Introduction to mathematics Introduction to mathematics

# Introduction to mathematics

## Introduction to mathematics

Mathematics results from the discovery, the formulation, the systematic development, and the application of patterns of inductive and deductive thinking. It consists of patterns of related ideas and patterns of thoughts. It is the queen of all sciences. It has continuously developed and changed with changing needs, it is known and expected that in the future it will continue to develop and change. Mathematics of the school curricula must also change since mathematics itself is growing. It must include new discoveries made in mathematics. The term modern mathematics is misleading to most readers. It leaves the impression that the entire contents of traditional mathematics have been replaced by something new. This introductory chapter presents the meaning of modern mathematics in various ways.

### Reasons for Studying Mathematics

The principal reason that mathematics is widely studied is because of its usefulness. It was easy in the past to identify a minimum amount of useful mathematics. New development in industry and science and new applications in different areas of human affairs call for a greater knowledge of mathematics than was required from early settlers of the world. Very little modern science can be learned without an understanding of mathematics. Physical science states its basic laws in mathematical language and uses mathematics continuously to formulate results and to make predictions. Social and behavioral sciences are using mathematics to an ever-increasing extent.

If the mathematics necessary to industry and science considered merely routine skills, the curriculum of mathematics would be simple and unchanging. Such is not the case. The needs are expanding too rapidly for an unchanging curriculum to suffice. New patterns that result in new routines are continuously being discovered; mathematics is not static. The nature of the subject must be learned so that the most useful and productive patterns may be selected and intelligently used. Mathematics should be learned in a way in which they are originally discovered. Mathematics is needed not only as a part of the language from effective everyday living but also as a language of industry and science.

### Pure and Applied Mathematics

The part of mathematics related to theoretical concepts is called pure mathematics and that related to practical affairs is called applied mathematics. Pure mathematics studies patterns of thought without regard to their application. Applied mathematics uses the patterns developed in pure mathematics to understand and control the world around us.

All pure mathematics rose from applied mathematics from the need to develop new methods to solve problems of everyday life. The developments in pure mathematics often were carried far beyond their original applications and led to further developments quite remote from practical experience. Later, what initially developed as pure mathematics often proved to be extremely useful in applied mathematics? Pure mathematicians do not know its wide application at the time of invention. For example, complex numbers were developed from the point of view of pure mathematics but now it finds intensive applications in electricity, radio, related fields of physics, and engineering. The reverse is also true. Mathematicians developed these methods with no provision of their use in modern electrical engineering. Indeed, at the time of their work, the knowledge of electricity was in a rudimentary stage.

Mathematics differs in no way from science in this respect. Discoveries in pure science are continuously being put to use in applied science, and problems in applied science, in turn, inspire investigations in pure science. Given the close relationship between pure and applied mathematics, it is unwise to consider one aspect without the other.

Applied mathematicians applied the discoveries made by pure mathematicians in daily life situations. When applied mathematicians face problems while applying the inventions made by pure mathematicians they again turned toward pure mathematicians. This process continues forever.

### The Generality of Mathematics

Another important aspect of mathematics is its generality. The more mathematics has developed the more general it became. All attempts to meet further demands must result in a more general system. Thus, ancient mathematics didn't provide models for games of chance but probability theory did so. Euclidean geometry failed to describe rather simple curves as ellipses until Descartes introduced coordinates. Cantor went further to meet more applications and introduced a theory of sets that admitted many more objects as geometrical figures. One may quote examples of the applications of mathematics by the hundreds and more that are still developed every day. But it can represent only a small portion of human activity. Mathematics is less general than the common language. This generality is limited to the structure of mathematics itself.

### Revolution in School Mathematics Programme

According to poet Goethe time and the world do not standstill. Change is the law of life. Change has not always been the law of life in mathematics education. History shows that certain consequences or events have inspired or compelled the people, societies, or nations for change. Changes were recommended by professional organizations in many countries of the world, but little or no change took place in the classroom anywhere around the world until about the time of World War II. For more than a third of this century, rote learning based on repetition formed the basic strategy for the teaching of mathematics at most levels and particularly at school levels.

After World War II, the world polarized into two superpowers. One was headed by the USA and the other by USSR. One of the major competitions between these two nations was to increase their military capabilities. As a result, USSR launched an artificial satellite into space in 1957. This incident focused the attention of Americans towards the reform of their education system. Accordingly, a great modification in school mathematics program was started fairly quickly and a modified program was implemented in school education in the 1960s in the United States. This modification created distinctive characteristics (involving organization, presentation, and content) between the school mathematics of the period 1900 through 1950 and that of the 1960s. Along with these changes in contents have come new insights into the way children learn and how best to use such natural developments in the teaching of mathematics. Because of the different characteristics of the mathematics of two periods (mathematics before the 1950s and that of 1960s), the mathematics of the 1960s is often referred to as contemporary, new, or modern mathematics while that of the earlier periods is usually called traditional mathematics.

### Factors to the Evolution of Modern Mathematics

Many factors have contributed to the evolution of modern mathematics and some of these factors are as follows:

1. In the first half of this century there was a lack of cooperation between school curriculum makers and mathematics professors, research mathematicians, and others who kept up with new developments in mathematics. Thus, a gap was created between mathematics programs and new developments in mathematics, and this gap widened considerably as the years passed. The effect of lack of cooperation between the two groups is more vivid when one considers that there have been more new developments in mathematics since 1850 AD than in all previous history. The realization of this gap between the two groups contributed to the evolution of modern mathematics when one tried to minimize the gap.

2. There has been a change in the concept of the nature of mathematics. The new look at the nature of mathematics began early in the nineteenth century but failed to gain momentum until the last half of the century. One factor which gave an impetus to the change in attitude about mathematics was criticisms on the lack of foundations and unity in mathematics. These criticisms were significant in that they encouraged many mathematicians to seek new concepts that would aid in establishing a foundation by which many of the rules and procedures could be validated. The new approach to mathematics originated with the ingenious idea to extract fundamental concepts from physical situations. The fundamental concepts served as a basis for the logical development of simple mathematical systems. In this way, many abstract mathematical systems were developed. The new approach to the conception and development of abstract mathematical systems promoted the enormous quantity of new inventions in mathematics in the last hundred years. The new approach to the nature of mathematics and the study of the structure of mathematical systems contributed to the reform in the school mathematics curriculum.

3 . There has been more time, experiments, and money spent in efforts to study, evaluate and improve the school mathematics curriculum. The result of these researches contributed to the evolution of modern mathematics.

4. Private and government agencies provided financial support, which encouraged thousands of teachers to return to college to acquire improved and up-to-date mathematical backgrounds.

### Place of Modern Mathematics in a Scientific World

We should plan our educational program only based on the needs of the citizens of today. The educational program must attempt to train a pupil or a future life. Many new scientific inventions, rapid developments in industrial fields, and changes in other nations have brought about and will bring about new and unforeseen questions and problems in our economic, social, and political life. We cannot teach the pupils the answers to the questions and problems that will arise in the future. The mathematics program must contain more than skills, things to do, and facts to memorize. It must provide an opportunity for each pupil to acquire and improve the abilities to perceive and develop a logical basis for considering and arriving at a satisfactory solution to each new question and problem that arises. Thus, the mathematical program must foster the recognition of fundamental mathematical concepts. It must offer the opportunity for each pupil to develop and improve the ability to deduce more complex notions and relations from the fundamental concepts. The acquisition of these abilities will provide a foundation for future citizens to derive the mathematical skills and knowledge demanded of them.

### What is Modern Mathematics?

The term modern mathematics is misleading to most of the students. It leaves the impression that the entire content of traditional mathematics has been replaced by something new. But this is not true. Certainly, some topics and skills of traditional mathematics are omitted in modern mathematics. However, there is an abundance of new concepts, language, and symbols whose functions are to provide a foundation for attaining the goals of modern mathematics. New language, new symbols, and new ways of doing the traditional skills are only a part of modern mathematics. Thus, modern mathematics can be regarded as mathematics with a new structure, new language and symbols, a new approach, and new contents.

Thus, modern mathematics is a mathematics-teaching program now widely used in many countries of the world. The United States introduced it into its schools during the 1960s. In the United Kingdom, it is sometimes called modern mathematics. Most countries, mainly in Europe and Australia followed the United States model and introduced new teaching programs fairly quickly. Mathematicians wrote new textbooks, and many mathematics teachers underwent training. Major undesirable features were eliminated from modern mathematics. The new concepts, symbols, new language, and new ways of teaching were introduced in the curriculum.

Educationists recognized for a long time that the methods of Euclid still used in most of the schools were outdated and even illogical. Tutors now approached geometry through the study of sets, or as transformational geometry. This form of geometry is the study of such transformations as reflections, rotations, enlargement, and symmetry.

Research into the study of mathematics especially that of the Swiss mathematician Jean Piaget influenced the teaching and choice of the subjects. Piaget's researches found that children pass through the stages, in which they understand the ways things are connected, but not distances, directions, or shapes. This led to the introduction of topology, a branch of mathematics popularly, known as rubber sheet geometry. The reform towards modern mathematics declined around the end of the 1970s. Criticism came from the parents who did not understand the new topics taught to their children.

Modern mathematics carries in it several new discoveries made since the seventeen century. Several new applications have come to light and consequently, several new branches of the subject have been developed. Some of these are set theory, topology, functional analysis, geometric transformation, linear programming, projective geometry, symbolic logic, probability theory, non-Euclidean geometry, etc.

In modern mathematics, fundamental laws are established which foster the discovery of new concepts by deductive reasoning. Thus, pupils may become aware of the unifying concepts and deductive nature of mathematics that received little attention in the traditional program of studying symbols, manipulations with symbols, telling, and drilling. Modern mathematics is an attempt to make the mathematics curriculum consistent with the demand for intelligence in living in the changing culture of our modern era. Modern mathematics introduces

A new point of view that attempts to redirect the emphasis in instruction to what and why of understanding as well how of manipulation. Attention is paid to the basic structure of the fundamental mathematical system as an effective means for developing background of understanding along with a facility in manipulation.

New subject matter that provides not only desirable and appropriate updating of the contents of the mathematics program in both the primary and the secondary schools, but also ways and means of clarifying, simplifying, and enriching its presentations.

New recognition of signification of application that directs attention to the basic structure of theorems to be proved or problems to be solved: The use of guess or conjectures, estimates, and other methods of intuition and induction are encouraged as an aid in approach to a theorem or problem situations. Appropriate techniques of deduction are carefully developed, and emphasis has been given to the necessity for their use in the development of a valid argument, whether in the proof of a theorem or solving a problem. This is true whether we are dealing with the myriad of solvable problems such as those which our environment creates daily, the unsolvable problems such as the three famous problems of antiquity (the trisection of an angle, the duplication of a cube, and squaring a circle), or the unsolved problems such as the discovery of proofs of Fermat's last theorem. The treatment is largely inductive rather than deductive because mathematics is more easily taught and learned inductively.

### Difference between Traditional Mathematics and Modern Mathematics

The impact of recent cultural, political, and technological revolutions highlights the inadequacies of the traditional approach to the teaching of mathematics. The traditional approach stresses mastery of skills, isolated facts, and technique before the understanding of concepts. The student is most highly rewarded for passive acceptance of that to which he is exposed. Topics and techniques were not those used currently in industry and commerce. Some mathematicians believed that the whole basis of school mathematics was somewhat illogical.

Recognizing these inadequacies, the curriculum planners, specialists, and others contributed to the development of a new mathematics program also called modern mathematics emphasizing that student must, from the beginning, be exposed to the structure of mathematics; they find concepts intensively interesting; they can discover and make use of patterns and relationships, they can think creatively and analytically; they are stimulated by and interested in new mathematical topics, and the learning process is shorter and more effective when based upon a conceptual approach that emphasizes the discovery and understanding of ideas and they use computers in studying mathematics.

### Components of Modern Mathematics

Modern mathematics has attempted to reorganize along more logical lines. There is greater stress on the role of definitions, axioms, logic, and foundations of the subject. The main components of modern mathematics are as follows:

#### 1. New Structure

Modern mathematics emphasizes mathematical structures. Every mathematical system starts from certain undefined terms, definitions, axioms, logic and laid the foundations of the subject. The student of modern mathematics looks for basic structures and general principles, rather than looking for unrelated tricks and processes he expected to memorize.

In a mathematical structure, we study the elements of a set together with the results of the operations on their elements. For example, the number system presents the simplest mathematical structure. In this system, we include the sets of natural numbers, integers, rational numbers, irrational numbers, real numbers, and complex numbers along with operations on them. Basic algebraic structures such as group, ring, filed and vector space also indicates such important properties that need to be known and understood. The geometric structure consists of the study of different types of geometries such as Euclidean, Non-Euclidean, projective, transformational geometries and topology, etc.

#### 2. New Language

Modern mathematics utilizes set notation and set language. It has presented set theory as a unifying branch for the entire subject. It is possible to adapt set theory as the base for all mathematics from arithmetic to the most advanced topics.

#### 3. New Contents

Modern mathematics constitutes new content. The main changes in the mathematics teaching program were in the choice of topics to be taught. Modern mathematics emphasized modern topics currently used in science and technology and economical sciences such as vectors and matrices, probability, and linear programming. Students used calculators and computers in the study of mathematics and sets provided a sound basis for learning the rest of the mathematics. Modern mathematics also included new topics in algebra such as Boolean algebra and group theory.

The basic philosophy that directs the shaping of the curriculum in secondary mathematics recognizes that there are different geometries, different algebra, different number systems, each with its individual postulation structure and techniques of investigation. Also, it is cognizant of the fact that the demands of a changing technological environment continually emphasize the necessity for modifications in the traditional contents within any instructional program, as well as for adaptations into the program.

#### 4. New Approach

A new approach to modern mathematics emphasizes understanding basic mathematical concepts rather than memorization of them. It introduces a change in the methodology of the teaching and learning of the subject matter. It places more emphasis on student's thinking and understanding, and the discovery mode of learning rather than reliance on the teacher's instruction and student's memorization. It also utilizes problem-solving techniques as suggested by Polya, Socrates, and others. The use of discovery and problem-solving methods is an important feature of the modern mathematics program.

### The emphasis in Modern Mathematics

Modern mathematics has placed special emphasis on understanding the mathematical structure and relationship of mathematical ideas, showing how these ideas are related to mathematics. The main emphases given in modern mathematics are:

1. Understanding the nature and role of deductive reasoning in algebra as well as geometry.
2. Understanding the language of sets which can be used to express all the different structures of mathematics.
3. Understanding basic arithmetical structures describing number systems such as properties of natural, rational, real, and complex numbers.
4. Understanding basic algebraic structures describing groups rings fields and vector space.
5. Understanding basic geometrical structures describing Euclidean, Non- Euclidean, projective, and transformational geometries and topology.
6. Understanding the basic structure of statistics describing the probability theory.
7. Understanding the basic structure of linear programming problems.

### Place of Modern Mathematics in School Curriculum

Modern mathematics deserves an important place in school education because of the following reasons:

1. It has made mathematics more enjoyable, more systematic, and more applicable.
2. It meets the demand of modern science and technology, industry and trade, and agriculture, etc.
3. Students learn mathematics meaningfully in less time.
4. It prepares students for more intelligent participation in the society of the future, which will be even more technological.
5. It offers clarification of mathematical ideas, and stimulating the students to learn mathematics.
6. It helps towards the unification and integration of different branches of mathematics.
7. Students develop a positive attitude towards mathematics.
8. It provides greater opportunities for thinking, reasoning, and discovery.

### Modern Mathematics in School Education of Nepal

The teaching of modern mathematical topics in schools of Nepal started with the introduction of the National Education System Plan (NESP) in 1971. School mathematics curricula of 1971 had been given greater emphasis on the new structure, new contents, and new approaches of teaching at all levels of school education. The school mathematics curriculum 1992 / 93 also had given the continuation of reform in the mathematics curriculum and included more advanced concepts of modern mathematics currently used in science and technology and economical sciences. Necessary modifications have been made in our secondary school mathematics curriculum by omitting old concepts and introducing new concepts. It has affected the existing mathematics contents in various ways. Its new approach to the teaching of the subject has introduced many new terms, symbols, concepts, and approaches, and demands of new mathematics teachers.

### Modern Mathematics in Bachelor in Mathematics Education

In Nepal, modern mathematical topics were introduced in school education since the introduction of the National Education System Plan 19711976. To meet the demand of mathematics teachers for teaching modern topics of mathematics in secondary schools of Nepal, the revised curriculum of the B. Ed. program, 2008 has included the modern mathematical topics in the course Foundation of Mathematics for three years B. Ed. and One Year B. Ed. in the mathematics education program of Tribhuvan University. Since then. Mahendra Sanskrit University, Purbanchal University have also introduced this subject in their year B. Ed. programs. The modern mathematics curricula of all universities have included the selected topics of modern mathematics such as set theory, symbolic logic, transformational geometry, non-Euclidean geometry, projective geometry, topology, probability theory, graph theory, number theory, correlation, regression, and linear programming into its newly reformed curriculum of modern mathematics of three years and one-year bachelor in mathematics education programs.

#### Conclusion

Modern mathematics has made teaching mathematics more meaningful, more realistic, more systematic, and more logical. Through its standard terminology, common symbols, sound structure, and logical approaches, it helps students to learn more mathematics in less time. It is the product of the realization that traditional mathematics is inadequate about the demands of the modern age of science and technology. It helps the student to prepare for intelligent participation in a society that will be much more technological than our own.