When things don't work...

There are days when classes just flow both you and the kids have fun, they learn and and with some formative assessment you even know that they have learnt. There are other days that build on what was done though seem uninspired. There are very rarely some classes that leave you drained out.

I have tried to relate this to preparing for the classes. I find that this is not the case. While some of my best classes were with preparation there were some lovely ones that were impromptu.

I then tried to relate it to the amount of freedom children had in the classes. As an experiment I had two sessions of no holds bar take anything you like apart in the electronics lab. A monitor a few adapters and other pieces of equipment were rendered unrecognizable. The children especially the girls got some confidence dealing with equipment, when I honestly sat down and analyzed the class with the kids they agreed that they did not learn much. Much freedom, but no progress.

I then turned my attention to the classes that left me drain me out. Generally these were classes when I was already tired/unwell or that went out of hand - be it behavior of kids in terms of being completely distracted or when I pitched things too high and missed something that the kids were unable to grasp. The children do carry a lot of their home or at school to the classroom, but a lot of boils down to how I handle it and handle myself. A lot had to do with how centered and calm I was in handling the classes.

One day a kid walked in and asked me if she could drink water, better at the beginning than in the middle so I asked her to go ahead, she went up till the door and came back and asked the same question and did the same loop 6 times. I can imagine that something like this would have thrown me off and I would have probably given a lecture on how valuable time is as a rookie even four months back. But, I was able to stay quiet and let it play out, by the 6th time the kids had lost their patience with her and she sat down. The class went well though I had not figured out what was going on with her.

A few days later I remembered that in one of the classes I had repeated an explanation of an algebraic solution to a problem every time I was interrupted and perhaps she was replicating something like that :). We live in interesting times...

More recently I have been able to have good and reasonable classes and avoid crash and burn classes mainly by working on myself. Hope I can sustain the self work.

Simulator

We have been trying to change how 'theory' is presented by the faculty and I had attempted to make some demos using the equipment they had. However, they felt that experiments were taking time and were unable to fit it in the schedule. They had a few doubts about the theory themselves and we were looking for solutions. One such solution was the introduction of a simulator both to them and the students.

This experiment made some simple gains:
- The questions they asked were related to design which they had not done and with a simulator we could design and iterate in a short time.
- Demos to students could be done in a shorter time by doing some amount of prep and making the schematics and testing without needing lab equipment.

Though we have had to pause all this due to the approaching examinations it may be something that can give a new twist to what we do in future.

Ah Oscilloscope...

I was able to borrow a single probe oscilloscope from the Auroville ITI for a few weeks. There is nothing like an oscilloscope to understand transient signals. I was able to use it with the 10th graders at Udavi who had been working on time changing circuits and with the 5-7th graders in Isai Ambalam (IA) who had been working on understanding Energy and had visited a solar energy company in Auroville and encountered AC/DC, but didn't know what this really meant.

The primary demonstration I gave was of a voltage regulator (AC mains to 12V 'DC' output) - transformer with center tap, half-wave, full wave rectification and what happens when we add a capacitor followed by what happens when you put a load across the capacitor.
Having only a single probe I could not show them that a center tapped transformer has in and out of phase components at its output. This had to be inferred by the full wave rectification.

I was able to couple it with a cute experiment I discovered of taking capacitors of varying values, charging it by touching them to a 9 V battery and connecting it across and LED to see that the time for which the LED is on increases as the value of the capacitance increases.

The 10th graders made a couple of astute observations at the end of their class:
1) Sun - Yes, Sanjeev this makes sense. I always wondered why on turning off the power in some devices the LED of its charger is still on for some time. I think this must be because the LED works off the output of the charger which has a capacitor. Is that right?
2) Des - If the output with a load always has dips then we can't use this directly as a good DC. Should we create a higher supply and feed it to a chip like 7805 (voltage regulator IC we had used in class) and then use its output? This way as long as we maintain the ripple to be beyond 5V it will give a 5V output.

Multi grade classroom (IA school)

I have been working with the seventh grade kids at Isai Ambalam (IA)since Jun 2013. Its a small class of 6 kids (actually, one kid is in 6th grade). Though IA is presently bottom heavy it is steadily changing with the strength of the kids in fourth grade onward jumping to 16. I had started working with the oldest kids in the school. This gave me a chance to work with kids as they were getting into abstract concepts in Math and Science. It also give me a chance to identify potential areas that teachers can provide support with in earlier grades.

I worked with these kids for a good part of the first term and we built a good learning environment even with the large variation in the skill levels in Math and language within the class. The Vth grade teacher was also supposed to join my class, but she was unable to find something for the kids to do and do so. Given the small strength of my class and the rapport I had built with them I merged the class with he Vth graders for this term. I had handled larger classes in Udavi and the ITI and so a multi grade classroom with 11 kids seemed possible.

Eevery activity we did together now needed to be graded so the fifth grade was able to follow and seventh could build on it. When this was not possible we split the classes through projects or separate activities. But, I'll focus on the journeys we were able to take together as a class and what I found.

Word problems:
The fifth graders had been doing multiplication and division 'sums', but 4/5 had difficulty in putting it in a story or understanding if a story was for multiplication or division. There was also a general discomfort to English which is an escalating problem as the grades progress. The first thing we did was work on a vocal discipline.

We told stories of multiplication and division. I quickly noticed that kids stick to the script of Rs.5 for one pen and cost of many pens. Money stories were quickly abolished. Its amazing how kids can't think of multiplication without money! 
It took some time, examples and prodding for the 2 pens in one box, how many pens in 5 boxes or the stuff they made up about 5 stones in one round how many in 10 rounds. We had already gone through this exercise with 7th grades and they started to get a little smug till I upped their level and asked them to tell such stories using speed, distance and time.

We didn't put pen to paper for these classes and this save enormous time. We just told stories based on who I pointed to and what I asked for division/multiplication. The mental stamina of the children who are not doing well in school is quite low and kids generally zone out in classes so the idea of picking up kids at random (not really  :)) kept them guessing who was going to be next and got them into the groove and pay attention to what was being said.

Within 3 days the 5th graders had moved to speed. distance and time and the 7th graders had graduated to stories of  mass, density and volume; power, time and energy and then to stories involving fractions. 
The 5th graders seem, to look forward to what was going to come next. The kids who were struggling in 7th grade got a chance to revisit the ideas and everyone started getting comfortable  with speaking English and the structure of word problems. 
The kids no longer blink an eyelid when a story is told and I have only one 5th grader who still is confused between multiplication and division stories.
In two weeks the English teacher informed me that the sentence construction of 7th graders had suddenly become much better in general.
They also got good at units. They heard and made so many stories that the speed was now automatically km/hr. They also finally started to get dimensional analysis using the units.

Of course, we kept making the games more complex, one person told a story and another inverted it (changing a multiplication story to a division one) by using the result. Then we started to record the stories with just numbers which essentially is the process of abstracting a story into math. The inversion was understanding the basic dual nature of multiplication and division (apparently, not so obvious to kids).

Fractions
Another set of classes that worked very nicely were fractions. I had already done fractions for the 7th graders in the first term and this was a refresher though we focussed on decimals and conversion of fractions to percentages (using 50% and 10% of the denominator) and creation of pie charts to understand the proportion of various objects. This helps kids who struggle with the whole LCM and other process of adding fractions.

I was able to let the 5th graders explore fractions with games and San and Ani really wanted to know how to add fractions. I asked them explore it with the games themselves and find every fraction they could add using the game. They found 10 things they could add already, they wanted more and I led them to equivalent fractions. They did that for some time and then said that they wanted to do it faster like the 7th graders and that led them to using LCM. Seeing them the other kids got into it and even Var our 6th grader 'one who shall not learn how to add fractions' decided that she would do it. It was interesting that adding fractions with LCM figured as the first thing for all the 5th graders in what they learnt well (even though it wasn't the last thing they did).

Algebra
Before everyone jumps on me for initiating Vth graders to algebra. I should clarify that I realized that at least a few puzzles that are solved using algebra can be solved without it as well algebra. Its just that you need to think of a new logic/methodology for each new kind of puzzle and it would have been easier to have just learnt algebra.
I also taught kids how to make their own puzzles and the fifth graders could make some to give to seventh graders.

Energy

We have been doing a project on Energy at Isai Ambalam. As with anything worthwhile it has been something we have been slowly working on, with discussions, experimentation, calculations and more recently a visit to Sunlit futures (a solar energy based company).

The most interesting discussion we had was regarding the law of conservation of energy. In 5th and 6th grade the children learnt that energy is the ability to do work. Work was implicitly understood through everyday activity and the idea that energy is used up in doing work. In 7th grade, however, the law of conservation of energy is introduced. 'Energy is neither created, not destroyed. It only converts from one form to the other (I added under normal conditions, without adding in E=mc^2).' The children seemed happy to have one more piece of info to rattle out, at which point I posed the following question.

Since energy is neither created nor destroyed how does any work, that apparently uses energy, get done? If they were getting out of confusion I gave sufficient examples of areas they thought energy is used up. Hmm...small pleasures of being a science teacher, let children mull over a gotcha and get them to a point they really want to know something :).

When energy is converted from one form to another, we call it work. Clear? Great when you clap kinetic energy gets converted to heat and sound energy. What happens to the sound energy that does not reach the ears of people when you clap...

We also did a bunch of Arvind Gupta's experiments that were quite informative and fun!

p^2 -1 = (p+1)x(p-1)

During preparation for my class with the idea that a p^2-1 (where p is prime > 3) is divisible by 24 I was trying to find a way to explain p^2-1 = (p+1)x(p-1).

I found the following way with the place value kit. Say you have 13^2. You have 13 rows of 13 each. When you remove one. You get 13 rows of 12 and one additional column of 12. Moving the column to a row you get 14 rows of 12. i.e. 13^-1 = 12x14. It can be seen from the picture that this would always hold.

In fact this is a nice way to see that any a^2-b^2 (where b
I didn't get a chance to use it in class, but i thought it was cute anyway.

The how of the two digit squaring method (2)

With the distributive property 'demonstrated' I continued splitting areas to get comfortable with the idea.

If we look at the square of a two digit number say 13. As shown below its 13x13. The 13 at the top and the left side are meant to act as rulers and are not to be added to the count.

The distributive property can be seen as a vertical split in this area

This can be written as
13x13 = 13x(10+3)
          = 13x10 + 13x3

We can now split the figure horizontally as well by splitting 13 as 10+3 again.

This is equivalent to 

13x10 + 13x3 = (10+3)x10 + (10+3)x3

                    = 10x10 + 3x10 + 10x3 + 3x3

All that remained is connecting the method to the madness :).

The how of the two digit squaring method (1)

The sixth graders I work with are (have become) a curious lot.

Following the exercise of p^2-1 (prime number square is divisible by 24) there were many questions of how things worked. My goal had been for them to look at the beauty of numbers and I had asked them which their burning question was why p^2-1 is divisible by 24 or how the technique of squaring works.

I had found the property of primes more interesting and as if to remind me that I am an adult the children overwhelmingly selected the squaring method deserved their attention.

There were three aspects that puzzled the kids about the method:
1) Why write the square of the units place is written as a two digit number even if the result is a single digit i.e. 1^2=01, 2^2=04 and 3^2=09.
2) Why the square of the tens place didn't need the same.
3) Why after multiplying the two numbers we needed to double it.
Of course, there was the addition of all these numbers, but apparently that was no puzzle :).

Distributive property of multiplication over addition
I needed to explain the distributive property of multiplication over addition to proceed. I realized that

the algebraic proof that came naturally to me was irrelevant as they had not encountered algebra meaningfully. I used that place value to show the same graphically.

A place value kit has blocks of ones, tens, hundreds (and thousands that I have now shown).

I had used the blocks to explain multiplication as the area of a rectangle i.e. 6x13 is the same as calculating the area of a shape having 6 rows and 13 columns (or 6 columns and 13 rows depending on how you look at it):

In that case, 6x(10+3) implies a split in the figure above of 13 into 10 + 3 i.e.

Which is the same as: 6x10 + 6x3

As a note, many aspects of algebra can be beautifully shown with the place value kit. I have been able to touch upon the following multiplication, squaring, decimals, fractions, area, volume with the same.

Prime square minus one

I had a few interesting classes with 6th and 7th graders around the idea that a prime number (beyond 3) square minus one is divisible by 24.

The biggest difficulty in approaching this topic is that many children get confused between double and squares. I indicated the difference using an area of a square of a side of a certain length (avoiding 2 so as not to add to the confusion) vs a rectangle of the same length and breath fixed as 2. This, however, was not enough for all children and I introduced a short cut to get squares of 2 digit numbers. Even though this was a diversion, it reiterated the squares of small numbers and helped them see squares of large numbers not just as random numbers.

The method is
1) To write the two digit number say 24 in two columns. Now you have single digit numbers in each column. 2) Take the units place which is 4 and get 4x4=16 and write it as a two digit number in the right side of the column.
3) Then you take the tens place 2 and square it to get 2x2 = 4 and write it on the other side of the column.
4) Then you take both the digits 2 and 4 multiply it to get 2x4 =8 and then double it to get 16 and write it in the bottom with the units digit in the first column.
5) Add the two rows you created in steps 2) though 4) as a regular addition.

2	4
4	16
1	6
5	76

One trick in the method is when the units place is 0,1,2,3 when the squares are also digit numbers. Here, you need to continue writing them as a two digit numbers - 0x0=00, 1x1=01 2x2=04 and 3^2 = 3x3 = 09. For example 63^2. 2) 3x3=09, 3)6x6=36, 4)6x3=18 double 36.

6	3
36	09
3	6
39	69

In the first class I tried this with the children had asked why this was. Being a rookie teacher I thought they meant they wanted to know why you write single digit squares as two digits and tried to explain that the '2' is 20 and its square is 400. Luckly, I also listen to the kids and realized that they only meant that I should repeat this step and give examples for them to master it. (Though it gave me an idea to build on the real why later).

The teacher had displayed prime number all around the class and we picked up prime numbers and squared them (subtracted one) and checked if this was divisible by 24. This is, of course, easier said than it was done. I tried to explain that the factors of 24 were 8 and 3 and we need to do a divisibility test of these two numbers and all these ideas fell flat on their but. Finally, we just wrote the 24 tables and did long division. The issue the children had with the divisibility test is that it doesn't tell you exactly by how much 24 divides the square and till they have this number the division is not real!

This itself was fun and helped as a way of learning squares and gave a practice of long division. What was more fun was to answer two questions:

1) Why does the squaring method work

2) Why is a prime number square minus one (p^2-1) divisible by 24.

I posed the two questions to the children and asked them which one they really wanted to know. If you are an engineer and not a teacher you would be surprised to know that the children only had interest in 1) and 2) was well, a nice side dish.

I realized that I think in algebra and it was a challenge to think of solutions that were visual and didn't require algebra. I was able to address 1) in this way, but that's for another blog.

Solving puzzles: Quiet time

I have been putting up mathematical puzzles at Udavi for about 3 months in the area for sections 7th to 10th grade. I have been getting responses mainly from a few kids in the 10th grade. The other kids have not been as involved in solving puzzles. I noticed that in the younger grades (6th) the teacher does put up puzzles occasionally and thought it would be interesting to pursue the solving puzzles there.

I created a couple of puzzles that were visual (match stick puzzles) and put them up in the 6th grade. Again I found that 2-3 children were keen on solving it, but the rest were not able to. We then had a class to solve the puzzles.

I had been working with these kids for a couple of months now and had noticed that the classroom gets pretty noisy as the kids who 'got it' were too eager to sprout out the answer and the kids who had 'not yet got it' were all to comfortable not having to think. I had been working with the kids to do individual work in the notebooks without having to raise their hands or talk allowing me and the other teacher to go around and look at their work. The classroom had been marginally quieter at times. Of course there is an equal mix of group activities (games) or discussion time (sometimes moderated) and hearing about each others work.

In this class, we took a step further. There were enough puzzles for someone to continue solving if they finished one and they were asked to work on this and try to create a quiet space to allow them to think. The teacher and I went around to see how the kids were doing. Occasionally, we encouraged the class that they could do it and just needed to relax and find their quiet space.

That class was magical, the children were able to find their space and over 80% of them were able to solve the puzzles on their own. The other 20% needed individual time from us to ask them questions that led them to think of the answer and elliminate the impossible that they were stuck with.

Unfortunately, with everything magical, it has been difficult to replicate. But, there are now more children that attempt puzzles and also ask for it when I am in class.

We have now been slowly moving from visual puzzles to ones with numbers and in a more recent class we all created puzzles of our own.

An exam that didn't end

One of the attitudes that some of the children have is 'finished', not an idea that something was completed, but just that something was done and effort can cease. This is particularly an issue with math when sums are finished and often nothing is learnt in the process. At the end of the first term exams are conducted at Isai Ambalam School. I decided to challenge this attitude...

A week before the exam I announced to the children that it would be a 'cheat sheet' examination i.e. they can bring one sheet of small/large notebook paper with whatever they needed help with to remember to the examination. To me this was groundwork for them to assimilate the information they had (which most of them didn't do).

I also created a question paper that favored word problems vs 'sums'. I created a paper for 120 marks to and in addition, I created bonus questions that would build on the understanding of a question. The children didn't attempt the bonus questions, but it gave me an opportunity to talk about it later and point to where our knowledge will grow in time.

I let them see the paper for a short time on a couple of days before the examination and I went over any questions with the English a day before the examination (as comprehension of English text is also a challenge).

The examination itself lasted for the standard 2-1/2 hrs, but many children were unsatisfied with this morning session, so they got another 1-1/2 hrs in the afternoon class as well. While the kids were happy they got extra time, I was smiling inside as this was the first time they did math for 4 hrs in a day and that too after they 'finished' with the examination! By the end of it though a child had reached the limit of his mental stamina and said he didn't need extra time even if it was available.

Nope, we were not done. Self-evaluation! I asked the children to grade their work and write down 'Yes', 'No' or 'Maybe' indicating they had confidence they had done it right, wrong or didn't know on their question papers. They would get an additional 2 marks for every correct assessment of their attempts with 'Yes' and 'No' and loose 2 marks for every incorrect assessment. They could play it safe with 'Maybe' and neither loose or gain marks.
This exercise was really good to understand attitude of children to their work. It also brought up some interesting introspection from the children regarding how they perceive their own work.

The examinations were week long and some of the days they would study for the next exam. But the next day I solved all the questions on the board and they did a self evaluation.

For the week long holiday after the exams their only homework was to understand and be in a position to answer the question papers (which they didn't do).

The context of word problems help revisit these questions even in the classes we do now when we revisit decimals or need to clarify a topic by making a story out of it.

Here is some of the feedback from the kids:
"I have never been able to solve word problems and this gave me the confidence that I can approach them."
"I guessed that I did things wrong when I got them right and right when I got them wrong. I don't trust myself enough."

Of course, two kids didn't do well and their performance was stark both in ability and in self assessment. One of them left her morning work of 2-1/2 hrs and restarted the paper in the 1-1/2 hrs I gave in the afternoon. Both did what they knew repeatedly without attempting something new. But, the fake confidence she displayed in class was broken and she has been more awake and engaged in class since...

Cooking with Arham

We made chana this morning. 

Flowers

Isai Ambalam School has many trees, most of the space around our home is under the shade of these trees. There is a Parijata tree that has started flowering and with the recent rain and wind, flowers get blown onto the ground. After one such rainy day these were the flowers and colours that I was drawn to.

Ask me anything...

I started volunteering at the Auroville ITI in Aug (2013) to work on delivery of Electronics curriculum. There is also concern about the standard of Math the youth come with. A glance at a math exam reveals familiar gaps in fractions, decimals, negative numbers, algebra, geometry from the 7th grade. Wish as we may, these doubts don't go away, it may become difficult to identify issues given everyone whips out a calculator and half the time get things right. Of course, to err is human and if you really want to screw up get a computer (but a calculator in the right hands is not to be trifled with :)).

The youth had come over one weekend for a Math class and I wondered how I should go about teaching 7th grade Math to kids in ITI. Why would they think its worth learning now when they haven't for the last 4-5 yrs. I have marked the replies from the students in italics

Electronics Class

1) Initial connection? - what would you like to do, what do you have difficulty in?

- Nothing. Everything is ok.

2) What is math?

Calculations - addition, subtractions, multiplication, division

probing

there is also geometry

more probing

ok, there is also algebra and all that other stuff in the book

How about puzzles?

What is that?

A man has to cross a stream with a goat, tiger and grass. The boat can only accommodate him and one of these. How does he get them across without leaving the goat with the grass or the tiger with the goat (as they will get eaten)? What puzzles do you know?

Never heard of these.

probing

No, really, this is the first we are hearing of these.

Puzzles are useful to help us exercise our brains for logical thinking, say you are building a torch and you need to figure out the order in which you do things. Lets try to solve this puzzle then

Can he tie the goat far away from the grass?

Generally, yes, but not for this puzzle, no rope.

Why doesn't the tiger just eat him?

Its a puzzle. Ok, its somewhat trained, it listens to him when he is around.

Can he ask the tiger to swim across?

No, not so well trained that it will do something on its own or not eat the goat when he is not around.

Can he swim across?

and have the tiger row the boat? (I got laughs for that one) No.

How do you write these things? Go (Goat), T(Tiger), Gr(Grass). How do you write out the steps so another person understands?

He can take a goat (Go) across first.

Ok. The T and Gr can coexist.

He can also take the T first.

Really, what happens to the Gr?

Oh, ok the Go eats Gr.

What next?

He take the T and leave it with the Go.

No, that's a problem.

He takes Gr and leaves it with Go.

time, time, time....

Can he sell the goat and get on with his life?

No.

Ok, fine we don't get it, tell us the answer.

Well, moving on

what is the answer?

I'm sure you will figure it out

He doesn't know the answer.

That is entirely possible depending on the puzzle, but in this case I do and so will you. This is your homework.

3) 

We talked algebra, it didn't go far so we settled down with

3x5=15

Tell me a story about it. [It took a little time to warm up, but once they got going pretty much everyone told stories.] 

We used the following as a template to go further:

We got into a bus and bought 3 tickets, each ticket was Rs.5. The total was Rs.15.

I followed up on Ohms law

V = I x R

(they didn't get confused though I did not write it as IxR=V.)

If 'I' is like tickets and 'R' is like the amount per ticket what if you know the total and amount per ticket, can you find the number of tickets i.e. V and resistance can you find the current?

Ok. Lets pick it up a notch and talk about something we could not address before. Same equation different context:

P = I x V (I left out the PF)

Watt = Amphere x Voltage

What is the Wattage of the ceiling fan? 60 W

What is the voltage applied to it? Its the AC mains? 230 V

I know yesterday you said that the current was 5A because you think of 5 A and 15 A fuses. But this only means that the current is smaller than 5 A.

60 W = I x 230V

Wait the total is smaller than one of two things being multiplied. Is this confusing? What will we get? Decimal. Yes, you need a number smaller than one.

[

Diversion:

The kids said that the mains was 230 W. W as we just discussed is the unit of power. What is power? Boy shows his biceps. Ok capacity to do work. But, what is energy? Hmm...

]

How much is the current then? Around 0.25 A. In 5 A you can then run 19 fans!

What is energy? Do you get electricity bill?

Yes, whether they give power or not they don't forget giving us a bill.

Do you understand the bill?

No, they just tell us some '200 units' and tell us how much to pay.

One of the kids looks in his wallet and pulls out the receipt.

But, this is a receipt, what about the bill that tells you how much power you used?

They don't tell us clearly. We are not able to understand what they say.

Being qualified in electricity for the outside world is ok, but don't you want to know what goes on at home? Do you really use so much power or are they putting any number they want?

This we want to know!

Ok. Energy is just how much power you use for how much time.

E = P x t

Again t is like tickets and P is like amount per ticket.

and 1 unit = 1 kWhr i.e. 1 kW for 1 hour.

What is 1 kW? 1000 W. Ok, great we are good to go.

Lets start with the energy required for the fans in this room for the time of the morning classes today

4 x 60 W  x 3 hr = 720 Whr

How many units is this? 720 units

Careful here we have Whr. How many Whr in one unit?

1000 Whr. Is this more than or less than 1000 Wh? Less.

So its less than one unit. How much?

0.72 units

Yes. How about the lights?

4 x 40 W x 3 hr = 480 Whr = 0.48 units

Together its more than one unit.

What if we now ask a different question how many How many fans can you run for one hour in 1 unit?

How do we do it....1000Whr/60W/1hr = 16.67

Yes, you need to run 16 fans to consume 1 unit of power in an hour. That's a lot isn't it?

How do you consume 200 units of power in a month? You should find out. If you can work out what appliance is on for how long you should be able to get a ball park estimate and then have a discussion with numbers with your electricity man.

That's the second homework.

One of the stories was about the 'efficiency' of 150 cc engine for 1 L of petrol. I think he just wanted to say fuel x fuel economy = distance. But, he had hit upon 'cc' which most people don't relate to ml so I talked about this further. It then turned into an IC engines class with talking about the volume of the cylinder and how you only put a bit of the fuel and how in a petrol engine you have a spark plug to ignite the mixture of fuel and air, an exhaust value.

Even the teacher got into the act now and was asking questions about how and why the choke works.

I realized it had become a sort of ask me anything class, but I guess if we want them to be engaged and learn democratic classroom needs to be part of the process.

Rigor

Educators talking about alternatives in education (including myself) lay a lot of emphasis on the understanding of the child about what he/she is doing. But, there is no alternative to rigor (repetition in various forms) to master a subject.

Often rigor is confused with rote. Rote is the process in which a child repeats the same thing without any understanding in an effort to learn it by heart. Rigor is repeating something after understanding or applying something learnt in multiple contexts in order to help the brain rewire to internalize something learnt or improve the understanding of what is learnt from different contexts.

Babies love repeating something they are getting a hang of. When they learn a new skill say crawling, they keep doing it. When they learn walking, they can't get enough of it. If they learn to say a new word, they will try to use it at every opportunity. As adults we get so busy with our jobs/lives that we forget what it means to learn something substantially different. It pays to think what it would take to learn a new musical instrument. Now imagine trying to do it without any rigor or repetition by understanding it...

Where does this all change? Why are some children unable to or apparently uninterested in rigor? Is it because they don't understand or didn't understand for so long that they have given up?

I have been trying to figure out how to reignite the desire to learn through understanding, but also to remind children that to master something rigor and independent work is indispensable.

This hits the children hard in 7th grade (in TN) where a multitude of abstract ideas really take off with algebra, geometry with algebra and the works. Children apparently understand something in class and think they have got it, do not work at home, because their teacher is so cool and they got it in class. Come a clean slate the next day and expect me to start from scratch.

Even though I start every class with what did we learn yesterday, it works well for experiments and ideas they saw and worked on, but not as well for abstract ideas if they didn't give it a look...

Pizza party

Many kids in grade 7 are unable to operate with fractions. Among them a big segment are unable to grasp what fractions are, a second smaller set are unable to proceed on the arithmetic even after they understand why they are supposed to factorize, take LCM, etc.

As I took supplementary classes for the 7th grade kids, it was obvious that they were comfortable operating fractions when the denominators were the same (5th grade). Of course this just meant that they had a system where they added/subtracted numerators without worrying themselves about what fractions are/were and this was a gotcha in 6th. I am also working with the 6th graders to address the issue here itself.

Numbers are abstractions, but fractions are more so given that they are parts of a whole and a fraction can take different avatars depending on what the whole is. E.g. 1/2 of 1 kg is 1/2 kg, but 1/2 of 1/2 kg is 1/4 kg. A nice abstraction of a whole is something circular. It makes it very obvious when pieces cut diagonally are extra or are missing.

Pizza Party
Most children around Auroville actually know what a pizza is (not all like it) and I spent quite some time with a teaching aid called pizza party (Creatives).  The kids that don't like pizzas assume that its a dosa. The game is fairly cheap (Rs.165) and is generally well done (though the suggested games need work).

What it has:
Base cards of fractions 1/2, 1/3, 1/4, 1/6, 1/8. Pieces of the same proportions. A die that has these fractions (5) and creative written on it.

Modified/invented games:
1) Getting familiar with the pieces using base cards. The idea was to roll the die and pick up the corresponding base card. Then in turns roll the die till the fractional piece on your base card comes to complete the pizza.
Well the kid getting 1/2 needs 2 pieces and the one with 1/8 needs 8 so the game if far from fair.
I tried to even out the odds by allowing them to check if the piece fits into the pie and taking as many pieces as it fits, so a kid with 1/8 base card can take 4 pieces of 1/8 when he/she gets 1/2. Now, it seems 1/8 has the advantage. But, we added a no overflow rule i.e. if you get a 1/2 and a 1/8, then you get another 1/2 i.e. you can't use it. The only disadvantage is for 1/3 base card who really do need to wait for 3 such cards to complete the pizza.
2) Complete the rest of the pizza: Roll the die make the rest of the pizza with the pieces you have.
3) Selecting pieces to make a pizza in turns, but you pick the piece for the next person to play. The one who completes a pizza gets a point. You can keep the full pizza as part of the game to see when it gets used and who gives it to who :). Nice game to get into the kids psyche.
4) Pizza delivery game: Group game, the next piece of the pizza is determined by the die roll and the team tries to build 5 pizzas for delivery.

Initially, the children who were getting it wanted to play the game with less luck, but given the mix of kids they ended up playing many different ones. Some kids also wanted new games and we introduced ones with subtraction of two fractions and finding pieces that match the difference, or a piece that is just larger or just smaller than the difference.

One often wonders if doing these fraction games is really worth the time and I started a conversation with the kids of what we learnt (not what we did) from the activity. With feedback starting from 4 pieces of 1/4 makes a whole. This helps reiterate 1/4 means you cut the pizza in 4 pieces and take one. Similar ideas continue for 2 pieces of 1/2 and  8 pieces of 1/8. Then we move on to expressing one set of pieces in terms of another, 1/4+1/4=1/2, 3x1/6=1/3 and my personal favorite 1/2+1/3+1/6 is a whole pizza. Wow, made my day.

A note of caution for teachers using mixed fruit to teach fractions. We ask children to treat everything as 'fruit', adding apples to oranges to make up a whole. The whole is not obvious as a fruit can be added or removed and it would still be a collection of fruits. We then ask them to remember the 'fruits' individuality by asking what fraction of the fruits are bananas. The children get comfortable adding grapes to watermelons, but they will also add 1/2 a pizza slice with 1/8 pizza slice to give 2 pizza slices.

Equivalent fractions
An often skipped section to work quickly towards factorization and LCM is the idea that a fraction can be expressed as equivalent fractions.
By now, most kids can tell stories about fractions.
What is the story of 1/4? You take a pizza and cut it into 4 pieces and take one piece.
The idea can be extended into the relm of 1 out of every 4 pieces. This helps build equivalent fractions. What if you had 8 pieces in the pizza then 1/4 would cover 2 pieces (from the pizza game). So
1/4=2/8=3/12=4/16=...

At this point you can reintroduce the idea of adding fractions with the same denominator say
1/8+3/8 = (1+3)/8 = 4/8
and remind them  that the denominator indicates the number of pieces you cut the pizza into. You can add the number of pieces as they are the same size.

1/2+1/4 the pieces are not the same size and can't be added directly. This can easily be seen from the pizza game. With equivalent fractions we can talk about what 1/2 will be if the pizza is cut into 4 pieces. One out of every two gives 2/4 pieces. Now adding:
2/4+1/4 = (2+1)/4 = 3/4

Most smaller fractions can be added by writing them in equivalent fractions and looking for a size that is common to both.
1/6+1/8
1/6=2/12=3/18=4/24
1/8=2/16=3/24=4/32
This gives 1/6+1/8 = 4/24+3/24 = (4+3)/24 = 7/24

I introduce LCM after I ask them to add
1/2+1/200 at which point most children start taking a short cut into
100/200+1/200.

Of course nothing works for every kid, but I was able to address 90% of the kids this way.

Sanjeev, where is your laptop?

Its interesting that sometimes we can't see patterns in what we do till someone points it out. This friday at the teachers meeting at Isai Ambalam I was organizing my thoughts when Stella said "Sanjeev, where is your laptop?" For a second I didn't understand what she was talking about and then I remembered that I have been showing something, a ted talk, videos of the kids experiments, what happens to rice when it is processed or some such thing in every teachers meeting. It appears that there is some anticipation that something different will happen as well.
I went ahead and did an activity of guessing their birthday (only month and day) by a number they produced after a bunch of operations (I blatantly copied this from this video) as an example of something that could be proved through Algebra. Its fun to get teachers to be surprised and happy. A couple of teachers had errors in their calculations, but went right back at it and punched in the air  "yes, I got it" in the end.
Teachers should have more spaces to be children. The best classes I have had are the ones I was myself curious about what was going to happen next.

Will this float? : 7th grade exploration on math and science

One of the areas we covered in the science/math classes was density. Here is a video of the experiments that the kids remember doing in class.

Remembering Archimedis
We started with toying with the idea of volume of an irregular shape and thinking of a way to measure it. We followed Archimedis story of having to figure out whether a crown which weighed the same as some amount of gold was in fact entirely made up of gold or not.
We spent some time thinking about it, or whatever it is that each child does when it wants to indicate that apparently he/she is thinking. It usually involves staring at a speck of paint on the ceiling. I insisted that what we were about to do they already knew, which was followed by even more intense staring.
Once we put it in water and I indicated that the displaced water has the same volume as the stone I immediately connected it to the crow story that everyone hears from when they are young. Our experiments were a bit of little kludge when we used the milkman's jars, the smallest being 100 ml to try to measure the volume of a small irregular stone. Luckly, it is also a math class and we measured that the height of the water was only around 1/4 the height of the cup (go fractions).
I looked around the house to find something that would make the measurements easier and found Ani's measuring jar and weighing scale. I also found some cups Arham has that fit into each other.
Does it float or sink?
I used the iron blots that I had collected for the pendulum experiments and we did a session of how much it takes to drown a small plastic cup. Shi mentioned that given iron sinks its surprising that ships that are made of iron float. I asked him to hold the thought. On placing the mass that sunk the smaller cup in the larger cup it would float. On loading this cup further...before doing each of these steps we would try to predict if this time the cup would float or sink.
Shi concluded that with the same mass if the volume is bigger things float, hence, ships float because their volume is so big.
Density of water
Since we had now encountered the impact of mass and volume I introduced the concept of density of an object being the mass per unit volume. We measured the density of water and confirmed that it was indeed close to one.
How much water is displaced when an object floats
Now things get interesting, I asked how much water is displaced when an object floats. Given the experience of a sinking object the children said it depends on the volume of the object. We tried putting the same mass in gradually increasing volumes and found that they were all displacing the same amount of water.
We then measured the weight of the cup and the objects in it and found that the number we get (in gms) is the same as the volume that is displaced (in ml).

Note 1: This would not be true if a liquid other than water is used (or if I added say copious amounts of salt to the water changing its density)
Note 2: Actually, the volume displaced is not 'exactly' the same. The larger cup has a slightly larger mass, but given the larger mass of the object placed in it the delta mass is not significant to alter our measurements given the accuracy we could do them with.
Assignment
They had to think and tell me:
1) Why it is that if an object sinks its volume matters and when it floats its mass matters?
2) Why does a solid like ice floats on water?
My head is going to explode
There are many interesting idioms in Tamil, one of them meaning I thought so much that my head is going to explode. Ok, we didn't get it, can you tell us now?
I think its these moments that teachers enjoy, when any word said is waited in anticipation and when what you don't say is going to have a bigger impact than anything you say. "Did you discuss your ideas? No, you should." After a minute or two of we really don't know followed by silence one kid finally says, the whole object doesn't go into the water so its whole volume can't matter. Now the kids start catching on, oh yeah, the amount of water displaced is only as much as the shape the object goes into water and this must depend on the weight (I haven't gotten around to differentiating weight from mass).
The second question is more of an opening for me to talk a bit about how we view solids and liquids in their atomic structure and the open lattice structure of ice.
Things to see
When I am short in time then instead of doing stuff we just watch some vides - one that showed hot water was less dense than cold water, that egg sinks in water, floats on salty water. How do you make it float mid way in water? Multi layered liquids (based on density). There was also a very nice animation of objects with different densities, masses and shapes that captured how the water in the tub increased when an object is dropped in as well.
Assessment Videos 
A couple of the kids had made the Bartons pendulum and coupled pendulum experiments for a friday teachers meeting. The kids were keen on making another video for the many things they had learnt over the week. I decided to use this as an assessment technique to see how they work alone and with others. The whole operation was done within 35 mins.

The kids had to discuss what they had liked and come up with what each wanted to talk about while giving the others also something to talk about. There was very little prep time, but mentioned what I found was missing the last time they made videos (mentioning the things that are used in their experiments). I tried not to interfere and even had to walk off in the middle of a video, I did prompt the youngest to say (ml) for the volume she found.

It also gave an opportunity for children think through and make a presentation logically, breaking down all they want to say in a step-by-step manner ( I should take the weight of the measuring jar before I fill water in it). This process is useful in their life no matter what they end up doing.

Class X intervention: Prove it!

Please read the other two posts
1) Why intervention?
2) First few classes
if you care for continuity.

I'm consolidating the key things that happened over the next two classes.

The next topic within geometry was on properties of chords, tangents with circles. For a transition from triangles to circles and I tried to connect up the two by showing them that the x coordinate of a cow going in a circle at constant speed is a sine wave :). This was way beyond the syllabus and had the teacher browse through the textbook to see where this 'stuff' was. It was a quick transition that touched on trigonometry and gave me a chance to plant a seed that I would water in my electronics class with them as I introduce AC signals...

In my preparation I realized that they were supposed to know some really neat stuff with chords and circles in 9th grade. An important one being given a chord any triangle it makes with any point on the circle (on one side of the chord) has the same angle. I tried proving it myself, I couldn't. I didn't find the 6th grade textbook either and finally went though the proof on khanacademy. It had a proof that the angle (subtended by the arc) with the center is double of the angle at any point. Nice, I didn't remember this.

In class I mentioned the theorem and the kids and went a step further and said that the angle with the center is double of the angle it makes, they said that they didn't know it.  It wasn't enough that they knew now they wanted it proved :). I could not believe these are the same kids who would take a formula without question and were afraid of proofs 5 classes back.

Its a cute proof that involves a specific case of a triangle made with the diameter to make an isosceles triangle and then generalizes it by splitting any angle as a sum (or difference) of two angles involving the diameter.

I proved the diameter and then gave an example of splitting a random angle as a sum of two angles. They were not convinced that this was general enough and came up with a point where the angles could not be added (point A - a limiting case of summation). I gave them the proof by subtracting the angles. The wheels in their heads were really whirling now. Then found a point on the other side of the chord (point B) and I told them that it was indeed not valid on this side o the chord and we would talk about it later.

There are only a few proofs that are going to come in the X class exam. The advantage was that I could still prove everything but didn't have to write it out in full on the board. We looked at the figure and I made my case by walking them through the thought process while marking on the figure. Its a real class and there are kids that zone out.

 The good part is that since I didn't need to write things out it doesn't take time to repeat the proof from where they didn't follow it. There may still be one child with a glazed look and when you ask him/her what he/she missed would come back with 'didn't understand anything'. In such cases you just start at the beginning, there is a circle and a chord. They come back with Dah, yes I know that, and then you build on it. I have done a proof at max 3 times orally before glazed looks are replaced by knowing ones.

I also needed the opposite angles of a cyclic quadrilateral add to 180 which I again mentioned. By now they were trying to get all that they didn't in 9th grade and asked me to prove it. I drew the angles required for proof and told them they could do it themselves.

We found similar triangles in the intersecting chord theorems (it would have been nice if they had asked them to prove it in the book). Here the two triangles APD and BPC are similar and their sides are proportional.

The atmosphere in the class is tremendously different from what we started in the beginning of the sessions. There is a lot of questioning regarding principles. The children are engaged and there is very little idle chatter. When it is there. I pause the class for the children to finish and I have reaffirmed that its not because they are disturbing the class, its because the rest of us don't want them to be left out. The children also listen when another child is talking and don't try to cut them off.

We are done with the 'theory', now to focus on application of the same.

Classroom intervention Grade X: Geometry

The problems:
1) Where do I start?
As I went through what they had for Geometry I realized that although they had done coordinate geometry for a couple of yrs, trigonometry for a couple of yrs, they were now going to encounter the idea that triangles can be similar and their sides would hold ratio between them! I also noted that there was no proof given in the text for the same :). Great, I can start with one of the most powerful concepts in geometry that allows me to link with everything they kida, sorta know of and I don't even need to prove it.

2) The kids may remember the concepts but not the corresponding names e.g. adjacent angles, corresponding angles. They would, of course, remember vertically opposite angles, but who doesn't. This gives rise to the issue that we can't talk the same language.

3) The kids also know me to generally have fun with puzzles and games and I need to have some connection with writing examinations.

Class 1
I started with asking kids to make a cheat sheet  (bit paper as the kids call it) for Geometry by putting in all that they know about it. The kids were excited by the prospect though I made it clear I was not encouraging them to take it to examination.

We started with a small list which was enough for our purposes and decided to add more as we went along:

I added one new one of similar triangles, with just one rule (AA) if two angles of two triangles are equal the triangles are similar. In similar triangles there is a constant ratio between any two corresponding sides.

I mentioned to them that the similarity of triangles is the most powerful of the geometrical theorems and now they can rule the world. We started with trying to prove the 'mid-point formula'.

Actually, if you can understand the picture the steps are simple, but I went through many iterations of building up this picture one piece at a time. The biggest bottleneck was that they didn't associate the coordinate in coordinate geometry with a real distance. In a point (x1,y1) they were unclear as to what x1 really was. It had just been a number they needed to substitute in an equation. Well, this was precisely what I wanted to address and I didn't mind going over it again and again. Once that is understood it was actually quite easy for them to take up the new concept of ∆CBD ~ ∆CAE and were even able to supply be the reasons (adjacent angle, right angle). As 

CA/CB=2 (mid point) 

 (x2-x1)/(x2-x)=2

=> x2-x1 = 2*x2 - 2*x 

=> x2-x1-2*x2 = -2*x

=> x2+x1 = 2*x or x = (x1+x2)/2

I proved it for x and I asked them to prove it for y at home. I completed the picture as below.

I asked them to complete a couple of problem sets in the book and using similar triangles prove Thales Theorem as well.

I worked out a couple of questions from two exercises and asked them to do the rest by themselves.

Class 2

I didn't expect everyone to do everything, but I expected someone to do something, and they did.

Four kids (in a class of 16) tried and succeeded in proving the mid point theorem, few others attempted the exercises in the book, as they now looked trivial. Few read the book by themselves for the first time to see what's really there in it :). However, there was still half the class that was waiting for the dust to settle and for me to solve the questions on the board.(ha)

I proved Thales theorem using similar triangles and asked them to add this to their cheat sheet as well.

I asked them to continue to solve the problems in their book. It didn't matter that they had done half the exercise or looked sheepish that they hadn't started. They all had to work and work independently.

Its surprising how noisy the classroom usually is, its not the kind of organized chaos the teachers permit its the kind that comes from boredom of kids who are able to solve faster and waiting time of those who don't follow for the answer to be written on the board.

They could not talk or borrow pencils/pens/erasers/scales or any of the other million things they seem to always need in the classroom. The instruction was, use what you have, do what you can and find out for yourself what you are capable of. It was not easy, they had not done this except in examinations. I needed a more supportive setting than an exam and I gave occasional individual attention, at times needed to remind a couple of kids about what we are doing, but we were able to get some peace and quiet in the class (another first). I went around to see how kids were doing and for once compromised and let kids know when they had got the answers. Almost 4-5 kids other than the ones who had already done some work were very kicked and would say, is that all there is to it.

This class was a very big achievement, kids worked and to their ability succeeded in solving some problems and started believing in their ability to do so. The most questions solved were close to 10, but even the least were 3. Usually 4-5 questions are covered in class when the teacher solves it on the board, not bad.

Class 3

It was much easier this time for the kids to settle and work this time. I addressed a couple of common issues encountered in the previous class. We talked about the common strategy in the questions to have multiple triangles and a common side that was divide in the same ratio as the others. We addressed ideas for cute questions like ratio of sides of a trapezium that needed an additional line to be drawn to see the triangles.

We also compared altitudes of similar triangles and proved that they are the same ratio (of course, using the idea of similar triangles). This brought us to right angled triangles and all the physics they had done of comparison of shadows of objects and their heights. We talked about how these triangles are the same and what would happen if the time or place of one measurement is different from another and what the assumption is to say that the angle of incidence is the same. We concluded that the rays of the sun need to be parallel between the two objects, times and places of the measurement. Our misconceptions are buried deep, so for yucks I drew the sun, as we drew as kids with rays coming out in all directions and pointed out that this in such a picture the rays don't seem parallel and any bright objects gives light in all directions :). It was a fun, after a couple of intense classes this whole exchange seemed just what we needed.

We then cover altitudes of right angled triangles that create smaller right angled triangles that (needs rotation carefully) are similar to the original triangle, which further can have altitudes giving similar and smaller right angled triangles. 

This was a fun exercise and we really needed to be sharp about our rotations. We finally figured out the algorithm to get things right. Start with the corner where the angle is unchanged, mention the corner with the right angle next and finally the corner left out.

I found a really cute problem in the book. Try it out:

Find the other diagonal AC given AB=BC, AD=CD and the right angled triangles marked as above.