recommendations of the “discussion meeting on school...
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Recommendations of the “Discussion Meeting on School
Mathematics” from 28 February to 2 March, 2013
The meeting was organised by the Ramanujan Mathematical Society (RMS) and was hosted
by the Indian Academy of Sciences. Meeting, held at the Orange County, Karnataka, was
attended by
President RMS:
1. Prof. M.S. Raghunathan, IIT, Bombay & President of RMS.
Representative of the Joint Science Education Panel of the science academies, namely Indian
Academy of Sciences, Indian National Science Academy and National Academy of Sciences,
India:
2. Prof. Parameswaran Sankaran, IMSc, Chennai.
Members of the Editorial Board of RMS series of Little Mathematical Treasures (LMT):
3. Prof. Phoolan Prasad (Editor-in-Chief, LMT), IISc, Bangalore.
4. Prof. Pradipta Bandyopadhyay, ISI, Kolkata.
5. Prof. Manjunath Krishnapur, IISc, Bangalore.
6. Prof. R. Ramanujam, Institute of Mathematical Sciences, Chennai.
7. Dr S. A. Shirali, Director, Sahyadri Education Centre (KFI), Rajgurunagar, Pune.
8. Prof. Geetha Venkataraman, Ambedkar University, Delhi.
9. Prof. A. Vijayakumar, Cochin University of Science and Technology.
Invitees:
10. Mr. Athmaraman Rajaratnam, Member, Office Bearers, Association of Mathematics
Teachers of India.
11. Mr. K. Chandran, Teacher, IISc Kendriya Vidyalaya, Bangalore.
12. Dr Jonaki B. Ghosh, Lady Shri Ram College for Women, New Delhi.
13. Mr. Sabyasachi Mitra, Calcutta International School, Kolkata.
14. Dr Lidson Raj J, Kerala SCERT, Trivandrum.
15. Mr. Rajkishore Patnaik, Educational Technology Design team, Azim Premji
University Resource Centre.
The agenda of the discussion meeting was
(1) Creation of an all India cadre of school teachers (along the lines of services like IAS
or IPS).
(2) Nurture programme for mathematics teachers.
(3) National level programme of quality mathematics education for bright students in
schools.
Each item of the agenda was discussed in great detail and there was a very intensive
discussion on all items over a period of 3 days. One important aspect of the meeting was the
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interest shown (and also confirmed later after the meeting) by the Azim Premji University in
working jointly with this group of mathematicians on teachers training. It was also interesting
to note the valuable work initiated by the Kerala SCERT on nurturing mathematics talent in
schools from 6th
standard, which may be followed by other states.
We first present the consolidated recommendation of the discussion meeting and then present
the detailed discussion on each of the three items, one by one, in form of a number of
Annexure.
1. Creation of an all India cadre of school teachers (along the lines of services like IAS
or IPS)
There was a discussion on the state of school education in the country. While there are
mathematics-specific issues, it was agreed that many of the problems are common to all the
subjects taught in school. In such a context, much of the remedial steps are not exclusive to
mathematics. In the light of this, the committee felt that a recommendation that would take
into its ambit school education at large needs to be made. One of the suggestions that came
up and was discussed was to start an all-India Educational Service along the lines of IAS, IPS
and other central services. The committee felt that it is an idea worth pursuing. A note on this
is attached to these minutes as Annexure 1.1.
2. Nurture programme for mathematics teachers
India faces a major challenge assuring quality education in Mathematics at school level
throughout the country. There is a problem of number and quality. The former is due to the
fact that crores of children are coming into secondary and higher secondary education within
a decade, thanks to the implementation of the Right to Education Act, and at that level,
mathematics teachers with specialized knowledge of the subject will be needed. The problem
of quality is severe, since the preparation that teachers have during their own school/college
education is often insufficient for developing competency and depth of understanding in
mathematics. Further the teacher education programmes usually strengthen neither their
content knowledge of the subject nor mathematics pedagogy.
The problem that the country faces is spelt out in some more detail in Annexure 2.1.
One may wonder at this juncture, ‗what can be done about it, and who is to do it‘? Perhaps
the shortage of teachers and quality is a problem of very large magnitude, but nevertheless
urgent. Clearly it is the government that needs to act, and with detailed action programmes.
But even as we urge the state to do so, and wait for it to do so, there is plenty that can and
must be done by the community of mathematicians, mathematics education researchers and
mathematics teachers at all levels. The Science Academies and Mathematics Societies have
the responsibility to highlight the situation and to provide ways of building capability to meet
the need. While the numbers may be too large, developing resources and means of delivering
them is well within the capability of this ―mathematics community‖, and indeed, nobody else
in the country can do so.
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The topic was discussed in great detail in the meeting and it was suggested that developing
resources for building teachers‘ capability should be the focus of our efforts. The meeting
resolved that we should:
• Build easily accessible educational resources by which teachers in the system can
acquire content knowledge in a demonstrable way.
• Develop curricula that strengthen the university system, perhaps re-envisaging it at the
undergraduate level, aimed at new entrants into mathematics teaching, and to meet the
large imminent need for mathematics teachers.
• In the process, identify and nurture a large pool of regionally distributed mathematics
teachers to act as potential resource persons for the programmes that the government
will/should take up to meet the need.
The second of the above calls for a major intervention with the university system and it was
suggested that the Academies should use their influence towards this.
For the other two, the meeting came up with the following recommendations:
1. In-service nurture: The single goal of this programme is to strengthen the content
knowledge of mathematics in teachers serving in schools so as to help them engage
with the mathematics curriculum and the students confidently and effectively. The
proposal is for an on-line course that teachers take at their own pace, eventually
accumulating enough credits for certification. It will have 4 components: Number
systems, Algebra, Geometry and Trigonometry, and Applications of Mathematics.
The treatment of each of these would be structured to include History of Concepts and
Ideas, Varieties of Problem Solving, Arguments and Visualization, and Use of
Computer Software in pedagogy.
Apart from these ―core‖ courses, there would also be short topical courses such as on
topology, graph theory, cryptography, etc. Teachers would work offline on assigned
problems/projects, eventually getting sufficient credits for a diploma, which could
lead some teachers into doctoral research.
A visit programme by which teachers visit mathematics research groups and interact
with mathematicians is envisaged to strengthen the programme.
Formation of teachers‘ networks and professional societies is suggested as a way to
nurture teachers in the later stages of their careers.
The details of the proposal are spelt out in Annexure 2.2, which is structured in two
parts: Annexure 2.2.1 gives the proposal for teachers at the elementary and secondary
stages, and Annexure 2.2.2 for teachers at the higher secondary stage.
2. Pre-service nurture: While the recommendations above pertain to teachers already
in the system, an urgent need is to ensure that new entrants are capable and
competent. Towards this, the meeting suggested the following measures:
a. Specialization for pedagogy: The Bachelor‘s and Master‘s programmes in
mathematics in Universities can introduce a stream that specializes in pedagogy of
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mathematics. For such students, an opportunity to revisit the content of school
mathematics is essential.
b. Nurture in colleges: The suggestion is to have a national competence test to
identify potential mathematics teachers and orient them strongly towards
mathematics teaching. Those that get selected by such a test would enter the
nurture programme. This will get them scholarships through their course of study,
but will also get them apprenticeships with teachers in different institutions.
The details of the proposal are spelt out in Annexure 2.3.
3. State-University initiatives in teachers training: Apart from certificate courses such
as the above, the meeting recommended that workshops for the professional
development of mathematics teachers be conducted. The role of mathematics
researchers and the government is crucial in this regard. When teacher education is
taken up by those with research experience in mathematics, it can be greatly
advantageous in strengthening their content knowledge and promoting a culture of
problem solving among teachers. The country has seen many efforts in training
mathematics teachers over the years. For instance, the Association of Mathematics
Teachers in India and the Tamil Nadu Science Forum have been conducting such
workshops for many years now, with the participation of research institutions such as
the IMSc in Chennai.
Annexure 2.4 details one example of such a programme successfully taken up at IISc,
Bangalore, which highlights the role that a partnership between a state government
and an institution of higher learning can play in training teachers. Such joint
initiatives need to be taken up on a larger scale in the country.
While many details need to be worked out for implementing the suggestions made above, it
was suggested that the formation of a resource group that can take up the online courses
envisaged above can be taken up immediately.
3. National level programme of quality mathematics education for bright
students in schools
A country aspiring to be a world leader in science and technology must have quality
education in Mathematics at the school level. We need to provide for full growth and for
utilization of complete capacity of good and talented students in mathematics in order to (i)
attract them to study mathematics deeply, and (ii) inspire them to work hard to learn and
enjoy it. One may wonder ―why do we need to have special programme of training in
mathematics alone and not in other subjects?‖ We provide some answers to this: (i) Majority
of the well trained students in mathematics would go in for a career in disciplines other than
mathematics and their enhanced capability in mathematics would enrich other disciplines: all
sciences, engineering, commerce etc. (ii) In contrast with other subjects, those who miss
training in good mathematics till Class 8 are not likely to learn mathematics later on. The
topic was discussed in great detail in the meeting and the following suggestions were made.
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1. Math Circles: The aim is to engage with mathematics teachers and students of high
school in towns and cities across the country, by forming ‗Math Circles‘ which meet
once every few weeks. ―Mathematical circles are a form of outreach that brings
mathematicians into direct contact with pre-college students. These students, and
sometimes their teachers, meet with a mathematician … in an informal setting, after
school or on weekends, to work on interesting problems or topics in mathematics. The
goal is to get the students excited about the mathematics they are learning; to give
them a setting that encourages them to become passionate about mathematics.‖
A detailed plan of Math Circle is given in Annexure 3.1 and some additional write
ups are available in Annexure 3.6 and 3.7.
Math Circles will be scattered all over the country. A coordinating agency is needed,
which will act as a central repository of materials and provide a platform for
networking, through dedicated pages on its website. Some financial commitment will
be required for the coordinating agency, which will fund the Math Circles in various
cities and towns.
2. Optional Mathematics: The mathematics curriculum is quite good in schools. But
the quality of examinations is of great concern since, both average and very good
students secure almost the same marks in these examinations. Hence, the examination
results are unable to distinguish a bright student from an ordinary one. This leaves no
motivation for talented students (whose number is very large in India) to learn
mathematics deeply; they simply practise for high marks. It would be very difficult to
change the present trend (which anyway is not any evil for average students) in
question paper setting and evaluation. But it is simple to nurture mathematics talent
by an OPTIONAL MATHEMATICS (OM) programme which will comprise (a)
providing excellent books in OM from class 6 to 12 (or even earlier); (b) having two
(or at least one) special classes, per week, in OM only for students good in
mathematics from these classes, in order guide them in learning and problem-solving
on their own; (c) and conducting final examinations at 10th
and 12th
level.
Since this is an OM programme, schools should be free to opt for this programme
depending on the availability of a competent teacher. We recommend that this
programme be started in all Navodaya Vidyalaya and along with some schools, which
may be recognised as Centres of Excellence in Mathematics Teaching. It is expected
that more and more schools will aspire for this recognition. All these schools will
have to have additional well qualified teachers in mathematics.
We urge that that CBSE also implements this programme starting with some schools
where it is able to appoint an additional teacher.
OM Programme is described in Annexure 3.2.
3. Sequence of Summer Schools for a Selected Group at State Level: This
programme is suggested based on ―Nurturing Mathematical Talents in Schools
(NuMATS)”, already run by SCERT Kerala since last year. In this programme about
100 students from 6th
standard are to be selected from all parts of a state through
nomination from schools, district level and finally state level rigorous aptitude tests.
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The process of selection may start from September and may be completed by the
middle of January next year. The nurture programme of this group of students will
start from February and will be continued for 5 years till they complete class 10. Each
year all 5 groups will be called for separate learning camps, which will consist of
lectures, problem solving and interaction with mathematicians. In each camp the
students will be given a sufficient number of books/ reading materials and problem
sets to last till the next camp. The constant help from experts will develop students‘
capacity and reasoning ability in such a way that at the age of 15 they will be in a
position to take the challenge of competitive examinations and further learning in
mathematics.
All states should implement this programme through their SCERT or State DST.
Details of the programme are available in Annexure 3.3.
We suggest that KV Sangathan also implements this programme as discussed in
Annexure 3.5.
4. Mathematics Laboratories in Schools: Mathematics has for years been the common
language for classification, representation and analysis. Learning mathematics forms
an integral part of a child‘s education. Yet, it is also the subject, which has
traditionally been perceived as difficult. The primary reason for this state of
mathematics learning today is the significant gap between content and pedagogy. In
the last few decades countries across the world have witnessed a major shift of
paradigm as far as mathematics teaching and learning is concerned. Mathematics
education is being revolutionized with the advent of new and powerful technological
tools.
In a vast country like India, large scale integrating of technology in classrooms is
fraught with numerous challenges. Above all, technology must be cost effective and
easy to deploy in order to achieve large scale integration. In this context, mathematics
laboratories may provide a solution. Every school may set up a mathematics
laboratory which can be the platform through which students are given access to
technology to explore and visualize mathematical concepts and ideas. Mathematics
laboratory has been very successfully run by one of the participants of the ―discussion
meeting on school mathematics‖. Most of the bright students of schools participate in
mathematics laboratories; this is one of the effective ways to nurture mathematics
talent.
Details are given in Annexure 3.4.
Acknowledgement: We thank Prof. M. S. Hegde, Department of Solid State and Structural
Chemistry Unit, IISc and Convener, Talent Development Centre (TDC), IISc Campus,
Kudapura, Karnataka for providing information on TDC and permitting us to include it in this
report (Annexure 2.4).
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Annexure 1.1
An all-India educational service for school teachers
The Right to Education Bill no doubt constitutes landmark legislation. However, in practical
terms the "right" is pretty much meaningless in the context of what is on offer to our children
by way of education. It is universally acknowledged that a large majority of our schools are
unable to meet minimal standards in imparting education. There is also a general perception
that the state of mathematics education is in particularly bad shape. We propose here some
drastic steps for improving the present state of affairs after identifying what we think is the
root cause of the problems facing us. These proposals are aimed at improving the state of
education across all disciplines, not just mathematics: at the school level, the problems in
different disciplines are not all that widely divergent.
The lack of proper physical infrastructure in most of our schools is no doubt one of the major
causes for the poor quality of our schools. This is of course an enormous problem, but it is
largely a matter of finding the money needed. It needs to be addressed with a sense of
urgency and finding the money is a matter of political will.
The more serious issue is the very unsatisfactory quality of the human resources in our
educational institutions. This is to some extent the result of faulty recruitment, including
some unhealthy practices. However, it is quite easy to see the reason for the quality of our
teachers being sub-standard: a large majority of the pool of applicants to teaching jobs itself
does not measure up to the requisite standards. This in turn is the result of the fact that the
teaching profession (especially at the school level) is far from attractive to most bright young
people. The emoluments of the school teacher do not compare favourably with those in other
professions open to intelligent youngsters, professions that often demand much less
commitment and dedication in shaping the future of India. Apart from emoluments, the
working conditions are far from satisfactory—they have to handle unwieldy classes and the
work-load of teaching is heavy giving them little time for serious preparation, leave alone
enhancing their knowledge; the infrastructure in most of our schools leaves much to be
desired and this of course affects the working conditions of the teachers. In such a context, it
is not surprising that the competence of much of the work-force in the teaching profession is
well below the minimum requirements: indeed the surprise is that there do exist many
teachers who are highly competent and strongly motivated; and that encourages one to think
that it is possible to remedy the situation.
One possible step towards raising the standards of our schools is the creation of an all-India
cadre of school teachers (which may be called Indian Educational Service (IES)) along the
lines of the IAS, IPS and other Central Services. The emoluments, perquisites and service
conditions should be comparable to those in these services. The recruitment too should be on
the basis of a similar competitive examination and in all government (or government aided)
schools at least one or two teachers or at least 20% of all teachers that are hired in each
school should be from among the recruits to this service. As with the Central Services the
candidates may indicate the choice of the state in which they want to work; however
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proficiency in the language of the state would be a pre-requisite for them to be assigned to
any particular state. The candidates for the examination should also indicate at least two
subjects that they would be willing to teach at the secondary level and at least one at the high-
school level. They are to be set examinations in 3 of these subjects (one at least from out of
the choices made for the high-school level). Those who do not want to be considered for
teaching at the high-school level need not specify any subject at high-school level and may be
examined in only two subjects. Apart from these, they need to be examined for proficiency in
English and the regional language. Each year the government may decide (on the basis of
vacancies as well as decisions regarding additional needs) the number of teachers needed.
The candidates for the IES examination need only have a Bachelors degree from a recognized
university. B Ed or any other similar education degree need not be demanded of them. Those
who are selected for the service should undergo a two year training programme which will
include one semester of course work where they can be taught something of what is done in a
B Ed or similar course as well as refresher courses on the subjects they are to teach. The rest
of the time is to be devoted for field work where the recruits will actually teach in schools.
Staff Colleges may be established in different towns all over the country for this purpose and
the teachers at this college are to be experienced teachers from renowned institutions (which
may be also institutions like schools run by the Krishnamurti Foundation) brought to the
college on deputation for limited periods of time.
Education being a concurrent subject, this centralised recruitment may not be welcomed by
many state governments. However the IAS and IPS are central services but work under the
state governments. It should therefore be possible to have a working arrangement for the IES.
Working conditions for these recruits cannot be different from those for the existing teachers.
This means that across the board conditions need to be improved for the existing teachers as
well. Also there should be a provision for the existing teachers to be also taken into the
service subject to their qualifying through a parallel competitive examination exclusively
among them. They may also compete in the direct recruitment to the IES. One important
reform that needs to be instituted is a drastic reduction in the number of teaching hours (15
hours a week would be a reasonable work load). This would of course immediately increase
the number of teachers needed and the recruitment should be geared to take care of it.
Teachers have enough on their hands (even with a reduced work-load recommended above):
they have exams and tests to set and papers to correct and sundry matters relevant to class-
room teaching. In fact this kind of load needs to be reduced and that can be done only with a
reduction in the number of students they have to handle: any number over 30 would make it
impossible for the teacher to have reasonable contact with individual students while at
present something like double that number seems to be the norm. A reduction of class size to
near 30 would again mean raising the number of teachers, nearly doubling it. The mechanism
of an Educational Service would probably be the agency that can take on best such drastic
changes. Evidently the kind of financial outlay needed is something that can be handled only
by the centre.
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This brief note is intended as a preliminary document proposing the establishment of an All
India Education Service as the means for improving our educational standards. Evidently
detailed discussions are needed before an action plan can be made for the IES envisaged in
this note. However early action is needed - we have really no time to lose in starting major
reforms of our educational system to make the Right to Education meaningful.
Annexure 2.1
The problem of teachers‟ training
For school teachers, the talk is typically of pre-service and in-service training. Why nurture
programme? To answer this, we should take a quick look at mathematics teachers in Indian
schools today.
What is the profile of a teacher of mathematics at the elementary school? A study of the data
provided by Sarva Shiksha Abhiyan (SSA) for 2011-2012 suggests that if we randomly select
such a teacher, the following profile fits with about 40% probability: she is in the age group
25 to 35, has passed higher secondary school and has undergone one year training in
education, typically for a diploma programme. For a secondary teacher, the age group would
be 35 to 45, male, with a bachelor‘s degree and a B. Ed. degree.
A more detailed look at the data reveals that there are nearly 20 lakh teachers in the country
who teach in elementary schools, perhaps at the primary level, who have not undergone any
educational training at all. Nearly a lakh teachers have not passed the higher secondary level.
The data further reveals a tremendous shortage of teachers at the secondary and higher
secondary level, especially in rural areas. With the implementation of the Right to Education,
several crores of children will be moving into secondary schools within the next decade, and
we will need nearly 40 lakh additional teachers of mathematics at the secondary and higher
secondary levels. Further, there are great regional imbalances. States in peninsular India,
especially in the coastal regions, have already met this challenge or are well on their way to
do so, and this holds for some of the states in the north as well. But nearly half the country is
entirely unprepared in this regard.
Of course, the data is generally on teachers, with no talk of subject expertise. But this is
especially important, since teachers‘ own knowledge of mathematical content has a critical
impact on their ability to teach mathematics, as has been repeatedly shown by educational
researchers, perhaps more so than in the case of other disciplines. An important factor is that
learning on the job may work very well for pedagogic techniques, but there is little scope of
learning the content of school mathematics all over again, in depth, and this is a problem.
Indeed, teachers‘ inadequate preparation in subject knowledge of mathematics can be (and
has been, many times) said to be one of the most important challenges of mathematics
education in India today.
Consider again the profile of the teacher we spoke of above. She would have had compulsory
mathematics in the ten-year school, but need not have gained more than passing (literally)
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familiarity with mathematical concepts. Board examinations emphasize calculations and
memory, accordingly much of the education she would have received might emphasize these
rather than conceptual clarity. The chances that she discussed any mathematical concept with
anyone all through school are rather slim. At the higher secondary stage, it is not necessary
that she took mathematics at all, and even if she did, it is quite unclear that the emphasis on
calculus that dominates higher secondary mathematics helped to clarify the concepts learnt
earlier in primary or secondary school. Then, in the one year diploma period, she learns next
to nothing related to mathematics or its pedagogy. Chances are that she may have worked
with some teaching aids for fractions during her practicum, but these are likely to be
constructions based on instruction, not thoughtful practice.
A similar story may be constructed around the teacher who got a B. Sc. in mathematics and
then the B. Ed. degree. (The number of mathematics teachers whose under-graduate degree
was in a subject other than mathematics is not trivial either.) In what way does the prevailing
undergraduate curriculum or the B. Ed. curriculum strengthen the basic mathematical
understanding of this person? Certainly, linear algebra, calculus and probability offer an
opportunity to revisit higher secondary mathematics, but he will be teaching secondary
school, not the higher secondary. The topics covered in secondary school (especially
trigonometry, coordinate geometry etc) are never formally revisited. The gap between school
mathematics and disciplinary mathematics at the university level is vast, and this
discontinuity has great impact on the college student of mathematics. Neither does B. Ed.
offer an opportunity to look at mathematics pedagogy from the perspective of educational
theories of learning and teaching. Thus the assumption that our extant university system will
produce, or is capable of producing, mathematics teachers of quality for secondary schools in
the numbers required seems to be rather unfounded on fact.
The eligibility for higher secondary teachers is a master‘s degree as well as a degree in
education, so the arguments above may not apply as such. However, only a few states have
teachers at this level in numbers close to the requirement. Note that today only a small
fraction of those in elementary education make it to science and mathematics stream at the +2
level: the national figure is close to 10%. With the advent of RTE, this number can be
expected to climb much higher within a decade. When it does, we have a double problem:
one is that we will need a large number of teachers, and the other is that in the existing
system, of the 10% who do reach this level, what percentage would eventually complete a
master‘s degree in mathematics and a degree in education and be ready to teach them?
There is much to be said in this vein, but in summary, the point being made is merely this:
For lakhs of mathematics teachers at school in the country, especially in the
elementary and secondary level, the preparation for mathematics, both in terms of
content knowledge and pedagogic techniques specific to mathematics, is woe-
fully inadequate. The situation for new entrants into the system is not much better.
The proposals made at the meeting take the problem above as their starting point.
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Annexure 2.2.1
In-service nurture at the elementary and secondary level
The single goal of this programme is to strengthen the content knowledge of mathematics in
teachers serving in schools so as to help them engage with the mathematics curriculum and
the students confidently and effectively. Problems of pedagogy, assessment, psychology and
sociology of children‘s learning and classroom management are all serious and important.
However, there is good reason to focus on one of the challenges, rather than trying to meet
them all: one is that this eases implementation, and the other is that the community discussing
the proposal (the RMS and the NBHM) is best qualified to concentrate on this aspect, while
having scant experience or clarity on the rest. Moreover, the range of mathematical processes
needed in the classroom (generalization, abstraction, making conjectures, searching for
counter-examples, multiple modes of representation, visualization, argumentation etc) are
best taken up in contexts provided by treatment of mathematical content. (This will be
clarified further below.)
While saying this, we focus on mathematics teachers at the middle school and secondary
school level. For teachers at the primary stage, the psychology of children‘s learning is of
paramount importance, and pedagogic considerations dominate. For teachers at the higher
secondary level who have master‘s degrees in mathematics from universities, revisiting
mathematical content takes on a different meaning.
The years from Class 5 (or Class 6) to Class 10 see mathematics emerging into its own.
Mathematics is a compulsory subject in the curriculum. Beyond the arithmetical operations,
children learn basic number theory, the number systems, transition to algebra, introduction to
geometry, visual and spatial reasoning, data handling, trigonometry, elementary probability,
and so on. All these aim to provide an introduction to mathematical ways of thinking rather
than any competence in mathematics as such. The idea is that the student develops a small
toolkit, but more importantly, a facility with mathematical means that helps not only with the
basics needed in its applications, but also helps her develop an interest in mathematics in
itself.
Seen from this viewpoint, it seems clear that the preparation teachers would have had in their
own schooling or undergraduate study, as well as during the educational training, is likely to
be very inadequate. The courses suggested below are envisaged to address this gap.
The proposal is for an on-line course that teachers take at their own pace, eventually
accumulating enough credits for certification.
Course content: The course will consist of four components:
1. Number systems
2. Algebra
3. Geometry and trigonometry
4. Applications of mathematics
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The first is to principally understand the integers, the rationals and the reals, in depth, but also
introduce some basic number theory. The reals are especially important, since they are
extensively used but little understood in school mathematics, and teachers have conceptual
difficulties with them. (For instance, understanding and explaining what 2. 3 means is hard
for one who is used to speaking of multiplication as repeated addition, or even as scaling.)
The components dealing with algebra, geometry and trigonometry are standard. Here, the
idea is mainly to revisit the content of school mathematics but in greater conceptual depth,
and provide an opportunity to strengthen foundations. (For instance, it is important for the
teacher to appreciate the difference between the use of x in the equation x + 3 = 3 + x from
that in x + 3 = 8; the inability to identify implicit universal or existential quantification leads
to a great deal of confusion in algebra.)
The last component is not only to take up topics like data analysis, probability etc but also
present teachers with an opportunity to connect up mathematics: integrate the different
substructures of mathematics such as algebra, combinatorics and geometry for general
problem solving, and to connect mathematics with other branches of learning, especially in
the sciences, economics and aspects of daily life. Providing a rich repertoire of such examples
would greatly strengthen teachers‘ own understanding of mathematics.
While the subject areas are as listed above, the treatment of these areas would be structured
as follows:
1. History of concepts and ideas
2. Varieties of problem solving
3. Arguments and visualization
4. Use of computer software
A child entering Class 6 meets negative integers and learns that the product of two negative
integers is positive whereas their sum is negative. This took a very long time for humanity to
arrive at, and learning a bit of the history would help the teacher not only to tell stories in
class but also help in appreciating children‘s learning difficulties.
It is problem solving that teachers and children see to be the distinguishing character of
mathematics at school, and yet, the only exposure they get to this is end of chapter exercises,
which hardly constitutes meaningful problem solving. Providing teachers with the experience
of exploratory problems, analytical problems, ones that motivate definitions and ones that
apply concepts with a twist, and so on, is important and enriching.
One problem with school mathematics is that most statements are seen as record of fact: that
the product of odd numbers is odd is indeed true, but clearly one that calls for arguments and
convincing. Very often such arguments require one to draw pictures, and move back and
forth between the statements and visual observations. This ability is needed in all forms of
mathematics.
The use of computer technology in the classroom can provide not only new pedagogical and
exploratory tools but also enable then to use the increasing collection of online educational
resources available on the Internet. This requires careful thought but computer use will
increase in the future and our teachers need to get on board very soon.
13
Modalities: Implementing such a course, especially with the advent of MOOCS (Massive
Online Open Courseware Systems), seems to be well within the realms of possibility. There
is expertise available for the design of these courses, though spread across the country.
Certification processes, evaluation procedures, and ascertaining integrity may be challenging,
but these need not stand in the way of initiating such effort.
Nurture in early years: Apart from the certification process, we also need to have a
nurturing mechanism that sustains interest and challenge, especially during the early stages of
service for young teachers (in the < 35 age group).
Here what we envisage is pretty much like the nurture programme for students currently
supported by the NBHM. Through this, young teachers visit universities and research groups,
where they attend some lectures and seminars, but also get to interact with researchers. This
exposure to institutions and networking will be greatly beneficial for their professional
development as teachers.
While visits are beneficial, participation by teachers will gain seriousness with a certification
process. We envisage short courses in specific subjects like topology, graph theory,
cryptography, etc that they participate in during these programmes and work offline on
assigned problems/projects, eventually getting credits for participation. Accumulating certain
amount of credits would entitle one to a diploma and this could be used to lead people
perhaps into doctoral research.
Nurture in later years: Continuing such interaction at later stages of teachers‘ careers would
be beneficial, but forming professional societies is perhaps the best answer to this. Some
associations exist, but the kind of nurturing links envisaged here tend to be lacking. The
powerful medium of the Internet needs to be better utilized to build teachers‘ communities
and the groups referred to in the sections above need to play a responsible role in building
communities, and in their turn, in nurturing them. Such leadership opportunities can, in fact,
generate enthusiasm for positive role as teachers.
Linking these societies with wide ranging mathematics circles (similar to the ones run in the
former Soviet Union) would be an excellent way of sustaining momentum, and reaching out
to new potential entrants to the teacher‘s nurture programme.
Identifying nodal groups that act as resource centres is important. These are locations where a
wide variety of mathematical material are made available and freely accessed, but in a guided
fashion (where needed).
14
Annexure 2.2.2
In-service nurture at the higher secondary level
Proposal for an in-service nurture programme for higher secondary mathematics
teachers
The goal of this programme is to enable higher secondary mathematics teachers to enhance
their content knowledge and to empower them by providing opportunities for professional
growth
The need for a nurture programme
One of the most pressing problems in mathematics education in our country is lack of teacher
preparation and the same has been highlighted in the position paper on the teaching of
mathematics of the National Curriculum Framework 2005. The document states ―More so
than any other content discipline, mathematics education relies heavily on the preparation
that the teacher has, in her own understanding of mathematics, of the nature of mathematics,
and in her bag of pedagogic techniques.‖ Typically secondary and senior secondary school
teachers enter the teaching profession with a Masters degree in mathematics followed by a
Bachelor of Education (B. Ed). However, once they join a school they get very few
opportunities to further their learning. Usually schools send teachers for workshops organised
by various agencies but these are sporadic and do not really contribute to the professional
growth of teachers. Very often it has been found that teachers lack strong fundamentals in the
subject and are unable to enhance their learning due to lack of professional development
opportunities or access to good resource materials. Also, having spent several years in the
profession without any professional growth, they begin to find it monotonous and lose their
motivation. Their inability to make connections between the different topics in mathematics
as well as between mathematics and other subject areas reflects negatively on the way they
teach mathematics. The overall effect is that students develop a ‗blinkered approach‘ to the
subject. Thus there is a tremendous need for sustainable in-service teacher nurture
programmes which not only help teachers to enhance their content knowledge in mathematics
but also develop their pedagogical skills.
The Vision
A nurture programme may be envisioned with the view to
Motivate the teacher with a sense of confidence and pride.
Strengthen the teacher‘s understanding of mathematics and its connections to other
subject disciplines.
Familiarise the teacher with various pedagogical skills.
Create a platform for teachers to interact with mathematicians, mathematics educators
and mathematics education researchers.
15
Structure
The nurture programme must be accessible to teachers across the country. It needs to be
sensitive to the conditions under which teachers work in schools. Constraint of time, large
syllabus, inflexible modes of assessment, preparing students for the board examinations are
some of the problems faced by teachers at the secondary and higher secondary stages.
The programme should have the following characteristics
It should be flexible and teachers should have the opportunity to complete the courses
within a given span of time.
The programme should be conducted in a distance mode. It may comprise of various
modules/courses which will have an online component as well as direct contact
sessions.
The contact sessions may be conducted during the summer and winter vacations
so as to ensure that these do not clash with the teacher‘s regular teaching schedule in
the school. These sessions may be conducted by college and university teachers.
Teachers will have access to the modules and courses of the programme through a
web portal. The same may also be made available in printed form. For large scale
dissemination the contact classes may be conducted using VSAT technology.
Content
The content of the nurture programme
should focus on the topics of secondary and higher secondary school mathematics
from an advanced standpoint, e.g., a course in linear algebra may refer to the concepts
covered in the topic ‗matrices and determinants‘ (usually taught in Classes 11 and 12).
An attempt should be made to provide a blend of content knowledge as well as
pedagogic approaches in transacting the content of various topics of the curriculum.
Care should be taken to provide linkages across topics.
Applications, mathematical models and technology enabled explorations may be
integrated into the topics wherever possible.
A historical background of concepts should form an integral part of the course
Some of the courses/modules offered in the programme may include
1. Nature of mathematical thinking
2. Learning theories in mathematics
3. Historical Development of Mathematics
4. Number Theory
5. Calculus and Applications
6. Linear Algebra and Applications
7. Coordinate geometry
8. Combinatorics
9. Probability and Statistics
10. Mathematics and Technology
16
The last course/module may focus on familiarising the teacher with computer software
specific to teaching and learning mathematics such as Computer Algebra Systems (CAS),
Dynamic Geometry Software (DGS), Spreadsheets and others. The emphasis should be on
exploring various mathematical concepts and ideas through these tools and on the
pedagogical opportunities they offer. The use of multimedia and internet resources for
teaching and learning may also form a part of this module.
Assessment and Certification
The programme should lead to certification and meritorious teachers should be given due
recognition. Assessment may be in the form of credits. Instead of conventional exams the
assessment may include components such as
1. Creation of lesson plans: Planning for a lesson is an important aspect of the
teaching-learning process. Teachers may prepare plans which document the process
of transacting a lesson in terms of pedagogical inputs, activities conducted, problem
solving etc.
2. Classroom teaching: Some classroom sessions conducted by the teacher may be
maintained in the form of video recordings.
3. Reflective journals: These will be maintained by the teacher for all classes conducted
by her and will focus on issues which emerge as a result of her classroom transaction.
4. Creation of professional development activities: Teachers who have undergone the
nurture programme must be equipped to create opportunities which help other
mathematics teachers to grow professionally.
Bright and meritorious teachers must be identified, who after having successfully completed
the nurture programme, may become resource persons who can further contribute to the
programme by conducting some of the modules. This will ensure that the pool of resource
persons for the nurture programme continues to grow. Also the programme should provide
opportunities to suitable candidates to further their academic growth by pursuing doctoral
research in mathematics education. National agencies such as the NBHM/RMS can play a
critical role in providing the resources for developing a teacher nurture programme of high
quality.
17
Annexure 2.3
Pre-service nurture
The major challenge in teacher education is for the university system to gear itself up to the
task of producing B Sc‘s, M Sc‘s and B Ed‘s in sufficient numbers and quality to meet the
need for secondary and higher secondary schools in less than a decade. While this seems
daunting, the following efforts seem worthwhile.
Specialization for pedagogy: The Bachelor‘s and Master‘s programmes in Universities can
introduce a stream that specializes in pedagogy. These are students that see mathematics
education as a career, rather than careers as mathematicians in research or industry. The
curricula would then include special courses for pedagogy. For instance, at undergraduate
level, we can conceive of two courses: one combining three of the four aspects listed above,
namely, History of concepts and ideas in mathematics, Varieties of problem solving, and
Arguments and visualization. Such a course would also be an opportunity to revisit the
content of school mathematics, through which these pedagogic components would be
illustrated. Another course can be on the use of computer software for mathematics pedagogy.
At the master‘s level, it would be important to offer these specializing students not only these
courses, but also an opportunity to revisit courses in Linear Algebra and Calculus, so that
foundations are sufficiently strengthened for higher secondary or even undergraduate
teaching. Revisiting mathematics for pedagogy is a critical approach to appreciating
difficulties in teaching/learning as well as design of problems and exercises and in
engagement with material. This can also offer room for making connections, crucially
missing in our curricula.
We note here that there are existing models for such programmes of Master‘s study in
Mathematics with specialization in pedagogy. For instance, the University of Waterloo in
Canada has been successfully conducting such programmes, and Utrecht University in the
Netherlands is experimenting with such ideas.
Nurture in colleges: While the challenges in influencing the University system may seem
daunting, we can yet take up the task of identifying potential mathematics teachers and orient
them towards Math teaching in a way that is forward looking, and foundationally strong.
Identifying such students is not an easy task. One simple approach to this is a national
competence test as a means of enlarging reach and expanding circles. This will also ensure
some dispersal of such efforts. While mass competitive examinations pose a myriad problem,
they still attach prestige to the outcome and attract talented youngsters.
Those that get selected by such a test would enter the nurture programme. This will get them
scholarships through their course of study, but will also get them apprenticeships with
teachers in different institutions. Including special courses in the curriculum such as the
above for such students would be important for this to work, over a period of time. These
courses can be made available through online accreditation mechanisms, though getting
universities to recognize the courses would yet be a difficult task.
18
Annexure 2.4
A Report on the PU College Teachers Training at Talent Development
Centre, Indian Institute of Science - Kudapura Campus, Karnataka
1. Introduction
Science Knowledge imparted to the students is proportional to Teacher‘s Knowledge. One
teacher teaches about 120 to 150 students per year. Therefore teaching and training the
teachers will have huge multiplier effect. Realizing this IISc decided to set up a Talent
Development Centre to train teachers at all level at its new campus in Kudapura.
IISc had started training high school teachers at TDC, IISc Kudapura. Karnataka Govt.
adopted CBSE syllabus with NCERT text books for PU level. At the request of the PU
Board, Karnataka, Training Need Analysis (TNA) was carried out with 100 representative
teachers, 25 each from PCMB subjects. Analysis showed that they need training and
upgradation of their knowledge for them to become true PU college teachers. IISc accepted to
train 1000 teachers during the first phase that began in March 2012. In this joint venture,
Government of Karnataka provides substantial financial support.
Taking advantage of the vast experience of IISc faculty, a rigorous training for PU College
Teachers has been developed with 60 % time devoted for laboratory experiments to
understand the theory subjects. Highlights of the program are given below.
1. PU College Teachers Training in TDC : Program
a. Objective: To make the teachers confident of teaching the subjects with full knowledge of
revised syllabus and more at the CBSE level with NCERT text books.
b. Participants: Govt. PU College Teachers of Karnataka.
c. Arrangement: PU board will send the teachers to TDC.
d. Subjects covered: Physics, Chemistry, Maths and Biology (PCMB)
e. Period of Training: 10 to 11 days Residential Training Program.
60 Physics Teachers + 60 Biology teachers in one batch.
60 Chemistry Teachers + 60 Mathematics teachers in one batch.
f. Time table: Science -Two lectures and two problem solving sessions in the morning (8 am
to 1-30 pm); Laboratory experiments (2–8 PM), (12 hours a day with breakfast, lunch and tea
breaks)
Mathematics One lecture – Tutorial based on the lecture; second Lecture + Tutorial; Third
Lecture + tutorial and so on for 10 to 12-hours in the Centre. Mathematics teachers are
required solve over 400 mathematics problems during the course.
19
g. Lectures: Lectures to cover the entire +2 syllabus and more.
Assignments/Lectures are on the basic concepts.
Demonstration/experiments in the classroom to make it easy.
Problem solving-assignment sessions to apply the concepts.
Assignment- writing in the class and not in hostel rooms
Laboratory Experiments to augment theory.
All the experiments are done by all the participants. Professors and tutors from IISc
are available all the time.
Submission of assignments is compulsory.
Assignments are corrected and returned immediately.
10 days continuous program without any break.
Experiments Involve measurements - employing an instrument.
Experiments done by the teachers are not those prescribed for the students in PU syllabus.
Experiments are designed to make the teachers understand the theory in each subject.
Determination of Plank constant, Rydberg constant, acceleration due to gravity g, Avogadro
no. N, Angle of minimum-deviation, transistor characters, diodes, Zener diodes, gas law,
absolute zero temperature, optics, spectroscopy, resistivity, earth's magnetic field, sound
velocity, density of solid, liquids, soluble and non-soluble salts in water, R vs. T are a few of
the experiments they do in Physics.
Over 50 experiments in Biology, 40 experiments in Physics, 40 in Chemistry are carried out
in 10 days. These experiments cover the entire syllabus they need to know for teaching PU
students. Most teachers have not performed these experiments during their B. Sc. or M. Sc.
Properly designed rigorous courses in Physics, Chemistry, Mathematics and Biology have
been developed with the help of experienced professors of IISc and from some of the
University professors of Karnataka. The program is not a sum of isolated lectures and
experiments. The teacher‘s interest is continuously kept up till the last hour so that they learn
the subjects with full component of experiments they have not done. Their participation and
involvement is continuously monitored. Motivation to study becomes contagious and also
competitive among the teachers due to unique residential atmosphere created in TDC. The
training has been made enjoyable too. Food and other facilities have been appreciated by all
the participants and there have been no complaints.
h. Tests and Examination
First day Morning: Examination 1. (On what they are supposed to know). Direct questions to
test the knowledge and no multiple choice questions.
10th
day morning (after the training): Examination 2 (Questions similar to first exam 1 but a
bit more difficult; different questions). Viva-discussion -interaction sessions to bring up the
bottom 20%. Question papers and answer papers are returned within two hours.
Each of the experiments done by the participants is examined and approved by the
Laboratory Instructors and professors. Approval is a must. They need to write the laboratory
20
report then and there and take the signature showing the correct results. In case the results are
not satisfactory, they need to repeat the experiments. No scope for copying or evasion of
doing experiments.
Tests and examinations became essential to make the program effective. In effect the test
after the training to compare what they did during the program is an indirect test for IISc
professors. All the lab manuals, assignments, test papers and evaluated answer papers have
been given back to the teachers. The system is fully open.
3. Feed back:
The teachers are unanimous in their feedback that this is a unique training program,
first and only of its kind and essential for their profession, a must for every teacher.
4. Results:
The result of the training program is in the form of marks the teachers received before and
after the training. The test is on their basic knowledge required for them to teach. The tests
have been standardized to have uniformity in all the subjects of PU, CBSE and a little higher
level. The figure below shows the bar diagram before and after the training for one batch of
Chemistry and Mathematics teachers. This is a typical result. Average marks scored before
the test is 25 to 33% in most of the batches. This means at least 50% fail the test on 35%
scaling. After the training they scored an average of 65 % and only 2 to 3% of the teachers
fail on 35% scaling. The standard of the test is such that if the teacher scores 40 and above,
he should be able to teach the subject reasonably well.
Those who scored 60% and above are very good and those above 80% are excellent who can
be utilized by the PU board to train other teachers. In our opinion the teachers who did not
pass 35% mark have not worked hard, took the program lightly in spite of our best efforts.
This number is less than 5%. Among them those scoring 20% and below in our test will
not be able to do justice to their profession and the students will suffer. It is for the PU
Dept. to do the needful to either persuade them to do well or utilize their services for
some administrative-alternative work.
21
7th batch PU Chemistry Average: First test 33%, Second test 67%
7th batch PU Mathematics Average: first test 34%, Second test 63%
5. An analysis of PU teachers:
There are three categories of teachers: (a) Regular M.Sc. degree holders from one of the
universities selected from KSPSC; (b) Regular M.Sc.; joined Govt. high school for a safe
Govt. job, promoted to PU colleges. (c) Distance education M.Sc. holders mainly from
Kuvempu University promoted from high schools.
Teachers in the first category are generally good and they can become excellent teachers.
Marks in our tests reflect this.
Those in the second category pick up the subjects fast. They will become good teachers soon.
22
The third category is the one that needs a discussion.
On principle, we have not made any grouping-distinction in our training program
because all the teachers need to be treated in only one way.
We have had a lot of difficulty to train the last category: their knowledge is truly poor
because of the way they are trained during their degree we believe. Most of them have not
done even the routine experiments a regular M. Sc. student does. It is not the fault of the
teachers. It is certainly the fault of the University that gave them the degree. These teachers
were diffident and apologetic. At times nervous. We believe that Kuvempu University
needs to do a far better job to propagate their science distance education program if
their students are to compete with the rest. PU Board should take note of our observations
and perhaps the facts be brought to the notice of the VC of Kuvempu University for him to
improve the science distance education program through appropriate channel such as Council
of Higher Education.
However, the promoted teachers coming from the distance M.Sc. are highly motivated and
also young. There is a huge human resource. Therefore we considered it our duty to bring
them up to the same level as others. We did an experiment by calling only the promoted
teachers in a batch. An average mark scored in Physics was 13 and 18 in biology – a result
that is expected. Realizing the difficulty, we extended the training by one more day. In the
second exam, they scored an average of 55 in Physics and of 73 in Biology- highest of all the
batches trained so far in comparison with all the three categories in a batch. It must be
mentioned here that a lot of effort was put in by the professors and instructors/tutors with
willing participation of the trainees to achieve this result.
6. Some observations and summary:
The teachers in general are highly talented and are highly motivated;
They work hard and improve their knowledge;
Highly enthused after seeing the Professors of IISc. ;
They pick up experimental skills faster than theory;
IISc training also inculcates discipline and truthfulness;
The teaches appreciate the proactive approach of the Govt. of Karnataka for this
continuing education;
Teachers are grateful to Govt of Karnataka & IISc for the opportunity.
Teachers expressed that all the Govt. and Govt. aided PU teachers should get this
opportunity to undergo training and they certainly feel the quality of teaching will be
far higher in the years to come.
23
They also commit themselves to study more, and teach the students to come up in
their life.
Feel committed to teach and train weaker sections of society.
Mathematics teachers are doing generally better than science teachers.
This is a unique training programme which does not exist anywhere else.
7. JNV PG Teachers training: Two batches of Navodaya School (JNV) PG teachers from
all over India have received training here in TDC. They are a shade better than our State PU
teachers. That may be because they were teaching PU students with CBSE syllabus following
NCERT text books for all the time. JNV PG teachers also found the programme highly
useful. Taking that point, the PU teachers trained at TDC should soon be comparable to JNV
PG teachers.
8. Conclusions: Two more batches, one Chemistry and Maths (Nov. 17-26) and Physics and
Biology (Dec. 2-11) will complete their training as planned. The training of the next 1000
teachers will start as soon as money is released.
Annexure 3.1
A PROPOSAL TO FUND AND COORDINATE A „MATH CIRCLES‟
INITIATIVE ACROSS THE COUNTRY
PROPOSAL
To engage with mathematics teachers and students of high school in towns and cities across
the country, by forming ‗Math Circles‘ which meet once every few weeks.
WHAT ARE „MATH CIRCLES‟?
―Mathematical circles are a form of outreach that brings mathematicians into direct contact
with pre-college students. These students, and sometimes their teachers, meet with a
mathematician … in an informal setting, after school or on weekends, to work on interesting
problems or topics in mathematics. The goal is to get the students excited about the
mathematics they are learning; to give them a setting that encourages them to become
passionate about mathematics.
―Math circles can have a variety of styles. Some are very informal, with the learning
proceeding through games or hands-on activities. Others are more traditional enrichment
classes, but without formal examinations. Some have a strong emphasis on preparing for
Olympiad competitions; some avoid competition as much as possible.
Models can use any combination of these techniques, depending on the audience, the
mathematician, and the environment of the circle. Athletes have sports teams through which
24
to deepen their involvement with sports; math circles can play a similar role for kids who like
to think. One feature all math circles have in common is that they are composed of students
who enjoy learning mathematics, and the circle gives them a social context in which to do
so.‖ [Mark Saul, NSF (USA)]
It seems to me that the notion of a Math Circle holds extremely rich possibilities, and is of
great relevance to us in India. It could be a way of reaching out to students in remote parts
and bringing together like-minded mathematicians scattered across the country. It could have
a hugely energizing effect on math education.
A coordinating agency is needed, which acts as a central repository of materials and provides
a platform for networking, through dedicated pages on its website.
Some financial commitment will be required, as travel is involved. The resource people will
want assistance with regard to purchase of materials, printing, photocopying, etc.
The role of the coordinating agency will be crucial for finding people who can anchor Math
Circles; finding institutions willing to offer their facilities for such activities; and making
such activities widely known through sustained publicity.
OBJECTIVES
1. To raise the level of awareness of mathematics as a human endeavour.
2. To raise the level of awareness of mathematics as a subject in which exploration can
be done.
3. To stimulate problem solving skills among students and math teachers.
METHODOLOGY
Problem solving in Euclidean geometry, number theory, combinatorics, finite
geometry, graph theory, cryptography and related topics, done collaboratively
Expository lectures on miscellaneous topics in mathematics, including applications in
medicine, economics, e-commerce, geography and so on
Expository lectures on the life and work of selected mathematicians
PROPOSED AUDIENCE
● High school and senior secondary school students
● High school and senior secondary school mathematics teachers
● Parents of students
LOCATION(S)
● High schools across the country
25
ADDITIONAL NOTES
The notion of a Math Circle originates from the former Soviet Union and countries
like Bulgaria, Hungary and Romania, where it has been highly successful because of
enthusiastic participation of the mathematicians of those countries.
It is not necessary to have a single monolithic model for a Math Circle. As noted in
the opening paragraph, there are a variety of ways in which such Circles can be held,
depending on location, personality of the anchor, and so on. So the Circles should be
allowed to develop their individual ‗personalities‘.
Annexure 3.2
OPTIONAL MATHEMATICS PROGRAMME FOR BRIGHT STUDENTS IN
SCHOOLS
A country aspiring to be a world leader in science and technology must have quality
education in Mathematics at school level. It is necessary to remember what V. I. Arnold
states: ―Mathematics training in Moscow usually begins before the school age‖. In India,
mathematics education up to class 10 is compulsory. The curriculum for this is largely quite
good. But the quality of examinations is of great concern since, both average and very good
students secure almost the same marks in these examinations. Hence, the examinations results
are unable to distinguish a bright student from an ordinary one. This leaves no motivation for
talented students (whose number is very large in India) to learn mathematics deeply; they
simply practice for high marks. Further, this situation drives institutions of higher education
and companies to hold their own examinations for admissions/employment. This results in
wastage of time and resources and creates tremendous physical and mental pressure on
students and parents.
It would be very difficult to change the present trend (which anyway is not any evil
for average students) in question paper setting and evaluation. Then it becomes evident that
we need to provide for full growth and for utilization of complete capacity of good and
talented students in order to (i) attract them to study mathematics deeply, and (ii) inspire them
to work hard to learn and enjoy it. It is quite simple to achieve both these aims and also grant
recognition to their talent by an OPTIONAL MATHEMATICS (OM) programme which will
comprise (a) providing excellent books in OM from class 6 to 12 (or even earlier); (b) having
two (or at least one) special classes, per week in OM, only for students good in mathematics,
from these classes, in order to provide appropriate learning material along with guidance in
learning and problem-solving on their own; (c) and conducting final examinations at 10th
and
12th
level.
Existence of mathematics Olympiad activity in India has ensured the availability of a
large number of excellent books, advocated in (a). To achieve (b) and (c) all examination
boards (e.g., CBSE, which can take a lead first) must introduce papers in OM, at 10th
and 12th
level, based on special enrichment material. These papers would test deeper understanding of
mathematics and ability to solve challenging problems. These papers would be meant for
26
those who have interest in learning deeper aspects of mathematics. The marks scored in these
papers must be recorded in transcripts. Problems set for these examinations must test the
competence in mathematical deductions and not in employing tricks and mere practice. The
responsibility of setting these papers and evaluation must be given to a special autonomous
cell (headed by an eminent mathematics Professor), set up for this purpose, in the
examination boards. The cell would choose college and university teachers along with some
school teachers for this task.
One may wonder ―why optional paper in mathematics alone and not in other
subjects?‖ We provide some answers to this: (i) Majority of the students taking OM would go
in for a career in disciplines other than mathematics and their enhanced capability in
mathematics would enrich other disciplines. (ii) In contrast with other subjects, those who
miss good mathematics till class 8 are not likely to learn good mathematics later on. (iii)
Success of bright students in OM programme in many schools would instil confidence and
will provide an excellent opportunity to the children of unprivileged members of the society
to compete with other students in admissions to national institutions, an opportunity which
probably no other existing programme can provide. (iv) USA‘s example, in rectifying and
revamping mathematics education, after USSR‘s launching of the sputnik, should not be lost
sight of.
Olympiads and Government supported KVPY programme are serious attempts to spot
and nurture talents. But there is no arrangement for students to learn various subjects
(mathematics included) deeply at an early stage. In developed countries there exist provisions
for bright students to learn subjects of their interest at their own pace. OM programme will
provide this opportunity (at least in one subject). One great advantage of the OM papers at
the 10th
and 12th
level will be a decrease in the number of examinations (like RMO; JEE and
KVPY at least in one subject; and admission tests of many institutions of higher education).
In view of OM scheme being implemented, there is no need to burden every student
with too many deeper and difficult concepts in mathematics.
Annexure 3.3
Nurturing Mathematical Talents in Schools (NuMATS)
The State Council Educational Research and Training (SCERT) Kerala has initiated a
programme for students called Nurturing Mathematical Talents in Schools (NuMATS).
We suggest a similar programme:
The aim of the Nurturing Mathematical Talents in Schools (NuMATS) should be to
attract the students to study mathematics by showing the challenge, thrill and beauty of
mathematics. This will prepare them not only to go for higher study in mathematics but also
to engineering and other sciences. Many of the students, through NuMATS, would eventually
choose a career in a discipline other than mathematics and the enhanced capability in
27
mathematics will help them to do better in their areas of specialization. One additional benefit
will be to prepare the students for Mathematics Olympiad.
This programme cherishes the student‘s talents from the age of 10. The constant help from
experts will develop students‘ capacity and reasoning ability in such a way that at the age of
15 they will be in a position to take the challenge of competitive examinations.
Each year, 74 students from 6th standard would be selected at the state-level. 3 Tier selection
process is adopted for this purpose.
School: Each school nominates 5 mathematically talented students (2 General, 1 SC, 1 ST
and 1 Differently abled) from those studying in 6th standard.
Sub district level: An aptitude test is conducted with Mathematics Quiz, Preparation of notes
on a selected topic, an activity among Drawing, Measuring or Construction, and Test on
Problem solving. The selection committee will select 9 students (6 General (3 urban 3 rural),
1 SC, 1 ST and 1 DA) based on the total score in the aptitude test.
State level: An Aptitude test is conducted for the selected students. From each of the 14
districts, 5 students (4 General (2 urban 2 rural), 1 SC/ST) will be selected. 4 differently
abled students are selected from the state list.
Activities:
After the selection is completed in the middle of January, the group of students will be given
a set of problems (containing simple mathematical creativity), which they will solve on their
own but with a little guidance of their school teacher or parents and relatives.
Finally a learning camp of 10 to 15 days will be organized for the selected students in the
month of April-May. Higher level camps according to the development of the children would
also be conducted. This continues till they complete class 10 (continuous 5 years). A 10 day
camp for classes 6 & 7 and a 15 day camp for classes 8, 9 & 10 are proposed. Students should
be guided by mentors. Finally, each year there will be 5 groups of students (each group
consisting of about 74 students from each of classes from 6 to 10). The camps will consist of
lectures, problem solving and interaction with mathematicians. In each camp the students will
be provided with a sufficient number of books/reading materials and problem sets to last till
the next camp. There will also be a cumulative record for each student.
The details of the NuMATS of SCERT can be seen on the website www.scert.kerala.gov.in.
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Annexure 3.4
Proposal for Setting up Mathematics Laboratories in Schools
The Rationale
Mathematics has for years been the common language for classification, representation and
analysis. Learning mathematics forms an integral part of a child‘s education. Yet, it is also
the subject, which has traditionally been perceived as difficult. The primary reason for this
state of mathematics learning today is the significant gap between content and pedagogy. In
the last few decades countries across the world have witnessed a major shift of paradigm as
far as mathematics teaching and learning is concerned. Mathematics education is being
revolutionized with the advent of new and powerful technological tools in the form of
dynamic geometry software (DGS), computer algebra systems (CAS), spreadsheets and
graphic calculators which enable students to focus on exploring, conjecturing, reasoning and
problem solving and not be weighed down by rote memorization of procedures,
computational algorithms, paper-pencil-drills and symbol manipulation which are often
characteristic of traditional classroom teaching.
However in a vast country like India, large scale integration of technology in classrooms is
fraught with numerous challenges. Above all, technology must be cost effective and easy to
deploy in order to achieve such large scale integration. In this context, mathematics
laboratories may provide a solution. Every school may set up a mathematics laboratory which
can be the platform through which students are given access to technology to explore and
visualize mathematical concepts and ideas. In fact the position paper on the ‗Teaching of
Mathematics of The National Curriculum Framework (NCF) 2005‘ for school education
developed by NCERT emphasizes that mathematics learning should be facilitated through
activities from the very beginning of school education. These activities may involve the use
of concrete materials, models, pattern charts, pictures, posters, games, puzzles and
experiments. The Framework strongly recommends setting up of a mathematics laboratory in
every school in order to help exploration of mathematical facts through activities and
investigations.
What is a Mathematics Laboratory?
In a mathematics laboratory, the students should be given the opportunity to explore and
visualize mathematical ideas and concepts by engaging them in various activities. These
activities should be designed to enhance their understanding of the subject as taught in the
classroom and also provide a glimpse of what is beyond. Activities conducted in a
mathematics laboratory may comprise of projects, experiments and modelling exercises
based on the mathematics taught in the curriculum. They must enable the students to make
connections across topics in the curriculum and at the same time enhance their problem
solving skills. The projects or exercises performed in a mathematics laboratory may be
designed to fulfil one or more of the following criteria. The exercise should
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• Highlight some known concept based on a well known mathematical theory.
• Shed new light on some aspect of the topic being studied.
• Lead to some original discovery on the part of the student.
• Focus on some interesting application of mathematics to a practical problem.
Activities of a Mathematics Laboratory
The primary objective of setting up a mathematics laboratory is to enrich mathematics at
school level and to transmit the beauty of mathematics as a discipline to students by
providing them with an environment, which encourages independent and original thinking
through the ‗learning by discovery‘ approach. The activities of the laboratory may be broadly
classified as follows
• To create and conduct projects and activities which focus on applications of mathematics to
practical problems thus relating school mathematics to situations outside the classroom.
• To integrate the use of technology, specifically graphing calculators, dynamic geometry
software, computer algebra and other software packages in mathematical modelling activities.
• To develop and enhance problem solving skills in students.
• To conduct hands-on activities which highlight some mathematical concept.
• To provide resources to students and teachers.
Hands on Activities (for middle school)
The use of manipulatives and hands on activities form an integral part of a Mathematics
laboratory. They support the constructivist view that students need to construct their own
understanding of any mathematical concept. The challenge is therefore to help students
develop their mental constructs through the use of manipulatives such as geoboards, tiles,
tokens, pipes and paper folding activities. Such activities emphasize the ‗learning by
discovery‘ approach where the students will be encouraged to explore concepts, discover
patterns and generalize results on their own while the role of the teacher will be that of a
facilitator guiding the students in the process of discovery.
Typically, an activity may have three components
1. A hands-on component: Here the student will be required to create a model or a drawing,
or perform a construction based on a mathematical concept. The model may be a physical
model or a computer-based model highlighting the concept.
2. An investigatory component: This part will require the student to understand and explore
the mathematics on which the activity is based. Technology may be used to support the
investigations if required.
3. A problem solving component: Here the students will attempt routine as well as non-
routine problems based on the concept and the model.
Technology enabled explorations for secondary and senior secondary school
One of the objectives of a mathematics laboratory is to enable the student to appreciate the
beauty of mathematics as a discipline and also to encourage a spirit of research among bright
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students. This can be achieved by introducing the student to problems and applications of
mathematics which they do not encounter in their school curriculum. Thus activities which
encourage the student to engage with the processes such as non-routine problem solving,
estimation, approximation, use of heuristics and generalising patterns, need to be designed. In
the laboratory, the teacher‘s role is primarily to facilitate students‘ explorations and lead them
to ‗discover‘ mathematical concepts for themselves. Technological tools such as computer
algebra systems, dynamic geometry software and spreadsheets can play a critical role in the
activities of a mathematics laboratory. They can help the student to explore, visualize and
compute. Technology enabled activities in a mathematics laboratory may be categorised
under the following heads:
1. Visualization and exploration of concepts using various technological tools.
2. Exploring geometrical ideas using dynamic geometry software.
3. Simulation of problems in probability using spreadsheets.
4. Investigatory projects based on mathematical modelling and applications of topics taught
in the curriculum.
Some suggested topics for exploration (in the form of investigatory projects) by students
in a mathematics laboratory
1. Mathematical modelling in genetics.
2. Modelling the growth of an epidemic.
3. Modelling AIDS and HIV Infection.
4. Mathematical Applications in Cryptography.
5. RSA: Public Key Encryption.
6. Modelling of Brand Switching and Weather Forecasting.
7. Queuing Problems at a Vehicle Service Station.
8. Arbitrating Disputes Using Utility Theory.
9. Application of Calculus to Radio Tuning.
10. Exploration of Gibbs Phenomenon.
11. Estimation of : The Buffon‘s Needle Problem and other Monte Carlo methods.
12. Simulating the Monty hall problem
13. Mathematics in games and gambling.
14. Volumes and hypervolumes.
Students‘ explorations may be mentored by bright college students, college teachers and
university professors along with their own teacher. The mathematics laboratory must be a
collaborative initiative within the school which not only provides the student with an
opportunity to explore and learn new mathematics but also nurtures the interest and talent of
the student. It should also help the mathematics teachers of the school enhance their
pedagogical knowledge as well as content knowledge in mathematics.
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Annexure 3.5
Nurture Programme in Mathematics for Secondary School Students in
Kendriya Vidyalaya Sangathan
There are 1090 Kendriya Vidyalayas under Kendriya Vidyalaya Sangathan, New
Delhi. These schools are doing very well in academics as well as co-scholastic activities. The
organization provides quality education to its students with a holistic approach through good
teachers. The teachers take utmost care of the slow learners and bring them up to Minimum
Level of Learning.
There are few programmes for nurturing Mathematics such as the prestigious KVS
Junior Mathematics Olympiad Examinations for the students of classes X and XI. The
toppers of JMO in each region are selected and given training to prepare for RMO and
INMO. However, no such programmes are available to the younger students of class VI
onwards till class X. Following are some of the suggestions to identify ―mathematically
talented‖ students of class VI and nurture their skills.
1. Every region can conduct a test and select talented students from that region in
Mathematics in class VI. The number can be restricted depending on the number of
Schools in the region.
2. Mathematics study camp can be organized for the selected students at the Regional level
during the summer holidays. The help of eminent Mathematics Educationists can be
sought to conduct such camps. Credits can be given to the participants for attending the
camps after appropriate assessment.
3. Such camps can be planned for these students in successive years at the end of each
academic year.
4. Finally a certificate can be issued to the successful students at the end of class X.
Around 5000 students can be trained in such camps in a span of 5 years and this can be
made a regular feature thereafter. This programme will create an awareness among the
students about Mathematics and motivate them to pursue education in Mathematics.
In each region of KV Sangathan one can also follow the model described in Annexure 3.3
on NuMATS.
Annexure 3.6
A note on Research in Schools Initiative
What started as a summer programme of lectures for school students has changed into
a Research in Schools Initiative in the last three years in Chennai. The main aim of the
programme is to attract school students to research in the basic sciences, convince them that
engineering and medicine are not the only options for bright youngsters. Led principally by
professors of IIT-Madras and support from faculty of IMSc, CMI, Anna University and
others, the programme is coordinated by Padma Seshadri Bal Bhawan group of schools.
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Towards the end of the year, a letter is sent to many schools requesting them to
sponsor students for the programme. Participating students fill in a questionnaire indicating in
detail their interest in science and mathematics (apart from their academic record). In
Feb/March, interviews are conducted for selected students to determine their aptitude and
interest, and about 50 students are selected (typically those completing Class XI in May).
For the selected 50, a month long programme of lectures is arranged, by scientists
from a range of institutions. Typically the lectures are held in the morning and students visit a
range of laboratories in the afternoons. Every student is assigned a mentor and for about two
months, the student meets the mentor regularly, and is assigned to research or advanced study
project. Often, these students keep in touch with the mentor throughout their Class XII year.
Assisting in labs and working on advanced level problems give them a taste of research, and
being physically in a research institution gives the students an idea of the processes involved.
Such early immersion in research is attempted in many leading American universities with
interesting results, and the Chennai experience has been positive.
Annexure 3.7
A note on Organized Math Circles
While Math Circles can and should run on individual effort and motivation, it is
worthwhile considering an idea to systematically set up Math Circles all over the county.
Ideally, we should have one in every district.
The proposal: Set up a mathematics circle in every district of the country, which
eventually creates a resource pool for mathematics education and mathematical talent search.
Pilot proposal: To test the feasibility of the idea, we could try this in 5 districts each
in 5 states: this should be eminently feasible in Kerala, Tamil Nadu, Maharashtra, West
Bengal and a northern state, coordinated from Delhi.
Activity: The circle would consist of 5 to 10 teachers, 50 to 60 children, who meet at
least once in a month, preferably twice. The activity of the circle would be to conduct non-
routine problem solving sessions, lectures on mathematical themes, mathematical design
activities etc.
Tamil Nadu Science Forum has had some experience in conducting such regular non-
routine problem solving sessions. The circle meets every month, and a set of problems in one
topic is chosen for each time: Algebra, Geometry, Number theory, Combinatorics etc. The
day is spent only on problem solving and discussion. Children and teachers bring their own
lunches, so expenditure is minimal. At the end of the day, a problem sheet is given to work
on until next time, which is reviewed at that meeting. Attendance is voluntary, and while half
the group comes and goes, about half are regular.
Structure: This would require some structure of coordination at state level, to plan
out activities, review programmes and organize resources (visits of lecturers, preparation of
problem sheets etc). However, the budgetary needs are small and can perhaps be met by
existing institutions.