EDU 04.7 Seminar [ Development of Mathematics ]

EDU 04.7: THEORETICAL BASE OF MATHEMATICS EDUCATION

 UNIT  I

NATURE AND DEVELOPMENT OF MATHEMATICS EDUCATION

Topic : 

Development of Mathematics 

• Human Needs as the Basis of Growth of Mathematics. 
• Mathematics - as a Structured Science.

1. Human Needs as the Basis of Growth of Mathematics. 

Mathematics, like everything else that man has created, exists to fulfill certain human needs and desires. It is very difficult to say at what point of time in the history of mankind, and in which part of the world, mathematics had its birth. More than 2,000 years before the beginning of the Christian era, both the Babylonians and the Egyptians were in possession of systematic methods of measuring space and time. They had the knowledge of basic geometry and basic astronomy. This rudimentary mathematics was formulated to meet the practical needs of an agricultural population. Their geometry resulted from the measurements made necessary by problems of land surveying.      Units of measurement, originally a stone or a vessel of water for weight, eventually became uniform over considerable areas under names which are now almost forgotten. Similar efforts occurred in early times in the southern part of Central Asia along the Indus and Ganges rivers and in Eastern Asia.

Projects related to engineering, financing, irrigation, flood control, and navigation required mathematics. Again a usable calendar had to be developed to serve agricultural needs.  Zero was defined and this at once led to positional notations for whole numbers and later to the same notation for fractions. The place value system which eventually developed was a gift of this period. These achievements and many more of a similar nature are the triumph of the human spirit. They responded to the needs of human society as it became more complex.

The men who shaped the stones in erecting the Temple of Mathematics were widely  scattered, a few in Egypt, a few in India, and yet others in Babylon and China. These workmen confronted nature and worked in harmony with it. Mathematics is something that the man has himself created to meet the cultural demands of time. Nearly every primitive tribe invented words to represent numbers. But it was only when ancient civilizations such as the Sumerian, Babylonian, the Chinese and the Mayan developed trade, architecture, taxation and other civilized contracts that the number systems were developed. Thus, mathematics has grown into one of the most important cultural components of our society. Our modern way of life would hardly have been possible without mathematics. Imagine trying to get through the day without using a number in some manner or the other !

 A degree of estimation, not only in money but in weights and measures, is very     important. Many of our daily routine chores involve sorting, ordering and organizing         processes. We handle many mechanized devices which require geometrical or spatial skills. For travel, reading of maps, diagrams, interpreting scales becomes an essential part of our intellectual equipment. A knowledge of methematics is useful to understand and interpret matters such as incoine tax and read information presented to us by the mass media in numerical form or in the form of graphs and understand the use of   phrases such as rising prices, index, per capita income, inflation, stock market index etc. in ordinary day to day language. It is not necessary to provide an exhaustive list to prove the case in favour of "mathematics for survival" or "useful mathematics". 

Conclusion

Majority of the basic branches of mathematics grew out of the daily life needs. For example, arithmetic and basic algebra grew out of the need for counting and other simple operations required to solve daily life problems. Human quest for possession of land and other properties needed measurement which led to the invention of geometry and trigonometry. Growth of geometry and trigonometry is also due to the curiosity and quest for understanding the Universe and happenings around us. Requirement for understanding the processes in physics led to the invention and growth of calculus. Many new branches of mathematics emerged out of necessity to solve problems faced by scientists, social scientists, commerce and trade organisations as well as warfare experts. Mathematical ideas emerging out of all these sources contribute towards the nature of mathematics.

2. Mathematics - as a Structured Science.

To understand more about nature of mathematics, we should perceive relationships between its constituents. For instance, we should know the meaning of the most basic terms called ‘undefined terms,’ then use these undefined terms to ‘define’ new ‘terms’ and then develop ‘axioms’ using these terms that form the foundations of mathematical theory. This way, we arrive at collection of mathematical terms and mathematical axioms. By applying inductive reasoning, we arrive at generalisations known as         propositions. We establish the truth of these propositions by using the rules of logic and call them ‘Theorems’. The method of establishing the truth of a proposition and making it a theorem is known as mathematical proof or simply a ‘proof.’ This nature of Mathematical process is known as deductive nature of mathematics.

• Undefined Terms

A minimal set of mathematical terms whose meaning we take for granted without explaining in terms of other mathematical terms, and give meaning of other mathematical terms in terms of members of this minimal set of mathematical terms are known as ‘undefined terms.’

For example, what is a rational number? ‘A rational number is of the form p/q  where p and q are integers and q is not equal to zero.’ Meaning will not be known unless you know what are integers. ‘So what are integers?’ ‘Natural numbers, zero and their negatives put together are integers.’ Here unless we know what are natural numbers, zero and the negatives of natural numbers, we will not understand the meaning of integers. Do we know atleast the meaning of natural members? We only give examples of a few natural numbers, say 1, 2, 3, etc… Do these answers give complete meaning of natural numbers? Either we have to assume that we know what natural numbers are or we have to give their meaning ‘axiomatically’ which we will discuss later. Thus, we take a ‘natural number’ as an undefined term.   In geometry, point, line, plane, etc. are taken as undefined terms.

• Definitions

The definition of a term is a characteristic explanation involving terms which are either undefined terms or terms which have already been defined. The word ‘explanation’ here is used in the meaning of common language usage. But what is a ‘characteristic explanation’? A characteristic explanation of a term is an explanation which characterises the given term, i.e., the explanation which is true for and only for the given term.

Example: 1. An equilateral triangle is a triangle in which all sides are equal.

                  2. Common divisor of two integers is a number which divides both the given

                      integers.

• Propositions

A ‘Proposition’ or a ‘Statement’ is a grammatically correct declarative sentence which makes sense, which is either true or false, but not both. If a ‘Proposition’ is true, we say that its truth value is ‘True’ and if it is false, we say that its truth value is ‘False’.

For example,

‘Sum of  2 and 5 is 7’

is a Proposition which is True, but

‘Sum of 2 and 5 is 6’

is also a Proposition which is False. However,

‘Is 7 the sum of 2 and 5?’

is not a Proposition, because it is not a declarative sentence as it does not declare anything, but is only an interrogation.

• Axioms

Those initial propositions whose truth we assume are known as Axioms. They were accepted as true because of their conformity with common experience. Euclid called them as common notions.

In his book, Elements, Euclid wrote a few axioms or common notions related to geometric shapes.

Axiom 1: Things that are equal to the same thing are equal to one another.

Axiom 2: If equals are added to equals, the wholes are equal.

Axiom 3: If equals are subtracted from equals, the remainders are equal. 

Axiom 4: Things that coincide with one another are equal to one another.

Axiom 5: The whole is greater than the part.

Axiom 6 and Axiom 7: Things that are double of the same things are equal to one another. Things that are halves of the same things are equal to one another.

• Postulates

Postulates are self-evident truths which are taken for granted without any necessity of proof or explanation. Postulates are the basic structure from which lemmas and theorems are derived. The whole of Euclidean geometry, for example, is based on five postulates known as Euclid's postulates.

Postulate 1: A straight line segment can be drawn for any two given points.

Postulate 2: A line segment can be extended in either direction to form a line.

Postulate 3: A circle can be drawn with any centre and any radius

Postulate 4: All right angles are equal to one another.

Postulate 5: Two lines are parallel to each other if they intersect the third line and the interior angle between them is 180 degrees.

• Theorems

A Mathematical theorem is a logically valid conclusion drawn from a set ofpremises, axioms and already established theorems of Mathematical system.

Examples : 

1. ‘If p is a prime number and p divides ab then either p divides a or p divides b’

2. ‘If a quadrilateral is a parallelogram, then its diagonals bisect each other’

References

• The Teaching of Mathematics  : Kulbir Singh Sidhu
• https://ncert.nic.in/
• https://egyankosh.ac.in/