So this other day, me and my daughter were watching a video where the author was talking about infinity and how if you divide a number by zero you get infinity. My daughter could not understand how anything can be divided into zero parts. Or how attempting to divide something into zero parts blows up into this huge number that we call infinity. Being the smart alec that I am, I thought I could explain her how division works and how something divided by zero gives you infinity. As you can imagine, I made a huge mess of it and it showed me how little I knew about teaching to kids. If you take pleasure in seeing a grown man fumble with math, go on.

To give you some context, my daughter is 11 years old and is being unschooled. So we typically never open a school book or math video and try to explain things when she is not ready. We like to see her struggle with daily problems and trying to solve them. When we feel she is ready to understand a concept that will solve her problem we explain the concepts. That is how she learned most of the things she currently knows. We taught her counting when she wanted to count how many beads she had in a box. She learned simple addition because she had to calculate how much money she had when she was playing Monopoly. Same with simple subtraction.

However, she does not know long addition or long subtraction work due to the simple fact that she never encountered those problems in her daily life for the longest time. Yet I tried to teach her long addition because I thought it would be helpful at some point. That attempt immediately exposed my teaching skills (or lack there of). She did not understand the concept of carry over and why it has to be carried to the higher place value. I was trying to teach her the way I learned in school where we were taught that when you add two numbers that results in a 2 digit sum, then we have a carry over to the higher place value. That did not go well. Not sure if she still remembers long addition and I have not tested her in a while.

That should already give you a glimpse of how poor my teaching skills are. Our daughter understands some multiplication and we explained how it works. Basically multiplication is just multiple addition of the same number (multiplicand) as many times as the multiplier. See that is the kind of jargon she hates. The concept of multiplication also came about as a result of some real life problem that she was trying to solve. Something like – “if she wants to gift 2 books to each of her 5 friends, then how many books does she need?” sort of a problem. That led to the introduction of 2 table.

We later explained how other number tables work. However, we never made her learn tables. So even to this day, when she has to do multiplication, she does “multiple addition” in her head to give an answer. So if she wants to know 3 times 8, she would add 3, eight times before she can answer. She could have done away with the simpler multiplication of 8 times 3 and only need to add 8, three times to save time if only she knew commutative property. I tried to tell her that 3 x 8 is the same as 8 x 3 without using the word “commutative”, but she tends to forget it. She would be like – “I am giving 3 books to 8 friends, not 8 books to 3 friends”. And I be like – “Yeah, but you could also think of it as 8 friends receiving 3 books”. I get a blank stare and then “good bye, off to play” response. I failed to teach as usual. May be someday she will realize it.

If you are with me so far, you can see how we let her learn things organically from problems she faces in real life. No crazy math symbols and carryovers or borrowing logic. So how does one teach the concept of infinity? I will get to it, but first let me tell you how she understood division and fractions. The same old story – if you have 4 friends, how many slices do you have to cut your cake / pizza / what have you, so that each of your friends get equal amount. You cut the pizza into 4 slices and each friends gets 1/4th piece of the whole. Thankfully she was able to understand the concept of fractions and division (cutting the pizza) but not from a math symbols perspective.

Anyway, coming back to the video, the author was explaining about infinity and how you cannot divide any number by zero. I mean “you could have 0 apples and you can divide it among six friends and each of them will get zero” says the author and my daughter nods in agreement. “What if I have 6 apples and I wish to share them among 0 friends?”. That does not make any sense. She wanted to understand how that concept works. Armed with all my math knowledge and zero teaching skills, I decided to explain it with calculus of all things. Seriously?

At this point I had to explain to her about math symbols and how division is represented. Then I had to tell her about what is called a numerator and denominator. As you can already guess, I am giving the kind of information that she does not like and she is no where close to learning the answer. Then I explained that take a pizza and slice it into 10 pieces. So each slice looks like 1/10. Now, there is a math concept which tells you that you can multiply numerator and denominator by the same number without changing the fraction. Complexity alert! Yet I continue. I tell her that you can multiply the 1/10 fraction with 10 in the numerator and denominator to get 10/1. At this point she was already pulling her hair.

Then I had to introduce her to decimal numbers. I told her that 1/10 = 0.1 and 1/100 = 0.01 and so on. Basically, you move the decimal places to the left of 1 by as many times as the zero in the denominator. Am I nuts? I proceeded to explain that the more zeros after the decimal, the smaller the number. Now, having learned all these concepts, if I divide a pizza by 0.1 then I have 1/0.1. Now multiply numerator and denominator by 10 and you get 10/1. It is like one person getting 10 pizzas. Now if I divide a pizza by 0.01 then it is 1/0.01 = 100/1, which means one person gets 100 pizzas and so on. As you can see, as the limit (mathematically written as lim – thanks calculus) of denominator tends to zero, which is basically smaller and smaller numbers, represented by more and more zero after the decimal, you get more and more pizzas. This elicited the usual blank face, “goodbye, have to play now” response.

Later in the day she comes back to me and says, “Oh now I get it. Basically, if you have a pizza and you become smaller and smaller, the pizza looks bigger and bigger to you. When you become so small like zero, then the pizza looks like it is huge, like an infinite pizza. Is that what you are trying to explain to me?”. I nodded in affirmation. She says “Why didn’t you say so instead of giving all that crazy explanation”. She ran away leaving me to wonder where I went wrong and scratching my head. Oh well, the modern day kids show so little respect to calculus :). Off to my work where I never need to use calculus.