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.