I responded to a questionnare sent by the NCERT regarding the teaching of mathematics and thought it would be worthwhile to share the questions and answers.

1. Role and purpose of mathematics education: NCF-2005 has envisioned that the main goal of mathematics education in schools is the mathematisation of the child’s thinking. Clarity of thought and pursuing assumptions to logical conclusions is central to the mathematical enterprise. There are many ways of thinking, and the kind of thinking one learns in mathematics is an ability to handle abstractions, and an approach to problem solving. Please comment on the level of importance you would attach to different stages of mathematics education and your views regarding its role and purpose(s).

Mathematical learning/teaching must deal with three components:

1. Development of the ability to carry out computations (often without worrying about the context). This is sometimes called Algebra.

2. Pinpointing one's doubts and analysing them through logical thinking. This is sometimes called Analysis.

3. Constructing new mathematical objects out of old ones and justifying these constructions. In school this is primarily in the context of Geometry but later on it occurs elsewhere as well.

All three aspects --- Algebra, Analysis and Geometry --- are essential to Mathematics. One neglects one or the other at one's own peril!

2. Concerns regarding mathematics: NCF-2005 has identified the following core areas of concern: (a) A sense of fear and failure regarding mathematics among a majority of children, (b) A curriculum that disappoints both a talented minority as well as the non-participating majority at the same time, (c) Crude methods of assessment that encourage perception of mathematics as mechanical computation, and (d) Lack of teacher preparation and support in the teaching of mathematics. (e) The emphasis on procedural skills rather than on the understanding of mathematics (f) There are concerns about the type and quality of the mathematics education that students experience in schools. Systemic problems further aggravate the situation, in the sense that structures of social discrimination get reflected in mathematics education as well. Especially worth mentioning in this regard is the gender dimension, leading to a stereotype that boys are better at mathematics than girls. Keeping these concerns NCF-2005 is in action since last 6-7 years. We would welcome your views on these issues or other concerns that you may wish to raise. Also comments on the practical experiences/ concerns on achievement via NCF-2005 recommendations are invited.

The point (d) about lack of trained teachers is the most important concern. The primary difficulty is not the lack of Mathematical training on the part of the teacher, though that is certainly lacking in many cases. Rather, the problem is that the teacher fails to see the classroom as one where she/he too can learn. Thus, the teacher (and hence the student) fail to learn to learn.

The second most important point is the focus on assessment (c) which is seen as a punishment and reward system rather than as a feedback mechanism by which the teacher and student figure out what needs to be studied next.

The remaining problems stem from these two sources.

3. Recent developments in mathematics education: In the past years, a revised mathematics curriculum has been implemented at different stages of schooling. NCF-2005 has recommended: (a) Shifting the focus of mathematics education from achieving ‘narrow’ goals to ‘higher’ goals, (b) Engaging every student with a sense of success, while at the same time offering conceptual challenges to the emerging mathematician, (c) Changing modes of assessment to examine students’ mathematization abilities rather than procedural knowledge, and (d) Enriching teachers with a variety of mathematical resources. The shift in focus NCF-2005 proposes is from mathematical content to mathematical learning environments, where a whole range of processes take precedence: formal problem solving, use of heuristics, estimation and approximation, optimisation, use of patterns, visualisation, representation, reasoning and proof, making connections, mathematical communication. Please comment on the impact of these changes and whether they go far enough to address the problems in mathematics that have been identified. Also comments on the practical experiences/ concerns on achievement via NCF-2005 recommendations are invited.

As mentioned above, no amount of twisting and turning with the curriculum and contents will have a significant impact unless such changes can help motivate the teachers better.

Also mentioned above is the need to retain the computational skill component of mathematics. One learns to ride a cycle before one learns how it is made and one learns to nurture and to grow plants in a garden well before one learns the chemical and biological processes behind agriculture. Of course, mathematics is distinct in that the process of computation can be analysed and re-built from the ground up. This does not mean that one should start with such a (de-)(re-)construction.

A bad teacher can teach computational skills (through hateful drill) which still will be useful, just as a bad programmer can still write ugly computer programs which work. However, one needs a good teacher if one is going to teach conceptualisation. While the suggested aspects like visualisation are important, one should ensure that one does not throw out the baby with the bath water.

4. Current trends in mathematics education: A crucial implication of recommended shift lies in offering a multiplicity of approaches, procedures, solutions. NCF-2005 see this as crucial for liberating school mathematics from the tyranny of the one right answer, found by applying the one algorithm taught. Such learning environments invite participation, engage children, and offer a sense of success. Please comment on the relative merits / concerns of applicability of such approaches in Mathematics classes at different stages.

The algorithms that one learns have been developed through the ages. This does not make them sacrosanct, but it does make it important to learn to carry them out. For learning is, in good part, the gathering of one's ancestral wisdom. At a certain age, (usually around the time one enters high school) one learns to question this wisdom --- and that should be encouraged, but one can only do that if one has it on hand.

In mathematics, there is almost always only one "right" answer. However, the emphasis should be on learning from the process by which one obtained the answer, rather than reward for the right answer and punishment for the wrong one.

It would be wrong to give students a sense of ambiguity about the correctness of mathematics. An important aspect of Mathematics (to all users of Mathematics) is its sense of universal correctness. This becomes more nuanced as one studies it deeper, but there is no doubt (amongst its practitioners) that this is its primary purpose.

5. Teaching of Mathematics:- A VISION STATEMENT: NCF-2005 envisioned that, school mathematics takes place in a situation where: • Children learn to enjoy mathematics: • Children learn important mathematics: • Children see mathematics as something to talk about, to communicate, to discuss among themselves, to work together on. • Children pose and solve meaningful problems: In school, mathematics is the domain, which formally addresses problem solving as a skill. • Children use abstractions to perceive relationships, to see structure, to reason about things, to argue the truth or falsity of statements. • Children understand the basic structure of mathematics: Arithmetic, algebra, geometry and trigonometry etc. • Teachers expect to engage every child in class. Please comment on these visionary statements and on the relationship between learning mathematics and these processes. Please comment on the relative merits / concerns of applicability of such approaches in Mathematics classes at different stages. We would also welcome your views on the students' learning intake and any suggestions you might have for improvement/ updating.

The kind of child-centred learning is difficult if there are a large number of students in each class. However, if the teacher involves students in teaching each other, a lot is possible. At the same time, the teacher needs to move from the role of giver (or imparter) and be more involved in being trained herself/himself. Adopting this role will help students see a live example of continuous learning and thus pick up the habit themselves Such a role will be difficult, if not impossible, in a traditional-minded society such as ours.

6. Influence of the assessment: In terms of assessment, NCF-2005 recommends that Board examinations be restructured, so that the minimum eligibility for a State certificate be numeracy, reducing the instance of failure in mathematics. On the other hand, at the higher end, it recommends that examinations be more challenging, evaluating conceptual understanding and competence. Please give us your views on the assessment of mathematics. Also, please comment on the relative merits / concerns of applicability of such approaches in Mathematics classes at different stages.

One of the greatest ills that plagues our education system is that it is largely geared towards certification and eligibility. Given societal needs this aspect of education may be unavoidable and perhaps even necessary evil. We need to think about mechanisms to give the students something more.

The primary purpose of in-classroom evaluation is as a feedback mechanism that helps the student and the teacher improve themselves and move forward. Focussing on grades at this stage is definitely counter-productive. I believe that using this assessment as part of the final grade has reduced its utility for this reason. Thus internal evaluation should remain internal!

Coaching/drilling for certification examinations may need to be separated from this process of classroom learning. This is already happening through coaching classes. While it may be true that coaching only helps students pass the "test of fire" (and then feel "burnt out"), the drill and stamina development is not unimportant if carried out in moderation. In today's society, one does need to learn how to give examinations, appear for interviews etcetera.

8. Syllabus levels and Curricular Choices: When it comes to curricular choices, NCF-2005 recommends moving away from the structure of tall and spindly education (where one concept builds on another, culminating in university mathematics), to a broader and well-rounded structure, with many topics “closer to the ground”. If accommodating processes like geometric visualisation can only be done by reducing content, NCF-2005 suggests that content be reduced rather than compromise on the former. Moreover, it suggests a principle of postponement: in general, if a theme can be offered with better motivation and applications at a later stage, wait for introducing it at that stage, rather than go for technical preparation without due motivation. As a practitioner, how you feel about these recommendations in current syllabus and its practices. Also, please comment on the issues, if any and on how they might be addressed within the current review.

At the shallow end of a pool one can learn not to fear water, and one can learn how to push it around to get an idea of the "theory" of swimming. However, one cannot learn swimming by wallowing in shallow waters!

Numerous examples of the above kind have already been provided before to say that there does not seem to be a reason to delay the development of skills until one has learned the concepts behind their operation. This does not mean that one should neglect analysis and construction. However, it is acceptable for the latter two to lag behind the pace at which one develops skills. Just like singing, dancing and playing football, it is easier to pick up a facility with numbers and symbols at an early age. It is only a few highly motivated individuals who show the courage and dedication required to learn these skills when they are beyond their adolescent years.

8. Student achievement in mathematics: In previous questions considers the evidences of learning Mathematics. How effective, in your view, would each of the following measures be in improving the performance of students in mathematics assessment?

Remark: In question (i) below, this additional class time should not be used to introduce more material!

(i) allocation of more class time to mathematics

effective

(ii) better pre-service and inservice education for teachers of mathematics

very effective

(iii) improved mathematics textbooks and other learning resources

effective

(iv) provision of learning support for students who are experiencing difficulties with the subject

very effective

(v) provision of ‘general’ as well as ‘specialist’ mathematics courses

effective

(vi) increased emphasis in examination questions on the application of mathematics to real-world problems

not effective

(vii) the introduction of additional forms of assessment, such as coursework

not effective

(viii) improving the perception of mathematics among parents and the general public

very effective

9. Teaching and learning in mathematics: What do you observe that currently Mathematics classrooms indicates that mathematics is taught and learned in a ‘traditional’ manner, mainly involving teacher exposition or demonstration of procedural skills and techniques for answering examination-type questions, followed by student practice of these techniques (in class or as homework) using similar questions. There appears to be little or no emphasis on students understanding the mathematics involved, or on its application in different or unfamiliar contexts. Please comment on the strengths and weaknesses of this approach. We would also welcome your views on change in teaching and learning approach in Mathematics, the degree to which syllabus change, assessment change, teacher professional development and support would contribute to bringing about changes in teaching and learning.

The current mathematics classrooms are limited, but what they are teaching is not unrelated to mathematical education. Thus the current approach needs to be supplemented with additional work on discussing the concepts, analysing doubts and attempting to construct new concepts. For this to work, the teacher must be also be encouraged to learn in the classroom along with her/his students. The teacher can learn to teach better and also learn to analyse and evaluate any mathematical ideas that are discussed.

A certain amount of change in the syllabus (say about 10%) is required to "keep up with the times". Even though Mathematics is eternal, tastes and utility of mathematical ideas changes with time.

In terms of assessment, it seems to me that involving internal assessment in the certification process (of big competitive examinations) is a failed experiment. It would be far better if the former is kept separate from the latter so that classroom assessment can serve its true goal of self-improvement.

10. Attitudes to and beliefs about mathematics: NCF-2005 – and research papers on international trends in mathematics education – raises, on a number of occasions, issues surrounding the perceptions, attitudes and beliefs that exist in relation to mathematics, such as • the view that mathematics is a difficult subject • negative attitudes towards mathematics including, for some, a ‘fear’ of the subject • the perception and advocacy of mathematics, particularly Higher level mathematics, as an elite subject for only the ‘best’ students • research findings that suggest a connection between teachers’ views of mathematics and their approach to teaching it. We would welcome your views on these or other issues associated with mathematics.

Since I have never feared mathematics myself, it is difficult for me to pronounce judgement on why someone may fear or hate it. However, I did fear and hate Biology and History in school since these subjects seemed to rely excessively on memorisation and elaborate descriptions, in place of analysis and accurate summaries. I have since come to realise that my vision of these subjects was myopic. I cannot with certainty say that it was due to bad teaching either!

Perhaps it is pessimistic to say so, but it seems there will always be some students who will fear and hate mathematics (or any subject) just as there will be some who love it. This will be independent of the quality of their teachers, the curriculum, the contents or the books.

Just as it would be wrong to focus on the best students, it is wrong to focus on the weak ("no child left behind").

Our primary job is to ensure that the large (97%) that sits around the middle of the class does get a good mathematical education---algebra, analysis and geometry/construction in roughly equal parts.

11. Other influences: The discussion paper draws attention to a range of other cross-cutting themes or issues that affect mathematics education in schools: • cultural issues related to the value of education in general and mathematics education in particular • equality issues (gender, uptake and achievement; socio-economic factors; educational disadvantage; students with disabilities or special educational needs) • recent developments in, and availability of, information and communications technology (ICT) in schools. Please comment on any of these issues, or on other factors that impact on mathematics education in schools.

To the extent that computers provide us with (yet another) example of the effectiveness of a mathematical approach, and to the extent that they can help us handle computations that cannot be done by hand, it would be good to involve them in mathematical learning.

At the same time, it must be said that excessive exposure to excellence and power can discourage. Just as it may be bad for a youngster to watch too much IPL during the hours when she/he should be playing cricket, it will hurt a young student of mathematics to use a computer to carry out calculations which are instructive to carry out "by hand".

 Conclusion
 The purpose of this review is to map out the direction that must
 be taken in planning curriculum and assessment provision for
 mathematics education at different school stages in the years
 ahead while reviewing the NCF-2005.  Please use the space below
 to make any additional comments on current issues in mathematics
 education or to give us your views regarding its future.

There is certainly scope for continuous improvement in the teaching of Mathematics. Some mathematical skills which were thought "advanced" 200 years ago could even be considered "essential" for the generations to come.

Elsewhere, I have pronounced that mathematics is "conscious" abstraction in the following sense. We make abstractions unconsciously all the time. However, these are often very individualised. In order to teach these abstractions to others we must understand our own thought process and go through the steps consciously. Only then can we turn these processes into theorems and algorithms ... and thus into mathematics.