# Pedagogy Of Mathematics: (B.Ed Notes In English)

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## What Is Mathematics?

Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers.

The word mathematics comes from the Greek word ‘máthema’, which itself derived from the word ‘manthano’ in the ancient Greek language or, ‘mathaino’ in the Modern Greek language both of which means "to learn".

The word ‘Máthema’ means

• "that which is learnt", or
• "what one gets to know", and its adjective is ‘mathematikós’ meaning "related to learning" or "studious", which likewise further came to mean "mathematical".

In Latin, and in English until around 1700, the term mathematics more commonly meant "astrology" or sometimes "astronomy" rather than "mathematics".

The meaning gradually changed to its present form about 1500 to 1800.

Mathematics is a tool specially suited for dealing with scientific concepts.

The unique nature of the mathematics language with its signs, symbols, terms, and operations, is that it can handle ideas with precision and conciseness that is unknown to any other language.

As per the formalist view, mathematics is the investigation of axiomatically defined abstract structures using logic and mathematical notation.

## Definition Of Mathematics By Different Authors

Mathematics is a tool specially suited for dealing with scientific concepts. As per the formalist view, mathematics is the investigation of axiomatically defined abstract structures using logic and mathematical notation.

### Definition Of Mathematics By Bacon, Aristotle, Lindsay, Kant, Augusta Comte's, And Gauss

Mathematics is the gateway and key to all sciences”  -  Bacon

"Mathematics the science of quantity"  -  Aristotle

Mathematics is the language of physical sciences and certainly no more marvelous language was created by the mind of man”  -  Lindsay

Mathematics is the indispensable instrument of all physical researches.” -  Kant

Mathematics is the science of indirect measurement” -  Augusta Comte's

Mathematics is the queen of sciences and arithmetic is the queen of all mathematics” -  Gauss

### Some More Definitions Of Mathematics

American Heritage Dictionary, 2000 defines mathematics as The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols”.

Oxford English Dictionary (1933) defines mathematics asThe abstract science which investigates deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations, and which includes as its main division's geometry, arithmetic, and algebra”.

Encyclopedia Britannica defines mathematics as The science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects”.

## Brief History And Evolution Of Mathematics

Mathematics starts with counting. It is not reasonable, however, to suggest that early counting was mathematics. It was only when some record of the counting was kept and, therefore, some representation of numbers occurred can mathematics be said to have started.

In Babylonia mathematics developed from 2000 BC.

A place value notation number system had evolved over a lengthy period with a number base of 60 which allowed arbitrarily large numbers and fractions to be represented and so proved to be the foundation of more high powered mathematical development like:

• Number problems such as that of the Pythagorean triples were studied from at least 1700 BC.
• Systems of linear equations were studied in the context of solving number problems.
• Geometric problems relating to similar figures, area, and volume were also studied and values were obtained for n.
• Quadratic equations were also studied and these examples led to a type of numerical algebra.

### History Of Greek Mathematics

The Babylonian basis of mathematics was inherited by the Greeks and independent development by the Greeks began from around 450 BC and the major Greek progress in maths was from 300 BC to 200 AD.

Some important mathematics breakthroughs in Greek history are:

• Zeno of Elea's paradoxes led to the atomic theory of Democritus.
• Studies of the area lead to a form of integration.
• The theory of conic sections shows a high point in the pure mathematical study by Apollonius.
• A more precise formulation of concepts led to the realization that the rational numbers did not suffice to measure all lengths.
• A geometric formulation of irrational numbers arose.
• Further mathematical discoveries were driven by astronomy, for example, the study of trigonometry started.

After this time, mathematics progress continued in many Islamic countries. Mathematics also flourished particularly in countries like Iran, Syria, and India though, this work did not match the progress made by the Greeks but in addition to the Islamic progress, it did preserve Greek mathematics.

### History Of European Math

Some important mathematics breakthroughs in European history are:

#### History Of European Mathematics (16th Century)

Following are some of the key progress made in mathematics during the 16th century in Europe:

• Major progress in mathematics in Europe began again at the beginning of the 16th Century Pacioli, then Cardan, Tartaglia, and Ferrari with the algebraic solution of cubic and quadratic equations.
• The progress in algebra had a major psychological effect and enthusiasm for mathematical research, in particular, research in algebra, spread from Italy to Stevin in Belgium and Viète in France.
• Copernicus and Galileo revolutionized the applications of mathematics to the study of the universe.

#### History Of European Mathematics (17th Century)

Significant mathematics findings and expansions throughout this period are:

Calculus was to be the topic of most significance to evolve in the 17th Century.

• The 17th Century saw Napier, Briggs, and others greatly extend the power of mathematics as a calculatory science with his discovery of logarithms.
• Progress towards the calculus continued with Fermat, who, together with Pascal, began the mathematical study of probability.
• Newton, building on the work of many earlier mathematicians such as his teacher Barrow, developed calculus into a tool to push forward the study of nature.
• Cavalieri made progress towards calculus with his infinitesimal methods and Descartes added the power of algebraic methods to geometry.

#### History Of European Mathematics (18th Century)

Some crucial mathematics discoveries and developments in European history during the 18th century are:

• Newton's theory of gravitation and his theory of light takes us into the 18th Century. His work contained a wealth of new discoveries showing the interaction between math, physics, and astronomy.
• Leibniz, whose much more rigorous approach to calculus (although still unsatisfactory) set the scene for the mathematical work of the 18th Century.
• Leibniz's influence on the various members of the Bernoulli family was important in seeing the calculus grow in power and variety of applications.

The most important mathematician of the 18th Century was Euler who, in addition to work in a wide range of mathematical areas, was to invent two new branches, namely the calculus of variations and differential geometry.

• Euler was also important in pushing forward with research in number theory begun so effectively by Fermat.
• Toward the end of the 18th Century, Lagrange was to begin a rigorous theory of functions and of mechanics.
• The period around the turn of the century saw Laplace's great work on celestial mechanics as well as major progress in synthetic geometry by Monge and Carnot.

#### History Of European Mathematics (19th Century)

The 19th Century saw rapid progress in mathematics, some of the critical mathematics findings and developments during this period are:

• Fourier's work on heat was of fundamental importance.
• Non-Euclidean geometry developed by Lobachevsky and Bolyai led to the characterization of geometry by Riemann.
• In geometry Plücker produced fundamental work on analytic geometry and Steiner in synthetic geometry.

Gauss, thought by some to be the greatest mathematician of all time, studied quadratic reciprocity and integer congruences. His work in differential geometry was to revolutionize the topic. He also contributed in a major way to astronomy and magnetism.

• The 19th Century saw the work of Galois on equations and his insight into the path that mathematics would follow in studying fundamental operations.

Galois' introduction of the group concept was to herald a new direction for mathematical research which has continued through the 20th Century.

• Cauchy, building on the work of Lagrange on functions, began rigorous analysis and began the study of the theory of functions of a complex variable. This work would continue through Weierstrass and Riemann.
• Algebraic geometry was carried forward by Cayley whose work on matrices and linear algebra complemented that by Hamilton and Grassmann.

The end of the 19th Century saw Cantor invent set theory almost single-handedly while his analysis of the concept of number added to the major work of Dedekind and Weierstrass on irrational numbers.

• Fredholm's work led to Hilbert and the development of functional analysis.
• Maxwell revolutionizes the application of analysis to mathematical physics.
• Statistical mechanics was developed by Maxwell, Boltzmann, and Gibbs. It led to ergodic theory.
• The study of integral equations was driven by the study of electrostatics and potential theory.
• Sophus Lie's work on differential equations led to the study of topological groups and differential topology.

## Characteristics of Mathematics

The key features and characteristics of mathematics are:

1. Logical sequence
2. Applicability
3. Mathematical systems
4. Generalization and classification
5. Structure
6. Mathematical Language and Symbolism
7. Rigor and logic
8. Abstractness
9. Precision and accuracy

### 1.  Logical Sequence

The earliest mathematics was firmly empirical (experiential), rooted in man’s perception of:

• Number (quantity)
• Space (configuration)
• Time, and
• Change (transformation)

Today mathematical fact can be established without reference to empirical reality. All this transpired with the gradual process of

• Experience,
• Abstraction, and
• Generalization in the field of math.

Mathematics is, today, built upon abstract concepts whose relationship with real experiences is useful but not essential. These abstractions mean that mathematical fact is certainly be influenced by reality, as it often is, but it is not considered mathematical fact until it is established according to the logical requirements of modern mathematics.

It is this evolution from empirical science to axiomatic science that has established derivability as the basis for mathematics and made logical sequence a main feature of mathematics.

### 2.  Applicability

Concepts and principles become more functional and meaningful only when they are related to actual practical applications. It is the natural instinct of man to

• Seek explanation,
• To generalize, and
• To attempt to improve the organization of his knowledge.
“Knowledge is power only when it is applied”.

Whenever knowledge is applied, especially, related to daily life situations it makes the learning of any discipline more meaningful and significant. Mathematical truth turns are applicable in very distinct areas of application from across the universe to across the street.

The study of math, requires the learner to apply the skills acquired to new situations.

### 3.   Mathematical systems

A typical mathematical system has the following four parts:

1. Undefined terms,
2. Defined terms,
3. Axioms and
4. Theorems.

#### a.  Undefined terms

In geometry or in any other mathematical system, we have to start with some terms, these terms are typically extremely simple and basic objects, so simple that they resist being described in terms of simpler objects.

Example: point, line, set, variable, plane, etc.

The choice of the undefined terms is completely arbitrary and generally facilitates the development of the structure.

#### b.  Defined terms

For example, A triangle having 3 equal sides is an equilateral triangle. Thus to define an equilateral triangle, one should have learned the terms

• Triangle,
• Equal, or
• Sides.

#### c.  Axioms

Axioms or postulates are a statement in a mathematical system that describes the relationships existing among the undefined terms of the system.
For example: To describe the relationship existing among undefined terms ‘line’ and ‘point’ some axioms that can be used are:
1. There can be one and only straight line joining two points.
2. Two lines meet at a point.
3. A line has one and only one mid-point.

#### d.  Theorems

A statement that we arrive at by successive application of the rule of implication to the axioms and statements previously arrived at is called a theorem.

For instance: The rule of implication states that

• the statement p implies the statement q and
• If the statement p is true, then the statement q will be true.

When we apply the rule of application to the axioms we generate new statements. Again we may apply this rule to these new statements.

### 4.  Generalization and classification

The generalization and classification of math are very straightforward in contrast to others fields of thought and activity. Mathematics unites numerous findings, conclusions, assumptions, etc. under one head, and from that makes schematic arrangements and classifications.

Some of the examples of successive generalizations in mathematics are:

• Number concept has itself widened from that of the whole number when it included successively negative numbers, fractional numbers, imaginary numbers, and irrational numbers.
• One of the significant traits of algebra is its generalized handling of the processes of arithmetic.
• In geometry, there are frequent occurrences for grouping and generating results.

When the students evolve there own concepts, theorems, definitions he/she is making generalizations.

### 5.  Structure

A mathematical structure is a mathematical system with one or more explicitly recognized (mathematical) properties.
Generally speaking, a structure denotes ‘the formation, arrangement, and articulation of parts in anything build-up by nature or art’.For example:
• If ‘S’ is a non-empty set on which one or more operations have been uniquely defined with respect to an equivalence relation, then the set S together with the operation(s) is called a mathematical system.
• Using one or more of the mathematical systems like commutative, associative, or distributive properties we may create a structure.

The mathematical structure has a variety of arrangements, formations, which results in putting parts together. For instance:

• A structure that comprises of a mathematical system <S; O> with one operation, in which the operation O is associative is called a semi-group.

Thus mathematics has got definite logical structures. These structures ensure the order and beauty of mathematics.

### 6.  Mathematical Language and Symbolism

Over the course of the past 3000 years, mankind has created sophisticated spoken and written natural languages which are tremendously efficient for expressing a variety of meanings, moods, and motives.

Man has the ability to assign symbols for ideas and objects.

The language in which Mathematics is developed is no less, and, when mastered, provides a highly effective and powerful tool for

• Mathematical Expression,
• Exploration,
• Reconstruction After Exploration, and
• Communication.

Usage of symbols constructs mathematics language and makes it more elegant and precise than any other language. Mathematical language and symbols

• Cut short the lengthy statements.
• Help the expression of ideas or things in the exact form.
• Mathematical language is free from verbosity.
• Math symbols help to form and clear exact expression of facts.

All mathematical operations, relations, statements are expressed using mathematical symbols. The training that mathematics provides in the use of symbols is excellent preparation for other sciences. For example:

• We can state the commutative law of addition and multiplication in a real number system in the verbal form as: ‘the addition and multiplication of two real numbers is independent of the order in which they are combined’.
• In concise form as: a + b=b + a (addition), and a * b=b * a (multiplication).
Mathematics is the language of physical sciences and certainly no more marvelous language was ever created by the mind of man”. - Lindsay

### 7.   Rigor and logic

Logic is essential in mathematics; logic regulates the pattern of deductive proof through which mathematics is developed. In modern times;

1. The primary pedagogical objective of Mathematics is that it must be understood institutively in geometrical or physical terms.
2. The secondary pedagogical objective of Mathematics is its rigorous presentation.
Argument concludes a question, but it does not make us feel certain, or acquiesce in the contemplation of truth, except the truth also be found to be so by experience”- AS Roger Bacon

### 8.  Abstractness

Everything in math cannot be learned through experiences with concrete objects the same way as other disciplines. Some mathematical concepts can be learned only through their definition and they may not have a physical matching part to be extracted from.

Mathematics is abstract in the sense that mathematics does not deal with actual objects in much the same way as physics. But, in fact, mathematics questions, as a rule, cannot be settled by direct appeal to experiment.

For instance: Our whole thinking is based on the belief or assumption that there are infinitely many numbers, there are infinitely many fractions between 0 and 1and therefore, counting never stops.

Infinity is something that we can never experience and yet it is a central concept of mathematics.

Man has no way of knowing, calculating, and justifying this as a man cannot observe and count all these which makes, Infinity, abstract concept, as it is not a concept corresponding to any object that man has seen or is likely to see.

Some other examples of abstract concepts in math are:

• Prime numbers,
• Probability,
• Limit and function,
• Continuous functions etc.

These all are abstract in the sense they can be learned only through their definitions and it is not possible to provide concrete objects to correspond to such concepts.

Even some of the concepts which one argues to be concrete are also abstract. For example:

• Concepts such as a line, a diagonal, a point, a circle, a ray, etc., which seen as concepts that are learned through observation of concrete instances, and as a result, they are concrete. But a figure of a circle, a dot (point), a line drawn on a board, are all mere representations of the concepts and they are not objects themselves.

### 9.  Precision and accuracy

Mathematics is known as an ‘exact’ science because of its precision. It is perhaps the only subject that can claim certainty of results. Even when there is an emphasis on approximation, mathematical results have some degree of accuracy.

• There is no midway possible in Mathematics. Mathematics is either correct or incorrect, right or wrong, it is accepted or rejected.
• Mathematics can decide whether or not its conclusions are right.
• Mathematicians can verify the validity of the results and convince others of their validity with consistency and objectivity.
• Mathematics true or false holds for everyone who uses maths, at any level, not only for the expert.

Mathematical culture is that what you say should be correct. What you say should have a definition. You should know the definition and limits of what you are stating, claiming, or saying.

Thus, the modern mathematical culture of precision arises because:

• Mathematics has developed a highly symbolic and precise language.
• Mathematical concepts have developed in a dialectic manner that allows for the adjustment, adaptation, and cumulative refinement of concepts based on experiences.
• Mathematical reasoning is expected to be correct.

## Aims Of Teaching Mathematics

The mathematics education is aimed for the learner:

1. To foster the good ability to solve problems and be able to critically analyze the data and arrive at precise conclusions.
2. To achieve a sound grasp of the:
1. Principles of mathematics and practice them.
2. Properties of mathematics and their real-life application.
• Mathematics symbols,
• Mathematics operations and their relationships and how to use them.
4. Practical application of mathematical
• Concepts,
• Facts,
• Theorems etc.
3. To develop the decision-making ability and maximize the ability of learners.
4. To attain a good understanding of Realization that there is always a justification for a mathematical rule.
5. To improve learning speed and form logical understanding.
6. To recognize the value of mathematics and its relationship with other school subjects.
7. To develop a good understanding of not just mathematical concepts but also surrounding.
8. To not just merely understand mathematics, but to speak mathematics too.
9. To succeed in rapid mental recall of facts and to nurture the ability to learn and think better.
10. To give concrete experiences which will lead to a generalization and helps to form an abstract concept among the minds of students.
11. To appreciate the order and beauty of maths.

## General Objectives Of Teaching Mathematics

Some of the common objectives of teaching mathematics are:

1. Teaching fundamental numeracy skills to all students.
2. Teaching heuristics and other problem-solving strategies to solve non-routine problems.
3. Teaching selected areas of mathematics such as:
• Teaching Euclidean geometry as an example of an 'axiomatic system' and a 'model of deductive reasoning'.
• Teaching Calculus is an example of the 'intellectual achievements' of the modern world.
4. Teaching abstract mathematical concepts at an early age such as
• Set,
• Function etc.
5. Teaching mathematics to students for future livelihood. For instance,
• Teaching 'practical mathematics' to prepare students to follow a trade or craft.
• Teaching 'advanced mathematics' to students who wish to follow a career in fields like
• Engineering,
• Science and Technology,
• Math etc.

## Need And Importance of Math Teaching

Mathematics has got much educational importance which shows the increasing importance of the subject in schools and in social life and determines the need of teaching the subject in schools.

This importance can be studied with the following headings mentioned below:

1. Practical Importance
2. Development Of The Power Of Concentration
3. Development Of Inventive Faculty
4. Disciplinary Importance
5. Vocational Importance
6. Teaching The Art Of Economic Living
7. Development Of Will Power
8. Cultural And Aesthetic Importance
9. Development Of The Decision Making Ability

### 1.  Practical Importance

Mathematics has great practical importance everyone uses some mathematics in every form of life. Math is needed by all of us whether wealthy or poor, high or low. Whoever earns and spends uses mathematics.

For example:

• An ordinary man on occasion can be short of reading or writing but he cannot go without calculating and counting.
• A housewife needs mathematics to look after her house; noting down numerous household transactions, forming family budgets and estimates, etc.
• Businessmen, engineers, accountants, bankers, planners, etc., even petty shopkeepers, humble coolies, carpenters, and laborers need maths not only for earning their livelihood but also to spend wisely and save for the future.
• We buy cloth, food items, fruit, vegetables, grocery, etc. We have to make purchases daily and for basic calculating we use maths.
Without mathematics, man's life is zero. There will be utter confusion and chaos without mathematics.

Just think if a fairy descends on earth and removes all mathematics there will be

• No industrial activity,
• No calendar,
• No accounts,
• No maps,
• No plans or projects,
• No fixations or measurements.

We live in a world of measurements. We have to measure areas, lengths, weights, and volumes. We need to set up wages, targets, timings, prices, percentages, exchanges, rates, etc.

In the nonappearance of all these fixations, measurements, counting the life in the present multifaceted society will come to a standstill and everyone will always be at the mercy of others.

Mathematics is critical for leading a flourishing, healthy social life, a person who is ignorant of mathematics can be easily fooled and cheated. Thus mathematics has tremendous importance in our daily life.

### 2.  Development Of The Power Of Concentration

Mathematics trains or disciplines the mind. The faculty to concentrate one's mind can only be learned by the study of math.

### 3.  Development Of Inventive Faculty

The study of mathematics develops the 'inventive faculty' of the students as the solving of a difficult problem in mathematics is just like making a discovery.

### 4.  Disciplinary Importance

According to Locke,

"Math is a way to settle in the mind a habit of reasoning".

Mathematics is a way to settle the habit of reasoning in the human mind. Mathematics trains and disciplines the mind, it develops

• Thinking and
• Reasoning power.

Mathematics is an 'exact' and 'definite' science. Reasoning in mathematics has the traits of

• Objectivity,
• Originality,
• Accuracy,
• Simplicity etc.

Every student of maths has to reason properly without any prejudices or unnecessary biases.

### 5.  Vocational Importance

Mathematical ability and knowledge are fundamental for a sound and productive vocational life.

Mathematics demands hard work from the learner which eventually make learner of mathematics be hard working in every aspect of their life. The mathematical study promotes and encourages in creating international understanding.

Study of mathematics aids in refining and developing the students, skills, and capabilities like:

• Generalization,
• Thinking and Reasoning,
• Discovery,
• Analysis & Synthesis etc.

### 6.  Teaching The Art Of Economic Living

The entire aspect of our life is interrelated with mathematics, there is no escape from mathematics of life and livelihood.

Math teaches economy in

• Time,
• Thought,
• Money and
• Speech.

### 7.  Development Of Will Power

Mathematics develops patience and perseverance in the students and strengths the willpower of learners.

### 8.  Cultural And Aesthetic Importance

Math has made a principal contribution to man's cultural advancement. The history of mathematics shows how mathematics has influenced civilization and culture at a specific time.

For example: Progress in mathematics, of Greeks and Egyptians in the past led to their cultural advancement and subsequently in the progress of their civilization.

Mathematics makes a direct or indirect contribution to the growth of all occupations. The progress of civilization has been mainly due to the progress of various occupations which build up culture such as

• Agriculture,
• Industry,
• Medicine,

Mathematics has got great cultural importance which steadily escalates day by day. Math is also a pivot for cultural arts such as Music, Poetry, Fine arts, and Painting.

For instance: Greeks, who were the greatest geometers of their times, were quite adept in fine arts.

It has been truly said by ‘Hogben’ that

“mathematics is the mirror of civilization”.

### 9.  Development Of The Decision Making Ability

Mathematical truths are definite and exact. The mathematical correctness of a human mind goes along with the accurate decision-making ability of the man.

Modern civilization is the result of the correct decision-making ability of human beings and is analogous to the decision maker's quantitative ability.

## Relation Of Mathematics With Other Subjects

Relationship between mathematics and other subjects:

1. Relationship Of Mathematics with Geography
2. Relationship Of Mathematics with Arts
3. Relationship Of Mathematics with Civics and Citizenship
4. Relationship Of Mathematics with History
5. Relationship Of Mathematics with English
6. Relationship Of Mathematics with Linguistics
7. Relationship Of Mathematics with Communication
8. Relationship Of Mathematics with Archaeology
9. Relationship Of Mathematics with Health and Physical Education
10. Relationship Of Mathematics with Computers
11. Relationship Of Mathematics with Philosophy
12. Relationship Of Mathematics with Insurance and Finance
13. Relationship Of Mathematics with Actuarial Science
14. Relationship Of Mathematics with Economics
15. Relationship Of Math with Social Science
16. Relationship Of Mathematics with Agriculture
17. Relationship Of Mathematics with Science
18. Relationship Of Mathematics with Biological Science
19. Relation Of Mathematics with Ecology
20. Relationship Of Math with Environmental study
21. Relationship Of Mathematics with Chemistry
22. Relationship Of Mathematics with Genetics
23. Correlation Of Mathematics with Physics
24. Relationship Of Mathematics with Engineering and Technology
25. Relationship Of Mathematics with Music
26. Relationship Of Math and Logic
27. Relationship Of Maths with Drug Kinetics

### 1.  Mathematics with Geography

Geography is nothing but a scientific and mathematical description of our earth in its universe.

There are numerous areas of geography which need the application of mathematics like:

• Lunar and solar eclipses,
• Maximum and minimum rainfall,
• The magnitude and dimension of earth, its position, and situation in the universe,
• Latitude and longitude,
• Formation of days and nights etc.

The surveying instruments in geography have to be mathematically accurate, in order to exercise desirable control over them. Some of the key areas where geography and mathematics work hand in hand are:

• Variations in the soil fertility,
• Differences in the distribution of forests,
• Changes in ecology etc.

### 2.  Math with Arts

"Mathematics and art are just two different languages that can be used to express the same ideas”.

Mathematics provides background and a basis for aesthetic appreciation. The arts and mathematics embrace students understanding of associations between rhythm, space, and time & line through the experience of these abstract concepts in countless mathematical ideas & art forms. Appreciation of balance, symmetry & rhythm postulates a mathematical mind.

Mathematics is in itself a piece of fine art. It is considered that the universe is written in the language of mathematics, and its characters are circles, triangles & other geometric figures.

The beauty of a piece of art depends on the manner in which it expresses truth. A mathematical mind can appreciate art with some sense of confidence.

• The old 'Gothic Architecture' is based on geometry.
• The 'golden ratio' is a mathematically related aesthetic consideration that is applied amongst numerous performing, visual multi-modal arts forms.
• Even the Egyptian Pyramids, based on mathematics.

Mathematics is knowledge of truth and realities.

### 3.  Mathematics with Civics and Citizenship

Mathematical structure and working play critical roles in so many key civics concepts, and are significant in various aspects of our society.

The ideas and concepts developed in the mathematics study are relevant to a variety of civic and citizenship understandings. Some parts of civics and citizenship which require mathematical understanding are:

• Majority rule,
• Absolute majority,
• One vote one value representation,
• Proportional voting systems etc.

### 4.  Maths with History

The concepts & competencies developed in mathematics are valuable in the study of a range of historical information and sources. The thoughts and skills acquired in mathematics are beneficial in demonstrating historical understanding and assist in translating and analyzing historical data.

The study of history involves the analysis and interpretation of a range of historical information which needs math’s like:

• Population charts,
• Diagrams,
• Statistical information etc.

### 5.  Mathematics with English

Mathematics, including the use of 'conjectures' and 'proof', has clear links to the development of structures and coherent argument in speaking and writing.

A mathematical structure is strongly related to

• Semantics syntax
• Propositions
• Quantifiers etc.

### 6.  Mathematics with Linguistics

Mathematics has had a great influence on research in literature. The notions of transformation and structure are as essential for linguistic as they are for mathematics.

Linguistics study needs

• Mathematical biology,
• Information theory,
• Mathematical psychologies etc.

In the development of machine languages and comparing artificial and natural language, all need extraordinary mathematical ability.

### 7.  Mathematics with Communication

Mathematics working and structure play key roles in understanding human and natural creations.

A range of communication equipment is utilized for representing various relationships and in displaying results. For Example:

• Venn diagrams,
• Tree diagrams etc.

### 8.  Maths with Archaeology

Archaeologists draw heavily upon various statistics & mathematical tools and techniques. From archaeological surveys, archaeologists collect and present the data, which shed light on past human behavior and attempt to differentiate patterns in their results.

Archaeologists utilize statistics and numerical methods, measures to

• Check the reliability of their analyses and interpretations.
• Monitor which pits are most successful during excavation, and with this data decide on further excavation.
• Spot patterns in their findings and compare them with past findings.

### 9.  Mathematics with Health and Physical Education

Mathematics tools and procedures are used in solving problems and to model situations in the field of health and physical education.

For Example:

• Sporting events maintain scores and scoring requires knowledge of distance, weight, number, and time as variables.
• Collection of data through a variety of fitness programs, testing, and analyzing athlete performance all need mathematical and statistical models and expertise.
• One main factor in any physical activity, sports is the interpretation of collected results and data like calculating percentage improvement, etc. all require mathematics skills.

### 10.  Math with Computers

The foundation of computer science is based only on mathematic. Every single mathematical process of use in applications has been quickly converted into computer package algorithms. There are computer packages for every mathematical concept such as

• Solution of linear, non-linear equations,
• Inversions of matrices,
• Solution of partial and ordinary differential equations,
• Symbolic differentiation and integration and the list never end.

Computer science has used every bit and piece of mathematics. From logic to relations & functions, to basic set and graph theory, to mathematical induction and discrete probability, to numerical analysis and Operation Research techniques, etc. etc. etc. there is no topic or sub-topic of math which computer science has not used or made a code.

Modern management techniques like Simulation, Monte Carlo program, Evaluation Research Technique, Critical Path Method, the study of Artificial Intelligence, Development of automata theory, etc. all require mathematics with computers.

### 11.  Mathematics with Philosophy

"The real finisher of our education is Philosophy, but it is the office of mathematics to ward off the dangers of philosophy.”- Herbart

Philosophy is defined as the science that investigates the ultimate reality of things, whereas, in mathematics, the philosophers find systematic and orderly accomplishments of unambiguous truths.

Mathematics eliminates irrationality, sets the philosophers on the right path of acquiring true knowledge.

Mathematical methodologies, approaches create realistic schools of thought in philosophy. It was this search of distinction between fiction and fact that led great thinkers like Plato and others underneath mathematics influence.

Therefore, there remains little doubt of the fact as to why mathematics occupies a central place between mental philosophy and natural philosophy.

### 12.  Mathematics with Insurance and Finance

Mathematics and statistics are applied quite a lot in insurance and finance, areas such as

• Banking,
• Making Insurance policy,
• Producing economic forecasts,
• Stock market trading etc. all require extensive mathematics use.

### 13.  Mathematics with Actuarial Science

Actuaries use statistics and mathematics to make financial sense of the future.

Any company or organization go onboard on a big project, they hire an actuary, and the job of an actuary is

• To study and analyze the venture (project).
• To evaluate and estimates the financial risks involved & model the future financial outcomes.
• To make the company or organization aware of his/her findings, and advise them about the decisions to be made.

### 14.  Mathematics with Economics

“The direct application of mathematical reasoning to the discovery of economic truths has recently rendered great services in the hand of master mathematicians.” - Marshall

Mathematical methods and language are used constantly in explaining economic phenomena.

Economics and mathematics are interlinked by the frequent use of mathematical models in wide-ranging topics of economics. Some of the examples illustrating are:

• We use statistical modeling and analysis in obtaining census data.
• To find expenditure of public money, sampling populations to predict election outcomes, etc.
• In modeling and forecasting economic indicators such as the consumer price index and business confidence mathematics are used.
• A great deal of mathematical thinking, mathematical models, planning goes into the task of national economic planning.

### 15.  Math with Social Science

Mastery of the fundamental processes is necessary for clear thinking.

Social science has a mathematics stamp all over it. Almost all personal and social activities today require mathematical literacy and this prerequisite is continually increasing.

### 16.  Mathematics with Agriculture

Agriculture relies heavily on mathematics. Agriculture as science needs a direct application of mathematics.

For example:

• Progress of the agricultural farm can be projected and judged by drawing graphs of different items of production.
• To calculate Time and work, Average investment and expenditure like the cost of labor, seed rate, etc. mathematics is used.
• To measure land or area, and to compute production per unit area all require mathematics knowledge.

### 17.  Mathematics with Science

Science is incomplete without mathematics. The expertise and abilities students absorb within the diverse dimensions of mathematics, support all aspects of science.

For Example:

Number concepts and measurement are predominantly used in science, be it, data collection, and analysis, estimation of error, and modes of reporting data.

The mathematics discipline supports science learners in developing critical knowledge and skills like:

• How to collect the records (data) and interpret them.
• To look for patterns and display data correctly.
• Teaches learner crucial number handling skills, drawing conclusions & making generalizations.
• How to estimates and do further investigations, extrapolations, and interpolations from their own experimental records or from reliable data, etc.

### 18.  Mathematics with Biological Science

Mathematics has made an enormous impact on Biological Science.

Mathematical models, concepts have played and will continue to play, an important role in Biological Science. For instance:

• When we study deterministic and stochastic models for the growth of the population of micro-organisms and animals, subject to given laws of birth, death, immigration, and emigration we use differential equations, difference equations, differential-difference equations, and integral equations.
• Mathematical models and applications are significant in
• ‘Neuroscience’,
• ‘Molecular biology'
• ‘Cellular biology’ and
• ‘Structural biology’.
• Computational and Mathematical methods complement experimental ‘structural biology’ by adding motion to molecular structure.
• Two additional regions of mathematics when mathematics successfully interacts with biology are:
• 2-dimensional differential geometry.
• 3-dimensional topology.

### 19.  Mathematics with Ecology

Mathematical concepts have played an important role in Ecology like the ‘prey- predator’ model of ecology which uses maths. Some more instances of this relation are:

• The mathematical theory of the ‘Spread of Epidemics’ solves ‘systems of differential equations' to determine the number of immunes, infected, and vulnerable peoples at any time.
• For monitoring, controlling epidemics in animals & plants numerous mathematical and epidemic models of pest control are critically examined.
• Dynamic programming and control theory are used in the control of epidemics subject to cost constraints.

### 20.  Mathematics with Environmental study

Mathematics is used to study and find solutions to many environmental issues. Some of these areas are:

• To study the problems of
• Waves,
• Tides,
• Cyclones flow in bays and estuaries.
• Mathematics is chiefly used in ‘Pollution Control Models’.
• To find out what proportion of pollutants is emitted from refineries, industries, chimneys, factories, etc. mathematics is used.
• To study how to limit and lower pollution be it water, noise, or air with maximum decrease and minimum cost possible.
• To study the diffusion of pollutants in the atmosphere mathematics knowledge is obligatory.
• To study the effect of leakages of poisonous gases.
• To understand the conditions that result in
• Avalanches,
• Volcanic eruptions,
• Ocean currents, etc.

### 21.  Math with Chemistry

Math is extremely important in physical chemistry.

For example:

• The basic step of production of any chemical is the mathematical ratio, which is needed to compute how many elements in which quantity have to be mixed.
• To calculate the molecular weights of organic compounds mathematics is used.
• The use of ratio and percentage are quintessential in the assessment and estimation of elements in organic compounds.

In advanced topics of chemistry such as quantum, statistical mechanics, etc., mathematical concepts form the core like:

• Probability theory forms a base of Statistical mechanics.
• To develop models of any complicated processes even in pure scientific research in biology and chemistry mathematicians are required, especially those with higher degrees in computer science.
• Quantum relies heavily on group theory and linear algebra and requires knowledge of mathematical topics such as Hilbert spaces and Hamiltonian operators.
• Biochemistry has some important topics which rely heavily on math, such as binding theory and kinetics.

Some other fields of chemistry which also use a significant amount of math are:

• Most modern Nuclear magnetic resonance (NMR), spectroscopy and Infra-Red (IR) machines rely on the Fourier transform to obtain spectra.
• Mathematical laws governed all chemical equations, their combinations.
• Mathematical calculations regulate the development and formation of chemical compounds.
• Pharmaceutical and cosmetics companies oblige to use teams of mathematicians to work on clinical data about the efficiency or dangers of new drugs and products.

### 22.  Mathematics with Genetics

Genetic engineering incorporates substantial mathematical modeling. Mathematics is used in areas like:

• Decoding genetic code and doing further research.
• To Study the inheritance of genetic characteristics from generation to generation.
• To find methods for genetically improving plant and animal species.

### 23.  Mathematics with Physics

In Physics, every principle and rule takes the mathematical form ultimately.

Mathematical tools and techniques are of particular use to physicists and engineers. Mathematical calculations occur at every step in physics like:

• Chare’s law of expansion of gases is built upon mathematical calculations.
• Application of mathematical concepts are frequently in use in
• Quantum mechanics,
• Fluid dynamic,
• Oceanography,
• Classical mechanics, and
• Electromagnetism.
• To obtain mathematical models for fluid and solid mechanics, a lot of physicists and mathematicians apply the basic laws of Newton.
• To study the effect of pollutants from nuclear and other plants in seawater, particularly on fish population in the ocean physicists use mathematics.
• To study atmospheric Sciences, dynamic meteorology, and weather prediction.
• In the problem of underwater explosions, we use mathematics with physics.
• Physics heavily focuses on mathematical concepts of complex variables, differential equations, vector spaces, integral transforms, matrix algebra, infinite series, integral equations, etc.
• The flight of torpedoes in the water, sailing of ships & submarines, etc. are also some of the main important areas where mathematics and physics work hand in hand specifically from a defense point of view.

Overall, we can say, Mathematics gives a final shape to the rules of physics.

### 24.  Mathematics with Engineering and Technology

Mathematics is considered to be the foundation of engineering.

Mathematics has played a vital role in the progress of chemical, civil, mechanical, and aeronautical engineering. It is well known that a maximum of the technological processes in engineering and technology are described effectually by means of a mathematical framework.

Some processes of engineering where the application of mathematics is crucial are:

• Mathematical sciences of plasma dynamics and magneto-hydrodynamics are used for experiments in controlled nuclear fusion, making flow meters & generate magneto-hydrodynamic.
• In dealing with designing, construction, surveying, verifying results, etc. engineers depend prominently on mathematics concepts and ideas.
• With the application of mathematics concepts like geometric principles during the design and constructions phase, the durability of the final product can be increased.
• The defense sector is an important employer of mathematicians.

### 25.  Mathematics with Music

Leibnitz, the great mathematician had said,

“Music is a hidden exercise in arithmetic of a mind unconscious of dealing with numbers”.

Music theorists frequently use mathematics to understand the musical structure and communicate new ways of hearing music.

• All music notes require rhythm, but few know that rhythm is the result of untold uncountable permutations and combinations.
• Fibonacci numbers and Golden ratio are incorporated into the works of many composers to understand musical scales.
• Most modern-day music is produced and fabricated by using digital processors and synthesizers. To add effects to the sound or even to correct pitch today’s music requires mathematics.
• Using a mathematical technique called Fourier analysis audio software engineers to manipulate the digital sound.
“Where harmony is, there are numbers”- Pythagoras.

### 26.  Mathematics and Logic

“Symbolic logic is mathematics, mathematics is symbolic logic”- C.J.Keyser

Mathematics assists the formulation and helps to develop logical laws. The methods and symbols used to survey the foundation of mathematics can be transferred to the study of logic.

D’Alembert says, “Geometry is a practical logic because, in it, rules of reasoning are applied in the most simple and sensible manner”.

### 27.  Mathematics with Drug Kinetics

Drug kinetics means the study of the spread of drugs in the various compartments of the human body. Some key areas where drug kinetics involves the concept of math are:

• Mathematical models are developed in the study of the comparative effects of various treatments used in many disease, especially used in cancer disease.
• Biomechanics deals with the stress & strain in bones and muscles, and with injuries & fractures in skulls etc. Solution of partial differential equations are used in solid Biomechanics.

## EDUCATIONAL OBJECTIVES

### Meaning of Objectives

Aims are sovereign in nature they are broken into specified objectives to provide definite learning experiences for bringing about desirable behavioral changes.

Objectives help in generating behavioral modifications in the learners for the ultimate realization of the aims of teaching.

The objectives are achieved within the specified classroom situation, and they

• Have swift purposes and are short-term.
• And they also Aid in accomplishing immediate goals.

### Definition of Educational Objectives

Objectives related to education as a whole, are known as 'Educational Objectives '.

“By educational objectives, we mean explicit formulations of the ways in which students are expected to be changed by the educative process, that is, the ways in which they will change in their actions”. - B.S.Bloom

### Need of Educational Objectives

Reasons for needing educational objectives are:

1. Educational objectives indicate the nature of the education system.
2. They serve as guides for teaching and learning.
3. Educational objectives provide guidelines in selecting teaching-learning activities.
4. It shows the direction in which education will act.
5. For teaching which aims at worthwhile behavior changes, a clear understanding of educational objectives is essential.
6. Educational objectives develop awareness among the teachers about the importance of their work.

### How to Form Educational Objectives

The formulation of educational objectives is a matter of careful choice on the part of the teachers and administrators.
1. In forming educational objectives following factors are involved:
• The needs and capabilities of the pupils.
• The specific demands of his social environment.
• The nature of the subject matter.
2. In order to save time and effort, it is very important that the objectives of a subject be clearly identified and defined.

### Criteria For Good Educational Objectives

The criteria of a good educational objective is that

1. The educational objectives should be ‘Unambiguous’.
2. The educational objectives should be ‘Useful’.
3. The educational objectives should be ‘Specific’.
4. The educational objectives should be ‘Feasible’.
5. The educational objectives should be 'In Accordance With General Aims of Education '.

### Objectives of Classifying Educational Objectives

Classification is a valuable system to group similar things under one heading based on common characteristics or common relationship that exists between groups and individuals.

The main objectives of classifying educational objectives are as follows:

1. Educational objectives are helpful in defining, translating, and exchanging educational thoughts in a uniform way.
2. These are vital in comparing curricular goals with wider educational objectives.
3. Educational objectives assist in preparing evaluation or testing materials.
4. They are highly valued in planning, teaching, and learning activities.
5. Educational objectives are significant in identifying desired behavioral outcomes among the learners.
6. It is crucial in the search for the relationships that exist among groups and individuals.
Educational objectives are fundamental in planning a curriculum.

Conclusion

The Pedagogy Of Mathematics Plays A Vital Role In Shaping The Next Generation Of Scientifically Literate Individuals. By Understanding The Principles And Strategies Discussed In This Blog Post, You Will Be Well-Prepared To Create Meaningful And Impactful Maths Learning Experiences For Your Students.

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