Workshops on Bhaskaracharya's Lilavati

         Bhaskaracharya was one of the great mathematicians who made important contributions to the rich tradition of Indian mathematics and astronomy. Born in Shalivahan shaka 1036 (1114 AD), he composed Lilavati, Bijaganita, Siddhantasiromani and Karanakutuhala. Of these the Lilavati became immensely popular throughout India. Since 2014 marked the 900th birth anniversary of Bhaskaracharya, the Vidya Prasarak Mandal, Thane, has embarked upon celebrating it in big way. As a part of a yearlong celebrations of this event workshops on Bhaskaracharya's Lilavati were conducted at various places in India. This write up gives a brief account of the strategy of workshops along with the first hand experiences gained in conducting these workshops.

The workshop was arranged in two sessions of 90 minutes each. The first session was devoted to acquainting the participants with the long and rich tradition of mathematics in India. Contributions of mathematicians like Boudhayana, Pingala and Katyayana before Christian era were discussed. Then the contributions of mathematicians like Aryabhatta, Varahmihira, Brahmagupta after Christian era were mentioned briefly. Bhaskaracharya, who was born in 12th century, was presented as a gem in the long tradition of mathematical scholarship existing in India. The influence of previous mathematicians on Bhaskaracharya's work was brought out clearly. His major contribution to astronomy and mathematics was highlighted through illustrative examples. Reference was also made to the inscription by his grandson Changdev at Patanadevi near Chalisgaon in Maharashtra. 

The second session of the workshop was devoted to the book Lilavati written by Bhaskaracharya. The salient features of the book like its poetic description, its didactic nature, its simplicity, etc. are highlighted. Major part of this session was used to solve the problems from Lilavati. These problems were selected taking into account the nature of the audience. For example, problems based on operations like addition, multiplication, division, subtraction, etc. were chosen if the audience was composed of primary school children. Problems related to fractions, permutations and combinations, mensuration quadratic equations, etc. were chosen if the audience was from high schools. For college level students problems related to Pythagoras theorem, trigonometry, mathematical progression were chosen. If the audience was composed of general public then problems based on interest, area, volume calculation etc. were chosen. The original problem in Sanskrit and its translation either in Marathi or English was displayed using LCD projector. The participants were encouraged to solve the given problem after giving appropriate hints. One of the successful candidates was then requested to come forward and solve the problem on the blackboard for the benefit of others in the group.In a span of a calendar year (2014) fifty workshops were conducted at different places in the states of Maharashtra and Andhra Pradesh and Madhya Pradesh catering to thousands of students (See Appendix 1 for the list of workshops and the number of participants). These workshops were arranged for students studying at different levels of education. Workshops were also arranged for students undergoing professional course like polytechnic, engineering and management. On some occasions the workshops were arranged for teachers, parents and members of industrial associations (See Appendix 2 for photographs). On the requests of educational institutions in India the task of conducting workshops on Lilavati has been continued in the calendar year 2015 also.  

The experience of conducting these workshops had been quite encouraging. It has been noticed that the students like the way Bhaskaracharya has posed problems taking situations from everyday life. The dialogue mode of raising a problem situation is appreciated by the students at all the ages. Moreover, they feel motivated to deal with a given problem as they are addressed as a lovely young girl, young mathematicians, etc. Majority of school students look at mathematics as a dull and dry subject to learn. When exposed to the problems in Lilavati they realise that their belief towards school mathematics needs to be changed. Teachers look at problems from Lilavati as guidelines for making mathematics interesting in the classroom. Bhaskaracharya has not solved any problem in Lilavati but has provided appropriate hints to deal with them and has provided freedom to deal with the situation differently. This open ended mode of problem solving, although different from their school teaching, is appreciated by a majority of the teacher participants. 

An important outcome of the workshops on Lilavati is the creation of awareness among students and teachers of India's contribution to the development of mathematics. Formal school education presents a Eurocentric approach to school and college mathematics. These workshops enable the participants to realise that India's contribution to mathematics has been substantial. This realization goes a long way in enhancing their confidence and changing their attitude towards mathematics.  

The workshops on Bhaskaracharya's Lilavati are being conducted by Dr. Sudhakar C. Agarkar, Professor and Dean, VPM's Academy of International Education and Research, Thane. Retired as Professor (H) from the Homi Bhabha Centre for Science Education (HBCSE), Tata Institute of Fundamental Research (TIFR), Dr. Agarkar has a long experience of conducting research in science and mathematics education. Working with slum dwellers in Mumbai and tribal students in western ghat he has developed Remedial Instructional Strategy that enhances the performance of disadvantaged students significantly. Anyone interested to arrange workshops on Bhaskaracharya's Lilavati should contact him at sagarkar@vpmthane.org OR +91-8097274097.

Appendix 1: List of Workshops

Sr. No.	Place	Date	Participants
01	English Medium High School, Akola Maharashtra State	25/01/2014	40 Students of standard 12
02	Bal Shivaji School, Akola, Maharashtra State	25/01/2014	60 Students of standards
03	SPM's High School, Wani, Dist. Yeotmal Maharashtra State	06/02/2014	55 Students of standard 10
04	Balaji Polytechnic, Wani, Dist. Yeotmal Maharashtra State	06/02/2014	300 Students of polytechnic Course
05	Sanmitra Sainik School, Bhiwakund Dist. Chandrapur, Maharashtra State	07/02/2014	320 Students of standards 5, 6, 7, 8, 9 and 10
06	Sau. A. K. Joshi English Medium School, Thane, Maharashtra State	13/02/2014	100 Students of standards 8 and 9
07	VPM's Polytechnic, Thane, Maharashtra State	14/02/2014	60 Students of First year Polytechnic course
08	Kaleshwar Vidyalaya, Moor, Dist. Ratnagiri, Maharashtra State	25/02/2014	150 Students of standards 8 and 9
09	Dadasaheb Tirodkar High School, Kudal, Dist. Sindhudurg, Maharashtra State	26/02/2014	150 Students of standards 7, 8 and 9
10	Shri L. S. Desai High School, Pat, Dist. Sindhudurg, Maharashtra State	26/02/2014	300 Students (stds 7, 8 & 9)
11	Dr. V. N. Bedekar Institute of Management Studies, Thane, Maharashtra State	15/03/2014	120 Students of First and Second year MMS course
12	Synergy School, Peddapuram, Dist. Rajahmundry, Andhra Pradesh	27/04/2014	70 Students (std. 12)
13	Synergy School Peddapuram, Dist. Visakhapatnam, Andhra Pradesh	28/04/2014	30 Primary school teachers
14	Sri Prakash School, Payakaraopeta, Andhra Pradesh	30/04/2014	40 Secondary school teachers
15	Thane Small Scale Industries Association (TSSIA), Thane, Maharashtra State	17/05/2014	30 Members of TSSIA
16	Sri Prakash Vidyaniketan, Visakhapatnam Andhra Pradesh	04/06/2014	60 Teachers (primary and secondary)
17	Solapur Science Centre, University Campus, Solapur, Maharashtra State	27/06/2014	150 Students of standards 7, 8, 9 and Parents
18	Suyash Gurukul and Junior College, Solapur, Maharashtra State	27/06/2014	320 Students standards 7, 8, 9 and 10
19	Rajaram Junior College, Pandharpur, Dist. Solapur, Maharashtra State	28/06/2014	90 (Students, teachers and parents)
20	VPM's B. N. Bandodkar College of Science, Thane, Maharashtra State	12/07/2014	150 Students of Second and Third year B.Sc.
21	Genesis Institute of Technology, Kasarwadi, Dist. Kolhapur, Maharashtra State	16/07/2014	200 Students of Second and third year Engineering
22	Ratnappa Kumbhar High School, R. K. Nagar, Kolhapur, Maharashtra State	16/07/2014	300 Students of standards 9 and 10
23	Smt. V. D. Shinde High School, Gadhinglaz, Dist. Kolhapur, Maharashtra State	17/07/2014	300 Students of standards 9 and 10
24	Jagruti Prashala, Gadhinglaz Dist. Kolhapur, Maharashtra State	17/07/2014	100 Students of standard 10
25	Shivaraj Junior College of Arts, Science and Commerce, Gadhinglaz, Dist. Kolhapur, Maharashtra State	18/07/2014	175 Students of standard 12
26	Sadhana Vidyalaya, Gadhinglaz Dist. Kolhapur, Maharashtra State	18/07/2014	100 Students of standard 9
27	Dudhasagar College of Science, Bidri Dist. Kolhapur, Maharashtra State	19/07/2014	30 Students of Final year B.Sc.
28	Dr. Bedekar Vidyamandir, Thane, Maharashtra State	28/07/2014	300 Students of standard 10
29	K. G. Joshi College of Arts and N. G. Bedekar College Commerce, Thane, Maharashtra State	31/07/2014	296 Students of Final year B. A.
30	Maharashtra Institute of Technology, Aurangabad, Maharashtra State	07/08/2014	190 Students of B.Tech. Course
31	MGM's Jawaharlal College of Engineering, Aurangabad, Maharashtra State	08/08/2014	260 Students of second and third year engineering
32	Ramanand Tirth Marathwada Research Centre, Aurangabad, Maharashtra State	09/08/2014	30 Mathematics teachers drawn from 10 districts
33	K. G. Joshi College of Arts and N. G. Bedekar College Commerce, Thane, Maharashtra State	13/08/2014	150 Students of standards 11 and 12
34	Saraswati Residential School, Sharda Vihar, Bhopal, Madhya Pradesh	17/08/2014	300 Students of standards 10, 11 and 12
35	New English School, Amravati Maharashtra State	19/08/2014	200 Students of standards 9 and 10
36	Saraswati Secondary School, Thane Maharashtra State	13/09/2014	300 Students of standards 8 and 9
37	Padmashree Annasaheb Jadhav, Secondary School, Bhiwandi, Dist. Thane Maharashtra State	22/09/2014	250 Students of standards 9 and 10
38	Gurunanak Junior College of Commerce Kalyan, Dist. Thane, Maharashtra State	17/11/2014	300 Students of standards 10, 11 and 12
39	Gurunanak English Medium High School, Kalyan, Dist. Thane, Maharashtra State	17/11/2014	250 Students of standards 7, 8 and 9
40	Vidya Pratishthan's New English School, Baramati, Dist. Pune, Maharashtra State	25/11/2014	300 Students of standards 9 and 10
41	Vidya Pratishthan's English School, Baramati, Dist. Pune, Maharashtra State	25/11/2014	60 Science and maths teachers from 12 schools
42	Vidya Pratishthan's New Balvikas Mandir, Pimpli, Dist. Pune, Maharashtra State	26/11/2014	300 Students of standards 9 and 10 from 3 schools
43	Vidya Pratishthan's Bal Vikas Mandir, Baramati, Dist Pune, Maharashtra State	26/11/2014	300 Students of standards 9 and 10 from 3 schools
44	Magarpatta City Public School, Hadapsar, Pune, Maharashtra State	27/11/2014	40 Teachers of all subjects
45	Biyani Military School, Bhusaval, Dist. Jalgaon, Maharashtra State	29/11/2014	250 Students of standards 7, 8, 9 and 10
46	Sanskrit Department, Tirupati University, Tirupati, Andhra Pradesh	10/12/2014	60 Adults and 80 students of Sanskrit Vidyapeeth
47	Lokpuram Public School, Thane (west) Maharashtra State	19/12/2014	280 Students of standard 8
48	Science College, Adoni, Dist. Kurnool, Andhra Pradesh	20/12/2014	55 Mathematics teachers from the region
49	Sri Prakash Vidyaniketan, Visakhaptanam, Andhra Pradesh	22/12/2014	350 Students of standard 10
50	Sri Prakash Vidyaniketan, Kappulpada, Andhra Pradesh	22/12/2014	200 students of standard 9
51	Arts College, Kowad, Tal. Chandgad, Dist. Kolhapur, Maharashtra State	09/01/2015	250 Students of FY, SY and TY B.A.
52	Shriram Vidyalaya, Kowad, Tal. Chandgad, Dist. Kolhapur, Maharashtra State	09/01/2015	200 Students of standards 8, 9 and 10
53	Jayaparkash Vidyalaya, Kini, Tal. Dist. Kolhapur, Maharashtra State	09/01/2015	50 Mathematics teachers (primary and secondary)
54	Sai International School, Sindhri, Tal. Gadhinglaz, Dist- Kolhapur, Maharashtra State	10/01/2015	65 Parents
55	New Madhyamik Vidyalaya, Uchagaon, Tal. Karveer, Dist. Kolhapur, Maharashtra State	11/01/2015	120 Students of standards 8, and 9
56	Mahatma Gandhi Vidyalaya, Varangaon Dist. Jalgaon, Maharashtra State	16/01/2015	300 Students of standards 8, 9 and 10
57	Bhusaval High School and Junior College, Bhusaval, Dist. Jalgaon, Maharashtra State	16/01/2015	100 Students of standards 8 and 9
58	K. Narkhede Vidyalaya and Junior College, Bhusaval, Dist. Jalgaon, Maharashtra State	16/01/2015	250 Students of standards 8, 9 and 10
59	Bari Samaj Madhyamik Vidyalaya , Shirsoli, Dist. Jalgaon, Maharashtra State	17/01/2015	200 Students of standards 8, 9 and 10
60	Bhagirath English School, Bhusaval, Dist. Jalgaon, Maharashtra State	17/01/2015	300 Students of standards 8, 9 and 10

Appendix 2: Photographs

A Session for High School students at Shiksha Prasarak Mandal's Vidyalaya, Wani Dist. Yeotmal, Maharashtra State	A Session for the polytechnic students of the Balaji Polytechnic Wani, Dist. Yeotmal, Maharashtra State
Workshop for High School students of Dadasaheb Tirodkar Vidyalaya Kudal, Dist. Sindhudurg, Maharashtra State	A workshop for the students of VPM's Polytechnic Thane, Maharashtra State
A workshop for the members of Thane Small Scale Industries Association Thane, Maharashtra State	Active participation of a graduate student at Sanskrit Vidyapeeth, Tirupati, Andhra Pradesh
A workshop for Junior College students at Gadhinglaz Dist. Kolhapur, Maharashtra State	A high school student attempting to solve the problem on the blackboard.
Students of Ratnappa Kumbhar High School Kolhapur engrossed problem solving