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.

Weighing in...math in a pencil box

A fair number of children who have difficulty with algebra have difficulty much earlier with understanding subtraction and division, two areas that we had covered through stories in class. The children felt that they had practiced and were confident that they can create stories, interpret my stories into numbers. I decided to use an assessment that would require application of these concepts in real life using a weighing balance.

I started with one of their pencil boxes, passed it around and asked them to guess its mass. Once they had all recorded their guesses in their notebook we measured it (52 g). I then took 10 pens and put in inside the pencil box and repeated the process. I asked them to guess the mass of one pen. Then scale it with 10 pens. They came up with 90g-100g as their guesses. We then made the measurement (102 g) and I asked for the weight of one pen. As with the previous process the children started guessing. I told them they had already guessed and this time I would like a correct answer. The children could not walk back the path they had come and correct for the measurement they came up with interesting answers including some that were over 10 g. I tried to help them by walking through how I would do it with what the mass of 10 pens would be. I was intrigued that some children were confused by this process and some even managed to add the two measurements (152g). On digging deeper I realized that these children could only think of subtraction when a clear word 'remove' was part of the construction.
I had to move from 
Pencil box mass is 52g. Pencil box with 10 pens mass is 102g. What is the mass of 10 pens 
TO
Pencil box with 10 pens mass is 102g. To find 10 pens mass I need to 'remove' the mass of the  pencil box, which is 52g. 

To open up the puzzle  (well it had become one) for children's participation, I asked the children to each list 10 kinds of items that they could put in the pencil box. With time I added clarifications that the items needed to be something they could find copies of and for a couple of children needed to reiterate that 'kinds' was different kinds of items and 10 pens didn't work. We then measured one of the item they had in mind and the children made their own stories with pencil-box+some number of objects * mass of one object that they had measured. This process involving addition and multiplication that all children find comforting. As they exchanged their stories and solved them they did get practice in handling what they had initially felt a very different problem.


Its actually interesting that I tried these out in different grades and how the numbers change appropriately. For children in 5th grade the number came out as a round number, for the children in 6th grade the objects we picked up had a touch of a fraction and when we did the same exercise in the 7th grades the weights gave some neat decimal numbers. 

Some activities

Here are some activities I have used in class either for refreshing the class, introducing some fun or doing a centering or calming activity.

Big fish, small fish

Just a quick activity to freshen the minds is to hold hands close and call it big fish, and far apart and call it small fish. Using this small twist you go fast and slow emphasize big and small at different times and keep going till the kids can keep up or realize that they made a few mistakes and stop.

1 to 20
Generally acts as a centering activity to let the group count from 1 to 20. The catch, if two people call out at the same time you start from the beginning. There can be no pre-decided format or calling order. You can lift your hand or give indicating that you are going to speak. If the kids start using a pattern you can just call out the number along with the kids and you begin again as a group.

Weighing balance

You can have unlimited fun with weighing balances. Starting with estimation and guesses, you can move to addition, subtraction, multiplication, division and then to algebra. You can also throw in puzzles of trying to find which ball is heavier/lighter in a given set.

Stetescope speaker

Measuing heartbeat can be a fun activity especially when coupled with some exercise like running. In addition getting a child (or adult) to hear heartbeat and act as a human loudspeaker calling out the lub-dubs quietens the entire class trying to count heartbeats.

Place value kits and tables

A neat activity we got going with place value kits (ones, tens and 100s) was to use a few of the blocks - 3x10 and 8 ones or so to come up with as many multiplication possibilities as possible (without repeats and without all in one row) with few or all of the blocks.

The activity really helped getting the idea of the area of a rectangle represents multiplication home for the children.