Empowering the Nation Through a Practical Scheme for Improving Mathematics in Schools Dinesh Singh In spite of the cautionary notes as well as the justifiable laments on the increasing divide between the rich and the poor, there is little doubt that India is on the road to economic prosperity. There are many pitfalls and lacunae on this road to well being but the right indicators seem to be falling in place. Quite naturally, any such economic boom, especially when the reasons are not fully comprehended, is bound to be accompanied by a cacophonic din of cautions and warnings. To this din, I would like to add my own note of concern. If India is to stay ahead and keep the momentum, it must have an economy that is powered by a knowledge base and the economy must be driven by competent and well trained humanpower. Thus, a well trained and capable human resource is a key component of this game. Any definition of a well trained human resource shall doubtless include quantitative and communication skills as principal features. It is my contention, nay, it is my firm belief that such a requirement is to be addressed through a sound exposure to mathematics through our schools. .. After a fair amount of thought and discussion, I have been led to the task of outlining a scheme that is to my mind as practical as it is comprehensive. In the next few pages, I shall endeavour to briefly outline this scheme with a healthy and brief dose of philosophy, coupled with a great deal of practical details. This scheme is based on the following two firm beliefs, which are in turn, derived from years of experience: Every child has mathematical ability. Gandhian premise of education, The 'what you do with your hands enters your heart'. The scheme is devised to cater to the enormous need for the provision of high quality mathematics teaching in a very holistic package for large sections of our nation. Why is 'Mathematics Teaching' Important? In addition to reasons outlined above, I would also like to mention that there is hardly any realm of human endeavour that has not been profoundly affected by mathematics. Given this omniscience of mathematics, it becomes imperative for the progress of any civil society to lay emphasis on the proper training of its youth, and particularly of its school children, in mathematics. Even if a child has no apparent intention of pursuing any career that relates to mathematics or uses mathematics, it still remains important for a child to have some mathematics training. In turn, mathematics shall train the child to think precisely and logically. Such training stays for the rest of one's life and enables one to enhance and empower oneself at every stage and in many ways. In general, mathematics education must be carefully developed and nurtured for the betterment of the citizens of a country and for the progress of civil society. The Situation as it Exists Today Unfortunately, the situation on the ground is not very reassuring; particularly for those schools that are involved in educating children who are drawn from the less privileged sections or more remote parts of society. There is not much by way of any worthwhile teaching. In any case, most of the teaching is done in a highly passive mode. It must also be borne in mind that school teachers are heavily burdened with little by way of rewards. Thus, there is no 'hands on' approach to the teaching of mathematics, which shall actively involve the child. Immediate Inference Given the above situation, we have to agree that there is enormous scope for improving the teaching of mathematics for schoolchildren. Characterizing the Problem The three problem areas that seem to characterize mathematics education in schools are: 1. The paucity of good teachers, especially at the level of primary schools. 2. The absence of good and meaningful textbooks/teaching materials and teaching aids that connect mathematics with the real world. 3. The almost complete lack of a 'hands on' approach to teaching
Remedies for the Situation It is obvious that there are no 'magic bullet' solutions that shall remedy the above problem at once. However, I base my prescription on my personal experience. This has been derived as much from my excursions into the field in a 'hands on' manner as from my understanding and insight as a mathematician. Target Group This is aimed principally at students (primary & secondary) and teachers of government schools or schools that cater to remote areas and similar schools in other parts of the country. The scheme is expected to be implemented in a manner that shall allow easy and wide replication. The Operating Levels Holistic training of teachers.
Infrastructure A training laboratory, which shall include:
At any suitable premises that allows the scheme to function in an unhindered fashion. Holistic Training of Teachers The ideal teacher is one who is a gifted creator of knowledge and has talents in the realm of knowledge dissemination. Note that all the great teachers of our land in the past were also great creators of knowledge. The next best situation is one where a talented creator of knowledge is closely linked with a gifted disseminator of knowledge. This translates into holistic training of teachers. Essentially, this means that teachers:
Will also be trained to use testing and examinations as a means of empowering the student to perform well and also to acquire knowledge. Care will be taken to ensure that examinations and testing are used in such a manner that they are not feared.
Creation of Teaching Materials These 'TV lessons' shall be encapsulated within a very holistic package that shall include such offerings as:
It is hoped that the same can be captured on CD's with voice and animation. These CD's can then also be used as teaching aids in those schools, neighbourhoods and homes where some very basic computers are available. Creation of a Small Corps of Teachers that shall Move from School to School Given all of the above, one of the easily achieved objectives of the programme shall be to create a very dedicated, well-trained, talented and small corps of teachers from amongst graduating college students and some carefully chosen full time school teachers. This corps shall be deployed in a phased manner from school to school to help implement the teaching methodologies that will have developed at the laboratory. Implementation of the Above Programme in Terms of Faculty and Budget The above programme can only be implemented if, as mentioned at the very beginning, a gifted creator of knowledge who is also a talented teacher heads this programme. This has to be a first-rate mathematician who has the dedication and experience to head the implementation of the programme. This mathematician must ever remain a creator, to keep skills and insight intact. However, a part of his/her time must be devoted entirely to this programme. This mathematician must be provided with all the necessary support in terms of resources and infrastructure. This will help create a team and set up the entire laboratory. Given the enormity of the task, the entire budget shall not be too unreasonable. It shall be used for supporting a full time chair for this mathematician and for some of the faculty that will get appointed. Some of the heads shall be: Full time chair for a distinguished mathematician; A few full time chairs for talented and experienced faculty; stipends /salary for assistants chosen from school teachers and college students; grants for travel and for visitors; grants for library and for various equipment and software as well as hardware; grants for the actual physical infrastructure of the laboratory; grants for the holding of workshops; grants for funding the stay of school teachers and for their training. All this shall be done under the aegis of the Mathematical Sciences Foundation. Evaluation and Testing: Empowering the Student It is an inviolable axiom with us that evaluation and testing should be used primarily for the empowerment of the student. This empowerment translates into:
Any process of evaluation is only as good as the person who devises and/or implements it. Given this premise, the limitations inherent in any evaluation mechanism must be recognized. It will be pertinent to mention here that at the beginning of the last century, one of the greatest mathematicians of all times, Henri Poincare, took the Binet tests (at the height of his creative powers) and was declared an imbecile! Thus, teachers must be trained and taught:
It will be pertinent to recall the style of learning as often portrayed in the Upanishads, where the teacher leads the student/disciple through a gentle series of questions in a logical sequence. This style actually makes the questioning (read testing) a process of self-evaluation for the student and culminates in a final insight where a significant amount of answering has been done by the student and the teacher has really only provided the insight when the student's mind was ripe through the questioning. So let us recognize that
Some Practical Steps First, an assumption: Let us assume that the delivery mechanism (teaching methodology) has been of a certain minimum required standard that incorporates the good practices of teaching mentioned elsewhere in this document. Then, the following practical steps are likely to spell out our methodology: The teacher must build simple but insightful oral questions into the lesson at critical steps. Such questions shall be implanted at the right places in the offerings that shall be prepared in the laboratory and be offered only via a 'teacher'. Each question must make the student pause and think. Written and oral questions must be framed in easy and clear terms. Applications of principles must be built into questions wherever possible, but these questions must not be seen as artificial or unreal applications as is often the case. The class must be broken up into smaller and reasonably homogenous groups that can be posed specific questions, both written and oral. The teacher must encourage the students to answer after collective reflection. The importance of oral testing and answering in which the whole class participates cannot be over emphasized. At the same time, the class must be taught to use language and communication skills effectively. Lessons must be broken up into key parts and the importance of each part must be demonstrated by preparing a series of questions and experiments related to the questions that will highlight the use of the key principle /part of the lesson in answering the questions. Written testing and evaluation should be done at two levels: For small groups that answer collectively and for individual children. and large, two different childrenBy should receive different sets of questions (in terms of different words/terms) that highlight the same principle/idea/lesson under discussion and are at the same level.
Objec t i v e t e s t i n g , s h o r t a n s w e r questions and rearranging jumbled groups of sentences in a logical sequence shall also be accorded their due importance. Overcoming Impediments in the Implementation of the Above Steps It is obvious that there are several impediments to implementing the above steps. The two most obvious ones are:
However, what we would like to emphasize here is that there seems to be a way to ensure that a certain minimum standard of teaching and learning can be achieved, which is far higher than what exists. The Large Class Size Problem I think that one of the major problems that our school teachers face today is the fact that class sizes are enormously large. The ideal situation is to have a much smaller student-teacher ratio than the one that exists. However, this is not a problem that can be wished away, nor will it be remedied for a long time. Yet, there are ways and means in which the situation can be rectified to a large extent. The solution that we suggest relies on the use of technology. A very enlightened teacher can easily deconstruct a lesson into key parts and ideas and map onto these parts, questions across several parameters. This is easy to do when implemented on a computer. A great deal of interactivity and personalization can also be put into the computer. Thus, it becomes easy for any other teacher to see the essential principles that are vital to a lesson. This teacher then accordingly produces a whole set of different question papers with various characteristics highlighting different important aspects of the lesson according to the needs of the class at the individual level and at the collective level. Simple interactive tools also help the teacher to produce several different questions highlighting the same idea or principle. The process of evaluation and grading also becomes very easy for the teacher with the use of information technology. The teacher can quickly and easily assess the standing of the individual pupil in percentile terms and even against the pupil's own performances. Several studies already exist that have demonstrated the effectiveness of such technologies and methods. At the proposed laboratory, we wish to implement our own version of these ideas and if needed, merge t h e m w i t h s o m e t r i e d a n d u s e f u l technology/method. The Poor Quality of Teaching We have already tackled the problem of the poor quality of teaching at a different place in our proposal. It is imperative that the kind of work that we wish to do should be executed in a knowledge environment with a practical outlook. This is precisely the case at the Mathematical Sciences Foundation. Here, I have at my disposal the committed expertise o f a d e d i c a t e d b a n d o f t a l e n t e d mathematicians, who are willing to put in time for the kind of activities that I have proposed. We also have a huge bank of bright and motivated undergraduates who shall be an immense help at various levels. The other point that I wish to make is that there is enormous technological expertise in such matters that already exists here. The use of technology to help free the teacher from mundane matters of teaching and for gaining insight into the level for individual students/for large classes/for across the board evaluation, is immensely beneficial. Implementing the Above Ideas: A Practical Road Map Objectives
Time Phase The plan shall be implemented over a 12year time period and shall require a 3-year preparation cum honing period. Thus, at the end of 15 years, there will be a comprehensive package that will have demonstrated on a large scale basis, a significantly better methodology (than the existing ones) for the teaching and learning of mathematics. Target Group This project is primarily meant for school teachers and children of all schools in various parts of the country. However, benefits shall accrue to other segments of society. A Special Word about the Girl Child and Mathematics Educators the world over are becoming increasingly aware of the special attention that must be paid to the needs of the girl child in many situations related to education. In India , this aspect acquires many dimensions and hues. There are obvious cultural needs and impediments. In addition, regular education beyond a certain age is not accessible to the girl child in large parts of our land. There is no special identifiable methodology that seems to have been developed, largely because the problem itself has not been very well defined. Thus, the problem becomes all the more difficult when it comes to the needs of mathematics education for the girl child. I have had some experience in this and am quite happy to report that after the initial hesitation, seemingly caused by cultural/social conditioning; girls in most situations have been as receptive and creative as boys. This is especially true when there are classes that comprise mostly girls. Of course, there seems to be some initial hesitation, which wears out after a while, and then the girls seem to take the same amount of interest and have the same kind of abilities and skills. These are of course just some preliminary findings. The concrete steps that need to be taken in this regard are:
The Five Tier Structure From the Top Downwards : 1. One or two very distinguished and committed mathematicians who are creative as researchers/teacher-thinkers, with a very practical and established track record. These mathematicians shall be hired on a full time basis and they shall devote at least a third of their time to guiding the effort at all levels. 2. Fi ve very g oo d a nd commi tted mathematicians who have a proven track record as mathematicians and as educators at the school level. These mathematicians shall also devote at least a third of their time to guiding the effort at all levels. 3. Ten lecturers who are coming in at knowledge level with a great deal of practical insight. They shall be working full time. 5. Twenty extremely bright and committed undergraduates. These twenty shall work for fixed hours each week. They may vary from year to year. They shall help in the implementation of the design and in issues related to preparation of the teaching material. They will work directly under the supervision of the mathematicians at level 1 and 2. In Addition: In addition to the above five tier structure, there shall also be the need to engage from time to time, consultants on issues related to education and transmission technology. The following milestones shall have been achieved at the end of first year:
A Word about Technology: Its Nature and Need It must be emphasized that there is a tendency on the part of some to bring in, or emphasize the need of technology without any proper justification. However, in our situation, we have identified certain kinds of technology that are imperative for the success of the plan. There are several advantages of technology: Interactivity and large areas can be covered with material of a very good quality. The ingredients of this material have been outlined above but are reproduced again over here. These TV/Computer/Internet 'lessons' shall be encapsulated within a very holistic package that shall include such offerings as:
This is done in an interactive way so that in addition to the text on the blackboard, the voice of the teacher; as well as the image of the teacher is also transmitted as and when required. The te c h no log y a l l o w s t h e st u d en ts to simultaneously view the lesson in all schools or anywhere else, as if in a classroom environment. In addition, with the provision of a camera at suitable times and places, students can be seen and heard by the teacher in the transmission room as and when required. The technology allows the use of the PC to transmit any part of a lesson, such as some pre-written text/animation/images etc. It allows the entire lesson to be stored on an inexpensive CD for future viewing. In addition to the above-mentioned technology, I would like to emphasize that there are simple ways of creating intelligent testing procedures (designed to improve learning), which require the power of the computer. This use of technology is proving highly successful and productive. Such gainful use of technology shall be actively encouraged. Schedule from Year 2 to Year 12 Each year, the entire teaching material, together with all support material, shall be created for a given standard starting from Standard I onwards. At the end of the twelfth year, the material for the entire school system is expected to have been created. Quality of the Material The quality of the material shall be of a high standard from several points of view. It shall conform to the requirements of the student and teacher alike in the sense that it shall be easy to use, shall generate interest and insight. It shall have a great deal of supplementary support material that is not found in regular classroom teaching. It shall lead to teacher training in a natural manner. 2. Two principal reasons why not much However, all experiments and simulations has been achieved in our country are: etc. shall be made available from the point of view of the student being able to relate the idea to his/her immediate environment. Testing shall be in a personalized mode to Not enough effort by good mathematicians to create authentic teaching material and methodologies at the school level. the extent possible and shall try and ensure We have always been in a hurry to that the student competes primarily against himself/herself. Testing shall also be used create material and methodologies. significantly for enhancing the learning of 3. Hence, if any thing significant and of the student. lasting value is to be achieved, then we must be ready for a long haul; we must Conclusion also have enormous patience. I would like to emphasize the following: 4. No compromises must be made in terms 1. Education, and in particular Mathematics education, is a serious and difficult business. 2. Two principal reasons why not much has been achieved in our country are:
3. Hence, if any thing significant and of lasting value is to be achieved, then we must be ready for a long haul; we must also have enormous patience.
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