional material‖ means content that conveys the essential knowledge and skills of a subject in the public school curriculum through a medium or a combination of media for conveying information to a student. The term includes a book, supplementary materials, a combination of a book, workbook, and supplementary materials, computer software, magnetic media, DVD, CD – ROM, computer courseware, on-line services, or an electronic medium, or other means of conveying information to the student or otherwise contributing to the learning process through electronic means, including open- source instructional material. Instructional materials are highly important for teaching, especially for in experienced teachers. Teachers rely on instructional materials in every aspect of teaching. They need materials for back ground information on the subject they are teaching. Young teachers usually have not built up their expertise whenever they enter into the field. Teachers often use instructional materials for less on planning. These materials are also needed by teachers to assess the knowledge of their students. Teachers often assess students by assigning tasks, creating projects, and administering exams. Instructional materials are essential for all of these activities. Teachers are often expected to create their own lesson plans. This can be difficult, especially if the teacher has limited back ground knowledge on the subject. Teachers are expected to have a wide variety of expertise in many different fields. Often, they need instructional aides to supplement their knowledge. Instructional materials can help provide back ground knowledge on the subject the teacher is planning for, and offer suggestions for less on plans. Lesson planning is often the most stressful aspect of teaching. Teachers are usually dependent on them to do their job properly. Assessing students correctly can sometimes be a challenge. There is some controversy about the effectiveness of exams in assessing the ability of students. Instructional materials can offer some insight into the best methods of creating exams. These materials can also help teachers create assignments and project ideas for students. Teachers are required to use several different methods for assess their students in order to provide the most accurate assessments. Instructional materials often provide innovative and creative ways to assess students‘ performance. It is hard to imagine any teacher who is capable of teaching effectively without the accompaniment of instructional materials. In addition to this, any teacher who is deprived of instructional materials most likely experiences stress and anxiety on a daily basis. 4.14 Web Based Learning Web based learning is often called online learning or e- learning because it includes online course content. Discussion forums via email, videoconferencing, and live lectures (video streaming) are all possible through the web. Web based courses may also provide static pages such as printed course materials. One of the values of using the web to access course materials is that web pages may contain hyperlinks to other parts of the web, thus enabling access to a vast amount of web based in formation. A ―virtual‖ learning environment (VLE) or managed learning environment (MLE) is an all in one teaching and learning software package. A VLE typically combines functions such as discussion boards, chat rooms, online assessment, tracking of students‘ use of the web, and course administration. VLEs act as any other learning environment in that they distribute information to learners. VLEs can, for example, enable learners to collaborate on projects and share information. However, the focus of web based courses must always be on the learner— technology is not the issue, nor necessarily the answer. The current focus of WBL development is on learning how to use the available tools and organize content into well- crafted teaching systems. Training designers are still struggling with issues of user interface design and programming for high levels of interaction. Unfortunately, there are few examples of good WBL design visible on the public Internet. As instructional designers and training analysts learn how to write and produce WBT, and as training vendors come to realize the overwhelming advantages of this delivery method, expect an explosion in training offerings available over the public Internet and private intranets. 4.14.1 Advantages of Web-based-Learning i) easy delivery of training to users ii) opportunities for group training ( asynchronous and synchronous) as well as individual training iii) multi- platform capabilities (Windows, Mac, UNIX, PDA, phone, other wireless devices) iv) easy updating of content v) quicker turnaround of finished product vi) requires less technical support vii) billing options by user ID, number of accesses, date/time of access viii) access is controllable ix) options for installations on private networks for security or greater bandwidth x) options to link with other training systems xi) multi tasking capability suitable for electronic performance support systems (EPSS) xii) vast market for distributed training xiii) growing level of acceptance

4.14.2 Disadvantages of Web-based-Learning i) bandwidth/ browser limitations may restrict instructional methodologies ii) limited bandwidth means slower performance for sound, video, and intense graphics iii) someone must provide server access, control us age, bill users. 4.15 Let Us Sum Up In this unit we have studied the learning strategies in mathematics. The group learning inside the class room is an attractive learning strategy. The group learning inside the class room develops a positive classroom environment. Students have more work on a mural, news letter, play or other project during the academic year. Computer – assisted instruction (CAI) refers to instruction or remediation presented on a computer. Many educational computer programs are available online and from computer stores and textbook companies. They enhance teacher instruction in several ways. Computer games are very useful in the development of logical thinking. Problem solving, field trips, surveys, projects, individual learning outside the class room, assignments are the several learning strategies in mathematics. Web – based learning (WBL) is an innovative approach to distance learning in which computer- based training (CBT) is transformed by the technologies and methodologies of the World Wide Web, the Internet, and intranets. Web- based learning presents live content, as fresh as the moment and modified at will, in a structure allowing self - directed, self – paced instruction in any topic. WBL is media – rich training fully capable of evaluation, adaptation, and remediation, all independent of computer platform. 4.16 Answers to Check Your Progress i) What is the CAI? ii) What is the group learning inside the classroom; get educational benefit of computer games? iii) What is the problem solving strategy? iv) What is the group learning outside the classroom? v) Write a short note on field trip, survey and project. vi) What do you mean by individual learning inside the classroom? vii) Write a short note on assignment. viii) What are the needs and importance of instructional materials? 4.17 References Ray, B., (2002). PDAs in the Classroom: Integration strategies for K – 12 Educators. The Access Center, Computer-Assisted Instruction and Mathematics. Computer- assisted instruction (CAI), (2012). In Encyclopedia Britannica. Retrieved on dated 8 may 2012 from http://www.britannica.com/EBchecked/ topic/130589/computer-assistedinstruction. Kaushik, R., (1996). ‗Effectiveness of Indian science centres as learning environments: a study of educational objectives in the design of museum experiences‘. Unpublished Ph.D. Thesis, University of Leicester, UK! Retrieved on dated 12.05.2012 from http://en.wikipedia.org/w/index.php?title=Field_trip &oldid=491340621. Dewey, John. (1910). How We Think. Boston: D. C. Heath & Co. Kilpatrick, William Heard (1918). ―The Project Method.‖ Teachers College Record, XIX (September). Knoll, Michael. (1995). ―The Project Method: Its Origin and International Influence.‖ In Progressive Education across the Continents. A Handbook, ed. Volker Lenhart and Hermann Rohrs. New York: Lang. Machisotto, Elena Anne, (1993), ―teaching Mathematics Humanistically: A New Look at an Old Friend‖, MAA Notes #32, edited by Alvin M. White. Teaching of Science, First Year Source Book (D. T. Ed.), Tamil Nadu Textbook Society, Chennai – 600006, p 47 – 60.

UNIT 5 Evaluations Structure 5.1 Introduction 5.2 Objective 5.3 Achievement Test in Mathematics 5.4 Subjective and Objective Tests 5.5 Teacher Made Test

5.5.1 Steps in Constructing Teacher Made Test 5.5.2 Types of Teacher Made Test 5.6 Standardized Tests 5.7 Criterion referenced and Norm Referenced Tests 5.8 Speed and Power Tests 5.9 Oral, Written and Performance Tests 5.9.1 Oral Test 5.9.2 Written Test 5.9.3 Performance Test 5.10 Diagnostic and Prognostic Tests 5.11 Characteristic of a Good Test 5.12 Interpretations Test Results 5.13 Measures of Central Tendency 5.13.1 Mean 5.13.2 Median 5.13.3 Mode 5.14 Standard Deviation (SD) and Rank Correlation 5.14.1 Standard Deviation (SD) 5.14.2 Rank Correlation 5.15 Continuous Evaluation 5.15.1 Characteristics of Continuous Evaluation 5.15.2 How Does Continuous Evaluation Help a Classroom Teacher? 5.15.3 Advantages of Continuous Evaluation 5.16 Cumulative record and Its Need 5.16.1 Need of the Cumulative Record Card 5.16.2 Content of a Cumulative Record Card (CRC) 5.16.3 Guidelines for Maintaining Cumulative Record Card 5.17 Let Us Sum Up 5.18 Answers to Check Your Progress 5.19 References 5.1 Introduction Evaluation, put simply, is the process by which people make value judgments about things. Evaluation is an integral part of any teaching and learning programme. Evaluation is a part of life. Evaluation is a systematic process of collecting evidence about students‘ progress and achievement in both cognitive and non-cognitive areas of learning on the basis of which judgments are formed and decisions are made. Evaluation is not always the end of a course. We not only want to know whether a student has developed a certain ability stated in the educational objectives or not but we also need to know about the progress during the course of teaching and learning. Thus, it is a continuous process. Even in small activities of human life like which dress to wear for work, what gift to buy or when to cross the road, evaluation has to be made. In education, evaluation is all the more important because only through evaluation a teacher can judge the growth and development of students, the changes taking place in their behaviour, the progress they are making in the class and also the effectiveness of her/ his own teaching in the class. Thus, evaluation has been an integral part of any teaching and learning situation. This unit discusses the different types of tests and emphasis the continuous evaluation. Measures of central tendency, standard deviation and rank correlation are also briefly explained in the present unit. 5.2 Objectives On completion of this unit students will be able to …. 18. understand the achievement test in mathematics; 19. understand the subjective and objective tests; 20. recognize criterion referenced and norm referenced tests; 21. describe the speed and power tests; 22. know the oral, written and performance tests; 23. comprehend the diagnostic and prognostic tests; 24. realize the characteristics of a good test; 25. interpret the test results; 26. illustrate the measures of central tendency; 27. understand the standard deviation and rank correlation; 28. explain the continuous evaluation; 29. elucidate the cumulative records. 5.3 Achievement Test in Mathematics Teachers teach the students to enable them to develop some abilities, skills and attitudes. After teaching, the learners need to be evaluated. It may be through monthly, half- yearly or yearly examinations. The teachers construct the tests to assess the achievement of students. The following steps are taken to develop these tests: i) Decision on units of content and their weightage ii) Identification of objectives and their weightage iii) Deciding types and number of items iv) Preparing Blueprint v) Preparation of test items/questions vi) Arrangement of items vii) Estimation of time viii) Preparing scoring key i) Decision on Units of Content and their Weightage Depending upon the time (half-yearly or yearly), we decide the topics of the syllabus and also fix their weightage in terms of marks. More topics are included in the yearly or pre-board type examinations. In quarterly or half year examinations, fewer topics are included. So each topic would be given more marks in comparison to the yearly examination. Sometimes the specifications of marks for topics are given in the syllabus and some topics are put within a section. ii) Identification of Objectives and their Weightage Normally, School Examination Board examinations are designed to test knowledge, skill, understanding and application objectives. We should consult the guidelines of the examining board or school. If such guidelines do not exist, the teacher may decide them on his own by consulting other teachers or seeing the pattern of previous ‗years‘ examination papers. For example, in a secondary class X examination in social sciences, the allocation for knowledge, understanding, application and skills was 35%, 90%, 15% and 10% respectively. iii) Deciding Types and Number of Items An analysis of previous year's examination question papers or model test papers of examining bodies may enable us to have an idea about the different types of items (essay, short answer or objectives), their numbers and marks for each type. It is only an estimate and should not be taken rigidly. Let us take two examples here. In an English model paper,, the long answer, short answer and objective type items were 5, 8 and 35 respectively and their respective total marks were 38, 27 and 35. Essay type of questions carried 6 – 10 marks each, short answer 2 – 4 marks and objective types 1 mark respectively. In a social sciences test paper, there were 7 essay type items (each of 6 marks), 9 short-answer types (6 items of 4 marks each and the remaining 13 of two marks each) and 8 parts of a question (one mark for each part). The word limit for answers was specified. Answers to questions of 2 marks, 4 marks and six marks were to be given within a range of 30, 60 – 80 and 100 – 125 words. The types of questions, number, marks and limit of words for answer can be thus properly decided and it should be considered tentative. iv) Preparing Blueprint A blueprint is a two-dimensional chart showing different types of items with marks for each topic/unit and each of the objectives. An outline of the blueprint is explained in Table 1 Examination Matrix (X) Max. Marks – 100 Time – Three Hours

Objective Knowledge Understanding Application Skill Total Total S. No. Content Units E SA O E SA O E SA O E SA O E SA O

1 Topic 1 3(2) 6(1) 1(2) 3(2) 6(1) 3(4) 1(2) 20

2 Topic 2 6(1) 1(5) 3(1) 3(2) 1(3) 1(2) 3(-)* 6(1) 3(3) 1(10) 25

3 Topic 3 3(1) 1(3) 3(1) 6(1)* 6(-)* 6(1) 3(2) 1(3) 15

4 Topic 4 6(1)* 3(2) 3(2) 1(2) 6(-)* 6(1) 3(4) 1(2) 20

5 Topic 5 3(1) 1(2) 6(1) 3(1)* 3(2) 3(-)* 6(1) 3(4) 1(2) 20

Sub- total 6(2) 3(4) 1(10) 6(2) 3(7) 1(5) 6(1) 3(6) 1(4) 6(-) 3(-) 3(-) 6(5) 3(17) 1(19)

Total 34 38(14) 28(18) 10(-) 100

Note: 1. Marks for one question are put outside brackets and number of questions within brackets. 2. * (Star) denotes an item covering two objectives of a topic and so a number of questions have been left blank as (-) It should follow the respective weightage of marks for the different objectives, and topics and various types of items as prescribed by the school or in the syllabus. These specifications have been discussed in the earlier steps of planning of this test. 3. Adjustment of total scores of skill has been done in starred common questions and so not included in total of 100. 4. No. of questions: Essay type (E) = 6 Short Answer type (SA) = 17 Objective (O) = 19 v) Preparation of Test Items/Questions Test items from the very basis of testing. A test constructor should have good knowledge of the subject. He / She should consult related literature of the subject, model test papers (of CBSE or others) and question banks (if available). The test items should be clear, unambiguous and according to the objectives. Different types of items - essay, short-answer and objective types - should be written out in sufficient numbers. Items of varying difficulty should also be prepared. Experienced teachers are able to estimate it by their judgments. Some items from model test papers or question banks can be taken up. After collection, a review of items is done on the basis of blueprint requirements and on quality of items. Only unambiguous and specific objective type items are retained. vi) Arrangement of Items Some examination boards and schools have divided the syllabus into parts/sections. Questions are to be set and answered from each section. Some within - question choices are allowed. After preparing items, they are arranged section-wise. Similar types of items - essay, short answer and objective - are put together and in increasing order of difficulty. If there is no sectional division, then we call take up objective, short answer and essay type items in a sequence and in increasing order of difficulty. vii) Estimation of Time For teacher-made achievement tests, only the experience of teachers should be enough for the estimate of time. We should try to analyze and estimate the time for different types of questions. Normally, in Board Annual Examinations, it is three hours but for a class test it may be 40 - 60 minutes. viii) Preparing Scoring Key The test constructor should give proper instructions for marking. Objective type tests have exact answers. Their answers and their marks should be given. Short answer questions are also quite specific in nature and possible points or ideas in answers should be mentioned with their marks for each. Essay type questions are lengthy and need specificity for uniform marking. Important steps or points of answer should be explicitly mentioned along with their corresponding marks. The above guidelines for marking questions from first to the last one should be able to make our testing more reliable. 5.4 Subjective and Objective Tests Mathematics education testing belongs to mathematical ability, and it is a scientific theory which estimates the learners‘ capability of mastering Mathematics Educational knowledge. With many years‘ practice and research, there have formed different theoretical systems and usable methods. Always, people ask the same question that mathematics education testing tests whether the usage of mathematical knowledge or the use of mathematical ability. Generally speaking, it is better to use the objective test to test the usage, while the subjective test is more suitable to test the use. But of course, that parlance is not absolutely right. Since the existence of the objective test and the subjective test, there is an argument about which one is better. Actually, we cannot make a simple comparison between the two because they have the unsubstantial effects on different stages respectively. Even at the moment or in the future, these two kinds of tests are surely to have great effects on testing. ‗Objective‘ and ‗Subjective‘ depend on the way of marking. The objective test usually has only one definite answer, and the way of marking is very mechanical. It gets the same objective result whether the test paper is marked by man or by the computer. On the contrary, the subjective test has an opposite way of marking. For it has no definite or standard answers, any answers may be the acceptable ones. As concerned to test papers, all the things are made by man subjectively. Even the objective test question such as multiple choices is worked out by the writer‘s subjective thinking. The examinee also has to distinguish each choice subjectively in order to make the right choice. And there is no objectivity. The objective test and the subjective test have their own advantages and disadvantages. First, we concern to the objective test. The objective test covers a wide range of linguistic knowledge which relates to every aspect of English. By and large, it can be more scientific, and can avoid the subjectivity in the process of marking. Especially, it suits the test of large size. Recent years, we have been using the computer to mark the paper; it not only accelerates the marking speed, but also improves the rate of correctness. And it has saved a lot of manpower and material resources. The familiar kinds of questions are multiple choice items and transformation. But disadvantages of the objective test are distinct. Always drawing circles and ticking is easy to take from students‘ ability to do written work. Furthermore, the computer may make mistakes, and cannot be one hundred percent correct. Next, we concern to the subjective test. The subjective test can test students‘ ability to use language directly; that is to say, can use different methods to test different abilities. And the forms are such as paragraph writing, answering questions, oral test and so on. The disadvantages of the subjective test are no less than that of the objective test. The marking of it needs much time and great vigor, and it is difficult to have a definite marking standard which embarrasses people a lot. So sometimes it‘s unfair to both the examinee and test. From my point of view, I prefer the objective test more. Because it is easier to work out the answer. When you do not really know which one is the correct answer, you also can make a guess instead of nothing filled in. Correspondingly, it needs less time to finish the questions. On the contrary, the subjective test is what students do not want to deal with, especially to those who are afraid of doing writing. To the students, they feel more depressed to do the subjective test such as paragraph writing. Because there is only a topic and a direction of only one or two sentences, and you must finish it in a given time. In case you got the misunderstanding, you will possibly get a very low mark, of fail in the test. Moreover, the teacher sometimes has different marking standards to different students. In fact, the subjective test is useful to improve students‘ abilities in practical use, and it is essential in high level tests. While the objective test trains students‘ abilities to deal with tests and it is useful for the low level students to enlarge their basic language knowledge. So it is very hard to say which test style is better because their functions are different, and we will continue use these two kinds of tests in the coming days. 5.5 Teacher-Made Test Teacher-made test is the major basis for evaluating the progress or performance of the students in the classroom. The teacher therefore, had an obligation to provide their students with best evaluation. This module presents topic on the steps in constructing teacher-made test, the types of teacher made test as essay and objective, and the advantages and disadvantages. Likewise, other evaluative instruments are being presented. After completing this module, the students are expected to: 1. identify the types of teacher-made test; 2. draw general rules/guidelines in constructing test that is applicable to all types of test; 3. explain how to score essay test in such a way that subjectivity can be eliminated; 4. discuss and summarize the advantages and disadvantages of essay and objective type of test; 5. enumerate and discuss other evaluative instruments use to measure students‘ performance; and 6. construct different types of test. 5.5.1 Steps in Constructing Teacher-Made Test 1. Planning the Test. In planning the test the following should be observed: the objectives of the subjects, the purpose for which the test is administered, the availability of facilities and equipments, the nature of the testee, the provision for review and the length of the test. 2. Preparing the Test. The process of writing good test items is not simple – it requires time and effort. It also requires certain skills and proficiencies on the part of the writer. Therefore, a test writer must master the subject matter he/she teaches, must understand his testee must be skillful in verbal expression and most of all familiar with various types of tests. 3. Reproducing the Test. In reproducing test, the duplicating machine and who will facilitate in typing and mimeographing be considered. 4. Administering the Test. Test should be administered in an environment familiar to the students, sitting arrangements is observed, corrections are made before the start of the test, distribution and collection of papers are planned, and time should be written on the board. One more important thing to remember is, do not allow every testee to leave the room except for personal necessity. 5. Scoring the Test. The best procedure in scoring objective test is to give one point of credit for each correct answer. In case of a test with only two or three options to each item, the correction formula should be applied. Example: for two option, score equals right minus wrong (S = R- W). For three options, score equals right minus one-half wrong (S = R-1/2 W or S= R-W/2). Correction formula is not applied to four or more options. If correction formula is employed students should be informed beforehand. 6. Evaluating the Test. The test is evaluated as to the quality of the student‘s responses and the quality of the test itself. Index difficulty and discrimination index of the test item is considered. Fifty (50) per cent difficulty is better. Item of 100 per cent and zero (0) per cent answered by students are valueless in a test of general achievement. 7. Interpreting Test Results. Standardized achievement tests are interpreted based on norm tables. Table of norm are not applicable to teacher-made test. 5.5.2 Types of Teacher Made Test I. Essay Examination Essay examination consists of questions where students respond in one or more sentences to a specific question or problems. It is a test to evaluate knowledge of the subject matter or to measure skills in writing. It is also tests students‘ ability to express his ideas accurately and to think critically within a certain period of time. Essay examination maybe evaluated in terms of content and form. In order to write good essay test, it must be planned and constructed in advance. The questions must show major aspect of the lesson and a representative samples. Avoid optional questions and use large number of questions with short answer rather than short question with very long answer.

Advantages of an Essay Examination i) Easy to construct. In terms of preparation, essay examination is easier to construct thus it saves time and energy. ii) Economical. Economical when it comes to reproduction of materials. It can be written on the board. iii) Trains the core of organizing, expressing and reasoning power. Encourage students to think critically and express their ideas. iv) Minimizes guessing. Guessing is minimized because it requires one or more sentences. v) Develops critical thinking. Essay exams calls for comparison, analysis, organization of facts, for criticism, for defense of opinion, for decision and other mental activity. vi) Minimizes cheating and memorizing. Essay test minimizes cheating and memorizing because essay tests are evaluated in terms of content and form and that an answer to question is composed of one or more sentences. vii) Develops good study habits. It can develop good study habits in the sense that students study their lesson with comprehension rather than rote memory. Disadvantages of Essay Examination i) Low validity. It has low validity for it has limited sampling. ii) Low reliability. This may occur due to its subjectivity in scoring. The tendency of the teachers to react unfavorably to answers of students whom he consider weak and give favorable impressions to answers of bright students. iii) Low usability. Time consuming to both teacher and students wherein much time and energy are wasted. iv) Encourage bluffing. It encourages bluffing on the part of the testee. The tendency of the students who does not know the answer is to bluff his answers just to cover up his lack of information. If bluffing becomes satisfactory on an easy examination, inaccuracy of the measuring instrument may occur and evaluation of the students‘ achievement may not be valid and reliable. v) Difficult to correct or score. Difficulty on the part of the teacher to correct or score due to an answer consisting of one or more sentences. vi) Disadvantages for students with poor penmanship. Some teachers react unfavorable to responses of students having poor handwriting and untidy papers. Scoring an Essay Examination i) Brush up the answers before scoring. ii) Check the students‘ answer against the prepared model. iii) Quickly read the papers on the basis of your opinion of their worthiness and sort them into five groups: 1) very superior, 2) superior, 3) average, 4) inferior, and 5) very inferior. iv) Read the responses of the same item simultaneously. v) Re-read the papers in each group and shift any that you feel have been misplaced. vi) Avoid looking at the names of the paper you are scoring. II. Objective Examination The two main types of objective tests are the recall and the recognition. The recall type is categorized as to: a. Simple recall b. Completion The recognition type is categorized as: a. Alternative response b. Multiple choice c. Matching type d. Rearrangement type e. Analogy f. Identification The first three recognition type are most commonly use. Recall Type 1. Simple Recall type: This test is one of the easiest tests to construct among the objective types where the item appears as a direct question, a stimulus word or phrase, or a specific direction. The response requires the subject to recall previously learned materials and the answers are usually short. This test is applicable in natural sciences subjects like mathematics, chemical and physical sciences where the stimulus appears in a form of a problem that requires computation. 2. Completion Test: This test consists of a series of items which requires the subject to fill a word or phrase on the blanks. An item may contain one or more blanks. Indefinite and overmultilated statements, keywords and statements directly taken from the book should be avoided Recognition Type

1. Alternative Response Test: This test consists of a series of items where it admits only one correct answer in each item. This type is commonly used in classroom testing particularly the two constant alternative test as true-false, plus-minus, right-wrong, yes-no, correct- incorrect, XY, etc. Others forms may use the three-constant alternatives as true-false-doubtful, constant alternative with correction and modified true-false type. Suggestion for the Construction of Alternative Response Type a. Items must be arranged in group of five and each group must be separated by two single spaces. b. Responses must be simple as TF, XY, etc. and if possible be placed in one column at the right. c. Avoid lifting similar statement from the test. d. Language to use must be within the level of students. Flowery words must be avoided. e. Specific determiners like all, always, none, never, not, nothing, no, are more likely to be false and so must be avoided. Moreover, determiners as may, some, seldom, sometimes, usually, and often are more likely to be true, hence, it should be avoided. f. Qualitative terms as few, many, great, frequent, and large are vague and indefinite and so they must be avoided. g. Partly right and partly wrong statement must be avoided. Consider statement that represents either true or false. h. Ambiguous and double negative statements must be avoided. 2. Multiple Choice Types: This test is made up of items which consists of three or more plausible options. It is regarded as one of the best form of test. Most valuable and widely used due to its flexibility and objectivity in scoring. In teacher-made test, it is applicable for testing vocabulary, reading comprehension, relationship, interpretation of graphs, formulas, tables, and drawing inferences from a set of data. Rules and suggestion for the Construction of Multiple Choices a. The main stem of the test item may be constructed in question, completion or direction form. b. Questions that tap only rote learning and memory should be avoided. c. Use unfamiliar phrasing to test students‘ comprehension, thus avoid lifting words from the text. d. Four or more options must be provided to minimize guessing. e. Uniform number of options must be used. f. Arrangement of correct answers should not follow patterns. g. Articles ―a‖ and ―an‖ are avoided as last word in an incomplete sentence. This word gives clues. h. Alternative should be arranged according to length. Varieties of Multiple Choice Types a. Stem-and-options variety. This is commonly used in the classroom and other standardized test. The stem serves as the problem and is followed by four or more options. b. Setting-and-options variety. The optional responses are dependent upon a setting or foundation which includes graphical representation, a sentence, paragraph, pictures, equation, or some forms of representation. c. Group-term variety. Consists of group or words or items in which one does not belong to the group. d. Structured-response variety. This makes use of structure response which is commonly used in testing natural science subjects. This test on how good the students are to judge statements which are closely related. e. Contained-option variety. This variety is designated to identify errors in a word, phrase, sentence or paragraph. 3. Matching Type: This type consists of two columns in which proper pairing relationship of two things is strictly observed. Column A is to be matched with column B. It has two forms: balanced and unbalanced, the latter being preferred. In balanced type the number item is equal to the number of option. In unbalanced type, if there are 5 items in column A there are 7 items in column B. Remember, the ideal number for matching type is 5 to 10 and maximum of 15. In constructing matching type, avoid using heterogeneous materials. Do not mix dates and terms, events and person and many others. The question item should be placed on the left and the option on the right option column should be in alphabetical order and dates in chronological order. 4. Rearrangement Type: This type consists of a multiple-option item where it requires a chronological, logical, rank, etc., order. 5. Analogy: This type is made of items consisting of a pair of words which are related to each other. It is designated to measure the ability of students to observe the pair relationship of the first group to the second. The kinds of relationship may be: according to purpose, cause and effect, part-whole, part-part, action to object, synonym, antonym, place, degree, characteristics, sequence, grammatical, numerical and associations. Advantages of an Informal Objective type a. Easy to score. It is easier to correct due to short responses involve. b. Eliminates subjectivity. This is due also to short responses. c. Adequate sampling. More items can be included where validity and reliability of the test can be adequately observed. d. Objectivity in scoring. Due to short and one correct answer in each item. e. Eliminates bluffing. Since the students only choose the correct answer. f. Norms can be established. Due to adequate sampling of test. g. Save time and energy in answering questions. Since the options are provided, time and energy may be utilized properly. Limitations of an Informal Objective Test a. Difficult to construct b. Encourages cheating and guessing. c. Expensive. Due to adequate sampling, it is expensive in terms of duplicating facilities. Questions cannot be written on the board. d. Encourages rote memorization. It encourages rote memorization rather than memorizing logically because an answer may consist only of a single word or a phrase. A student‘s ability to thick critically, express, organize and reason out his ideas is not developed. e. Time consuming on the part of the teacher. 5.6 Standardized Tests The test is designed to measure test takers against each other and a standard, and standardized tests are used to assess progress in schools, ability to attend institutions of higher education, and to place students in programs suited to their abilities. Many parents and educators have criticized standardized testing, arguing that it is not a fair measure of the abilities of the test taker, and that standardized testing, especially high-stakes testing, should be minimized or abolished altogether. A Standardized test is a test that is given in a consistent or ―standard‖ manner. Standardized tests are designed to have consistent questions, administration procedures, and scoring procedures. When a standardized test is administrated, is it done so according to certain rules and specifications so that testing conditions are the same for all test takers? Standardized tests come in many forms, such as standardized interviews, questionnaires, or directly administered intelligence tests. The main benefit of standardized tests is they are typically more reliable and valid than non-standardized measures. They often provide some type of ―standard score‖ which can help interpret how far a child‘s score ranges from the average. Standardized tests can either be on paper or on a computer. The test taker is provided with a question, statement, or problem, and expected to select one of the choices below it as an answer. Sometimes the answer is straightforward; when asked what two plus two is, a student would select ―four‖ from the list of available answers. The answer is not always so clear, as many tests include more theoretical questions, like those involving a short passage that the test taker is asked to read. The student is instructed to pick the best available answer, and at the end of a set time period, answer sheets are collected and scored. There are some advantages to standardized tests. They are cheap, very quick to grade, and they allow analysts to look at a wide sample of individuals. For this reason, they are often used to measure the progress of a school, by comparing standardized test results with students from other schools. However, standardized tests are ultimately not a very good measure of individual student performance and intelligence, because the system is extremely simplistic. A standardized test can measure whether or not a student knows when the Magna Carta was written, for example, but it cannot determine whether or not the student has absorbed and thought about the larger issues surrounding the historical document. Studies on the format of standardized tests have suggested that many of them contain embedded cultural biases which make them inherently more difficult for children outside the culture of the test writers. Although most tests are analyzed for obvious bias and offensive terms, subconscious bias can never be fully eliminated. Furthermore, critics have argued that standardized tests do not allow a student to demonstrate his or her skills of reasoning, deductive logic, critical thinking, and creativity. For this reason, some tests integrate short essays. These essays are often given only short attention by graders, who frequently vary widely in opinion on how they think the essay should be scored. Finally, many concerned parents and educators disapprove of the practice of high-stakes testing. When a standardized test is used alone to determine whether or not a student should advance a grade, graduate, or be admitted to school, this is known as high-stakes testing. Often, school accreditation or teacher promotion rests on the outcome of standardized tests alone, an issue of serious concern to many people. Critics of high-stakes testing believe that other factors should be accounted for when considering big issues including classroom performance, interviews, class work, and observations. 5.7 Criterion Referenced and Norm Referenced Tests The essential characteristic of norm-referencing is that students are awarded their grades on the basis of their ranking within a particular cohort. Norm-referencing involves fitting a ranked list of students‘ ‗raw scores‘ to a pre-determined distribution for awarding grades. Usually, grades are spread to fit a ‗bell curve‘ (a ‗normal distribution‘ in statistical terminology), either by qualitative, informal rough- reckoning or by statistical techniques of varying complexity. For large student cohorts (such as in senior secondary education), statistical moderation processes are used to adjust or standardize student scores to fit a normal distribution. This adjustment is necessary when comparability of scores across different subjects is required (such as when subject scores are added to create an aggregate ENTER score for making university selection decisions). Norm-referencing is based on the assumption that a roughly similar range of human performance can be expected for any student group. There is a strong culture of norm-referencing in higher education. It is evident in many commonplace practices, such as the expectation that the mean of a cohort‘s results should be a fixed percentage year-in year-out (often this occurs when comparability across subjects is needed for the award of prizes, for instance), or the policy of awarding first class honors sparingly to a set number of students, and so on. In contrast, criterion-referencing, as the name implies, involves determining a student‘s grade by comparing his or her achievements with clearly stated criteria for learning outcomes and clearly stated standards for particular levels of performance. Unlike norm-referencing, there is no pre-determined grade distribution to be generated and a student‘s grades are in no way influenced by the performance of others. Theoretically, all students within a particular cohort could receive very high (or very low) grades depending solely on the levels of individuals‘ performances against the established criteria and standards. The goal of criterion-referencing is to report student achievement against objective reference points that are independent of the cohort being assessed. Criterion-referencing can lead to simple pass-fail grading schema, such as in determining fitness-to-practice in professional fields. Criterion-referencing can also lead to reporting student achievement or progress on a series of key criteria rather than as a single grade or percentage.

5.7.1 Comparison between Criterion-Referenced and Norm- Referenced Tests Dimension Criterion-Referenced Norm- Referenced

Purpose To determine whether each student To rank each student with respect to has achieved specific skills or the achievement of others in broad concepts. areas of knowledge.

To find out how much students To discriminate between high and low know before instruction begins and achievers. after it has finished.

Content Measures specific skills which Measures broad skill areas sampled make up a designed curriculum. from a variety of textbooks, syllabi, These skills are identified by and the judgments of curriculum teachers and curriculum experts. experts.

Each skill is expressed as an instructional objective.

Item Each skill is tested by at least four Each skill is usually tested by less Characteristics items in order to obtain an adequate than four items. sample of student performance and Items vary in difficulty. to minimize the effect of guessing. The items which test any given skill Items are selected that discriminate are parallel in difficulty. between high and low achievers.

Score Each individual is compared with a Each individual is compared with Interpretation preset standard for acceptable other examinees and assigned a score- achievement. The performance of usually expressed as a percentile, a other examinees is irrelevant. grade equivalent score, or a stanine. A student's score is usually Student achievement is reported for expressed as a percentage. broad skill areas, although some

Student achievement is reported for norm-referenced tests do report individual skills. student achievement for individual skills.

Which of these methods is preferable? Mostly, students‘ grades in universities are decided on a mix of both methods, even though there may not be an explicit policy to do so. In fact, the two methods are somewhat interdependent, more so than the brief explanations above might suggest. Logically, norm- referencing must rely on some initial criterion-referencing, since students‘ ‗raw‘ scores must presumably be determined in the first instance by assessors who have some objective criteria in mind. Criterion-referencing, on the other hand, appears more educationally defensible. But criterion- referencing may be very difficult, if not impossible, to implement in a pure form in many disciplines. It is not always possible to be entirely objective and to comprehensively articulate criteria for learning outcomes: some subjectivity in setting and interpreting levels of achievement is inevitable in higher education. This being the case, sometimes the best we can hope for is to compare individuals‘ achievements relative to their peers. Norm-referencing, on its own — and if strictly and narrowly implemented — is undoubtedly unfair. With norm-referencing, a student‘s grade depends – to some extent at least – not only on his or her level of achievement, but also on the achievement of other students. This might lead to obvious inequities if applied without thought to any other considerations. For example, a student who fails in one year may well have passed in other years! The potential for unfairness of this kind is most likely in smaller student cohorts, where norm-referencing may force a spread of grades and exaggerate differences in achievement. Alternatively, norm-referencing might artificially compress the range of difference that actually exists. Criterion-referencing is worth aspiring towards. Criterion-referencing requires giving thought to expected learning outcomes: it is transparent for students, and the grades derived should be defensible in reasonably objective terms – students should be able to trace their grades to the specifics of their performance on set tasks. Criterion-referencing lays an important framework for student engagement with the learning process and its outcomes. Recognizing, however, that some degree of subjectivity is inevitable in higher education, it is also worthwhile to monitor grade distributions – in other words, to use a modest process of norm- referencing to watch the outcomes of a predominantly criterion-referenced grading model. In doing so, if it is believed too many students are receiving low grades, or too many students are receiving high grades, or the distribution is in some way oddly spread, then this might suggest something is amiss and the assessment process needs looking at. There may be, for instance, a problem with the overall degree of difficulty of the assessment tasks (for example, not enough challenging examination questions, or too few, or assignment tasks that fail to discriminate between students with differing levels of knowledge and skills). There might also be inconsistencies in the way different assessors are judging student work. Best practice in grading in higher education involves striking a balance between criterion-referencing and norm-referencing. This balance should be strongly oriented towards criterion-referencing as the primary and dominant principle. 5.8 Speed and Power Tests A pure speed test is homogeneous in content, the tasks being so easy that with unlimited time all but the most incompetent of subjects could deal with them successfully. Speed tests are suitable for testing visual perception, numerical facility, and other abilities related to vocational success. Tests of psychomotor abilities (e.g., eye–hand coordination) often involve speed. Speed tests are tests that have a fixed time limit, at which point everyone taking the test must stop. Speed tests contain more items than power tests although they have the same approximate time limit. Speed tests tend to be used in selection at the administrative and clerical level. Power tests tend to be used more at the graduate, professional or managerial level. Although, this is not always the case, as speed tests do give an accurate indication of performance in power tests. In other words, if you do well in speed tests then you will also do well in power tests. In a speed test the scope of the questions is limited and the methods you need to use to answer them is clear. Taken individually, the questions appear relatively straightforward. Speed test are concerned with how many questions you can answer correctly in the allotted time. For example:

For example: 139 + 235 =? A) 372 B) 374 C) 376 D) 437 Power Test: A power test contains items that vary in difficulty to the point that no subject is expected to get all items right even with unlimited time. In practice, a definite but ample time is set for power tests. Power tests tend to be more relevant to such purposes as the evaluation of academic achievement, for which the highest level of difficulty at which a person can succeed is of greater interest than his speed on easy tasks. Power tests are tests that have no time limit; applicants are allowed as much time as needed to complete the test. A power test on the other hand will present a smaller number of more complex questions. The methods you need to use to answer these questions are not obvious, and working out how to answer the question is the difficult part. Once you have determined this, arriving at the correct answer is usually relatively straightforward. For example: Below are the sales figures for 3 different types of network server over 3 months.

Server January February March

Units Value Units Value Units Value

ZXC4 3 32 480 40 600 48 720

ZXC5 3 45 585 45 585 45 585

ZXC6 3 12 240 14 280 18 340

In which month was the sales value highest? A) January B) February C) March What is the unit cost of server type ZXC 53? A) 12 B) 13 C) 14 Examples: For those who are competent at Math, at ask where people are asked to solve a page of simple addition problems as fast as possible would be a speed test. A test where a person is asked to press a button with their finger as many times as possible in 30 seconds would be a second example.

5.8.1 Conclusion Speed tests consist of problems designed to be very easy for the test taker, but with far too little time to complete all of them. Power tests, on the other hand, are used to determine mastery of the material. Questions range from very easy to very difficult with plenty o f time to focus on every problem. If any test taker can finish all the problems in either test, the examiner would not know how much more they could have accomplished had the test been beyond their limits. Most standardized tests include a sampling of both types of testing. Many tests have both varying levels of difficulty and a time limit. Often the time limit is not actually a factor as all students finish well within the time limits. Choosing a test depend heavily on the purpose of the examination. If testing for the best prospects for college admission, one would choose a standardized group aptitude test, probably with both speed and power elements. To try to pick out gifted students one would prefer a group achievement test. If speed of performance was not relevant it would be a pure power test. A power test often contains more difficult items than a speed test. Large-scale testing programs often include speed tests because all test forms can be collected at the same time. For some jobs, working speed is an important component of successful performance. A power test would not be able to evaluate this skill properly. 5.9 Oral, Written and Performance Tests 5.9.1 Oral tests Oral tests are conducted to formally evaluate if an applicant has the knowledge, skills and abilities that are important for success in a job. (An oral test is not the same as an employment interview.) In an oral test, a panel of subject matter or job experts asks applicants a set of pre-determined oral test questions and listens and evaluates their responses to these questions. Panel members take detailed notes of each applicant‘s responses, usually using rating sheets that contain the answers to the questions. A structured procedure is used to score applicants‘ answers to the oral test questions. In an oral test, all applicants are asked exactly the same questions in exactly the same order. The specific questions asked in an oral test are often similar to essay questions asked in a written test. A monitor is present to ensure that the test is administered and scored in a standardized way to all applicants. Communication in an oral test is highly structured and mostly one-way; applicants are not given an opportunity to present information not specifically requested or to ask questions about the agency or department. Oral tests are designed to assess the knowledge, skills and abilities that are important for success in a job. Oral test questions are not intended to assess information that can be learned quickly on the job or by reading a book or brochure. Test questions are not developed from specific reading materials, and we do not provide applicants with specific information about what to study in order to prepare for oral tests. How to do Preparation for Oral Tests There are several things applicants can do preparation for oral tests. Applicants should keep in mind the following points in series of preparation and faced problems during oral tests: i) Read the job announcement carefully. Job announcements normally include a description of the purpose of the job class and a list of the knowledge, skills and abilities that are required in the job class. Use this to determine the subject areas that are likely to be represented in the oral test. ii) Read the job specification for the job class. (This is available on the DAS web site). Job specifications include a list of the duties performed by employees in a job class. This may also be useful in suggesting areas that are likely to be represented in the oral test. iii) You may want to visit your local library for test preparation materials related to individual jobs or professions. Please keep in mind that these materials are not designed to prepare you for any one state employment examination. DAS does not endorse any specific study materials, test preparation publishing organizations or test preparation courses. iv) Practice answering questions verbally and/or giving a presentation in front of others. Applicants often find it helpful to simulate the environment that they will face at the oral test. It is important to note that an oral test is not the same as an employment interview. Employment interviews are conducted, with the purpose of making a final selection of an applicant to fill a specific position. Although interviews are structured, they are often more informal than oral tests. Employment interviews are typically conducted so that the employer can meet with applicants to ask questions about their prior work history, to provide applicants with information about the organization and work unit, and to assess the level of fit between the applicants and the specific position being filled. Employment interviews are also conducted so that applicants can determine if the position and the agency are of interest to them. 5.9.2 Written Tests Written tests are tests that are administered on paper or on a computer. A test taker who takes a written test could respond to specific items by writing or typing within a given space of the test or on a separate form or document. In some tests; where knowledge of many constants or technical terms is required to effectively answer questions, like Chemistry or Biology - the test developer may allow every test taker to bring with them a cheat sheet. A test developer‘s choice of which style or format to use, when developing a written test is usually arbitrary given that there is no single invariant standard for testing. Be that as it may, certain test styles and format have become more widely used than others. Below is a list of those formats of test items that are widely used by educators and test developers to construct paper or computer- based tests. As a result, these tests may consist of only one type of test item format (e.g., multiple choice test, essay test) or may have a combination of different test item formats (e.g., a test that has multiple choice and essay items). 5.9.3 Performance Tests A performance test is an assessment that requires an examinee to actually perform a task or activity, rather than simply answering questions referring to specific parts. The purpose is to ensure greater fidelity to what is being tested. An example is a behind - the- wheel driving test to obtain a driver‘s license. Rather than only answering simple multiple- choice items regarding the driving of an auto mobile, a student is required to actually drive one while being evaluated. Performance tests are commonly used in work place and professional applications, such as professional certification and licensure. When used for personnel selection, the tests might be referred to as a work sample. A licensure example would be cosmetologists being required to demonstrate a haircut or manicure on a live person. The Group - Bourdon test is one of a number of psychometric tests which trainee train drivers in the UK are required to pass. Some performance tests are simulations. For instance, the assessment to become certified as an ophthalmic technician includes two components, a multiple- choice examination and a computerized skill simulation. The examinee must demonstrate the ability to complete seven tasks commonly performed on the job, such as retinoscopy that are simulated on a computer. 5.10 Diagnostic and Prognostic Tests Diagnostic tests may serve multiple educational objectives that benefit the individual student. The educationalist may use tests to : a) identify predispose risk factors to modify risk; b) identify early problem – associated mathematical or psychological changes prior to test; and c) determine which specific type of problems is involved to guide selection of the most effective remedy. The term ―diagnosis‖ is used in mathematics education to describe the determination of the nature of mathematical problems. A ―diagnostic‖ refers to tools, procedures, or technologies that are used in that determination. A diagnostic test or technology is used to identify elements of an existing mathematical understanding process. Diagnostic tests differentiate whether or not a person has a specific field of mathematical concept at the time. The accuracy of a diagnostic test in detecting a specific mathematical problem is calculated using certain criteria. These criteria are based on whether the mathematical problem does or does not exist at the time of the diagnosis. The accuracy of the diagnostic test is calculated when the specific mathematical problem is present and when it is not present. Various diagnostic test performance criteria have been established, including sensitivity, specificity, and predictive values. The application of performance criteria to the remedial use of mathematical diagnostic tests has been extensively described. A ―prognosis‖ is the prediction of the future course and/ or outcome of a problem or a mathematical understanding, given current information. Because there are no facts about the future, a prognosis is described in terms of probabilities. Prognostic tools or technologies are used to make the prediction. The strength of a specific prognostic test is often described in terms of the relative risk for a future event when the test parameter is present compared to when it is not present. Studies on the value of prognostic tests provide a confidence interval that gives the user some indication of how useful the test may be in a broad population. 5.11 Characteristic of a Good Test The main function of education is to bring about desirable change in the education of learners (objectives). We conduct tests/examination to assess the progress in achievement of set objectives. A good test should enable us to measure the achievement or ability accurately and economically. Here we take three main characteristics of a good test – validity, reliability and usability. i) Validity A test is said to be valid if it measures what it is supposed to measure. Cronback defines validity as ―the extent to which a test measures what it purports to measure‖. Oral communication ability in a language can be properly assessed through speech or discussion techniques. So these are valid tools here. But written examination cannot be a valid tool for assessment of oral communication. We cannot say that one who writes good answers in the examination would be efficient in oral communication. A class test in arithmetic can be valid to know the achievement in arithmetic and not for diagnosis of weakness of the pupils. Practical and theoretical examinations conducted in science subjects judge the different types of abilities and objectives. One cannot be used for the other. They are valid for their related content and objectives. Normally teachers prepare achievement tests for their classes. A test should cover the given content and the specified objectives (like knowledge, kill and application) for accurate measurement. If these two conditions are satisfied, then it is said to have content validity. There are some other types of validity also. One easily understandable is predictive validity. This indicates predications of success of an individual in future in accordance with the criterion of a test. National talent Search Tests for school students are conducted in India. These are supposed to select students who can excel in specified fields in future selected ones are awarded scholarships. Banks also conduct tests to select clerks on the predictive validity concept. We, as teachers, are generally concerned about content validity of achievement tests conducted by the school or state school boards of education. So such tests should try to cover the prescribed syllabus and the specified objectives. Their proportionate weightage is also given sometimes. If not, the teacher may decide weightage (assigning marks) to each selected topic and the each objective. ii) Reliability A test is reliable if it gives consistent results of measurement on its administration at different occasions. Pupils should score similar scores if the test is repeated. Its administration may be after some gap of time and the condition is that pupils do not receive any further training of related test content. Reliability may be affected by vague wording of questions/items, general attitude of particular teachers, lack of instructions about marking, circumstances of examinee (as ill health) etc. Some precautions may be taken to increase the reliability of a test. Length of test increases reliability and reduction in size decreases it. Language of instructions in items should be clear. This helps the testee to know the specific scope and desired length of the answer. ―Write a short note on population problem‖ is not a good item. It can be improved as, ―Write a short note of 200 words on the problem of over- population in the Indian context‖. Further, in an objective question, multiple choice type, there should 4 -5 alternatives which resemble the correct answer. True-false type should preferably be avoided. Items of varying difficulty level should be included but a large number of items should be of moderate difficulty. One relation between validity and reliability is worth noting. A valid test is reliable. High reliability does not necessarily imply validity. Estimates of Reliability We consider three simple methods to estimate reliability of tests. a) Test-retest Method b) Parallel form method c) Split – half method a) Test-retest Method In this method, a test is conducted twice on the same group of pupils after a gap of some time. Thus each individual gets two score. Here we calculate Pearson's correlation coefficient between two sets of scores of the individual. This is known as test-reliability or measure of stability. This is useful for the tests which contain a large number of items to minimize the carry-over effect of memory. The reliability of some intelligence and personality tests can to be calculated by this method. b) Parallel Form Reliability An equivalent test (or parallel form) is also prepared in addition to the original test. These two tests are administered to the same group of pupils on the same day. Then Pearson‘s coefficient of correlation between these pairs of scores of individuals gives the reliability measure. This is called parallel form or equivalent reliability. It is commonly employed for teacher-made or achievement tests. c) Split-half Method of Reliability A single test is divided into two equivalent parts or sub-parts, say A and B. A may contain odd numbered items (1, 3, 5...) and so B should contain the even numbered ones (2, 4, 6…). The test is administered only once. The total scores for half part A and total score for half part B are separately noted for each pupil. A correlation coefficient is calculated for these sets of scores. This gives the reliability of half test (rhh.). The (rhh.) is used to calculate reliability of the full test reliability (r,) using the Spearman- Brown proficiency formula as follows:

2 x √r hh 2 x reliability of half test (r,) = ------= ------1 + √r hh 1 + reliability of half test Some other methods of estimating reliability are also available using Kuder-Richardson and Cronback alpha formulae. But these involve some complex statistical calculations. The teachers and administrators interested in intensive study of the topic are advised to consult books like Educational and Psychological Measurements and Evaluation by Stanley and Hopkin or some other ones on the subject. iii) Usability Usability refers to practicability - ease and economy in terns of energy, time and money. Usability means a test should be:  easy to construct;  easy in administrating;  easy in scoring;  easy in interpreting; and  economic in terms of money and time. The School Head and the teachers should take care that cost, economy and facility in administration should not dilute the accuracy, purpose and reliability of the test. 5.12 Interpretations of Test Results Many local factors should be taken into account when interpreting the standing of pupils according to norms derived on a nationwide basis. Among these factors are (1) the legal age for entering school, (2) the average age of actually entering school, (3) promotion and detention policies, (4) rate and selectivity of elimination from school, (5) efficiency of the teaching personnel, (6) the grade placement and time allowances, (7) general nature of the curriculum, (8) the standing of local pupils in mental ability and other aspects related to the one being evaluated, (9) the relative emphasis in the local school on academic, social, and vocational development, and (10) the home background of pupils. The meaning of the derived scores for a given group of pupils can then be interpreted in the light of these factors. When interpreting the performance of individual pupils or of a class as a whole, the teacher should take into special consideration differences in the cultural background of families and communities. There are wide variations in the kind of experiences pupils have. We can expect that differences in language background, richness of home resources, and intensity of the desire for an education will be reflected in pupil performance. Performance of pupils also varies with varying emphasis on different aspects of the school curriculum. In some subject matter areas, such as Arithmetic, the teacher usually cannot expect his/her pupils to go much beyond instructional materials. In other areas such as reading, there are many opportunities for students to develop skill and knowledge on their own outside the school program. Thus the performance of individuals and groups should be judged, in part at least, on the basis of the curriculum to which they have been exposed. When the performance of a class or an individual deviates considerably from the norms on standardized tests, a need for reappraisal of the school curriculum and of teaching emphases may be indicated. In many practical situations it is necessary to use norms whose applicability to local conditions is questionable or unknown. To the extent that it is so, the data obtained from them cannot he interpreted meaningfully for individual pupils. A final precaution is to avoid using tests to punish pupils or to foster a spirit of rivalry among teachers or schools. Teachers and administrators must keep the welfare of pupils uppermost in their minds and be sensitive to the requirements of adequate human relations. Failure to do this in administering and interpreting a testing program has produced negative feelings about tests in both pupils and teachers, so much for qualitative interpretations. 5.13 Measures of Central Tendency A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. As such, measures of central tendency are sometimes called measures of central location. They are also classed as summary statistics. The mean (often called the average) is most likely the measure of central tendency that you are most familiar with, but there are others, such as, the median and the mode. The mean, median and mode are all valid measures of central tendency but, under different conditions, some measures of central tendency become more appropriate to use than others. In the following sections we will look at the mean, mode and median and learn how to calculate them and under what conditions they are most appropriate to be used. 5.13.1 Mean The mean, or ―average‖, is the most widely used measure of central tendency. The mean is defined technically as the sum of all the data scores divided by n (the number of scores in the distribution). The mean (or average) is the most popular and well known measure of central tendency. It can be used with both discrete and continuous data, although its use is most often with continuous data. The mean is equal to the sum of all the values in the data set divided by the number of values in the data set. So, if we have n values in a data set and they have values x1, x2, ..., xn, then the sample mean, usually denoted by (pronounced x bar), is:

This formula is usually written in a slightly different manner using the Greek capitol letter pronounced ―sigma‖, which means ―sum of...‖:

You may have noticed that the above formula refers to the sample mean. So, why call has we called it a sample mean? This is because, in statistics, samples and populations have very different meanings and these differences are very important, even if, in the case of the mean, they are calculated in the same way. To acknowledge that we are calculating the population mean and not the sample mean, we use the Greek lower case letter ―mu‖, denoted as µ:

The mean is essentially a model of your data set. It is the value that is most common. You will notice, however, that the mean is not often one of the actual values that you have observed in your data set. However, one of its important properties is that it minimizes error in the prediction of any one value in your data set. That is, it is the value that produces the lowest amount of error from all other values in the data set. An important property of the mean is that it includes every value in your data set as part of the calculation. In addition, the mean is the only measure of central tendency where the sum of the deviations of each value from the mean is always zero.

When not to use the mean The mean has one main disadvantage: it is particularly susceptible to the influence of outliers. These are values that are unusual compared to the rest of the data set by being especially small or large in numerical value. For example, consider the wages of staff at a factory below: Staff 1 2 3 4 5 6 7 8 9 10 Salary 15k 18k 16k 14k 15k 15k 12k 17k 90k 95k The mean salary for these ten staff is Rs 30.7 lacks. However, inspecting the raw data suggests that this mean value might not be the best way to accurately reflect the typical salary of a worker, as most workers have salaries in the Rs 12 lacks to 18 lacks range. The mean is being skewed by the two large salaries. Therefore, in this situation we would like to have a better measure of central tendency. As we will find out later, taking the median would be a better measure of central tendency in this situation. Another time when we usually prefer the median over the mean (or mode) is when our data is skewed (i.e. the frequency distribution for our data is skewed). If we consider the normal distribution as this is the most frequently assessed in statistics when the data is perfectly normal then the mean, median and mode are identical. Moreover, they all represent the most typical value in the data set. However, as the data becomes skewed the mean loses its ability to provide the best central location for the data as the skewed data is dragging it away from the typical value. However, the median best retains this position and is not as strongly influenced by the skewed values. This is explained in more detail in the skewed distribution section later in this guide. 5.13.2 Median Technically, the median of a distribution is the value that cuts the distribution exactly in half, such that an equal number of scores are larger than that value as there are smaller than that value. The median is by definition what we call the 50th percentile. This is an ideal definition, but often distributions cannot be cut exactly in half in this way, but we still can define the median in the distribution. Distributions of qualitative data do not have a median. The median is most easily computed by sorting the data in the data set from smallest to largest. The median is the ―middle‖ score in the distribution. Suppose we have the following scores in a data set: 5, 7, 6, 1, 8. Sorting the data, we have: 1, 5, 6, 7, 8. The ―middle score‖ is 6, so the median is 6. Half of the (remaining) scores are larger than 6 and half of the (remaining) scores are smaller than 6. To derive the median, using the following rule. First, compute (n+1)/2, where n is the number of data points. Here, there are 5, so n = 5. If (n+1)/2 is an integer, the median is the value that is in the (n+1)/2 location in the sorted distribution. Here, (n+1)/2 = 6/2 or 3, which is an integer. So the median is the 3rd score in the sorted distribution, which is 6. If (n+1)/2 is not an integer, then there is no ―middle‖ score. In such a case, the median is defined as one half of the sum of the two data points that hold the two nearest locations to (n+1)/2. For example, suppose the data are 1, 4, 6, 5, 8, 0. The sorted distribution is 0, 1, 4, 5, 6, 8. n = 6, and (n+1)/2 = 7/2 = 3.5. This is not an integer. So the median is one half of the sum of the 3rd and 4th scores in the sorted distribution. The 3rd score is 4 and the firth score is 5. One half of 4 + 5 is 9/2 or 4.5. So the median is 4.5. Here, notice that half of the scores are above 4.5 and half are below. In this case, the ideal definition is satisfied. Also, notice that the median may not be an actual value in the data set. Indeed, the median may not even be a possible value. The median number of sexual partners last year is 1. Here, n = 177, and (n+1)/2 = 178/2 = 89, an integer. So in the sorted distribution, the 89th data point is the median. In this case, the 89th score is a 1. Notice that this does not meet the ideal definition, but we still call it the median. It certainly is not true that half of the people reported having fewer than 1 sexual partner, and half reported having more than 1. Violations of the ideal definition will occur when the median value occurs more than once in the distribution, which is true here. There are many ―1‖s in the data. Computing the median seems like a lot of work. But computers do it quite easily. In real life, you should rarely have to compute the median by hand but there are some occasions where you might, so you should know how. 5.13.3 Mode The mode is the most frequent score in our data set. On a histogram it represents the highest bar in a bar chart or histogram. You can, therefore, sometimes consider the mode as being the most popular option. An example of a mode is presented below:

Normally, the mode is used for categorical data where we wish to know which is the most common category as illustrated below:

We can see above that the most common form of transport, in this particular data set, is the bus. However, one of the problems with the mode is that it is not unique, so it leaves us with problems when we have two or more values that share the highest frequency, such as below:

We are now stuck as to which mode best describes the central tendency of the data. This is particularly problematic when we have continuous data, as we are more likely not to have any one value that is more frequent than the other. For example, consider measuring 30 peoples' weight (to the nearest 0.1 kg). How likely is it that we will find two or more people with exactly the same weight, e.g. 67.4 kg? The answer, is probably very unlikely - many people might be close but with such a small sample (30 people) and a large range of possible weights you are unlikely to find two people with exactly the same weight, that is, to the nearest 0.1 kg. This is why the mode is very rarely used with continuous data. Another problem with the mode is that it will not provide us with a very good measure of central tendency when the most common mark is far away from the rest of the data in the data set, as depicted in the diagram below:

In the above diagram the mode has a value of 2. We can clearly see, however, that the mode is not representative of the data, which is mostly concentrated around the 20 to 30 value range. To use the mode to describe the central tendency of this data set would be misleading.

Skewed Distributions and the Mean and Median

We often test whether our data is normally distributed as this is a common assumption underlying many statistical tests. An example of a normally distributed set of data is presented below:

When you have a normally distributed sample you can legitimately use both the mean or the median as your measure of central tendency. In fact, in any symmetrical distribution the mean, median and mode are equal. However, in this situation, the mean is widely preferred as the best measure of central tendency as it is the measure that includes all the values in the data set for its calculation, and any change in any of the scores will affect the value of the mean. This is not the case with the median or mode. However, when our data is skewed, for example, as with the right-skewed data set below:

we find that the mean is being dragged in the direct of the skew. In these situations, the median is generally considered to be the best representative of the central location of the data. The more skewed the distribution the greater the difference between the median and mean, and the greater emphasis should be placed on using the median as opposed to the mean. A classic example of the above right- skewed distribution is income (salary), where higher-earners provide a false representation of the typical income if expressed as a mean and not a median. If dealing with a normal distribution, and tests of normality show that the data is non-normal, then it is customary to use the median instead of the mean. This is more a rule of thumb than a strict guideline however. Sometimes, researchers wish to report the mean of a skewed distribution if the median and mean are not appreciably different (a subjective assessment) and if it allows easier comparisons to previous research to be made.

Summary of when to use the mean, median and mode

Please use the following summary table to know what the best measure of central tendency is with respect to the different types of variable.

Type of Variable Best measure of central tendency

Nominal Mode

Ordinal Median Interval/Ratio (not skewed) Mean

Interval/Ratio (skewed) Median

Some Important Questions about Measures of Central Tendency Please find below some common questions that are asked regarding measures of central tendency, along with their answers. These important questions are in addition to our article on measures of central tendency. i) What is the best measure of central tendency? There can often be a ―best‖ measure of central tendency with regards to the data you are analyzing but there is no one ―best‖ measure of central tendency. This is because whether you use the median, mean or mode will depend on the type of data such as nominal or continuous data; whether your data has outliers and/or is skewed; and what you are trying to show from your data. Further considerations of when to use each measure of central tendency are found. ii) In a strongly skewed distribution, what is the best indicator of central tendency? It is usually inappropriate to use the mean in such situations where your data is skewed. You would normally choose the median or mode with the median usually preferred. This is discussed under the subtitle, ―When not to use the mean‖. iii) Does all data have a median, mode and mean? Yes and no. All continuous data has a median, mode and mean. However, strictly speaking, ordinal data has a median and mode only, and nominal data has only a mode. However, a consensus has not been reached among statisticians about whether the mean can be used with ordinal data and you can often see a mean reported for Likert data in research. iv) When is the mean the best measure of central tendency? The mean is usually the best measure of central tendency to use when your data distribution is continuous and symmetrical, such as when your data is normally distributed. However, it all depends on what you are trying to show from your data. v) When is the mode the best measure of central tendency? The mode is the least used of the measures of central tendency and can only be used when dealing with nominal data. For this reason, the mode will be the best measure of central tendency (as it is the only one appropriate to use) when dealing with nominal data. The mean and/or median are usually preferred when dealing with all other types of data but this does not mean it is never used with these data types. vi) When is the median the best measure of central tendency? The median is usually preferred to other measures of central tendency when your data set is skewed (i.e. forms a skewed distribution) or you are dealing with ordinal data. However, the mode can also be appropriate in these situations but is not as commonly used as the median. vii) What is the most appropriate measure of central tendency when the data has outliers? The median is usually preferred in these situations as the value of the mean can be distorted by the outliers. However, it will depend on the how influential the outliers are. If they do not significantly distort the mean then using the mean as the measure of central tendency will usually be preferred. viii) In a normally distributed data set, which is greatest: mode, median or mean? If the data set is perfectly normal then the mean, median and mean are equal to each other (i.e. the same value). ix) For any data set, which measures of central tendency have only one value? The median and mean can only have one value for a given data set. The mode can have more than one value. 5.14 SD and Rank Correlation 5.14.1 Standard Deviation (SD) To overcome the problem of dealing with squared units, statisticians take the square root of the variance to get the standard deviation. The standard deviation (for a sample) is defined symbolically as

So if the scores in the data were 5, 7, 6, 1, and 8, their squared differences from the mean would be 0.16 (from [5-5.4]2), 2.56 (from [7-5.4]2), 0.36 (from [6-5.4]2), 19.36 (from [1-5.4]2), and 6.76 (from [8-5.4]2). The mean of these squared deviations is 5.84 and its square root is 2.41 (if dividing by N), which is the standard deviation of these scores. The standard deviation is defined as the average amount by which scores in a distribution differ from the mean, ignoring the sign of the difference. Sometimes, the standard deviation is defined as the average distance between any score in a distribution and the mean of the distribution. The above formula is the definition for a sample standard deviation. To calculate the standard deviation for a population, N is used in the denominator instead of N – 1. Suffice it to say that in most contexts, regardless of the purpose of your data analysis, computer programs will print the result from the sample sd. So we will use the second formula as our definitional formula for the standard deviation, even though conceptually dividing by N makes more sense (i.e., dividing by how many scores there are to get the average). When N is fairly large, the difference between the different formulas is small and trivial. Using the N – 1 version of the formula, we still define the standard deviation as the average amount by which scores in a distribution differ from the mean, ignoring the sign of the difference, even though this is not a true average using this formula. 5.14.2 Rank Correlation According to http://oxforddictionaries.com rank correlation defines as ―An assessment of the degree of correlation between two ways of assigning ranks to the members of a set‖. According to http://en.wikipedia.org ―In statistics, a rank correlation is the relationship between different rankings of the same set of items‖. Correlation coefficient defines as ―A measure that determines the degree to which two variable's movements are associated‖. The correlation coefficient will vary from –1 to +1. A –1 indicates perfect negative correlation, and +1 indicates perfect positive correlation. Charles Spearman established the method of finding correlation coefficient by using rank order and generally this method is known as Spearman's Rank Correlation method. Spearman's Rank Correlation is a technique used to test the direction and strength of the relationship between two variables. In other words, it is a device to show whether any one set of numbers has an effect on another set of numbers. In statistics, Spearman's rank correlation coefficient or Spearman's rho, often denoted by the Greek letter (rho) or as , is a non-parametric measure of statistical dependence between two variables.

Example

In some cases your data might already be ranked but often you will find that you need to rank the data yourself (or use SPSS to do it for you). Thankfully, ranking data is not a difficult task and is easily achieved by working through your data in a table. Let us consider the following example data regarding the marks achieved by students in a maths and English exam: English 12 14 12 17 19 12 14 13 15 Maths 15 14 10 11 9 11 16 18 12 The procedure for ranking these scores is as follows: First, create a table with seven columns and label them as below: Table 1 Rank Rank English Maths Students English maths d d2 (mark) (mark) (Rx) (Ry) 1 12 15 8.0 3.0 5.0 25.00 2 14 14 4.5 4.0 0.5 0.25 3 12 10 8.0 8.0 0 0 4 17 11 2.0 6.5 4.5 20.25 5 19 9 1.0 9.0 8.0 64.00 6 12 11 8.0 6.5 1.5 2.25 7 14 16 4.5 2.0 2.5 6.25 8 13 18 6.0 1.0 5.0 25.00 9 15 12 3.0 5.0 2.0 4.00 ∑ d2 = 147

You need to rank the scores for maths and English separately. The score with the highest value should be labelled ―1‖ and the lowest score should be labelled ―9‖ (if your data set has more than 9 cases then the lowest score will be how many cases you have). Look carefully at the two individuals that scored 14 in the English exam (highlighted in bold). Notice their joint rank of 4.5. This is because when you have two identical values in the data (called a ―tie‖) you need to take the average of the ranks that they would have otherwise occupied. We do this as, in this example, we have no way of knowing which score should be put in rank 4 and which score should be ranked 5. Therefore, you will notice that the ranks of 6 and 7 do not exist for English. These two ranks have been averaged ((4 + 5)/2 = 4.5) and assigned to each of these ―tied‖ scores. To calculate Spearman's rank-order correlation coefficient the formula is:

Where, di = difference in paired ranks and n = number of cases. From table, d2 = 147 and n = 9 From the above equation 6 * 147 = 1 – ------= – 0. 23 9 (9 * 9 – 1) as n = 9. Hence, we have a of –0.23. This indicates a strong negative relationship between the ranks individuals obtained in the maths and English exam. That is, the higher you ranked in maths, the higher you ranked in English also, and vice versa. 5.15 Continuous Evaluation The term ‗Continuous Evaluation‘ involves two terms ‗continuous‘ and ‗evaluation‘. The term ‗continuous‘ refers to regularity in assessment. The growth of a child is a continuous process. Therefore, the students‘ progress should be evaluated continuously which means that evaluation has to be completely integrated with the teaching and learning process. ‗Evaluation‘ is the process of finding out the extent to which the desired changes have taken place in the pupils. It, therefore, requires collection of evidence regarding growth or progress, so as to use that information for decision making. In this way, information gathering, judgment making and decision taking are the three phases of the process of evaluation. Thus, continuous evaluation means a regular assessment of the pupils‘ development in school. In Mathematics teaching – learning, continuous evaluation would involve a regular assessment of the progress of students in developing various mathematical operations. It is the process in which students are evaluated in the four skills along with grammar and literature at regular intervals over the school year. 5.15.1 Characteristics of Continuous Evaluation There are following some basic characteristics of continuous evaluation: i) Careful examination of the course, and specification of competencies to be attained by the learners in terms of knowledge, understanding, application (analysis, synthesis, evaluation for higher grades) and skill performance. ii) Requirement of knowledge and skills of evaluation, commitment, and assistance to provide remedial teaching on part of the teacher. iii) Continuous evaluation is basically formative in nature and is school based. It: is to be carried out by the teachers teaching a particular class. iv) The purpose of continuous evaluation is mainly improvement in learning. For this the learning gaps and weaknesses are diagnosed so that remedial measures can be provided, v) Continuous evaluation is informally carried out in the classroom. There is no need of making lengthy arrangements required for a formal examination like printing of question papers, seating arrangement, etc. vi) Multiple techniques of evaluation need to be used for continuous evaluation. These include not only written tests but oral tests, assignments, projects, observation, peer evaluation, self appraisal, etc. vii) Continuous evaluation is built into the total teaching-learning programme and is a part of the daily routine for a teacher.

5.15.2 How Does Continuous Evaluation Help a Classroom Teacher? Yes, in the following ways the continuous evaluation helps a classroom teacher: i) To identify learning difficulties in mastering certain competencies and the intensity of such learning difficulties. ii) To improve students‘ learning through diagnosis of their performance. iii) To plan appropriate remedial measures to enable he students who have learning difficulties in mastering the competency. iv) To improve or alter instructional strategies to enhance the quality of teaching. v) To decide up on the selecting of various media and materials as a supportive system in mastering the competencies. vi) To strengthen evaluation procedure itself. 5.15.3 Advantages of Continuous Evaluation There are several advantages in continuous evaluation: i) Continuous evaluation and improvement in learning The purpose of continuous evaluation is to improve learning by making evaluation an integral part of the teaching-learning progress. Traditionally, the evaluation of students has been used for a number of purposes like classification, ranking, grading, selection, certification and promotion of the students. Evaluation has been used to discriminate between good achievers and low achievers, but in continuous evaluation the focus is on three Ps- progress, process and product. Through this evaluation the teacher continuously monitors the progress of the student in different areas of learning. The process of learning is also monitored and, if need be, altered according to the requirement of the individual student. Continuous evaluation also assesses the quality of the product of learning. To operationalize continuous evaluation, different steps are to be followed. On the basis of the feedback from the tests, diagnosis of the hard spots of learning is made. After diagnosing the gaps in learning, remedial instruction is provided so that weaknesses can be removed. Sometimes this process may demand the re -teaching of the whole unit or a change in the strategy of teaching. The students are then retested to find out the improvement in learning. Thus evaluation also helps the teacher in taking a relook at his/her transactional strategies and redesigning them in order to be more efficient and effective. ii) Continuous evaluation and techniques of assessment Continuous evaluation allows for the use of multiple techniques of assessment. Multiple methods are best to improve the reliability of evaluation. In the traditional system students are assessed mainly through written technique at the end of the term. The one shot examination is not enough to test all the skills and activities developed during the session. Continuous evaluation using varied techniques makes it possible to evaluate the process of learning in all the four language skills through oral testing, observation as well as written technique. For assessing the process of learning, it is important to involve the learners in the assessment of their own work. While the assessment of product is very often undertaken by a third person or the teacher, assessment of process necessarily involves those involved in that process. Therefore self assessment and peer assessment motivates the students to do better. Self -appraisal helps increase pupil's awareness of their strengths and weaknesses. 5.16 Cumulative Records and Its Need A cumulative record is a systematic account of information about a student. It is an evaluation tool which presents a comprehensive record of the achievement or otherwise of each student in different aspect -physical academic, moral, social and health. It begins as soon as the child enters the school and continues still he leaves the school. The information written in the cumulative record cards is collected from different sources over a period of time. 5.16.1 Need of the Cumulative Record Card The cumulative Record card of a student is needed for the following important points. i) To give a complete picture of a pupil's all round progress in different areas i.e. physical, academic, moral, social and health. ii) To assess the child, his potentialities, interest, aptitude and talents. iii) To provide proper guidance and counselling taking into consideration of students curricular and co - curricular achievements. iv) To help parents and teachers in the placement of pupil after the completion of schooling / study. 5.16.2 Content of a Cumulative Record Card (CRC) The cumulative record card records the following information regarding different aspects of pupils‘ development. i) Personal data — Personal data give introductory information about a child like his name, sex, date of birth, age, permanent address, parents name and family back ground. ii) Academic data — It deals with the information about the previous schools at tended earlier, present grade or class, roll number, examinations appeared, results, division and percentage of marks in each examination failures, percentage of attendance etc. iii) Health data — It reveals information regarding height, weight, blood pressure, communicable diseases if any, treatments given, food habits, exercise parental disease if any, care taken if handicap etc. iv) Co -curricular activities data — the child's participation in different co -curricular activities, leadership qualities, certificates awarded, prizes and medals received are recorded in it. v) Personality characteristics — this reveals the psychological aspects like intellectual ability self - confidence, emotional stability, leadership qualities, tolerance, initiative and sense of responsibility etc. vi) Record of counseling and guidance — the problems found with the child, date of interview, reasons discovered, remedial measures taken, following programmes etc. are mentioned in the card. vii) General over all Remarks — General remarks by the class teacher and Headmaster on the performance and talent s of the pupil. 5.16.3 Guidelines for Maintaining Cumulative Record Card i) The teachers should in-charge of maintaining cumulative records he should make the entries up -to -date. ii) Required information should be collected from various people like parents, friends, subject teachers and child himself by different techniques. iii) Different techniques should be used to collect information about a child i.e. psychological tests, observations by teachers, Examination result etc. iv) The teachers‘ in-charge of maintaining cumulative records should be given proper orientation and training about how to maintain it. v) Secret or confidential matters should find a place in it but a separate file may be made for such entries. vi) The guidance worker of the school should be in overall charge of maintenances of cumulative records. vii) Cumulative record cards maintenance should be supervised by the Headmaster and Inspector of schools. 5.17 Let Us Sum Up Our instructional programme is designed to bring about desired changes in students behavior. If the evaluation programme should be successful there should be a realistic statement of objectives. The teacher has to ascertain at every stage of the teaching process to what extent his pupils have realized the educational objectives. To do this, the teacher uses a number of tests. The teacher has to choose the proper tool in order to make his evaluation perfect. Every teacher therefore should possess a sound knowledge of evaluation. 5.18 Answers to Check Your Progress i) Describe the achievement test in mathematics. ii) What are the subjective and objective tests? iii) What do you mean by criterion referenced and norm referenced tests? iv) Describe the speed and power tests. v) Write an essay on oral, written and performance tests. vi) What are the diagnostic and prognostic tests in teaching mathematics? vii) What are the characteristics of a good test? viii) Interpret the test results. ix) Illustrate the measures of central tendency. x) Write a short note on standard deviation and rank correlation. xi) Explain the continuous evaluation. xii) Elucidate the cumulative records. 5.19 References Bond, L. (1996). Norm- and criterion-referenced testing. Practical Assessment, Research & Evaluation, 5(2). Retrieved September 2002, from http://ericae.net/pare/getvn.asp?v=5&n=2. Linn, R. (2000). Assessments and accountability. ER Online, 29(2), 4-14. Retrieved September, 2002, from http://www.aera.net/pubs/er/arts/2902/ linn01.htm. Sanders, W., & Horn, S. (1995). Educational assessment reassessed: The usefulness of standardized and alternative measures of student achievement as indicators for the assessment of educationaloutcomes. Education Policy Analysis Archives, 3(6). Retrieved September 2002, from http://olam.ed.asu.edu/epaa/v3n6.html. Glaser R: Instructional technology and the measurement of learning outcomes: Some questions. Am Psychol 18:519-521, 1963 Boehm A: Criterion-referenced assessment for the teacher. Teachers College Record 78:117-126, 1973 Patton, M. (1997) Utilization-focused evaluation, London: Sage.

UNIT 6 CURRICULUM DESIGNING IN MATHEMATICS Structure 6.1 Introduction 6.2 Objective 6.3 Concept of Curriculum 6.4 Content Selection for Mathematics Curriculum 6.5 Principles of Curriculum Designing in Mathematics 6.6 Updateness in Mathematics Curriculum 6.7 Interdisciplinary Treatment for Curriculum Designing in Mathematics 6.7.1 The Goal of Interdisciplinary Studies Programme 6.8 Content organization of Mathematics Curriculum – 6.8.1 Topical Plan 6.8.2 Logical Plan 6.8.3 Psychological Plan 6.8.4 Spiral or Concentric Plan 6.9 Homogeneous Grouping 6.9.1 Types of Homogeneous Grouping 6.10 Dolton Plan 6.11 Supervised study 6.12 Curricular models 6.13 SMSG 6.13.1 Criticism of SMSG 6.14 SCERT 6.14.1 Structure of SCERT 6.15 Assessment of various school curricula 6.15.1 Gathering Data 6.15.2 Interpreting Data to Improve the Curriculum 6.16 Let Us Sum Up 6.17 Answers to Check Your Progress 6.18 References

6.1 Introduction In the previous unit you have formed a general idea of the learning strategies in mathematics and different types of test for the purpose of evaluation of teaching mathematics. In this unit we are going to discuss the various principles that govern curriculum construction and organization in mathematics. It is true that the rapid advances in science and technology and the phenomenal social changes in of recent times make this task somewhat complicated. Added to these is the revolution in the modern educational theory and practice in terms of learning experiences, behavior patterns and evaluation of instruction. These factors clearly bring out the need for justifying the selection of the organization of the content material not merely on traditional lines, but on various commitment considerations as well. 6.2 Objectives On completion of this unit students will be able to …. 1. understand the content selection for mathematics curriculum; 2. understand the principle of curriculum designing in mathematics; 3. recognize the content organization for mathematics curriculum; 4. describe spiral and concentric plan for mathematics curriculum; 5. explain the Dalton plan; 6. know the SMSG and SCET; 7. aware the assessment of various school curricula. 6.3 Concept of Curriculum It is a term that is often used to denote ―the range or list of subjects‖ on a school‘s programme; ―a group‖ of school subjects meant to be taught to the learner‖; the ‗course of study‘ or a ‗syllabus‘ which is only part of the total curriculum – all these are parts of what we mean by ‗curriculum‘. The modern concept of curriculum, therefore, emphasizes not only the academic subjects but other activities which are planned and guided by the school. However, here are some of the definitions of the word ‗curriculum‘: i) A programme or course of activities which is explicitly organized as the means whereby pupils may attain the desired objectives (Hirst and Peters 1980, p.60). ii) A Curriculum is a plan for learning (Wiles and Bondi 1984, p. 19). iii) The planned experiences offered to the learner under the guidance of the school (Wheeler 1967, p.11). iv) By ‗curriculum‘ is meant the sum total of all the experiences a pupil undergoes (Bishop 1985, p.1). v) All that is taught in a school including the time-table subjects and all those aspects of its life that exercise an influence in the life of the children (Farrant 1980, p.24). vi) All the learning which is planned or guided by the school, whether it is carried on in groups or by individuals, inside or outside the school (Kerr 1968, p.16). vii) In the words of Cunnigham, ―It is a tool in the hands of the artist (the teacher) to mould his material (the pupil) in accordance with his ideals in his studio (the school)‖. 6.4 Content Selection for Mathematics Curriculum Curriculum is a plan for learning which is as comprehensive as our Constitution. The word ‗curriculum‘ means a racing chariot or wagon which has to run a course to reach a goal. Curriculum is the instructional and educative programme by following which the pupils achieve their goals, ideals and aspirations of life. It is curriculum through which the general aims of School education finds concrete expression. So, curriculum does not mean only the academic subjects of instruction or a course of study having a list of contents and indicating activities, which are only a part of the curriculum. The curriculum must include the totality of experiences that a pupil receives through the manifold activities that go on in the school, inside the classroom as well as outside, at the play ground and in the numerous informal contacts between teachers and pupils. In this sense, the whole of school life becomes a school curriculum which can touch the life of the students at all points and helps in gradual unfolding of a balanced personality. The curriculum must consist of contents and activities which the school employs for the purpose of training the students. A curriculum is not static but dynamic. It is constantly changing according to the changing needs, demands and aspirations of the society. Therefore, in order to develop a suitable curriculum some important points should have to be kept in mind. i) Aims to be achieved ii) Learners‘ need, interest and capacity iii) Civic needs iv) Demand of the society in global context v) Learn ability, utility vi) Leisure and recreation vii) Variety and flexibility viii) Local specificity ix) Students‘ real life situation x) Time span etc. 6.5 Principles of Curriculum Designing in Mathematics The following principles are used to guide the evolution of the aims and objectives, the structure of the curriculum and the identification of objectives in each module and unit in the syllabus. i) Target-oriented To ensure that learners will spend their time and effort meaningfully and for maximum benefits, there must be a plan for them to work according to specific Learning Targets and Objectives which are geared towards the Aims and Objectives of the school mathematics curriculum. They are organized progressively across four Key Stages in primary and secondary schooling: Key Stage 1 (Primary 1 to 3), Key Stage 2 (Primary 4 to 6), Key Stage 3 (Secondary 1 to 3) and Key Stage 4 (Secondary 4 to 5). The learning targets and objectives for Key Stages 1 and 2 can be found. Continuing the learning in primary schooling, the overall Aims and Objectives for Key Stages 3 and 4 are stated in Unit 2. The Learning strategies for each dimension in each learning stage are further elaborated in Unit 4 to spell out the specific learning objectives for each learning area. All learning and assessment activities fulfill the learning objectives of that particular unit and are geared towards the maximum learning effectiveness for achieving the Aims and Objectives. ii) Catering for Learner Differences Besides the various ways of organizing students‘ activities in the class to cater for learner differences, the foundation part of the curriculum is identified. The Foundation Part is the essential part of the Syllabus which all students should strive to learn. Apart from the Foundation Part, teachers can judge for themselves the suitability and relevance of other topics in the Whole Syllabus for their own students. For more able students, teachers can adopt some enrichment topics at their discretion to extend these students‘ horizon and exposure in mathematics. In order to provide further flexibility for teachers to organize the teaching sequences to meet individual teaching situations, the learning units and modules for each dimension are subdivided into key stages (KS), i.e. KS3 for S1-S3, KS4 for S4-S5. Teachers are free to design their school based mathematics curriculum for each year level with all learning areas suggested for each key stage in mind. iii) Relevance of Study to Students In order that mathematics learning efforts are effective, the knowledge and skills to be learnt should be determined by the activities deemed suitable for the age-group concerned. Great care is taken to ensure that the curriculum is organized with a cognitive developmental perspective. For instance, exposure to concrete objects and personal experience is planned to support abstract discussion as far as possible. Students who find the study relevant to their experience will be motivated to learn the subject. Daily life applications are emphasized in the curriculum. Stories of historical development of mathematics knowledge are included to enable students to understand mathematics knowledge evolved from real life problems and refined after years. A new module ―Further Applications‖ which includes the application of mathematics in more complex real-life situations requires students to integrate their knowledge and skills from various disciplines to solve problems. iv) Impact of Information Technology The tools for solving mathematical problems change from time to time. The introduction of electronic calculators in 1980‘s has influenced the teaching of secondary school mathematics. There are different roles electronic calculators can play. A general worry among teachers and parents is that the unwise use of calculators by students would hamper their development of computational skills. With years of experience in the classrooms of various countries, the positive role that calculators could play in the mathematics learning is generally aware of. Today we are confronted with a similar situation. The popularity of graphing calculators, the availability of computers and other information technology aids in the classrooms will have impact on the mathematics curriculum in terms of contents and strategies for teaching and learning of mathematics. There are ranges of ways in which information technology may be used in mathematics classes, including data analysis, simulation device, graphical presentation, symbolic manipulation and observing patterns. The appropriate use of information technology in the teaching and learning of mathematics becomes one of the emphases in the mathematics curriculum. v) Fostering General Abilities and Skills Knowledge is expanding at an ever faster pace and new challenges are continually being posed by the rapid changes in technology and in the way society evolves. It is important that students need to develop their capabilities to learn how to learn, to think logically and creatively, to develop and use knowledge, to analyze and solve problems, to access information and process it effectively, and to communicate with others so that they can meet the challenges that confront them now and in the future. Acquiring mathematics knowledge has always been emphasized, but fostering these general abilities and skills are strongly advocated for all students in the revised curriculum. In the curriculum, fundamental and intertwining ways of learning and using knowledge such as inquiring, communicating, reasoning, conceptualizing and problem-solving are considered important in mathematics education. On the one hand, students are expected to learn mathematics to enhance the development of these skills. On the other hand, students are expected to use these learning strategies to construct their mathematics knowledge. A variety of learning activities should be planned and geared towards the development of these general abilities and skills. 6.6 Updateness in Mathematics Curriculum No system of education has up to now succeeded in evolving the ‗ideal curriculum‘ as defined by educational theorists. However, models which closely approximate to this ideal have been evolved in many parts of the world. Ideal curriculum models are to be developed using an accepted sequence of procedures – steps like identifying the final outcomes to be attained through the curriculum, defining the outcomes of learning in terms of expected behaviour changes, or changes in certain critical mental processes and skills to be acquired, and then by deducing or generating the content areas which would help to achieve these outcomes, once appropriate instructional strategies are adopted for making such changes on the part of the learners. This approach also presumes the use of a properly spelt-out epistemological framework in terms of which the curriculum has to be conceived. The operational aspect of the approach mentioned above can be illustrated with the help of a few models. Take for example the procedure to be used for developing a process-centered school curriculum for science. The outcomes expected of a process-centered curriculum are best defined in terms of a wide range of scientific processes (e.g. observing the world around, drawing generalizations about what is happening, hypothesizing, designing experiments to test the truth of the hypothesis, interpreting what is observed, judging the validity of hypotheses, etc). When such processes are defined as expected outcomes, the curriculum designers present the learning context using illustration from different scientific disciplines (physical science or biosciences or other areas of school science) in a manner which will help to illustrate and develop such process skills on the part of the learners. An illustration of the process model of defining curriculum is presented in a subsequent section of this Report. Similar specialized novel approaches are available for developing the curriculum for language teaching (e.g. functional language teaching) and for teaching of subjects like mathematics (e.g. ‗situated cognition approach‘). Curriculum defined using such models will help to achieve higher outcomes in language or mathematic learning with greater effectiveness. In the ‗situated-cognition‘ approach for mathematics teaching, the methods used are different from what is used in the conventional approaches intended for peripheral transfer of mathematical knowledge. The situated- cognition approach for example, is developed around the educational aim that ‗knowledge is embedded in the context of real life situations‘. The use of this approach for curriculum development would imply the structuring of a mathematics curriculum which is conceived around real-life situations, with instructional content drawn from ‗real problem solving situations‘ present in day- to-day experiences of children. There are a number of other new approaches around which curriculum materials can be organized. Enquiry-based ‗generic social studies‘ is yet another new approach of this kind used for teaching social studies. This approach takes the stand that in teaching social studies, one has to go beyond an understanding of historical facts to develop concepts and generalizations about society and culture. The content has to be defined to conform to this approach, if this assumption about the curriculum is accepted for curriculum framing in the subject. The ―generic and targeted moral education for democratic functioning‖ is yet another new approach used for teaching of moral education in schools. A number of such new approaches are available to curriculum framers. We have to identify what are the models most appropriate for designing the new curriculum in different subjects. The examples discussed above are illustrative of some of the recent movements in curriculum reform and the theoretical-pedagogic framework used for the purpose. The curriculum reform in the state should be entrusted to an Expert Curriculum Committee of ‗real specialists‘ in different curriculum areas, who are also familiar with the professional developments, in curriculum designing in their chosen subject areas. While it is advisable to have a few subject experts in such committees, the majority of the members should be those with professional training in teacher education with specialization in methodologies of teaching the concerned subject. The Expert Curriculum Committee should be entrusted with the task of integrating the differing new approaches within a common operational framework. The SCERT should play the lead role in this exercise, within a common operational framework set by the SCSE. The Head of the Curriculum Department of the SCERT should take up the responsibility for preparing the final document. A very popular method widely used for curriculum designing in different parts of the world is the use of validated curricular models developed by the influential professional groups like the SMSG or the PSSC as the basic framework for a curriculum design. The fact that such models have been developed by experts using the latest scientific principles and using empirically validated procedures, add to their credibility. Only precaution to be observed in using these models is that they have to be suitably adapted for use in the school context of the State. 6.7 Interdisciplinary Treatment for Curriculum Designing in Mathematics Our world is increasingly interconnected and interdependent. Communications networks exchange information around the globe, creating new forms of collaboration and transforming the nature of work and learning. New areas of study develop to advance human knowledge and respond to the challenges of our changing world with insight and innovation. These include areas that often combine or cross subjects or disciplines, such as space science, information management systems, alternative energy technologies, and computer art and animation. Students today face an unprecedented range of social, scientific, economic, cultural, environmental, political, and technological issues. To deal with these issues, they first need competencies derived from discrete disciplines. The following are some examples: i) An interdisciplinary studies course in hospitality management would integrate studies in marketing and hospitality to help students understand the relationship among marketing practices, the local economy, and standards and innovative practices of the hospitality industry. In such a course, students might prepare a research report comparing successful and unsuccessful ventures into regional, national, and international tourist ventures, and analyze the impact of quality improvement on the financial health of a hotel organization. ii) An interdisciplinary studies course that introduces students to information studies would integrate studies in history, philosophy, and science to develop an understanding of the human need to use information to communicate knowledge, scholarship, and values in a global society. In such a course, students would use a variety of inquiry and research methods to analyze the evolution and impact of information and information technologies on society, and to report on effective ways to use knowledge institutions such as libraries and postsecondary institutions to support community involvement, future employment, and lifelong learning. iii) An interdisciplinary studies course that introduces students to biotechnology would integrate studies in biology and chemistry that are relevant to biotechnology to investigate biotechnology developments and careers in such diverse fields as health care, agriculture, forestry, and marine life. In such a course, students would research key trends and evaluate the economic, political, social, cultural, environmental, and ethical issues raised by biotechnology. They would apply their findings to assess the impact of biotechnological products on their local community. iv) An interdisciplinary studies course in small business operations would integrate studies in technological design and business entrepreneurship to enable students to address the specific needs of an identified market. In such a course, students would analyze needs, design and develop both prototypes and finished products (e.g., a business plan in electronic format that uses arresting graphics and effective hypertext links similar to business plans used by international enterprises), and apply entrepreneurial and design skills either to school and community projects or to potential employment ventures. Both their academic and applied work would help students recognize their strengths and skills for current and future employment. v) Skills that focus on the issues themselves, especially skills related to the research process, information management, collaboration, critical and creative thinking, and technological applications. Students need to know new methods and forms of analysis, interpretation, synthesis, and evaluation that will allow them to build on skills acquired through the core curriculum. Interdisciplinary practitioners can use modern systems-thinking and systems- design vi) To deal with today‘s issues, students also require interdisciplinary approaches to investigate how lasting solutions take into account all external and internal factors. Using models and prototypes, students can simulate ideas and test variables to produce new products or perspectives or find and implement solutions that go beyond established disciplines. vii) To make sense of the growth and often disparate nature of data and information, students must become information literate. To do this, they must be able to combine diverse models of research and inquiry, integrate a range of information-management skills and technologies, and apply the processes of information organization, storage, and retrieval to new situations and across many disciplines. Consequently, it is important to recognize that the skills, knowledge, insights, and innovations of the discipline of information studies are central to interdisciplinary work. viii) Students with well-developed information studies skills and knowledge will have increased marketability in a variety of careers. For instance, biology and chemistry graduates who know how to use global networks for scientific research to retrieve information and manage data will have greater opportunities for work in research labs. In the same way, graduates of economics, history, and political science who have taken courses requiring them to use information systems, online databases, and advanced research methods should have increased employment opportunities. In interdisciplinary studies courses, students consciously apply the concepts, methods, and language of more than one discipline to explore topics, develop skills, and solve problems. These courses are intended to reflect the linkages and interdependencies among subjects, disciplines, and courses and their attendant concepts, skills, and applications, and are more than the sum of the disciplines included. In an unpredictable and changing world, interdisciplinary study encourages students to choose new areas for personal study and to become independent, lifelong learners who have learned not only how to learn but also how to assess and value their own thinking, imagination, and ingenuity in decision- making situations. 6.7.1 The Goal of the Interdisciplinary Studies Programme The goal of the interdisciplinary studies program in Grades 11 and 12 is to ensure that students: i) build on and interconnect, in an innovative way, concepts and skills from diverse disciplines; ii) develop the ability to analyze and evaluate complex information from a wide range of print, media, electronic, and human resources; iii) learn to plan and work both independently and collaboratively; iv) are able to apply established and new technologies appropriately and effectively; v) use inquiry and research methods from diverse disciplines to identify problems and to research solutions beyond the scope of a single discipline; vi) develop the ability to view issues from multiple perspectives to challenge their assumptions and deepen their understanding; vii) use higher-level critical- and creative-thinking skills to synthesize methodologies and insights from a variety of disciplines and to implement innovative solutions; viii) apply interdisciplinary skills and knowledge to new contexts, real-world tasks, and on-the- job situations and thus develop a rich understanding of existing and potential personal and career opportunities; ix) use interdisciplinary activities to stimulate, monitor, regulate, and evaluate their thinking processes and thus learn how to learn. Interdisciplinary studies courses are appropriate for students with diverse abilities, interests, and learning styles, ranging from those who may need assistance in meeting diploma requirements to those enrolled in specialized programs of study such as technology or the arts. They will help students who are preparing to enter the workplace, as well as those who are planning to go on to study at a college or university. These courses reinforce students‘ general skills in a wide range of academic and applied contexts. 6.8 Content Organization of Mathematics Curriculum Cuoco, A.A, et.al. (1996) asserted that for many years school students have studied what is called mathematics. But this mathematics has very little connection with the way mathematics is created, applied outside of schools. Cuoco continued one of the reasons for this is a view of curriculum in which mathematics is seen as device for communicating established results and methods. That is the objective is preparing students for life after school by giving them a lot of facts for the students and mathematics was not considered as part of life. For instance students learn solving equations; find areas and calculate interest on a loan. In this view Curriculum reform simply means replacing one set of established results by another one. For instance, in steady of studying Algebra, students study Statistics. In this type of reform in curriculum the methods used are the same in which they learn some properties, work some problems in which they apply the properties and move on. The properties discussed in the above paragraph reflect the characteristics of traditional curriculum. There is another way of looking at mathematics Curriculum. In this method the priorities are turned around. Much more important are the habits of mind by the people who create those results (Cuoco A. A. and Goldberg E. P. 1996). In this new approach instead of asking what content is good to be included in the Curriculum the question now turns to ―what habits of mind are core? What are good mathematical habits of mind? In this type of Curriculum organization, the method by which mathematics is created and the techniques used by researches is highly emphasized. The goal of mathematics teaching is not to train school students for to be University mathematicians but it is to help students learn and adopt some of the ways that mathematicians think about problems. The objective is to give students the tools they will need in order to use, understand and even make mathematics that does not yet exist and to close the gap between the habits of mind of inventors of mathematics and the students. Cuoco A. A. and Goldenberg E. P. (1996) address a series of mathematical ―habits of mind‖, arguing that students should be pattern sniffers, experimenters, describers, tinkerers, inventors, visualizers, conjecturers, and guessers. The authors‘ discussed that the materials for teaching and learning school mathematics must provide students with problems and activities that develop these habits of mind and put them into practice. The authors‘ further pointed that every problem in a mathematics curriculum, organized around the habits of mind needs to: 1. Contain important, useful mathematics. 2. Have multiple ways, using different solution strategies. 3. Have various solutions or allows different decisions or positions to be taken and defended. 4. Engage students and encourages discourse. 5. Requires higher-level thinking and problem solving. 6. Contribute to students' conceptual development. 7. Connect to other important mathematical ideas. 8. Promotes the skillful use of mathematics. 9. Provides an opportunity to practice important skills. 10. Creates an opportunity for the teacher to assess what students are learning and where they may be expert. The lessons that constitute the content of syllabus for the entire school courses should be properly arranged and correlated for the different standards of the school so that teaching may be in the right lines and progress of the pupils may be studied. Any order of development should be based on the convenience of the teachers and the mental development of the pupils. We have four recognized procedures the topical and the spiral, the logical and the psychological orders. 6.8.1 Topical Plan According to the topical plan, the lessons are classified under various topics. The topical plan pays exclusive attention to one topic for a considerable amount of time and the topic is treated exhaustively. Then only the next topic is taken up for consideration. The four fundamental operations are taken up in the order addition, subtraction, multiplication and division and each one is taken up in it‘s entirely before the next topic is taken up. If we take up the number system as a topic, we continuously deal with the whole numbers, natural numbers, integers, rational numbers, irrational numbers and finally the real number system. In arithmetic we have topic like fractions, rational proportion, interest, percentage, average, areas, volume etc. In algebra the different topic will be symbolic representation, generalization, fundamental operations simplification, factorization, identities, equation, inequation, graph, etc. in theoretical geometry we have angles, polygons of triangles, circles, quadrilateral, etc. If the curriculum material is divided into topics, we are following logic, overlooking child psychology. No doubt it is desirable to aim at a comprehensive treatment of a topic until mastery is achieved before proceeding to the next. Advantages of Topical Plan i) The pupil is kept conscious that he is learning a particular topic and is satisfied that he has mastered it fully. ii) Pupil will be given opportunities to learn its several aspects as part of a whole. iii) Pupil‘s weakness in respect of it can be located and remedied once for all. Limitations of Topical Plan i) Exclusive attention to each one of the topics in arithmetic, algebra or geometry tends to mechanize the subject. ii) The understanding of pupil cannot be clear because of their difficulty to learn some of the later lessons in each topic which will be too much for the pupil. The pupil may not have the mental maturity to grasp the concepts. iii) Pupils will develop a sense of boredom when the same is dealt with for a long time. That is why rationalization which is very important for clear understanding may not be possible when this plan is strictly followed. iv) The richness derived from a course which contains a variety of application is lost. v) The various branches of mathematics are mutually supplementary. When properly inter correlated, it contributes to clear understanding. But under this plan the child cannot understand that they have all the same value. vi) There is every possibility for the earlier topics to be forgotten by the pupils. 6.8.2 Logical Plan A set of lessons are said to be arranged in a logical order the proofs are based on topics that are covered earlier. No theorem should be proved with the help of those that have not been already proved. Similarly no principles in algebra or geometry should involve rules and procedures that have not been already taught. The sequence of lessons based on this principle is the logical order. Mathematics is based on logic and the principles and procedures are interrelated by logical deductions. Naturally no sequence can be considered satisfactory unless they are logically arranged. Generally the logical development of any subject matter begins with a set of definitions and assumptions which are called axioms or postulates. In geometry the logical arrangement would involve the concept of a point, the formation of a line, the formation of a surface and ultimately the concept of three dimensional solid. Similarly in algebra we begin with the use of symbols, fundamental operation and then precede the explanation of technical terms like expression, degree, factor, product, etc. and then only we talk of functions and generalization. In arithmetic we must teach all about numbers before dealing with operations like addition or subtraction. Further logic demands that we should deal with decimal notation, after the teaching of integers. Limitations of Logical Plan i) This arrangement takes into consideration only the development of the subject matter in a logical manner but not the difficulties and interests of the pupils. ii) Some of the difficult lessons are bound to appear too soon in the list and the pupils may find difficulty in learning them. 6.8.3 Psychological Plan This plan refers to the arrangement of a set of lessons to suit the developing interest of the pupils. This is based on the psychological laws of learning. The organization of syllabus on the logical basis helps us in maintaining the link between the topics while the psychological basis will enable the pupils to easily grasp the subject matter. We proceed from easy to difficult, from the concrete to abstract and from practical to theoretical. While the logical order is based on the scientific development of subject matter, the psychological order concerns itself with the abilities and needs of the growing pupils. Taking geometry for illustration the solid which can be perceived by our senses is certainly more concrete than the surface. The surface is less abstract than the line. The point is the most abstract concept. Our school geometry begins with solid like cube, cuboid, cylinder, cone, etc. through which pupils get fairly good concept of solids and surfaces. Then the concept of the line is introduced as the intersection of the two plan surfaces. At the end of the notion of the point the intersection of two lines is introduced. In algebra we begin with generalizations than proceed to symbolic expression and then only fundamental operations. In arithmetic we take up decimals, only after the introduction of fractions. Advantages of the Psychological Plan i) It helps to create interest in the lesson. ii) It supplies the necessary motivation. iii) It relates teaching to the pupil‘s needs and purposes. iv) Integration of many mathematical processes and ideas become possible. Correlation gets its importance. In reality there is no opposition between what is logical and what is psychological. Logic tells us how to consider all the major premises and select one according to the need. The selection of premise is left to psychologists. In this way a thing which is logical also becomes psychological. Psychology takes into consideration the power of understanding and grasping of the pupils at a particular age level. The organization of the content on the logical basis helps us in maintaining the link between the topics while psychological basis will enable the pupils to easily grasp the subject matter. 6.8.4 Spiral or Concentric Plan Under this plan of organization the various branches are taught simultaneously and in a spiral fashion i.e. repeated in successive grades but in widening circles. A topic is taken first in its simple form and elementary ideas about it are given. Gradually as the pupils grow in age and knowledge the treatment of the topic becomes more and more exhaustive. A topic is chosen and is divided into convenient units say four or five. They are introduced in different classes in the order or their difficulties. The pupil‘s mental capacities are taken into consideration while we introduce the units in various classes. Each unit is introduced when it is needed or when the pupils have reached a stage of development which would make them appreciate its use. A look into the present syllabus helps one to understand how the spiral plan is used in introducing various units of the topics in different standards. We will illustrate a topic from the present syllabus of the school mathematics. Spiral Plan: (Topic 1) – Average Standard V: Average, Simple realistic problems involving application of the fundamental process to average. Standard VI: Average – Arithmetic mean, simple problem Standard VII: ………….. Standard VIII: ………… Standard IX: …………… Standard X: Measurement of central tendency, Arithmetic mean of discrete data, continuous data presented in frequency distribution, weighted arithmetic mean, its uses – median – mode. Spiral Plan: (Topic 2) – Mensuration (Tamil Nadu State Board Syllabus) Standard VI – Square, rectangle, four walls – area, perimeter, cube and cuboid – volume. Standard VII: area square and rectangular paths, running outside, inside and cross paths, right angled triangle – area, circle – circumference, area. Standard VIII: area of triangle and quadrilateral field, parallelogram and circular ring area, right prism triangular base, right triangle base, square base, surface area and volume, cylinder, curved surface and volume. Standard IX: area right angle triangle, equilateral triangle area of square in terms of diagonal, right, isolated triangle, regular hexagon, triangle when three sides are given area – sector, length of arc, perimeter and area. Standard X: prism, surface area – cross section equilateral triangle, hexagon etc. volume, cylinder – CSA, TSA and volume cone slant height, CSA and volume sphere – volume and surface area, hemisphere, cone sphere. Standard XI: trigonometry – properties and solutions of triangles, analytical geometry – triangle, circle, conics properties etc. Standard XII: vectors, geometrical applications of integration, area under curves, volume of solid revolution and area of a surface of revolution, isometries etc. While breaking a topic into units, we should not separate things which are best learnt together (a + b)2 = a2 + b2 + 2ab. The geometrical proof for the same and the method of finding out the squares of numbers are best learnt together. This plan is more natural and tiring to the pupils. Advantage of Spiral Plan i) It is a psychological plan child‘s interest, child‘s familiarity, mental maturity, spaced learning are all possible here. ii) Correlation is possible and hence learning is purposeful, meaningful and lasting. iii) This plan provides scope for revision at every stage of learning. Variety is introduced in teaching. New ideas are frequently placed before the pupils. iv) The risk of forgetting the concepts and principles learnt in various classes is very little. It provides for their repetition not only in a particular class but also from one class to another in the shape of spiral learning. Disadvantage of Spiral Plan While using this method no topic should be split up into meaningless fragments. This will isolate and make it difficult for the pupils to see the relations. The teacher has to be resourceful enough to make correlation natural. The spiral or concentric curriculum organization turns out to be the soundest and most effective method. 6.9 Homogeneous Grouping Definitions According to www.about.com ―Homogeneous Groups are groups organized so that students of similar instructional levels are placed together, working on materials suited to their particular level, as determined through assessments‖. According to Kulik (1992) ―This term refers to the presence of ability grouping in which students of like ability are placed into smaller learning groups‖. Grouping students based on their ability levels is generally referred to as Homogeneous Grouping. The basis of this method is to keep students who are at or around the same ability level together so that they can be challenged based on their individual educational needs. Their level of readiness for the tasks and challenges put forth to them is very similar, making learning a more stream lined assignment. Examples: When organizing reading groups, the teacher puts all of the ―high‖ students together in their own group. Then, the teacher meets with all of the ―high‖ readers at the same time and read a ―higher‖ book with them, and so on, through the various reading levels that exist in the class. 6.9.1 Types of Homogeneous Grouping There are several different ways that students can be grouped homogeneously. i) Within – Class Grouping This refers to student‘s staying in the same class, but being divided into small groups based on their ability levels. This is usually done in a math or reading class. Example of Within – Class Groupings: A reading teacher has a class of students with different reading abilities and comprehension levels. For a book assignment she divides the class into two different groups. One of her groups is a more advanced group of students who are above the reading level for their grade. The other group has students who are below the reading level for that grade. She assigns each group a book that is an appropriate level for them and then works with each group during class on their particular assignments. ii) Between – Class Grouping In this method the student‘s will be separated into different classes based on their ability level. It can be as simple as taking a different class or as complex as having a completely different course track within the school schedule. This is sometimes called ―tracking‖. A more specialized way to do this is sometimes called ―a school within a school‖. Example of Between – Class Groupings: A school system uses a standardized test that covers multiple subjects to discern the high achieving students in a school. Those students are then grouped together in a separate class (or group of classes) and assigned to specific teachers who will work with them at a more advanced level. Advantage of Homogeneous Grouping Homogeneous grouping made up of people from the same race, gender, social background and age, often provide equal access and participation of members, as there is less chance for exclusion. Members of a homogeneous grouping will have an easier time of comprehending each other‘s verbal and non verbal communications and will have more shared experiences in common. The similarities can, to some extent, avoid misunderstandings, prejudices and, arguably, speed up work processes and the completion of tasks, although this is not always the case as personality conflicts can occur within homogeneous grouping as easily as within heterogeneous grouping. There is evidence, such as within educational programs for gif ted students, that homogenizing groups on the basis of intelligence provides a good environment for high achievers to progress at a faster pace than is possible in mixed ability groups. Homogeneous Team Disadvantages Many studies, for example ―Cognitive effects of racial diversity‖ by Samuel R. Sommers, et al, in ―The Journal of Experimental Social Psychology,‖ show that the lack of diversification in a homogeneous group stifles creativity and information processing. It is, perhaps rightly so, very difficult to form homogeneous teams without causing feelings of exclusion to minorities, be those racial or gender. 6.10 Dalton Plan Dalton Plan is known as secondary-education technique based on individual learning. Developed by Helen Parkhurs in 1919, it was at first introduced at a school for the handicapped and then in 1920 in the high school of Dalton, Mass. The plan had grown out of the reaction of some progressive educators to the inadequacies inherent in the conventional grading system, which ignored individual variables in learning speed. Parkhurst developed a three-part plan that continues to be the structural foundation of a Dalton education—the House system, the Assignment, and the Laboratory. i) The House is a social community of students. ii) The Assignment is a monthly larger goal which students contract to complete. iii) The Laboratory refers to the subject teachers and subject-based classrooms intended to be the center of the educational experience from fourth grade through the end of secondary education. Students move between subject ―laboratories‖ (classrooms) exploring themes at their own pace. The Dalton Plan divided each subject in the school‘s curriculum into monthly assignments. Although pupils were free to plan their own work schedules, they were responsible for finishing one assignment before starting another. Pupils were encouraged to work in groups. Although popular for a time in the United States, Great Britain, Europe, and the Western colonial world, the Dalton Plan was criticized for being too individualistic and was finally given up. The Dalton Plan is designed i) to give pupils a training in handling a job ii) to teach a pupil to manage time and to plan his work iii) at each step of the way take himself and his needs into account in order to assure individual development at each point. 6.11 Supervised Study Hall Quest defined the term thus: ―Supervised study is that plan of school procedure whereby each pupil is so adequately instructed and directed in the methods of studying and thinking that his daily preparation will progress under conditions most favorable to a hygienic, emotional and self reliant career of intellectual endeavor‖. The term supervised study refers to the attempt on the part of the individual who supervise not alone to give directions for study, but also to see that such directions are specially applicable to the subject matter in hand and are followed out by the pupil. Supervised study involves a change from mass teaching to individual or smaller group instruction. The tendency in most of the supervised study plans is to ―relief‖ the individual, to set him apart from the group so as to give him a better opportunity to employ his type of learning to an efficient achievement according to his rate of progress. The objection often urged that supervised study prevents the pupil from relying on himself, is answered sufficiently by suggesting that without it he does not rely upon himself, but instead enlists family, friends, and classmates in his behalf. Home study is harmful, not only because unhygienic, but also because it tempts the pupil to claim as his own work what others have done for him. Problems in mathematics, theme writing, and outline work are referred to others whenever feasible to do so. This is natural. The difficulties are severe, the assignments are unreasonably long, and class marks are usually impersonal. What is the pupil to do? Let him without offense in this type of home study cast the first stone. Supervised study means working with the pupil but not for him. It is preventive. It deals not merely with subject matter, but with methods of learning it, and for this reason it is necessary that the teacher for a considerable time devote part or most of the class period to this fundamental of learning. The high school pupil must be taught how to think, how to organize, and how to apply. The main points of supervised study are: i) The supervision of individual students who are studying silently at their desks should replace a considerable part of the time now spent on recitations and home study. ii) Poor students especially fail to profit under the system of recitations based on home study. iii) Precisely measured, experimental investigations show that supervised study improves the work of poor students. iv) Divided periods, part for recitation and part for supervised study should be arranged as regular parts of the daily programs in most high school subjects. v) Conditions favorable to study are those favorable to concentration of attention. a) Physical conditions and certain routine habits may be easily improved. b) Spontaneous interest and concentrated thinking are more difficult to secure, but are essential. vi) A special technique of supervising study should be mastered by teachers. It should include. a) Skill in determining the character of the progress being made by students while they are studying. b) Skill in stimulating and aiding this progress by means of questions and suggestions without assisting too much. 6.12 Curricular Models How does a program move from the purpose of helping learners develop key abilities to actually identifying and designing courses? It may be helpful to examine two of the most common curriculum models are ―content-based‖, ―knowledge-based‖ and ―developmental‖ models. We will then describe ways in which a developmental model differs from and goes beyond both of these models. i) Content – Based Model In a strictly content-based model of curriculum, the faculty focuses almost entirely on selecting content. Their main task is to prepare lectures. Such an approach can be appropriate if the course is responsive to the needs of the potential learners, if the faculty members teach so that the content is relevant to learners, and if the experience enables learners to better perform their courtroom responsibilities. And sometimes that is indeed the case. Often this approach is poorly executed, however. The central question for the faculty becomes ―What topics shall I cover?‖ rather than ―What can I do to help learners develop the abilities they need?‖ Teaching can become merely ―telling,‖ where the faculty member only conveys information to passive learners. ii) Knowledge – Based Model Judicial education programs can take a giant step by moving from a content-based curriculum to a knowledge-based curriculum. This is the model advocated by many colleges of education and followed by most elementary and secondary schools, some college programs, and many continuing professional education programs. A knowledge-based curriculum centers on the knowledge gained from the perspective of the learner. It places great emphasis on developing clear and sound learning objectives. By organizing the information to be learned into manageable objectives, planners can then design activities that help learners master the objectives. The planners can sequence the topics to be dealt with, can choose the appropriate teaching methods, and can select the necessary instructional resources. Through designing courses in terms of learning objectives, a curriculum becomes oriented to learning outcomes rather than simply ―covering content‖. However, a knowledge-based curriculum done poorly has major shortcomings. The central question becomes ―What are the objectives of this course?‖ This emphasis on the development of objectives sometimes leads to reductionistic thinking on the part of planners (and learners). Knowledge becomes broken down into smaller and smaller pieces. Such a process conjures up the image of ―gathering‖ knowledge similar to squirrels gathering nuts. The squirrel that gets the most nuts has the most knowledge. Falling into reductionistic thinking frequently happens, even though all faculty and learners agree that programs should help learners to think more deeply, interpret data, and solve problems, rather than ask them to accumulate information. As Cervero (l989) phrased it, good continuing educational programs are those which help professionals learn ―to take wise action‖. It is important, however, not to caricature knowledge-based curriculum and to denigrate its accomplishments. Thinking in terms of objectives and learning outcomes is a leap forward compared with content-based curriculum planning. A well-developed curriculum, with carefully devised objectives, can and does help learners do more than learn discrete facts. It helps them develop ways of thinking required for high performance. iii) Developmental Model A developmental curriculum model -- the approach discussed in this chapter -- in no way minimizes the need judges have for specific information and skills. It, too, is based on an assessment of learners‘ needs, and it emphasizes helping participants learn the information they need in order to perform day- to-day tasks. It differs from content-based models and knowledge-based models in that it seeks to help participants learn practical information in the service of further development of abilities deemed necessary for outstanding performance. In a developmental curriculum, the central question is ―How can I help the learners engage with content in ways that help them develop the abilities they need to be excellent performers?‖

6.13 SMSG The School Mathematics Study Group (SMSG) and various committees which have been undertaking the revision of mathematics curriculum from time to time on the basis of research findings and also on the basis of the changing needs of society have worked out the content in detail from grades 1 to 12. There is considerable emphasis on commutative, associative and distributive properties: i) Number systems with different bases. ii) Inverse operation of addition iii) Emphasis on number systems which are treated algebraically. iv) No artificial distinction between plane geometry solid geometry and a considerable amount of solid geometry is included. v) Introduces coordinate geometry. vi) More emphasis on the understanding of logarithms than on computations. vii) Placement of topics one or two grades earlier than they were previously placed. viii) Replacement of some absolute subject matter. ix) Use of concepts to organize number line, relation, function, sets, etc. x) The deductive discovery method develops interest in mathematics, skill in mathematical thinking and it develops versatility in applied mathematics.

6.13.1 Criticism of SMSG i) The SMSG programme has been abstract and remote from the children‘s experiences, their conceptual abilities and their skill in performing numerical operations. ii) It lacked inherent devices to arouse motivation. iii) It contained insufficient recapitulation of separate topics and transitions between topics. iv) The modern mathematics programme seems to make the obvious complicated. v) Approaches the same concept from too many angles. 6.14 SCERT The State Council of Educational Research and Training (SCERT) were set up in each state on the pattern of NCERT. As education is a state subject, NCERT cannot do much to improve education in the state. The SCERT has a Programme Advisory Committee under the Chairmanship of the State Education Minister. SCERT is also the affiliating body for Elementary Teacher Education (ETE) Course and Early Childhood Care and Education (ECCE) Course. SCERT is responsible for preparing the curriculum, prescribing syllabi, course of study, academic calendar for these Courses. SCERT also conducts the Entrance Examination for its affiliate institutes and admits candidates for the Course. SCERT awards the Diploma to successful students in ETE and NTT. The functions of the SCERT are to: i) act as an agent of change, and quality improvement in school education, non-formal education and teacher education as well as carry out institutional assessment and accreditation. ii) arrange for the in-service education and orientation of supervisory / inspecting officers dealing with pre-school, elementary, secondary and higher secondary stages of education. iii) organize in-service education of teacher-educators at all stages. iv) plan and implement programmes (including distance education) for the professional development of teachers and teacher-educators. v) produce instructional materials for teacher training institutes, specially at the preprimary elementary stage. vi) provide extension service centers in teacher training institutions and coordinate the work of these centers. vii) develop curriculum for all stages of school education. viii) produce text books, teacher-handbooks and other materials for the use of educational institutions from preschool to the higher secondary stages. ix) conduct studies and investigations in various aspects of curriculum development and transaction at the school and teacher education levels. x) coordinate the work of different associations concerned with school education (subject, teacher and school). xi) interact with the state, national as well as the international organizations in the field of education like the. 6.14.1 Structure of SCERT The SCERT has the following Departments and units:

i) Department of Curriculum Development

ii) Department of Teacher Education and in-Service Education

iii) Department of Educational Research

iv) Department of Science and Mathematics Education

v) Department of Educational Technology

vi) Department of Evaluation and Examination Reforms.

vii) Department of Non-formal Education

viii) Department of Population Education

ix) Department of Pre -school and Elementary Education

x) Department of Adult Education and Education for Weaker Sections

xi) Department of Extension Service and School Management

xii) Public action Unit

xiii) Library Unit

xiv) Administrative Unit SCERT is serving as the academic wing of the Department of Education. It is trying to bring about the qualitative improvement of school education and teacher education. It is maintaining close links with the national and international organizations of education. It is organizing a large number of workshops and training courses for various categories of teachers and teacher educators. 6.15 Assessment of Various School Curricula Having completed the teaching portion of the curriculum model, our focus shifts back now to the curriculum level and evaluation. A variety of procedures can be employed to judge quality and to improve the curriculum. Educators have two important tasks in assesment: (a) to gather data on the effectiveness of the courses and of the curriculum and (b) to interpret the data to improve the curriculum. 6.15.1 Gathering Data At the course level, there is an array of data to gather. Frequently, learners are asked to complete a written evaluation at the conclusion of a course. Questions may be included to evaluate: i) The relevance of the content. ii) The appropriateness of the course design. iii) The effectiveness of the faculty. iv) The adequacy of the logistical arrangements such as registration, facilities, and food service. There may also be questions concerning the learning itself. Participants can be asked to rate the extent to which they believe course objectives have been met. Judicial educators may also want the faculty to rate the extent they believe they were able to help the learners achieve each objective. In a curriculum oriented to helping learners learn ―in order to do‖ rather than simply learn ―in order to know‖, the impact of the curriculum on job performance should be assessed. One technique for this is to ask a sample of learners to complete a follow-up questionnaire, perhaps six months after the course. They could be asked to describe ways the course has been useful to them in their work. 6.15.2 Interpreting Data to Improve the Curriculum The purpose of gathering these kinds of data is to enable judicial educators to make judgments about the effectiveness of the curriculum and to improve the planning and implementation of the courses. For example, the data may indicate that some of the curriculum objectives are inappropriate or that additional ones should be developed. The design of the courses may be deemed inappropriate to achieve the purpose and objectives of the curriculum. Some of the faculty may be replaced or encouraged to engage in professional development activities. 6.16 Let Us Sum Up This unit deals with the curriculum designing in mathematics. A good syllabus leads to better mathematical learning outcomes in pupils. Teacher may be able to select a good content for mathematics curriculum. The principles of curriculum designing in mathematics are used to guide the evolution of the aims and objectives, the structure of the curriculum and the identification of objectives in mathematics. We have basically four organizational plan of mathematics curriculum namely topical, logical, psychological and Spiral or Concentric plan. The Dalton Plan is designed (1) to give pupils training in handling a job (2) to teach a pupil to manage time and to plan his work (3) at each step of the way take himself and his needs into account in order to assure individual development at each point. Finally this unit provides an opportunity to educators have two important tasks in assessment of school curricula: (a) to gather data on the effectiveness of the courses and of the curriculum and (b) to interpret the data to improve the curriculum. 6.17 Answers to Check Your Progress i) How to select content for mathematics curriculum? ii) Explain the principle of curriculum designing in mathematics. iii) Describe the content organization for mathematics curriculum. iv) Describe spiral and concentric plan for mathematics curriculum. v) Explain the Dalton plan. vi) Write a short note on the SMSG and SCET. vii) How to assess the various school curricula? 6.18 References Cuoco, A. (1995). Some worries about mathematics education. Mathematics Teacher, 88(3), 186-187. Cuoco, AA, Goldeberg, EP & Mark, J. 1996: Habits of Mind: An organizing Principle of Mathematics Curriculum. In Journal of Mathematics Behavior, 15:375 – 402. National Research Council (1989). Everybody counts: A report to the nation on the future of mathematics education. Washington, DC: National Academy Press. Kleiner, I. (1986). The evolution of group theory: A brief survey. Mathematics Magazine, 59(4): 195- 215. Dalton Plan. (2012). In Encyclopedia Britannica. Retrieved from http://www.britannica.com/EBchecked/topic/150307/Dalton-Plan Evelyn Dewey, (1922).The Dalton Laboratory Plan, E.P. Dutton Alfred Lawrence Hall-Ques (1917). Supervised Study: A Discussion of the Study Lesson in High School. The Macmillan Company, New York. G. W. Willett (1979). Supervised study in High Scholl. The University of Chicago Press. Chicago.

UNIT VII EQUIPMENTS AND RESOURCES FOR MATHEMATICS TEACHING Structure 7.1 Introduction 7.2 Objective 7.3 Text-books 7.3.1 Advantages of Textbooks 7.3.2 Disadvantages of Textbooks 7.4 Blackboard 7.5 Instruments and apparatus 7.6 Reference-books 7.7 Hand-books 7.8 Work-books 7.9 Library 7.10 Mathematics club 7.11 Radio 7.11.1 Major Educational Radio Projects in India 7.12 TV 7.12.1 The Indian Beginning of TV 7.12.2 Major Educational Projects through TV in India 7.13 VCR 7.14 Computer 7.14.1 The advantages of Computers in Education Primarily Include 7.15 Let Us Sum Up 7.16 Answers to Check Your Progress 7.17 References 7.1 Introduction In the process of teaching mathematics, the teacher has often to report to certain devices besides observing certain principle, methods and procedure. Of course, the use of text books, work books, blackboard, the classroom conditions, mathematics club and library are used to develop the mathematical knowledge and interest in pupils. This unit describes the different types of classroom resources and equipment which would enhance teaching in mathematics lessons. In addition, where there are resources which would be useful to achieve the specific objective of mathematics teaching. 7.2 Objectives On completion of this unit students will be able to …. 8. understand the equipments and resources for mathematics teaching; 9. understand the advantages and disadvantages of text books; 10. recognize the various teaching instruments and apparatus; 11. describe the reference books and hand books; 12. know the use of library and work books; 13. aware the educational utility of radio and TV; 14. use the computer and VCR. 7.3 Textbooks As the time you visit class rooms, you probably notice and wonder that most, if not all, of those classrooms use a standard textbook series. The reasons for this are many, depending on the design and focus of the curriculum, the mandates of the administration, and/or the level of expertise on the part of classroom teachers. 7.3.1 Advantages of Textbooks i) Textbooks are especially helpful for beginning teachers. The material to be covered and the design of each lesson are carefully spelled out in detail. ii) Textbooks provide organized units of work. A text book gives you all the plans and lessons you need to cover a topic in some detail. iii) A textbook series provides you with a balanced, chronological presentation of information. iv) Textbooks are a detailed sequence of teaching procedures that tell you what to do and when to do it. There are no surprises — everything is carefully spelled out. v) Textbooks provide administrators and teachers with a complete program. The series is typically based on the latest research and teaching strategies. vi) Good textbooks are excellent teaching aids. They‘re a resource for both teachers and students.

7.3.2 Disadvantages of Textbook A textbook is only as good as the teacher who uses it and it‘s important to remember that a textbook is just one tool, perhaps a very important tool, in your teaching arsenal. Sometimes, teachers over-rely on text books and don‘t consider other aids or other materials for the classroom. As a teacher, you‘ll need to make many decisions, and one of those is how you want to use the textbook. As good as they may appear on the surface, textbooks do have some limitations. 7.4 Blackboard Any teacher can use Blackboard, whether teaching in a traditional classroom, online, or in a hybrid format. Blackboard allows teachers to set up their course websites to provide relevant resources for their students. Because teachers create their own Bb websites, the sites match the students' needs. Teachers can even choose a color scheme for their Blackboard website and place a welcome banner on the website, as well as their photos and other graphics. Each teacher chooses which features to use or not use in Blackboard so that a class may include as much or as little as the teacher wishes. One teacher may post the course assignments syllabus and use a discussion area in a Blackboard class. Another teacher might also use the Grade book and Small Group work areas, in addition to the other features. One teacher might have a different Blackboard class for each section taught each semester. Another teacher might use one Blackboard class for every section of the same course taught in a semester. The options with Blackboard seem endless based on the teacher‘s preferences and the students‘ needs. Blackboard is an online tool that can be used to disseminate information and exchange course materials with students. It is ―a Virtual learning environment (VLE) designed to facilitate teachers in the management of educational courses for their students, especially by helping teachers and learners with course administration.‖ Blackboard can help you monitor your students‘ progress by keeping track of assignments with the use of a grade book. You can also use blackboard to create online test materials. Ideas for use in the classroom: i) Posting assignments of any kind. ii) Disseminating course documents of any kind. iii) Administering online surveys, quizzes or tests. iv) Conducting discussions on various topics related to the curriculum. v) Conducting virtual chats about daily assignments, for review work, conferencing with students, etc. vi) Posting important links to use in the classroom. vii) Posting readings for students. viii) Creating podcasts and videos for curriculum information, enrichment or review. ix) Exchanging or sharing documents with students. x) Facilitating group discussions. 7.5 Instruments and Apparatus A mathematical instruments and apparatus are a tools or devices used in the study or practice of mathematics. Mathematical instruments and apparatus have been in use since the beginning of civilization. They were in use at the time of building of Babylon and were well known to the Egyptians. The earliest surviving examples date from roman times. Here some popular mathematical instrument and apparatus are mentioned below. Compasses: A piece of equipment used for drawing circles, consisting of two thin parts joined in the shape of the letter V. Dividers: A piece of equipment used for measuring or drawing lines and angles, consisting of two pieces of metal with pointed ends that are joined together at the top. Plumb line: A piece of string with a metal object attached to the bottom, used for checking whether a wall is straight or for measuring the depth of water. Protractor: An object that is shaped like half a circle and is used for measuring and drawing angles. Ruler: an object used for measuring or for drawing straight lines, consisting of a long flat piece of plastic, wood or metal marked with units of measurement. Set square: A triangle for drawing lines and measuring angles. Slide rule: A simple piece of equipment like a ruler with a piece in the middle that slides along used for calculating. Theodolite: A piece of equipment that a surveyor uses for measuring angles. Triangle: A small object with three sides and one angle that is 900 used for drawing straight lines, triangles and right angles. T-square: A tool in the shape of the letter T that can help you to draw lines that are exactly parallel or in the shape of square. A T-square is used in making detailed plans for building, etc. 7.6 Reference Books Reference books are the ideal starting place for any research you conduct in the library. These are sources that can provide back ground information on a topic or even help you to decide if the topic is one that you wish to continue researching. They can also provide you with facts, definitions, statistics, biographies, quotations and more. You will usually find reference books in the library's Reference Collection located on the main floor behind the Reference Desk. Reference books are for library use only and cannot be taken outside the library. There are several types of reference books which you should know about: i) Dictionaries ii) Encyclopedias iii) Almanacs iv) Atlases v) Bibliographies vi) Directories vii) Handbooks viii) Yearbooks We call these types of reference books General Reference Works. i) Encyclopedias Encyclopedias provide compressed, factual overviews on a number of subjects. They may be composed of many volumes of information and are usually found in the reference section under the A call number section. The Hartnell College Library has the following types of general encyclopedias: a) Encyclopedia b) Americana c) Encyclopaedia d) Britannica e) Hispanica f) World Book One of the most important parts of an encyclopedia is the index. The index allows you to find multiple parts of information related to the topic you are researching. These may be located in more than one volume of the encyclopedia. The index is usually located in the last volume of a multi -volume encyclopedia or in the back of a single volume encyclopedia. There are also subject-specific encyclopedias. Some examples within the Hartnell Library are: a) Encyclopedia of philosophy b) The Encyclopedia of popular music c) An encyclopedia of religion ii) Dictionaries The library offers several different types of language dictionaries. They provide definitions of words in alphabetical format. Some provide information about pronunciation and etymology of the word, as well as examples of how the word might be used within a sentence. Here are some examples of general dictionaries: a) The American Heritage college dictionary b) The Random House dictionary of the English language c) Webster‘s dictionary of English usage The library also contains specialized dictionaries such as the following titles: a) APA dictionary of psychology b) The complete film dictionary c) The Facts on File dictionary of classical, biblical, and literary allusions d) Merriam-Webster‘s collegiate thesaurus e) The new Partridge dictionary of slang and unconventional English. iii) Atlases Atlases allow you to find the location of places, directions how to get to it, and information about its geographic features. Some of the atlases the Hartnell Library ha s in its reference collection includes the following: a) Atlas of North America b) Hammond world atlas c) Rand McNally commercial atlas & marketing guide, 2006 d) The world book atlas iv) Yearbooks A yearbook is a reference which gives facts and other historical information as having occurred during a particular year. A yearbook may be of general scope or limited to one subject e.g. Yearbook of World Affairs and Yearbook of Agriculture. A general encyclopedia is usually updated by publishing a yearbook e.g. The Britannica Book of the Year. v) Handbooks A handbook is a reference which contains basic and brief information on specific subjects e.g. Handbook of Business. vi) Dictionaries A dictionary is a reference work which gives the meaning of words, their origin, and the way they are pronounced. A dictionary may be: a) Monolingual: listing words and their meanings in the same language e.g. Webster's Dictionary of the English language. b) Bilingual: lists words in one language and gives their meaning in another. e.g. - A1 -Mawrid English -Arabic Dictionary. c) Multilingual: lists words and their meanings in two or more languages e.g. Dictionnaire Trilingue: Arabe -Franca is -Anglais. There are also dictionaries that list the terms of a subject and their meanings e.g. A Dictionary of the Physical Sciences A Dictionary of the Social Sciences: English -French -Arabic. vii) Biographical Dictionaries These dictionaries give information on the lives of important people. There are 3 major types of such dictionaries in relation to the people included: a) People in general, international in scope e.g. Dictionary of International Biography. b) People in general but of the same nationality e.g. Who's Who in the Arab World. c) People belonging to a profession or a certain field of knowledge e.g. Dictionary of Scientific Biography.

7.7 Hand Books A handbook is a type of reference work, or other collection of instructions, that is intended to provide ready reference. A handbook is sometimes referred to as a vade mecum (Latin, ―go with me‖) or pocket reference that is intended to be carried at all times. Also it can be referred to as an enchiridion. Hand books may deal with any topic, and are generally compendiums of information in a particular field or about a particular technique. They are designed to be easily consulted and provide quick answers in a certain area. For example, the MLA Hand book for Writers of Research Papers is a reference for how to cite works in MLA style, among other things. ―Hand book‖ is sometimes applied to documents that are produced within an organization that are not designed for publication— such as a company hand book for HR, for instance. In this case, the term is used nearly synonymously with ―manual‖. The name ―hand book‖ may sometimes be applied to reference works that are not pocket- sized, but do provide ready reference, as is the case with several engineering hand books such as Perry‘s Chemical Engineers‘ Hand book, Marks Standard Hand book for Mechanical Engineers, and the CRC Hand book of Chemistry and Physics. Hand books are widely used in the sciences as quick references for various kinds of data. 7.8 Work Books The workbooks will also be helpful for teachers who have to teach more than one grade in a class. Because the workbooks are available for learners from grade 1 to 6, it is possible for the teacher to work separately with different grade sin the same classroom – giving each grade its own grade specific workbooks. They will also be useful for teaching mixed ability groups – with each group working on a different worksheet. Work books in the American education system, are cheap, paperback textbooks, issued to students. Work books are usually filled with practice problems, where the answers can be written directly in the book. The workbooks will assist teachers to identify learners‘ needs for extra support from early in the school year. The Department‘s new workbooks all start with revision of the previous grade level with grade 1 giving attention to school readiness and cognitive development. The startup of each grade with a review/ revision of previous grade‘s work will enable the teacher to use this to identify learners in need of extra support or remedial action. In the workbooks each alternate lesson also requires the teacher to make a judgment on which learners require additional or remedial support and which high achievers many need extended activities. They will provide organized work in the form of worksheets for every child in mathematics and language. The aim of the workbook project is provide every child with two books of worksheets – one for numeracy/mathematics and one for literacy / language in the child‘s mother tongue. Each book contains 128 worksheets (two pages each) – one a day for four days of the week. In the third term, learners will be provided with another two books – one for mathematics and one for language. Learners will use the books to do written exercises in language and mathematics. They will be a great help to teachers. The worksheets are also intended to assist teachers who have large classes and who won‘t necessarily have resources like photo copiers or stimulating reading materials for children to read. The workbooks will help teachers who teach multilingual classes. The worksheets will also be useful for teachers (mainly in urban areas) where they have to teach multilingual classes. Sometimes teachers have up to five different mother tongue languages in one class. Because the books are available in all 11 languages, teachers will find the books useful in mixed language. Work books are often used in schools for younger students, either in middle school or elementary school. They are favored because students can work directly in their books, eliminating the need for loose leaf and copying questions from a textbook. In industry, they may be customized interactive manuals which are used to help provide structure to an otherwise complex problem. Work books also hold an advantage because they are usually smaller and lighter than textbooks, which equates to less trouble when the student brings the book home to complete their homework.

7.9 Library The school library is central to learning and plays a key role as a place for encouraging innovation, curiosity, and problem solving. Your library is a catalyst for literacy and reading and for teaching and scaffolding inquiry learning. School libraries make a difference to students‘ understanding and achievement and provide support for teaching and learning throughout the school. The school library is an important part of the school community and reflects and welcomes this community. The later part of the 19th century marked the beginning of the modern American library movement with the creation of the American Library Association (ALA) in 1876 by a group of librarians led by Melvil Dewey. At these beginning stages of development, the school libraries were primarily made up of small collections with the school librarian playing primarily a clerical role. 1920 marked the first effort by the library and education communities to evaluate school libraries with the publication of the Certain Report, which provided the first yardstick for evaluating school libraries. The school library media center program is a collaborative venture in which school library media specialists, teachers, and administrators work together to provide opportunities for the social, cultural, and educational growth of students. Activities that are part of the school library media program can take place in the school library media center, the laboratory classroom, through the school, and via the school library's online resources. Many school librarians also perform clerical duties. They handle the circulating and cataloging of materials, facilitate interlibrary loans, shelve materials, perform inventory, etc. School library plays a key role in the cultural and social life of the school. It can be a central point for engagement with all kinds of reading, cultural activities, access to information, knowledge building, deep thinking and lively discussion. School library can also play a key role in building a learning community. A school library reflects students‘ identities through ensuring that the languages and cultures of the school community are an integral part of the library‘s collection, services, and environment. The library is a place for inclusiveness. The International Federation of Library Association‘s (IFLA) Manifesto states: ―The school library provides information and ideas that are fundamental to functioning successfully in today‘s information and knowledge - based society. The school library equips students with lifelong learning skills and develops the imagination, enabling them to live as responsible citizens‖ (2008). A school library (or a school library media center) is a library within a school where students, staff, and often, parents of a public (state) or private (fee paying) school have access to a variety of resources. The goal of the school library media center is to ensure that all members of the school community have equitable access ―to books and reading, to information, and to information technology.‖ A school library media center ―uses all types of media... is auto mated, and utilizes the Internet as well as books for information gathering.‖ School libraries are distinct from public libraries because they serve as ―learner- oriented laboratories which support, extend, and individualize the school's curriculum... A school library serves as the center and coordinating agency for all material used in the school.‖ In many schools, school libraries are staffed by librarians, teacher- librarians, or school library media specialists who hold a specific library science degree. In some jurisdictions, school librarians are required to have specific certification and/or a teaching certificate. The school librarian performs four leadership main roles: teacher, instructional partner, information specialist, and program administrator. In the teacher role, the school librarian develops and implements curricula relating to information literacy and inquiry. School librarians may read to children, assist them in selecting books, and assist with schoolwork. Some school librarians see classes on a ―flexible schedule‖. A flexible schedule means that rather than having students come to the library for instruction at a fixed time every week, the classroom teacher schedules library time when library skills or materials are needed as part of the classroom learning experience. 7.10 Mathematics Club Every school can have a mathematics club. The mathematics club is an association of student‘s society for mathematics who are interested in mathematics. It aims to provide a beneficial social outlet for the needed students. In addition to its many social opportunities the mathematics club also provides mathematics resources, such as a reference library and exam package. The mathematics clubroom is typically open during the day; many students come and hang out and study there. Motivating children to learn mathematics with interest and involvement through appreciation of its intrinsic worth poses a challenge to practicing teachers of mathematics and parents. One of the ways of solving this crucial problem is to expose children to mathematics in the environment and help them to engage themselves in manipulating ordinary objects and numbers associated with them so as to experience the mathematics inherent in such manipulations without resorting to a formal study of mathematics. It is the birthright of every mathematically gifted child to be given opportunity to develop his research potential to become junior mathematicians before blossoming into senior mathematicians later. Provision of a mathematics club with a small library stocked with enrichment books is the least the gifted have a right to expect in a school, if not at home. 7.10.1 Aims of Mathematics Club i) Create an easy to learn concept of mathematics. ii) Develop techniques for critical thinking. iii) Influence students to like math and explore their hidden potentials in math. iv) Find paths to introduce math in creative and innovative ways. v) Encourage the study and appreciation of mathematics. vi) Help students with difficulties in math. vii) Explore students‘ potential as being mathematicians. viii) Promote Exceptional Work Habits and Behaviours ix) Create a supportive learning environment for mathematics. 7.11 Radio Radio is a powerful mass medium used in education for disseminating information, imparting instruction and giving entertainment. It serves with equal ease in both developed and developing countries. It spreads information to a greater group of population thereby saving time, energy, money and man- power in an effective way. Radio is a simple and cheap medium readily available as a small toy. Now small and handy transistors are available with even poorest of people. A small transistor can carry the message to any place on - the earth. It needs very little for maintenance and cheaper production can be taken up with more and more resources. Radio speaks to an individual so also to millions at a time. Hence, any listener can think the broadcast is meant for him whereas when listened in group all think the massage directed towards them. Each student takes the broadcast as very intimate to him. Due to its portability and easy accessibility radio could found its place everywhere whether it was a field, a school, a kitchen or a study room. Radio is a blind man‘s medium and is meant for ears only. It plays with sound and silence where the sound can be anything like voice or word, music and effect. When one hears radio, simultaneously one can imagine happenings in his/her mind. So it is called as theatre of blind or a stage for the mind. Radio can be listened to simultaneously along with another work like reading also. In United Kingdom, education was taken up through radio just after two years of starting of broad casting in 1922 with initiation of British Broad casting Company. This company became British Broad casting Corporation (BBC) after 5 years. Then educational radio was controlled by an Educational Council. Twenty local radio stations are now in operation in England, each of them broad casting locally devised programmes. Australian Broad casting Corporation introduced educational broad cast in 1929 where representatives from schools assisted in their earlier attempt. Radio came to India through amateurs with educational purpose first in1923 in a small way and after four years it could find its root here. In India, then it was used for educational purposes in almost all the possible fields. Being the only instrument to reach to masses in this country for a long time, its educational role was exploited thoroughly. All India Radio was a government medium and had the opportunity of covering the entire Country. It has been mostly used for developmental activities after independence. As such All India Radio has an objective to broad cast education with information and entertainment. So in most of its broad casts the educational element used to be there. While the accent of all the programmes whether for the general listener, or specific groups like farmers, women, children, students, teachers or industrial workers, is on education in the widest sense, some programmes planned with a specific educational objective.

7.11.1 Major Educational Radio Projects in India The main projects that describe the growth of educational radio are: i) School Broadcast Project This project was commissioned in 1937 and the target group was School students. This programme started from Delhi, Calcutta, Madras and Bombay. In the beginning the school programme were not strictly governed by the curriculum. With the passages of time and acquisitions of more experience, the AIR tried to make its radio broadcasts more curriculums oriented, but in absence of common syllabi and time tables in schools, even within the same state, it could not succeed in its aim. ii) Adult education and community development project (Radio Forum) Commenced in 1956, the Villagers of 144 villages in the vicinity of Poona (in Maharastra state), were the main beneficiaries of this project. This was agriculture - based project, which was originally designed and tried out in Canada. With the help of UNESCO, it was tried in 144 villages of Poona and was named as ‗Radio forums Project‘ (defined as a listening cum- discussion- cum action group). The members of the forum could listen thirty - minute radio programme on some a gricultural or community - development programme, then discuss and decide regarding its adoption in their own village. This project was a great success. Many action programmes were planned and put into practice. iii) Farm and Home Broadcast Project This project was commenced in 1966 and again targeted at Farmers and villagers. These broadcasts were designed to provide information and advice on agricultural and allied topics. The aim was to educate the farmers and provide them assistance in adopting innovative practices in their fields as per the local relevance. The experts also conducted occasional farm radio schools, which proved to be very effective. iv) University broadcast project This project for University students was initiated in 1965, with an aim to expand higher education as widely as possible among the different strata of society. The Programme consisted of two types – ‗General‘ & ‗enrichment‘. The general programmes included topics of public interest and enrichment programmes supported correspondence education offered by universities in their respective jurisdictions. School of Correspondence studies, University of Delhi and the Central Institute of English and Foreign Languages, Hyderabad is well known for preparation and broadcast of their programmes through AIR. v) Language Learning Programme The project, popularly known as 'Radio Pilot project' was started in 1979- 80 jointly by AIR and Department of Education Government of Rajasthan, with an aim to teach Hindi to School going children as first language in 500 primary schools of Jaipur & Ajmer districts on experimental basis. The project was found useful in improving the vocabulary of children. With its success, similar project was repeated in Hoshangabad district of Madhya Pradesh with some modifications but had limited success. vi) IGNOU- AIR Broadcast In collaboration with IGNOU, AIR stations of Mumbai, Hyderabad and Shillong started radio broadcasts of IGNOU Programmes from January 1992. Main target group of this project were students of Open / Conventional Universities. Although Shillong started this but discontinued later on. Therefore presently it is being broadcast from AIR Mumbai (Every Thursday and Saturday from 7: 15A M- 7:45A M) and AIR Hyderabad (Every Tuesday, Thursday & Saturday from 6: 00 AM - 6:30 AM) only. This programme is still popular in the respective region. vii) IGNOU- AIR Interactive Radio Counseling (IRC) Started in 1998 for students of Open / Conventional Universities, this project is also very successful. In order to bridge the gap between Institutions and learners by instantly responding to their queries and also to provide Academic Counselling in subject area, IGNOU in collaboration with AIR Bhopal started this project in May 1998 as an experimental programme for one year (Sharma, 2002a). With the success of the experiment, it was extended to 8 other AIR stations (Lucknow, Patna, Jaipur, Shimla, Rohtak, Jalandhra, Delhi and Jammu). Presently Interactive Radio counseling is being provided one very Sunday for one hour (4:00 PM - 5:00 PM) from 186 radio stations of All India Radio. This includes two Sundays on the National hook- up. Toll- free telephone facility is available from 80 cities (effective from February 2001) enabling the learners to interact with experts and seek clarification, without paying for their telephone calls. The first and third Sundays of the month, AIR stations of Delhi (Hindi) and Kolkata (in English) broadcast from national hook-up, which 186 radio stations relay either of them. The 2nd and 4th Sunday are slotted for programmes of various regional centers of IGNOU and State Open universities respectively. The slot of 5th Sunday (if any) has also been given to region based programmes of IGNOU. This programme is gaining popularity day by day. viii) Gyan- Vani (Educational FM Radio Channel of India) This project is recently launched (in year 2001) and again the target group is students of Open / Conventional Universities. Gyan Vani (Gyan = Knowledge, Vani = aerial broadcasting) is Educational FM Radio Channel of India, a unique decent ralised concept of extending mass media for education and empowerment, suited to the educational needs of the local community (Sharma, 2002b). It is operating presently through Allahabad, Banglore and, Coimbatore FM stations of India on test transmission mode. The network is slotted to expand to a total of 40 stations by June - 2002. Gyan Vani stations will operate as media cooperatives, with day - to- day programmes contributed by different Educational Institutions, NGO's and national level institutions like IGNOU, NCERT, UGC, IIT, DEC etc. Each stations will have range of about 60- KM radius, covering the entire city / town plus the surrounding environs with extensive access. It serves as ideal medium addressing the local educational developmental and socio cultural needs (IGNOU, 2001). Gyan Vani is not only for the conventional educational system but also a main tool in making available the dream of education for all come true. Gyan Vani‘s main intention is to take education to the door steps of the people. Gyan Vani, in addition to giving the hardcore education will also deal with awareness programmes including the ones for Panchayati Raj Functionaries, Women Empowerment, Consumer Rights, Human Rights, the Rights of the Child, Health Education, Science Education, Continuing Education, Extension Education, Vocational Education, Teacher Education, Non- formal Education, Adult Education, Education for the handicapped, Education for the down trodden, education for the tribal and soon. Gyan Vani is available through commercial FM radio set.

ix) Radio - Text Radio has been used a long with textual data transfer via computer networks simultaneously to create a ‗radio- text‘ environment. The teaching end is normally a FM radio station having data broadcast facility through a computer network. The main points of the radio broadcast are sent through textual mode to the receiving end via a computer network. The learning end has radio listening facility as well as a computer screen to receive the textual data. Since both audio and text are broadcast simultaneously, the learner at the receiving end gets high quality and low cost teaching. An experiment on the use of radio- text at Yashwant Rao Chavan Maharashtra Open University, Nasik, India resulted in the satisfaction of more than 80 percent the learners. It also used for peer group discussion at the receiving end after the broadcast, which indicates radio- text could be used for varieties of objectives (Chaudhary, 1996). 7.12 Television (TV) Television, which has an important place in mass communication, has a significant role in distance education with its special position, the way of presentation and qualities peculiar to itself. Technological developments in the field of communication can be adapted in the field of education as it is adapted to many fields of life. Thanks to the new technologies available in this field and the advantages they provide, television can already be seen as an outdated tool. Yet as long as the opportunities it provides still keep its validity, television technology is not far from the new developments. Use of television as an instructional medium was first reported in 1932 by State University of IOWA in USA on an experimental basis in a world fair. Later on, due to the World War II the introduction of television was slowed down; and as a result by 1948 there were very few educational institutions involved in using television as an instructional medium in spite of great interest in television by the educationists. 7.12.1 The Indian Beginning of TV Television first came to India [named as ‗Doordarshan‘ (DD)] on Sept 15, 1959 as the National Television Network of India. The first telecast started on Sept 15, 1959 in New Delhi. After a gap of about 13 years, second television station was established in Bombay in 1972 and by 1975 there were five more television stations at Shrinagar (Kashmir), Amritsar (Punjab), Calcutta, Madras and Lucknow. For many years the transmission was mainly in black & white. Television industry got the necessary boost in the eighties when Doordarshan introduced colour TV during the 1982 Asian Games (http://www.indiantelevision.com/indianbrodcast/ history/historyoftele.htm). The second phase of growth was witnessed in the early nineties and during the Gulf War, that foreign channel like CNN, Star TV and domestic channels such as Zee TV and Sun TV started broadcast of satellite signal. This changed the scenario and the people got the opportunity to watch regional, national and international programmes. Starting with 41 sets in 1962 and one channel (Audience Research unit, 1991) at present TV in India covers more than 70 million homes giving a viewing population more than 400 million individuals through more than 100 channels (http://www.indiantelevision.com). Easy accessibility of relevant technology, variety of programmes and increased hour of transmission are main reasons for rapid expansion of TV system in India. 7.12.2 Major Educational Projects through TV in India In India, since the inception of TV network, television has been perceived as an efficient force of education and development. With its large audience it has attracted educators as being an efficient tool for imparting education to primary, secondary and university level students. Some of the major educational television projects are discussed as here under: i) Secondary School television project (1961) This project was designed for the secondary school students of Delhi. With an aim to improve the standard of teaching in view of shortage of laboratories, space, equipment and dearth of qualified teachers in Delhi this project started on experimental basis in October 1961 for teaching of Physics, Chemistry, English and Hindi for students of Class XI. The lectures were syllabus - based and were telecasted in school hours as a part and parcel of school activities. According to Paul (1968) ‗by and large, the television schools did somewhat better in the test then did the non- television schools‘.

ii) Delhi Agriculture Television (DATV) Project (Krishi Darshan) (1966) The project named Krishi Darshan was initiated on January 26, 1966 for communicating agricultural information to the farmers on experimental basis for the 80 selected villages of Union territory of Delhi through Community viewing of television and further discussions among themselves. Experiment was successful and that there was substantial gain in the information regarding agricultural practices. (IGNOU, 2000). iii) Satellite Instructional Television Experiment (SITE) (1975) This project, one of the largest techno- social experiments in human communication, was commissioned for the villagers and their Primary School going children of selected 2330 villages in six states of India. It started on August 1, 1975 for a period of one year in six states Rajasthan, Karnataka, Orissa, Bihar, Andhra Pradesh and Madhya Pradesh. The main objectives of this experiment were to study the process of existing rural communications, the role of television as new medium of education, and the process of change brought about by the community television in the rural structure with following two type of telecast: a. Developmental education programmes in the area of agriculture and allied subjects, health, family planning and social education, which were telecast in the evening for community viewing. b. The school programmes of 22 ½ minutes duration each in Hindi, Kannada, Oriya and Telugu were telecast on each school day for rural primary school children of 5- 12 years age group to make the children realize the importance of science in their day to day life. SITE experiment showed that the new technology made it possible to reach number of people in the remotest areas. The role of television was appreciated and it was accepted in rural primary schools as an educational force (IGNOU, 2000). iv) Post - SITE project (1977) The target group for this post SITE project was the villagers of Rajasthan. This was a SITE continuity project and was initiated in March 1977 when a terrestrial transmitter was commissioned at Jaipur. The main objectives of SITE continuity project were to: a. Familiarize the rural masses with the improved and scientific know how about farming, the use of fertilizers and the maintenance of healt hand hygiene; b. Bring about national and emotional integration; and c. Make rural children aware of the importance of education and healthy environment. This project was also successful. v) Indian National Satellite Project (INSAT) (1982) The prime objective of the INSAT project was aimed at making the rural masses aware of the latest developments in the areas of agricultural productivity, health and hygiene. It was initially targeted at villagers and their school going Children of selected villages in Orissa, Andhra Pradesh, Bihar, Gujarat, Maharashtra and Uttar Pradesh. As a part of INSAT of Education project, ETV broadcasts were inaugurated and continued through terrestrial transmission from 15 August 1982 in Orissa and Andhra Pradesh. Later, other states namely Bihar, Gujarat, Maharashtra and Uttar Pradesh were covered under INSAT service using INSAT- 1B in June 1983. In each state, a cluster of 3- 4 districts were selected on the basis of backwardness of the area, availability of suitable developmental infrastructure and utilization of existing production facilities. Besides developmental programmes for community viewing, educational programmes (ETV) for two different age groups of school children (5- 8 years and 9- 11 years) are telecast daily. A capsule of 45 minutes duration consisting of two separate programmes - one for the lower age group and the other for the upper age group - were telecast regularly. Each programme runs for a duration of 20 minutes with five minutes change over time from one age group to the other. As of today, these ETV programmes are offered in five languages Oriya, Telugu, Marathi, Gujarati and Hindi- for a large population of primary school children. Programmes telecast in Hindi are being received in all Hindi- speaking states in the northern belt (IGNOU, 2000). vi) UGC- Higher Education Television Project (HETV) (1984) University students were the beneficiaries of this project. The University Grants Commission in collaboration with INSAT started educational television project, popularly known as ‗Country wide Classroom‘ on August 15, 1984 with the aim to update, upgrade and enrich the quality of education while extending their reach. Under this programme, a one - hour programme in English on a variety of subjects is presented with the objective of general enrichment for under graduates, educated public and the teachers as well. An inter - university Consortium for Education Communication (CEC) a long with a chain of about 20 audio- visual media Mass Communication Research Centres were set up by the UGC at different institutions in the country, to as certain high quality of programming for this project. Besides producing programmes at these centers, some programmes are imported from other countries, and are edited to suit the requirements of the Indian students. This project is very popular among students, teachers and other learners. vii) IGNOU- Doordarshan Telecast (1991) The IGNOU- Doordarshan telecast programmes, designed mainly for Distance learners started in May 1991. Initially they were telecast on Monday, Wednesday and Friday from 6.30 to 7.00 A.M through the national network of Doordarshan with an aim to provide telecounselling to students of open universities in remote areas. Owing to the encouraging response from viewers, the frequency of this project was increased to five days a week. This programme is very popular. viii) Gyan- Darshan Educational Channel (2000) Ministry of Human Resource Development, Information & Broadcasting, the Prasar Bharti and IGNOU launched Gyan Darshan (GD) jointly on 26th January 2000 as the exclusive Educational TV Channel of India. IGNOU was given the responsibility to be the nodal agency for up-linking/ transmission. It started out as a two- hour daily test transmission channel for students of open and conventional Universities. This duration was increased in February to nine hours a day. The times lot transmission was further increased due to good response up to 16- hours by 1st June and by 1st November it turned out to be 19- hours channel. Within one year of its launching, 26th January 2001, it became nonstop daily 24 hours transmission channel for educational programmes. ―The programming constitutes 23 hrs of indigenous programmes sourced from partner institutions and one hour of foreign programmes. Transmission of 12 hrs each for curriculum based and enrichment programmes is being made. The programmes of IGNOU CIET- NCERT including NOS are telecast for four hours each, IIT programmes for three hours, CEC- UGC programmes for two and a half hours and one hour each for TTTI and Adult Education.‖ (IGNOU Profile – 2002) The signal for Gyan Darshan transmission are uplinked from the Earth Station (augmented as one plus one system for redundancy) set up at IGNOUHQs New Delhi, and down linked all over the country through INSAT 3C on C Band Transponder. Although Gyan Darshan has made its presence felt in all Open Universities and most of the prominent conventional Universities / schools, it still has the potential to reach to the door steps of learners through cable TV network. At present Gyan Darshan through the cable transmission covers about 90% in Kerala, most parts of Tamil Nadu, a few pockets in the North East, Nashik, Ahmedabad and Pune. Asia Net has been providing it free of cost in Kerala. Efforts are being made to make Gyan Darshan available through terrestrial transmission. ix) Teletext in India The teletext service in India, popularly known as ‗INTEXT‘ (Indian teletext) was started by the Doordarshan Delhi on November 14, 1985. Similar to other teletext system, in INTEXT also the data are organized into pages in the form of text and graphic symbols. The information is pooled and transmitted on a few predetermined lines in vertical‘ blanking‘ interval of television signals. The information is in the form of magazines, each of which contains about 100 pages with details of contents of the magazine appearing on the first page, like news items, sport events, financial trends, timings of arrival and departure of important trains, weather fore cast, city engagements, AIR and TV programmes to be telecast, etc. Though, teletext has the potential for delivering educational instructions, no such experiments have been reported in India. 7.13 Video Cassette Recording (VCR) The history of the video cassette recorder follows the history of video tape recording in general. Ampex introduced the Ampex VRX- 1000, the first commercially successful video tape recorder, in 1956. Ampex VRX- 1000 could be afforded only by the television networks and the largest individual stations. In 1963, Philips introduced their EL3400 1″ helical scan recorder (aimed at the business and domestic user) and Sony marketed the 2″ PV- 100, their first reel- to - reel VTR intended for business, medical, airline, and educational use. In 1970 Philips developed a home video cassette format. Confusingly, Philips named this format ―VCR‖ (although it is also referred to as ―N1500‖, after the first recorder‘s model number). The format was also supported by Grundig and Loewe. It used square cassettes and half- inch (1.3 cm) tape, mounted on co - axial reels, giving a recording time of one hour. The first model, available in the United Kingdom in 1972, was equipped with a crude timer that used rotary dials. At nearly £600 ($2087), it was expensive and the format was relatively unsuccessful in the home market. This was followed by digital timer version in 1975 — the N1502. In 1977 a new (and incompatible) long- play version (―VCRLP‖) or N1700, which could use the same tapes, sold quite well to schools and colleges. A third format, Video 2000, or V2000 (also marketed as ―Video Compact Cassette‖) was developed and introduced by Philips in 1978, and was sold only in Europe. Grundig developed and marketed their own models based on the V2000 format. Most V2000 models featured piezoelectric head positioning to dynamically adjust the tape tracking. V2000 cassettes had two sides, and like the audio cassette had to be flipped over halfway through their recording time. Users switchable records protect levers were used instead of the break able lugs found on VHS/BetaMax cassettes. These early VCR‘s were expensive and generally owned by the wealthy. By the 1980‘s prices on VCR‘s were reasonable and began to be used in classrooms. Teachers used VCRs to show their students educational and entertainment videos. 7.14 Computer Computers have changed the way we work, be it any profession. Therefore, it is only natural that the role of computers in education has been given a lot of importance in recent years. Computers play a vital role in every field. They aid industrial processes, they find application in medicine; they are the reason why software industries developed and flourished and they play an important role in education. This is also why the education system has made computer education a part of school curriculum. Considering the use of computer technology is almost every sphere of life, it is important for everyone to have at least the basic knowledge of using computers. Let‘s look at what role computer technology plays in the education sector. Computer technology has had a deep impact on the education sector. Thanks to computers, imparting education has become easier and much more interesting than before. Owing to memory capacities of computers, large chunks of data can be stored in them. They enable quick processing of data with very less or no chances of errors in processing. Networked computers aid quick communication and enable web access. Storing documents on computers in the form of soft copies instead of hard ones helps save paper. 7.14.1 The advantages of Computers in Education Primarily Include Computer teaching plays a key role in the modern education system. Students find it easier to refer to the Internet than searching for information in fat books. The process of learning has gone beyond learning from prescribed textbooks. Internet is a much larger and easier-to -access storehouse of information. When it comes to storing retrieved information, it is easier done on computers than maintaining hand -written notes. i) Computers are a brilliant aid in teaching Search Online education has revolutionized the education industry. Computer technology has made the dream of distance learning, a reality. Education is no longer limited to classrooms. It has reached far and wide, thanks to computers. Physically distant locations have come closer due to Internet accessibility. So, even if students and teachers are not in the same premises, they can very well communicate with one another. There are many online educational courses, where by students are not required to attend classes or be physically present for lectures. They can learn from the comfort of their homes and adjust timings as per their convenience. ii) Computers have given impetus to distance education Computers facilitate effective presentation of information. Presentation software like PowerPoint and animation software like Flash among others can be of great help to teachers while delivering lectures. Computers facilitate audio -visual representation of information, thus making the process of learning interactive and interesting. Computer-aided teaching adds a fun element to education. Teachers hardly use chalk and board today. They bring presentations on a flash drive, plug it in to a computer in the classroom, and the teaching begins. There‘s color, there‘s sound, there‘s movement - the same old information comes forth in a different way and learning becomes fun. The otherwise not -so - interesting lessons become interesting due to audio -visual effects. Due to the visual aid, difficult subjects can be explained in better ways. Things become easier to follow, thanks to the use of computers in education. iii) Computer software helps better presentation of information Internet can play an important role in education. As it is an enormous information base, it ca n be harnessed for retrieval of information on a variety of subjects. The Internet can be used to refer to information on different subjects. Both teachers and students benefit from the Internet. Teachers can refer to it for additional information and references on the topics to be taught. Students can refer to web sources for additional information on subjects of their interest. The Internet helps teachers set test papers, frame questions for home assignments and decide project topics and not just academics; teachers can use web sources for ideas on sports competitions, extracurricular activities, picnics, parties and more. iv) Computers enable access to the Internet which has information on literally everything Computers enable storage of data in the electronic format, thereby saving paper. Memory capacities of computers to rage devices are in giga bytes. This enables them to store huge chunks of data. Moreover, these devices are compact. They occupy very less space, yet store large amounts of data. Both teachers and students benefit from the use of computer technology. Presentations, notes and test papers can be stored and transferred easily over computer storage devices. Similarly, students can submit homework and assignments as soft copies. The process becomes paperless, thus saving paper. Plus, the electronic format makes data storage more durable. Electronically eras able memory devices can be used repeatedly. They offer robust storage of data and reliable data retrieval. v) Computer hard drives and storage devices are an excellent way to store data This was about the role of computers in education. But we know, it‘s not just the education sector which computers have impacted. They are of great use in every field. Today, a life without computers is unimaginable. This underlines the importance of computer education. Knowledge of computers can propel one‘s career in the right direction. Computers are a part of almost every industry today. They are no longer limited any specific field. They are used in networking, for information access and data storage and also in the processing and presentation of information. Computers should be introduced early in education. 7.15 Let Us Sum Up This unit provides the teacher and opportunity to introduce and use of text books, work books, blackboard, the classroom conditions, mathematics club and library that are used to develop the mathematical knowledge and interest in pupils. Every school can have a mathematics club. A well organized group of student‘s society for mathematics who are interested in mathematics is called as mathematics club. The mathematics club functions under the guidance of the mathematics teacher. Mathematics club provides a beneficial social outlet for the needed students. This unit describes the different types of classroom resources and equipment which would enhance the knowledge of mathematics pupils. It is well known fact, where there are some educational resources like Radio, TV, VCR and Computer which would be useful for learners to achieve the specific objective of mathematics teaching. 7.16 Answers to Check Your Progress i) What do you mean by the equipments and resources for mathematics teaching? ii) What are the advantages and disadvantages of text books? iii) What are the various teaching instruments and apparatus? iv) Describe the reference books and hand books. v) What are the uses of library and work books? vi) What are the educational utility of radio and TV? vii) What are the uses the computer and VCR? 7.17 References IGNOU Profile (2002), Indira Gandhi National Open University, New –Delhi, India pp 51. Morris, 2004; Thomas, M. J. & Perritt, P. H. (2003, December 1). A Higher standard: Many states have recently revised their certification requirements for school librarians. School Library Journal. Available online at http://www.schoollibraryjournal.com/article/CA339562.html?industryid= 47056 Board man, Edna (September – October 1994). ―The Knapp School Libraries Project: The Best $ 1,130,000 Ever Spent on School Libraries.‖. Book Report 13 (2) :17 – 19. ISSN 0731 - 4388 (//www.worldcat.org/issn/0731- 4388). ERI C # EJ489785. The goals of the school library program should support the mission and continuous improvement plan of the school district. Standards for the 21st Century Learner (http://www.ala.org/ala/mgrps/divs/aasl/aaslproftools/learningstandards/ standards.cfm) Morris, B. (2004). Administering the School Library Media Center. West port, CT: Libraries Unlimited. (p .32).

UNIT VIII AUDIO – VISUAL AIDS IN TEACHING MATHEMATICS Structure 8.1 Introduction 8.2 Objective 8.3 Audio-Visual Aids 8.4 Types of Audio-visual Aids 8.5 Non-Projected Aids 8.5.1 Charts 8.5.2 Graphs 8.5.3 Models 8.5.4 Radio 8.6 Projected aids 8.6.1 Episcope 8.6.2 Epidiascope 8.6.3 Slides 8.6.4 Filmstrips 8.6.5 OHP 8.6.6 Transparencies 8.7 Software for Computer and Video Programme – 8.8 Need for Improvised aids 8.9 Let Us Sum Up 8.10 Answers to Check Your Progress 8.11 References 8.1 Introduction Effectiveness of teaching – learning process does not depend only on teacher but also upon the different types of equipments available in the classroom. The different equipments generally called audio- visual aids makes teaching – learning process more interesting, more stimulating, more reinforcing and more effective. According to Indian Education Commission (1964 – 66), ―the supply of teaching aids to every school is essential for the improvement of the quality of teaching. It should indeed bring about an educational revolution in the country. ―These are those instructional devices which are used in the classroom to en courage learning and thereby make it easier and interesting. Albert Duret rightly said, ―It is easier to believe what you see than what you hear, but if you both see and hear, then you can understand more readily and retain more lastingly.‖ They are called so because they call upon both the auditory and visual senses of the learners. Briefly discussion on instructional methods and strategies are given in this unit. 8.2 Objectives At the end of this unit, you should be able to: i) define audio-visual aids ii) list the various types of audio-visual aids iii) understand various software for computer and video programme iv) explain how audio visual can be used effectively v) aware about the need for improvised aids. 8.3 Audio – Visual Aids Audio-Visual aids or instructional materials are different forms of information carriers which are used to record, store, preserve, transmit or retrieve information for the purpose of teaching and learning. They also transmit information in such a fashion that will modify the attitude, habits and practices of students. In a general way audio-visual aids facilitate learning. We will examine how some of these aids are used to enhance learning in the classroom. In so doing, it must be emphasized that audio- visual aids are supplements and should not replace teaching by the subject teacher. Definitions Some important definitions are given below: i) According to Carter V. Good:- ―Audio – visual aids are those aids which help in completing the triangular process of learning i.e; motivation, classification and stimulation.‖ ii) According to Edger Dale:- ― Audio – visual aids are those devices by the use of which communication of ideas between persons and groups in various teaching and training situations is helped. These are also termed as multisensory materials.‖ iii) According to Burton:- ―Audio – visual aids are those sensor y objects or images which initiate or stimulate and rein force learning.‖ iv) According to Mcknown and Roberts: - ― Audio – visual aids are supplementary devices by which the teacher, through the utilization of more than one sensory channels keeps to clarify, establish and correlate concepts, interpretations and appreciations.‖ v) According to S.P. Ahluwalia:- ―Audio – visual materials reinforce the spoken or the written words with concrete images and provide rich perceptual experiences which are basis of learning. These materials make learning less non -verbalistic and reduce the boredom of mere verbal ism.‖ Thus audio – visual aids are those instructional devices which make teaching – learning process more interesting and effective. They use multi – sensory organs like hearing, seeing in order to make the process more vivid and impressionable. It reduces the rate of verbalism by providing content material in the form of concrete forms. 8.4 Types of Audio – Visual Aids The audio – visual aids have been classified in a number of ways according to different approaches, some areas: 1) Technical Approach:- They have been classified in to two types viz , audio aids and visual aids. a) Audio – aids:- The aids involving the sense of hearing are called audio – aids e.g.; radio, tape- recorder, records player etc. b) Visual aids:- Those aids which use sense of vision are called as visual aids, e.g.; models, pictures, maps, bulletin board, slides, epidiascope, over head projector etc. 2) According to 2nd approach, the audio – visual aids have been classified into two types viz ; projected and non - projected teaching aids. a) Non –Projected aids:- Teaching aids which do not help in their projection on the screen are called non-projected teaching aids. For example, chalk board, charts, actual objects, models, taps – recorder, radio etc. b) Projected aids:- Teaching aids which help in their projection on the screen are called as projected aids. For example, film strips, slides, film projector, over head projector, epidiascope etc. 8.5 Non – Projected Aids Teaching aids which do not help in their projection on the screen are called as non -projected teaching aids. It includes the following:- 8.5.1 Charts A chart is an information on a large sheet of paper or hard cardboard. This is usually mounted on the chalkboard or similar surface in front of the class. A necessary tool to use with the chart is the pointer. A chart is a combination of pictorial, graphic, numerical or vertical materials which presents a clear visual summary. The most commonly used types of charts include outline charts, tabular chart, and organization charts. Charts represent desirable permanent equipment for teaching purposes. To achieve the best results, they should be in a natural color, large enough and sufficiently clear to be seen easily from all parts of the classroom. Charts are in common use almost everywhere. A chart is a diagram which shows relationships. The organizational chart is one of the most widely used. This chart shows the various branches of a particular organization. Air and sea maps that are used for navigation purposes are also charts. Charts usually refer to displays on large sheets of paper or cloth that are designed to be shown to a class or group in the course of lesson. Good charts may be constructed rapidly by using either of the following methods:- i) Tracking a magnified image ii) Using a photograph to magnify a diagram map or sketch, filling filmstrips and slides. Films, filmstrips and slides should be closely integrated with standard lesson procedure and not used merely as embroidery or entertainment. For a successful use the teacher should be thoroughly familiar with the filmstrips or slides and should indicate to the class the specific areas to be observed. The relevance of these features to the lesson or lessons should be pointed out. It is customary to show films without interruption and the showing with discussion. With the answering of questions asked prior to the showing or with the types or recapitulation since filmstrips or slides represent forms of ―still pictures frequently, they requires preference by teachers for their flexibility and adaptability. 8.5.2 Graphs A graph is a diagrammatic treatment or representation of numeric or quantitative data. They are considered as pictures which are self – explanatory and tell their story at a glance. They are used for analysis, interpretation and for comparison. The different types of graphs include line graph, bar graph, circle or pie graph, pictorial graph and flannel graph. 8.5.3 Models Model is a recognizable three – dimensional representation of the real thing. These are concrete objects, some considerably larger than the real objects. They are mostly 3 – dimensional and sometimes sectional to explain clearly the cross-section or functions of the original. Working scale models – specific action of the original is duplicated and could be explained easily. Models are objects that duplicate as accurately as possible the real objects. Sometimes they are smaller versions of the real objects. Some models can be commercially bought for teaching purposes and some can teacher-made. Some useful models that you might consider having in the language classroom: model telephone, globe, clock Uses of Models in Lessons i) Ensure that the model will work ii) Make sure that it is big enough for your students to see iii) If you pass the model around in the class, make sure that your students are given enough time to examine the model. Purpose of Models in Instruction i) Models simplify reality. As they are three – dimensional they evoke curiosity and interest. ii) Models concretize abstract concepts. They simplify complex objects and accentuate important features with color and texture. iii) Models are of compact dimension. They enable us to reduce or enlarge objects to an observable size. iv) Models give the correct concept. It may not be possible or practicable to make students see the whole of a large industrial unit or even a large machine unit. v) A large process could be easily demonstrated by a model as they provide interior views of objects and machines vi) Preparation of models in itself could form a topic for project work. Simple paper or clay or plasticize models can be used for younger children. A working model will secure immediate attention and will serve as a motivation. The interest stimulated could be utilized to get the real thing or when the real thing will not be helpful to give a better explanation. Materials for Preparation of Models i) Cardboard ii) Plastic materials (clay, bee wax and plasticene) iii) Sand/glass paper iv) Plaster of Paris (Duplicate Copies of Objects) v) Wood vi) Metal (sheets, roads, tubes, angles, wire etc.) vii) Acrylic (Perspex) and PVC Some Specific Applications of Models i) They can be used to reduce very large objects and enlarge very small objects to a size that can be conveniently observed and handled. ii) They can be used to demonstrate the interior structures of objects or systems with a clarity that is often not possible with two - dimensional representations and at a cost that is not yet matched by virtual- reality products. iii) They can be used to demonstrate movement - another feature that it is often difficult to show adequately using two - dimensional display systems and that is more expensive in virtual- reality experiences. iv) They can be used to represent a highly complex situation or process in a simplified way that can easily be understood by learners; this can be done by concentrating only o n essential features, eliminating all the complex and often confusing details that are so often present in real- life systems. Making Your Own Models i) The range of methods available for making models for instructional purposes is enormous, but readers may find some of the following standard techniques useful. ii) Use of commercially – available kits of parts, such as the ball – and – spring systems that are used to make models of molecules and the various types of tube – and – spigot systems that can be used to make models of crystals. iii) Use of construction systems such as ‗Meccano‘ and ‗Fischer – Price‘ to make working models. iv) Use of inexpensive materials such as card board, hard board, wood and wire to make up static models of all types (models of buildings, geometrical bodies, three- dimensional shapes, and so on). v) Use of materials like modeling clay and plasticine to produce realistic models of animals, anatomical demonstrations, and so on. vi) Use of materials like Plaster of Paris and papier mache to produce model landscapes. Advantages of Models i) They draw students‘ attention to the subject or topic ii) They stimulate or motivate students‘ interest in a lesson iii) They provide opportunities for class participation in-groups or individual. iv) They make clear and understanding process and constrictions, which are not possible with two- dimensional aids. v) The models appeal to senses more than sight and sound vi) The handling and construction of models in an essential part of learning process which help in registration and retention of learning task. vii) They can be used to replace real objects or specimen, which are not readily available viii) it is not costly as compared to the real object Disadvantages of Models With models, learners are deprived of feeling the real texture, smell, sound etc. as compared to the real object. Models if not handled carefully cannot last. 8.5.4 Radio The aids which use sense of hearing are called audio aids. These include human voice, gramophone records, audio tapes/ discs, stereo records, radio broadcast and telephonic conversation. Radio is a very common type of hardware teaching aid. The use of radio for educational purpose was tested in England in 1924. School broadcasting was started in 1932 from Kolkata. Institutions such as Central Institute of Educational Technology, New Delhi, State Institute of Educational Technology, cells of SCERT produce need based auto – programmes for school children. Radio brings subject experts and other great men in the classroom. R. G. Reynolds writes, ―Radio is the most significant medium for education in its broadest sense that has been introduced since the turn of the Century. As a supplement to classroom teaching its possibilities are almost unlimited. Its teaching possibilities are not confined to the five or six hours of the school day. Radio brings subject experts programmes for school children. It is available from early morning till long after midnight. By utilizing the rich educational and cultural offerings of the radio, have access to the best of the world`s stores of knowledge and art. Some day it s use as an educational instrument will be as common place as textbooks and blackboards.‖ The radio broadcasts are generally used to introduce a new lesson, to present a complete lesson, to review the previous lesson and to solve major problems occurring in a lesson. The National policy on education 1986 and modified policy, 1992 has observed the media has profound influence on the minds of children. The mass media make the constraints of time and distance manageable. Modern educational technology must reach out to the most distant areas and the most deprived section of beneficiaries simultaneously with the areas of comparative affluence and ready availability. 8.6 Projected Aids Projected teaching aids are those aids help in their projection on the screen. When a projected aid is used, an enlarged room is either totally or partially darkened. It includes the following aids. 8.6.1 Episcope As stated by Longton Gould-Marks (1966), Episcope is a type of projector that can project any non- transparent or opaque object on to a screen. Such as pictures, book illustration text, flat Specimens like coins piece of textiles. Some modern episcopes can project three-dimensional objects such as tools and is often referred to as an opaque projector. The principle is simple. The object to be projected is placed on a tray usually at the bottom of the projector. A powerful light is reflected onto a mirror and the image is reflected through a lens onto the screen.

Advantages of Episcope i) It can project illustrations, photographs, maps, technical diagrams and almost every form of graphic material. ii) It can give enlarged copy of diagrams which can be simplified, coloured or modified to meet your need. iii) The projector has the ability to compel attention of the learners. iv) The preparation of the materials is simple and its preview for the purpose of lesson preparation presents no difficulty. v) It is good for large and small audience. vi) Its usage does not require any special training. Disadvantages of Episcope The Episcope has so many disadvantages and among those stated by Ralph Cable. (1979) i) The range at which materials can be projected is restricted. ii) Most episcopes are bulky and unwieldy and are thus not easily transportable. iii) The heat generated by the powerful lamp sometimes has a damage effect on flimsy material. iv) Episcope projection requires very good blackout. v) Notes taking is difficult since the room is darkened during viewing How to use the Episcope for Effective Teaching i) Arrange the seats and prefaces the object to be projected so that amendments, alterations or modifications can be made before the learners are seated. ii) The projected time should be well scheduled so that materials will not be kept on the platform of the projector for a long time since this might have effect on the material due to excessive heat generated by the lamp. iii) The room must be totally dark since the light output of an Episcope projector is relatively low, so that objects will be seen clearly by the learners. iv) The order of materials you intend to project must be pre-arranged.

8.6.2 Epidiascope The epidiascope is a type of opaque project or developed in the early years of the 20th century. Unlike the episcope or epidiascopes, which have the ability to project opaque images only, epidiascope can project images of both transparent and opaque images. This quality made the device especially useful in educational circles for most of the century. Definitions i) An optical device for projecting on a screen a magnified image of an opaque or transparent object. ii) A projector by which images are reflected by a mirror through a lens, or lenses, onto a screen, using reflected light for opaque objects and transmitted light for translucent or transparent ones. iii) A projector for images for both transparent and opaque objects. The basic functionality of the epidiascope involved harnessing the power of light to create the images. In the earliest models of the epidiascope and other similar projectors, limelight was used as the medium. The light would be directed downward onto an object, creating the image. To focus the light and create a viable image, a series of lenses or mirrors would be used to direct the image onto a screen. While somewhat costly to produce at first, the epidiascope became more affordable as the device was refined. Along with commercial models, low powered versions were produced and marketed as toys for school age children. By the middle of the 20th century, the typical epidiascope was produced using incandescent light as the source for creating the image. Desktop models of the device were in common use in schools and colleges across the globe. Within a few years, halogen lamps began to replace the incandescent bulbs, providing an even sharper projected image. An epidiascope is a project or for showing both transparent slides and opaque objects. This combination of functions made the epidiascope the ideal projector for schools. The epidiascope is a larger and more complex version of the opaque projector. Epidiascopes are capable of projecting the image of a page from a book, a document, picture or a three-dimensional object. A circular from The Scientific Shop, circa 1906, describes an early epidiascope. The device was large and had to be pushed about on wheels like a cart. The device uses a powerful electric lamp, a series of focusing mirrors and lenses and generates quite a bit of heat. Vents and a cooling tower are included in the construction to eliminate excess heat. The device cost more than $400 and was available in several models. As the era of the personal computer dawned in the 1980‘s, new technology began to replace the epidiascope. Utilizing projectors that would attach easily to desktop and laptop computers, it became possible to create images using software and project the results onto an overhead screen. The combination of a laptop and a project or made it possible for sales persons to take along presentations and other documents to meetings with new clients or show presentations at trade shows with much greater ease. While not used as extensively as in times past, the epidiascope is still sometimes used in schools and other learning settings. Because the transparencies used with an epidiascope can be created using computer programs or by hand, the device has remained in service and is still offered for sale by a number of manufacturers. Applicability of Epidiascope The epidiascope is an instrument which can project images or printed matter or small opaque objects on a screen or it can project images of a 4‖ x 4‖ slide. With the help of any epidiascope, any chart, diagram, map, photograph and picture can be projected on the screen without tearing it off from the book. No slide is needed for this purpose. An epidiascope serves two purposes. It works as epidiascope when it is used to project opaque objects. It works as epidiascope when it is used to project slides (by operating a lever). It works on the principle of horizontal straight line projection with a lamp, plane mirror and projection lens. A strong light from the lamp falls on the opaque object. A plane mirror placed at an angle of 45‖ over the project reflects the light so that it passes through the projection lens forming a magnified image on the screen. 8.6.3 Slides A slide is a piece of film in a frame for passing strong light through or to show a picture on a surface. It is a piece of transparent surface like cellulose acetate film, translucent paper, glass etc. of a specific dimension with drawings or pictures which can be mounted individually for use in a projector or for viewing by transmitted light. Fred A Teague (1989) defines slides as small positive films mounted in a study frame. The slide projector is the device used to transfer the information on the slide to the screen. Picture Mounting frames are typically white cardboard that comes in several sizes. The 50mm*50mm slide is the most common type, for instructional purpose. Types of Slide Projection Systems The three types of slide projector are i) slide projector ii) sound slide programmes iii) multi image presentation i) Slide Projector A set of slides is arranged in the carousel tray of the slide projector to suit the lecture, so that the lecturer starts lecturing whilst projector transfers the information in line with how he talks. ii) Sound Slides Programmes This type of slides is operated with an audio recorder. A cable is connected from the special sync (synchronizer) tape recorder to the slide projector. This helps in listening and viewing programs at the same time. The teacher is virtually free here until the slide show is over. iii) Multi Image Presentation This also uses an audio recorder, but the difference is that this can project two or more images onto a screen at the same time. Advantages of using the Slide Projector i) The slide projector brings out a vivid image on a screen, which portrays realism. ii) This is a good media for individual learning since slides can be synchronized with sound. iii) In using slides, it can be arranged according to how one wants it. Arrangement can equally be changed at any time to suit the lectures. iv) Slides are easy to operate and can be stored conveniently. v) Slides are inexpensive; teachers and students can provide their own slides for programs. Educational Value According to Hass and Parker (1954), the following are the advantages of slides: i) Attract attention ii) Arouse interest iii) Assist lesson development iv) Test student‘s understanding v) Review instruction vi) Present next lesson or subject vii) Facilitate student teacher participation. 8.6.4 Film Strips A film – strip is 35mm wide and has a series of 12 to 48 picture frames arranged in a sequence so that they develop a theme. A film – strip can be prepared by taking a series of photographs using a 35mm camera and then by taking a positive print of the negative film on another 35mm film. They are then projected on a slide projector or a film strip projector. According to Good‘s dictionary of education, a film – strip is, ―a short length of film containing a number of positives, each different but usually having some continuity, in tended to be projected as series of still pictures by mean s of film strip projector. These are available in market and in libraries. A wonderfully inspiring and in formative series of twenty film strips, entitled ‗Bring India to your class – room‘ has been released by the Al Mervyn studio, Mumbai. The material of this series ranges from India‘s land and people to her agriculture, industries, hill stations and cultural activities etc. Educational Importance i) They are helpful for composition lessons. ii) It allows maximum participation of the students. iii) All subjects‘ content material can be made interesting and effective with the help of film – strips. iv) It is economical in terms of time and energy. It saves the time and energy of both teacher and taught. v) It develops the habit of discussion, explanation, argumentation among learners by projected different content materials in a logical sequence on a screen. vi) They are good substitutes for direct experience of the learners including different topics like atomic energy, sulphur and its compounds, citizenship etc. 8.6.5 Over Head Projector (OHP) The overhead projector is a device, which projects large transparencies from a horizontal table through a prism and a lens to form a brilliant image on to a screen behind teacher. It is taught of as the modern equivalent of a chalkboard. Seymour (1937) and foster (1968) conclude that dark lettering displayed on a light-colored board is more easily seen by students than the traditional white on a dark board. Over – head projector is a device that can project a chart, a diagram, anything written on transparent sheet etc upon a screen on the white wall in front of students in a class. The name ‗over -head projector‘ comes from the fact that the projected image is behind and over the head of the speaker / teacher. In it, a transparent visual is placed on a horizontal stage on top of light source. The light passes through this transparency and then is reflected at 90° angle on the screen at the back of the speaker. Educational Significance i) It helps us to make teaching more illuminative, illustrative and impressive. ii) It is economical in terms of saving teacher‘s time used in drawing or writing. iii) Transparent sheets once prepared can be used for future displays while taking up the same topic. iv) The teacher can maintain complete class control and interest in a lesson by turning a switch on or off. v) It can be used for large group of students in any type of classroom. vi) Problems like writing on blackboard, rub the written material occasionally etc. have been overcome by the use of over head projector. vii) Whether in a teaching or a presentation situation, the audience sees the visualization from the same point of view as the communicator. The feeling of oneness with the communicator is created. viii) The teacher can always face the class, maintaining eye contact with the pupils. 8.6.6 Transparencies Definition An overhead transparency is an image usually 8 ½‖ * 11‖ on clear acetate or plastic which has been prepared for use on an overhead projector. An overhead projector is a device which throws an image on a screen. It is placed In front of an audience and may be used in a completely lighted on semi – darkened room: it utilized 3¼‖*4‖, 7‖*7‖, 10*10‖, or most commonly, 8½‖ *11‖ transparencies. Characteristics of Transparencies The lens system of the overhead projector is designed so that the projector can be placed in the front of a room. The projection angle often causes the image to appear in a trapezoid form sometimes called ―key stoning‖ (see illustration). This is corrected by tilting the screen as shown in the accompanying figures. This distortion is not always bothersome and does not measurably change the amount of light on the screen. The projector should be placed as low as possible, as illustrated, so that the body of the equipment does not interfere with the line of vision between the students and the screen. Advantages of Transparencies The overhead transparency and projector have become extremely popular since the early 1960‘s. The cost of the projector has decreased and the potential for local production of transparencies has increased. In some instructional settings the overhead projector makes an excellent replacement for the chalk board. There are many advantages to this simple projector and its associated materials. i) The equipment is used in the front of the room. The teacher can maintain eye contact with the class while using the equipment. ii) A bright image can be projected in a fully room. This permits the teacher and learns to see each other. iii) Materials to be used on the overhead projector are easily produced by the teacher in the local school. The teacher in the local school. The teacher may write on a piece of acetate during the class as he would use the chalk board. He may use transparencies which have been produced through heat or chemical processes. In either case the transparency is a highly accessible and easily used medium of instruction. iv) The equipment is simple to operate and has no maintenance problems save the occasional replacement of the lamp. Limitations of Transparencies In their enthusiasm for use of the overhead projector, teacher often overlook some of the limitations. i) Unless the equipment is properly positioned the base and head of the projector may obstruct the student‘s line of Sight. The projector should never be placed on the teacher‘s desk in the front of the room. A low projector stand or a table with a recessed out -out is essential. ii) Since the overhead projector becomes such a vital part of the teaching learning process, it cannot easily be scheduled for use in one classroom one hour and in another the next. Therefore it may be necessary to assign an overhead projector to each room if teachers have extensive needs for this piece of equipment. This may create an economic burden. 8.7 Software for Computer and Video Programme To be able to use the computer in your teaching, you need to have sufficient knowledge about how it looks like and how it works. Then you can use it to your best advantage. The computer is an electronic device capable of receiving data, processing or manipulating the data, storing and retrieving the data at will. This device makes it possible to reach out for information that may not be available to you in your immediate environment either by buying software that has such information or connecting the device to a global network in the same way a telephone is connected word wide. Software for Computer Software is the programme that runs the PC system. The software distributes information to all areas of the system and directs each area to complete various functions. It is available in various programmes from simple text programmes to detailed graphics and page layout programmes. The computer code or language that the software uses varies depending on the type of software. When choosing software the type of hardware that will support the software and whether the software is compatible with the output devices must be known. For example, if you intend to use a laser printer, you will have to install the software so the computer can understand its operation and take commands from it. Some types of computer software programmes are the word processing software, the graphics software and the page layout software. The computer has such basic components to make it function as a central processing unit, a monitor, a typewriter keyboard, a mouse and a printer. The scanner, digital camera, microphone, output speakers, liquid crystal display unit, disk writer and other accessories can be attached for better performance or special functions. All these physical components (machines) mentioned here are referred to as the hardware of the computer system. The computer works with programmes that are installed in them. Without these programmes, the computer is useless and cannot function. These programmes give instructions that tell the computer how to perform tasks. Software products are written in special codes by programmemers. They are either copied onto a storage medium that can be installed by the user or are stored on the computer before they are sold out. With both hardware and software, computers are used to speed up problem solving issues, improve quality in administration of information and increase productivity. There are two types of software and these are systems software and applications software. System Software helps the computer to manage its own internal functions that control other hardware devices and to manage the applications software that we use in our daily lives. Applications Software are those programmes that users use to perform a specific task, for example, Microsoft Word is a word processing software used to prepare and process documents and Power point which is used to project information unto a screen may be for lecture purposes or report presentations. 8.8 Need for Improvised Aids The following are some needs for improvised aids. i) They should be relevant to the curriculum. ii) They should be previewed or tried out in advance before use in the class. iii) They should be taught, not merely shown. They should be useful not as mere decorations. iv) Provision should be made for definite follow-ups. v) Records should be kept of the results obtained, evaluation should be made. vi) Too many teaching aids should not be used at a time. vii) The types of materials used should be within the knowledge and experience of children. viii) They should be used in the classrooms or laboratory. Some schools have geography room, history room etc. ix) They should be available when and where needed. x) All teaching aids should be tactically and technically correct. xi) No one type or materials is best for all living situations. Each has a specific role in order to provide maximum effects. 8.9 Let Us Sum Up Audio visual aids transmit ideas and concepts of mathematics through auditory and visual senses. Learning experiences occur in three levels. In direct experiencing, first hand experiences like seeing, teaching, tasting, feeling, smelling etc. provide experience is given by using charts, pictures, diagrams etc. thirdly symbols are used to give experience. According to Cominius ―the foundation of all learning consists in representing clearly the senses, sensible objects so that they can be appreciated easily‖. Audio visual aids are visual aids, auditory aids and audio visual aids. Instructional aides like chalkboard, graph, diagram, pictures, posters, models, films, slides are visual aids. Radio, tape recorder are some of the auditory aids. Sound motion pictures, T.V. dramatization are some of the chief audio visual aids. 8.10 Answers to Check Your Progress i) Define audio-visual aids. ii) What are the various types of audio-visual aids? iii) Describe the software for computer and video programme. iv) How audio visual can be used effectively? Explain it. v) What are the needs for improvised aids? 8.11 References Webster's New World College Dictionary Copyright © 2010 by Wiley Publishing, Inc., Cleveland, Ohio. Used by arrangement with John Wiley & Sons, Inc. http://www.webdictionary.co.uk/definition http://www.medilexicon.com/medicaldictio Kinney and Purdy, ―Teaching of Mathematics in the Secondary School‖, Rine Hart and Company. Dr. S. Packiam, Teaching of Modern Mathematics. B. N. Chandra, The Teaching of Mathematics.

UNIT IX CONTENT PRESCRIBED FOR MATHEMATICS IN STANDARDS VI TO XII OF TAMIL NADU TEXT COMMITTEE Structure 9.1 Introduction 9.2 Objective 9.3 Content Prescribed for Mathematics in Standard VIII of Tamil Nadu Text Committee 9.4 Content Prescribed for Mathematics in Standard IX of Tamil Nadu Text Committee 9.5 Content Prescribed for Mathematics in Standard X of Tamil Nadu Text Committee 9.6 Content Prescribed for Mathematics in Standard XI of Tamil Nadu Text Committee 9.7 Content Prescribed for Mathematics in Standard XII of Tamil Nadu Text Committee 9.8 Let Us Sum Up 9.9 Answers to Check Your Progress 9.10 References 9.1 Introduction Mathematics being one of the most important subjects which not only decides the career of many young students but also enhances their ability of analytical and rational thinking and forms a base for Science and Technology. So, it is very urgency to teach students the appropriate knowledge of mathematics according to the age and understanding level. In this unit there are mentioned the prescribed content by the Tamil Nadu Text Committee for the mathematics in standards VIII to XII. 9.2 Objectives On completion of this unit students will be able to 15. understand the level of content prescribed for mathematics in standards VIII to XII of Tamil Nadu Text Committee 16. understand the content prescribed for mathematics in standards VIII to XII of Tamil Nadu Text Committee 9.3 Content Prescribed for Mathematics in Standard VIII of Tamil Nadu Text Committee Mathematics plays a vital role in the modernization of this civilization. It is everywhere and affects the everyday lives of people. Although it is abstract and theoretical knowledge, it emerges from the real world. Mathematics is one of the essential and basic areas of the college curriculum which has a wide field of subject matter. In education, mathematics plays an important role. The content prescribed for mathematics in standard VIII is well designed within the frame work of NCF 2005 by Tamil Nadu Text Book Committee. The content prescribed for mathematics in standard VIII of Tamil Nadu Text Book Committee is given below. A. Real Number System i) Introduction ii) Revision: Representation of Rational Numbers on the Number Line iii) Four Properties of Rational Numbers (Closer Property, Commutative Property, Associative Property, Additive Property). iv) Simplification of Expressions Involving Three Brackets v) Powers: Expressing the Numbers in Exponential Form with Integers as Exponent vi) Laws of Exponents with Integral Powers vii) Squares, Square Roots, Cubes, Cube roots viii) Approximations of Numbers ix) Playing with Numbers B. Measurements i) Introduction ii) Semi-Circles and Quadrants (Perimeter of a Semicircle, Area of a Semicircle, Quadrant of a Circle, Perimeter of a Quadrant, Area of a Quadrant). iii) Combined Figures (Polygon, Regular Polygon, Irregular Polygon, Concave polygon, Convex Polygon).

C. Geometry i) Introduction ii) Properties of Triangle (Kinds of Triangles, Angle Sum Property of a Triangle). iii) Congruence of Triangles (Congruence among Line Segments, Congruence of Angles, Congruence of Squares, Congruence of Circles, Congruence of Triangles). iv) Conditions for Triangles to be Congruent (SSS Axiom (Side-Side-Side Axiom), SAS Axiom (Side-Angle-Side Axiom), ASA Axiom (Angle-Side-Angle Axiom), RHS Axiom (Right Angle-Hypotenuse-Side Axiom)). v) Conditions which are not sufficient for Congruence of Triangles (AAA (Angle-Angle- Angle), SSA (Side-Side-Angle)). D. Practical Geometry i) Introduction ii) Quadrilateral iii) Area of a Quadrilateral iv) Construction of a Quadrilateral (Construction of a Quadrilateral when Four Sides and one Diagonal are given, Construction of a Quadrilateral when Four Sides and one Angle are given, Construction of a Quadrilateral when Three Sides, one Diagonal and one Angle are given, Construction of a Quadrilateral when Three Sides and Two Angles are given, Construction of a Quadrilateral when Two Sides and Three Angles are given v) Trapezium vi) Area of a Trapezium vii) Construction of a Trapezium: Construction of a Trapezium when Three Sides and One Diagonal are given, Construction of a Trapezium when Three Sides and One Angle are given, Construction of a Trapezium when Two Sides and Two Angles are given, Construction of a Trapezium when Four Sides are given viii) Isosceles Trapezium ix) Area of a Isosceles Trapezium x) Construction of Isosceles Trapezium xi) Parallelogram xii) Area of a Parallelogram xiii) Construction of a Parallelogram (Construction of a Parallelogram when Two Adjacent Sides and One Angle are given, Construction of a Parallelogram when Two Adjacent Sides and One Diagonal are given, Construction of a Parallelogram when Two Diagonals and One Included Angle are given, Construction of a Parallelogram when One Side, One Diagonal and One Angle are given). 9.4 Content Prescribed for Mathematics in Standard IX of Tamil Nadu Text Committee The Government of Tamil Nadu has decided to evolve a uniform system of school education in the state to ensure social justice and provide quality education to all the schools of the state. With due consideration to this view and to prepare the students to face new challenges in the field of Mathematics, the content prescribed for mathematics in standard IX is well designed within the frame work of NCF2005 by Tamil Nadu Text Book Committee. The content prescribed for mathematics in standard IX of Tamil Nadu Text Book Committee is given below. A. Theory of Sets: Introduction, Description of Sets, Representation of Set, Different kinds of sets, Set Operations, Representation of Set Operations using Venn Diagram). B. Real Number System (Introduction, Decimal Representation of Rational Numbers, Irrational Numbers, Real Numbers, Surds, Four Basic Operations on Surds, Rationalization of Surds, Division Algorithm C. Scientific Notations of Real Numbers and Algorithms: Scientific Notation, Converting Scientific Notation to Decimal Form, Logarithms, Common Logarithms. D. Algebra: Introduction, Algebraic Expressions, Polynomials, Remainder Theorem, Factor Theorem, Algebraic Identities, Factorization of Polynomials, Linear Equations, Linear Inequalities in One Variable E. Coordinate Geometry: Introduction, Cartesian Coordinate System, Distance Between Any Two Points. F. Trigonometry: Introduction, Trigonometric Ratios, Trigonometric Ratios of Some Special Angles, Trigonometric Ratios for Complementary Angles, Method of Using Trigonometric Tables. G. Geometry: Introduction, Geometry Basics, Quadrilateral, Parallelograms, Circles. H. Mensuration: Introduction, Sectors, Cubes, Cuboids. I. Practical Geometry: Introduction, Special Line Segments within Triangles, The Points of Concurrency of a Triangle. J. Graphs: Introduction, Linear Graph, Application of Graphs. K. Statistics: Introduction, Graphical Representation of Frequency Distribution, Mean, Median, Mode. L. Probability: Introduction, Basic Concepts and Definitions, Classification of Probability, Probability – An Empirical Approach. 9.5 Content Prescribed for Mathematics in Standard X of Tamil Nadu Text Committee Mathematics, the queen of all sciences, remains and will remain as a subject with great charm having an intrinsic value and beauty of its own. It plays an indispensable role in sciences, engineering and other subjects as well. So, mathematical knowledge is essential for the growth of science and technology, and for any individual to shine well in the field of one‘s choice. In addition, a rigorous mathematical training gives one not only the knowledge of mathematics but also a disciplined thought process, an ability to analyze complicated problems. The summery of the content prescribed for mathematics in standard X of Tamil Nadu text committee is given below. A. Sets and Functions: Introduction, Sets, Operations on Sets, Properties of Set Operations, De Morgan‘s Laws, Cardinality of Sets, Relations, Functions. B. Sequences and Series of Real Numbers: Introduction, Sequences, Arithmetic Sequence, Geometric Sequence, Series. C. Algebra: Introduction, System of Linear Equations in Two Unknowns, Quadratic Polynomials, Synthetic Division, Greatest Common Divisor and Least Common Multiple, Rational Expressions, Square Root, Quadratic Equations. D. Matrices: Introduction, Formation of Matrices, Types of Matrices, Operation on Matrices, Properties of Matrix Addition, Multiplication of Matrices, Properties of Matrix Multiplication. E. Coordinate Geometry: Introduction: Section Formula, Area of a Triangle, Col-linearity of Three Points, Area of a Quadrilateral, Straight Lines, General form of Equation of a Straight Line. F. Geometry: Introduction, Basic Proportionality and Angle Bisector Theorems, Similar Triangles, Circles and Tangents. G. Trigonometry: Introduction, Trigonometric Identities, Heights and Distances. H. Mensuration: Introduction, Surface Area, Volume, Combination of Solids. I. Practical Geometry: Introduction, Construction of Tangents to a Circle, Construction of Triangles, Construction of Cyclic Quadrilaterals. J. Graphs: Introduction, Quadratic Graphs, Some special Graphs K. Statistics: Introduction, Measures of Dispersion L. Probability: Introduction, Classical Definition of Probability, Addition theorem on Probability. 9.6 Content Prescribed for Mathematics in Standard XI of Tamil Nadu Text Committee The content prescribed for mathematics in standard XI is designed in accordance with the new guidelines and syllabi – 2003 of the Higher Secondary Mathematics – First Year, Government of Tamil Nadu by the textbook committee of subject experts and practicing teachers in schools and colleges. In the era of knowledge explosion, prescribing content for Mathematics in standard XI is challenging and promising. Mathematics being one of the most important subjects which not only decides the career of many young students but also enhances their ability of analytical and rational thinking and forms a base for Science and Technology. The content prescribed for Mathematics in standard XI would be of considerable value to the students who would need some additional practice in the concepts taught in the class and the students who aspire for some extra challenge as well. A. Matrices and Determinants i) Matrix Algebra – Definitions, types, operations, algebraic properties ii) Determinants – Definitions, properties, evaluation, factor method, product of determinants, co-factor determinants B. Vector Algebra: Definitions, Types of Vectors, Operations on Vectors, Position Vector, Resolution of a vector in two and three dimensions, Direction Cosines and Direction Ratios. C. Algebra i) Partial Fractions – Definitions, linear factors, none of which is repeated, some of which are repeated, quadratic factors (none of which is repeated) ii) Permutations – Principles of counting, concept, permutation of objects not all distinct, permutation when objects can repeat, circular permutations iii) Combinations iv) Mathematical induction v) Binomial theorem for positive integral index – finding middle and particular terms D. Sequence and Series: Introduction, Sequences, Series, Some special types of sequence and series, Mean of Progression, Binomial theorem for rational number other than positive integer, Binomial Series, Summation of Binomial Series, Exponential series, Logarithmic series. E. Analytical Geometry: Locus, Straight Lines (Normal form, Parametric form, General form, Perpendicular Distance from a Point), Family of Straight Lines, Angle between Two Straight Lines, Pair of Straight Lines, Circle – General Equation, Parametric Form, Tangent (Tangent Equation, Length of the Tangent, Condition for Tangent, Equation of Chord of Contact of Tangents from a Point), Family of Circles (Concetric Circles, Orthogonal Circles). F. Trigonometry: Trigonometrical Ratios and Identities, Signs of T-ratios, Compound Angles A ± B, Multiple Angles 2A and 3A, Sub multiple (half) Angle A/2, Transformation of a Product into a Sum or Difference, Conditional Identities, Trigonometrical Equations, Properties of Triangles, Solution of Triangles (SSS, SAA and SAS types only), Inverse Trigonometrical Functions.

G. Functions and Graphs: Introduction i) Function: a) Graph of a Function – Vertical Line Test b) Types of functions – Onto, One-to-One, Identity, Inverse, Composition of Functions, Sum, Difference Product, Quotient of Two Functions, Constant Function, Linear Function, Polynomial Function, Rational Function, Exponential Function, Reciprocal Function, Absolute Value Function, Greatest Integer Function, Least Integer Function, Signum Function, Odd and Even Functions, Trigonometrical Functions, Quadratic Functions ii) Quadratic Inequation – Domain and Range H. Differential Calculus i) Limit of a Function – Concept, Fundamental Results, Important Limits ii) Continuity of a Function – At a Point, In an Interval, Discontinuous Function. iii) Concept of Differentiation – Derivatives, Slope, Relation between Continuity and Differentiation iv) Differentiation Techniques – First Principle, Standard Formulae, Product Rule, Quotient Rule, Chain Rule, Inverse Functions, Method of Substitution, Parametric Functions, Implicit Function, Third Order Derivatives I. Integral Calculus: Introduction, Integrals of Functions Containing linear functions, Methods of Integration (Decomposition Method, Substitution Method, Integration by Parts), Definite Integrals – Integration as Summation, Simple Problems. J. Probability: Introduction, Classical definition of Probability, Some Basic Theorems on Probability, Conditional Probability, Total Probability of an event, Baye‘s theorem (statement only)

9.7 Content Prescribed for Mathematics in Standard XII of Tamil Nadu Text Committee The 21st century is an era of Globalization, and technology occupies the prime position. In this context, the content prescribed for Mathematics assumes special significance because of its importance and relevance to Science and Technology. The content prescribed for mathematics in standard XII by the Tamil Nadu text committee is presented in an innovative way to facilitate the students for easy approach. The content consists of applications of matrices and determinants, vector algebra, complex numbers and analytical geometry which is dealt with a novel approach. Solving a system of linear equations and the concept of skew lines are new ventures. It also includes Differential Calculus – Applications, Integral Calculus and its Applications, Differential Equations, Discrete Mathematics (a new venture) and Probability Distributions. The summery of the content prescribed for mathematics in standard XII of Tamil Nadu text committee is given below. A. Applications of Matrices and Determinants: Introduction, Adjoint, Inverse, Rank of a Matrix, Consistency of a System of Linear Equations. B. Vector Algebra i) Introduction ii) Scalar Product – Angle between Two Vectors, Properties of Scalar Product, Applications of Dot Products iii) Vector Product – Right Handed and Left Handed Systems, Properties of Vector Product, Applications of Cross Product iv) Product of Three Vectors – Scalar Triple Product, Properties of Scalar Triple Product, Vector Triple Product, Vector Product of Four Vectors, Scalar Product of Four Vectors v) Lines – Equation of a Straight Line Passing through a given Point and Parallel to a given Vector, Passing through Two given Points (derivations are not required), Angle between Two Lines vi) Skew Lines – Shortest Distance between Two Lines, Condition for Two Lines to Intersect, Point of Intersection, Collinearity of Three Points vii) Planes – Equation of a Plane (derivations are not required), Passing through a given Point and Perpendicular to a Vector, Given the Distance from the Origin and Unit Normal, Passing through a given Point and Parallel to Two given Vectors, Passing through Two given Points and Parallel to a given Vector, Passing through Three given Non-collinear Points, Passing through the Line of Intersection of Two given Planes, The Distance between a Point and a Plane, the Plane which contains Two given Lines, Angle between Two given Planes, Angle between a Line and a Plane viii) Sphere - Equation of the Sphere (derivations are not required) Whose Centre and Radius are given, Equation of a Sphere when the Extremities of the Diameter are given. C. Complex Numbers: Introduction, Complex Number System, Conjugate of Complex Numbers, Ordered Pair Representation, Modulus of Complex Numbers, Geometrical representation, Solutions of polynomial equations, De Moivre‘s Theorem and Its Applications, Roots of a complex number. D. Analytical Geometry: Introduction, Definition of a Conic, Parabola, Ellipse, Hyperbola, Parametric Form of Conics, Chords, Tangents and Normals, Asymptotes, Rectangular Hyperbola. E. Differential Calculus (Applications I): Introduction, Derivative as a rate measure, Related rates, Tangents and Normal, Angle between Two Curves, Mean value theorem and Its Applications, Evaluating Indeterminate Forms, Monotonic Functions, Maximum and Minimum values and their Application, Concavity, convexity and points of inflexion. i) Differential Calculus (Applications II): Differential: Errors and Approximations, Curve Tracing, Partial Derivatives - Euler‘s Theorem. F. Integral Calculus and Its Applications: Introduction, Simple Definite Integrals, Properties of Definite Integrals, Reduction Formulae, Area and Volume, Length of Curve, Surface area of a Solid. G. Differential Equations: Introduction, Order and Degree of a Differential Equation, Formation of differential equations, Differential Equations of First Order and First Degree, Second order linear equations with constant coefficients, Applications. H. Discrete Mathematics i) Mathematical Logic: Logical statements, connectives, truth tables, Tautologies ii) Groups: Binary Operations, Semi groups, Monoids, Groups (Problems and Simple Properties only), Order of a Group, Order of an Element I. Probability Distributions: Introduction, Random Variable, Mathematical Expectation, Theoretical distribution. 9.8 Let Us Sum Up The development of curriculum in mathematics is a complex process. The scope of the subject is wide and almost all branches of study need the application of fundamental mathematics. The aim of mathematics in schools is to develop practical, disciplinary, cultural and social values. While making mathematics as a compulsory subject up to high school level the interests of students who are leaving the school and joining some work and those who continue their studies must also be safe guarded. The syllabus must contain useful material for both the group of students. In the higher secondary level it is made as an optional subject and only those students who are aiming for higher studies related with mathematics need to study it. 9.9 Answers to Check Your Progress i) Describe the content prescribed for mathematics in standards VIII of Tamil Nadu Text Committee ii) Describe the content prescribed for mathematics in standards IX of Tamil Nadu Text Committee iii) Describe the content prescribed for mathematics in standards X of Tamil Nadu Text Committee iv) Describe the content prescribed for mathematics in standards XI of Tamil Nadu Text Committee v) Describe the content prescribed for mathematics in standards XII of Tamil Nadu Text Committee 9.10 References Mathematics VIII Standard, Government of Tamil Nadu, Tamil Nadu Textbook Corporation, College Road, Chennai – 600006. Mathematics IX Standard, Government of Tamil Nadu, Tamil Nadu Textbook Corporation, College Road, Chennai – 600006. Mathematics X Standard, Government of Tamil Nadu, Tamil Nadu Textbook Corporation, College Road, Chennai – 600006. Mathematics XI Standard, Government of Tamil Nadu, Tamil Nadu Textbook Corporation, College Road, Chennai – 600006. Mathematics XII Standard, Government of Tamil Nadu, Tamil Nadu Textbook Corporation, College Road, Chennai – 600006.

UNIT X RECENT TRENDS IN MATHEMATICS EDUCATION Structure 10.1 Introduction 10.2 Objective 10.3 Integrated Treatment for Subject Matter 10.3.1 Defining Integrated Curriculum 10.3.2 Multidisciplinary Integration 10.3.3 Interdisciplinary Integration 10.3.4 Transdisciplinary Integration 10.3.5 Comparing and Contrasting the Three Approaches to Integration 10.3.6 Importance of Integrated Treatment of Subject Matter 10.4 New Mathematics 10.5 New look at Mathematics 10.6 Contribution of Piaget, Gagne and Bruner to the Teaching of mathematics 10.6.1 Contribution of Piaget to the Teaching of Mathematics 10.6.2 Contribution of Robert Gagne to the Teaching of Mathematics 10.6.3 Contribution of Jerome Bruner to the Teaching of Mathematics 10.7 Diagnostic and Remedial Teaching 10.8 Let Us Sum Up 10.9 Answers to Check Your Progress 10.10 References 10.1 Introduction The functions of education are changing from time to time. Originally the purpose of mathematics was taught of as just counting and measuring. Slowly its role entered in almost all branches of knowledge. The idea of developing critical thinking and scientific reasoning were the main purposes of mathematics some time back. In mathematics education importance is given to both quantitative expansion and qualitative improvement. Piaget, Gagne and Bruner have contributed much too cognitive development in children. Their views have a good bearing on the teaching of mathematics. The study of the history of mathematics particularly the contribution of Indian mathematicians is quite stimulating and revealing. 10.2 Objectives On completion of this unit students will be able to: i) identify Gagne‘s contribution to the study of mathematics; ii) distinguish between Bruner‘s and Gagne‘s contributions to the learning of mathematics; iii) give three stages according to Burner, when a child attains cognitive learning; iv) watch questions, and problems on mathematics learning and teaching with the appropriate field of educational psychology research; v) understand the meaning, nature and purpose of diagnostic testing; vi) describe of diagnostic testing in the classroom teaching learning process; and vii) conduct remedial teaching in mathematics in classroom situations. 10.3 Integrated Treatment of Subject Matter 10.3.1 Defining Integrated Curriculum What exactly is integrated curriculum? In its simplest conception, it is about making connections. What kind of connections? Across disciplines? To real life? Are the connections skill-based or knowledge-based? Defining integrated curriculum has been a topic of discussion since the turn of the 20th century. Over the last hundred years, theorists offered three basic categories for interdisciplinary work; they defined the categories similarly, although the categories often had different names. Integration seemed to be a matter of degree and method. For example, the National Council of Teachers of English (NCTE) offered the following definitions in 1935: Correlation may be as slight as casual attention to related materials in other subject areas . . . a bit more intense when teachers plan it to make the materials of one subject interpret the problems or topics of another. Fusion designates the combination of two subjects, usually under the same instructor or instructors. Integration: the unification of all subjects and experiences. We joined this conversation in the early 1990s. At the time, we were unaware of the long history of educators with similar concerns. In our separate locations, we defined three approaches to integration—multidisciplinary, interdisciplinary, and transdisciplinary. Our definitions of these categories emerged from our personal experiences in the field. We noticed that people seemed to approach integrating curriculum from three fundamentally different starting points. In looking back, we see that our definitions closely aligned with the definitions proposed by other educators over the decades. The three categories offer a starting point for understanding different approaches to integration. 10.3.2 Multidisciplinary Integration Multidisciplinary approaches focus primarily on the disciplines. Teachers who use this approach organize standards from the disciplines around a theme. Figure 1 shows the relationship of different subjects to each other and to a common theme. There are many different ways to create multidisciplinary curriculum, and they tend to differ in the level of intensity of the integration effort. The following descriptions outline different approaches to the multidisciplinary perspective. Figure 1: The Multidisciplinary Approach

English

Music Family Studies

Science History Theme

Design & Drama Technology

t Math Geography Physical Education

10.3.3 Interdisciplinary Integration In this approach to integration, teachers organize the curriculum around common learnings across disciplines. They chunk together the common learnings embedded in the disciplines to emphasize interdisciplinary skills and concepts. The disciplines are identifiable, but they assume less importance than in the multidisciplinary approach. Figure 2 illustrates the interdisciplinary approach. Figure 2: The Interdisciplinary Approach

10.3.4 Transdisciplinary Integration In the transdisciplinary approach to integration, teachers organize curriculum around student questions and concerns (see Figure 3). Students develop life skills as they apply interdisciplinary and disciplinary skills in a real-life context. Two routes lead to transdisciplinary integration: project-based learning and negotiating the curriculum. Figure 3: Transdisciplinary Approach

Subject Areas Theme Concepts Life Skills Real World Context Student Questions

10.3.5 Comparing and Contrasting the Three Approaches to Integration Figure 4 shows the relationships among the three different approaches. Some differences in intent are apparent. We found, however, that the educators who actually implement integrated approaches are the same educators who are interested in the most effective ways to teach. They are the ones who constantly ask, ―How can I engage all of my students in this learning?‖ They also are the ones who use the most effective planning strategies, such as a backward design process, and are concerned with authentic assessment practices. Therefore, despite some differences in the degree and the intent of integration, the three approaches share many similarities. The centrality of standards and the need for accountability bring the three approaches closer together in practice. Figure 4: Comparing and Contrasting the Three Approaches to Integration

Multidisciplinary Interdisciplinary Transdisciplinary Interdisciplinary Standards of the  Real-life skills and concepts Organizing disciplines context embedded in Center organized around a  Student disciplinary theme questions standards

 All  Disciplines  Knowledge knowledge connected by best learned interconnected and common concepts through the interdependent Conception and skills structure of the  Many right of  Knowledge disciplines answers Knowledge considered to be  A right  Knowledge socially constructed answer considered to be  Many right  One truth indeterminate and answers ambiguous

 Procedures of discipline Disciplines identified Interdisciplinary Role of considered most if desired, but real- skills and concepts Disciplines important life context stressed  Distinct emphasized skills and concepts of discipline taught

 Coplanner  Facilitator Role of Facilitator  Colearner  Specialist/ge Teacher  Specialist  Generalist/sp neralist ecialist

 Student  Interdisciplin Disciplinary questions and Starting ary bridge standards and- concerns Place  KNOW/DO/ procedures  Real-world BE context Degree of Moderate Medium/intense Paradigm shift Integration Interdisciplinary Interdisciplinary Assessment Discipline-based skills/concepts skills/concepts stressed stressed Concepts and Concepts and Concepts and essential essential essential Know? understandings understandings understandings across disciplines across disciplines across disciplines

 Disciplinary  Interdisciplin Interdisciplinary skills as the focal ary skills as the focal skills and point Do? point disciplinary skills  Interdiscipli  Disciplinary applied in a real-life nary skills also skills also included context included

10.3.6 Importance of Integrated Treatment of Subject Matter An integrated approach to teaching and learning college mathematics offers an approach that is different from the traditional approach of chalk-talk-homework-exam. In this integrated approach we give more stress on the following ten points: i) Conceptual understanding rather than only computations. ii) Relational understanding rather than just instrumental understanding. iii) Exploring patterns and relationships rather than just memorizing formulas. iv) Variety of pedagogical strategies rather than just chalk and talk. v) Variety of non-traditional assessments rather than just traditional tests/exams. vi) Effective and meaningful learning rather than just learning for test/exams. vii) Listening (hearing, interpreting) to students‘ thinking rather than only telling (speaking, explaining). viii) Cooperative learning rather than just individualistic learning. ix) Making sense of mathematics using real life applications rather than just explaining abstract concepts. x) Helping students to develop an appreciation of the power of mathematics rather than a negative view of math. 10.4 New Mathematics According to merriam-webster.com ―basic mathematics taught with emphasis on abstraction and the principles of set theory — called also new mathematics‖. According to thefreedictionary.com new mathematics is an approach to mathematics in which the basic principles of set theory are introduced at an elementary level. According to dictionary.reference.com ―new mathematics is a unified, sequential system of teaching arithmetic and mathematics in accord with set theory so as to reveal basic concepts: used in some U.S. schools, especially in the 1960s and 1970s‖. New Mathematics or New Math was a brief, dramatic change in the way mathematics was taught in American grade schools, and to a lesser extent in European countries, during the 1960s. The name is commonly given to a set of teaching practices introduced in the U.S. shortly after the Sputnik crisis in order to boost science education and mathematical skill in the population so that the intellectual threat of Soviet engineers, reputedly highly skilled mathematicians, could be met. One focus of the new math was set theory, where students were encouraged to think of numbers in a new, hopefully more concrete way. New Math emphasized mathematical structure through abstract concepts like set theory and number bases other than 10. Beginning in the early 1960s the new educational doctrine was installed, not only in the USA, but all over the developed world. Other topics introduced in the New Math include modular arithmetic, algebraic inequalities, matrices, symbolic logic, Boolean algebra, and abstract algebra. Most of these topics (except algebraic inequalities) have been greatly de-emphasized or eliminated since the 1960s. The new math is now viewed as a failure, and many of the ideas that were its hallmarks — ideas like the use of sets and the teaching of understanding by going into mathematical theory — are now viewed as silly ideas. But at the time they were viewed as wonderful. What happened? Are there any ideas that we are promulgating today that will be viewed as silly 30 years from now? It is obvious that in any successful movement there is an up-cycle, followed by a time in which the idea is in vogue, followed by a down-cycle. In the case of new math, the original pioneers were in a group called the University of Illinois Committee on School Mathematics, UICSM. There were three principals in that group: Gertrude Hendrix, a professor of education who wrote about what was called un-verbalized awareness; Herbert Vaughan, a professor of mathematics who felt that if mathematics were made rigorous by the precise use of language and notation, then children would better be able to learn it; and the person who put it all together, Max Beberman, who believed fervently that one learned better if one was led to discover the mathematics rather than being told it. The ―new math‖ movement lasted in most countries from the late 1950s to the early 1970s. In many countries this reform was replaced with stressing the basic skills, i.e. arithmetic/ computation. Also in many countries these changes resulted in strong discussions and heated debates among mathematics educators. The ―new math‖ movement, however, was not one single reform. It had many different forms, but there were common elements. We can use concepts such as ―structure‖ and ―logic‖ to talk about these common elements, but the ―new math‖ movement was much more. Also it should be noted that many other reforms e.g. individualized math instruction used new math to gain momentum. New Math was an attempt to emphasize important mathematical concepts — concepts that would help the young mathematical mind mature. Unfortunately, basic skills took a back seat to concepts. And the concepts chosen for emphasis were themselves questionable. Example 1: What comes to mind when I say ―twenty- six‖? You may very well visualize the digit 2 followed by a 6. Proponents of New Math wouldn‘t be happy. ―Twenty- six‖ isn‘t a string of symbols, it‘s a concept. Better to visualize a collection of twenty- six marbles. So New Math placed emphasis on sets: the set of block son a table or the subset of red blocks. To make sure students really understood arithmetic concepts, addition, subtraction and the like were taught in number systems other than our usual base ten (Twenty- six, by the way, is represented by the string of digits ―32‖ in base eight). Example 2: Multiplication is not all that difficult if one learns the multiplication tables and the logical, precise algorithm for the process. Using the traditional algorithm for multiplication, carefully explaining the steps how to multiply and carry; keep columns in line; keep track of which numbers have already been used in the process; multiply numbers containing a zero; mark place value with commas. It is an interesting example of New- Math multiplication process.

Figure 1 (An Example of Mathematical Table) Making the New Mathematics Worth While In the ―new‖ mathematics, then, first there must be freedom of thought; second, we do not want to teach just words; and third, subjects should not be introduced without explaining the purpose or reason, or without giving any way in which the material could be really used to discover something interesting. I don‘t think it is worth while teaching such material. 10.5 New Look at Mathematics Mathematics classes today should be an active learning environment for students instead of just time spent. Today mathematics is not taught as a subject developing mental discipline. Useless unattractive and complicated puzzles are not given to the pupil. Skill in solving practical problem should be developed. The important objective of teaching mathematics is to encourage and impart modern practical and useful knowledge for pupils. At the time of leaving the school, every pupil must be exposed to basic ingredient knowledge in mathematics with which they must be able to get as well in life. Mathematics should be so taught that it develops a scientific bend on mind and ability to solve the problems. Problem based approach should be adopted. Using Heuristic methods pupils must be given training in identifying, analyzing and solving problems. Mathematics teaching should be based on work experience. The knowledge of simple interest, compound interest, profit and loss, mensuration, accountancy, discount, average, graphs and simple geometrical ideas can all be given through properly organized work experience. Today mathematics is taught taking into account the aptitude and interest of children. This is called child centered approach. Pupils must be enabled to appreciate the different aspects of mathematics. Today we want to make provision of gifted children to grow fully by engaging them in skillful activities, slow learners are encourage to develop at least fundamental study in mathematics. Today we make available self learning materials for children to study and make their own rate of progress. Programmed learning technique, computer assisted instruction and laboratory method of learning are some of the self learning processes. For effective learning, aids are of vital importance. We talk of improvised teaching aids. In mathematics we encourage children to make such aids. As a result they have very clear knowledge about the concepts and principles in mathematics. Film strips are available today to teach many mathematical concepts. Magic squares, puzzles, beauty of numbers are studied as leisure time activities. Such recreation based activities develop creative talents, leadership qualities, cooperation and group sprit. Today we encourage student to participate in talent search competitions. This goes to a long ways in developing interest in the study of mathematics. 10.6 Contribution of Some Famous Psychologist to the Teaching of Mathematics Educational psychology is generally concerned with the study of human behavior. For class teachers, educational psychology will enable to them cope with problem of how children learn and under what conditions maximum learning can take place. In this unit you will see how educational psychology can be applied to gainfully teaching and learning mathematics. Also, in this unit you will study the cognitive aspect of learning mathematics and the contributions of three psychologists-Piaget, Bruner and Gagne to the learning of mathematics. They are, not the only contributors but they represent a selection whose choice for further reason will become apparent as you read the unit. 10.6.1 Contribution of Piaget to the Teaching of Mathematics Jean Piaget‘s work on children‘s cognitive development, specifically with quantitative concepts, has garnered much attention within the field of education. Piaget explored children‘s cognitive development to study his primary interest in genetic epistemology. Upon completion of his doctorate, he became intrigued with the processes by which children achieved their answers; he used conversation as a means to probe children‘s thinking based on experimental procedures used in psychiatric questioning. Jean Piaget was a French-Swiss psychologist who was originally trained as a biologist. For more than fifty years he studied and analyzed the growth and development of children‘s thinking. His school in Geneva is noted for the study of psychological problems underlying the learning of mathematics. His work has the greatest significance for teachers of mathematics especially at the primary level. One contribution of Piagetian theory concerns the developmental stages of children‘s cognition. His work on children‘s quantitative development has provided mathematics educators with crucial insights into how children learn mathematical concepts and ideas. This article describes stages of cognitive development with an emphasis on their importance to mathematical development and provides suggestions for planning mathematics instruction. This applies to his study of children‘s general intellectual development and specifically to the development of mathematical concepts. You will now study this in more detail. Piaget view cognitive development in terms of well-defined sequential stages in which a child‘s ability to succeed is determined partly by his biological readiness for the stage and partly by his experiences with activity and problems in earlier stages. The age 0-2 years is known as the sensorimotor stage where the child relates to its environmental through its senses only. Towards the end of the second year of life, children ‗have rudimentary understanding of space and are aware that objects have an existence apart from their immediate experience of them. The pre-operational stage, (2-7 years) which generally cover the cognitive development of children during the pre-school (KINDERGARTEN) years, is marked by the ability to deal with reality in symbolic ways. The thought processes of children in this stage are, however, limited by centering (inability to consider more than one characteristics of an object at a time). Children at this stage also have difficulty with reversibility (The ability to think back to the causes of events). Because of these deficiencies, they cannot conserve (retain) important characteristics of objects and event, and cannot engage in logical thinking in any concrete sense. The child is said not to posses the concept of conservation of number, volume, quality or space. Piaget demonstrated the lack of conservation in two experiments. The following is an account of the experimental procedure: The child was presented with two rows of five plastic squares each row arranged (as show below) in one-one correspondence. All the squares are equal in size. □ □ □ □ □ Child‘s □ □ □ □ □ Examiner‘s He then repeats the statement and question to the child. The child responded by saying he (the child) has more-that is they are no longer both the same. The child at the pre-operational stage is influenced by the perceptual features of the stimulus (i.e. plastic squares). He lacks the ability to see that movement has not altered the plastic squares. Conservation of numbers is a fundamental requirement in the understanding of numbers, for without it a child cannot match sets by pairing to establish equivalent sets. For instance, for the child who is learning a number sentence such as: 3 + 2 = 5 000 + 00 = 00000 and forms a union of disjoint set of three counters and another set two counters, the physical movement of the counters into a new pattern may mean: they change in their number characteristics; and though they may recite 3 plus 2 equals 5, he lacks understanding. The implication for the mathematics teacher is that it is a waste of time (and probably harmful to children) to try to tell children things that cannot be experienced through their senses, that is, through seeing, feelings as well as hearing. Abstract mathematical ideas should therefore not be introduced at this stage. Children at this stage must be permitted to manipulate objects and symbols, so as to be able to appreciate reality. Mathematically oriented recreational facilities (e.g. games, play blocks, counters, marbles etc) are important tools for learning mathematics at this stage. The period of concrete-operational stage (7-12 yrs) is particularly important to the primary school teacher because most primary school children are in this stage of development. This stage marks the beginning of what is known as logico-mathematical aspect of experience. It is illustrated by the case of the child who learn that counting a set of objects leads to the same result whether he counts from front to back, back to front, or whatever configuration in which the objects are arranged. Also logico- mathematical experience underlies the physical act of grouping and classifying in what is known as the algebra of sets. Piaget studies the concrete operational stage using the concept of conservation of invariance, which is a basic characteristic in this stage. For example a child is shown two identical glasses containing the same amount of water as illustrated in Figure (a). The water in one glass is then poured into a taller glass with smaller diameter as shown in Figure (b). If the child understands Figure (a) Figure (b) the amount of water is still the same no matter the nature of the glass container and rejects what perception tells him (one look like it contains more water), he is using logic and has arrived at the concrete operational thought level for his concept. The amount of water is conserved and remains invariant (unchanged) after the transfer into another container. This is referred to as the concept of conservation of invariance. The child realizes that the process can be reversed- that if the liquid is poured back into the initial container, the amount should remain the same. Another example is the process of adding one number to another number. In thought, this would be: if I want to add 4 to 3, I would take a collection of three marbles, physically and four more and then determine the result. But in practice I would represent the process internally. Reversibility would imply that if I recognize that 3+4 = 7 then I would recognize that 7- 4 = 3. The psychological condition for a reversible operation is that of conservation, i.e. + A is reversed by – A. The concrete-operational stage is therefore important for mathematics learning because many of the operations a child is able to carry out at this stage are mathematical in nature for example the operation of classification, ordering, construction of the idea of nature, spatial and temporary operation. This includes all the fundamental operations of elementary logic of classes and relations of elementary mathematics, geometry and even physics. There is however one limitation at this stage. Children have difficulty from hypothetical assumptions. Care must be taken in the type of materials that is included in their mathematical curriculum. Children can reason abstractly if they are not affected by the limitations of the concrete-operational stage. They can proceed to the fourth stage of Piaget‘s process of cognitive developments. This is known as the formal-operational stage, from about 12 years. Only one-fourth of adolescents and one- third of adults, however, ever fully functions at the formal operational level as shown by results of research. At this level, the child now reasons or hypothesizes with symbols or ideas rather than needing objects in the physical world as a basis for his thinking. He can use the procedures of the logician or scientist, a hypothetic-deduction procedure that no longer ties his thoughts to existing reality. He has attained new mental structures and constructed new operations. 10.6.2 Contribution of Robert Gagne to the Teaching of Mathematics As a behaviorist psychologist Gagne devoted his time to study the conditions of learning. He believes that learning occurs as a result of interaction between the learner and the environment. Learning is known to have taken place when we notice (observe) Gagne maintains that the stages described by Piaget are not necessarily the inevitable result of an inborn ―timetable‖ but are instead a consequence of children having learned sets of rules that are progressively more complex. How do children acquire these sets of rules? According to Gagne children are ―taught‖ the rules by their physical and social environment. Notice the differences in what Piaget and Burner are claiming on one hand and what Gagne is saying. If we follow Piaget‘s (and Burner) assertion we will assume that children will develop complex concepts‘ understanding and problem solving skills when they are ready. That is when their nervous systems have matured sufficiently and they had enough experiences with simpler more elementary problems. Mathematics teachers who see learning as a process of discovery are likely to borrow heavily from Piaget and Burner. Others who will see learning as produced primarily by children environment are likely to take their cues from Gagne. The following illustrates Gagne contribution to the learning of mathematics. We have already mentioned that he emphasized the idea of pre-requisite knowledge in learning mathematics. That is the idea that one cannot master complex concepts without mastering the fundamental concepts necessary for such complex concepts. For instance, a child cannot successfully add fractions without the knowledge of finding common denominator of fraction. That is, A child cannot do 1/3 + 2/5 unless he has been Lead to learn that 1/3 = 5/15 and 2/5 = 6/15 Therefore 1/3 + 2/5 = 2/15 + 6/15 = 11/15 Besides there is an intermediate step which we have ignored in this sequence: That is the fact that fractions with same denominator could be added as like terms. That is n/A + m/A = (m + n)/A for A‘s Which are real numbers for instance; 2/3 + 5/3 = (2 + 5)/3 The problem of learning by discovering methods has been mentioned the earlier that Piaget and Burner supported it. Some other psychologists hold a different view. Gagne held a compromising view which he referred to as guided discovery. Guided discovery has been shown to be best in the field of elementary mathematics. 10.6.3 Contribution of Jerome Bruner to the Teaching of Mathematics Jerome Burner worked on the process of thought in general. Later he applied this to the process of learning mathematics. He devised experiments to help him observe how mathematical thinking in children develops. The investigation concerned the individual strategies by which a child tries to discover a given logical relationship. The procedure in most of the experiments was to present a number of cards to the child. Each card has its diagrams of triangle, circle or square separately or a combination of these. Each card was red or green or blue. So there were three variable- number, shape and colour-each with three values. A concept such as red triangles was thought of by the experimenter and the subject chose cards to which the experimenter answered either Yes or No: if the card was red and had triangles on it and No if not. Subjects were asked to find the concept, which the experimenter had in mind in the least number of trials. Sometimes more variables were used; sometimes the numbers of choice were restricted. From this single procedure, Bruner was able to claim that learning in general depended on four factors. i) the structure of the concept that is to be learnt: ii) the nature of the learner‘s intuition: iii) the desire of the learner to learn: iv) the readiness for learning- (biological readiness ). Thus Bruner considered adequacy of both the subject matter and the learner himself necessary for the leaning of Mathematics. By this he meant that the learner must be intuitively ready to learn and the materials to be learnt must be presented in a form (or structure) that matches the learners ―readiness stage‖ This led to his controversial, but yet popular, assertion that‖ Any concept can be taught effectively in some honest form to any child at any age provided such a concept is introduced at the child‘s language level‖. This sort of reasoning let him to attempt a classification of these levels or stages. The following are the three stages through which Brunner says a child goes through in cognitive learning. i) The Enactive Stage: At this stage the child thinks only in terms of action. This stage is characterized by the mode of representing past events through motor responses. The child enjoys touching and manipulating objects as teaching proceeds. Specifically no serious learning occurs at this stage. Topics can however be introduced to a child at this stage using concrete materials. The child‘s methods of solving problems are limited because he cannot ―act the solution‖ – he cannot solve problems. ii) The Iconic Stage: This is the stage of manipulation of images. Here he builds up mental images of things already experienced. Generally, such images are composite being formed from a number of experiences of similar situations. Learning at this stage is usually in the form or in terms of seeing and picturing in the mind any object which transforms learning. The child uses thinking thereby making transfer of learning considerably easy. Bruner emphasized that before any image is formed to represent a sequence of acts, certain amount of motor skills and practices have to take place. iii) The Symbolic Stage: At this stage child possesses the ability to evaluate learning. Logic, language and mathematical symbols are used to discuss what has been learnt. Acquired experiences are translated into symbolic form the three stages can be illustrated this way, using the concept of addition of positive whole numbers. Consider the problem: 3+2=5. First the child must work with block, marbles, counter or other real objects. Take the three first, take another two mix them up and then count the mixture (or union). At the second stage he will be able to work with worksheets containing pictures of objects (images). Instead of the physical objects he is now able to recognize their image and can solve something like it while not necessarily requiring the production of the ducks physically. At the final stage, he can solve the real problem 3 + 2 =5 using symbols 3 and 2 and 5. Bruner‘s three stages, in brief of the sequence Action | Image | Word and correspond approximately to Piaget‘s sensor motor | Perceptual | Abstract modes of cognitive functioning. Also Burner, like Piaget, believe that all mathematics could be learnt by discovery approach provided that search is started early enough in the life of the child by presenting to him concrete materials relevant to the concept we want him to learn at a higher stage. For example properties of a triangle could be taught by making sure children play with triangular shape object at the pre-school (Kindergarten ) stage, draw enough image of triangles at the at the primary school and by the junior secondary school they would have been sufficiently equipped with various terms to discover for themselves some, if not all the properties of a triangle. 10.7 Diagnosis and Remedial Teaching Diagnostic testing is individually administered tests designed to identify weaknesses in the learning processes. Usually these are administered by trained professionals and are usually prescribed for elementary, sometimes middle school, students. Diagnostic testing is an important tool for educators who want to know where their students are academically in order to bring those students to where they need to be. Teacher should have to first identify and locate the area where the error lies. The process adopted for this purpose in educational situations is known as Diagnostic Testing. We may say that Diagnostic Testing implies a detailed study of learning difficulties. In diagnostic testing the following points must be kept in mind: i) Who are the pupils who need help? ii) Where are the errors located? iii) Why did the error occur? Let us take the following examples to make this point more clear. Example 1 Suppose you have taught the simple method of subtraction of two-digit numbers without borrowing and then conducted a test which indicates the solutions as follows: Student‘s name—Mr. ‗X‘ 45 34 35 58 49 - 34 - 23 - 15 - 38 - 27 ------11 11 25 28 22

54 94 72 57 67 - 34 - 43 - 32 - 32 - 24 ------24 51 42 25 43 After assessment of the whole group you have to find out about each individual the area of difficulty or the concept where the learner commits errors. For example, Student ‗X‘ has solved all the questions of subtraction of two-digit numbers without borrowing correctly except for the subtraction of one digit from another one-digit number. You find that his answers are 5 – 5 = 5, 8 – 8 = 8, 4 – 4 = 4, 2 – 2 = 2. You are in a position to diagnose the particular concept which Mr. ‗X‘ could not understand. This is known as Diagnostic Testing. The main aim of Diagnostic Testing is to analyze not to assess. Its main aim is to find the weakness of learner which appears after the assessing the diagnostic test. In another word, the process of identification of the area of learning difficulties and testing the problem is called Diagnostic Testing. 10.8 Let Us Sum Up Today mathematics is not taught as a subject developing mental discipline. Useless unattractive and complicated puzzles are not given to the pupil. Skill in solving practical problem should be developed. The important objective of teaching mathematics is to encourage and impart modern practical and useful knowledge for pupils. Today mathematics is taught taking into account the aptitude and interest of student. This is called child centered approach. Pupils must be enabled to appreciate the different aspects of mathematics. Today we want to make provision of gifted children to grow fully by engaging them in skillful activities; slow learners are encouraged to develop at least fundamental study in mathematics. In this context, Jean Piaget studies the process of intellectual development that takes place in a child through his early ages. He concentrated on the cognitive development of a child till he reaches the period of adolescence. Jerome S. Bruner contributed also to the cognitive development of child. He says that the language play a dominant role in learning activity. According to Bruner oral learning, learning through pictures and learning through signs are the three types of learning. One another famous behaviorist psychologist Robert Gagne believes that learning occurs as a result of interaction between the learner and the environment. According to Gagne children are taught the rules by their physical and social environment. 10.9 Answers to Check Your Progress i) Identify Gagne‘s contribution to the study of mathematics. ii) Distinguish between Bruner‘s and Gagne‘s contributions to the learning of mathematics. iii) Give three stages according to Burner, when a child attains cognitive learning. iv) Explain the meaning, nature and purpose of diagnostic testing. v) What is diagnostic testing in the classroom teaching learning process? vi) How to conduct remedial teaching in mathematics in classroom situations. 10.10 References Definition of new mathematics http://dictionary.reference.com/browse/new+math. Definition of new mathematics http://www.merriamwebster.com/dictionary/ new%20math. Definition of new mathematics http://www.thefreedictionary.com/new+maths N. Kuppuswami Aiyangar: The Teaching of Mathematics in the New Education. Joseph Crecimbem: Teaching of New Mathematics. Sundararajan: Theory & Principle and Methods of Teaching of Mathematics. Kulbir Singh Sidhu: The Teaching of Mathematics. Butler and Wren: The Teaching of Mathematics.