Post 4: The Shape of Numbers 1: Addition is commutative

An important idea in math is commutativity. Addition of numbers is commutative. In the simplest of terms, a + b = b + a. And though this may sound obvious, and a child may "know" it, it may not be so obvious as to why it is true. So, my son and I explored this idea a little. 

You pick a number A and you add it to number B. You get a number C. 

Now you do the reverse. 

You take the number B, and you add it to number A. You again get the number C. Why?

There could be several ways to understand it. (Also prove it formally). But intuitively, here is some of the stuff we did, as "geometric" proofs. 

First, we look at the "shape". Here is 4+3 on the first two rows, and 3+4 on the next 2 rows on an abacus. 

And we see that the "shape" of 4+3 is the same as the "shape" of 3+4, except that one is the flipped or mirrored or reflected version of the other. So, we see that the "shape" has not changed, it is the same shape, we have only somehow moved it, and therefore, 3 + 4 = 4 + 3.

And what if now we increase the number of numbers being added? Here is 1+2+3 and 3+2+1. Again, the "shape" is flipped, but since it is the same shape, we can say 1+2+3 = 3+2+1. 

Of course, this does not immediately show that 2+1+3 will also be the same "shape", so we need to explore it in another way too. 

What if we look at another shape way? Here are 7 rectangles forming a large rectangle. 

And now, you can cut up 3 rectangles leaving 4, or cut up 4 rectangles leaving 3, like this. 

And now it becomes clear: because to make the larger rectangle, you have to put together smaller numbers of rectangles, but the order of how you put them together does not matter, you can either put 3 first and then 4, or 4 first and then 3, both make the same "shaped" larger rectangle of 7 smaller ones. And this idea scales now to more numbers than two: using this idea of cutting up a large rectangle into several pieces, you can show that 1+3+3 = 3+3+1 = 3+1+3. And so on. 

Another similar "proof" is that you have a bucket that needs to be filled up by say a certain number of balls. Say the bucket can contain 10 balls altogether. Then, it does not matter whether you throw in 4 balls followed by 6 balls, or throw in 6 balls followed by 4 balls. The end result is that the bucket gets filled up. 

This property says that addition of numbers is commutative. 

And before we as adults start saying, "this is obvious", we should think up some counter-examples, where changing the order of operations does not lead to the same final result, and we find that most of our everyday processes are actually non-commutative.  

Eating your food and then digesting it is not the same as digesting the food and then eating it. In fact, it is not possible. In the same way, putting a ball in front of you and then kicking it, is not the same as kicking a ball and then putting it in front of you. However, when you are in Manhattan, going down 1 block east followed by 1 block north gets you to the same point as going 1 block north followed by 1 block east. This is commutative.

I asked my son to think up of one commutative example and one non-commutative example. He said, putting the engine on and then flying a plane is not the same as flying the plane and then putting the engine on, because you cannot fly a plane without first putting the engine on. Whereas, if you have a plane and a helicopter, putting on the engine of the plane followed by putting on the engine of the helicopter, is the same as putting on the engine of the helicopter and then the engine of the plane, because now you have both engines on.  

So, the important idea to grasp is that when things commute, the order does not matter. In whichever order you choose to do things, you reach the same end result. Whereas, in things that do not commute, the order matters. If you change the order of doing things, you get different end results. 

We had good fun discussing examples and counter examples of commutative behaviour: including this one that made us both laugh. He said, you know, at school yesterday, the computer had my name as Last name, First name, while I know my name is First name, Last name. And I asked him, so does your name commute? He said yes, because it is the same name! :) 

Post 3: 2 ones are 2, 2 twos are 4, ...

I grew up in India. We had in school what were called "mental math" tests. A teacher would rapidfire at us 50 questions in quick succession, and we, ready with our pencils and paper, would have to quickly jot down the answer, and whoever did the most correct answers got the highest marks. The questions went like this:

Teacher, reading out: 

19 4s are ... (2-3 seconds pause)

1000 - 564 (2-3 seconds pause)

Does november have 30 or 31 days, yes or no (2-3 seconds pause) ...

... and so on. 

So, obviously, the only way you could do well in these tests was practice, practice, practice. And my grandfather (blinded by cataract, he would hold his transistor radio, and walk up and down listening to the news all day) took responsibility. Every morning before school was what we called "Tables time": Times Table Time!

I would sit and sing away the tables, from 2 to 20, and then there would be the harder practice: how you could use the tables and place value to quickly figure out higher numbers multiplying each other without resort to pen and paper. 

I started school at 3 years of age (in those days there were no rules on being too young or old, there was a test, if you passed, they let you sit in class :)). And this game continued in simple form till about the time I was in Year 2, and then became a lot harder and more pleasurable, because the nature of my grandfather's questions changed to: "Is this number X prime, or not".

So, of course, armed with this experience, as a parent I introduced the times tables to my son when he was 3. He loves his (ever-growing) set of hot wheels cars, and I thought this is perfect. So, the 2s, 5s, and 10s happened quickly (little cars arranged in rows of 2s, then skip count verbally, then the same with 5s, and then 10s, maybe a post on this later with pictures). Worked like a charm. Before he went to pre-school, he was chiming away at questions like, if there are 3 wires and 2 birds on each wire, how many birds altogether? 

By the time he was starting formal school, kindergarten at 5 years of age, (the age for starting school in Australia), he knew his tables from 2 to 10, maybe not the 8 or 9, but if he did not "remember" he knew how to work it out, and he knew the importance of "remembering" the answer, when say we are walking on the road, mum asks a question, and I have to quickly work out the answer.

And later, he did not even need to memorize or learn by externally provided parental support, because he began to figure out the number patterns. For example, here are multiples of 11: I did not make him learn it, he was not counting by groups of 11, he simply figured out the pattern of progression, especially after 110, all by himself. He knows he is counting by groups of 11, but that is not the "algorithm" used, the algorithm emerged from number play and pattern recognition. And to be honest, I realize, even today, the delight of this. To date, I discover a new sequence that will make me go, WOW! (sequences and series post later). And my hope is that it all starts here, in elementary school math.

So, to me, maths is as much about discovery and play and conceptual understanding, as it is about practice and making memory stronger. Only when you dribble a ball for hours, you master a game. Only when you run everyday or go to gym everyday, do you master fitness. Only when you run your fingers again and again on the piano, do you master playing it. Only when you sing repetitive patterns of notes again and again (the term in Indian classical music is called "alankars"), do you master melody. Only when you put in practice, do you master anything. 

By the way, I completely agree, that every child will have their own pace, and practice without conceptual understanding is not a good idea at all. But, in working with my child, I found that the reverse is also true: conceptual understanding without practice will never take root and flower in the brain. And sometimes, like my post yesterday, pattern based understanding comes not only from learning the formal rule or concept or schema or prototype and its application (deduction), it comes from doing some pattern over and over again, and then discovering the general rule or schema from the examples (induction). In other words, deductive learning is one way, inductive the other, (there is a third, but maybe later on that), and mastering something comes from both the deductive AND the inductive ways. 

Finally, I found a couple of days ago, that this seems to be the case not only in math, but also in language (see specially his answer to the last question by the interviewer): Steven Pinker's Sense of Style. 

Post 2: The importance of being visual and verbal

When my son was quite young, about 3-4 years old, there was a lot of scribbling and drawing. There still is. The main story in this post is that: we never know when and how they are doing and learning very important stuff in all that scribbling. It may not be all scribbles to them even if that is what it appears as to us. 

So, here is the story. He had learnt to count up to 10 and 20 quite quite quite early. And by simply talking about things, I think that he somehow latently and implicitly knew two very important things in learning to count things: 

1. Numbers and their symbols are not the same thing: The symbol is representational. Meaning every language writes the symbol differently, but the particular number in question, the concept of 1 thing, 2 things, etc. persists. (So, for example, we have three languages in the house, but another post on that). 
2. Numbers are used to count things: so, the larger the number of things, somehow the bigger or more complex the representation of that number, and smaller the thing, simpler the representation. 

So, imagine my surprise, when at 3 or 4 years of age, I found him "drawing" the following (this picture is a much-later version, and hence much clearer and the handwriting much nicer, the original versions were on a little blackboard and chalk, when he pretended he was "teaching" mum): 

The thing is, he is 6 now, and he can work easily with tens, hundreds, and thousands. He does not yet formally know place value beyond that (in the sense that I have not yet worked with him beyond that or introduced anything formally, but you never know how they surprise you!), but in this drawing play, I learnt something very important: almost always, pattern based, partial understanding precedes formal and deeper understanding, for children as well as adults. And we should encourage this play. 

I also think he had captured the idea that numbers can grow as big as you want them to, and so his question on the previous post, came after a lot of this kind of play, both on paper and verbally. For example, he would say random and wrong numbers, as follows: 

Me: How many rotis is mumma going to make today? (Rotis are Indian bread that we make fresh everyday, so dough making time is the Indian version of kids doing play-doh in the kitchen). 

He: 5 hundreds and 4 thousands, and 95 hundred. (followed by giggles). 

Me: Ha, funny funny, but seriously, how many should mumma make? How many do you eat? 

He: 1. 

Me: How many does mumma eat? 

He: 2. 

Me: How many for papa? 

He: 4. 

Me: How many should I make then? 

He: hmm, 7. 

So, when this kind of conversation happened, I never said, you are wrong, that is not a right number.  I never think, it is too early either. I let him babble away, and to my surprise, several days later, he would start saying the correct stuff. Statistical pattern recognition, statistical human learning, pattern based learning. Something to totally encourage. 

Post 1: What is the biggest number?

Children ask the most difficult questions. So, it is important that they don't get lost. I have this belief: If you can break down a difficult concept into stuff that the child can relate to, then many of the concepts that are acknowledged as too difficult for a young child ("it is too early for him to know"), can be made not only understandable but enjoyable. And the child learns it in a way that they will never forget. 

When my child was in Kindergarten, he asked one day:

Mum, What is the biggest number?

Me: What is the biggest number you can think of?

He: A million?

Me: And what happens when you have a million lollies? J

He: Wow, can I? J 

Me: Ok, so, imagine, just like Charlie and Mr. Wonka, that you have a million lollies.

He: Well, that would take YEARS to eat J

Me: Ok, what happens when I give you just 1 more lolly after the million? How many do you have?

He: I have a million and one lollies J

Me: So, you said 1 million is the biggest, but now you have 1 million and 1, which is bigger!

He: Oooooh, yes…so if you give me 1 more lolly I will have a million and 2, which is even bigger?

Me: Yes, you are getting it!!

He: So, if I have 2 million lollies, and you give me 1 more, that is bigger than 2 million?

Me: Yes! So, no matter how many lollies I give you, you can always add 1 more lolly to make a bigger number of lollies J

He: SO, there is no biggest number, because there is always 1 more than the biggest?

Me: Yes, and that is why the word we use to describe this is called “Infinite”. We say there are infinite numbers, because they go on and on and on and on…no matter where you stop, you take one more and one more from there, and so on.

He: So, the biggest number is called infinite?

Me: no, there is no number with the name “Infinite”. Infinite means this idea that the numbers go on forever. And the way we write it in math is we take the number 8 and ask it to lie down, like this (at this point, I write an 8, followed by a lying down 8, the sign for infinity).

He: Can I try it?

Me: Yes, sure (and he draws for a while).  And by the way, going from lollies to stars, how many stars do you think there are in the universe?  

He: Ooh, got it J So, that is why my book said the universe and stars and galaxies and all that stuff is infinite, because it too goes on forever?

Me: well yes, maybe yes J

He: Ooh this is mind boggling (he has learnt this new word, so he uses it liberally).

And yes, this was not a one-off conversation. We continue to explore other such pictures of infinity in the universe. He is in Year 1 now, and he has realized that infinity can go the other way too, meaning you can make a number as small as possible, by going the other way (negative numbers)!

Post 0

I want to write on a topic I am passionate about: the education of young ones. My son started school in Sydney last year. Newspaper reports and books tell me that later performance in life (in every sense, be it academic, intellectual, social, empathetic, human, economic, or psychological…) has deep correlations with the quality of education received. However, the debate is focused on how the higher economic tiers in society have access to better education and resources, and so continue to perform better in life. The middle and lower economic tiers don’t. And this leads to re-distributive policy changes, where the focus usually is on devoting more money (or technology, or iPADs) to more disadvantaged schools. You get the idea, given the enormous attention this topic has received in Australia (and in the world) in these few years. I am not denying the importance of this debate.  

However, as a parent, it is rare for me to witness debates, discussions, or thoughts on the content and quality of how things are taught. My son goes to a great public school! I am not really bothered at all about the resource question here: compared to my class in India with 70 students to a teacher in class, my child’s class has about 20. They have smartboards, and iPADs and lovely chairs and tables and stationery, and to me that looks like luxury! The teachers are super helpful, and my son is happy at school. So, my focus is on the question: as a parent, how can I support and engage with my child at home?

And I strongly believe that sending the child off to extra tutoring classes at such a young age is a poor solution (but sadly all the rage right now). Academics and education is not just about finishing off that course and building skill sets - children need to build a lifelong relationship with parents, and as an adult fully functioning member of the society, can I not guide my child through the basics of what I myself learnt at the primary school level? Is it that hard?

So, I am going to write this blog with two focal questions in mind:

1.     Can I unpack some of the deeper pedagogical questions and debates around this question, as my child goes through school?

2.     Can I devise ways to come up with interesting old and new content and ways of working with my child?

This last aim is truly important to me: since it is something that I believe most parents are constantly engaged with anyway. So, as I spend time daily with my child and his friends and their parents, this would be an effort to document the learning experiences and critical insights into processes of learning, that hopefully will reinforce and support what they are learning at school.  If you feel you have something to share, you are very welcome to provide those insights and experiences, I would love to know about them.

Warning: Due to personal bias, this blog will have more math content.