A Child Thinking About Infinity (2001) [pdf]

Interesting read. Though Socratic method much? The writer uses leading questions[1] as a primary expository method, directing the kid towards conclusions that "should" be made within the proposed system, rather than merely exposing a small set of rules and letting the child freely explore their consequences. All the talk about the different infinities felt that way.

This leading style may be good for teaching the history of mathematics, i.e. introducing concepts that have been relevant to solve problems in the past. But for teaching math skills I prefer methods where a problem is stated, and the student is left to build their own abstractions about it; the IOUs and thermometer examples were more in this line. This competence will prove more useful when solving new problems than merely following the trodden path of already invented mathematics.

[1] https://en.wikipedia.org/wiki/Leading_question

I'm not sure my son was even 6 yet (he may have been five) when he learned about subtraction. We started by talking about things: to illustrate 4 minus 2, you start with four marbles and remove 2 of them, leaving 2. For a couple of weeks he would ask me to subtract numbers, then he moved on to asking me to ask him numbers (and writing down simply arithmetic problems to solve).

Then, one day, out of the blue, he asked with a look of incredulity in his eye: "Daddy. What is one minus TWO???"

He clearly realized that he had come up with a question which had no representation in the marble-based construct.

I was so proud.

I sat down with him and drew a number line...and then showed him how we could extend the number line to the other side of zero.

He does know about infinity now (he's 7) -- and his most recent revelation that there is no such thing as an infinite number of any physical object. Even germs. Even atoms -- no matter how tiny, an infinite number of them can't exist in the universe (which he certainly pictures as finite).

So this resonates with me. In general I don't have a particularly detailed memory of my early childhood. However, I do recall one day when I was 4 years old sitting at the end of the driveway waiting for my dad to get home from work.

I had picked up 2 digit counting from my listening to my mom work with my older sister. As I am sitting there by myself counting and then I got to 99. At that moment something partly clicked and I realized that 3 digits let me continue counting and I could just carry on the pattern. I remember being pretty excited at this and counted for awhile longer ... maybe another 20 or 30 digits and then in a flash I realized that the same thing works when I run out of 3 digits and I intuited the fact that there is really no end to counting. And, even though I didn't have a name for it I remember being kind of awestruck at something that could just go on and on without end.

It makes me smile today at 54 to be reminded of that rush of discovery.

Sounds like a child was picking up on OPs intonation when he asked the questions. That's how children answer. They pick on body language about what the correct answer is and basically tell what the one who asks wants them to tell. This was made especially easy by the binary format of questions. Yes/no answers are very easy to guess based on the way you ask. And people tend to ask extra suggestively when talking with children.

I vaguely recall having a "night terror" centered around infinity when I was very young. I think I was trying to truly grasp how tiny and insignificant I am and how immensely huge everything else is and it just freaked me out.

Just a few days ago I had an interesting conversation with my 4 year old son that went something like this:

"Daddy, what's the biggest number?"

"There is no such thing as the biggest number; numbers keep getting bigger and bigger forever. This is a concept called infinity."

"Which number is the biggest?"

And so on; he's only 4, so I don't expect him to grasp the concept of infinity, but I was impressed that he's already asking about it.

My son is in his first year of Montessori education, and has learned to count into two digit numbers already, and understands zero, but doesn't yet understand negative numbers. Montessori is fairly interesting because they teach math in a different way, by learning powers of 10 and using a bead system similar to an abacus, so Montessori educated kids can actually do simple arithmetic with large numbers (in the thousands) by around age 5.

I would like to see fMRI studies on how human experience the infinite or perhaps near infinite. There is an instinctual appeal of vastness (near infinite, perceptually infinite?) - mountains are climbed, sunsets into far away horizons over seemingly infinite oceans, skyscrapers are built and their height and subsequent view are valued,... - These are times when humans first experience vastness which may trigger pathways in the brain related to comprehending / thinking about the infinite. I would like to see is the parts of the brain associated with self-awareness are active during these experiences.

On a side note: In the photo given of 10^100 - 999...(99 9s) = 1, there should be 100 9s, not 99 9s, for the result to be 1.

I was born in communist Poland and back then everybody was catholic there, and so was I. So everyone had to go to the church, so I did too. I listened what the preast had to say about the heaven and the hell, but particularly that he said one would end up there for eternety.

That sunday lying in my bed I couldn't sleep. I was thinking about this infinity thing. I thought to my self, going to hell is bad because you get tortured and stuff, but going to heaven for eternity isn't really much better either, because it never ends! This was so frightening to me I couldn't sleep.

I was ok with going to hell for some really long amount of time, say 10k years, as long as I knew it would end some day. But being in heaven for ever? I still get a bad feeling when I remember my thoughts from back then, that would be unbearable, you couldn't even commit suicide, nothing would change anything, you'd be trapped forever with no hope of escape.

That was when I was seven or something, when I was eleven we moved to Germany where not everybody was catholic, half of the people were protestants. Then later when I was 27, so 10 years ago, I moved to Sweden where most people are atheists, and with time I also liberated myself from this scary religion.

Now I feel really happy about the fact that there most probably is no god and thus no inifinite amount of time in heaven either.

Amusing typesetting error:

    > We also discussed powers of ten, so that he knew that 
    > 102 was 100, 103 is 1000, and 10100 is a 1 followed by a
    > hundred noughts.

Some guy brags about his son..

I know this isn't the point of the article, but did anyone else LOL at the bottom of page 3 when the difference of 10^100 - (ninety nine 9's) was given as 1? The author seems proud to attest "There really are ninety nine 9s!" The problem though is that this doesn't equal 1. You would need to subtract a number with one hundred 9's in a row to get an answer of 1.