
A Child Thinking About Infinity (2001) [pdf] - Phithagoras
http://homepages.warwick.ac.uk/staff/David.Tall/pdfs/dot2001l-childs-infinity.pdf
======
TuringTest
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](https://en.wikipedia.org/wiki/Leading_question)

~~~
jmilloy
I think it's noteworthy that the article you point to pertains to common law
and testimony, rather than to philosophy, pedagogy, or education. When a
student freely explores a concept, that clearly demonstrates a higher ability
or skill in mathematics. But "leading questions" are scaffold to show the
student how to explore and build abstractions, and most will have to learn how
to do that from example.

~~~
TuringTest
The Wikipedia article may be primarily written about legal cases, but leading
questions appear in many other contexts. In psychology and interviews they're
often advised against, as they induce a precise answer and thus make it hard
to learn about the subject of study, instead reinforcing the interviewer
preconceptions; in user interface design they can produce the wrong
conclusions about how easy an interface is to use.

Leading questions have their use in education, but as the other commentator
below points out, the child might be _giving the answer that the questioner
wants to hear_ , rather than showing a true understanding of the topic. Having
a stated problem to solve may prevent that, as the child can instead _answer
the question in a way that solves the problem_.

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oaktowner
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).

~~~
sethammons
I had a similar experience when my oldest was still young. The conversation
was a little flipped. We live in a cold climate and she had started taking
note of the thermometer. I had asked her "what is 5 - 5?" > "zero!"; "what is
that minus 5?" > "you can't do that!" > "What happens if it is 0 degrees out
and it gets 5 degrees colder?" > * eyes open wide * "there are all the same
numbers below zero!?"

Some time later, when she was learning algebra, I told her "0.999... == 1, can
you tell me why?". I was expecting an algebra lesson to show her how that was
true. She surprised me when, a second or two later, she looked up and said,
"1/9 is 0.111... so 9/9 is 0.999... and 9/9 is 1". I was very proud.

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mswen
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.

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cLeEOGPw
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.

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CookieCats
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.

~~~
geierdmtr
I still fall into a rabbit hole every-time i imagine infinity. it started when
i was 5 or so. brain goes into a loop and imagines the infinite. for me it was
and is connected to space. you have to hit a wall at some point, but then
there is something behind the wall.. there can never be something that isn't
there. the limited donut universe theory framed the space imagination. but the
donut has to be surrounded by something. here goes the infinite loop again.

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illumin8
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.

~~~
knicholes
I'm not sure if this is normal or expected, but my two-year-old can count to
20. I'm not trying to brag-- maybe just seeing if maybe I should start
bragging? ;)

~~~
irrational
Not really. We lived in an apartment from the time our youngest daughter was
born until she was 2 or so. There was no elevator so we would count the stairs
when we went up. I don't recall how many stairs there were, but she could
count higher than 20 because of that. Of course she was a bit precocious. We
were at university at the time and the only people she interacted with were
other university students (no other kids around). By the time she was 2 she
was fully fluent and could carry on lengthy conversation with any adult as if
she was a college student. It was quite amusing.

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dpflan
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.

~~~
pbhjpbhj
Mathematical concepts like nothing, straight lines, circles, are also
abstractions that have no complete reference in nature - is it just infinity
that you consider needs self-awareness? I'm not quite sure what you're hinting
at?

~~~
dpflan
I was just musing, no concrete connections at the moment. I don't know if
infinity needs self-awareness, but I was thinking that it could prompt one to
call into question his/her self relative to the infinite that is being
experienced, and in that perhaps adjust one's concept of self through
comparison. Or even just thinking about how one can conceptualize such a thing
based upon all life experiences where almost everything encountered is finite.

~~~
hodwik
In my experience, children (and most people) just conceptualize infinity as a
very large set, just as they perceive a tire a "circle", and think a square
drawn on paper is made up of "lines".

It requires years of thinking about these mathematical entities before people
get a first glimpse of what they really mean, beyond a simplistic qualitative
experience.

To most people, children included, infinity is conceptually no different from
the amount of sand on a beach. They just allow that it is simply not
quantifiable. That turns off the brain.

The more interesting thing to test, I would argue, would be the human mind
conceptualizing a very, very large but ultimately _finite_ set of something.
That requires that we make a distinction between the discreteness of something
and our ability to quantify it.

That is a much more difficult concept to comprehend, and it must be understood
before we can even begin to think about teaching a child about mathematical
entities like infinity, circles, or lines.

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paulvs
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.

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jeena
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.

~~~
tn13
For the same reason I have found the notion of Christian God to be very scary.
A father who does not die and keeps judging his children and is always
powerful.

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OJFord
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.

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thrownear
Some guy brags about his son..

~~~
estrabd
Maybe, but it's interesting because I had quite a few discussions with my kids
about infinity. It came up when playing the "how much do I love you" game. I
introduced infinity, and from there I was able to teach them a little about
it. They were very young and I am still impressed with how easily they grasped
some of the concepts.

~~~
Phithagoras
Do you think it's possible that they accepted your ideas of infinity in a
dogmatic sort of way, similar to how many young children accept god? I
remember first having discussions with my father about infinity as young as 6,
but I didn't really start exploring infinity for myself until my mid teens. It
would be interesting to see how your kids thoughts change as they mature.

~~~
estrabd
I do think they accepted it in a dogmatic way. Incidentally, I am raising our
children Catholic so it's an idea that fits pretty well with the idea of God,
the Holy Trinity, and all that. Three in One, yet one is the Father and the
other is the Son? Yes, dear. Oh, ok. <grin>

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Wile_E_Quixote
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.

