I am sure all you mathematics teachers out there have had the following experience. You meet a stranger and they ask you what your profession is; you say you teach mathematics and immediately they respond by either saying that they were very good at mathematics or by becoming apologetic about having been terrible at mathematics! In short, very few are neutral towards mathematics – they either love it or hate it! Perhaps this is an indicator that mathematics education needs a lot of attention from educators.
Nowadays there is a great push towards improving mathematics education, and in my opinion for the wrong reason: that the world is becoming increasingly technological, and in order to survive one needs to be numerate. While this may be so, I think we need to pay attention to the teaching of mathematics for other reasons. For one, it is a subject that one encounters very early in life and studies for at least 10 years. Mathematics (as well as music) often throws up child prodigies, and proficiency in mathematics is often used to gauge a child’s intelligence. So if a child has had years of bad experience in doing mathematics, this seems to make serious inroads into a his self-confidence and sense of well-being. I also strongly feel that apart from making children competent at mathematics, we must strive to convey to them a flavour of its beauty, depth and order.
In the first week of July 2000, the Rishi Valley School organized a conference for mathematics teachers from the Krishnamurti schools and Centre For Learning. Like most things Rishi Valley School does – it was very well organized! We also had the good fortune of having in our midst Shri P.K. Srinivasan, the renowned mathematics educator, who is known for his penchant for keeping mathematics teachers on their toes. The conference could be said to have had four major strands:
- teachers as students of mathematics
- issues concerning the teaching of mathematics
- new ideas/techniques in the teaching of mathematics
- an exploration of the notion of ‘order in mathematics’.
To become students of mathematics again or to simply play with a difficult problem was indeed great fun. We were exposed to sessions on ‘Transformation Geometry’ and ‘Problem Solving’, where we were actually expected to work on fresh problems and rediscover the anxieties and joys of struggling towards a solution. Of course, once a teacher always a teacher – so while doing problems we also used the opportunity to put ourselves in the place of students and see what insights we may derive about teaching and learning. We also saw some excellent video programmes related to mathematics, among them ‘The Mathematical Mystery Tour’.
It never ceases to amaze me how similar the questions and issues in the teaching of mathematics that many of us face are. Some of these are:
- how does one address the general ‘math phobia’ that exists in all societies?
- how do we ensure that every child acquires the minimal core competency in mathematics to function in everyday life?
- how does one provide remedial help to the child with difficulties in mathematics ?
- do we tend to cater too much to children with a lot of difficulty and not give enough support to those who are really interested and talented?
- can we find or develop a curriculum and learning material which addresses all our needs, that is, it is user-friendly, ensures that there is sufficient drill to help master key skills, and also helps children discover the beauty of mathematics?
We spent a few sessions discussing many of these issues. Opinions regarding whether mathematics was for everyone ranged from an affirmative ‘yes’ to a more humble ‘perhaps not!’ I understand that attempts to develop and organize curricular materials are already under way in many of our schools. Perhaps some further meetings of concerned teachers towards this end of sharing and working on curricular materials would help in the creation of a richer learning programme. We did come up with two tentative goals at the end of the conference:
- to create a compilation of nonstandard problems, graded according to age.
- to create a ‘laboratory’ manual for activities and projects, again graded according to age.
An exposure to new ideas and techniques in maths education was initiated by a delightful visit to the Rishi Valley Rural School, where we took part in the ‘metric mela’: a demonstration by rural children of measurement skills and techniques that they had mastered in non-formal and practical situations. Later that day we had a presentation by CFL teachers about a ‘maths mela’ that they had conducted and, in general, the role of a mathematics laboratory in schools (look out for a book based on this experience, ‘Maths Alive!’). We were treated to a coupleof fascinating sessions on paper folding and mathematics. We also participated in playing mathematical games; this, apart from being instructive, proved to be a truly fun session. There was a session on investigative techniques in the teaching of mathematicsand one of the teachers talked about his experiences and suggestions regarding ICSE projects. All these exposures clearly reflected various attempts to bring life to a subject otherwise considered arid. However, it is not yet clear how we can integrate these innovations as part of our regular curriculum and meet the demands of mastering skills and obtaining certification.
Shri P.K.Srinivasan had a couple of sessions where he exposed us – in a hands-on manner - to his idea of ‘pattern language’as a basis for the teaching of algebra, and alsomade several suggestions for project ideasthat draw directly upon the concepts of themainstream curriculum. He also shared withus his views on mathematics education in general. He had a valid point when he said that, rarely are the practitioners of mathematics also educators of mathematics, and hence while great mathematics is being done all over the world, mathematics education remains in quite a pitiful state. Shri Srinivasan himself has devoted his life to developing new pedagogic approaches to school mathematics, and has been a consultant for many schools and textbookwriters. Late in his life, he continues to attract maths lovers – students and teachers - to the Ramanujam Museum in Chennai. It would benefit us all if someone did a systematic study of Shri P. K. Srinivasan’s work and methodologies (in fact, a documentary filmmaker who attended the conference seemed to be doing just that!).
Finally there were two expositions that approached from two different angles the central notion of order in mathematics. One talked of the surprising occurrence of patterns in nature that have a mathematical basis. The many occurrences of the Fibbonaci sequence of numbers was especially highlighted. The second talk focused on experiments and investigations in number theory that yielded patterns and results that were truly astonishing. Krishnamurti’s oftquoted statement ‘mathematics is order’, however, remains as enigmatic as ever. The‘is’ juxtaposed between mathematics and order perhaps needs the greatest attention and dialogue, and may indeed be a fruitful subject for a seminar for practicing mathematicians.
Without question, we all enjoyed being in the beautiful valley for the conference. What struck some of us who have been to other professional meetings was that there was no sense of competition present and no one was trying to prove how clever they were. We were just a group of adults with a common concern sharing and probing various issues. I do think we should have more such meetings and build on many of the ideas initiated in this conference. Perhaps, in addition to all that was attempted at this meeting, we could also take up as a workshop theme one area of mathematics that most children find difficult to master (for example, addition of fractions). If we can come up with various ways of teaching the topic and actually prepare some material that can be used later on, it would be very worthwhile. We could also attempt to keep in mind a central concern of our schools – is it possible to bring about attention in a child in the context of learning mathematics? For no matter how innovative and comprehensive our curriculum, unless we understand and convey a quality of attention in the process of learning, mathematics would have only a limited significance in our lives as students and teachers.
We welcome contributions from all teachers. Please send any interesting problems (mentioning the age you think they are suitable for) to Shailesh Shirali, Rishi Valley School, Rishi Valley, Chithoor district, A.P. 517 352, or an abstract of any mathematical activity, project or game (again mentioning the age you think they are suitable for) to Shashidhar Jagadeeshan, Centre For Learning, 462, 9th Cross, Bangalore 560 011.
The birds have vanished into the sky, and now the last cloud drains away. We sit together, the mountain and me, until only the mountain remains.