Math BEd. Chapter No. 1,2,3,4,5,6

CHAPTER 1

INTRODUCTION TO TEACHING OF MATHEMATICS

1.1 NATURE OF TEACHING MATHEMATICS:
(i) VIEWS ABOUT MATHEMATICS:
According to American Heritage Dictionary
“Mathematics is the study of measurement, properties and relationships of quantities using number s and symbols.”
According to Longman Dictionary
“Mathematics is the science of numbers and shapes including Algebra, Geometry and Arithmetic.”
According to Mathematicians
“ Mathematics is a science which deals with space, numbers, Arithmetic, Algebra, Geometry and trigonometry.”

The term “Mathematics” has been interpreted in different ways.
(i) Mathematics may also be defined as the science of abstract form.
(ii) It has also been defined as the science of numbers and space.
(iii) Mathematics is a science which deals with space, numbers, Arithmetic, Algebra, Geometry and trigonometry.
(iv) Numbers relating to quantities, scales figures and their inter-relationship is called Mathematics.
(v) Mathematics is a way of logical thinking as reasoning.
(vi) It deals with quantity and relationships as well as the problems involving space and form.
(vii) It deals with the relationship between magnitudes.
(viii) Its Hindi or Punjabi name is ‘Gintia’, which means the science of calculations.
(ix) According to “Locked”. “Mathematics is a way to settle in the mind a habit of reasoning.
(x) Mathematics is an independent world created out of intelligence.
(xi) Mathematics is the cheapest science in which needs only pencil and paper.

(ii) BASE OF MATHEMATICS
Knowledge of mathematics is very vast. It is progressing day by day. So it is very difficult to draw its limits.
Mathematics relies on both logic and creativity. It is used both for a variety of practical purposes and for its intrinsic interest. For some people, and not only professional mathematicians, the essence of mathematics lies in its beauty and its intellectual challenge. For others, including many scientists and engineers, the chief value of mathematics is how it applies to their own work.
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(iii) NATURE OF TEACHING MATHEMATICS
ACCORDING TO ITS BRANCHES

‘Mathematics is a science which deals with space, numbers, Arithmetic, Algebra, Geometry and trigonometry’. So we discuss the nature of Mathematics according to its bravches.

1. Nature of Teaching Arithmetic at Elementary Level
Arithmetic is the science of numbers and the art of computation. Historically the arithmetic is developing out of a need for a system of counting. It is considered to be essential for successful living.
It is needed for every person in his daily life. House wife, modern former, merchant, worker, professional man etc. need mathematics to run their affairs.
The teaching of arithmetic has to fulfil their major responsibilities.
(i) To give proper understanding of number system and intelligent proficiency in its application.
(ii) The socialization of number experiences method for teaching Arithmetic.
(iii) The Hindu Arabic system of numbers is used frequently. Now a days,, Roman system of numbers is used for a few purposes
(iv) Addition, subtraction, multiplication, division are the basics in teaching Arithmetic, fractions, decimal, Real numbers, L.C.M., H.C.F., square rout, ratio proportion, profit, average etc, are taught in Arithmetic.

2. Nature of Teaching Algebra
Algebra is the generalization of arithmetic. In Algebra we can solve different difficult problems in the form of equation easily. Symbols are used in Algebra for saving time and energy. We use generally x,y,z for variables and a,b,c or numbers for constants.

The world Algebra is Arabic in origin. It is modified name of the word “Al-Gaber” Al-Muqabula, where Jabr refers to transferring a quantity from one side of an equation to another after changing the sign, while “muqabula” means the process of subtracting similar quantities from both sides of an equation.
Algebra deals with Algebraic sentences, basic formulas, factors, one degree, two degree, three degree equation with one, two or three variables, matrix equation, in equation elimination ,Formula and factors, quadratic equation etc..

3. Nature of Teaching Geometry at Primary Level
Geometry is the science of space and extent. It deals with the position shape and size of bodies.
Practical arts have become the primary and permanent source of geometric learning.
At primary level the emphases is given on understanding of fundamental concepts and techniques such as drawing lines, singles, triangles and polygons. The teacher should depend on visual arts. The pupil should be asked to observe things themselves by actual measurement. The hand of the pupil should be trained to draw figures.
Some fundamental concepts of Geometry at primary level are solid, surface, line, point, angle, obtuse angle, complimentary angle, supplementary angle, polygon, triangle, perimeter, areas, different types of triangle, square, parallel on rectangle, rhombus, trapezium circle etc.
In Geometry we have to deal with measurement of two dimension, or three dimensional things, we have to measure length, width, perimeters etc.

1.2 PHILOSOPHY OF TEACHING MATHEMATICS

Philosophy of Teaching Mathematics can be explained by the following steps:
(i) Basic Philosophy Of Teaching Mathematics:
Philosophy of teaching mathematics indicates basic knowledge of mathematics and its importance in life. Basic of philosophy mathematics lies in three things.
(a) Simplicity
(b) Continuity
(c) Reasoning ( Logic)
(ii) Philosophical foundations of mathematics
1-The foundational philosophy of Mathematics proposes the existence of a world of mathematical concepts; the truths about these concepts are discovered by humans. In this view, the laws of nature and the laws of mathematics have a similar status. Not our axioms, but the very real world of mathematical concepts forms the foundation of Mathematics.
. 2- Some theories tend to focus on mathematical practice, and aim to describe and analyze the actual working of mathematicians as a social group. Others try to create a cognitive science of mathematics, focusing on human cognition as the origin of the reliability of mathematics when applied to the real world.
3- The obvious question, then, is: how do we access this world?
The foundational philosophy of formalism, as exemplified by David Hilbert, is based on axiomatic set theory and formal logic. Virtually all mathematical theorems today can be formulated as theorems of set theory. The truth of a mathematical statement, in this view, is then nothing but the claim that the statement can be derived from the axioms of set theory using the rules of formal logic.
Merely the use of formalism alone does not explain several issues: why we should use the axioms we do and not some others, why we should employ the logical rules we do and not some others, why do "true" mathematical statements (e.g., the laws of arithmetic) appear to be true, and so on. In some cases these may be sufficiently answered through the study of formal theories, in disciplines such as reverse mathematics and computational complexity theory.
(iii) Problem Solver:
We come across many problems in daily life by means of Mathematics. Mathematics can solve easily those problems in a systematic way. Mathematics is a part of our life. So teaching of mathematics is very necessary. To understand this universe, we need the philosophy of arithmetic.
(iv) Key of Sciences:
Mathematics is the key of science. Mathematic s give base to science subjects we can understand this universe on the basis of mathematics from morning till down our daily life problem are solved by mathematic.
(v) Basic Thing for the Technological Age:
Due to importance of Maths it is taught in school as a compulsory subjects. We are now life in the age of science and technology, so the importance of math is increasing day by day. Computer works in base 2. Today data communication has become very easy due to use of computer.
(vi) According to Some Thinkers:
‘Plato’ had written at the door of his study room;
“The man who does not know mathematics especially geometry is not allowed to come in any room.”
According to ‘Abna Khaldoon’:
“Mathematics cleans the mind as soap cleans the dirty clothes.”
According to ‘Berthlod’:
“Mathematics is an indispensable instrument of all physical research.”

(vii) Active Involvement:
Learning mathematics is a process the requires active involvement. As a teacher, you must provide opportunities for students to become engaged in the learning process. Mathematics is best learned through activities that allow students to explore and understand the mathematics on their own.

Now we explain philosophy according to the types of Mathematics.

(viii) Philosophy of Teaching Arithmetic:
Arithmetic is the science of numbers and the art of computation. Historically, arithmetic developed out of a need for a system of counting. It is considered to be essential for efficient and successful living.

The following are some of the philosophies of teaching arithmetic:
To teach the learner the mathematical type of thought to understanding statements, to analyse them and to arrive at right conclusions.
To arouse the child’s interest in the quantitative side of the world around him and its use as a simple tool in business.
To give accuracy and facility in simple computations of the fundamental processes.
To impart a working knowledge of practical arithmetical applications which are useful in life.
To prepare the way for higher mathematics.


(ix) Philosophy of Teaching Geometry:
Geometry is the science of space and extent. It deals with the position, shape and size of bodies but has nothing to do with their material our physical properties.

The following are some of the philosophies of teaching Geometry
1. It enables to learner to acquire a mass of geometrical facts.
2. The geometric principles of equality, symmetry, and similarity are implanted in the very nature of things.
3. It is important in a person’s cultural development.
4. It develops the ability to draw accurate plans.
5. It provides a content that is objective and non-controversial.
6. It is useful in engineering, machine shop, construction industries, landscape architecture, interior decoration and other areas of appreciation.
7. It demonstrates the nature and power of pure reason.
8. It is the key to mathematical thinking.

(x) Philosophy of Teaching Algebra:
It is a generalization of arithmetic. In teaching and learning, its principles are frequently needed to return to corresponding situations in arithmetic. With the introduction of algebra, the student learns to extend the number system of arithmetic, so that the four fundamental operations can be performed in all cases. The first extension is the introduction of the negative numbers, to provide for the subtraction of larger from smaller numbers.
As it has been called a generalized arithmetic, so it may be related to geometry by saying that algebra is only written geometry and geometry is merely pictured algebra.

The following are some of the philosophies of teaching Geometry
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1. It is useful in other branches of mathematics. It has especially simplified for the learner, many problems of arithmetic.
2. It gives compact formulae or generalization to be used in all cases. The solution of problems by equations and by factorisation are example of this.
3. It has a practical value in many of the trades and industries.
4. it provides an effective way for expressing complicated relations.
5. It gives a new, good approach to be study of abstract mathematical relationship though the use of a new language and a new symbolism.
6. It inculcates the power of analysis.
7. Verification of results in simpler an more satisfactory in algebra than in any other branch of mathematics. It develops confidence among the students.
8. It helps in the generalisation of scientific truths into simple and compact formulae.



1.3-HISTORY OF TEACHING MATHEMATICS

(1) History of teaching mathematics is as old as the history of human beings. In the beginning knowledge of mathematics was started with natural numbers. Later on man started writing “zero”, which was discovered by Muslim Mathematician Musa Khuaramzi.
Different method were used to country the things. One stone was used to count one sheep, two stones were used to count two sheep, so on. A Sphifered was quite satisfied when he compared the stones with number of sheep.
In early day Egyptian Romans & Greek used figures and symbols number.
(2) Greeks were against to write any thing in fraction. After natural a whole numbers fractions were used in Rome. According to them a year was equal 365¼ days. One of that sense common year had 365 days and “leap year” has 366 days.
(3) Musa Alkhuarmzi introduced decimal system. Alkhuarmzi was librarian in the period of Mamamon Rashid.
He wrote a book “Algeber-o-Almuqabla” in Mathematics. In which he introduced zero, fractions, decimal numbers, natural number, whole number, place value of digits in numbers square root, power of a number logarithm etc.
(4) Geometry was introduced by the mathematicians of Egypt. Due to flood in river Neil it was decided to divide the land.
They divided the land in rectangles, squares, triangles, etc. Concept of area was created by them. They knew well about the measurement of line, line segment, triangle, rectangle etc.
(5) Word “Geometry” is the combination of two words Geo and Metron mean measurement. Therefore wordy meaning of “Geometry” is measurement of Earth.
(6) In 16th Century a French Mathematician introduced positive and negative numbers.
(7) Greeks introduced p. According to them, ratio between circumference and diameter of a circle is called p. Its values is or 3.14……
(8) Muhammad Bin Musa Khuarmzi had written book “Algeb-0-Muqabla” in Mathematics. This was the first book about equations and basic operations on numbers. When thus book was translated in European language it was given name “Algebra”.
(9) Arabs knew the basic concepts of Geometry. They were aware how to find the area of a triangle and volume of cylinder and cuboids.
(10) Muhammad Bin Musa Alkhuarmzi translated a book “sidintha” written by “Berham Gupta”. In which Sinq, Cosq, tanq are introduced.
(11) Albaroni had shown the trigonometric ratios in his Book “Qanoona-Masodi”.
(12) Ahmes was an Egypt mathematics. He wrote mathematics manual. In which he his introduce fraction, ratio, simple equation and arithmetic progression. Had also written in the book how to find the area of triangle, rectangle, trapezium and circle. He had introduced that ratio in the sides of right-angled triangle is 3:4:5.
(13) Pythagoras proved that in a right-angled triangle square of hypotenuse is equal to the sum of square of its other two side.

ANCIENT CIVILIZATION AND MATHEMATICS:

1. Babylonian Mathematics:
The early Babylonians drew isosceles triangle on wet clay plates with needles. In this way, they made wedge-shaped letters. After making cuneiform the baked the plates to keep them for a long time.
These plates were excavated at the Dynasty of King Hammurabi’s era, about 1600 B.C. After deciphering the wedge-shaped letters, we can know that the Babylonians used very high system of calculation in commerce and agriculture with the sexagesimal positional system.
Babylonians already know the solution of quadratic equations and equations of second degree with two unknowns and they could also handle equations of the third and fourth degree.

2. EGYPTIAN MATHEMATICS:
Using a kind of reed, papyrus Egyptians made papers. About 1650 B.C. in Ahmes Papyrus which was written Ahmes, we can see how to calculate the fraction and the superficial measure of farmland.
Ancient Egyptians say that the area of a circle is repeatedly taken as equal to that of the square of 8/9 of the diameter.
They also extracted the volume of a right cylinder and the area of a triangle but they handled only a simple equation.

MAKING OF NUMBER:
Probably the earliest way of keeping a count was by some simple tally method, employing the principle of one-to-one correspondence. In keeping a count on sheep, for example, one finger per sheep could be turned under. Counts could also be maintained by making collections of pebbles or sticks, by making scratches in the dirt or on a stone, by cutting notches in a piece of wood, or by tying knots in a string.
As the way of counting, people should learn how to mark the numbers. Each nation, therefore, used its peculiar making of numbers.

3. THE ROMAN NUMERAL SYSTEM:
Numeral system was decimal system or quinary, the subtractive principle, in which a symbol for a smaller unit placed before a symbol for a larger unit means the difference of the two units, was used only sparingly in ancient and medieval times.

4. THE HINDU ARABIC NUMERAL SYSTEM:
1,2,3,4,5,6,7,8,9,0
The Hindu-Arabic numeral system is named after the Hindus, who may have invented it, and after the Arabs, who transmitted it to western Europe; The Persian mathematician al-Khowarizmi describes such a completed Hindu system used position value or 0(zero) in a book of A.D. 825.
It is not certain when this numeral system transmitted to Europe but this system was used all over the Europe about 13th century.
The dispute between the abacist and the allegorist went on. Finally, the abacus disappeared in 18th century.
Our word Zero probably comes from the Latinised form Zephirum of the Arabic Sifr, which in turn is a translation of the Hindu Sunya, meaning “void” of “empty”.
By virtue of the symbol of ‘0’ the decimal system was established. And so we can use four operations more freely than ever.

(5) THE GREEKS
About the seventh-century B.C., an active commercial intercourse sprang u between Greece and Egypt. There naturally took place an exchange of ideas as well as knowledge. Almost all the great Greek philosophers and mathematicians visited the land of the pyramids. To Egypt, Greece is indebted, among other things, for its elementary geometry. But this does not lesson the glory of the Greek mind. “Whatever we Greeks receive, we improve and perfect”, says Plato. The Greeks felt a craving to discover the reasons for things.

(i) THE IONIC SCHOOL:
To Thales (640-546 B.C.) goes the honour of having introduced the study of geometry in Greece. He is said to have studied physical science and mathematics with the Egyptian priests; but the soon excelled his masters, and amazed King Amasis by measuring the heights of the pyramids from their shadows. It was done by using the knowledge of proportion. He is also know as the inventor of theorems on the equality of vertical angle, the quality of the angles at the base of an isosceles triangle, the bisection of a circle by any diameter, the congruence of two triangles having a side and the two adjacent angles equal respectively, the sum of the three angles of a triangle to be equal to two right angles and that all angles inscribed in a semicircle are right angles. With Thales begins also the study of scientific astronomy. He acquired great celebrity by making the prediction of a solar eclipse in 585 B.C.

(ii) THE SCHOOL OF PYTHAGOREANS:
Having failed to found a school in his own country. Pythagoras (5807-500? B.C.) quit his home and founded the famous Pythagorean school at Croton in South Italy. The school was like a brotherhood, its members were forbidden to divulge the discoveries and doctrines of their school to any stranger. This secrecy caused it to become an object of suspicious. People in Lower Italy revolted and destroyed his school. Pythagoras fled Methapontum, where he was murdered.
At this school, mathematics was the principal study, Pythagoras raised mathematics to the rank of a science. His geometry is much concerned with areas (the famous Pythagoras theorem). He was especially fond of those geometrical relations which admitted of arithmetical expression. He was familiar with the construction of a polygon equal in area to a given polygon and similar to another given polygon. Pythagoras called thee sphere the most beautiful of all solids, and the circle the most beautiful of all plane figures. The star-shaped pentagram was used as symbol of recognition by the Pythagoreans, and was called Health by them.

(iii) THE SOPHIST SCHOOL:
In the year 480 B.C. the Greeks introduced democracy in their country, and therefore every citizen became a politician. Education had to be provided to every common man. Thus there arose a demand for teachers. The supply came principally from Sicily, where Pythagorean doctrines had spread. These teachers were called “Sophists” or “Wisemen”. They taught geometry, astronomy, philosophy and language.
The geometry of the circle, which had been entirely neglected by the Pythagoreans, was taken up by thw Sophists. They made many attempts to solve the following three famous problems.
i) To “Trisect an arc or an angle”
ii) To “double the cube”
iii) To “square the circle”
Hippo crates, a sophists, showed that circles are to each other as the squares of their diameters. Antiphon, Bryson, Zeno and Democrats were other famous mathematicians of this period.

(iv) THE PLATONIC SCHOOL:
Plato was born at Athens in 429 B.C. and died in 384 B.C. He travelled widely and came in contract with great philosophers and mathematician of other countries. He established his school in 389 B.C. and devoted the remainder of his life to teaching and writing.
He sought in arithmetic and geometry the key to the universe. He place the ascription over his porch. “Let on one who is unacquainted with geometry enter here.” He always stressed the continuous relationship between mathematics and philosophy. His school produced a large number of mathematicians. One of the greatest achievements of Plato and his school is the invention analysis as a method of proof. Plato has been called a maker of mathematicians. Manaechmus, Dimostratus, Eudoxus, Leodsmes and many others were his students, who did research work in mathematics. Their researches pertained to prism, pyramid, cylinder, cone, parabola, ellipse, hyperboles, solid geometry, foci and astronomy.

(v) THE FIRST ALEXANDRIAN SCHOOL:
Ptolemy founded the University of Alexandria in about 338 B.C. It soon became a great centre of learning. Euclid was invited to open the mathematical school here. Ptolemy once asked him if geometry could not be mastered by an easier process, then by studying the elements. Euclid answered, “There is no royal road to geometry.” It is a remarkable fact in the history of geometry that the Elements of Euclid, written over two thousand years ago, are still regarded by some as the best introduction to the mathematical sciences. Another prominent a few more books on: spherical geometry, astronomy, the hypothesis that light process from eye and not from the object seen; division of plane figures into parts in a given ratio. Some of his works on other subjects have been lost.

(vi) ARCHIMEDES:
(287 B.C. – 212 B.C.) was another famous and celebrated mathematician of those days. He wrote many books, namely:
1) Centres of Plane Gravities
2) Quadrature of Parabol
3) The Method
4) Two books on the Sphere and Cylinder
5) The Measurement of the Circle
6) One Spirals
7) Conoids and Spheroids
8) The Sand Counter
9) Two books on Floating Bodies
10) Fifteen Lemmas
In the book The Measurement of the Circle, he proves first that the area of a circle is equal to that of a right angled triangle having the length of the circumference for its base, and the radius for its altitude. Aristotle knew the properly of the lever, but could not establish its true mathematical theory. Archimedes proof of the property of the lever holds its place in many textbooks to this day. His estimate of the efficiency of the lever is expressed in the saying attributed to him. “Give me a fulcrum on which to rest, and I will move the earth”. His attention was first drawn to the subject of specific gravity when King Hieron asked him to test whether a crown, professed by the maker to be pure gold, was not alloyed with silver. His book Floating Bodies is a treatise on hydrostatics. He rote on a very wide range of subjects, and is known as the Newton of antiquity.

(6) THE ARABS:
“The Arabs were destined to be the custodians of the torch of Greek science, to keep it ablaze during the period of confusion and chaos, and afterwards to pass it on to the Europeans.” This remark applies in part also to Hindu science. Thus science passed from Aryan (Hindu and Greek both) to Semitic races, and later came back to the Aryan.
The Abbasids at Baghdad encouraged the introduction of the sciences by inviting able specialists to their court, irrespective of nationality or religious belief. Medicine and astronomy were their favourite sciences. In the year 772 there came to the court of Caliph, a Hindu astronomer with astronomical tables rals with zero and the principle of position also were introduced along with these astronomical tables among the Arabs. About the form and shape of the Arabic numerals, the statement of the Arabic writer Al-Biruni, who spent many years in India, is of interest. He says that the shape of the numerals different in different localities, and that the Arabs selected from the various forms the most acceptable.
But better information is now available about the way in which Greek science, dashed upon and penetrated Arab soil. In Syria the sciences, especially philosophy and medicine, were cultivated by Greeks. From there, Greek physicians and scholars were called to Baghdad. Caliph Al-Mamun (813-833) secured a large number of Greek manuscripts for translation into Arabic. At the beginning of the tenth century, the important Greek words could all be read in the Arabic tongue. The first important translations into Arabic were the works of Apollonius, Archimedes, Heron and Diophantus. It is thus evident that in the course of a century the Arabs gained access to the vast treasures of Greek science.
In astronomy there was great activity in original research as early as the ninth century. The religious observances demanded by Mohammedenism presented to astronomers several practical problems. The prayers had to take place at definite hours during the day and night. The Mohammedan feast had to be held on ?? days. For these reasons astronomical tables and instruments were perfected, observatories erected and connected series of observations instituted.
Important to the student of mathematics is Al-Khowarizmi’s work on algebra and arithmetic. He lived during the reign of Caliph Al-Mamun. An Arabic writer says. “The arithmetic of Khowarizmi, being based on the principle of position and the Hindu method of calculated, excels all others in brevity and easiness, and exhibits the Hindu intellect and sagacity in the grandest inventions. “His work the Algebra is very important. The title of this book is self-explanatory. It means “restoration and reduction”. By “restoration” was meant the transposing of negative terms to the other side of the equations, by “reduction” the uniting of similar terms. It explains the elementary operations, and the solutions of linear and quadratic equations, and contains little that is original. It contains also a few meagre fragments on geometry. He gives the theorem of right angled and calculates the areas of the triangle, parallelogram and circle. He prepared astronomical tables containing not only the sine function, but also the tangent function.
Next to be noticed is Tabit ibn Korra (836-901) who was born at Harran in Mesopotamia. He was proficient no only in astronomy and mathematics, but also in the Greek, Arabic and Syrian languages, His translation of Apollonian, Archimedes, Euclid, Ptolemy rank among the best. His dissertation on ‘amicable numbers’ (of which each is the sum of factors of the other) is the first known specimen of original work in mathematics on Arabic soil. He also trisected an angle. He is the earliest non-Chinese writer to discuss magic squares.
Foremost among the astronomers of the ninth century ranked Al-Battani. He was the first to prepare a table of cotangents. He dealt with horizontal and also vertical dials. Probably he knew the law of sines.
Abu’l Wefa (940-998) was another scholar of the region. He made the brilliant discovery of the variation of the moon. He translated Diaphanous. He introduced also the secant and cosecant.
Al-Kuhi, another astronomer, was a student of Archimedes and Apolonius. He along with others made a study of the trisection of angles.
Creditable work in the theory of numbers and algebra was done by Al-Karkhi of Baghdad, who lived in the beginning of the eleventh century.

CHAPTER 2
OBJECTIVES OF TEACHING MATHEMATICS


2.1 MATHEMATICS PROBLEM SOLVING

2.11 MATHEMATICS PROBLEM SOLVING

According to an American Educationist John Devi:
“Problem is that matter or situation (may be less important) which creates complication in mind, and stresses the mind to think over it to get any solution.”

According to a Mathematician,” A problem is a sort of obstruction or difficulty which has to be overcome to reach the goal.”
According to Yoakum and Simpson “a problem occurs in a situation in which a felt-difficulty to act is realised”.
L.A. Averill has said, “The only worthwhile life is a life which contains its problems; to live without any longings and ambitions is to live only half-way.”
The 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 set to the students in a natural way and it is ensured that the students are genuinely interested to solve them.
Mathematics is a subject of problems. Its teaching and learning demands solving of innumerable problems. It pre-supposes the existence of a problem in the teaching-learning situation. Man has to face new problems at any moment, in life. Some problems are solved automatically. But in some cases we have to deeply think to reach any solution. There may be more than one solutions of the problem. Then we have to choose the right and suitable solution.

It is a difficulty that is clearly present and recognised by the thinker. It may be a purely mental difficulty or it may be physical and involve the manipulation of data. The distinguishing thing about a problem however is that it impresses the individual who meets it as needing a solution. He recognises it as a challenge.”
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A human child has to meet and solve problems as he grows-problems which present themselves in his physical surroundings, his intellectual associations and in his social contacts. These problems grow in number and complexity as he grows older and older. His success in life is in large measure determined by the individual’s capacity and competence to solve them. Problems exist for him at every step; his growth, development and living lies in their solution. In school, the child is to be trained in the art and craft of problem-solving.
Mathematics differs from other school subjects, as it provides an opportunity of solving problems which evoke thinking. Training children to solve problems is meant for training them to meet and surmount difficulties.

Problem-solving is a suitable approach in the teaching of mathematics. It develops in the learners the ability to recognise, analyse, solve and reflect upon the problematic difficulties. Problem-solving helps us at every step in our teaching-learning process. The only precaution about its use is to select the problems which have definite educational values and to set up an intellectual atmosphere in the class for proper problem-solving.

Merits
1. Problem-solving in schools prepares the pupils to solve the problems of life. It approximates to life. Facing and solving problems is the true nature of life itself.
2. The method involves reflective thinking. Therefore it stimulates thinking, reasoning and critical judgement in the students.
3. The pupils learn by problem-solving which is the method of learning by self effort.
4. It develops qualities of initiative and self-dependence in the students as they are to face the problematic situation themselves.
5. It is a stimulating method. The problem is a challenge. Once it is properly recognised, it acts as a great motivating force and directs the students’ attention and activity.
6. It is especially suitable for mathematics which is a subject of problems.
7. In it there is strong motivation, tension and mental activity which are the conditions for effective learning.
8. It serves individual differences. There are no limits on student achievement. He can solve any number of problems in a specified time and make progress accordingly.
9. It develops desirable study habits in the students. They get engaged in analysis of the problem, reflective thinking, systematic data gathering, verification and critical study.
10. It is a method of experience-based learning. Problem-solving by self-effort is an experienced of its own type. Such an experience is found missing in the lectures and reading from the textbooks.
11. There is possibility of close contact between the teacher and the taught. Every student needs individual guidance from the teacher. The teacher comes to know the difficulties which the students face and helps them accordingly.
12. The students get valuable social experience like patience, cooperation, self-confidence, etc.

Demerits:
1. Its limitations are largely due to its ineffective use. There are, otherwise, no limitations inherent in it. “If the teacher is not able to think reflectively, does not have an attitude of critical enquiry; or when the classroom situation is dominated by him, and the atmosphere is that of recitation and of readymade answers, the problem-solving method is going to fail.”
2. It is difficult to organise the contents according to the requirements of this method. It is difficult to frame really good problems and to introduce them at every step.
3. It is a time-consuming method. The progress of the students is bound to be slow.
4. All the topics and subject areas cannot be covered by this method.
5. The method does not suit the students of lower classes. They do not possess enough background for scientific approach to problems.
6. When the structure of the problem itself is not up to the mark, or when the problems chosen are unreal and artificial, the method will not be applicable.
7. Textbooks written in the traditional style do not help in the use of this method. There is absence of suitable books for reference and guidance.
8. Teacher’s burden becomes heavier. Real, scientific approach to problem necessitates a lot of study and preparation on his part.
9. It is an intellectual approach in learning. Mental activity dominates in this method and there will be neglect of physical and practical experience.
10. In case the assigning of problems or proposing of problems becomes the teacher’s main job, then the procedure obviously smacks of authority, spoon-feeding and artificiality.

. 2.1.2 CRITERIA FOR SUCCESSFUL PROBLEM SOLVING IN ELEMENTARY MATHEMATICS

The criteria of problem method are almost like that of the project method. It can also take the form of inductive-deductive methods. The procedure involves the following steps:
1. Sensing the problem/ Interpreting, defining the problems.

2. Gathering, Organising and evaluating the data.
3. Formulating tentative solutions.
4. Arriving at the true or correct solution.
5. Verifying the results.
It is a research-like method. It involves scientific thinking as a process of learning. Its relationship with inductive-deductive method is very intimate.
According to Mathematical inductive approach:
The method has the form of the steps
1- Sensing the problem
2- Analysing &organising Data/information
3- Framing solutions
4- Elimination
5- Verification.
Now we apply these steps in examples:
EXAMPLE-1:
Find the ‘Volume of a Cylinder’.
Solution:
1-Sensing the Problem:
Finding the volume of a cylinder is a problem before the class. To find: What is required? We have to find volume of a cylinder.
2-Analysing & Organising Data/information:
We know a similar sort of problem. Formula of volume of a cylinder has to be developed on the basis of the earlier formula for the volume of a cuboid.
volume of a cuboid=Length x Width x Height
Also we know
Area of base =Length x Width
The area of the base of the cylinder is found by an already known formula
Area of the base of the cylinder= ¶ r2
3-Framing Solutions
volume of a cuboid=Length x Width x Height
=Area of base x Height
. = ¶ r2 x h
= ¶ r2h
4- Elimination
volume of a cuboid = ¶ r2h
5-Verification:
For the purpose of verification it is applied to a number of similar problems or situations and the results are checked. The solutions to the problems always come from the students. The teacher remains in the background and directs or guides the student activity from that position.
.
EXAMPLE-2
Given (63)(54) = (N)(900), find N.
1-SENSING THE PROBLEM
What are we asked? The value of N that satisfies an equation.
2-Analysing & Organising Data/information:
Will any particular strategy help here? Yes, factor each term in the equation into primes. Then, solve the equation noting common factors on both sides of the equation.
3- Framing Solutions:
Break down the equation into each term’s prime factors.
63 = 6 ´ 6 ´ 6 = 2 ´ 2 ´ 2 ´ 3 ´ 3 ´ 3
54 = 5 ´ 5 ´ 5 ´ 5
900 = 2 ´ 2 ´ 3 ´ 3 ´ 5 ´ 5
Now (63)(54) = (N)(900)
6 ´ 6 ´ 6 X 5 ´ 5 ´ 5 ´ 5=(N)(900)
2 ´ 2 ´ 2 ´ 3 ´ 3 ´ 3´ 5 ´ 5 ´ 5´ 5= (N)( 2 ´ 2 ´ 3 ´ 3 ´ 5 ´ 5)

Two 2’s and two 3’s from the factorisation of 63 and two 5’s from the factorisation of 54 cancel the factors of 900.
The equation reduces to
2 ´ 3 ´ 5 ´ 5 = N
So N = 150
4- ELIMINATION
Asked value of N is 150.
5- Verification:
Did you answer the question? Yes.
Does our answer make sense? Yes.
If we put value of N in given question, the equation is satisfied. i.e
L.H.S= R.H.S

EXAMPLE-3
How many degrees are in the Celsius equivalent of -22°F?
1-SENSING THE PROBLEM
What are we trying to find? We want to know a temperature in degrees Celsius instead of degrees Fahrenheit.
2- Analysing & Organising Data/information:
Since we have a formula which relates Celsius and Fahrenheit temperatures, let’s replace F in the formula with the value given for degrees Fahrenheit.
F = 1.8C + 32
3-Framing Solutions:
The formula we’re given is F = 1.8C + 32. Substituting -22 for F in the equation leads to the following solution:
– 22 = 1.8C + 32
– 22 – 32 = 1.8C
– 54 = 1.8 C
The answer is -30°C
4- - ELIMINATION
Degrees in the Celsius are -30°C
5- Verification/LOOK BACK
Did you answer the question? Yes.
Does our answer make sense? Yes.
If we put value of N in given question, the equation is satisfied. i.e
L.H.S= R.H.S

2.1.3 ROLE OF TEACHER TO PROMOTE THESE
CRITERIA IN ELEMENTARY STUDENTS

1-Role of teacher to promote these criteria in elementary students is to make certain problems according to the topic of the subject. He gives that problems to the students and ask them to solve them.
2-Teacher role is to guide the students. He should constantly look upon the difficulties of the students. He should see weather the students are going on the right path and try to give hints to the students if necessary.
3- Make up problems based on everyday experiences.4- Solve problems using a variety of strategies (for example: make a list, draw a picture, or guess and check).5- Formulate and solve real-world problems. 6- Verify and interpret results with respect to the original problem. 7- Generalize solutions and apply strategies to new problem situations. 8- Solve multi-step problems. 9- Use problem solving approaches to investigate and understand new mathematical content, both independently and in groups. 9- Demonstrate that a problem may be solved in more than one way. 10- Exhibit confidence in their ability to solve problems independently and in groups. 11- Display increasing perseverance, and persistence in problem solving. 12- Write about problem solutions and solution processes.
13- Formulate problems from everyday and mathematical situations. 14- Solve problems that require the use of strategies (for example: making a list,drawing a picture, looking or a pattern, or acting out). 15- Solve problems with and without using manipulatives and calculators


2.2 MATHEMATICAL REASONING

2.2.1 Meaning of Mathematical reasoning
Mathematics is also called the science of reasoning. In it , we approach every thing with a question mark in our mind. Thus every step needs reasoning. Why it is so is the first need of Mathematics. Here the results are developed through a process of reasoning. Mathematical steps need simplicity, accuracy, certainty, originality and verification.
Locke has said,” Mathematics is a way to settle a habit of reasoning”.
Mathematics may also be defined as the science of abstract form. Abstract form needs concrete things to be explained.
At the simplest level, mathematics is a vast symbolic logic system possessing a few simple postulates from which number of statements can be formed. The symbols are commonly used to express the entities of mathematics and the rules for manipulating them have also been arbitrarily selected. These symbols are truly abstract; they represent nothing. "2" is not two sheep, two random objects, or anything we can describe; it is simply a "2". Likewise, the rules we use with these symbols mean nothing--how do we "+" things? What does "=" mean? Even in combination, as in "2+2=4", we are not referring to anything in our world, simply to constructs of the human mind. Through common agreement, we all know that saying, "Two sheep plus two sheep equal four sheep," is shorthand.
Most of the symbols used in mathematics have been inspired by the real world. Numbering probably arose when someone decide that this business about hands and fingers was a mess, and that naming these quantities would be very useful. At some later point, someone thought of using these names for quantities without anything being quantified, and became the first true mathematician. Other concepts, such as Euclidean lines and perfect circles, also came from reality, but did not, do not, and cannot really exist. Even imagining a one-dimensional line, not to mention a zero-dimensional point, is arguably beyond that capability of our minds. No matter what we dream up, it's still too thick.

2.2.2 Drawing Logical Conclusions about Mathematics:

Mathematics is a subject in which each step needs logic. In this subject conclusions are drawn after a lot of thinking and logic approach. For logic conclusions the flexible activities are organized by the teachers and students. We encourage each class to participate in one introductory activity and one activity linking math to a social concern or issue of equality.

For logic conclusions in Mathematics we see some examples:
EXAMPLE.1: How much long a stick can exactly measure 10,12,20 meters lengths.
SOLUTION: Required length of the stick should divide exactly these given lengths.
So we have to find H.C.F. This is our first conclusion.
Factors of 10 = 1,2,5
Factors of 12 = 1,2,3,4,6
Factors of 20 = 1,2,4.5,10
Common factors = 1,2
Highest common factor=H.C.F =2
SECOND METHOD
So 2 meter long stick can measure all the lengths exactly.
This is our second conclusion.
EXAMPLE.2: How much long distance can exactly be measured with 15m, 20m and 25m long ropes.





SOLUTION: The required long distance which can be exactly measured with these three ropes can be found by L.C.M. This is our first conclusion.
Thus 300m long distance can be exactly measured with these three ropes. This is our second conclusion.
Logical Conclusions by Some Puzzles:
These puzzles require both logical and mathematical reasoning. Can you solve them?
1--- A high school has a strange principal. On the first day, he has his students perform an odd opening day ceremony:
‘There are one thousand lockers and one thousand students in the school. The principal asks the first student to go to every locker and open it. Then he has the second student go to every second locker and close it. The third goes to every third locker and, if it is closed, he opens it, and if it is open, he closes it. The fourth student does this to every fourth locker, and so on. After the process is completed with the thousandth student, how many lockers are open?’
Solution for #1
The only lockers that remain open are perfect squares (1, 4, 9, 16, etc) because they are the only numbers divisible by an odd number of whole numbers; every factor other than the number's square root is paired up with another. Thus, these lockers will be "changed" an odd number of times, which means they will be left open. All the other numbers are divisible by an even number of factors and will consequently end up closed.
So the number of open lockers is the number of perfect squares less than or equal to one thousand. These numbers are one squared, two squared, three squared, four squared, and so on, up to thirty one squared. (Thirty two squared is greater than one thousand, and therefore out of range.) So the answer is thirty one.
2--- ‘You must cut a birthday cake into exactly eight pieces, but you're only allowed to make three straight cuts, and you can't move pieces of the cake as you cut. How can you do it?’
Solution for #2
Use the first two cuts to cut an 'X' in the top of the cake. Now you have four pieces. Make the third cut horizontal, which will divide the four pieces into eight. Think of a two by two by two Rubik's cube. There's four pieces on the top tier and four more just underneath it.

2.2.3 Using models, facts, properties and relationships to explain thinking and justifying solutions:


Using models to explain thinking and justifying solutions:


1- THE GEOBOARD
The geoboard is a device used in elementary schools to aid in the teaching of basic geometric concepts. Geoboards may be purchased commercially from the usual supply houses or they may be constructed out of common household materials using common tools. A simple geoboard can be made from a square piece of wood and 25 finishing nails. A grid of 5 vertical lines and 5 horizontal lines evenly spaced are drawn on the square piece of wood. Nails are placed at thee intersections of the lines .











Figures are made on the geoboard by stretching rubber bands from one nail to another until the desired shape is formed. Segments can be shown by connecting only two nails. The first task to perform is to determine how many different segments may be constructed on the geoboard. As with most of the problem solving situations we will encounter, we will first simplify the problem, then look for patterns to help solve it, and then attempt to generalize the solution. It is easier to work with an actual geoboard then to work on paper alone, but if a geoboard is not available, dot paper can be used. The dot paper below shows some figures drawn as if on a geoboard.











2- 100 SQUARE BOARD






































































































100 square board is a device used to clear the concepts of common fractions, decimal fractions, percentage, graph, area , square , equality and inequality , square root etc.


3- CONE:
A Solid of the shape shown in the figure is known as a cone. The position of its point ‘O’ is known as its vertex.
The distance between the centre of the base and the vertex is known as its height. The distance of the vertex from any point on the boundary of the circular base is known as its Slant Height.
The lower circular top is called its Base.
The radius of the base is called it Radius.









4- VOLUME OF A CUBE:
The cube shown in the figure has length, width and height of 1cm. Its volume can be found by multiplying its length, width and height.
Thus, the volume of a cube
With an edge of 1cm = 1cm ´ 1cm ´ 1cm
= 1cm3



It is read as 1 cubic centimetre.
To find out the volume of any cube means that how many cubes of volume 1cm3 can fit in it.
If the length of an edge of a cube is 3cm then it can hold 3 ´ 3 ´ 3 = 27 cubes of 1cm3 in it.
Thus,
Do you know?

1 litre is equal to 1000cm3.






5- VOLUME OF A CUBIOD:
The given cuboid has a length of 4cm, height of 3cm and width of 2cm. It can hold 24 cubes of volume 1cm3.
Thus,
Remember that
Volume of Cubiod = (Length ´ width ´ height) (units)3
2- Using ‘facts’ to explain thinking and justifying solutions:

1. Solid:
Anything that occupies space is called a solid. A solid has three dimensions, viz, length, breadth and thickness.
The teacher should illustrate a solid with the examples of a brick, a pillar, a stock, etc.
2. Surface:
A boundary of solid is called a surface. A surface has length and breadth but no thickness. This idea can be illustrated with the examples of a tabletop, a floor, a human body, a swimming tank.

3. Plane Surface:
A surface is plane if it is perfectly flat, level and even; such as tabletop, floor etc.
4. Curved Surface:
If a surface is not perfectly that and even, it is a curved surface; such as, that surface of an uneven heap of wheat, the surface of a football, etc
5. Line:
It is a boundary of a surface. It has only one dimension, length. A line is that which has length
6. Straight Line and line segment:
It is that which has the same direction from point to point throughout its whole length. The shortest distance between any two points is called Line segment.
Place a foot rule and draw a line with it to show the shape and nature of a line segment. The name of a line segment can also be indicated by writing two letters, one at each end.
7. Curved Line:
A line which does not have the same direction from a point to point throughout is whole length is called a curved line.
This concept should be explained by drawing an arc or a line with bends.
8. Point:
It is the boundary or the extremity of a line. It has, therefore, position but no magnitude. A point is that which has position, but has no length, no breadth, no thickness. It is named by a single capital letter.
It should be shown by a dot, or by two intersecting straight lines. It is another name for the full stop.
9. Angle:
An angle is the inclination of a straight line on another. It is formed by the rotation of a straight line about its end. The amount of rotation, or the angle, does not depend upon the length of the line (arm).
The idea of angle should be given with the help of a part of compasses, the hand of a toy watch, the opening of a door, a pair of scissors, and the opening of a book. Its actual diagram on a chart or blackboard will be a more effective demonstration.

3- Using ‘Relations’ to explain thinking and justifying solutions:
Activity:
A boy wants to find the height of a tree without climbing the tree. He stands another boy near the tree in the sun. He measures the height and shadow of the other boy. Also he measures the shadow of the tree to find the height of the tree.
Let
Shadow of the tree = 10 meter
Shadow of the boy = 1 meter
Height of the boy = 1.5 meter
Height of the tree = x = ?
He used the ratio formula as given below:
Height of the tree : Height of the boy =
Shadow of the tree : Shadow of the boy
Product of the extremes = Product of the means
2.3 MATHEMATICAL CONNECTIONS:

2.3.1 Link conceptual and procedural understanding
Mathematics is mostly in abstract form. Therefore concept formation in Mathematics is utmost necessary. As a theoretical discipline, mathematics explores the possible relationships among abstractions without concern for whether those abstractions have counterparts in the real world. The abstraction can be anything from strings of numbers to geometric figures or sets of equations..
Mathematicians, like other scientists, are particularly pleased when previously unrelated parts of mathematics are found to be derivable from one another, or from some more general theory.
Mathematics may also be defined as the science of abstract form. For concept formation Abstract form needs concrete things to be explained.
At the simplest level, mathematics is a vast symbolic logic system possessing a few simple postulates from which number of statements can be formed. The symbols commonly used to express the entities of mathematics and the rules for manipulating them have also been arbitrarily selected. These symbols are truly abstract; they represent nothing.

Most of the symbols used in mathematics have been inspired by the real world. For concept formation Numbering probably arose when someone decide that this business about hands and fingers was a mess, and that naming these quantities would be very useful. At some later point, someone thought of using these names for quantities without anything being quantified, and became the first true mathematician.
Other concepts, such as Euclidean lines and perfect circles, also came from reality, but did not, do not, and cannot really exist. Even imagining a one-dimensional line, not to mention a zero-dimensional point, is arguably beyond that capability of our minds. No matter what we dream up, it's still too thick.
Examples of ‘Concept Formation’
"2" is not two sheep, two random objects, or anything we can describe; it is simply a "2". We can write ‘2’ in any form for concept formation.
Likewise, the rules we use with these symbols mean nothing--how do we "+" things? What does "=" mean? Even in combination, as in "2+2=4", we are not referring to anything in our world, simply to constructs of the human mind. Through common agreement, we all know that saying, "Two sheep plus two sheep equal four sheep," is shorthand.

2.3.2 Recognizing relationships among different topics in Mathematics
For many years, mathematics was thought of and taught as a series of isolated topics. Students in elementary and middle grades learned number concepts and skills, geometry concepts, and measurement skills. There was little discussion of the relationship among these topics, and little, if any, discussion of connections to the world outside the classroom, other than the contrived word problems usually found at the bottom of the textbook page! In high school, formal algebra, geometry, and statistics courses made up the mathematics curriculum -- usually as separate courses. In today's mathematics classroom, helping students see the connections among the content standards and how mathematics connects to their informal experiences enables us to introduce some mathematical ideas much earlier, and helps students develop a greater understanding of those ideas.

In this section, we will take a look at some aspects of connections that we can use in the early grades to help students gain a better understanding of mathematics.


2.3.3 Using Mathematics in other subject areas and in real world applications

In real world applications:
When you buy a car, follow a recipe, or decorate your home, you're using math principles. People have been using these same principles for thousands—even millions—of years, across countries and continents. Whether you're sailing a boat off the coast of Japan or building a house in Peru, you're using math to get things done.
How can math be so universal? First, human beings didn't invent math concepts; we discovered them. Also, the language of math is numbers, not English or German or Russian. If we are well versed in this language of numbers, it can help us make important decisions and perform everyday tasks. Math can help us to shop wisely, buy the right insurance, remodel a home within a budget, understand population growth, or even bet on the horse with the best chance of winning the race.
Join us as we explore how math can help us in our daily lives. In this exhibit, you'll look at the language of numbers through common situations, such as playing games or cooking. Put your decision-making skills to the test by deciding whether buying or leasing a new car is right for you, and predict how much money you can save for your retirement by using an interest calculator.

IN OTHER SUBJECT AREAS:

Science and mathematics:
The alliance between science and mathematics has a long history, dating back many centuries. Science provides mathematics with interesting problems to investigate, and mathematics provides science with powerful tools to use in analysing data. Often, abstract patterns that have been studied for their own sake by mathematicians have turned out much later to be very useful in science. Science and mathematics are both trying to discover general patterns and relationships, and in this sense they are part of the same endeavour.
Mathematics as an applied science Mathematics is also an applied science. Many mathematicians focus their attention on solving problems that originate in the world of experience. They too search for patterns and relationships, and in the process they use techniques that are similar to those used in doing purely theoretical mathematics. The difference is largely one of intent. In contrast to theoretical mathematicians applied mathematicians, in the examples given above, might study the interval pattern of prime numbers to develop a new system for coding numerical information rather than as an abstract problem. Or they might tackle the area/volume problem as a step in producing a model for the study of crystal behaviour.
Physics & Mathematics
Most of the Laws and theories of Physics are written in Mathematics. All numericals are solved and elaborated in Mathematics. For example. Newton Laws Of Motion, Gravitational theory etc.
Every rule and principal of Physics takes ultimately Mathematics form. Units of measurement of quantities used in Mathematics are also applied in Physics. All the topics of Physics need Mathematics. e.g. Sound, Heat, Electricity, Magnetism, Properties of matter etc needs Mathematics.

Chemistry & Mathematics
Symbols , atomic numbers, atomic weights , tables are written in Mathematics. Atomic models are explained in Mathematics.
Similarly Mathematics is used in Biology ,Statistics, Psychology etc.


2.4 MATHEMATICAL COMMUNICATIONS

2.4.1 Relate physical materials, pictures , Mathematical language, symbolic representation to Mathematical ideas and situations:

Physical materials, pictures, Mathematical language, symbolic representation to Mathematical ideas and situations is explained by some activities below.
All the branches of Mathematics e.g., Topics of Algebra, Geometry, Trigonometry need symbols, pictures to clarify Mathematical ideas. e.g. Area ,Volume, Perimeter need formulas, pictures. original things, symbols to elaborate them.
(i)Using Pictures For Mathematical Idias
1-. Ask students to draw a few sets of parallel lines with two lines in each set and draw a transversal to cut these lines. Let them measure the alternate and corresponding angles in each case.
They will find them equal in all the cases. This conclusion in a good number of cases, will enable them to formulate the relevant generalisation.
Result: Alternate angles are equal, also corresponding angles are equal always.
2- EQUIVALENT FRACTIONS:
We know that if the denominator and the numerator a fraction are multiplied by the same non-zero number then the fraction remains unchanged.

e.g.






i.e. each of the practice is presenting half of a circle.
These fractions are called “Equivalent fractions” or “Like fractions”.
3- MIXED FRACTION:
An improper fraction also be written as a sum of a whole number and a proper fraction.
For Example:






This sum can briefly be written as . The fraction written in this form is called a mixed fraction.



(ii)Using Physical Materials for Mathematical Ideas

1-We use match sticks for counting.
2- We use cardboards, sticks and models to give the ideas of Geometrical basic concepts like Triangle, Trapezium, Rectangle etc.
3- Base 2 and base 5 concepts can be easily be explained by physical material.

4-Activity of Addition Subtraction:
Draw on the board three parallel lines then two intersecting lines. Place a "+" or "-" sign next to the second parallel line. What you have made is a grid of empty boxes, with three boxes in each of the three rows.
Explain to the students that you are going to roll a die and the number that is rolled is to be placed into one of the squares in the top two rows. The bottom row is for the answer. The die will be rolled until the empty boxes in all the rows, except the bottom row, are filled.
While the students are putting their numbers onto their paper you are also playing by putting your numbers into the squares on the board. (I found this to help the slower students, and the quick students try to beat you.)
Then ask if anyone beat your answer. The best answer is written on the board and anyone with that answer receives a point.
Create smaller or larger grids to adapt to your students level.
The object of the game is to get the highest number if adding or the lowest number if subtracting.

(iii)Using Mathematical Language for Mathematical Ideas
As mentioned, mathematics is also a kind of language. Very unlike our everyday languages, it can unambiguously describe situations involving its numbers, figures, and operations, yet it is limited. Not in the sense that it cannot talk about our world, only its abstractions. Mathematics cannot talk about itself; it can only express mathematical concepts. In technical terms, it lacks the capacity of a met language. Mathematics is of particular value when used in combination with normal languages to provide a sort of shorthand, allowing us to relate mathematical abstractions to certain physical concepts, letting us to count sheep easily, as well as build bridges that don't fall down.
(iv) Mathematical Symbolic representation for Mathematical ideas and situations:
Mathematics has its own Sumbolic language for explaining ideas.
For example
1-For combining two sets A & B, we write
( A U B )
If 2 is element of set A, we write
2 € A
2- Multiplication in Mathematics language:

Use Jodo blocks / marbles / pebbles for the following sequence:
* Give me 2 + 3.
With Jodo Blocks : Children will make :
With Pebbles : Children will give :
2 pebbles of one colour and 3 of other colour.
OR 2 in one hand and 3 in the other hand and then
joined together.
* Give me 3 + 3.
* Give me 3 + 3 + 3 + 3 + 3
* When the same number is added (joined) to itself repeatedly we can also
join the groups of 3 blocks differently, one below the other to make a
rectangle. 3+3+3+3+3
* Bhajiwala has made piles of 4 lemons each.
This also has a multiplication :
Three times four : 3 x 4
The lemons can also be arranged thus to make a rectangle :
THUS WE CAN SAY THAT
EVERY MULTIPLICATION GIVES A RECTANGLE
AND
EVERY RECTANGLE IS A MULTIPLICATION.

2.4.2 Representing, discussing, reading, writing and listening to Mathematics as vital part of learning and utilizing Mathematics

The students interview a relative or other adult in their community about how they use mathematics in their job, daily life, or studies. Encourage students to ask questions about the specific ways in which math is used, e.g. making calculations, handling money, creating budgets, taking measurements, analyzing numeric data, etc. Students then write a report or create math word problems for their peers based on the information they gathered.
A group of students tours the school to collect and graph data on the images that appear on the school and classroom walls. Whose pictures and words are portrayed? Students analyze the data on the basis of gender, race, class or disability, comparing percentages of voices and images represented in each category with the population in their class, at their school site, in their state, and in their country.
Students analyze entire newspaper stories. They can outline in one color all the stories about violence and crime, for example, and use another color to outline stories about people working for justice and peace. Similarly, one can highlight how many times people of color are featured in stories of crime or drug-addiction, and how many times they are portrayed positively.
Students look at front-page photos for one month in three major dailies to record what percentage of front-page photo subjects are women or people of color and when they do appear how they are represented, i.e. as athletes, criminals, victims, or representatives of government or business.
In each case, encourage students to use math skills of simple computation, averages, percents, and graphing to create displays on bulletin boards. Be sure to ask students to consider how these images affect the way they feel and how the decisions are made about which images or stories appear. Students can then take action against any inequities they might discover by writing to the newspapers or publishers and using their findings to teach younger children about the bias they detected.
CHAPTER 3


THEORETICAL FOUNDATIONS FOR MATHEMATICS EDUCATION


3.1 COGNITIVE THEORIES

Some Important Cognitive Theories are given below:

1- Theory of cognitive development--Jean Piaget
2. Cognitive Load Theory
(J. Sweller)
3- Cognitive Flexibility Theory (R. SPIRO,
P. FELTOVITCH & R. COULSON)
4. Cognitive Dissonance (L. FESTINGER)


1-Theory of cognitive development--Jean Piaget

Although there is no general theory of cognitive development, one of the most historically influential theories was developed by Jean Piaget, a Swiss psychologist (1896–1980). His theory provided many central concepts in the field of developmental psychology and concerned the growth of intelligence, which for Piaget, meant the ability to more accurately represent the world and perform logical operations on representations of concepts grounded in the world. The theory is considered "constructivist", meaning .For his development of the theory, Piaget was awarded the Erasmus Prize.

“The theory concerns the emergence and acquisition of schemata—schemes of how one perceives the world—in developmental stages, times when children are acquiring new ways of mentally representing information. It asserts that we construct our cognitive abilities through self-motivated action in the world.
Piaget divided schemes that children use to understand the world through four main periods, roughly correlated with and becoming increasingly sophisticated with age:
· Sensorimotor period (years 0–2)
· Preoperational period (years 2–7)
· Concrete operational period (years 7–11)
· Formal operational period (years 11–adulthood) “
·
(i) Sensorimotor period

According to Piaget, this child is in the sensorimotor period and primarily explore the world with senses rather than through mental operations.
Infants are born with a set of congenital reflexes, according to Piaget, in addition to a drive to explore their world. Their initial schemas are formed through differentiation of the congenital reflexes (see assimilation and accommodation, below).
The sensorimotor period is the first of the four periods. According to Piaget, this stage marks the development of essential spatial abilities and understanding of the world in ‘six sub-stage’s:
· The first sub-stage, known as the reflex schema stage, occurs from birth to six weeks and is associated primarily with the development of reflexes. Three primary reflexes are described by Piaget: sucking of objects in the mouth, following moving or interesting objects with the eyes, and closing of the hand when an object makes contact with the palm (palmar grasp). Over these first six weeks of life, these reflexes begin to become voluntary actions; for example, the palmar reflex becomes intentional grasping. (Gruber and Vaneche, 1977).
· The second sub-stage, primary circular reaction phase, occurs from six weeks to four months and is associated primarily with the development of habits. Primary circular reactions or repeating of an action involving only one's own body begin. An example of this type of reaction would involve something like an infant repeating the motion of passing their hand before their face. The schema developed during this stage inform the infant about the relationships among his body parts (e.g., in passing the hand in front of his eyes he develops a motor schema for moving his arm so that the hand becomes visible.) Also at this phase, passive reactions, caused by classical or operant conditioning, can begin (Gruber et al., 1977).
· The third sub-stage, the secondary circular reactions phase, occurs from four to nine months and is associated primarily with the development of coordination between vision and prehension. Three new abilities occur at this stage: intentional grasping for a desired object, secondary circular reactions, and differentiations between ends and means. At this stage, infants will intentionally grasp the air in the direction of a desired object, often to the amusement of friends and family. Secondary circular reactions, or the repetition of an action involving an external object begin; for example, moving a switch to turn on a light repeatedly. The differentiation between means also occurs. This is perhaps one of the most important stages of a child's growth as it signifies the dawn of logic (Gruber et al., 1977). However, babies still only have a very early rudimentary grasp of this and most of their discoveries have an "accidental" quality to them in that the initial performance of what will soon become a secondary circular reaction occurs by chance; but then operant conditioning causes the initial "accidental" behavior (which was followed by an 'interesting' pattern of stimulation) to be repeated. And the ability to repeat the act is the result of primary circular reactions established in the previous stage. For example, when the infant's hand accidentally makes contact with an object he is looking at the infant receives both visual and tactile feedback about the object; and his subsequent ability to bring his hand into contact with other objects in his field of vision is based on the primary circular reaction of briniging his hand into his field of vision. Thus the child learns (at the level of schemata) that "if he can see it then he can also touch it" and this results in a schema which is the knowledge that his external environment is populated with solid objects.
· The fourth sub-stage, which occurs from nine to twelve months, is when Piaget (1954) thought that object permanence developed. In addition, the stage is called the co-ordination of secondary circular reactions stage, and is associated primarily with the development of logic and the coordination between means and ends. This is an extremely important stage of development, holding what Piaget calls the "first proper intelligence." This stage marks the beginning of goal orientation or intentionality, the deliberate planning of steps to meet an objective (Gruber et al. 1977).
· The fifth sub-stage, the tertiary circular reactions phase, occurs from twelve to eighteen months and is associated primarily with the discovery of new means to meet goals. Piaget describes the child at this juncture as the "young scientist," conducting pseudo-experiments to discover new methods of meeting challenges (Gruber et al. 1977).
· The sixth sub-stage, known as the invention of new means through mental combinations stage, is associated primarily with the beginnings of insight, or true creativity. In this stage the trial-and-error application of schemata, which was observable during the previous stage, occurs internally (at the level of schemata rather than of motor responses), resulting in the sudden appearance of new effective behaviors (without any observable trial-and-error.) This marks the passage into the preoperational stage.

The role of imitation:
Piaget postulated that imitative activity is the forerunner of mental symbolism.[1] Bodily activity, imitating the action of perceived phenomena, actually builds bodily/behavioral signifiers that stand for phenomena in a comparable way to that by which mental symbols will later stand for these phenomena. Such imitative formations provide the basis upon which mental symbolic activity can later build. The symbol is, according to Piaget, an internalized imitation.
For Piaget, even perception of an object is an imitative activity; the eye tracing the shape of an object is forming a pre-symbolic concept of the object. Piaget suggests that the motions experienced here may be repeated by the child in an abbreviated fashion when recalling the object; this bodily image symbolizes the object that was perceived earlier.

(ii)Preoperational stage
The Preoperational stage is the second of four stages of cognitive development. By observing sequences of play, Piaget was able to demonstrate that towards the end of the second year a qualitatively new kind of psychological functioning occurs. (Pre)Operatory Thought in Piagetian theory is any procedure for mentally acting on objects. The hallmark of the preoperational stage is sparse and logically inadequate mental operations.
According to Piaget, the Pre-Operational stage of development follows the Sensorimotor stage and occurs between 2–7 years of age. It includes the following processes:
Symbolic functioning—characterised by the use of mental symbols, words, or pictures, which the child uses to represent something which is not physically present.
Centration—characterized by a child focusing or attending to only one aspect of a stimulus or situation. For example, in pouring a quantity of liquid from a narrow beaker into a shallow dish, a preschool child might judge the quantity of liquid to have decreased, because it is "lower"—that is, the child attends to the height of the water, but not to the compensating increase in the diameter of the container.
Intuitive thought—occurs when the child is able to believe in something without knowing why she or he believes it.
Egocentrism—a version of centration, this denotes a tendency of a child to only think from her or his own point of view. Also, the inability of a child to take the point of view of others. Example, if a child is in trouble, he or she might cover her eyes thinking if I cannot see myself my mom cannot either.
Inability to Conserve—Through Piaget's conservation experiments (conservation of mass, volume and number) Piaget concluded that children in the preoperational stage lack perception of conservation of mass, volume, and number after the original form has changed. For example, a child in this phase will believe that a string of beads set up in a "O—O—O—O" pattern will have a larger number of beads than a string which has a "OOOO" pattern, because the latter pattern has less space in between Os; or that a tall, thin 8-ounce cup has more liquid in it than a wide, short 8-ounce cup.
Animism The child believes that inanimate objects have “lifelike” qualities and are capable of action. Example, a child plays with a doll and treats it like a real person. In a way this is like using their imagination.

(iii)Concrete operational stage:
The Concrete operational stage is the third of four stages of cognitive development in Piaget's theory. This stage, which follows the Preoperational stage, occurs between the ages of 7 and 11 years and is characterized by the appropriate use of logic. Important processes during this stage are:
Seriation'—the ability to arrange objects in an order according to size, shape, or any other characteristic. For example, if given different-shaded objects they may make a colour gradient.
Classification—the ability to name and identify sets of objects according to appearance, size or other characteristic, including the idea that one set of objects can include another. A child is no longer subject to the illogical limitations of animism (the belief that all objects are animals and therefore have feelings).
Decentering—where the child takes into account multiple aspects of a problem to solve it. For example, the child will no longer perceive an exceptionally wide but short cup to contain less than a normally-wide, taller cup.
Reversibility—where the child understands that numbers or objects can be changed, then returned to their original state. For this reason, a child will be able to rapidly determine that if 4+4 equals 8, 8−4 will equal 4, the original quantity.
Conservation—understanding that quantity, length or number of items is unrelated to the arrangement or appearance of the object or items. For instance, when a child is presented with two equally-sized, full cups they will be able to discern that if water is transferred to a pitcher it will conserve the quantity and be equal to the other filled cup.
Elimination of Egocentrism—the ability to view things from another's perspective (even if they think incorrectly). For instance, show a child a comic in which Jane puts a doll under a box, leaves the room, and then Jill moves the doll to a drawer, and Jane comes back. A child in the concrete operations stage will say that Jane will still think it's under the box even though the child knows it is in the drawer.

(iv)Formal operational stage:
The formal operational period is the fourth and final of the periods of cognitive development in Piaget's theory. This stage, which follows the Concrete Operational stage, commences at around 11 years of age (puberty) and continues into adulthood. It is characterized by acquisition of the ability to think abstractly, reason logically and draw conclusions from the information available. During this stage the young adult is able to understand such things as love, "shades of gray", logical proofs, and values. Lucidly, biological factors may be traced to this stage as it occurs during puberty (the time at which another period of neural pruning occurs), marking the entry to adulthood in Physiology, cognition, moral judgement (Kohlberg), Psychosexual development (Freud), and social development (Erikson). Some two-thirds of people do not develop this form of reasoning fully enough that it becomes their normal mode for cognition, and so they remain, even as adults, concrete operational thinkers.

2-Cognitive Flexibility Theory (R. Spiro, P. Feltovitch & R. Coulson)
Cognitive flexibility theory focuses on the nature of learning in complex and ill-structured domains. Spiro & Jehng (1990, p. 165) state: "By cognitive flexibility, we mean the ability to spontaneously restructure one's knowledge, in many ways, in adaptive response to radically changing situational demands...This is a function of both the way knowledge is represented (e.g., along multiple rather single conceptual dimensions) and the processes that operate on those mental representations (e.g., processes of schema assembly rather than intact schema retrieval)."
The theory is largely concerned with transfer of knowledge and skills beyond their initial learning situation. For this reason, emphasis is placed upon the presentation of information from multiple perspectives and use of many case studies that present diverse examples. The theory also asserts that effective learning is context-dependent, so instruction needs to be very specific. In addition, the theory stresses the importance of constructed knowledge; learners must be given an opportunity to develop their own representations of information in order to properly learn.
Cognitive flexibility theory builds upon other constructivist theories (e.g., Bruner, Ausubel, Piaget) and is related to the work of Salomon in terms of media and learning interaction.
Scope/Application:
Cognitive flexibility theory is especially formulated to support the use of interactive technology (e.g., videodisc, hypertext). Its primary applications have been literary comprehension, history, biology and medicine.
Example:
Jonassen, Ambruso & Olesen (1992) describe an application of cognitive flexibility theory to the design of a hypertext program on transfusion medicine. The program provides a number of different clinical cases which students must diagnose and treat using various sources of information available (including advice from experts). The learning environment presents multiple perspectives on the content, is complex and ill-defined, and emphasizes the construction of knowledge by the learner.
Principles:
1. Learning activities must provide multiple representations of content.
2. Instructional materials should avoid oversimplifying the content domain and support context-dependent knowledge.
3. Instruction should be case-based and emphasize knowledge construction, not transmission of information.
4. Knowledge sources should be highly interconnected rather than compartmentalized.
References:
Jonassen, D., Ambruso, D . & Olesen, J. (1992). Designing hypertext on transfusion medicine using cognitive flexibility theory. Journal of Educational Multimedia and Hypermedia, 1(3), 309-322.
Spiro, R.J., Coulson, R.L., Feltovich, P.J., & Anderson, D. (1988). Cognitive flexibility theory: Advanced knowledge acquisition in ill-structured domains. In V. Patel (ed.), Proceedings of the 10th Annual Conference of the Cognitive Science Society. Hillsdale, NJ: Erlbaum.

3-Cognitive Load Theory (J. Sweller)
This theory suggests that learning happens best under conditions that are aligned with human cognitive architecture. The structure of human cognitive architecture, while not known precisely, is discernible through the results of experimental research. Recognizing George Miller's research showing that short term memory is limited in the number of elements it can contain simultaneously, Sweller builds a theory that treats schemas, or combinations of elements, as the cognitive structures that make up an individual's knowledge base. (Sweller, 1988)
The contents of long term memory are "sophisticated structures that permit us to perceive, think, and solve problems," rather than a group of rote learned facts. These structures, known as schemas, are what permit us to treat multiple elements as a single element. They are the cognitive structures that make up the knowledge base (Sweller, 1988). Schemas are acquired over a lifetime of learning, and may have other schemas contained within themselves.
The difference between an expert and a novice is that a novice hasn't acquired the schemas of an expert. Learning requires a change in the schematic structures of long term memory and is demonstrated by performance that progresses from clumsy, error-prone, slow and difficult to smooth and effortless. The change in performance occurs because as the learner becomes increasingly familiar with the material, the cognitive characteristics associated with the material are altered so that it can be handled more efficiently by working memory.
From an instructional perspective, information contained in instructional material must first be processed by working memory. For schema acquisition to occur, instruction should be designed to reduce working memory load. Cognitive load theory is concerned with techniques for reducing working memory load in order to facilitate the changes in long term memory associated with schema acquisition.
Scope/Application:
Sweller's theories are best applied in the area of instructional design of cognitively complex or technically challenging material. His concentration is on the reasons that people have difficulty learning material of this nature. Cognitive load theory has many implications in the design of learning materials which must, if they are to be effective, keep cognitive load of learners at a minimum during the learning process. While in the past the theory has been applied primarily to technical areas, it is now being applied to more language-based discursive areas.
Example:
In combining an illustration of blood flow through the heart with text and labels, the separation of the text from the illustration forces the learner to look back and forth between the specified parts of the illustration and the text. If the diagram is self-explanatory, research data indicates that processing the text unnecessarily increases working memory load. If the information could be replaced with numbered arrows in the labeled illustration, the learner could concentrate better on learning the content from the illustration alone. Alternatively, if the text is essential to intelligibility, placing it on the diagram rather than separated will reduce cognitive load associated with searching for relations between the text and the diagram (Sweller, 1999).
Principles:
Specific recommendations relative to the design of instructional material include:
1. Change problem solving methods to avoid means-ends approaches that impose a heavy working memory load, by using goal-free problems or worked examples.
2. Eliminate the working memory load associated with having to mentally integrate several sources of information by physically integrating those sources of information.
3. Eliminate the working memory load associated with unnecessarily processing repetitive information by reducing redundancy.
4. Increase working memory capacity by using auditory as well as visual information under conditions where both sources of information are essential (i.e. non-redundant) to understanding.
References:
Sweller, J., Cognitive load during problem solving: Effects on learning, Cognitive Science, 12, 257-285 (1988).
Sweller, J., Instructional Design in Technical Areas, (Camberwell, Victoria, Australia: Australian Council for Educational Research (1999).

3.2 THEORIES INTO PRACTICE

Some Theories into Practice are given below:
1-Aptitude-Treatment Interaction
2-Bandura, Social Learning Theory
3-Behaviorism
4-Contiguity Theory
5-Cooperative Learning
6-Dual Coding Theory
7-Gestalt Theory
8-Piaget
9- The Learning Classroom: Theory Into Practice
10-Situated Learning
The Learning Classroom: Theory Into Practice
Stanford University professor Linda Darling-Hammond hosts a professional development series examining specific learning theories and their applications to classroom practice. The programs include discussions, classroom examples, follow-up assignments, and suggestions for background reading and are designed for teachers of all grade levels as well as college education students.
19 episodes:
Workshop 1. How People Learn: Introduction to Learning Theory
Teacher interviews and classroom footage illustrate why learning theory is at the core of good classroom instruction and demonstrate the broad spectrum of theoretical knowledge available for use in classroom practice.
Workshop 2. Learning as We Grow: Development and Learning
Examines the concept of readiness for learning and illustrates how various developmental pathways—physical, cognitive, and linguistic—play their parts in learning. A 1st-grade teacher, a 7th- and 8th-grade science teacher, and a senior physics teacher are featured, and University of California at Santa Cruz professor Roland Tharp and Yale University professor James P. Comer offer expert commentary.
Workshop 3. Building on What We Know: Cognitive Processing
How prior knowledge, expectations, context, and practice affect processing and using information and making connections. Features a 1st-grade teacher, a 9th- and 10th-grade mathematics teacher, and a special education teacher, with expert commentary from Stanford University professor Roy Pea.
Workshop 4. Different Kinds of Smart: Multiple Intelligences
Harvard University professor Howard Gardner leads an exploration of his own theory of multiple intelligences, describing how people's learning skills differ in significant ways. Teachers who share a class of 5- through 8-year-olds, including several mainstreamed special-needs students, and a 9th- and 10th-grade social studies teacher are featured.
Workshop 5. Feelings Count: Emotions and Learning
Introduces ways to create an emotionally safe classroom to foster learning and to deal effectively with emotions and conflicts that can be obstacles. Features a 5th-grade teacher and an 8th-grade band teacher, with expert commentary from Daniel B. Goleman, author of the book Emotional Intelligence, and Yale University professor James P. Comer.
Workshop 6. The Classroom Mosaic: Culture and Learning
How culturally responsive teaching enables students to create connections, access prior knowledge and experience, and develop competence. A 6th-grade teacher and two 9th-grade teachers are featured along with expert commentary from University of Wisconsin professor Gloria Ladson-Billings and University of Arizona professor Luis Moll.
Workshop 7. Learning from Others: Learning in a Social Context
Based on Lev Vygotsky's work, this session explores how learning relies on communication and interaction with others as communities of learners. It features a 5th-grade teacher and a 9th- through 12th-grade teacher, with expert commentary from Tufts University professor David Elkind, Yale University professor James P. Comer, and University of California at Santa Cruz professor Roland Tharp.
Workshop 8. Watch It, Do It, Know It: Cognitive Apprenticeship
Demonstrates how teachers help their students develop expertise and accomplish complex tasks by modeling, assisted performance, scaffolding, coaching, and feedback. Features a 5th- and 6th-grade teacher and an 11th- and 12th-grade English and social studies teacher, with expert commentary from University of Michigan professor Annemarie Sullivan Palincsar.
Workshop 9. Thinking About Thinking: Metacognition
How thinking about thinking helps students better manage their own learning and learn difficult concepts deeply. The featured teachers are a senior English teacher and a 6th-grade teacher. University of Michigan professor Annemarie Sullivan Palincsar and Lee S. Shulman, president of the Carnegie Foundation for the Advancement of Teaching, provide expert commentary.
Workshop 10. How We Organize Knowledge: The Structure of the Disciplines
Covers the ways in which the organization of knowledge and understanding can influence learning and introduces Bruner's and Schwab's ideas about the structure of the disciplines. Features a 4th-grade teacher, a 10th-grade biology teacher, and a 9th- through 12th-grade teacher, with expert commentary from Lee S. Shulman, president of the Carnegie Foundation for the Advancement of Teaching.
Workshop 11. Lessons for Life: Learning and Transfer
Describes the conditions needed for knowledge and skills learned in one context to be retrieved and applied to a novel situation and what teachers can do to increase the possibility of such transfer. A 4th-grade teacher and a 7th- and 8th-grade teacher are featured; the expert commentator is Lee S. Shulman, president of the Carnegie Foundation for the Advancement of Teaching.
Workshop 12. Expectations for Success: Motivation and Learning
Teachers can enhance their students' motivation by encouraging them to be thoughtfully and critically engaged in the learning process, by supporting their drive for mastery and understanding, and by helping them become self-confident. This program takes a second look at classrooms seen previously to show how motivational techniques work in concert with other learning theories. Stanford University School of Education Dean Deborah Stipek adds her insight.
Workshop 13. Pulling It All Together: Creating Classrooms and Schools That Support Learning
Discusses how schools can organize for powerful learning through a coherent, connected approach to teaching and learning that is reinforced and supported by structural features. Real-life examples feature the staff and students of two schools: a public school in Michigan serving grades 3 through 8 and a first-year charter school in California.

CHAPTER 4


SCOPE AND SEQUENCE OF ELEMENTARY MATHEMATICS CURRICULUM


4.1 EXPECTATIONS FROM ELEMENTARY SCHOOL
CHILDREN
Expectations from Elementary School children are given below:
1. To produce understanding of basic concepts of Mathematics education in the students.
2. To able the students to get information of Mathematics and apply them.
3. To produce sense to study the phenomenon of the world.
4. To cerate the curiosity in the students.
5. To create the understanding and Mathematics behaviour in students.
6. To produce ability of observation.
7. To create ability that they can evaluate result from observation.
4.2 DEVELOPMENT OF MATHEMATICS CURRICULUM THROUGHOUT THE ELEMENTARY LEVEL
Development of Mathematics curriculum throughout the elementary level is explained below:
(A) Knowledge and understanding expectations:
1. Develops the knowledge and understanding of mathematical concepts like number, units of measurement, size, shape, direction, distance, grouping, sub-grouping and fractions.
2. Develops the knowledge and understanding of mathematical facts and processes like place value of numbers, meaning and significance of zero, four fundamental operations, percentage, unitary method, mensuration, etc.
3. Develops the knowledge and understanding of mathematics terms and symbols like digits and numbers, fractions, percentages, etc.
4. Develops the knowledge and understanding of fundamental mathematical relationships.

(B) Skill Expectations:
The student develops the following skills.
1. Ability in counting, reading and writing of numbers.
2. Skill in four fundamental operations dealing with integral numbers and fractions.
3. A reasonable speed, accuracy and neatness in oral and written computational work.
4. Technique of solving problems, which involve elementary mathematical processes and simple calculations.
5. Skill in the use of mathematical tables.
6. Proficiency in making quantitative estimate of size and distance.

(C) Application Expectations:
1. He is able to solve both oral and written mathematical problems independently.
2. He applies elementary mathematical concepts and processes in every day life.

(D) Attitude Expectations:
1. Develops self-confidence for solving elementary mathematical problems.
2. While solving a problem, he tries to read it carefully, analyses it, collects all the known evidences and then draws proper inferences.
3. Develops the habits of neatness, regularity, honesty and truthfulness.
4. Develops the habits of logical thinking and objective reasoning.

(E) Appreciation and Interest Expectations:
1. Develops interest in the learning of mathematics.
2. Appreciates the contribution of mathematicians and gets inspiration from their work.
3. Appreciates the power of computational skills.
4. Appreciates and takes interest in using his knowledge of mathematics in solving problems of daily life.

CHAPTER 5


WRITING BEHAVIORAL OBJECTIVES WITH SPECIAL REFERENCE TO BLOOM’S TAXONOMY

5.1 BEHAVIOURAL OBJECTIVES WITH SPECIAL
REFERENCE TO BLOOM’S TAXONOMY:
36 research worked under the supervisions of Bloom and worked abut educational objectives, In 1956 Bloom divided educational objectives into three domains.

(A) COGNITIVE DOMAIN:
This domain is concerned with mental abilities. Bloom divided it into six subgroups.

1. Knowledge:
Student recognize or recall information, ideas and principles, knowledge means to know about certain material.
Q. What is the capital of Pakistan?
Q. Who wrote “Bangedara”?
Q. What is the Formula to find volume of a Cuboid?
Words typically used: define, recall, recognize, remember, who, what, where, when.

2. Comprehension:
The student has sufficient understanding to organise and arrange material mentally. He can translates, comprehends, interprets informations in his own wads.
Q. What is Cuboid?
Q. If a basket contains 20 apples and another contains 60 apples Then what is the comparison of apples in the two baskets.

Comprehension may be as under:
(i) There is 1:3 in the apples
(ii) First basket contains 40 less apples then that of second basket.
(iii) Second basket contains 40 less apples then that of first basket.
Words typically used: describe, compare, contrast, rephrase, put in your own words, explain the main idea, summarises, illustrate.

3. Application:
A question that asks a student to apply previously learned information to reach an answer, solving math world problems in an example.
Words typically used: apply, classify, use, choose, employ, write, and example, solve, how many, which, what is.
Application of a formula
Q1. Apply the formula of Volume of Cuboid, to find Volume of room having 20m length,12m width and 10m height.
Q 2. Apply the formula A=¶R2 ,to find the area of a circle of radius 7 cm.

Solution:
R = 7 cm
A = ?
We know the formula of area of a circle. We will apply this formula to find area of the circle..


4. Analysis:
The students think critically and in depth about certain material or thing. They may divide the material into simpler parts to justify the meterial. Unless students can be brought to the higher levels of analysis, synthesis, and evaluation, it is unlikely that transfer will take place. If teachers don’t ask higher level questions, it is unlikely that most students will transfer school work to real life. They may not even be able to apply it to school situations other than the one in which it was “learned.” In analysis questions, students are asked to engage in three kinds of cognitive processes:
1. Identify the motives, reasons, and /or causes for a specific occurrence
2. Consider and analyse available information to reach a conclusion, inference, or generalization based on this information.
3. Words typically used: identify motives/ causes, draw conclusions, determine evidence, support, analyse why.

(5) Synthesis:
Higher order question that asks the student to perform original and creative thinking, Synthesis questions ask students to:
1. Produce original communications.
Q. What’s a good name for this invention? Write a letter to the editor on a social issue of concern to you. Make a collage of pictures and words that represents your beliefs and feelings about the issues.
2. Make predictions:
3. Solve problems – although analysis questions may also ask students to solve problems, synthesis questions differ because they don’t require a single correct answer but, instead allow a variety of creative answers. (How could we determine the number of pennies in a jar without counting them? How can we raise money for our ecology project?
Words typically used in synthesis questions: predict, produce, write, design, develop, synthesise, construct, how can we improve, what would happen if, can you devise, how can we solve.
(6) EVALUATION:
A higher level question that does not have a single correct answer. It requires the student to judge the merit of an idea, a solution to a problem, or an aesthetic work. The student may also be asked to offer an opinion on an issue.
Q. Do you think schools are too easy? Is bussing an appropriate remedy for desegregating schools? Differing standards are quite acceptable and they naturally result in different answers. This type of question frequently is used to surface values or to cause students to realize that not everyone sees things the same way, It can be used to start a class discussion. It can also precede a follow-up analysis or synthesis question like, “why?”

(B) AFFECTIVE DOMAIN OF THE TAXONOMY OF
EDUCATIONAL OBJECTIVES:
The Affective Domain addresses interest, attitudes, opinions, appreciations, values and emotional sets.
The original purpose of the Taxonomy of Educational Objectives was to provide a tool for classifying instructional objectives. The Taxonomy is hierarchical (levels increase in difficulty/ sophistication) and cumulative (each level builds on and subsumes the ones below). The levels, in addition to clarifying instructional objectives, may be used to provide a basis for questioning that ensures that students progress to the highest level of understanding. If the teaching purpose is to change attitudes/behaviour rather than to transmit/ process information, then the instruction should be structured to progress through the levels of the Affective Domain.

1. Receiving:
The student passively attends to particular phenomena or stimuli (classroom activities, textbook, music, etc. The teacher’s concern is that the student’s attention is focused. Intended outcomes include the pupil’s awareness that a thing exists.
Sample objectives: listens attentively, shows sensitivity to social problems, Behavioural
Terms: asks, chooses, identifies, locates, points to, sits erect, etc.

2. Responding:
The student actively participates. The pupil not only attends to the stimulus but reacts in some way. Objectives: completes homework, obeys rules, participates in class discussion, shows interest in subject, enjoys helping others, etc. Terms: answers, assists, complies, discusses, helps, performs, practices, presents, reads, reports, writes etc.

3. Valuing:
The worth a student attaches to a particular object, phenomenon, or behaviour. Ranges from acceptance to commitment (e.g., assumes responsibility for the functioning of a group).

Attitudes and appreciation:
Objectives: demonstrates belief in democratic processes, appreciates the role of science in daily life, shows concern for others welfare, demonstrates a problem solving approach, etc. Terms: differentiates, explains, initiates, justifies, proposes, shares etc.

4. Organization:
Bringing together different values, resolving conflicts among them, and starting to build an internally consistent value system – comparing, relating and synthesizing values and developing a philosophy of life.
Objectives: recognizes the need for balance between freedom and responsibility in a democracy, understands the role of systematic planning in solving problems, accepts responsibility for own behaviour, etc.
Terms; Arranges, combines, compares, generalizes, integrates, modifies, organizes, synthesizes, etc.
(C) PSYCHOMOTOR DOMAIN OF EDUCATIONAL
OBJECTIVES:
Instructional objectives and derived questions/ tasks typically have cognitive/affective elements, but the focus is on motor skill development. The suggested areas for use are speech development, reading readiness, handwriting and physical education. Other areas include manipulative skills required in business training (e.g., key boarding), industrial technology, and performance areas in science, art and music, American education has tended to emphasize cognitive development at the expense of affective and psychomotor development. The well rounded and fully functioning person needs development in all three domains. In the psychomotor domain, performance may take the place of questioning strategies in many cases.

1. Reflex Movement:
Segmental, inter-segmental, and supra-segmental reflexes.

2. Basic fundamental Movement:
Locomotor movements, non locomotor movements, manipulative movements.

3. Perceptual Abilities:
Kinesthetic, visual, auditory and tactile, discrimination and coordinated abilities.

4. Physical Abilities:
Endurance, strength, flexibility, and agility.

5. Skilled Movements:
Simple, compound and complex adaptive skills.

6. Non-discursive Communication:
Expressive and interpretive movement.
General objectives: writes smoothly and legibly, accurately reproduces a picture, map, etc. ; operates a (machine) skilfully; plays the piano skilfully; demonstrates correct swimming form; drives and automobile skilfully; creates a new way of performing (creative dance); etc.
Behaviour terms: assembles, builds, composes, fastens, grips, hammers, makes, manipulates, paints, sharpens, sketches, uses, etc. (See Anita Harrow, 1792, for more detail on the psychomotor domain.
CHAPTER 6

METHODS/APPROACHES OF TEACHING MATHEMATICS

6.1 EXPOSITION TEACHING

The exposition teaching method, generally identified as lecture, is considered to be the most efficient way to convey large amounts of information in a short time frame. Exposition with interaction teaching contains both a lecture (or other information dissemination method) and a questioning session.

The textbook-driven lecture is probably the most common teaching method among middle and secondary school teachers, because it requires relatively little teacher preparation, time, or content mastery. Problems are inherent in this method: (1) it allows little flexibility and (2) it is extremely boring.
Exposition teaching, to be effective, must be well planned and carefully timed. The younger and/or less motivated the student, the shorter the lecture portion of the lesson should be. The time limit for lecture should not exceed 20 minutes, and that length should be used only with older and more academically motivated students. The remainder of the lesson should include other techniques: discussion, demonstration, guided practice, peer-teaching, group work.
In preparing the lesson, begin with your specific objectives in mind. These objectives should be shared with the students at the outset; no useful purpose is served by requiring students to guess what the lesson will cover. Consider what the students should be doing while you present the lecture portion of the lesson. Time your activities carefully and be prepared for effective closure to the lesson.
As you present the lesson, you should rely on student feedback (facial expression, body language, general attention to the class) and learn to adjust your timing and presentation style. Organization is crucial to an effective presentation: if you have visuals, they must be accessible; equipment should be set up in advance and in good working order; a backup plan will help you overcome unexpected problems such as a blown projector bulb.* Create a stimulating introduction to the lesson.
* Have visual aids to supplement all lectures; they should guide the learning process, not take the place of instruction.
* Use vocal inflection for emphasis and variety, and speak in standard English.
* Maintain eye contact with students.
* Move about the room.
* Use enthusiasm in your teaching; if you don't express interest in your topic, your students aren't likely to, either.
* Plan an effective closing activity.
MERITS
it has its strengths. It allows the teacher
(1) to present background information,
(2) to create a frame of reference for a unit of study
(3) to summarize an activity, a lesson, or a unit.
DEMERITS
Conversely, exposition teaching can be abused. Its weaknesses include the tendency
(1) to encourage passive learning,
(2) to become boring,
(3) to address only the lowest level of understanding,
(4) to engender discipline problems.

6.2 GUIDED DISCOVERY METHOD

The discovery method is a teaching technique that encourages students to take a more active role in their learning process by answering a series of questions or solving problems designed to introduce a general concept (Mayer 2003 Jerome S. Bruner).
In this method teaching is arranged such that the students derive the principles, formulas, concepts themselves. In this method students collect the information and data themselves. By activities and experiments they reach on certain results.
This method is the developed form of ‘Heuristic Method’ and ‘Inquiry Method’.
Three main principles guided Bruner’s development of this approach:
1. Consideration should be given to “experiences and contexts that motivate the student’s interests.
2. There should be a spiral organization of the material forcing students to build upon previously acquired information.
3. The instruction should “facilitate extrapolation constructivist theory.

Method Outline
A. Examples Manipulation
B. Practice with ore examples
C. New concepts explained/ defined (Hopkins 2002)
Mayer describes these as pure discovery, guided discovery, and expository, respectively.
The Discovery method refers to how much guidance a teacher should give their students. There are three levels of guidance in teaching.
1. Pure Discovery – The student receives representative problems to solve with minimal teacher guidance (Mayer, 2003).
2. Guided Discovery – The student receives problems to solve, but the teacher provides hints and directions about how to solve the problem to keep the student on track (Mayer, 2003).
3. Expository – The final answer or rule is presented to the student (Mayer, 2003).

IMPLICATIONS OF THE DISCOVERY METHOD
Pure Discovery
Pure discovery methods often require excessive amounts of learning time, result in low levels of initial learning, and result in inferior performance on transfer and long term retention (Mayer 68). When the principle to be learned is obvious or when a strict criterion of initial learning is enforced, pure discovery students are likely to behave like guided discovery students. Apparently, pure discovery encourages learners to get cognitively involved but fails to ensure that they will come into contact with the rule or principle to be learned (Mayer 68).

Guided Discovery
Guided discovery may require more or less time than the third, expository instruction, depending on the task, but tends to result in better long term retention and transfer (Mayer 68). Guided discovery both encourages learners to search actively for how to apply rules and makes sure that the learner comes into contact with the rule to be learned (Mayer, 68).

Expository Instruction
Expository Instruction may sometimes result in less learning time than other methods and generally results in equivalent levels of initial learning as compared to guided discovery (Mayer 69). If the goal of instruction is long-term retention and transfer, expository methods seem inferior to guided discovery. Apparently expository instruction does not encourages the learner to actively think about the rule but does ensure that the rule is learned (Mayer, 69).

Merits
· In this method as the students get information themselves, therefore knowledge gained is every lasting.
· Students are motivated to discover more.
· Students acquire learning skills.
· Students attitude towards Science is created.
· Students become active. The teacher only guides.
· Students learn the methods of Science Learning.
· In this method useful knowledge is gained.
Demerits
· Enough trained teachers of this method are not available.
· This method differs from traditional methods.
· In this method most of the time is wasted.
· A wrong discovery may lead to frustration and discouragement.
· Science Laboratories are not fully equipped with instruments.

6.3 CO-OPERATIVE LEARNING

‘Cooperative learning’ is a technique of teaching which has been taken from the educational group activities of ‘John Davie’. In this method students work in small groups on a Project or Assignment.
It is a successful teaching strategy in which small teams, each with students of different levels of ability, use a variety of learning activities to improve their understanding of a subject. Each member of a team is responsible not only for learning what is taught but also for helping team-mates learn, thus creating an atmosphere of achievement. Students work through the assignment until all group members successfully understand and complete it.
This technique develops students’ social skills and promotes student self-esteem.
Cooperative efforts result in participants for mutual benefit so that all group members:
· Gain from each other’s efforts. (Your success benefits me and my success benefits you.)
· Recognize that all group members share a common fate. (We all sink or swim together here.)
· Know that one’s performance is mutually caused by oneself and one’s team members. (We cannot do it without you.)
· Feel proud and jointly celebrate when a group member is recognized for achievement. (We all congratulate you on your accomplishment!)

Elements of Cooperative Learning
It is only under certain conditions that cooperative efforts may be expected to be more productive than competitive and individualistic efforts. Those conditions are:

1. Positive Interdependence (sink or swim together)
· Each group member’s efforts are required and indispensable for group success.
· Each group member has a unique contribution to make to the joint effort because of his or her resources and/or role and task responsibilities.

2. Face-to-Face Interaction (promote each other’s success)
· Orally explaining how to solve problems
· Teaching one’s knowledge to other
· Checking for understanding
· Connecting present with past learning


3. Individual & Group Accountability (no hitchhiking! No social loafing)
· Keeping the size of the group small. The smaller the size of the group, the greater the individual accountability may be.
· Giving an individual test to each student.
· Randomly examining students orally by calling on one student to present his or her group’s work to the teacher (in the presence of the group) or to the entire class.
· Assigning one student in each group the role of checker. The checker asks other group members to explain the reasoning and rationale underlying group answers.
· Having students teach what they learned to someone else.

4. Interpersonal & Small-Group Skills
· Social skills must be taught:
· Leadership
· Decision-making
· Trust building
· Communication
· Conflict-management skills.

5. Group Processing
· Group members discuss how well they are achieving their goals and maintaining effective working relationships.
· Describe what member actions are helpful and not helpful.
· Make decisions about what behaviours to continue or change.

AN EXAMPLE OF GROUP WORK
We give students a project to grow vegetables in the school. We have to make three or four groups of students on their choice. Assign work according to the taste of students. Every group works to complete the assignment. Students consult each other. At the end teacher checks their work. Remedy is taken at the spot to give them courage. In this way work is completed in a team work.

SOME EXAMPLES CLASSROOM COOPERATIVE LEARNING ACTIVITIES
1. Jigsaw – Groups with five students are set up. Each group member is assigned some unique material to learn and then teach to his group members. To help in the learning students across the class working on the same sub-section get together to decide what is important and how to teach it. After practice in these “expert” groups the original groups reform and students teach each other. (Wood, p.17) Tests or assessment follows.
2. Think-Pair-Share – Involves a three step cooperative structure. During the first step individuals think silently about a question posed by the instructor. Individuals pair up during the second step and exchange thoughts. In the third step, the pairs share their responses with other pairs, other teams, or the entire group.
3. Three-Step Interview – Each member of a team chooses another member to be a partner. During the first step individuals interview their partners by asking clarifying questions. During the second step partners reverse the roles. For the final step, members share their partner’s response with the team.
4. Round Robin Brainstorming – Class is divided into small groups (4 to 6) with one person appointed as the recorder. A question is posed with many answers and students are given time to think about answers. After the “think time,” members of the team share responses with one another round robin style. The recorder writes down the answers of the group members. The person next to the recorder starts and each person in the group in order gives an answer until time is called.
5. Three-minute review – Teachers stop any time during a lecture or discussion and give teams three minutes to review what has been said, ask clarifying questions or answer questions.
6. Numbered Heads – A team of four is established. Each member is given numbers of 1,2,3,4. Questions are asked of the group. Groups work together to answer the question so that all can verbally answer the question. Teacher calls out a number (two) and each two is asked to give the answer.
7. Team Pair Solo – Students do problems first as a team, then with a partner, and finally on their own. It is designed to motivate students to tackle and succeed at problems which initially are beyond their ability. It is based on a simple notion of mediated learning. Students can do more things with help (mediation) than they can do alone. By allowing them to work on problems they could not do alone, first as a team and then with a partner, they progress to a point they can do alone that which at first they could do only with help.
8. Circle the Sage – First the teacher polls the class to see which students have a special knowledge to share. For example the teacher may ask who in the class able to solve a difficult math homework question, who had visited Mexico, who knows the chemical reactions involved in how salting the streets help dissipate snow. Those students (the sages) stand and spread out in the room. The teacher then has the rest of the classmates each surround a sage, with no two members of the same team going to the same sage. The sage explains what they know while the classmates listen, ask questions, and take notes. All students then return to their teams. Each in turn, explains what they learned. Because each one has gone to different sage, they compare notes. If there is disagreement, they stand up as a team. Finally, the disagreements are aired and resolved.
9. Partners – The class divided into teams of four. Partners move to one side of the room. Half of each team is given an assignment to master to be able to teach the other half. Partners work to learn and can consult with other partners working on the same material. Teams go back together with each set of partners teaching the other set. Partners quiz and tutor team-mates. Team reviews how well they learned and taught and how they might improve the process.
MERITS
Research has shown that cooperative learning techniques are:
1. To help students to develop skills in oral communication
2. To promote student learning and academic achievement
3. To increase student retention
4. To enhance student satisfaction with their learning experience
5. To develop in students social skills
6. To promote self-confidence.
7. To help the students to promote positive4 relations with class fellows.
8. To develop a sense of knowledge sharing activities.
9. For training of decision making and problem solving.
10. To learn good behaviours of trust building, working relations.
11. To produce sense of feeling responsibilities

6.4 GAMES

Games in the Classroom - Centre for Innovation in Mathematics TeachingWhat research has been done into the use of games in mathematics seems to have been concerned with games that were devised for the teaching, practising and/or reinforcement of specific skills such as negative numbers, coordinates, fractions etc., which are referred to as 'educational games'. Generally, that research has reported good results both in terms of motivation and in the measured progress in learning that took place. The interest here is in games of a type that could well be played by choice to occupy moments of leisure which, to distinguish them from 'educational games', are here referred to as 'mathematical games'.
Computer Games for Mathematical Empowerment Girls and boys play mathematical computer games in mathematical. Sample dialogues of children playing and interacting with games
3-Teaching Mathematics through Games:
I hear, I forget
I see, I remember
I do, I understand
It is fact that knowledge should not be forcefully given to the students but they should obtain knowledge by their own effort.
Students should not be passive listener but should actively take part in the activities arranged by the teachers or directly involved in the activities. Some activities are given below:


GAME No. 1
Sum of the three angles of a triangle is equal to 180°. (Class IV)
Class – V
Aids: i. Triangular cards of different sizes.
ii. Meter rod
iii. Protractor
iv. Scissors
Students should be divided in the groups of ‘5’ students in each group:
Give some triangular cards of different size to each group and ask to measure their angles by the hep of protractor. Result should be written on the Blackboard.


mÐA

mÐB

mÐC
Sum of the three angles
1st group
45°
45°
90°
180°
2nd group
60°
60°
60°
180°
3rd group
60°
40°
80°
180°

Result: Sum of the three angles of a triangle is 180°.

GAME No. 2
Puzzle Game of counting






































There are more than 25 squares of different sizes in the large square shown above. How many squares can you find?
(i) Start counting 1 ´ 1 squares
No. of squares …………
(ii) Now start counting 2 ´ 2 squares
No. of squares …………
(iii) Then start counting 3 ´ 3 squares
No. of squares …………
(iv) Now write sum of squares = …………

GAME No. 3
Less or greater (Class 3)

Multiply by 12 (L.C.M. of 3 and 4)
Both sides of inequality we will get 8 < 9
(i) Take two equal strips.
(ii) Divide one strip into three equal parts and shade first two parts
(iii) Now divide other strip into four parts. Shade first three parts.
(iv) Now compare the two strips. You will differentiate between the two strips and conclude










GAME No. 4
Area of Rectangle (Class V)
In two dimensional figure, measurement of the space in a closed figure is called area. Generally we take unit of area 1 square centimetre i.e. 1 cm2.

1 cm2
1 cm

1 cm
Step – 1 Take a rectangle shown below:
2 cm









6 cm
Count 1 square centimetres in this rectangle.
There are 12 sq. cm.
Hence its area is 12 sq. cm.
2 cm ´ 6 cm = 12 cm2
Step – 2 Take a following rectangle:







3 cm













7 cm
Count 1 square centimetres in this rectangle.
There are 21 sq. cm.
Hence its area is 21 sq. cm.
3 cm ´ 7 cm = 21 cm2


6.5 INVESTIGATION

Inquiry Teaching Model encourages students active investigation. Dictionary meaning of “Inquiry” is “to investigate or to inquire”. In the “Inquiry Teaching Model” students are asked to know or investigate about certain things themselves. This the model of learning through inquiry. Objectives/goals are chosen by the teachers and the students try to achieve them by their own effort. This model is according to the thinking way of “Socrates”. It includes discussions and questioning – answering process. Students collect material/data through different sources. First of all they collect facts or information. Then according to these information they try to reach about the realities. In this way a continuous effort, research and investigation process is carried on. In this way the inquiry habits are created in the students. They always have a critical view of any fact themselves and then admit it. In this way students pass through educational process of getting results.
Learning through inquiry is an integral part of discovery learning, in which the children are given the opportunity to discover for themselves the answers to problems. In the process, the children develop desirable behaviours and learn key concepts and conceptual schemes. When learning through inquiry, the children use their skills (both physical and mental) and previous science background to actively search for and collect information, using whatever methods of investigation that are available or can be devised, which will help them discover the solutions to problems they are attempting to solve.
Learning through inquiry follows a general pattern. First, a question (or series of questions) is raised, the answers to which the children do not know. Though discussion a problem is identified and narrowed until it seems likely that the children can investigate and possible solve the problems. With the help of the teacher the children then propose ways of investigating the problem and of gathering the data, using their mental resources and whatever materials (laboratory, printed, audio-visual, etc.) those are available. Working either individually, in small groups, or as a class, the children now conduct investigations, gather data which they interpret and summarize, and come to conclusions which they evaluate. All this leads to new questions which raise new problems which require new investigations which produce new conclusions and this procedure is then repeated over and over again.
The teacher plays an equally important role while the children are in the process of learning through inquiry. The teacher provides a variety of thought-provoking questions that will start inquiry learning and keep it moving along. The teacher directs the learning activities so that the children can discover for themselves the answers to their questions and problems. This means that the teacher is a guide and a counsellor, not a source or dispenser of knowledge. When inquiry learning bogs down, the teacher does not supply answers, but offers cues instead that will help the children continue with the investigation. The teacher is constantly on the alert to keep the children from jumping to hasty conclusions or getting side-tracked into other problems that are either barely related or unrelated to the original problem being investigated. Finally, the teacher participates actively in the enthusiasm that prevails during the investigations, and shows both excitement and delight when the children discover the answers to their problems.

Steps of “Inquiry Teaching Model”
(i) Information collection
(ii) Sorting out
(iii) Going further
(iv) Making conclusions
(v) Taking action

Principles of “Inquiry Teaching Model”
According to J.S. Broner, there are four principles of Inquiry Teaching Model which are as follows:
1. Acquainting with the aptitude of student
2. The expression should be according to the mental level of student
3. Maintaining the continuity in teaching process
4. Acquiring the knowledge of result.

(1) Acquainting the Aptitude of Student
Purpose of teaching is children growth in every aspect. So it is very important for the teacher to know about his cultural background and his aptitude. Teacher cannot prepare them for study or inquire about the problems without knowing necessary information.

(2) The expression should be according to the mental level of student
Information or problem given to the students should be according to the their mental level. According to “Jarom S. Broner”, we can give knowledge about anything by three methods as follows:
(i) By activity
(ii) By picture and model
(iii) By discussion

(3) Maintaining the continuity in teaching process
Maintaining the continuity in teaching process is very important. If we try to reach on the result at once by leaving some chain steps. Then students will become mentally upset.

(4) Acquiring the knowledge of results
Teacher should encourage the students for searching correct results. Encouragement of the students can be done by any praising remark or by giving them any reward in cash or giving anything.

Merits:
There are several distinct benefits and advantages to be derived when children learn through inquiry. First, the child becomes a participant rather than a spectator. The focus of attentionis on the child, not on the teacher. This eliminates boredom and encourages the children to rely more on their own resources. It also gives children a satisfying sense of accomplishment and promotes their self-confidence. Also, since this technique is highly activity-oriented, it tends to develop competency in the use of process skill, and it encourages the promotion of desirable scientific attitudes. Finally, learning through inquiry is in keeping with the theories of such educational psychologists as Piaget, Bruner, and Gagne on how children develop intellectually and learn.

Demerits:
Learning through inquiry also has its problems as well as advantages. Some psychologists and educators believe that learning through inquiry is very difficult for slow learners because such learners find it hard to persist in tasks that are not immediately fruitful. Others maintain that learning through inquiry is not appropriate for younger children, especially those below the age of nine, because these children do not have the high motivation to master intellectual tasks and because they tend to be impulsive, jump prematurely to conclusions, and fail.

6.6 LABORATORY APPROACH

1. Laboratory is an important place where the students perform experiments which produce working abilities in the students.
2. Practicals lead them to confidence and discoveries.
3. In laboratories students get a chance to work with the instruments themselves.
4. Scientific attitude is produced in the students.
5. Practical activities lead to understanding. Understanding leads to comprehension.
6. Practical activities produce analysis and synthetic qualities in the students.
7. Laboratory work is helpful for the future life of the students.
8. Students become familiar to the instruments used in the laboratories.
9. Science students perform practical activities in problem solving situation.
10. Students collect data and information and try to reach any conclusion.
11. Investigation and inquiry abilities are produced in the students while performing practicals.
12. Laboratory activities lead to students towards realities.
13. Students acquire habit of testing scientific principles, theories and laws.
14. Students are able to look after instruments used in houses and daily life.
15. Students get awareness how to avoid injuries during the laboratory activities.
16. Practicals produce cooperativeness in the students. Because in some practical activities students work in groups.
17. Practical activities produce healthy atmosphere in the laboratories. Students get courage and confidence.
18. Students become habitual of observations. So they work in a systematic way.
19. Students learn scientific skills.
20. Students learn how to take accurate measurements.
21. Research activities are produced by the students.
22. Students are able to make report of practical result.
23. Practical work produce interest in the students.
24. Struggle habit is produced in the students.

STIMULATION STUDENTS DURING LABORATORY
ACTIVITIES
1) Laboratory activities produce learning habits in the students.
2) Practical activities produce interest in the students to work continuously and efficiently.
3) Students should be instructed before starting practical.
4) Students can apply the information and concepts according to the need.
5) Practicals produce the ability of analysis and synthesis in the students. They can obtain correct results.
6) Teacher should always supervise the students during practical work, so that they provide guidance to the student at the spot.
7) Practical activities give the student courage.
8) Students can weigh and measure the things correctly. This gives confidence to the students.
9) Teachers should appreciate the students while performing activities in the laboratory.
10) Prices should be given to the students on their best work.

6.7 PROBLEM SOLVING METHOD

The problem 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 set to the students in a natural way and it is ensured that the students are genuinely interested to solve them.
Mathematics is a subject of problems. Its teaching and learning demands solving of innumerable problems.
Efficiency and ability in solving problems is a guarantee for success in learning this subject.
It pre-supposes the existence of a problem in the teaching-learning situation. A problem is a sort of obstruction or difficulty which has to be overcome to reach the goal.
According to Yoakam and Simpson “a problem occurs in a situation in which a felt-difficulty to act is realised”. It is a difficulty that is clearly present and recognised by the thinker. It may be a purely mental difficulty or it may be physical and involve the manipulation of data. The distinguishing thing about a problem however is that it impresses the individual who meets it as needing a solution. He recognises it as a challenge …”
Life is an arena of problems. L.A. Averill has said, “The only worthwhile life is a life which contains its problems; to live without any longings and ambitions is to live only half-way.”
A human child has to meet and solve problems as he grows-problems which present themselves in his physical surroundings, his intellectual associations and in his social contacts. These problems grow in number and complexity as he grows older and older. His success in life is in large measure determined by the individual’s capacity and competence to solve them. Problems exist for him at every step; his growth, development and living lies in their solution. In school, the child is to be trained in the art and craft of problem-solving.

Procedure:
The procedure of problem method is almost like that of the project method. It can also take the form of inductive-deductive methods. The procedure involves the following steps:
1. Recognising the problem or sensing the problem.
2. Interpreting, defining and delimiting the problems.
3. Gathering data in a systematic manner.
4. Organising and evaluating the data.
5. Formulating tentative solutions.
6. Arriving at the true or correct solution.
7. Verifying the results.
It is a research-like method. It involves scientific thinking as a process of learning. Its relationship with inductive-deductive method is very intimate.
According to exclusive inductive approach:
The method will take the form of the steps – sensing the problem, analysing the situation, organising information, framing solutions, elimination and verification.
According to exclusive deductive approach:
The steps will be – understanding the problem, collecting information, reviewing, drawing inference and verification.
But how to employ this method in the teaching of mathematics?
Example: Finding the volume of a cylinder is a problem before the class.
Its formula has to be developed on the basis of the earlier formula for the volume of a Cuboid
Volume of a Cuboid=Length X Width X Height
= Area of base X Height
While analysing the problem, it gets connected with the previous knowledge that volume of any regular solid can be found by multiplying area of its base with the height of the object. The given information is so organised that it becomes the required information. The area of the base of the cylinder is found by an already known formula and method
A=¶R2
Then the required formula is obtained by multiplying this area with the given height.Therefore:
Volume of cylinder=Area of base X Height
= ¶R2 X h
=¶R2 h
Verification: For the purpose of verification it is applied to a number of similar problems or situations and the results are checked. The solutions to the problems always come from the students. The teacher remains in the background and directs or guides the student activity from that position.

PROBLEM SOLVING STRATEGIES
The problem-solving method that will be used in the following discussion consists of four basic steps:
FIND OUT Look at the problem.
Have you seen a similar problem before?
If so, how is this problem similar? How is it different?
What facts do you have?
What do you know that is not stated in the problem?
CHOOSE A STRATEGY
How have you solved similar problems in the past?
What strategies do you know?
Try a strategy that seems as if it will work.
If it doesn’t, it may lead you to one that will.
SOLVE IT
Use the strategy you selected and work the problem.
LOOK BACK Reread the question.
Did you answer the question asked?
Is your answer in the correct units?
Does your answer seem reasonable?
Specify strategies may vary in name. Most, however, fall into these basic categories:
· Compute or Simplify (C)
· Use a Formula (F)
· Make a Model or Diagram (M)
· Make a Table, Chart or List (T)
· Guess, Check and Revise (G)
· Consider a Simpler Case (S)
· Eliminate (E)
· Look for Patterns (P)
To assist in using these problem-solving strategies, the answers to the Stretches, Warm-Ups and Workouts are coded to indicate possible strategies. The single-letter codes above for each strategy appear in parentheses after each answer.
In the next section, the strategies above are applied to previously published MATHCOUNTS problems. The following examples model possible approaches to teaching problem solving.

EXAMPLE I
Given (63)(54) = (N)(900), find N.
FIND OUT What are we asked? The value of N that satisfies an equation.
CHOOSE A STRATEGY
Will any particular strategy help here? Yes, factor each term in the equation into primes. Then, solve the equation noting common factors on both sides of the equation.
SOLVE IT Break down the equation into each term’s prime factors.
63 = 6 ´ 6 ´ 6 = 2 ´ 2 ´ 2 ´ 3 ´ 3 ´ 3
54 = 5 ´ 5 ´ 5 ´ 5
900 = 2 ´ 2 ´ 3 ´ 3 ´ 5 ´ 5
Now (63)(54) = (N)(900)
6 ´ 6 ´ 6 X 5 ´ 5 ´ 5 ´ 5=(N)(900)
2 ´ 2 ´ 2 ´ 3 ´ 3 ´ 3´ 5 ´ 5 ´ 5´ 5= (N)( 2 ´ 2 ´ 3 ´ 3 ´ 5 ´ 5)

Two 2’s and two 3’s from the factorisation of 63 and two 5’s from the factorisation of 54 cancel the factors of 900.
The equation reduces to
2 ´ 3 ´ 5 ´ 5 = N
So N = 150
LOOK BACK Did you answer the question? Yes.
Does our answer make sense? Yes, since 900 = 302 = (2 ´ 3 ´ 5)2, we could have eliminated two powers of 2, 3 and 5 to obtain the same answer.

EXAMPLE - 2:
Finding the volume of a cylinder is a problem before the class.
To find: What is required? We have to find volume of a cylinder.
CHOOSE A STRATEGY:
We know a similar sort of problem.Its formula has to be developed on the basis of the earlier formula for the volume of a cuboid.
Solve it:
volume of a cuboid=Length x Width x Height
=Area of base x Height
While analysing the problem, it gets connected with the previous knowledge that
Volume of any regular solid can be found by multiplying area of its base with the height of the object.
The given information is so organised that it becomes the required information.
The area of the base of the cylinder is found by an already known formula ¶ r2.
Then the required formula is obtained by multiplying this area with the given height. Thus:
Volume of cylinder= Area of base x Height
= ¶ r2 x h
= ¶ r2h
Verification: For the purpose of verification it is applied to a number of similar problems or situations and the results are checked. The solutions to the problems always come from the students. The teacher remains in the background and directs or guides the student activity from that position.

Merits:
1. Problem-solving in schools prepares the pupils to solve the problems of life. It approximates to life. Facing and solving problems is the true nature of life itself.
2. The method involves reflective thinking. Therefore it stimulates thinking, reasoning and critical judgement in the students.
3. The pupils learn by problem-solving which is the method of learning by self effort.
4. It develops qualities of initiative and self-dependence in the students as they are to face the problematic situation themselves.
5. It is a stimulating method. The problem is a challenge. Once it is properly recognised, it acts as a great motivating force and directs the students’ attention and activity.
6. It is especially suitable for mathematics which is a subject of problems.
7. In it there is strong motivation, tension and mental activity which are the conditions for effective learning.
8. It serves individual differences. There are no limits on student achievement. He can solve any number of problems in a specified time and make progress accordingly.
9. It develops desirable study habits in the students. They get engaged in analysis of the problem, reflective thinking, systematic data gathering, verification and critical study.
10. It is a method of experience-based learning. Problem-solving by self-effort is an experienced of its own type. Such an experience is found missing in the lectures and reading from the textbooks.
11. There is possibility of close contact between the teacher and the taught. Every student needs individual guidance from the teacher. The teacher comes to know the difficulties which the students face and helps them accordingly.
12. The students get valuable social experience like patience, cooperation, self-confidence, etc.

Its Limitations/Demerits:
1. Its limitations are largely due to its ineffective use. There are, otherwise, no limitations inherent in it. “If the teacher is not able to think reflectively, does not have an attitude of critical enquiry; or when the classroom situation is dominated by him, and the atmosphere is that of recitation and of readymade answers, the problem-solving method is going to fail.”
2. It is difficult to organise the contents according to the requirements of this method. It is difficult to frame really good problems and to introduce them at every step.
3. It is a time-consuming method. The progress of the students is bound to be slow.
4. All the topics and subject areas cannot be covered by this method.
5. The method does not suit the students of lower classes. They do not possess enough background for scientific approach to problems.
6. When the structure of the problem itself is not up to the mark, or when the problems chosen are unreal and artificial, the method will not be applicable.
7. Textbooks written in the traditional style do not help in the use of this method. There is absence of suitable books for reference and guidance.
8. Teacher’s burden becomes heavier. Real, scientific approach to problem necessitates a lot of study and preparation on his part.
9. It is an intellectual approach in learning. Mental activity dominates in this method and there will be neglect of physical and practical experience.
10. In case the assigning of problems or proposing of problems becomes the teacher’s main job, then the procedure obviously smacks of authority, spoon-feeding and artificiality.

6.8 PROBLEM POSING METHOD

A problem is an exercise and something novel. Sometimes it is compared with a project. Mathematics differs from other school subjects, as it provides an opportunity of solving problems which evoke thinking. Training children to solve problems is meant for training them to meet and surmount difficulties, and for enabling them to solve problems offered by life itself. But the proposed approach to the nature and scope of problems has been misdirected to some extent. The problems so far framed and proposed have served no better purpose than making trapped-in-the-school boys solve them.
Suitability of the problem is very important. Generally the text-book writers have followed the traditional line in this respect. Some problems have become a “must” in every text-book, and those are very important also for examination purposes. Since the text-book writers have failed to deliver the goods therefore the organiser has to show them the way. The syllabus should contain model problems on its topics and text-book must be written accordingly. Every problem should be a novelty and should pose a new challenge to the thinking capacity of the pupil.

Teacher to Teacher: Using Problem-Posing Dialogue in Adult Literacy Education by Sarah Nixon-Ponder
A group of women are gathered around a table, deep in discussion. Spread sheets with schedules, open notebooks with lists, copies of government documents, and a diagram with measurements of a living space are spread before them. The women are discussing several options, looking earnestly at the pros and cons of each, and speaking in detail on specific aspects of one option. While two women are searching for a specific reference in the government documents, another is rapidly taking notes on the discussion at hand. In all aspects, this appears to be a professional business planning meeting, right? Close, but not quite. This is a group of women in an adult literacy class who have arrived at a solution to their childcare situation by using a process called problem-posing dialogue.
Problem-posing is a tool for developing and strengthening critical thinking skills. It is an inductive questioning process that structures dialogue in the classroom. Problem-posing dialogue is rooted in the works of Dewey and Piaget who were strong advocates for active, inquiring, hands-on education that resulted in student-centered curricula (Shor, 1992). Freire (1970) expanded on the idea of active, participatory education through problem-posing dialogue, a method that transforms the students into "critical co-investigators in dialogue with the teacher" (p. 68).
Learners bring to adult education programs a wealth of knowledge from their personal experiences, and the problem-posing method builds on these shared experiences. By introducing specific questions, the teacher encourages the students to make their own conclusions about the values and pressures of society. Freire (1970) refers to this as an "emergence of consciousness and critical intervention in reality" (p. 68).
So how is this done? What does it look like? What is the final outcome? Let's take a look at these questions as we walk through the process of problem-posing.
HOW TO CONDUCT PROBLEM-POSING DIALOGUE
Problem-posing begins by listening for students' issues. During breaks, instructors should listen to students' conversations with one another and make notes about recurring topics. Based on notes from these investi-gations, teachers then select and bring the familiar situations back to the students in a codified form: a photograph, a written dialogue, a story, or a drawing. Each situation contains personal and social conflicts that are of deep importance to the students.
Teachers begin by asking a series of inductive questions (listed below) which moves the discussion of the situation from the concrete to the analytical. The problem-posing process directs students to name the problem, understand how it applies to them, determine the causes of the problem, generalize to others, and finally suggest alternatives or solutions to the problem. The "responsibility of the problem-posing teacher is to diversify subject matter and to use students' thought and speech as the base for developing critical understanding of personal experience, unequal conditions in society, and existing knowledge" (Shor, 1992, p. 33).
FIVE STEPS OF PROBLEM-POSING
Auerbach (1992) has simplified the steps of problem-posing. Problem-posing is a means for teaching critical thinking skills, and many adult learners need the initial structure these steps provide in order to build confidence and esteem in their ability to think critically. When beginning to problem-pose, it is important to spend time on each step, for these are all essential components in learning how to critically think about one's world.
The teacher presents the students with a code. Codes are a vital aspect of problem-posing. They must originate from the students' concerns and experiences, which makes them important to the students and their daily lives. According to Wallerstein (1983), codes can be written dialogues, taken from a variety of reading materials, that directly pertain to the problem being posed. role-plays adapted from written or oral dialogues. stories taken from the participants' lives and experiences. text from newspapers, magazines, com-munity leaflets, signs, phone books, wel-fare or food stamp forms, housing leases, insurance forms, school bulletins, etc. pictures, slides, photographs, collages, drawings, photo-stories, or cartoons.
After the students have studied the code, the teacher begins by asking questions, such as: What do you see in the picture (photograph, drawing, etc)? What is happening in the picture (photograph, drawing, etc)? or What is this dialogue (story, article, message) about? What is happening in the dialogue (story, article, message)?
The students uncover the issue(s) or problem(s) in the code. Teachers may need to repeat the following questions: What is happening in the picture (photograph, drawing, etc)? What is happening in the dialogue (story, article, message)? Students may identify more than one problem. If this occurs, the teacher should ask the students to focus on just one problem (especially with beginning problem-posers), using the other problem(s) for a future problem-posing idea. Students may identify two problems or issues that cannot be separated and must be dealt with together. This, too, is acceptable just as long as it is the students' decision to work with the two problems together.
At this point, the teacher becomes the facilitator of the discussion, thus guiding the students to talk about how this problem makes them feel and what the problem makes them think about, so that they can internalize the problem. Through discussion, the students will relate the issue(s) or problem(s) to their own lives and cultures. The facilitator should assure that all students are given the chance to share their experiences, understanding as well that some may choose not to share. No one should be made to speak if she/he does not feel comfortable doing so. Learning that others have been in similar situations is very important; this experience will serve as an affirmation to their experiences, lives, and cultures; as an esteem builder; and as a means for bonding with other learners and the facilitator.
The facilitator guides the students toward a discussion on the social/economic reasons for the problem by asking them to talk about why there is a problem and how it has affected them. During this step, it is critical for the facilitator not to expound upon personal and political beliefs. This temptation may be very strong during problem-posing dialogue, but resistance to do so is absolutely vital to the growth of the students. Because students' beliefs may differ greatly from those of the facilitator's, students will be more apt to take risks and openly share their beliefs if they believe that this is their dialogue and they have ownership in its process.
Problem-posing is more than a technique that teaches critical thinking; it is a philosophy, a way of thinking about students and their ability to think critically and to reflect analytically on their lives. Eduard Lindeman, one of America's founding fathers of adult education, firmly believed that the responsibility of adult education was to teach learners how to think analytically and critically; this, too, is the role of problem-posing.

PROBLEM-POSING AS A NEW CONCEPT
This was a new concept for most of my students. Some students had a difficult time with the non-traditional format of the class structure. Most were not used to being asked their opinions or beliefs. They did not believe in themselves; they did not believe that they were capable of helping to build the curriculum of a class, of their class. And of course, a few grasped this idea whole-heartedly and ran with it from the beginning. They became leaders, and they accepted the challenge to change their education. They also helped the others to see the benefits of problem-posing dialogue and the importance of learning to think critically.
As the instructor, I had to learn the art of facilitating the discussions and the cooperative groups. I had to learn to let go of power and control and turn it over to my students, thus becoming a facilitator who guides, shares, and coaches. Problem-posing taught me to trust students, to trust in their abilities, to rely on their resourcefulness and experiences, and to make learning meaningful to them.
Problem-posing enables students to bring to the program their experiences, cultures, stories, and life lessons. Their lives are reflected in the thoughtful, determined, and purposeful action that defines problem-posing dialogue. Moreover, problem-posing is a dynamic, participatory, and empowering philosophy that teaches students how to critically think and analytically examine the world in which they live.
MERITS
1-Problem-posing delves deeply into any issue or problem, demonstrating the extent of its social and personal connections.
2- Problem-posing "focuses on power relations in the classroom, in the institution, in the formation of standard canons of knowledge, and in society at large"
3- It challenges the relationship between teacher and student and offers students a forum for validating their life experiences, their cultures, and their personal knowledge of how their world works. Problem-posing is dynamic, participatory, and empowering.
4- Through the use of problem-posing, student-led debates, small group cooperative learning projects, journaling, field experiences, peer-teachings, and electronic portfolio assessment, a methods class can be transformed into an empowering vehicle for student learning as well as help students foster a greater sense of ownership in their future profession.
5- problem-posing builds confidence and community among learners
6- Problem-posing enables students to bring to the program their experiences, cultures, stories, and life lessons.
7- This is problem-posing in action. It is exciting and educational; it is cross-cultural and multicultural because it draws from all of the students' cultures. Additionally, problem-posing builds confidence and community among learners.


6.9 SIMULATION METHOD

SIMULATION
Simulations are a useful teaching strategy for illustrating a complex and changing situation. Simulations are (necessarily) less complex than the situations they represent.
In a simulation, the learner acts, the simulation reacts, the learner learns from this feedback. Examples of simulations: car and flight simulators, SIM City, Monopoly, mock elections, model UN. Note that in each of these cases, the “game” involves rules, and the students must make decisions. Each decision a student makes affects the outcome of the game.
For the students to learn what you intend for them to learn from the simulation, you must hold a discussion during and/or after the game. This is integral to the students' learning. There is so much we could have learned from playing Monopoly that went right through our heads because there was no discussion about what it all meant. (Not that there isn't time to play without focusing on learning. But that kind of play takes place outside of classrooms, not in them.)
Practicalities
Phase One: Orientation
- Explain to your students what simulations are about and for. (If you mention some common games they play which are simulations, they might start thinking about what real life complex situations the games model, and might learn something about them.)
- Describe the particular simulation.
- Ensure the students understand the purpose of the simulation.
- Outline the rules for the students. I put the rules on an overhead, and leave the overhead on during the simulation. You could also write the rules on bristle board, and hang this in a conspicuous place during the activity.
- Assign roles to the students.
Phase Two: The Simulation
- The students participate in the game, playing their roles as assigned. You, are the coach and referee. You should stay uninvolved, except when you notice that you can facilitate the educational opportunities the simulation presents.
- While your students are playing, you could make anecdotal records, or fill in checklists.
Phase Three: Debrief
For every teaching strategy involving a debrief, I will suggest a different method. There are a number of ways in which debriefs can be done. Please mix and match the different forms of debriefs you use.
- Put the students into small groups.
- Choose three or four learning objectives for the simulation. Write up these learning objectives as questions for discussion. One question should be about how the students think the simulation is like the real thing and how it is not like the real thing. Give each small group of students one question to discuss.
- Tell the students how much time they have to discuss the questions.
- Five minutes before the time is up, visit each group with a card which has written on it: Five minutes until presentation. “Choose a speaker and write a summary of your discussion for the speaker to present to the class.”
- An alternative to the above method would be to put groups who have discussed different question together to discuss their different questions and answers. This way, each group has an opportunity to discuss at least two of the questions.
- If you use this second method, you could have students write answers to the questions in a learning log instead of having them present to the class.
Examples of Simulations:
To illustrate the complexity of scientists at work constructing knowledge, have small groups of students assemble different parts of a jig saw puzzle.
To illustrate the variety of factors involved in animal survival, there are many simulations available in the Project Wild Activity Book. For example, one involves bears preparing for winter. The teacher drops a number of food cards around the area where the students will be playing. Some bears are given handicaps. For example, one bear is blind, so is blindfolded. One is lame, so must never run - only walk. Some bears have young, so must collect twice as much food as others. At the end of the game, tally up how many food points each bear has collected. The blind, and lame bears are unlikely to have as many points as the healthy ones, but they might. Etc.
MERITS
1- simulation based training generally focuses on applying knowledge
2-, simulation-gaming can be more effective than traditional methods of instruction in facilitating positive attitude change toward the subject and its purposes.”
3- There are many reports that simulation games significantly
Increase the motivation and interest level of student players
4- simulation/game seminar outperforms
a conventional seminar with respect
to all aspects of [Kolb’s] learning cycle
5- Simulators can effectively identify errors and appropriateness of decision making
6- Students learned as much from a simulation as they did
from actual hands-on experience.
7- Social psychology students learn more with simulations than they do with lectures
8- Simulation led to better performance than training using a traditional textbook approach

6.10 INDUCTIVE METHOD
It leads from concrete to abstract, particular to general and from examples to formula. It is the method of constructing a formula with the help of a sufficient number of concrete examples. It is based on induction which means proving a universal truth by showing that if it is true for a particular case and is further true for a reasonably adequate number of cases, it is true for all such cases. A formula or generalisation is thus arrived at through a convincing process of reasoning and solving of problems. After a number of concrete cases have been understood, the student successfully attempts the generalisation.

Procedure:
Example 1. Ask the students to construct a few triangles. Let them measure and sum up the angles in each case.
Result should be written on the Blackboard.


mÐA

mÐB

mÐC
Sum of the three angles
1st group
45°
45°
90°
180°
2nd group
60°
60°
60°
180°
3rd group
60°
40°
80°
180°

The sum will be the same in all the cases. Thus they can safely conclude that the sum of angles of a triangle is equal to two right angles.
Result: Sum of the three angles of a triangle is 180°.

Example 2. Give them a number of cases e.g.,
a + b, x + y, l + m and p + q,
and ask them to find the squares in each case, by the method of multiplication. They can further be helped to generalise on the basis of these conclusions that
(1st term + 2nd term)2 = (1st term)2 + (2nd term)2
+ 2(1st term) ´ (2nd term)
Example 3. Give 3, 5 and 7 things to three students individually, and then ask them to divide the things equally among themselves. For this purpose they will first calculate the total number of things and then divide the total number by the number of students. Again give 3, 6, 7 and 8 things to four students individually. To divide the things equally among themselves, they will first calculate the total number of things and then divide this total by the number of students. Another similar concrete case may be taken. The generalisation can be made on these cases. It may be introduced to them that this equal quantity is known as the average. To calculate this average they have to first calculate the sum of the given quantity, and then to divide this sum by the number of quantities. Thus the following formula or generalization is arrived at:
There can be many more examples to illustrate the procedure to be followed in this method.
Merits of the Inductive Method
1. It helps understanding. It is easy to understand a mathematical principle established through a number of simple examples. Any doubts about the “how and why” of a formula are clarified in the very beginning.
2. It is a logical method. So it suits mathematics.
3. It gives the opportunity of active participation to students in the discovery of formula.
4. It is based on actual observation, thinking and experimentation.
5. It curbs the tendency to learn things by rote, and also reduces homework.
6. As it gives freedom from doubts, and helps in understanding, it suits the child.
Drawbacks of the Inductive Method
1. It is limited in range. It contains the process of discovering the formula with the help of a sufficient number of cases, but “what next?”, is not provided in it. The discovery of a formula does not complete the study of the topic. A lot of supplementary work and practice is needed to fix the topic in the mind of the learner.
2. Inductive reasoning is not absolutely conclusive. Three or four cases are picked up to generalise an observation. Therefore the process establishes a certain degree of probability which can, of course, be increased and made more valid by increasing the number of cases.
3. It is likely to be more laborious and time consuming.
4. At the advanced stage, it is not so useful as some of the unnecessary details and explanations may make teaching dull and boring.
5. Its application has to be restricted and confined to understanding of rules in the early stage. Once a formula has been established, time should not be wasted in rediscovering it for every subsequent problem.

6.11 DEDUCTIVE METHOD

It is the opposite of Inductive Method. Here the learner proceeds from general to particular, abstract to concrete, and formula to examples. A pre-constructed formula is told to the students and they are asked to solve the relevant problems with the help of that formula. The formula is accepted by the learners as a pre-established and well-established truth.

Procedure:
Immediately after announcing the topic for the day, the teacher gives the relevant formula. To explain further the application of the formula to problems, he solves a number of problems on the blackboard. The students come to understand how the formula can be used or applied. Then a few problems are given to the students. They solve them on the same lines as have been explained by the teacher. A few important formulae are given below:
and profit or loss is always calculated on the cost price. The students apply these formulae to solve the problems, and then memorise them for future use.
Examples of Deductive Method
Example 1 Find the area of a circle of radius 7 cm.

Solution:
R = 7 cm
A = ?
We know the formula of area of a circle. We will apply this formula to find area of the circle. Therefore this method of finding area will be deductive method.
Example 2 The price of 12 books is Rs.336. Find the price of 20 books.
Solution:
We know unitary method. We shall the apply formula of unitary method to find the price of one book.

Merits of the Deductive Method
i) It is short and time-saving. The solving of problems by pre-determined formulae takes little time. Authors and teachers, therefore like to adopt it (give it preference over others).
ii) It glorifies memory, as students have to memorise a considerable number of formulae.
iii) At the “practice and revision” stage, this method is adequate and advantageous.
iv) It combines with the inductive method to remove the incompleteness and inadequacy of the later.
v) It enhances speed and efficiency in solving problems.
vi) Its Drawbacks
1. It is very difficult for a beginner to understand an abstract formula if it is not preceded by a number of concrete instances.
2. Pure deductive work requires a formula for every type of problems and an extensive use of this method will demand blind memorisation of a large number of formulae.
3. It will thus cause an unnecessary and heavy burden on the brain. It may even result in brain fag.
4. Memory becomes more important than understanding and intelligence, and that is educationally unsound.
5. If the pupil forgets the memorised formula, which is very likely to happen in case of blind cramming, he is at a loss and cannot recollect and reconstruct the formula easily.
6. The students cannot become active learners.
7. It is not suitable for the development of thinking, reasoning and discovery.

Comparison and Contrast of Inductive and Deductive Method
As we know the scope, merits and drawbacks of these two methods. It leads us to conclude that Inductive Method is very important for the concept formation and deriving formulas. It is necessary for the learners. This method is predecessor of Deductive Method.
Inductive method takes enough time but we cannot ignore it. After deriving any formula we use Deductive method which is short and time saving.
Teachers and students like to adopt it. We can say that the Deductive method gives a good follow up. Any loss of time due to slow speed of inductive method is covered by the Deductive method.
There may be many arguments against deductive method but it cannot be ignored. It is to serve as the complement of inductive method.
The two methods are such good partners that the demerits of the one are offset by the other.
Deductive method is a process suitable for final statement and Inductive method is most suitable for the exploration of new fields.
Inductive method is the work of induction and the Deductive method is the work of applying the induction. Application and practice should be preceded by understanding. Blind practice is dangerous.
Thus the teachers should begin with Inductive method and end by Deductive method.

6.12 ANALYTIC METHOD

The Analytical Method
Analysis means literally to break a complex problem down into smaller, more manageable "independent" parts for the purposes of examination. To begin from ‘To PROVE’ and to proceed towards ‘Given’. This method of solution is called ‘ The Analytic Method’.
This method is used in the class room to find solution of any problem. Result is attained by connecting different parts. This method of solution is lengthy but clear having logic of solution.
It proceeds from unknown to the known facts. It is a process of thinking. It answers satisfactorily any question.

SOLTION METHOD
GIVEN: ‘A’ is true.
TO PROVE: ‘D’ is true.
Analytical method will be as under:
‘D’ is true if ‘C’ is true.
‘C’ is true if ‘B’ is true.
‘B’ is true if ‘A’ is true.
But it is given that ‘A’ is true. Hence ‘D’ is true.

DPD addressed this problem of conceptual pluralism by using what it called the analytical method: a method for making decisive, rational choices among alternative conceptualizations. The analytical method makes the process of conceptualization systematic because it involves thinking clearly about fundamental theoretical requirements

Merits:
1- In this method every step is clearly explained.
2- This method is just like ‘ Discovery Method’.
3- In this method solution quite obvious.
4- Every part has logic.
5- Difficult results become easier.
6- Confidence is developed.
7- It is process of thinking.

6.13 SYNTHETIC METHOD

Synthetic is usually used in the sense of synthesis, the combination of two or more parts, whether by design or by natural processes. To combine different parts is ‘Synthesis’. To begin from ‘Given’ and to proceed towards ‘To Prove’. This method of solution is called ‘ The Synthetic Method’. Solution by this method is always short but not clear and not logical.
It proceeds from known to unknown. It is process of presentation of the previously discovered facts. It is quick and straightforward method. It does not satisfies the doubts. It is a method of crammer. It is not easy to recall. It develops memory.

SOLTION METHOD
GIVEN: ‘A’ is true.
TO PROVE: ‘D’ is true.
Analytical method will be as under:
Because ‘A’ is true so ‘B’ is true.
Now because ‘B’ is true so ‘C’ is true.
Now because ‘C’ is true so ‘D’ is true.

Merits:
1- This method is short.
2- It is easy to grasp.
3- Labour is required to get result.
4- No risk of continuity.
5-It is time saving method.
6- It develops memory.
7- It is quick and straightforward method.


DEMERITS
· In this method solution is not obvious.
· Solution is not logical.
· If solution is forgotten. It is not easy to recall.
· It is a method of cramming.
· It does not satisfies the doubts.