Change in culture

Four years back when I moved to Auroville I would post one or two puzzles a week on the school notice board at Udavi. I had announced them in the classes I taught and there were always two children U and D in the school (in 10th grade) who would attempt to answer the puzzles. They would discuss the puzzles among themselves and often submit a solution to the puzzles. Sometimes they would have doubts and come and explain their logic to me and ask me to find a flaw.
I did some visual puzzles with arranging match sticks and these some of the 6th graders were interested in, but by and large, I got one or two solutions for the puzzles with barely any discussion about the puzzles.

Last week Naveen took up the exercise of putting up a puzzle a week in STEM Land. There has been quite some discussion about the puzzle - the 9th graders applying algebra to solve it and the younger attempting to do it without algebra. I even noticed a volunteer who has come to learn STEM has been attempting to solve it. 

It remains to be seen how much of this buzz translates to children taking the initiative to submit their queries, especially as we have not marked any prizes for their submissions. Nonetheless, I can see a change in culture, of attempting puzzles, that was missing in the children before.

Deep learning...

Someone had recently asked me to give them an example of what I considered deep learning.

The STEM Land (@ Udavi) display that the children in the 9th grade had broken down and a couple of letters were not lighting up. A couple of children in the 7th grade had expressed interest to fix the display. I had walked them through how powering a leg of the display lights up one of the 7 segments (and the decimal point). They had been fascinated by being able to fix something and over a couple of classes went ahead and fixed it.

After fixing the board and seeing it work one of the children was interested in doing something more with the Arduino. What the 9th graders had done was the treat a digit as a unit and light up one digit at a time, etc without changing the essential message of what was being printed. But this child wanted to do more. He wired up the 7 segments to a separate Arduino pin and wanted to control them individually.

He managed to light up a 1 by end of a class. I mentioned to him that this will soon get out of hand if he did not start organizing the code (in a language he was learning) into functions.

Later that day one of my youth asked me to give them an example of deep learning in STEM Land.

The next morning B didn't have a class, but during lunch, he came by and asked me to show him what these functions are. I told him that it was similar to the setup and loop that are already present, but with any name he wanted and they could be called into action when needed, similar to the blocks in Scratch. His face immediately lit up and he said ok, I understand blocks.

He had an activity class after lunch and he sat down to implement what he had in mind. He ran into a couple of errors of syntax (defn of functions needs matching '{}', calling functions in C needs '()' ), but then he did not ask for me for about 45 minutes. Then he came to me and said that he needs some help in extending the functionality. I went up and saw what he had done and realized that he had made a single digit counter that went from 1 to 9 every one second. He had now decided to make a clock out of the arduino and had added a second 7 segment display to get the second digit of the clock he needed. I thought he would have missed the logic of the first digit continuing to run when the second was set, but he had got that logic right and implemented this as well. 

I could find nothing wrong with his program and I think was a little surprised how much he was able to progress in a single session starting from wiring up getting the logic of an entire set of digits right and moving on to the next logically. Perhaps, the issue is with the hardware and we are attempting to draw too much current from the USB and none of the displays were lighting up. The debug continues
...but it gave me an answer to what I presently consider an example of deep learning, taking a concept in an entirely different context from blinking light->fixing a 7segment display->individual control of segments and putting it all together to shift between digits in the hope of creating a clock with very little help from an adult.

Of course, there are not enough outputs in the Arduino to create all the digits of a full-fledged clock and I let him know, but it seemed he was happy with what he could get.

Multiplication tables with children who never learnt it

There are children who are able to hold their fort without knowing the multiplication tables in the calculations in Mathematics. Some are quick at addition, some remember specific tables and extrapolate quickly. They survive till 9th grade where powerful mathematical concepts are introduced with milder calculations and are able to generally manage.

However, for most children for whom the number sense has not landed and are slow with calculations, the lack of knowing the tables is a serious hindrance in survival and any possibility of enjoying mathematics.

We took up the challenge of identifying 7th, 8th (and 9th) graders who did not know their tables. We worked with the children in identifying what they already knew - generally the 2, 5 tables. For a couple of children, the short-cut for the 9 tables was quick and they were comfortable with using that quickly. Some have the hand rules for 6x6 onwards, but it appears too slow for practical use. The table tricks were kludgy and not used e.g. some knew methods to write down the 7 tables, but it was too easy and not practiced enough to be usable in action and certainly not at random.

The common issues for children were 3x6 to 3x9, 4 tables, 6x6 to 6x9, 7x7 to 7x9, 8x8 and 8x9. I looked at a demo of the Vaughn Cube. Now, the demo is public domain and looking at the video it is easy to figure out how it works. They have given enough in the video (unknowingly?) to figure out not just how it works, but work out the details of the method. The rest of the post is about our Indian hack of the same and what we did with it.

The idea is simple, replace mathematical facts with objects that you need to remember in a 3-D room. The sounds t(1), n(2), m(3), r(4), l(5), ch(6), ka(7), f(8), p(9) are memorized in simple ways. An advantage of an object between two number is that it both 3x7 = 7x3 are the same. Iya one of the children who had a lot of difficulty with learning the tables announced that other than the 8 and 9 tables he knew everything. When I drew a pic of the 9 tables on his request he suddenly realized 9x3 = 3x9. Hey Anna, 9x3 is the same as 3x9. Then he noticed that all the other objects were also familiar, this makes the table so much easier he added. I had attempted to convey the same to him earlier and when I shared his Aha moment with his teacher she remarked that she had told him as much many times. I guess, it doesn't matter how many times someone tells me something, it only matters when I get it.

What we did with it. 
1: Game: Figure out as many of these combinations by watching the video. As it gets tedious, use the skip forward and back feature to make it a little interesting.
2: Exercise: Practice with what you know and find out what is missing.
3: Project: Make a mini project in Scratch that shows the objects as you type in the questions.
4: Practice, Practice, Practice
5: Project: Make a game that tests rigor by asking tables at random, build on it to time yourself. By now I have quite a few kids who want us to test them because they have gotten so good at it.

Some notes, 
- With some practice, the children who had given up on the tables in the past were able to remember tables.

- However, these are mathematical facts and it lacks a logical framework that children can work with when things go wrong. e.g. if the child remembers the wrong object it can be an absurd result.
- I'm still sticking with when all else fails (flash cards, logic of knowing 5 tables, knowing squares and adding and subtracting) perhaps give visualization a try.

From '15 does not have a square root' to 'what is the square root of 0.05'

After such a long break I though I will start with something simple. A couple of questions from children. A couple of weeks back one child said 15 does not have a square root (only 16 does and that is 4). I wondered what kind of creatures children thought surds are. I had done some square root work with 10 graders using graphs and it seemed to give them some idea that these are also numbers of some sort a month back, but it was a passing comment and I let it go.

Recently, a volunteer started coming into STEM Land and I asked him as I ask all adults to work on some projects to inspire children. I suggested creating some sort of graph in scratch for children to see the squares of non-whole numbers. He started it and got as far as creating squares for whole numbers (similar to what the 7th graders with Sundar did last year) and then gave up, preferring to use his time more productively by finding children who wanted to be taught.

Anyway, another week went by and I decided to take up this project and made a rudimentary squaring version with visual boxes (my hope is always to make something functional, but so terrible that anyone who sees it will want to modify it :-)). I went in yesterday and showed it to the volunteer. One child came by and asked what we were up to (never underestimate the curiosity of a child), I showed him, then another, then another kid. They played with it for some time. Then I modified the program to calculate a square root. This seemed to garner a little more interest. They started playing with the program and one of the 10th graders had an activity class and connected it to the graphs he had drawn and started explaining it to the younger kids. My work was done and I walked off leaving the computer to them.

Towards the later half of the session one girl came to me and asked me what the square root of 60 was. I walked her through my guess and she concurred, but murmered something about off screen. A little later she came back and asked me what the square root of 0.05 was. Hey where did that come from? She explained that she was tying to figure out how small a figure she can get and saw the really small square of just 4 of 1/100 and she triggered the software by asking it to calculate the closest square to 0.05. 

I went back and put 60 into the program and realized she was testing where the program ran out of space to fill the boxes. Boundary conditions, interesting...

[Thanks Swathi for suggesting graph notebooks and that areas up to 2 decimal places can naturally be represented in a graph notebook.]

Boring digression from the story:

If we look at a regular graph sheet it is divided into large 1 cm squares. Within these the squares are split vertically 1 mm lines and horizontally in 1 mm thin lines. There are also helpful guide lines at 5 mm that are a little thicker than the 1 mm lines. These lines divide a single square into 10 vertical pieces (0.1) and 10 horizontal pieces (0.1) and together into 100 smaller 1mm^2 squares. The big square is whole and a smaller square is 0.01.

Here is a picture:

Children are comfortable with decimal numbers on the scale. They can easily mark out 5.4 cm on the graph paper and soon realize that its 5 dark lines and 4 light lines. 

If we marked 5 cm in both x and y axis and completed the square we will get 25 full squares. No surprise, area is multiplication.

The same happens if we mark 5.4 cm in both x and y axis. What is the area of this? The following image may be a helpful... 25 big squares (=25), 4 rods x 5 (4x0.1x5 = 2) vertical and same horizontal (=2) and 4x4 (1/100) (=0.16). 

Now playing with modified program of a square root.

Here are the programs to play, break and make something better!

Squaring program     : https://scratch.mit.edu/projects/115474246/

Square root program: https://scratch.mit.edu/projects/115470991/

16 Dec 2015 epiSTEME6 conference presentation

Presentation that goes with the paper on epiSTEME6 conference.

STEM Land inauguration on Friday 9 Oct 2015

STEM (Science, Technology, Engineering, Mathematics) Land is an space exploring Science and Mathematics holistically with active use of Technology and Engineering (deriving principles from experience), its also a resource center with Math and Science materials, a maker space and a puzzles and games library. 
Its impossible to describe what STEM Land is, you need to experience it. Come join the adventure and start exploring STEM Land at the opening and a small fair at STEM Land @Udavi school at 10:45 a.m. on 9th Oct [Main Building], Auroville.
Please RSVP so we can have enough snacks (healthy and unhealthy) for everyone.

If you want to read more about STEM Land, please take a look at http://www.auraauro.com/stemland/

Why STEM Land?

Though STEM is an acronym it can also refer to an approach to science and math education that is oriented towards application and making, tinkering and engineering. Its Land and not Lab because we are targeting a culture of learning rather than teaching and just as in France children learn French just fine and in Tamil Nadu children learn Tamil just fine, we hope that children will learn STEM naturally in STEM Land. 

Team:

The center is being initiated by the team (Bala, Sundar, Sanjeev, Vaidegi) of Aura Auro Design. But, we are well aware that the use of the center will involve a lot more and be driven by the interest of the children to learn.

Thanks to:

- Big Sanjeev and Udavi school for providing us space and up-ing it based on enthusiasm of the children.

- SAIIER who have provided us with the funding at just the right time to set up the center.

- Many other friends Ram, Gaurav, Shree, freecharge (company) for supporting materials that are being used at STEM Land.

- Many of our friends who helped us set up the center voluntarily, we spent no money on labor and did all the work ourselves and with friends. 

- Naturally, thanks to Aura for supporting Aura Auro Design without which this would have never become a reality.

Completed paper on children learning using computers

Many things have kept me away from blogging. This post is about the first, refining a paper that has been accepted at epiSTEM6. epiSTEM is a biennial Conference Series, to Review Research in Science, Technology and Mathematics Education conducted by Homi Baba Center of Science Education and IIT Bombay.

As Heidi Watts who helped me rethink the paper put it the paper now is about how children learn rather than about mathematics or computers. It has case studies on how children have learnt by trying to explain something to the computer through programming. 
Thanks also to Vaibhav who proof read the first version that got accepted, but both he and I knew it wasn't up to the mark then :).

Below is the abstract. Here is the link to the paper.

Using programming with rural children For Learning to think mathematically

Sanjeev Ranganathan, Bala Anand, Sundranandhan Kothandaraman, Vaidegi Gunasekar

Aura Auro Design, SAIIER (Auroville, India), sanjeev@auraauro.com

Is it possible to use computer labs in a rural setting that encourages reasoning, visualization, abstraction in children (as envisioned in NCF 2005) while at the same time addressing curricular needs?

This paper addresses the question through the use of programming in two rural schools including integration of curricular areas for fractions, cube roots, algebra, compound interest, data handling, geometry, etc.

We explore three styles of instruction - projects for children to demonstrate their understanding, challenges to visualize abstract concepts, and games created by children themselves for mastery.

Copy-Paste (the hard way)...

The children had been thrilled with being able to animate the moon going round the earth and wanted to show the solar system with the earth going round the sun. 

Thanks to Arham I had recently seen a video called Earth's motion around the Sun that indicated some of the complexity from the simplistic info that children had:
https://www.youtube.com/watch?v=82p-DYgGFjI

We looked at it and the children felt that it would be interesting to get an object to move in an elliptical path and replicate the earth going round the sun.



The circle was more or less straight forward with moving and turning in each step, but an ellipse...I talked to them of using Geogebra to draw the curve they wanted e.g. an ellipse and then "copy-pasting" it in scratch. Of course the issue was not drawing a shape, but to get an object to travel along the curve so it needed to be true to the curve.

I started with a simple case if x+3 = 5, we went over the story and I reminded them that the answer was not important, it was a trivial case and we were trying to get a method that would help the computer do a repeated simple task till it reached a condition. Also to have a simple condition we changed this to the expression x+3-5 and substituted values of x from -3 to 3 to find when the value of the expression goes to zero giving us the correct x.

We then attempted to solve 2x+3=6 i.e. 2x+3-6 and check when it becomes 0. But, it did the following:
For x=-3-->-9 ; x=-2-->-7 ; x=-1-->-5 ; x=0-->-3; x=1-->-1;  
      x =2-->1  ; x=3-->3    ; x=4-->5    ;x=5-->7 ; basically missing 0. What would then be the way of capturing where the answer lies. The idea that came to the children was that the result of the expression changes direction. Naturally, the last result will need to be remembered, but how do we check if something has changed direction. 
I asked them what they had learnt from integers that might help them do so crisply.They could not quite figure out how to go about it and when they couldn't quite connect after struggle I guided them into the product of integers and checking if the result it positive or negative. Of course here we get the closest integer number, but if the equation is large even though we don't have the decimal parts the curve looks good enough.

We then moved to a real equation x+y=3 and for each value of y we swept x...Once it worked for a line we just moved to the curve and only needed to handle the boundary conditions. 
    

Where is the moon now?

One interesting conversation I had recently was asking the children where they expect the moon to be. When we talk about how disruptive technology can be an app line startracker that shows the position of the moon 'right now' gets into a lot of how children think.

From the first answer in the sky to where in the sky then conversations on what happens during the day and night. I asked the same question over time to different children and with the changing position of the moon it enlightned different aspects of learning. Here are some of them:

1) In an initial thought most children thought the moon rises and falls consistently at more of less the same time. So it should be the other side of the earth. But on looking for the moon it was noticed in a very unexpected place. 

We then decided to for the moon and record where we see it and the time we saw it. That was a week before the new moon and a lot of children came back with there was no moon this week, but some others noticed that they saw it in the morning. We talked about sometimes seeing moon even after sunrise and sometimes even before sunset. What this all this mean?

2) Drawing the expected position of the moon and trying to represent meaningful information of a 3 Dsetting in 2D that led to frame of reference direction and directions. This led to discussions of which plane we can draw aka where does the moon rise and set. It was interesting that there was a child in each class who through that once the sun sets in the west the moon rises from the west.

3) While most children disagreed to the above they could not give any convincing argument to their peer other than that's the way it is.
There was some discussion of how the moon rises and sets. We have been talking about differentiating science from information i.e. being able to understand a phenomenon so we can use our understanding to explain another phenomenon.
One of them was understanding the revolution of the earth which explained both the sun and the moon rising from the east.

4) The children got some clue of which of their classmates were conveying useful information regarding their sightings and which were just making stuff up :).

5) Discussions on fractions! Based on where the moon we talked about where the moon would have been (w.r.to earth) 12 hrs before 6 hrs before and then the more open ended question of when would the moonrise have been. I say open-ended because it brought up the question of at what degree do we consider it moonrise and what the present angle is...

6) It brought up interest on how to create an animated model of the moon around earth, in fact about the moon and earth around Sun.

7) And then the question, how does the camera see that far in the sky :). 
I moved the discussion to possible ways of doing it including a three axis accelerometer that helps is determine where we are looking + gps + the star map + calculation of the moon and planet movements and how that was a cool way to use math. The children immediately got the wiff that I have switched from anna mode to 'teacher' mode and they started to loose interest. One of them even remarked that then we can't see the man-made satellites. 

I learnt my lesson and decided to keep my agenda of teaching something aside to avoid messing up with their learning. When asked this question the second time by another set of kids  just stuck with, 'no clue, but isn't it cool!'

Hmm...upgrades...

When you get around to doing something the second time...there is always something new that happens to keep things interesting.

Last year I had worked with some 7th graders on integer subtraction by making something move from its current position to a target position. Here is a note on the same

"Children initially tend to understand negative numbers in integers as a loan. However, even a simple situation like 5-(-5)=5+5=10 becomes hard to comprehend. A number line is a better visualization for introducing negative numbers. This happens implicitly when using Scratch as it incorporates the Cartesian coordinate system. A character (sprite) when placed at (0,0) is at the center of the display. Changing x with a positive number takes it to the right correspondingly changing x with a negative number takes it to the left.

As children use a change x function they get a handle of the integers. In time they get curious and check what happens with change y with integers to get a sense of the Cartesian system.

As the children felt more confident they created a simple game with integers: the user needs to give the number to add to take a character from a start location to a target location. Though initially the children put numbers and situations they already knew e.g. two positive numbers start:5, target:10, to add:(10-5)=5. The program would then move from the start to start+to add and check that the result is the target. Once the children are comfortable enough to randomize one and then both the starting point and destination, all possibilities of subtraction scenarios arise e.g. start:10, target:5, to add:(5-10)=-5. They also clearly see the need to move left to get to the target. Similarly, start:-5 target:5 to add:5-(-5)=10 also makes sense as the target is to the right of the object."

The emphasis last time was a bit on programming and getting the strings and numbers to match up well. 

But this time the children were implicitly unconvinced about the motivation of the character (in one case) the crab to move. The crab must have had a reason to have a target, right. So they came up with their reasons. The crab was tired and wanted to sit on a rock!

There were other stories of a diver catching a fish, of a cat catching a mouse, someone looking for a key, etc). The character was at the current location and there was a visible target that the children could see is behind or ahead of the character. This helped them re-look at the result they got to see if it made sense i.e. positive number means the rock should be in ahead and a negative result is a rock behind me.

A simple square rooting game

At Isai Ambalam a couple of the children I was working intensely with programing graduated, it left a small class of 6th graders and a couple of 8th graders. The youth with AuraAuro took up various classes including my 6th grade. I was left shuttling between classes and the 8th graders. The 8th graders who are still here are not ready for their next grade both in language and math. Taking a leaf from my being independent book they were keen on taking the initiative and learning on their own and asking for help when they needed it. They were working on squares and square roots for a week on their own..., but I realized that they were not quite getting the sense of the numbers. They were doing much drill, making mistakes, but unable to notice their mistakes or having a ball park estimate. 

They are hesitant to program themselves and had relied heavily on their partners who had left school. I asked them if they wanted to the computer as a calculator...they readily agreed (ha!).

Looking around and starting visualization

The sixth graders were drawing the representation of linear expressions like 5x+10 (and then changing the constant or the slope and getting various staircases for positive values of x. I asked the 8th graders if they would like to use scratch as a calculator and do the same for x^2 = x*x. They plotted it along with the linear curves and soon realized the much faster rate of its growth to exceed the screen size. They felt that x^3 should perhaps increase even faster and went ahead and implemented it to find out.

We talked about the possibility of using the computer to find the square roots of numbers by repeated calculating x^2 for numbers starting from 1 and stopping when they reached the numbers as given in the book.

Starting to look at the consecutive squares

We made a very simple program which started with x=1 kept incrementing x it by one and calculating a square (one multiplication per second). Scratch can automatically show the variables of your choice making it easier since we didn't need to program the printing.

They started looking at the results of squares of numbers and basic internalizations that the squares of numbers do increase as numbers increase and start to notice a few basic squares that they knew.

Efficiency of calculation

As we got to 4 digit numbers the time taken by the program was quite a lot and I asked them what could be done to speed up the program without changing 1 multiplication per second. My argument for the constraint was that that much time is required to register the square of a number and I did want them to connect a number to a square.

It took them a little while, but they concluded that we could start with squares of every 10 numbers 10, 20, 30, 40, etc and stop when the number was overshooting, go back one level and then increment by 1 to get to the result.

This reduced the square of 69 from 69 seconds to 7+9=16 seconds. They also started getting a clearer understanding of the ball park of the squares since these were the same as the squares single digit numbers in 100s.

Guessing the result
This brought us to the most interesting part of the book that talked about how you could guess the square root of a perfect square (of a 2 digit number). It talked about getting the 10s place of the number. They could see this from the program the optimized code a number like 2209 needed to be in the 40s as 40^2=1600 and 50^2=2500. It was then a matter of which of the 40s gives the result. 

This brought us to the ones/units place and the fact that there is a clear mapping (two to one at times) between the units of the number and the units of its square, e.g. 2/8 have 4 in the units place, 3/7 have 9, 4/6 have 6, 5 is just 5 and 1/9 have 1. 2,3,7,8 do not come in the units place of a perfect square of an integer. 

Looking at 2209, it could have a square of either 43 or 47. The next choice is determined by whether it is closer to 40^2 (1600) or 50^2 (2500).

Permeating
The fun with being able to guess something marginally larger than they thought they were originally capable of was that they showed what they could do to the 6th graders who were promptly curious on how this was all done and learnt it too.

This gave them endless fun of resolving the solution and pitting themselves against their program and getting the solution before it did.

The finch robot : Programming (1)

Programming with Scratch has been getting more interesting with interacting with the real world with the makey-makey that was able to let the computer receive signals from bananas and vegetables. Gaurav's gift of a Finch robot gave a way for children to control something that happens in the real world.

The setup on Ubuntu 14.04/Scratch 2.0 worked with the given guide. 

I explained and demonstrated the sensors, played with its nose light and then we got down to making the finch move. In the initial exercise I was trying to get the children to move the finch around in a certain order, but what made it interesting was to try to move a crumbled paper ball around and put it in a basket without the Finch falling off.

The finch has two wheels that are controlled independently with (two) motors. The only input possible is a speed from -100 (backward) to +100 (forward). In order to rotate an object, for example, you need to move one wheel faster than the other. Also, since we had not figured a direct relation between the speed and a specific distance there was some guess work involved in the time that it needs to be done for.

The phases involved in solving the problem involved in drawing out what they wanted and what speeds in which wheels for how long will be involved in doing so. The wheels were a little wobbly and even when the two speeds were exactly the same the finch had a bit of a drift to one side. But, that just added to the challenge :).

I split the students in groups of 3 or 4 to allow for discussion and team work. The game was to be able to do it in the least number of attempts. I noticed that groups were not that keen on observing what another group was doing as they wanted to only follow their own program to avoid confusion. But, since my purpose was for them to start predicting the outcome I provided an incentive of an extra attempt if a group predicted the outcome accurately. This got the groups interested in examining each others work as well.

After a couple of attempts the children started to really get into the idea of predicting what a set of instructions is going to do and how to go about solving the puzzle. The reality of a paper crushed into a ball is that it didn't always move with the finch and kept things interesting. Its interesting that while children keep track of the number of tries, etc none of this really matters and the first time a ball falls into a box the children want to do the next puzzle.

The ideas of rotation, positive negative numbers, speed, timing were reiterated. In addition problem solving by prediction, verification and correcting what they were doing and the idea of learning from what happens rather than it just being right/wrong was an aspect looked into.

Makey, Makey...(1)

At Isai Ambalam school we had set up the computer center and this year I was looking at how to best utilize this for the older children (6th and 8th graders) at the school. We have been working with Scratch programming for math concepts and more recently for creating games with English.

Regine is volunteering with us at the school and observed some of the classes got a very interesting gadget called the makey, makey from her friends to see what we could do with it in the school.

The idea of makey, makey is so simple that it adds to elegance of thought to come up with it. It notices if a circuit is complete and sends a signal to the computer that it knows well - the space bar, left, right, up, down arrow keys and a mouse click. Not very interesting in itself, but the subtle part is that it can detect even a large resistance closing the circuit. This makes it possible for it to detect a human body, a banana, a leaf, etc to close the circuit and lets it interact with objects from the outside world not associated with the computer.

The picture above is me playing a drum and a guitar string with each banana using Scratch.

Technically you need to hold one wire in your hand and play with the other hand. 
But, in our class the grounding of the computer room was quite good and since we leave our footwear out we were completing the circuit by simply letting our feet touch the ground. this meant that anyone could just touch the banana and get a space registered which meant in scratch could make a drum beat.

Of course the children were not convinced that it was the ground completing the loop and they pulled up their feet sitting on the plastic chairs and checked. Luckly the internet had been shut down due to heavy rains and we have to open scratch and create our own programs, choice of music and notes.
Once the kids were comfortable with what they were doing they asked the other teachers to come and try out the musical instruments.

Now, we will build on it and see what else is possible. One child has promised to work on a burglary alarm, another is making a full piano, lets see.

Seeds...

With the 6th grade at Isai Ambalam we have did some experiments with measuring the pendulum as an application of decimal numbers (and division by 10). I use my watch as a stop watch for the measurements and kids were having a little bit of fun trying to find out the shortest time they can start and stop a watch. We were talking about time and measurement when Saj asked how often I change the battery in my watch. 
I mentioned that I hadn't changed the battery since I bought the watch so it should easily last a year. Saj immediately got back with the fact that I have been around for 1-1/2 yrs and he has always seen me with this watch. Observation (check).
Then he says, wow, that watch must have "very low Watt". Wait where did this come from?
Vij who is new to the class enquires what is Watt. Saj replies that we did it last year, its the energy that the watch takes. Ani corrects him that its the power. Saj concurs and reminds himself that it is indeed the power. He goes on to tell Vij that the light consumes 40W and the fan 70W of power and its only when it is multiplied by time that it becomes energy, actually Wh. Recall (check), application (check).

Last year I had asked the 5th graders to go around the school and look at the number of lights and fans in each class and estimate the power and look at the time it was on for in a day to estimate the energy (as application of multiplication). 

I said the power is very, very low and you would need to split a Watt many times. Oh said Saj, does this mean it will be a decimal point followed by many zeros? Decimals (check).

I actively look to connect concepts that we have done in the past, but it was interesting to see the seeds that were planted a while back come forward as a regular conversation without any planning.

Government teaching training workshop

Isai Ambalam School has been partnering with the government schools for many years now. Their partnership has been useful for the school in spreading what they learnt and to review any training material that is received by the government schools. They are also informed of any ongoing teacher training. Last month there was a government school teacher training for 6-8th graders math teacher. These are the grades that I have been working with for the last year or so. Subash suggested that I attend the workshop and (if opportunity presented itself) train the teachers. At that time I wasn't very convinced exactly how an opportunity would present itself. I skipped the first day of the three day workshop. But, Kavitha attended it and built a rapport with the coordinating trainers who thought it would be interesting to have me there the next day.

I took along all the TLM that we had used, pizza party, dienes blocks a mini weighing balance and also some of the computer related stuff - geogebra and scratch programs made by children.

The teacher training happening was interesting, on the one hand teachers were told that they need to connect what they teach with everyday life of children most of the time was spent on providing teachers tit-bits of trivia that would help them make the classes interesting e.g. different antiquated units of measurements used in India for land measurement, their conversions (beyond cent, acre and hectare), the number of years it took to build certain temples and some puzzles - two numbers multiply to give a third number and all the digits need to be used only once. Finally, we got to using 10- 5 to get certain numbers and there were many possibilities and I started having fun :).

The teachers were curious as to who this new kid on the block was and I got my turn. I gave then a background of the work with low cost materials as well as the computer based stuff that we had done in the schools and let them choose what they wanted me to talk about. The teachers wanted to see it all. In the first day, for the rest of the morning, I presented the work of the children (on scratch), the tools we used and how we can creatively combine the tools e.g. the denise blocks along with a weighing balance. I also talked about how many of the areas are connected to each other and that they could be linked to each other when they are introduced. They asked for another session the second day to concentrate on areas that they had difficulty with primarily fractions, place values, decimals and algebra.

We talked about fractions can be introduced with pizza party games and addition of at least a set of 10 fractions can be explored with this without formally getting into it. They progressing through equivalent fractions and only then moving to LCM. I also showed some of the work the children had done in explaining how fractions can and cannot be added. The teachers seem to find the approach different and interesting.

The teacher were quite surprised with the work of the children on scratch and while this was appreciated and four teachers even copied the software and the work of the children to view it for themselves at home. They made it amply clear, that they did not consider it possible to take this to their children even though they had a computer lab and an instructor.

The teachers showed most interest with use of materials like the denise blocks, in volume measurements (cc = ml, how much is 1L in number of cubes), algebra and a mini weighing balance. The idea of introducing place value by weighing bunch of blocks on one side and a set of tens and ones on the other to show that the decimal system is more convenient (for humans).

However, when it came to using a program like Geogebra some of the teachers were up in arms. Why would we teach four digit multiplication if it can be done easily with a computer? Good question, why do we teach children four-digit number multiplications? These discussions helped me get deeper into the purpose of math as interpreted from the NCF 2005 document. If 4 digit multiplication is taught it is to help the child's procedural mind. We also talked about the idea of an approximate solution and a feel for the numbers that children do lack which would certainly be worth working on.
Of course, its also a tool that can help in developing an intuition into something that would take them much longer to do e.g. discovering the relationship between the radius and the area of a circle, or a series of lines like x+y=constant.

Given the apparent confusion of children between fractions and percentages/decimal I walked them through the method of looking at the denominator to estimate 50% (1/2 of the denominator), 10% (.1 of denominator) and 1% (.01 of denominator) to compare and build up the numerator. But, by an large the teachers had difficulty in grasping it and told me that even though they had difficulty and children are unable to get a handle on the sense of a fraction it was out of syllabus :). Ah well...

It was nice to see that the teachers were able to notice what the children had difficulty with and when they felt that someone could help them help their children they were quite involved. Teacher training is a mandatory program and teachers get back by taking 1 hr tea breaks, but the same teachers moved their lunch by almost 45 mins to accommodate my session which I much appreciated.

Probability for 6th graders?

As an introduction to fractions the children were playing the pizza party game. They read the rules and the first game they played helps them get accustomed to the pieces (1/2, 1/3, 1/4, 1/6, 1/8). There are fraction sheets corresponding to each of the fractions as shown below.

The children throw a dice with these fractions on it, they pick up the fraction sheet that faces up. They then try to fill the sheet by waiting for their fraction to come up in their turn.

The game is of course terribly unfair for the children who get smaller fractions (and yet children do enjoy playing it), but I wondered if we could use this as an opportunity to see probability in action. I asked each child to write down the fraction (not person) who won the game in their group. Once the game was played a few times I made a tally of how frequently a fraction won.


As each student read our her/his findings it soon became clear that most of the time 1/2 was winning, once in a while 1/3 won and very infrequently 1/4 won. With these rules 1/6 and 1/8 did not win any games played. 

We had a conversation of whether the rules were fair to each fraction. Then we moved to the question of why it is not fair. It was nice that some of the children were able to think this through. The first child who got it said that its because 1/8 fraction needs 8 pieces and you need to get 1/8 8 times, vs 1/2 where you only need to get 1/2 2 times. Getting something two times is 'easier' than getting something 8 times. In a few minutes most children were also giving their explanations along the same lines.

I proposed a different set of rules for the next game to try to help the 1/6 and 1/8 (and get them comfortable with equivalent fractions). If you get any piece that lines up with the lines on your sheet you can take that many pieces i.e. on a 1/8 fraction sheet, if you roll 1/2 you can take 4 pieces, if you roll 1/4 you can take 2 pieces and if you roll 1/8 you can take one piece. But, if you roll 1/3 or 1/6 you need to pass. I asked if this set of rules would even the odds...the children were unsure so we went for a few rounds of the game with the new rules.

We had just enough time to come together and tally the results. 1/8 followed by 1/6 were the most common winners, 1/4 was next followed by 1/2. 1/3 was the least common fraction to finish first.

I asked them to think up game rules that would both be interesting and fair to the fractions. A couple of days later, they came up with some games. Some of the games  proposed were repeats (apparently arrived at independently). I asked the children to rate the games in two parameters, their interest in playing the game and if they thought the game was fair. 

Game 1: Like the first game, but you can take a piece of your fraction even if you get a nearby fraction e.g. 1/6 for 1/8. 
Q & A: What about 1/3 & 1/2? Yes you can take these pieces as well.
Interesting - 13, Fair - 1

Game 2: Have all pizza common, you place the fraction you get. The person who finishes a pizza first wins.
Interesting - 11, Fair - 8

Game 3: Like first game, but if you get a fraction you can't use, you give it to another person who has that card.
Interesting - 11, Fair - 11

Game 4: You just need to make a full pizza based on whatever your dice gives.
Interesting - 14, Fair - 10

Game 5: You can give and take pieces from others. 
Q & A: On what basis? Not clear, it needs to be figured out. (Calvin Ball!)
Interesting - 16, Fair - 6

5 children were not participating in the fair/not fair question as they were in doubt, but a large number of the remaining were able to guess that 1 was unfair; 2 was fair (to people); got tricked in 3 because it seemed that you were being nice (so it must be fair!); that 4 is fair. I am unclear about the rules of 5, clearly the children found the lack of clarity appealing :), but were less sure of its fairness!

Need to update this blog when we play the games again.

Guided learning: Using computers

One of the fun aspects of learning while having access to a computer center is the exploration style of learning that is possible accurately with Geogebra on the computer. To learn about shapes like triangles, angles, circles, etc through measurements. Looking or patterns and making observations and deriving some sort of a generalization made for an excellent exercise.

Here are some of the things that worked well us:
1) Using geogebra as a drawing tool in practical geometry to draw various equilateral triangles. A line segment is drawn with A as center passing through B. Then a circle is drawn with center of A passing through B and then a circle with center at B drawn through A. The meeting point of the two circles is equidistant from both A and B for the same lenght thus giving the equilateral triangle. The children can then measure the angles and distances. As they create and measure many more triangles using different lengths for a side they realized that all equilateral triangles have the same internal angle of 60 degrees.

2) A triangle with three sides can be drawn with a line of one of the lengths and then circles with radius of the other two sides. The point of intersection of the circles gives the third point of the triangle.
Picking up three random sides of a triangle in a class brings up the triangle inequality, when children are unable to find a point of intersection and naturally moves one to the realization that two sides have to be greater than a third to make a triangle. I encourage the children to try using the first side as any of the other sides and see what happens e.g. 1 cm, 2 cm, 4 cm has the following three ways of looking at it. Interestingly, I only thought of the inequality as the third figure, but some of the children seemed to find one of the other two more sensible.

3) Some other observations about triangles  also came about by this process

- Isosceles triangle has two angles the same

- Sum of the angles was always 180' no matter what the triangle

- The largest side is opposite the largest angle.

Then we did some games using a combination of these e.g.  would the central angle of a triangle 4, 5, 5 be less or more than 60'.

The children were starting to get a feel that sides and angles are not independent quantities.

4) Circles - circumference in relation to circle diameter, area relation to the area of a square with one side as radius. These were fun exercises that I had attempted by getting the kids to do physical measurements last year. It took quite some time and due to measurement inaccuracies (especially when measuring small objects like ear rings) could throw the kids off. The ratio of pi (that they presently know as 3.14) came out like magic as geogebra could be used to measure these quantities of interest no matter how small or how large the circle was. I skipped the calculations at this point and stuck to creating a spreadsheet of the circles that children drew.

The children were each able to document 10 circles or more that we were unable to last year. The number has really stuck with them as an assessment recently (with circumference given and radius/diameter to be calculated) showed with over 75% of the class guessing these right.

5) Algebra

The notion of what lines like x+y=10, x+y=20 look like coupled with stories like you and I share 10 chocolates, if I get one more you should get one less (negative slope) had done. Again an assessment later indicated that most children were able to get this.

An interesting detour...

I usually go over addition of fractions with equivalent fractions. The idea being that you can only add fractions when the pieces of the pizza (so to speak) are the same size...

The children had worked with this for three days and learnt to notice that when the denominators are the same they could use that fraction to replace the original fraction. In order to reiterate that doing equivalent fractions was to get pieces of the same size and look at the number of pieces we would have for that size I started comparing fractions using a similar method i.e. do equivalent fractions till the denominators are the same (same size of pieces) and then compare the numerators (number of pieces).

We did a few examples in class and I asked the class to write fractions of their own and do a similar comparison at home. Homework is always interesting, it gives you a look into how children are processing the material and often the kind of short cuts they have thought of. The children came back with fractions that they knew how to compare. Other than the pieces they had learnt well about by playing fraction games (1/2, 1/3, 1/4, 1/6, 1/8). None of them had used the equivalent fractions. We started looking at the kind of fractions that are easy to compare and don't need an equivalent fraction.

1) Fractions with the same denominator (obvious)
2) Fractions with the same numerator
This started with fractions with a numerator of 1 and it was nice that the children were able to connect with division stories and notice that if the denominator was larger than the size of the piece was smaller and the larger fraction was the one with a smaller denominator!
This was then generalized to other numerators as long as they were the same. 
3) I decided to push the envelope and introduce another rule to get the kids to think about it. How about comparing 4/5 and 5/6 and coming up with a logic for that.
I spelled it out each of them is a pizza with one piece missing, the first one has a bigger piece missing (1/5) the second a smaller one (1/6). What does that mean?
Two kids in class got it immediately, I use peer learning significantly in class and I asked the kids who got it to reiterate what they understood, after a couple of them said what they understood (even if the words they use are the same as what I used, more kids get it each time!). A good part of the class had got it and I asked children to practice 10 comparisons with each of the rules.

An alternative I use for 'did you do your homework' is asking 'if anyone wants to show me their homework'. This time when the kids got back, I asked the kids what fraction pair they still found difficult to compare. The pair they put on the board gave an indication of how much they were able to assimilate. Each student said they still had a pair that they found difficult to compare. One by one they put 17 fractions on the board. Thankfully, there were no fractions of the 1st and 2nd kind. 

I asked the class if they could find a fraction pair on the board that they did know how to compare. If they could they would come and confirm with me that their logic was ok and then explain it to the class. There were some of the third rule, then things got interesting.
- They were able to extend the third category with fractions like 6/8 and 13/15. Both are missing two pieces, but the second is a smaller piece. Neat now they were thinking are there others that can be compared like that? There were 5 of this kind.
- The next set that the children noticed noticed where the fraction numbers where the denominator was large, but the numerator was smaller i.e. The piece is smaller and there are less of them 3/25 vs 2/27.
- The last was special, one child pointed out that one of the fractions was less than half and the other more than half. 17/42 he pointed out was less than 21/42 which was half. The other fraction 9/16 was greater than 8/16 which was half. So 9/16 was greater than 17/42 even though there were more pieces of a smaller kind.

With these rules 15 of the 17 fractions on the board were knocked off. There were only two left these were pretty close to call with and the children who could compare them had done so using equivalent fractions.

Then we moved on to sorting...it was nice to have started from something that I was required to do - teach addition of fractions  and move to something that I wanted to do, a feel for fractions, creating abstract rules from real situations and being able to think creatively.

On confusions and clarity...

Its interesting how small things can confuse or clarify things for children. At times the children dwell in their confusion and come out with clarity and growth. At other times they reach the end of their patience and look for adult intervention. I often relying on giving time, peer learning, but sometimes peers are unable to fix the problems because they don't have the same problem and can't find the patience to understand the problem. Here something that happened this week at Udavi (6th Grade) and Isai Ambalam (5th Grade)...

We had been working with the weighing balance to apply addition, subtraction, multiplication and division with using things that could be put into pencil boxes. As I was weighing some working with different blocks I noticed that the hand made blocks were not all the same and for a change we went into a puzzle to identify a heavy block among three blocks. In time, I talked about the use of math to document what we learn and create processes. We numbered the blocks so we could distinguish between them and know what you did with them. When the children were comfortable having observed the process with real blocks we thought of ways of writing it out and came up with the following flowchart.

This chart confused quite a few kids. I asked what they had not understood and if they would like me to repeat with the blocks. The children said that they understood the physical measurements. I spent some time revisiting >, < signs and a couple of children did have difficulty with that, but for most that was not the problem. They just felt they didn't understand and something was off. I stared at the picture myself and then some clarity dawned and I drew this one...

Logically it was the same flowchart. Only it had when one is used on the left and when 2 was used on the right with the corresponding >, < signs reversed. This clarified the issue for almost all children instantly. A couple of children go it once they were able to discuss it with their classmates who had now understood.

It was interesting that when the children were working on finding the lighter block they actually switched the signs again to get 1 continuing on the left and 2 on the right. 

Its also interesting that as they started dwelling into more complicated problems of not knowing if the ball was lighter or heavier and with 4 balls they got quite good at the symbols and were able to play around with any order.

I took the same problem with the 5th graders in Isai Amlalam. Having learnt from my experience I did not want them to fall into the same trap and once they were comfortable with what physically happens with 3 balls I went to the pic that I found was flowing in class. 

Apparently the pic didn't make sense to some kids. I went over the signs >, < but still there were 4-5 kids who looked puzzled and few others who seemed to have got it were also a little uncomfortable applying it to another puzzle. Finally, one child articulated that he didn't understand how 2 could be greater than 1. I told him that its only a name to keep track of the balls, but then I took the names of three of the kids in the first row and used these for the name of the balls. I wrote their initials V, D, A and this put the children at ease to work on these puzzles.