My colleague Baskar and I conducted a math camp at DAV Boys and DAV Girls school recently. I have jotted down a few of our thoughts on the topic here.

The kids were always respectful, always ready with a Namaste, always on time and largely interested in learning something new. They did not grudge the fact that some holiday time had been taken up by math and seemed genuinely disappointed when the math camp drew to a close. Quite a few of them had taken up 2-3 other activities and seemed to enjoy all of them.

There is a certain raw enthusiasm about them that is endearing, that made me want to design more classes for them, even though some of their over-exuberance drove me crazy on occasions.

They have registered whatever processes they have been taught, and know most of the toolkit that they should know by class V or VI. They can all add, multiply, divide merrily and have the ability to take in more formulae and details if these are thrown at them.

Now, I am going to spend some time on this list. Mainly because of two reasons -1) No point saying kids know everything already. All of us are looking to identify the gaps and plug them. So, we need to be brutally honest about these gaps and 2) I have a high expectation of what I think kids should know. The challenges we faced 25 years ago are nothing compared to what these kids will face globally 10 years from now. If they are not equipped, they will struggle.

I am going to be giving a few math examples here. Nothing too technical, but some bits of what I learn from class are best described by using the same examples.

We discussed a question USA + USSR = PEACE, where U, S, A, R, P, E, C stand for digits from 0 to 9 and this addition holds good. This is a fabulous question for understanding the idea of “carry over” conceptually. We had done quite a few exercises on “carry over” prior to this, so they knew the idea. About 90% of the class did not know how to go about this, which is alright. About 80% of the class did not try to crack it until they were prodded and pushed. Anything out of the ordinary makes the majority of the class wind down and wait for the method to be unveiled.

Once they are prodded, they gave it a good go. But they are unused to the idea of figuring out and drawing inferences. That is a giant gap in their learning attitude that will need to be plugged.

This is particularly true of boys. The boys are overly keen to answer a question without ever pausing to think “Am I missing something here?” It is almost a scenario of any-answer-is-good-enough. Their mind jumps to the first possibility, they scream it out and then they are done with the question. Frequently (very frequently), their first answer is incorrect. But since they feel this enormous pressure to shout out an answer, they do not pause to think.

I had asked the students to add all the numbers from 1 to 100. One of them took some time, developed a fabulous method and gave the answer correctly as 5050. I asked all the boys to add numbers from 1 to 200. About 60% of the class did not try this seriously. Of the remaining who tried, quite a few confidently wrote down 10100 (twice of 5050) and stopped thinking after that.

They are happy to receive formulae but less ready to receive ideas

On the same question of adding all natural numbers from 1 to 100, I noticed a range of responses

1. Some tried to “brute-force” this. They just added numbers merrily, happily. I love this group. In my view, they will go on to learn lots of great things. Sooner or later, they will realize that they will have to do better than brute-force, and then their brute-force experience will help them come up with a method.

2. Many quit. Sad, but true.

3. Some tried and came up with new techniques. These were the brightest kids.

4. Some knew the formula. A great many of those who knew the formula were keen to know the new method I was teaching. So, their attitude was correct

5. Some knew the answer without knowing the approach or the formula ( do not ask me how. Apparently they were taught this in Abacus). These switched off the moment they wrote down the answer.

The method I taught them was the method that apparently Gauss had used in school. The story goes like this –

When Gauss was in the equivalent of 6th standard, his teacher had been called by the principal for a few minutes. Since the teacher did not want the class to erupt, he gave them all a task – to add numbers from 1 to 100. Gauss pretty much jumped up immediately and said 5050. When quizzed about the method, Gauss said 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101. If we pair up the numbers from the extremes, each pair adds up to 101. There are 50 such pairs. So the total would be 5050.

I outlined this method and the story to the students and then asked them to add from 1 to 200. Students in groups 1, 3 and 4 were receptive. But the rest were not. One can add numbers from 13 to 98 easily with Gauss’s method, and that is the beauty in learning the idea.

If there is one clear take-away from this entire math camp it is this – The peer pressure and group dynamic seen in boys classrooms is very counter-productive to learning. They are keen to shout out an answer, deeply conscious of who is seen to be doing well, cannot contain themselves when they know a correct answer and frequently switch off when something is not in their comfort zone.

The way the course was designed, we would have 75% of the questions to be simple ones. These are built for teaching an idea. The remaining 25% would be the trickier ones, designed to push the students to think. Only 20% of the class tried the second 25%. The rest were either not too keen or intimidated by the top 20% to try the question.

There is a lot of research in the west that says that men occupy the very top slots in companies and academia, but women as a group outperform men as a group substantially. Our classes were a microcosm of this. We taught about 120 students – 90 boys and 30 girls. The top 10 would have probably had 6-7 boys and 3-4 girls, which is about par. But probably 28 of the 30 girls would have been in the top 60 overall. The girls have the patience to keep trying when things do not fall in place, and enjoy the challenge of pushing themselves to solve newer types of questions. When they work in teams, all students contribute and they learn from each other. When boys work in teams, the smart one tries and the other 4-5 just accept their ideas. Boys learn very little from their peers. They are probably too young for learning in teams.

We enjoyed large parts of the math camp. We ended the camp with enormous respect for teachers who handle kids of this age-group. One needs loads of patience to be able to handle this age-group and consistently deliver value. A big round of thanks is due to the teachers who do this well. We can state unequivocally that it is not easy.

To put it bluntly, the variance in intellectual ability across the student group is vast. Some are really sharp, some far less so. The school system simply cannot do justice to the entire group. If the course is pegged close to the lowest level (as is often done), the brightest kids end up largely twiddling their thumbs. The top 10% of the kids learn in an average school day what a sharp teacher can teach them in 40 minutes. If the pedagogy is pegged at a higher level, then the bottom one-third gets left out.

It is politically incorrect to have break classes into sections of “bright” and “not-so-bright” students. Although if I had to be very objective about it, I would encourage this kind of breaking up. In the current set-up, the brightest kids are not deriving value from the school. And the slower ones end up being intimidated by the brighter ones from early on and end up not trying new ideas. So, with our current system we are helping neither bunch.

This is probably the most critical question facing us. I would argue that two things need to be done

1. Parents need to play a role in learning and teaching: This is non-negotiable. Our schools are stretched. They simply cannot handle the vastness of this school. Parents will have to learn new ideas and teach their kids. If the kid is bright, parents will have to find avenues to keep the kid intellectually stimulated.

2. We have to go online aggressively. The pace at which learning is imparted in schools will be correct only for perhaps 20% of the class. For the rest of the class, it will be too slow. Students need to be provided avenues for pushing themselves hard.

**The kids are smart, well-behaved and keen**The kids were always respectful, always ready with a Namaste, always on time and largely interested in learning something new. They did not grudge the fact that some holiday time had been taken up by math and seemed genuinely disappointed when the math camp drew to a close. Quite a few of them had taken up 2-3 other activities and seemed to enjoy all of them.

There is a certain raw enthusiasm about them that is endearing, that made me want to design more classes for them, even though some of their over-exuberance drove me crazy on occasions.

**What are they good at?**They have registered whatever processes they have been taught, and know most of the toolkit that they should know by class V or VI. They can all add, multiply, divide merrily and have the ability to take in more formulae and details if these are thrown at them.

**What are they less good at?**Now, I am going to spend some time on this list. Mainly because of two reasons -1) No point saying kids know everything already. All of us are looking to identify the gaps and plug them. So, we need to be brutally honest about these gaps and 2) I have a high expectation of what I think kids should know. The challenges we faced 25 years ago are nothing compared to what these kids will face globally 10 years from now. If they are not equipped, they will struggle.

I am going to be giving a few math examples here. Nothing too technical, but some bits of what I learn from class are best described by using the same examples.

**Students have very little practice of “figuring out” stuff, and too little patience for “hanging in” there.**We discussed a question USA + USSR = PEACE, where U, S, A, R, P, E, C stand for digits from 0 to 9 and this addition holds good. This is a fabulous question for understanding the idea of “carry over” conceptually. We had done quite a few exercises on “carry over” prior to this, so they knew the idea. About 90% of the class did not know how to go about this, which is alright. About 80% of the class did not try to crack it until they were prodded and pushed. Anything out of the ordinary makes the majority of the class wind down and wait for the method to be unveiled.

Once they are prodded, they gave it a good go. But they are unused to the idea of figuring out and drawing inferences. That is a giant gap in their learning attitude that will need to be plugged.

**Students do not have a ‘pause’ button, where they take in stuff and internalize them**This is particularly true of boys. The boys are overly keen to answer a question without ever pausing to think “Am I missing something here?” It is almost a scenario of any-answer-is-good-enough. Their mind jumps to the first possibility, they scream it out and then they are done with the question. Frequently (very frequently), their first answer is incorrect. But since they feel this enormous pressure to shout out an answer, they do not pause to think.

I had asked the students to add all the numbers from 1 to 100. One of them took some time, developed a fabulous method and gave the answer correctly as 5050. I asked all the boys to add numbers from 1 to 200. About 60% of the class did not try this seriously. Of the remaining who tried, quite a few confidently wrote down 10100 (twice of 5050) and stopped thinking after that.

They are happy to receive formulae but less ready to receive ideas

On the same question of adding all natural numbers from 1 to 100, I noticed a range of responses

1. Some tried to “brute-force” this. They just added numbers merrily, happily. I love this group. In my view, they will go on to learn lots of great things. Sooner or later, they will realize that they will have to do better than brute-force, and then their brute-force experience will help them come up with a method.

2. Many quit. Sad, but true.

3. Some tried and came up with new techniques. These were the brightest kids.

4. Some knew the formula. A great many of those who knew the formula were keen to know the new method I was teaching. So, their attitude was correct

5. Some knew the answer without knowing the approach or the formula ( do not ask me how. Apparently they were taught this in Abacus). These switched off the moment they wrote down the answer.

The method I taught them was the method that apparently Gauss had used in school. The story goes like this –

When Gauss was in the equivalent of 6th standard, his teacher had been called by the principal for a few minutes. Since the teacher did not want the class to erupt, he gave them all a task – to add numbers from 1 to 100. Gauss pretty much jumped up immediately and said 5050. When quizzed about the method, Gauss said 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101. If we pair up the numbers from the extremes, each pair adds up to 101. There are 50 such pairs. So the total would be 5050.

I outlined this method and the story to the students and then asked them to add from 1 to 200. Students in groups 1, 3 and 4 were receptive. But the rest were not. One can add numbers from 13 to 98 easily with Gauss’s method, and that is the beauty in learning the idea.

**Boys’ group dynamic is hurting their learning.**If there is one clear take-away from this entire math camp it is this – The peer pressure and group dynamic seen in boys classrooms is very counter-productive to learning. They are keen to shout out an answer, deeply conscious of who is seen to be doing well, cannot contain themselves when they know a correct answer and frequently switch off when something is not in their comfort zone.

The way the course was designed, we would have 75% of the questions to be simple ones. These are built for teaching an idea. The remaining 25% would be the trickier ones, designed to push the students to think. Only 20% of the class tried the second 25%. The rest were either not too keen or intimidated by the top 20% to try the question.

**The girls are streets ahead attitude-wise, are ahead in mathematical ability also**There is a lot of research in the west that says that men occupy the very top slots in companies and academia, but women as a group outperform men as a group substantially. Our classes were a microcosm of this. We taught about 120 students – 90 boys and 30 girls. The top 10 would have probably had 6-7 boys and 3-4 girls, which is about par. But probably 28 of the 30 girls would have been in the top 60 overall. The girls have the patience to keep trying when things do not fall in place, and enjoy the challenge of pushing themselves to solve newer types of questions. When they work in teams, all students contribute and they learn from each other. When boys work in teams, the smart one tries and the other 4-5 just accept their ideas. Boys learn very little from their peers. They are probably too young for learning in teams.

**A great experience for the two of us, it opened our own minds a lot**We enjoyed large parts of the math camp. We ended the camp with enormous respect for teachers who handle kids of this age-group. One needs loads of patience to be able to handle this age-group and consistently deliver value. A big round of thanks is due to the teachers who do this well. We can state unequivocally that it is not easy.

**Schools cannot do much – this is a truth that we have to accept**To put it bluntly, the variance in intellectual ability across the student group is vast. Some are really sharp, some far less so. The school system simply cannot do justice to the entire group. If the course is pegged close to the lowest level (as is often done), the brightest kids end up largely twiddling their thumbs. The top 10% of the kids learn in an average school day what a sharp teacher can teach them in 40 minutes. If the pedagogy is pegged at a higher level, then the bottom one-third gets left out.

It is politically incorrect to have break classes into sections of “bright” and “not-so-bright” students. Although if I had to be very objective about it, I would encourage this kind of breaking up. In the current set-up, the brightest kids are not deriving value from the school. And the slower ones end up being intimidated by the brighter ones from early on and end up not trying new ideas. So, with our current system we are helping neither bunch.

**What can be done to plug the gaps?**This is probably the most critical question facing us. I would argue that two things need to be done

1. Parents need to play a role in learning and teaching: This is non-negotiable. Our schools are stretched. They simply cannot handle the vastness of this school. Parents will have to learn new ideas and teach their kids. If the kid is bright, parents will have to find avenues to keep the kid intellectually stimulated.

2. We have to go online aggressively. The pace at which learning is imparted in schools will be correct only for perhaps 20% of the class. For the rest of the class, it will be too slow. Students need to be provided avenues for pushing themselves hard.

Marvelous post this. Keen eye for patterns one can find in people. Power to you Sir! :D

ReplyDeletePertinent. Interesting. I guess, my evolution into being a parent also forces me to think about this a lot more closely. I have usually been worried about the usual straight-jacketedness of most curriculum (designed for mass broadcasting).

ReplyDeleteThe careful breaking of groups is something I would support too. I think this could be similar to how some of the sports prodigies are treated by school through extra coaching sessions/ waivers on certain aspects (like mandatory attendance ;)), etc.

Loved reading this one.

Gauss method is the proof for the formula isn't it?

ReplyDeletei.e.

x = 1 + 2 + ... + 100

x = 100 + 99 + ... + 1

2x = n * n+1

I have observed the part about the students blurting out the first thing that comes to mind, and more importantly, stopping there. My term for this is limited intellectual stamina. How does one increase intellectual stamina? Any suggestions?

ReplyDelete