Introducing Place Value System

How do you introduce place value system to a child for the first time?

What is place value system?

Why is it only 1, 10s, 100s, 1000s and not any other grouping?

How do you meaningfully convey the concept and its usage? Why do we use it, in the first place?

From the wiki: https://en.wikipedia.org/wiki/Positional_notation

Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations (such as Roman numerals) for its use of the same symbol for the different orders of magnitude (for example, the "ones place", "tens place", "hundreds place"). This greatly simplified arithmetic, leading to the rapid spread of the notation across the world.

With the use of a radix point (decimal point in base-10), the notation can be extended to include fractions and the numeric expansions of real numbers.

The Babylonian numeral system, base-60, was the first positional system developed, and its influence is present today in the way time and angles are counted in tallies related to 60, like 60 minutes in an hour, 360 degrees in a circle. The Hindu–Arabic numeral system, base-10, is the most commonly used system in the world today for most calculations. The binary numeral system, base-2, is straightforwardly implemented in digital electronic circuitry and used by almost all computer systems and electronics for calculations and representations.

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Much earlier to even talking about the place value system, I saw myself answering to DD1's questions like 'How do we write 42?' 'How do we write 67?' etc as just telling out the number 4-2 and 6-7 out. This was the time when I was thinking of how should the child be introduced to number system so that she herself will know how to write numbers with more than 1 digit?

I could recall my childhood at school, where all of us used to recite hundreds of times the following:

1-1  Eleven

1-2 Twelve

1-3 Thirteen

2-1 Twenty One

2-2 Twenty Two

2-3 Twenty Three 

and so on

Somewhere inside my mind, I knew this kind of recitation and rote learning about what the numbers actually signify was going against the philosophy of self-discovery. 

And, I didn't think much about it then. 

Recently, we took up the study of Place Value System. Let's see what is needed from a perspective of child's ability to understand the PVS.

1) Counting/writing 0-10

2) Grouping objects in 10s

3) Counting 10, 20, 30, 40 objects

4) Simple Addition/Subtraction

Let's talk more about each of the points:

1) Counting/Writing 0 - 10 

Knowing the names to numbers and ability to write 0 to 10 is the minimum prerequisite and for this, there are various tools that can be used to introduce writing to child. Tracing in sand, flour, rawa, on sandpaper and last comes pen on paper.

2) Grouping objects

Grouping can be introduced with any number of objects. You could use sticks, toys, seeds, stones as tools for this. This can be taken up as a fun game and count the number of groups of each number in the end. Some examples:

1. Make groups of 2
2. Make groups of 4
3. Make groups of 7
4. Make groups of 10
5. Count number of each of the groups
6. Just writing down the count 

3) Counting 10, 20, 30, 40 objects

4) Familiarity of where we write what

This is, in general, knowing that we read/write from left to right.

Now, what is the main aim of using the place value system? Why are we even introducing it to children?

PVS was introduced to make counting easier. To have a system in place of how objects can be represented in numbers. Remember all this while, the child did not know the way to write double digit numbers. So, this, in a way, is introducing the concept of representing objects, in number,  whose count is more than 10.

Before even starting to introducing PVS, the child has to be introduced to lot of fun games like, grouping, counting, simple addition/subtraction and the like.

The following is the way I did with DD1:

We had a box full of seeds beside us.

I asked DD1 to pick like a handful of seeds and keep them in another box 2.

Now, let's count the seeds in box 2.

DD1 began counting and reached the number Thirty Six.

I wrote Thirty Six in words and said, now, let us see how we can write thirty six using numbers. 

Group the seeds in 10s, each group placed in a small cup.

Group the remaining seeds in groups one 1, i.e., placed each one separately.

Place the cups containing groups of 10 in the square shown below.

Place the individual seeds (groups of 1) in the corresponding square below.

The first time we did this, we wrote and drew using up the entire floor around us. Couldn't click pics as I got too much involved and forgot about clicking pictures. 😇😇😇

10 (groups of 10)	1 (groups of 1)

Next count the number of groups (or cups) present in the square corresponding to 10. ---> 3

count the number of seeds present in the square corresponding to 1 ----> 6

3

6

This is the number that represents the total quantity of seeds, that was earlier counted to be Thirty Six. In other words, a quantity of Thirty Six is represented as 3-6, that is, 36.

Following this, lot of other maths could also be done, like:

1. Counting the total seeds that were grouped as 10 = 30
2. Counting seeds in each cup = 10
3. Counting the number of cups = 3
4. Understanding that 10 + 10 + 10 = 30 
5. Understanding that 10 * 3 = 30
6. Understanding that when you add 10 three times, you get 30

DD1 just loved to do this activity and wanted to play another time. 

After couple of times, you can just sit back and watch as your child explains to you the concept. She becomes the teacher and you the student!!

Is teaching mandatory?

Every child naturally starts learning from his/her environment. Starting from birth, any child learns to roll over, sit, crawl, stand, walk, learns the primary language of his home, learns to do most of the house hold activities by just observing how his mother/father is doing. Every parent would definitely experience at least once in their child's young age,  that AHA moment, where, they realise that their child has picked up something even without being taught about it specifically. True?

Yes. Learning happens naturally. Where there is intentional living, heartful living, learning is bound to happen naturally.

As with most of the things that the child has managed to self-learn, the child will continue doing so with very little effort from the parent(just facilitate or guide) if the environment is set so. But, starting from when ABCs, 123s and other so called 'subject-related' matter is introduced at school, the parents and teachers, both begin to feel the pressure of whether the child would be doing whatever was supposed to be done. Every adult has her own targets to meet. But, unfortunately, this kind of set-up doesn't work for a child to learn. A child learns best naturally, when they are set free, when they are not bound by rules and restriction, when they are not confined to a specific place, specific time to do certain things and many other factors.

Recently, I heard a 4 year old saying to her friend 'Let me teach you, this is how we do it, and then you can do it that way'. Why should children be 'taught' only in a particular way. When you use the word teach, it is kind of, one tells and other follows. When both child and parent are involved in the process of learning together, the environment is totally changed. The teaching that happened in a monologue accompanied with frustrated replies at the umpteen number of questions that children ask is replaced with conversational set-up where parents learn how to actually learn and the child follows simply the intuition while learning.

To quote an example , DD1 loves to take the lead, form her own questions and solve it or pose question to me for me to answer. It had always happened that way. Once I ask her, 'we are on the third floor, we got down one floor, which floor did we reach now? (we were actually getting down stairs)' and she answers it, I tell her to question me for which she frames the question and knows its answer.

Children love to take the lead. Whether, ample opportunities are provided either at school or home is upto the teacher/parent, which usually does not happen, and cannot happen, because each one has their own targets to meet. The teacher has to finish a lesson in prescribed time, mother has to get the homework done in limited time. Where is the time left for exploration, questioning, learning ?

The one who explains(teaches) a concept understands better than the one who merely listens to what is being told(taught). The one who poses questions is in a better state of understanding than the one who merely listens to answers. For this kind of exploration to happen, children should be given ample time. Rushing things will not lead to meaningful living. Children should be provided with ample opportunities where they are encouraged to question, to learn by self, to be self-motivated to do something.

Mentioning another example, DD1 usually counts adds, subtracts quite quickly. In fact, every child is capable of this if the right environment is provided. By a mere look at dot cards or any objects, DD1 tells out the number. While playing, she plays with toys, sticks, cups, spoons, whatever comes in her sight, she keeps some of them in each room saying, one room is school, one outdoor play, one home and the like. Then, she quickly says, 'there were 5, one is in p room, I had 4, I put 1 in q room, now i have 3. Or, she would count the people at home at any moment and then say, we are 6 right, to which, I talk in my mind 'we are actually 3 here, should I say or just listen to what she says?', and boom she talks fast 'thatha at p place, mamma at q place, naanna at r place, we 3 here at home, so total we are 6, 3 at home and 3 outside home'. My face brims with a huge smile.

If you observe, a child has very powerful intuition and all their learning is intuitive, right from the moment of their birth. The same addition/subtraction when introduced in school, would begin with 'count 4 on your fingers, add 1, count all'. The intuitive learning is clearly replaced with the rote learning.

The joy of learning is lost when living is rushed. 

Basic Arithmetic

Be it writing, drawing, tracing, or any other - I have always tried to introduce concepts with playful things rather than using paper, pen.

Let's see how we can introduce the concept of addition and subtraction to a toddler.

I first wrote '2 + 1 = ' and asked DD1 to place the corresponding number of seeds under each number and then add the total number of seeds. When doing that way, in case of subtraction (p-q), she could not place seeds under the q, rather, she had to remove seeds from p. This might cause slight confusion, hence, changed our way of doing it.

Then we drew a big box where she keeps putting/removing seeds accordingly and then just count the total number of seeds present in the box at the end.

I started with just adding/subtracting two numbers and later moved to three.

Not too long into it, DD1 wanted to form the question herself and solve it in her way.

In the process, DD1 happened to write '2+1-3', for which, she put 2 seeds, put 1 seed, removed 3 seeds and realised there were 0 seeds at the end. She usually exclaims 'Zero MEANS empty!!!!'


All this while, DD2 was happily picking seeds from one box and dropping into another, and even, picking the seeds from floor and dropping into a box.

Where is the BOX?

There's a new colouring book at home. With pictures of the completely coloured and the outline(to be coloured) by its side. Personally, I have never developed any liking towards such books. Because, if the end product is shown to a child and it mandates to be coloured in the same manner as shown, there is no creativity in the activity, we are not welcoming new thoughts from the child. 

I never insisted DD1 to colour the picture as shown, nor, did she colour it the same way as mentioned in any of her colouring books. 

As I put DD2 to sleep and stepped out of the room, I noticed DD1 happily colouring in the book that she just got from her school. I just watched her for some time. She started telling about what she was colouring, saying, there is a tree here and I'm colouring it black , here there is a building and I'm coloring it brown, there's another tree here, I am yet to colour it, I will be colouring it green. 

Me: Oh, so this tree has black leaves? 

DD1: Yes, because those leaves are all getting burnt, they just turned black .

Me: Is it?

DD1: Yes, there was a fire in the forest and the tree got burnt. There was absolutely no air at all. Hence, it got burnt and turned black.

Me: Oh! Fire? What about this tree? Does it have green or black leaves?

DD1: This tree will have green leaves. Because, this did not get burnt. This tree is getting lot of air, hence, it is not burnt. So, I will colour it green. 

For us, the parents, it was just a black and white picture, which had to be coloured the routine way. But, for a child, who is left free to her imagination, who is not bound by rules and instructions, who is not dictated about what has to be done and how, the same black and white picture opens up as a creativity play ground. 

Entire childhood is pressed upon to follow rules and instructions, to do as the teacher/parent says, to learn as told, to not question the existence, and, as a developing teen/adult, the person is expected to think OUT OF THE BOX. Where was this box during the childhood? Who drew the box upon a child's thinking? Is it not the teacher/parent who, in the first place, drew a box around the child's mind as to what she has to think and what/how/when she has to do certain things? Did the child ask for the box? Can a child who is set free, who is not bogged down with rules and instructions draw a box around her mind? NO. 

If you wish your child to think out of the box, then, in the first place, do not draw the box for her/him.

Thinking is an individual experience, learning is an individual experience. Set the child free and see for yourself, the beauty that blossoms as days progress. 

Related articles: Dear Creative Child and What is Creativity

Why do wet things stick?

3. 5 yr old asks:

The pant becomes sticky, tight and difficult to pull out when it is wet. Why does the pant stick tight to skin when it is wet?

On another instance,

DD1 measures the rice, washes, pours water and places the vessel in cooker - all of which take a total 15 minutes to complete.

She first runs her fingers in the rice can. She just loves scooping some rice, emptying it back in the can and repeat.

She then washes the rice with water. I can't even count how many times she dips her fingers into the rice+water mixture and takes it out. She keeps doing it again and again. She presses all her fingers down the vessel, takes them out, on to which she observes, there is some rice stuck. She repeats. Now, she presses just one finger, takes it out, to which there is no rice, but drops of water still dripping down the finger. So, water was stuck and then falling off the finger.

She asked me, why is the rice sticking to my hands?

Me: Isn't it? Even I see it. Why do you think it is sticking?

DD1: Because it is wet?

Me: How was the dry rice? Did it also stick to your hand?

DD1: No

Me: I think, only that which is wet seems to stick, right?

DD1: Yes

Me: So, why is it sticking? Why is something that is wet sticking?

DD1: I don't know. You say?

Me: I too don't know. Let's both together think about it?

DD1: OK.

Me: So, how can we know about it?

DD1: I'm done with washing, let's put the vessel in cooker.

The conversation has paused, definitely, not ended. I am sure she will come back to it as her mind is developing to constantly understand the world around in a better way.

Learning is a journey. Learning cannot be confined to school, college, university. Learning is a continuous process that happens all through our life.

Peek-a-boo

This a play where there is a hidden image and it is revealed in 3-4 steps. I drew images of just animals/characters like mickey, cat etc.

 

Fold a piece of paper step wise. You could make 2-3 folds. Draw the image. Now open the paper completely and show it to your child asking him to guess the character, fold one part and throw some interesting questions like, where are the eyes, where is the face etc? Then fold the second part until the image is fully shown. This would make an interesting play for the toddlers.

Another idea is to draw an image on the inside (as well as outside) of an empty crayon box/match box. Asking your child what surprise was waiting in the box, you could slowly slide open the box to show the image.

Hidden Treasures in Ice

Just make ice with some tiny things included in the ice tray. Once set, empty all the cubes in a bowl and see how happily your child plays with them.

Below pictures are from two different days:

Pass it down - 2

This one  was a very interesting activity that DD1 did a couple of years back. And, it is time for me to redo now, for both DD1 and DD2.

This time, DD2 has suggested ways of how to stick the boxes on wall and at what angles so that the ball drops freely.



Reference link for the one that DD1 did: IS THIS

Making Gulab Jamun

Every moment in life gives an opportunity to learn. Let your child accompany you in your tasks and give them chances to learn on their own. 

DD1 insisted we make gulab jamun for the evening, She rolled the jamuns while I fried them. She was on my side all through the task. Saw how sugar syrup is made, how we deep fry the jamun, and how we soak them in the syrup.