My students – most of the grade-8 students found it difficult to add and subtract fractions. Though it was easy to make them understand addition and subtraction of simple fractions using the pie diagrams, it became difficult to do the same when the denominator was a large number and/or there was more than one pie involved (in case of improper fraction). Also, the concept of why we should find the LCM of the denominators and then add/subtract fractions became difficult for them to grasp. At this juncture of their age, it is important that the students first get the concept for their operational use and are confident to apply it rather than understanding the concept from first principle method. So i first made them get the concept operationally through a method I call the Gaddha(Donkey),Ghhorra (Horse) and Haathi (Elephant) method.

So let us say that we have to solve the following question –

3/4 – 4/6 = ? —> Change this to 3 gaddhas – 4 ghhorras = ?

Symbol for Gaddha = /4 —> so this is how you identify gaddhas; All gaddhas have denominator as number 4.

Symbol for Ghhorra = /6 —> so this is how you identify ghhorras; All ghhorras have denominator as number 6.

Now, i ask the students – if they can take out 4 ghhorras from 3 gaddhas — the answer is a resounding ‘No’. So, i tell them that now we need to make them same so that we can solve this question.

How we do it is to USE ONLY the multiplication operation on the gaddha and the ghhorra respectively and find ONE resulting number that will make the gaddha and ghhorra into a haathi.

Easiest way to do it is by multiplying gaddha and ghhorra with each other.

Therefore, /4 X 6 = /24 —> Haathi

/6 X 4 = /24 —> Haathi

One thing we need to remember is that whatever we multiply in the denominator, we must multiply the same number in the numerator of this fraction (otherwise the value of the fraction will change).

So, we finally get our new baby elephants (resulting numbers) through cross-breeding horse with a donkey.

3/4 becomes equal to 18/24 (18 haathis or elephants)

4/6 becomes equal to 16/24 (16 haathis or elephants)

=> 3/4 – 4/6 = 18/24 – 16/24 = 18 elephants – 16 elephants = 2 elephants = 2/24.

I have tried this method couple of times with my students and it works wonders.


It’s 8:15 AM on a Monday morning when the school bell rings and you usher in the students into the assembly hall. 15 minutes down you enter your class with the class register only to be greeted by your students with the familiar “We will get 100% in mathematics” or occasionally “13 ones are 13, 13 twos are 26…” After a summary of what will be discussed today, you move on to the instruction and 15 minutes later you ask five of your students, who you are coaching for the NTSE exam, to solve the worksheet given to them the day before. As the day goes by, you take a 15 minute break at 11:00 AM and then meet with your four-member math team for the cluster meeting. Having discussed the analysis of the weekly tests, the target for the next week and the plan of action for achieving the target you set to discuss with a team-member a solution to a problem they are facing in their classroom.

45 minutes later, you find yourself working on the teaching-plans and assessments for the next month. The bell rings and you hear the familiar noises of children signaling the beginning of the lunch break, since you head the teacher-participation-team you discuss the staff issues which need to be raised to the school management while you eat the lunch with the other staff members.

Two-and-half hours later, after the school ends you take an extra class and finally leave the school at 530 PM for home.

At 6pm, you check the emails sent to you, only to learn that you have been asked to give an inaugural address at an event two days later to welcome Dr. APJ Abdul Kalam to the Teach For India movement and you have received a confirmation to attend the mathematics conference at Homi Bhabha Center for Science Education.

After working on your speech for an hour-and-a-half and calling up and troubling all your friends who were good in elocution; you finally come up with the first draft of your speech. You meet your colleague for dinner at 8 pm to discuss the update on the community radio station project you are doing with her. You get-back home to work on your computer-based-test software that you have created for the school team to conduct weekly tests, while an email pops-up telling you that on Friday there is a leadership forum with Mr.Nachiket More (Head, ICICI foundation for Inclusive growth)!

Each day like this gives you an opportunity to learn, create and share with others. Occasionally you pay a visit to your students’ communities after school just to invest the parents in their child’s education or there are days when you attend focus groups on social entrepreneurship and at other times as a member of TFI’s funding team you travel to Mumbai just to have a dinner with the US counsel general and meet with other venture capitalists to invest them in the Teach For India movement.

And yes, on weekends you write a blog and have initiated a venture called TechPeeth where you share your teaching experiences with the entire world.

From bringing about a gradual-change in the way math is taught in school to creating standard operating procedures such as the student intervention team for helping the bottom 20% students in the school, you have done all you could and have endeavored to make a change.

My Reflections on Teaching Higher Grade Levels

Last year, I taught sixty students each from Grades 4th (Social Studies & Science), 5th (Math) and 7th (Intelligence). There were two broad challenges that I faced last year; creating the content for the lesson plans and tracking students’ progress.

Managing the content for the lesson plans felt like a challenge bigger than any project I managed at Citibank. I had to study, make notes and then create a lesson plan that would “incept” the concept in the students’ minds. Each LP took me 2 hours to complete! I also got teachers in my school to observe my classes frequently. The observations made me more accountable and provided me with valuable feedback which culminated into different ideas that I could use to plan better. I soon realized that I was running out of creative ideas to teach, so I went and met professors at Delhi University and picked their brains on the same. Having conquered the planning menace, I addressed the next big challenge of regularly tracking my 180 students. I couldn’t track students’ homework regularly given my daily workload and hence couldn’t identify individual learning gaps. To manage this, I created computer-based-testing software to gauge my students’ performance every week. The regular data tracking coupled with the class feedback, helped me become more efficient.

This year, I am teaching Mathematics to grades 6th and 8th. It’s only when you start teaching the grade-8 level do you realize how deep and wide the achievement gap is. The class average on the last year’s EOY assessment for current grade-8 was 34%. In June this year, while some of my students hated mathematics, the others had an inherent math-phobia. I realized that the fear of math was deeper when I met their parents and learnt that even though the kids sincerely studied the kids performed poorly on the day of the test.
I began this year by telling my Grade-8 students their big goal will be to score 80% in the term-end exams and at least 20 students will take the National Talent Search Exam (NTSE). If the goals were achieved I promised to take my students for two days to Goa! Ever since, the NTSE has many takers! To cater to these students accelerated needs I started the “one-chapter-a-day” group. The students in this group stay back after school everyday to understand SSC (Math) chapters and solve all exercises from these chapters on their own. These students also do extra work on their own at home and clarify their doubts when they are faced with one. A corporate could look at this and nominate it as a case study for capacity building!

I started “one-chapter-a-day” group with one student and in a matter of a week, 15 students joined this group. Not only do the students inspire each other by achieving feats like completing 80% of the exercises in the math text-book for this year but also have helped raise my Grade-8 class average by 26% in a matter of one month! Some of the students in this group are the same students who hated math.

With regular weekly tests and small successes, I get immense joy watching my students gradually achieve the milestones towards their big goal. As FDR said,

Happiness lies not in the mere possession of money; it lies in the joy of achievement, in the thrill of creative effort… These dark days will be worth all they cost us if they teach us that our true destiny is not to be ministered unto but to minister to ourselves and to our fellow-men.

Lesson Objective: YWBAT – Describe the meaning of simple interest.

During olden days, the banking industry marked its beginning with what you all are studying today as SIMPLE INTEREST. In those days, the rich gave money to the poor but what was their interest in giving money to the poor?

What was that a rich man got if he lent out some money to a poor man?

The reason a rich man lent money to a poor man was his INTEREST in the money that he would get OVER-AND-ABOVE the money that he has lent to the poor man.

For example: If I gave Amruta – Rs. 1000 then after 1 year she not only has to return to me my Rs.1000, but also has to return some more money (because I gave Amruta the loan and let her keep my money for 1 year) – I called this additional money I got and that I called as my INTEREST.

Why did the rich man charge or take additional money from the poor man?

So the intelligent rich man had a good reason to justify this. According to him, suppose he had NOT given this money to a poor man and instead he had purchased seeds of mangoes worth Rs.1000 from the market and he had put them in his field. After 1 year, these seeds would have grown into mango trees which would give him mangoes worth Rs. 1200 (if he sold the mangoes from these trees in the market). This means, over-and-above his Rs.1000 he was earning Rs.200 as profit/ getting Rs.200 extra, then if he Now lends this same money to a POOR man then he (the rich man) must earn Rs. 200 extra from this deal. And hence the rich man charges what he calls as the interest.

Today, I asked my Grade-6 students to solve the following problem –
-15 + 20 = ?

Though I have only given them a brief idea of integers in Grade-5, i expected that they would be able to solve this question with the help of a number line.
Well to my surprise,I got 4 different answers to this question and they were : – 20, 5, -35 and 35.

I, therefore, decided to change course and gave the students five important rules that they needed to learn if they were to be good in the world of numbers.
Rule-1: a + b = b + a
Rule-2: a – b is not = b – a
Rule-3: a X b = b X a
Rule-4: a/b is not = b/a
Rule-5: a X 0 = 0, a X 1 = a, a + 0 = a and a – 0 = a

Now, i asked the students to solve the problem again, but still some students weren’t able to solve the problem, so i decided to take help of mathematical induction.Here is what I did –
I said,
In the question -15 + 20 = ?
If we take a = -15 and b = 20, then
-15 + 20 can also be written as 20 + (-15) — Using Rule No.1

Next I told them that now there are 2 possibilities of opening the bracket in case we need to find out the value of 20 + (-15),
Option-1: -15 + 20 = 20 + (-15) = 20 – 15 OR
Option-2: -15 + 20 = 20 + (-15) = 20 + 15

Now, let us assume that option-2 is correct, in which case, we can write,

-15 + 20 = 20 + (-15) = 20 + 15
-15 + 20 = 20 + 15

But we know that,
20 + 15 = 15 + 20 — Using Rule No.1

Therefore, we can replace 20 + 15 with 15 + 20
-15 + 20 = 15 + 20
Now, let us subtract 20 on both sides, we get,
-15 = 15
But, we know that this is not possible, therefore option-2 is wrong and hence Option-1 is correct.

This means that,
-15 + 20 = 20 + (-15) = 20 – 15 = 5 (Answer)

Pune, 4th November 2009

We are attempting to democratize education. Socratic is same as upanishad type model. father’s dialogue to sun – nachiketa’s dialogue with yama in katho upanishada. Yudhishtra with his dad. Upanishada means sitting at the feet. Oxford and Cambridge – they wouldn’t let people into univ. England has a Socratic way of teaching. There system is called the tutorial system – one on one – it was v.interesting. Homoerotic relation between teacher n student. Whitehead n Bertrand Russel – student pupil relation – principia mathematica. Williams college in MA – only college in USA to follow Socratic or tutorial method of teaching. How to deal with bad kids – drunken German univ. Type of kids – how are you going to deal with this? Find out some rules of thumb on you can find out – you all have to empirically work on. Ideal is Socratic because we will not always have s pupil like Plato. Under Elizabeth in England they achieved 60 pc literacy. Many Nobel laureates collaborate with students to publish great work. Jerry didn’t have an answer and urges us to engage with the question. Keep daily dairies. Coming generations of teachers would be able to us the paradigm developed by us.

you are going to teach Shaheen and me by telling us what works and what doesn’t matter. keep the diaries with Socratic and upanishadic method as the baseline..don’t maintain slipshod diaries. It is not like corporates u can measure the outcome quickly it will happen in 10 years. You also have a whole system that is not working.

There is volumes of Current research against charter school – what people don’t have a historical background. Teacher is a woman usually historically. These are things that have worked elsewhere. One teacher multiple subjects is best model. If there is a general ambiance for learning by and large those kids have progressed.

How to deal with parents?

Create an atmosphere of respect for knowledge.

We just don’t know what works. The Indian middle class is the best exam taking class.

Learning and exam taking technique to take exams – u have to teach the students and parents this – we have to remain competitive.

Learning by rote is important for math and poetry.

Through your experience create case studies, and a body of knowledge to crack this problem of canning.

Korea vs. India – avg korean is much better than avg. India. there per capita gdp is 25 times ours.

Problems with higher education in our country. – we have just 3 Nobel laureates – that are British educated. More Nobel laureates in Israel which is the size of ratnagiri district.

So i am troubled with the outcomes of our education. We need to produce newtons.

Maharaja states paid lot of money to teachers than under the British commune. British didn’t want to democratic education.

Teachers believe they come from a higher social class and students come from slums. They think that students are karmically supposed to do this.

‘you know where they come from’.. Book – govt brahmin -kannada book- he is now a sociology prof.. written by a dalit. All of you should read it.

Combined expenses of 6 IITs = revenue from coaching classes.

If your kids are slightly more educated they have a huge differential to the GDP.

Why we don’t produce Nobel laureates?

1. Creativity is squeezed out

2. Amount of resources spent on research.

Koli, chairman emeritus TCS – said there are more phds in sanskrit in US than in India.

What jerry suggested sibal?

Give institutions of excellence full autonomy and grants. And make them multiple disciplinary. Impose condition they will adhere to reservations, second no poor kid should go away without a scholorship. The govt shudnt tell what the college should do.

All of them were given charters of how to run themselves – and slowly a culture of excellence will come.

In our anxiety to go downside, why forget the upside.

No country has become rich until and unless it has achieved 90% literacy. Really to create wealth we must increase female literacy to at least 85%.

You can tell them you can become richer in other ways such as BPO.

Most 10th fails are useless – most of them can be very useful citizens but they create social tensions.

Particularly in education there is no immediate feedback. Govt wants to spend money but its not clear on what. Give autonomy. Some may do scams or frauds but it doesn’t mean that you wont let companies work freely.

Yes certain minimum marks are important and that is okay, but lets look at child’s creativity.

There is a lot of focus now on class 7 and 8.

About 60 pc of staff in cruise ships in Florida are Indians. Assistant chef on a ship is not a menial job. Hdfc bank is doing data entry in villages – attrition rates are low and cheaper.

We need to get them away from low rate dead end jobs. For them the present income is more imp. than income later. their discount rates are very high-35% and ours are 12%.

T.S.Elliot’s poem – wasteland. God is dead but we are alive and we are dead.

*Disclaimer: Please note this is the version of our discussion with Jerry Rao I recall of my memory so it may have some discrepancy between what he said and what i recollect.

I ‘taught’ Intelligence (logical thinking and reasoning skills) to the classes Shantiniketan & Nalanda (the two sections in the current grade-8 in the school where I teach) last year. Since my classes were scheduled at the end of the day and that too once a week, It was difficult to engage the students and the students at times made attempts to test my knowledge (given that I taught core subjects to grades 4th and 5th ONLY :)). I, for the first time witnessed the shocking manifestation of achievement gap in these classes – at grade-7 level while most students didn’t know how to divide, some of them didn’t know how to ‘correctly’ perform the other basic operations on numbers. It felt like students spoke or did math more out of their memory than with understanding.

On the flip-side, during these classes in Nalanda, I also met a student who never seemed to pay attention and later when asked a question used to give out the answer in a matter of seconds. I was impressed with this, as I was somewhat like him in school, so I asked him to share his method of approaching these problems with the class but to which he refused. Well the only way to get him engaged was, i realized, to increase the level of rigor for him and surprisingly this worked for him! So while, i facilitated grade-appropriate teacher-student interactions with Nalanda, i gave higher-order thinking questions to Vishal. One day we finally became friends (but he still didn’t share his methods – though he learned mine, well I thought that was the teacher’s curse :() and he walked up to me during the lunch break to “request” me to teach grade-8 mathematics in the next academic year. I wished I could teach them.

This year, I have felt privileged to be given these challenging grade-8 classes, where students thought-felt-believed that they hated math. I felt privileged because –

  • a) As this was an opportunity to show them how cool it was to be a math-wizard and to also teach them an important life-skill that of ‘self-empowerment through positive personal choices’.
  • b) More than this, for me it is a challenge to achieve this turn-around because this would help my students move this huge block in their minds to their advantage.
  • c) When I was in grade-7, even I hated math and circumstances taught me valuable skills that now I was on a perfect platform to impart to the next generation of my country.

And so – In the beginning of this academic-year, in my second math class while other students attempted a question I had given to the class, I walked up to Vishal to tell him that I wanted him to appear for the N.T.S.E exam. To do this,therefore, I told Vishal to start completing one-chapter-a-day while I taught in the class, with the result that he would finish the entire grade-8 syllabus in 2 months time. He agreed! And well, this marked my first successful step towards my achieving vision for my students. Within, 15 days Vishal has inspired 12 other students to follow-suite and now not only the ones who were average want to achieve ‘super-brilliance’ in math by taking this challenge, but also students like Raju, who struggle with basic math sit with these 13 other students and attempt to solve the easier chapters first. Together, on an average, these students have finished on their own (with little help from me after school or during the breaks) 6 chapters (out of 24 in grade-8) in ~15 days.

One-chapter-a-day my students are taking giant strides towards realizing their potential!