# A Holistic Approach To Creative Education Through Ancient Indian Mathematics - A New Paradigm

INTRODUCTION

The philosophy of education is to acquire right knowledge of different aspects and dimensions of the world around us and to disseminate the same for the benefit of the individual, society, nation and world.

The present day's education, though has undergone a radical change, with the development of modern science and technology, especially the computers, much needs to be done on the holistic approach to education, which can evolve personality. One might get tremendous amounts of information by logging on to the Internet, and the glut of information will lead to information garbage. In spite of all these we are lagging in imparting right education to right people thereby, creating disharmony. Therefore, one needs to have proper guidance and an evolved system which will lead to evolution of personality. This system should have a positive impact on the thinking process, beliefs, attitudes with its holistic approach to concepts, methods and techniques; thus creating harmony and ushering in peace.

Today, in majority of the cases, mathematics is considered to be a dry subject and still notion exists that it is only for the intelligent. Many dread mathematics, because the teachers themselves teach in an unimpressive way and the joy of making learning of mathematics a pleasurable experience is almost lacking. This is found right from the school to the college level.

In this regard, Vedic/Ancient Indian Mathematics (VM/AIM) with its wonderful concepts, methods and techniques, which have stood the test of time comes in handy for imparting a holistic, creative education in mathematics.

The Vedic Seers and Ancient Indian Scientists and Mathematicians perceived mathematics in Nature, in the form of primordial concepts, the different shapes and patterns symbolising the cosmic truths in the Micro and Macro aspects.

Mathematics was developed right from the fundamental concepts of Numerals, Place value, Zero and Infinity to the advanced concepts finding their applications in the Artificial Intelligence, computer based numerical methods, philosophy, science, arts, etc. In fact, one will be baffled by the fertile imagination of the ancient seers and scholars, who perceived mathematics in nature in all its glory and expressed the great mathematical truths and concepts through hymns, theorems and postulates which are lyrically beautiful, yet mathematically precise which speak volumes about them. It goes without saying that these methods, concepts and techniques of Mathematics in the form of Slokas and Sutras (Aphorisms) were conceived by the Ancient Indian Seers and scholars, centuries before their modern counterparts.

OBJECTIVE

(a) The present objective is to create an awareness about the vast potential of the Vedic/Ancient Indian Mathematics (V.M./A.I.M.) for the curriculum development and R & D works.

(b) To demonstrate with practical examples chosen from different levels of mathematics education; the benefits of learning V.M./A.I.N. like faster computing time, alternative and creative approach to problem solving, etc. This is substantiated by the results of number of workshops and case studies conducted.

(c) To motivate scientists scholars, teachers, philosophers and others to take up an in-depth study of VM/AIM in proper perspective and adopt it to the modern times, so that a great amount of this wonderful traditional knowledge is revived and developed which would be a true homage to the Seers and Scientists of yore.

(d) Last but not the least, to pave way for the adoption of time tested VM/AIM in the school/college curriculum and establish dedicated RBD centres.

APPROACH

An Integrated approach for the study and learning VM/AIM is made:

(i) Mathematical concepts embodied in the Vedas, Upanishads.

(ii) 'Vedic Mathematics' as expounded by H. H. Bharatikrishna Teerthaji consisting of 16 Aphorisms and 16 Corollaries, covering different topics in Mathematics, right from the fundamental Arithmetic operations to the higher level of Mathematics.

(iii) Ancient Indian Mathematics - contribution by various mathematicians of yore up to the celebrated genius Ramanuja. This integrated approach is unique of its kind. Various methods techniques are culled out, analysed and are expressed in modern mathematical notations, making it easily adaptable.

SELECTION CRITERIA

The methods and techniques are selected based on the following criteria. If the VM/AIM methods are better than the conventional ones in terms of step size, length, computation time, elegance and novelty, either individually or collectively, then these are selected in place of the conventional ones.

TECHNIQUE

The teaching of VM/AIM is done following the usual techniques like lecturing, using Audio-visual equipments like OHP, slide projector, computer aided learning, etc.

Before we go into the subject of VM/AIM in terms of examples, it is imperative to know at least some fundamental facts about the Vedas, their nature, structure, etc., which would facilitate in understanding and appreciating the mathematical concepts embodied in the Vedas.

Vedas are the earliest systematic literature in the entire world existing since thousands of years, which have stood the test of time and continue to challenge the mankind with renewed freshness and vitality. Vedas, which are the fountain head of knowledge and wisdom, are built upon sound structure which are definite, clear, unambiguous, generative and complete, thus satisfying the properties of an algorithm! This amply proves the fact that the Vedas are the highest revelations to the Rsis (Seers) of Yore and are not a mere compendium and collection of literature from different parts of the world, and no second version of Vedas exists anywhere. Hence, the Vedas have to be studied and analysed in proper perspective, in order to bring out the truths hidden in them. Now let us take some stock illustrations from Vedas and other ancient Indian sources in Mathematics at different levels i.e. right from the fundamental level of Arithmetic to Algebra, Calculus, Quadratics, Polynomials, Astronomy, Geometry, etc. which will give a glimpse of the vast scope of the same.

Now coming to the Vedas, the most outstanding, fundamental contribution for which the entire world is beholden to ancient India is the invention of decimal numerals (with place values), zero and infinity, for example: 576, 685, 1998, etc. Can there be a more elegant, better method where the digits placed in the units, tenths, hundredths or thousandth place, indicate their respective values and magnitude.

In arithmetic operations one finds use of complimentary numbers very effectively and finds 10 different varieties of multiplication based on the Vedic mathematical Sutras propounded by H. H. Bharatikrishna Teertha (1880-1964). He being a brilliant scholar and Saint discovered 16 aphorisms and 16 corollaries after 8 years of intense penance, covering variegated topics (both elementary and advanced) in mathematics. Consider the following examples:

(i) Complimentary numbers Here the objective is to convert all the numerals greater than 5 to less than 5 in order to facilitate speedy calculation and reduce carry over. For ex. 9,8,7,6 are expressed as 1,2,3 & 4 respectively with base as 10.

(ii) Using complimentary notation known as 'Vinculum' following examples will make clear, about the usage and benefits of this:

(a) 28 -> 32 [eq. to 30-2]

(b) 19 -> 21 [eq. to 20-1]

(c) 278 -> 322 [eq. 300-22]

(d) 289621 -> 310421

Note: whenever complimentary notations are used, the digit preceding, gets augmented by 1 and while normalising it gets reduced by 1. In the example (d) we see how only the digits greater than 5 are converted and the others remain as they are.

Before we take up other topics in Mathematics we cannot help but quote some examples on the philosophical aspect of numbers, which the ancient seers very effectively utilised.

Now let us consider this very interesting example. The idea of counting nine numbers and zero which is an Indian system is closely connected with nine months of development of human embryo. Man develops in nine months through nine successive stages, like the nine number and completes his birth in the 10th and this is the 10th Avatara of 'MAN' to put it figuratively. Similarly the body has nine openings (Navadvara) and the 10th being the Brahmadvara which can be located in the soft region in the head in a new born infant. This small micro-opening called the Brahmarandhra. At the top of the head through which a yogi leaves the body. Thus the 10th opening is the gate to the highest stage of development, similar to his birth from the womb in the 10th month. In the former he is delivered downwards and in the lateral upwards! This is the essence of story of deliverance of man from the imprisoning restrictions of matter which has been beautifully delineated in many philosophical Tantric and musical texts.

Similarly the concept of Sets and cardinal numbers are conceived by the Seers and scholars. For example we have deep significance for numbers like seven as Saptaswara (seven basic notes in Music), Sapta Rsis (seven great sages), Saptavarna (seven colours), nine planets (Navagrahas), Navanidhi (nine treasures), Dasadik (ten directions), Ekadasa Rudra (eleven Rudras), Dvadasamasa (twelve months), Saptavimsat Nakshatrah (27 stars) etc.

Similarly we can quote many examples from different branches of mathematics, even in geometry: the Sulva Sutras (which are the origin of geometry) make use of this branch of mathematics to construct varieties of altars for the purposes of sacrifices. The magnificent temples which stand testimony to the skills and designs of the workers are the living examples of applied mathematics.

Mathematics was also brought into worship by use of Mantrabandhas which are nothing but mystical geometric diagrams inscribed with digits or cosmic energy capsules (beejaksharas). These diagrams are of various sizes, shapes and colours. The word of Mantra and Tantra is indeed a network of energies, forces and vectors and are very dynamic. These geometric forms indicate outward manifestation of the spirit so as to exert its influence in the desired manner. The power possessed by the Yantra (graphical representation) is sought to be stepped up by their inscription and infusion of the appropriate Mantra. In short these can be termed in the modern context as the printed circuit boards which vibrate into light through the electrical energy passed and performs the desired functions independently and collectively. The Yantra is the basic hardware and the Mantra is the software.

CONCLUSIONS

1. Vedic/Ancient Indian Mathematics (VM/AIM) with their novel methods, techniques and approach help the learners to develop their aptitude and creative rational thinking.

2. The VM/AIM which have stood the test of time for centuries with their novel creative approach and techniques help the students to develop their intuitive capacity, which in turn has a positive impact on the right half of the brain, paving way for the holistic development.

3. Many workshops and courses conducted for students of different classes, be it urban or rural, have proved the efficacy of these and the results are very encouraging.

4. It has been found from practical trials that one year portion of school curriculum in mathematics can be completed in about 60% of the time, thus making learning faster and enjoyable. In fact VM/AIM has been popularly known as 'Maths with Smiles'.

5. This will prove to be a boon for all the competitive exams, with its much faster and accurate results and can be adopted globally.

6. Learning of VM/AIM leads to education in different dimensions and appreciate philosophy, religion in better perspective which in turn has a positive impact on the personality development, leading to harmony and peace.

7. VM/AIM open up new vistas for R&D, especially on computer based numerical methods, development of new algorithms apart from opening up of frontier areas of interdisciplinary research. Example: Nyayasastra (one of the six Darshanas has wonderful scope in Artificial Intelligence). The author has already culled out and developed about 15 computer based numerical methods based on VM/AIM which are unique of their kind.

8. The Ancient Indian Mathematics not only kept up the curriculum development, but also made mathematics interesting to house wives, common people and people from non-mathematical areas of study. They composed aphorisms, slokas and hymns, which were lyrically beautiful yet mathematically precise, thus contributing to the holistic development of the subject.

9. Last but not the least, study and practice of VM/AIM with their creative and novel approach can aid in holistic development of personality. This will not only revive the ancient treasure of knowledge and wisdom, but also will be a true homage to the great Seers and Scholars of Yore.

The philosophy of education is to acquire right knowledge of different aspects and dimensions of the world around us and to disseminate the same for the benefit of the individual, society, nation and world.

The present day's education, though has undergone a radical change, with the development of modern science and technology, especially the computers, much needs to be done on the holistic approach to education, which can evolve personality. One might get tremendous amounts of information by logging on to the Internet, and the glut of information will lead to information garbage. In spite of all these we are lagging in imparting right education to right people thereby, creating disharmony. Therefore, one needs to have proper guidance and an evolved system which will lead to evolution of personality. This system should have a positive impact on the thinking process, beliefs, attitudes with its holistic approach to concepts, methods and techniques; thus creating harmony and ushering in peace.

Today, in majority of the cases, mathematics is considered to be a dry subject and still notion exists that it is only for the intelligent. Many dread mathematics, because the teachers themselves teach in an unimpressive way and the joy of making learning of mathematics a pleasurable experience is almost lacking. This is found right from the school to the college level.

In this regard, Vedic/Ancient Indian Mathematics (VM/AIM) with its wonderful concepts, methods and techniques, which have stood the test of time comes in handy for imparting a holistic, creative education in mathematics.

The Vedic Seers and Ancient Indian Scientists and Mathematicians perceived mathematics in Nature, in the form of primordial concepts, the different shapes and patterns symbolising the cosmic truths in the Micro and Macro aspects.

Mathematics was developed right from the fundamental concepts of Numerals, Place value, Zero and Infinity to the advanced concepts finding their applications in the Artificial Intelligence, computer based numerical methods, philosophy, science, arts, etc. In fact, one will be baffled by the fertile imagination of the ancient seers and scholars, who perceived mathematics in nature in all its glory and expressed the great mathematical truths and concepts through hymns, theorems and postulates which are lyrically beautiful, yet mathematically precise which speak volumes about them. It goes without saying that these methods, concepts and techniques of Mathematics in the form of Slokas and Sutras (Aphorisms) were conceived by the Ancient Indian Seers and scholars, centuries before their modern counterparts.

OBJECTIVE

(a) The present objective is to create an awareness about the vast potential of the Vedic/Ancient Indian Mathematics (V.M./A.I.M.) for the curriculum development and R & D works.

(b) To demonstrate with practical examples chosen from different levels of mathematics education; the benefits of learning V.M./A.I.N. like faster computing time, alternative and creative approach to problem solving, etc. This is substantiated by the results of number of workshops and case studies conducted.

(c) To motivate scientists scholars, teachers, philosophers and others to take up an in-depth study of VM/AIM in proper perspective and adopt it to the modern times, so that a great amount of this wonderful traditional knowledge is revived and developed which would be a true homage to the Seers and Scientists of yore.

(d) Last but not the least, to pave way for the adoption of time tested VM/AIM in the school/college curriculum and establish dedicated RBD centres.

APPROACH

An Integrated approach for the study and learning VM/AIM is made:

(i) Mathematical concepts embodied in the Vedas, Upanishads.

(ii) 'Vedic Mathematics' as expounded by H. H. Bharatikrishna Teerthaji consisting of 16 Aphorisms and 16 Corollaries, covering different topics in Mathematics, right from the fundamental Arithmetic operations to the higher level of Mathematics.

(iii) Ancient Indian Mathematics - contribution by various mathematicians of yore up to the celebrated genius Ramanuja. This integrated approach is unique of its kind. Various methods techniques are culled out, analysed and are expressed in modern mathematical notations, making it easily adaptable.

SELECTION CRITERIA

The methods and techniques are selected based on the following criteria. If the VM/AIM methods are better than the conventional ones in terms of step size, length, computation time, elegance and novelty, either individually or collectively, then these are selected in place of the conventional ones.

TECHNIQUE

The teaching of VM/AIM is done following the usual techniques like lecturing, using Audio-visual equipments like OHP, slide projector, computer aided learning, etc.

Before we go into the subject of VM/AIM in terms of examples, it is imperative to know at least some fundamental facts about the Vedas, their nature, structure, etc., which would facilitate in understanding and appreciating the mathematical concepts embodied in the Vedas.

Vedas are the earliest systematic literature in the entire world existing since thousands of years, which have stood the test of time and continue to challenge the mankind with renewed freshness and vitality. Vedas, which are the fountain head of knowledge and wisdom, are built upon sound structure which are definite, clear, unambiguous, generative and complete, thus satisfying the properties of an algorithm! This amply proves the fact that the Vedas are the highest revelations to the Rsis (Seers) of Yore and are not a mere compendium and collection of literature from different parts of the world, and no second version of Vedas exists anywhere. Hence, the Vedas have to be studied and analysed in proper perspective, in order to bring out the truths hidden in them. Now let us take some stock illustrations from Vedas and other ancient Indian sources in Mathematics at different levels i.e. right from the fundamental level of Arithmetic to Algebra, Calculus, Quadratics, Polynomials, Astronomy, Geometry, etc. which will give a glimpse of the vast scope of the same.

Now coming to the Vedas, the most outstanding, fundamental contribution for which the entire world is beholden to ancient India is the invention of decimal numerals (with place values), zero and infinity, for example: 576, 685, 1998, etc. Can there be a more elegant, better method where the digits placed in the units, tenths, hundredths or thousandth place, indicate their respective values and magnitude.

In arithmetic operations one finds use of complimentary numbers very effectively and finds 10 different varieties of multiplication based on the Vedic mathematical Sutras propounded by H. H. Bharatikrishna Teertha (1880-1964). He being a brilliant scholar and Saint discovered 16 aphorisms and 16 corollaries after 8 years of intense penance, covering variegated topics (both elementary and advanced) in mathematics. Consider the following examples:

(i) Complimentary numbers Here the objective is to convert all the numerals greater than 5 to less than 5 in order to facilitate speedy calculation and reduce carry over. For ex. 9,8,7,6 are expressed as 1,2,3 & 4 respectively with base as 10.

(ii) Using complimentary notation known as 'Vinculum' following examples will make clear, about the usage and benefits of this:

(a) 28 -> 32 [eq. to 30-2]

(b) 19 -> 21 [eq. to 20-1]

(c) 278 -> 322 [eq. 300-22]

(d) 289621 -> 310421

Note: whenever complimentary notations are used, the digit preceding, gets augmented by 1 and while normalising it gets reduced by 1. In the example (d) we see how only the digits greater than 5 are converted and the others remain as they are.

Before we take up other topics in Mathematics we cannot help but quote some examples on the philosophical aspect of numbers, which the ancient seers very effectively utilised.

Now let us consider this very interesting example. The idea of counting nine numbers and zero which is an Indian system is closely connected with nine months of development of human embryo. Man develops in nine months through nine successive stages, like the nine number and completes his birth in the 10th and this is the 10th Avatara of 'MAN' to put it figuratively. Similarly the body has nine openings (Navadvara) and the 10th being the Brahmadvara which can be located in the soft region in the head in a new born infant. This small micro-opening called the Brahmarandhra. At the top of the head through which a yogi leaves the body. Thus the 10th opening is the gate to the highest stage of development, similar to his birth from the womb in the 10th month. In the former he is delivered downwards and in the lateral upwards! This is the essence of story of deliverance of man from the imprisoning restrictions of matter which has been beautifully delineated in many philosophical Tantric and musical texts.

Similarly the concept of Sets and cardinal numbers are conceived by the Seers and scholars. For example we have deep significance for numbers like seven as Saptaswara (seven basic notes in Music), Sapta Rsis (seven great sages), Saptavarna (seven colours), nine planets (Navagrahas), Navanidhi (nine treasures), Dasadik (ten directions), Ekadasa Rudra (eleven Rudras), Dvadasamasa (twelve months), Saptavimsat Nakshatrah (27 stars) etc.

Similarly we can quote many examples from different branches of mathematics, even in geometry: the Sulva Sutras (which are the origin of geometry) make use of this branch of mathematics to construct varieties of altars for the purposes of sacrifices. The magnificent temples which stand testimony to the skills and designs of the workers are the living examples of applied mathematics.

Mathematics was also brought into worship by use of Mantrabandhas which are nothing but mystical geometric diagrams inscribed with digits or cosmic energy capsules (beejaksharas). These diagrams are of various sizes, shapes and colours. The word of Mantra and Tantra is indeed a network of energies, forces and vectors and are very dynamic. These geometric forms indicate outward manifestation of the spirit so as to exert its influence in the desired manner. The power possessed by the Yantra (graphical representation) is sought to be stepped up by their inscription and infusion of the appropriate Mantra. In short these can be termed in the modern context as the printed circuit boards which vibrate into light through the electrical energy passed and performs the desired functions independently and collectively. The Yantra is the basic hardware and the Mantra is the software.

CONCLUSIONS

1. Vedic/Ancient Indian Mathematics (VM/AIM) with their novel methods, techniques and approach help the learners to develop their aptitude and creative rational thinking.

2. The VM/AIM which have stood the test of time for centuries with their novel creative approach and techniques help the students to develop their intuitive capacity, which in turn has a positive impact on the right half of the brain, paving way for the holistic development.

3. Many workshops and courses conducted for students of different classes, be it urban or rural, have proved the efficacy of these and the results are very encouraging.

4. It has been found from practical trials that one year portion of school curriculum in mathematics can be completed in about 60% of the time, thus making learning faster and enjoyable. In fact VM/AIM has been popularly known as 'Maths with Smiles'.

5. This will prove to be a boon for all the competitive exams, with its much faster and accurate results and can be adopted globally.

6. Learning of VM/AIM leads to education in different dimensions and appreciate philosophy, religion in better perspective which in turn has a positive impact on the personality development, leading to harmony and peace.

7. VM/AIM open up new vistas for R&D, especially on computer based numerical methods, development of new algorithms apart from opening up of frontier areas of interdisciplinary research. Example: Nyayasastra (one of the six Darshanas has wonderful scope in Artificial Intelligence). The author has already culled out and developed about 15 computer based numerical methods based on VM/AIM which are unique of their kind.

8. The Ancient Indian Mathematics not only kept up the curriculum development, but also made mathematics interesting to house wives, common people and people from non-mathematical areas of study. They composed aphorisms, slokas and hymns, which were lyrically beautiful yet mathematically precise, thus contributing to the holistic development of the subject.

9. Last but not the least, study and practice of VM/AIM with their creative and novel approach can aid in holistic development of personality. This will not only revive the ancient treasure of knowledge and wisdom, but also will be a true homage to the great Seers and Scholars of Yore.

About the Author

This Article is by P.K.Srivathsa, Expert at the Vedic Maths Forum India, an expert authority in the field of Vedic Mental Mathematics who revolutionizes the way children learn math. The tutorials on the Vedic Math Methods can be found at http://www.vedicmathsindia.org

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