Mathematical heritage of india: some remedial issues: mathematics history in teaching
Yüklə 56,33 Kb.
|
- Bu səhifədəki naviqasiya:
- Mathematics in India-Some Remarks
- Root Cause of Lacking of Indian Mathematics Expansion
- 2) Chronological order had not been strictly adhered to.
- 3) There was no tradition of educating enmasses through proper writings.
- 4) Till the middle of eighteenth century A.D. Europeans were unaware of Indian culture and its contributions, though they knew it little from various sources
- Why do Research on Indian Mathematics
- The Necessary Steps for Upliftment of History of Indian Mathematics
- History of Mathematics as a Course of Study
- Course Outline Progressive series
- Triangles and Quadrilaterals
- Incircumscribing triangles and Cyclic quadrilaterals
- Combinatorics and Binomial Theorem
- History of Mathematics in Teaching-Some Issues
- Experimentation in Teaching
- Acknowledgement
|
MATHEMATICAL HERITAGE OF INDIA: SOME REMEDIAL ISSUES: MATHEMATICS HISTORY IN TEACHING Vinod Mishra Sant Longowal Institute of Engineering & Technology, India vinodmishra.2011@rediffmail.com
Mathematics has the largest history of more than 5000 years developed in isolation in the beginning of different cultures-later exchanged the ideas somehow till the emergence of modern era-and what we observe today is the synthesis of fruitful concepts of ancient, medieval and, of course, later modern periods in its major share. Post independent India has paid special attention towards the better understanding of history of Indian science by initiation of Government of India by constituting a National Commission (1969) functional under the Indian National Science Academy (INSA, New Delhi) and establishing National Institute of Science, Technology and Development Studies (New Delhi). In this context it is worthy to mention that the first major activity in the form of a symposium was held at University of Delhi in 1950. Since then a number of theme oriented programs (lectures/seminars/symposia/workshops) on history of mathematics in global perspective have been organized. Doubtless to say that the first International Congress on History of Science (July 8-14, 1929), Mexico City, under the auspices of International Commission on History of Science proved to be a pace-setter for the whole world. INSA has brought record publications of books and project based monographs on history of mathematics. It also brings a journal Indian Journal on History of Science. Yet another organization Indian Society for History of Mathematics constituted in 1978, successfully brings out journal Ga
In Indian subcontinent, mathematics flourished under the banner of Hindu astronomy in the early centuries and after the emergence of Jaina philosophy in about 300 B.C., Indian mathematics grew interacting each other. Interfusion of Indian mathematics with other cultural mathematics after the siddh  ntic age cannot be ruled out, in fact on small scale. The concept of permutation and combination, infinity among others are of course some of the notable contributions of the Jaina sects.
The ancient civilization of India differs from those of Egypt, Mesopotamia and Greece, in that of its traditions has been preserved without a break down to the present day. Until the advent of the archaeologist, the peasant of Egypt or Iraq had no knowledge of the culture of his forefathers, and it is doubtful whether his Greek counterpart had any but the vagvest ideas about the glory of Peri-clean Athens. In each case there had been an almost complete break with the past. On the other hand, the earliest Europeans to visit which indeed exaggerated that antiquity, and claimed not to have fundamentally changed for many thousands of years. To this day legends known to the humblest India recall the names of shadowy chieftains who lived nearly a thousand years before Christ, and the orthodox Brahman in his daily worship repeats hymns composed earlier. India and China have, in fact, the oldest continuous cultural traditions in the world [2, p. 4]. Regarding transmission of Buddhist culture and Indian scientific culture outside India, Gupta's view is worth noticeable (cf. Ga The rock edicts of king A The tradition of Buddhist education system gave birth to large-scale monastic universities. Some of these famous universities were Nalanda, Valabhi, Vikramsila, Jagaddala and Odantapuri. They attracted students and scholars from all parts of Asia. Of these, the Nalanda University was most famous with about 10000 students and 1500 teachers. The range of studies covered both sacred and secular subjects of Buddhist as well as Br Today we know more about western contribution than Indian. What are the reasons behind? Why did they impose their superiority? and put stamps even on the problems / ideas which were well-known and well-defined in India hundreds year ago. A few instances are:
had been obtained by M
for a cyclic quadrilateral, wherein r = circum-radius, a, b, c, d, the lengths of sides and s = (a+b+c+d)/2, has been given by Parame Others record correctly. A few quotes are: 1. “I shall not now speak of the knowledge of the Hindus,------ of their subtle discoveries in the science of astronomy –discoveries even more ingenious than those of the Greaks and Babylonians – of their method of calculation which no words can praise strongly enough –I mean the system using nine symbols. If these things were known by the people who think that they alone have mastered the sciences because they speak Greek they would perhaps, be convinced, though little late in the day, that other folk, not only Greeks, but men of a different tongue, know something as well as they”. The Syrian astronomer–monk Severus Sebokht (writing A.D.662) (cf. [2, p. VI ]) 2. “Most of the popular history of mathematics texts gives the impression that contemporary mathematical though stems directly from resurgence of western creativity after the dark age. However, mathematical ideas continued to develop in India, the Middle East and Orient from their ancient beginnings. There is evidence that some of modern mathematics travelled from these sources. Examples of Hindu mathematics appear in extant manuscripts from 8th and 9th centuries. Many of the permutation and combination formulas attributed to Cardano, Tartaglia and Pascal were known to the Hindus-----. It is clear that the algebra of combinations in the 12th century and perhaps earlier was considerably more advanced in Hindu mathematics than in Western Circles”. Sharon Kunoff (1990/91) 3. “The development of our system of notation for integers was one of the two most influential contributions of India to the history of mathematics. The other was the introduction of an equivalent of the sine function in trigonometry to replace the Greek tables of chords.” C.B. Boyer [p. 241] 4. “The Rule of Three which originated among the Hindus is a device used by oriental merchants to secure results to certain numerical problems.” Lam Lay Yong (p.329)(cf. Ga Other few examples are the (so-called) Pascal’s triangle, the notion of differential calculus, the basic trigonometry, Pell’s equation etc. Behind these discoveries they sometimes claim that Indians took inspiration from a common source, though it might not be.
Let us see the root cause of lacking of Indian mathematics expansion during ancient and medieval periods. 1) The proofs were not explicitly mentioned through they were understood well. In Hayashi’s view, “Neither the  ryabhat ya nor the Brahmasphutasiddhnta contains proof of their mathematical rules, but this does not necessarily mean that their authors did not prove them. It was probably a matter of the style of exposition. In fact, later prose commentaries contain a number of demonstrations or derivations, together with underlying principles ------. The recognition of the importance of proofs dates back at least to the time of Bhskara I (around A.D. 600), who, in this commentary on the ryabhatya, rejected the Jain value of  , , saying that it was only a tradition and there was no derivation of it. He also emphasized the importance of verifications of solutions to mathematical problems. ------ verifications are still found in Si hatilakas ri’s commentary (in the thirteenth century) on Srpati's Ga itatilaka, but occur very rarely thereafter, in contradiction to the growing popularity of derivations”
2) Chronological order had not been strictly adhered to. “Chronology is the back bone of history and its knowledge is essential for a historian dealing with any period, culture-area or subject. There cannot be a coherent history without a chronological order. Proper historical writing is not possible unless there is a sound chronology”, says Gupta (1989). For the case of India he further emphasis that “the problem of chronology continues to be very serious especially with regard to the prehistoric and ancient periods. The dates of most of the important events and literacy sources are full of serious controversies and divergent opinions. What to say about the absolute chronology, even a relative chronology is not free from challenges.” Chronology problem has arisen because whatever Indian writers wrote they dedicated to God and did not want credit by mentioning authorship, date etc. and more so in some cases they generally attributed as if the things (matters) were enunciated in antiquity. 3) There was no tradition of educating enmasses through proper writings. While writing proto- historic period of India, Balshaw (p. 30) argues that “among the many people who entered India in the 2nd millennium B.C.# was a group of related tribes whose priests has perfected a very advanced poetic technique, which they used for the composition of hymns to be sung in praise of their gods at sacrifices. These tribes, chief of which was that of the Bh ratas, settled mainly in East Punjab and in the region between the Satlaj and the Jamna which later became known as Brahmavarta. The hymns composed by their priests in their new home were carefully handed down by word of mouth, and early in the first millennium B.C. were collected and arranged. They were still not committed to writing, but by now they were looked on as so sacred that even minor alternations in their text were not permitted, and the priestly schools which preserved then devise the most remarkable and effective system of checks and counter checks to ensure their purity. Even when the art of writing was widely known in India the hymns were rarely written, but, thanks to the brilliant feats of memory of many generations of Brahmans and the extreme sanctity which the hymns were thought to possess, they have survived to the present day in a form which, from internal evidence, appears not to have been seriously tampered with for nearly three thousand years. This great collection of hymns is Rg Veda, still in theory the most sacred of the numerous sacred texts of the Hindus.”
In connection with “s 4) Till the middle of eighteenth century A.D. Europeans were unaware of Indian culture and its contributions, though they knew it little from various sources. The main reason was the lack of knowledge of Sanskrit, the principal Indian language, in which most of the oriental books had been written. “Until the last half of the 18th century Europeans made no real attempt to study India’s ancient past, and her early history was known only from brief passages in the works of Greek and Latin authors. A few devoted missionaries in the Pennisula gained deep understanding of contemporary Indian life, and a brilliant mastery of the vernaculars, but they made no real attempt to understand the historical background of the culture of the people among whom they worked. They accepted that culture at its face value, as very ancient and unchanging, and their only studies of Indian’s past were in the nature of speculations linking the Indians with the descendants of Noah and the vanished empires of the Bible” [2, p. 4]. Meanwhile Father Hanxleden was one of a few Jesuits who succeeded in mastering Sanskrit in the year from 1699 to 1732 and compiled the first ever Sanskrit grammar in European language. Since then more scholars including Father Coeurdoux (1767) (a Jesuit) and Sir William Jones (1783) (an Englishman) took keen interest and recognized the kinship of Sanskrit. [2, pp. 4-5]. “Of the little band of Englishmen who administered Bengal for the Honourable East India Company only one, Charles Wilkins (1749-1836), had managed to learn Sanskrit. With the aid of Wilkins and friendly Bengali pandits Jones began to learn the language. On the first day of 1784 the Asiatic Society of Bengal was founded on Jones’ initiative and with himself as president. In the Journal of this society, Asiatic Researches, the first real steps in revealing India’s past were taken. In November 1784 the first direct translation of a Sanskrit work into English, Wilkins’ Several important translations of works appeared in successive issues of Asiatic Researches and elsewhere. Jones and Wilkins were truly the fathers of Indology. They were followed in Calcutta by Henry Colebrooke (1765-1837) and Horace Hayman Wilson (1789-1860). To the works of these pioneers must be added that of the Frenchman Anquetil-Duperron, a Persian scholar who, in 1786, published a translation of four Upani Interest in Sanskrit began to grow in Europe as a result of those translations. In 1795, the government of the French Republic founded the Ecole des Languages Orientales Vivantes, and there Alexander Hamilton (1762-1824), one of the founding members of the Asiatic Society of Bengal, held prisoner on parole in France at the end of the Peace of Amiens in 1803, became the first person to teach Sanskrit in Europe. It was from Hamilton that Friedrich Schlegel, the first German Sanskritist, learnt the languages. The first university chair of Sanskrit was founded at the College de France in 1814 and held by Leonard de Chezy, while from 1818 onwards the larger German universities set up professorships. Sanskrit was first taught in England in 1805 at the training college of the East India Company at Hertfort. The earliest English chair was the Boden Professorship at Oxford, first filled in 1832, when it was conferred upon H.H. Wilson, who had been an important member of the Asiatic Society of Bengal. Chairs were afterwards founded at London, Cambridge and Edinburgh, and at several universities of Europe and America” [2, pp. 5-6]. Afterwards the establishment of French Societe Asiatique (1821), Paris and Royal Asiatic Society (1823), London beginnings the work of editing and study of ancient Indian literature went on space throughout the 19th century [2, p. 6].
Is it obligatory for an Indian to do research on Indian mathematics if one makes a mind of research in history of mathematics and its pedagogical advantages in teaching/education? The reason is obvious because:
For the benefit of mathematical community the following steps may be initiated/taken care of:
Hardly a University in India has adopted a course of study in History of Mathematics at undergraduate and postgraduate level. Might be due to shortage of expertise, due to lack of interest or non-availability of resource materials. The best course could be (1) History of Ancient and Medieval Mathematics, (2) History of Modern Mathematics or (3) Combination of (1) & (2). Below is a proposed outline syllabus of ‘History of Indian Mathematics’ course from the exhaustive content. Its parts could be suitably adopted after minor changes as per need at B.Sc. and M.Sc. programmes with mathematics as a major subject. Each chapter has to contain origin and development of the concepts, their necessasity, proofs /rationale, parallelism with other culture areas and relation to modern mathematics. Sufficient problems / examples should be included. Course Outline Progressive series Arithmetic progression. Geometrical and symbological interpretation. Complex series. Geometric progression and their geometrical interpretation.
Rule of three terms and their inversion. Rules of seven and nine. Series for Pie Values of Pie. Various types of Pie series and their convergence. Indeterminate Equations Indeterminate equations of first order of the type Second degree indeterminate equations: Theorem of square on the diagonal. Geometrical proof. Some geometrical constructions. Application of right triangle in Mah Convex quadrilaterals.
Incircumscribing triangle. Area. Circum-radius. Cyclic quadrilateral. Area. Third diagonal concept.
Theorem of square on half chord. Examples: Problems of broken bamboo, Lotus, Hawk and rat, Crane and fish, Two bamboos. Brahmagupta’s theorem. Shadow problems. Combinatorics and Binomial Theorem Combinatorics and Meru representation. Permutation of unlike and some as like things. Sum of permutations. The so called Binomial theorem. Meru way of presentation. Plane Trigonometry and Calculus  ,  . Sines and cosines of multiple and submultiple arcs. 
Notion of differential and integral calculus. The other topics which could be included are: quadratic equations, frustum like solids, circles and spheres, spherical trigonometry and many more.
The recent years have witnessed a growing attraction towards the role of history of mathematics not only in improving the teaching and learning of mathematics in its many facets at all levels but also in training of teachers and educational research. How to utilize history of mathematics in teaching? obviously becomes an important component of research in mathematical education, and forces us to evolve a method or a program that develops “a deeper understanding of the factors involved in the relations between history and pedagogy of mathematics, in different areas of mathematics, and with pupils at different stages with different environments and backgrounds”.
I would like to share teaching experience using history of mathematics in a Semester Course “Numerical Methods” for Bachelor of Engineering class at my parent institution. Students were to be taught the concept of numerical solutions of algebraic and transcendental equations particularly using Newton-Raphson (N-R) method for finding the nth root of N,  , a positive non-square integer. Instead of providing the students only the modern theory and applications of N-R method, I thought to equip them right from the effort of computing  in the Indian as well as other culture areas and their necessasity through algebraic and geometrical means. And ultimately relating the evaluation of to that of N-R method as a particular case of it. Even they were taught some extended modern generalization of . The class started with the basic introduction to algebraic and transcendental equations followed by computational efforts for evaluating  and in various cultural areas of India, Babylonia and Greek. Then followed the Lemma as [see Vinod Mishra, A Study in Vedic Geometry and Its Relevance to Science and Technology, Ph.D. Thesis, GKU Hardwar, 1998, Chapter II]:
Lemma 1: Let be a non square integer such that  where  is the largest positive integer and  the smallest integer. Then
are the successive convergents to the series  . (1)
Where The above Lemma can be easily extended to Yet another concept from Nar The following Lemma due to Brahmagupta is less known:
Are known then any number of other solutions can be found. For example if If The second approximation would be The above two results are exactly derived from Newton-Raphson iterative method. Let At the end they were explored some examples in old as well as modern areas of concern, not much beyond the scope of syllabus. With the little effort, the outcomes were:
Every teaching has different and unique strategy. The origin of concepts, their necessasity and development in various cultures, various proof of a particular aspect, their applications in varied areas and efforts made by various mathematicians are the major components to be included. The effort of introducing some glimpse of old theory and applications, wherever possible, must continue in the class in order to maintain their interest towards mathematics course. This in turn will strengthen their base. Acknowledgement Author feels gratefulness to Dr. S.L. Singh, retired Professor of Gurukula Kangri University (Hardwar) for some fruitful suggestions. REFERENCES 1. Bag, A.K., Mathematics in Ancient and Medieval India. Chaukhamba Orientalia, Varanasi, 1979. 2. Basham, A.L., The Wonder That Was India, Sendwick and Jackson Limited, London, 1985. 3. Boyer, C.B., A History of Mathematics (revised by Uta. C. Merzbach), John Wiley & Sons, 1989.
1(1979), 36-38.
Mishra, Vinod and S.L. Singh Theorem of Square on the Diagonal and its Application, Indian Journal of History of Science 31(1996) 157-166. Mishra, Vinod and S.L. Singh First Degree Indeterminate Analysis in Ancient India and its Application by Virasena, Indian Journal of History of Science 32(1997), 127-133. Mishra, Vinod and S.L. Singh Values of p from antiquity to Ramanujan, Sugakushi Kenkyu, No. 157, 1998, 12-25. Mishra, Vinod and S.L. Singh, Incircumscribing Triangles & Cyclic Quadrilaterals in Ancient & Medieval Indian Geometry, Sugakushi Kenkyu, No. 161, 1999,1-11 Mishra, Vinod, Geometry Relating to Circles and Spheres, Studies in History of Medicine and Science 18 (2002), 47-106, No. 1. Mishra, Vinod, Combinatorics and the (So-called) Binomial Theorem, Studies in History of Medicine and Science 18 (2002), 59-121, No.2. Mishra, Vinod, Similarity of Plane Figures: A Reflection in Indian Geometry, Studies in History of Medicine and Science 19 (2003), 95-114, No. 1-2. # This argument is rejected by many Indian and foreign scholars through archaeological findings and other ways, and concludes that Aryans were actually Indians. The report recently appears in the National News Paper The Hindustan Times, New Delhi Edition, Sept.2, 1998, front cover, under the title: ‘Aryans Were Not Invaders from Central Asia: Archaeologists Debunk Earlier Theories’. Vinod Mishra Department of Mathematics Sant Longowal Institute of Engineering & Technology Longowal 148106 Punjab. INDIA vinodmishra.2011@rediffmail.com Yüklə 56,33 Kb. Dostları ilə paylaş:
|
ta Bh
oka (3rd century B.C.) show that he had already paved the way for the expansion of Buddhism outside India. Subsequently, Buddhist missionaries took Buddhism to Central Asia, China, Korea, Japan and Tibet in the North, and to Burma, Ceylon, Thailand, Cambodia and other countries of the South. This helped in spreading Indian culture to these countries. It is well said that “Buddhism was, in fact, a spring wind flowing from one end of the garden of Asia to the other and causing to bloom not only the lotus of India, but the rose of Persia, the temple flower of Ceylon, the zebina of Tibet, the chrysanthemum of China and the cherry of Japan. It is also said that Asian culture is, as a whole, Buddhist culture”. Moreover, some of these countries received with Buddhism not only their region but practically the whole of their civilization and culture.
ika) dates back to Veda
Kuttaka method. Linear simultaneous indeterminate equations.
Cakraval method. Evaluation of 
and 
stand for the sum of denominator and numerator up to 
.
(2)
and 
then 
is also a solution of (2). If we take 
in this Lemma, then 
is also a solution. For finding an approximate value of square root, Nar
is a solution of (2), then 
(
), the first approximation. From (2), 
. Nar
is a close approximation of 
.
Then 
and 
So the first and second approximations of 
are respectively 
and 
Continuing of above technique yield third and subsequent approximations.