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How to explain infinity to kids

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Re: How to explain infinity to kids

#51

Post by keithnz »

mncharity wrote: post #40
Boulth wrote: post #2

> A professor of mine once said to me that all teaching was a process of lying, and then of replacing the lies with successively better approximations of the truth. Well put! I'm in the process of explaining my 3yo daughter different molecules (there are some cheap kits on Aliexpress) and cells and life on a micro scale and this description of moving from simple but inaccurate models to more complex and accurate is s…

>> all teaching was a process of lying > Well put! Shudder. Consider a thought experiment - a military briefing. A captain briefing generals. One must necessarily simplify. But imagine a briefing that is grossly incomplete, assortedly incorrect, very misleading, written without understanding and without mentioning and characterizing that lack, and pervasively incompetently bogus, and that captain later on the carpet…

a military briefing is different from teaching. A military briefing should be communicating things the generals already understand.

teaching is a process of lying is kind of provocotaive, but teaching is a set of progressions, each progression necessarily leaves the edges blury while trying to make one aspect clear.

Re: How to explain infinity to kids

#52

Post by lwansbrough »

marcelluspye wrote: post #47
kylec wrote: post #38

Earlier quoted context omitted.

They might also take it literally. Like, they might think that infinity is less than a million because they know they couldn’t physically count to a million.

>couldn’t physically count to a million. This is only because they/you haven't tried. Source: counted to a million once.

How long did that take you? Seems like it take at least a week.

Re: How to explain infinity to kids

#53

Post by lwansbrough »

kylec wrote: post #38
duncancarroll wrote: post #29

I like the concept, but if I may editorialize, I feel the phrasing needs work. "The smallest number you can't count to" is a negative statement, which makes it confusing right off the bat. What do you mean, a number I can't count to? If I'm 8 years old this is like throwing a null pointer exception in my brain.

They might also take it literally. Like, they might think that infinity is less than a million because they know they couldn’t physically count to a million.

I feel like this is the most likely answer, and stood out immediately when I read the sentence the first time.

Re: How to explain infinity to kids

#54

Post by jackgolding »

scandox wrote: post #24

But before you do that it's worth just asking them what they think comes after everything else and see what they say. Because kids often have really interesting ideas on those kinds of topics and once you tell them something, then their ideas get pushed out.

This is one of my mottos for life - you can only be naive once so it can be beneficial to let your imagination run wild based on your unique background before you seek out best practices.

Re: How to explain infinity to kids

#55

Post by zhte415 »

antidesitter wrote: post #26
romwell wrote: post #11

Picking the first infinite ordinal as the infinity to explain to kids might not be the best choice for every kid, though. Oridnals are tricky. I am not too comfortable with them myself, and I say that as an adult with a degree in math! The thing is, this kind of infinity just doesn't come up that often when dealing with other objects in math. Even though, as Tom Lehrer sang, one can count up to infinity - or somewher…

I agree. Cardinal numbers (and bijections) are probably easier to start with.

Given Cantor made it for the battle field.. it needed to be easy to deal with.

Re: How to explain infinity to kids

#56

Post by romwell »

throwawaymath wrote: post #33
romwell wrote: post #11

Picking the first infinite ordinal as the infinity to explain to kids might not be the best choice for every kid, though. Oridnals are tricky. I am not too comfortable with them myself, and I say that as an adult with a degree in math! The thing is, this kind of infinity just doesn't come up that often when dealing with other objects in math. Even though, as Tom Lehrer sang, one can count up to infinity - or somewher…

I agree with you about Cantor's diagonalization. I think that really gives a "tactile" conception of infinity. If I had to explain infinity I'd probably choose to explain injections, surjections, and bijections; followed by countability and uncountability. If the child is old enough to understand basic addition and multiplication, you can probably run through a short explanation of the different (elementary) number s…

The cool thing is that you can introduce these concepts (bijections, etc) without calling them by their overly formal names, and yet maintaining rigor.

E.g.: on a pasture, there are black and white sheep. How can you know, without counting, whether there are more black sheep than white sheep, the other way around, or there's the same number?

Well, start taking them out in pairs, black and white sheep in each pair. If at some point you have a black sheep, but no white one to pair with, you know there at least as many black sheep as there are white ones. Same for the other way around. And if all sheep can come out in pairs like that, you conclude there must be the same number of them!

My wording is not the clearest here, but you can get the idea. The notion of "same size" for sets via putting things side-by-side is something kids can get before they learn numbers.

Re: How to explain infinity to kids

#57

Post by peter30744 »

I am not a mathematician, but I know that at least in standard mathematics it is fairly ill-advised to treat infinity implicitly as a number. Doing so can result in various contradictions (two different seemingly valid solutions to a problem). It should be thought of as a property of a process, i.e. for any number x you can change the process so that it results in a number greater than x. One example of infinity might be: "Think of a number which you are allowed to change after each number I suggest. If for any number I suggest you change it for a bigger one, then what you think of is infinity."

Chapter 15: Paradoxes of probability theory in Jaynes's "Probability Theory: Logic of Science" is a great reading on the topic (you can find a pdf easily on google). It starts with a quote from Gauss:

"I protest against the use of infinite magnitude as something accomplished, which is never permissible in mathematics. Infinity is merely a figure of speech, the true meaning being a limit." -- C. F. Gauss

Anyway, there are plenty of theories in mathematics which use infinity implicitly, but one should perhaps be cautious.

Re: How to explain infinity to kids

#58

Post by romwell »

gowld wrote: post #42
romwell wrote: post #11

Picking the first infinite ordinal as the infinity to explain to kids might not be the best choice for every kid, though. Oridnals are tricky. I am not too comfortable with them myself, and I say that as an adult with a degree in math! The thing is, this kind of infinity just doesn't come up that often when dealing with other objects in math. Even though, as Tom Lehrer sang, one can count up to infinity - or somewher…

Get a sheet of graph paper. Imagine that it goes on forever in 2 of the 4 directions. Name the intersection points in order -- left to right, top to bottom. What's the name of the first point in the first row? 0 What's the name of the first point in the second row? \omega How is that more complicated or less interesting than cardinals?

Well, Cantor's diagonalization argument can be used to prove the Halting theorem, which is another problem that is easy to state and is very interesting to think about.

The notion of the cardinality of sets comes up everywhere in mathematics, and often enough you end up showing that something holds up to a countable number of exceptions. These two kinds of infinities - cardinality of naturals and reals - are so pervasive, you can't get away from them.

But you can do a lot of math without ever having to deal with the ordinal numbers.

For that reason, cardinals are more interesting to me - not just as a concept in and of itself.

Re: How to explain infinity to kids

#59

Post by vilhelm_s »

oska wrote: post #46
mncharity wrote: post #40

Earlier quoted context omitted.

>> all teaching was a process of lying > Well put! Shudder. Consider a thought experiment - a military briefing. A captain briefing generals. One must necessarily simplify. But imagine a briefing that is grossly incomplete, assortedly incorrect, very misleading, written without understanding and without mentioning and characterizing that lack, and pervasively incompetently bogus, and that captain later on the carpet…

> Shudder. I consider the "teaching is lying" meme to be vile . Couldn't agree more. And when I read this in the article I also thought of high school chemistry even before I read your comment. I was put off chemistry in high school precisely because of its incoherence. The best thing to do, as always, is to be honest. Tell your students that what you are teaching them is a simplification; a model that is useful at t…

The approach in this article is to give a "technically unimpeachable" definition. It doesn't involve any lies.

Re: How to explain infinity to kids

#60

Post by Sniffnoy »

romwell wrote: post #11

Picking the first infinite ordinal as the infinity to explain to kids might not be the best choice for every kid, though. Oridnals are tricky. I am not too comfortable with them myself, and I say that as an adult with a degree in math! The thing is, this kind of infinity just doesn't come up that often when dealing with other objects in math. Even though, as Tom Lehrer sang, one can count up to infinity - or somewher…

> The other kinds of infinity - cardinals, for example - are encountered early on, and there are things you can do with them.

There's plenty you can do with ordinals too! Being able to interate a function transfinitely many times can be quite useful.

> So, I might be biased in that, but I think that the cardinals are the most playful type of infinity.

Definitely disagree. Once you know the basics, doing things with cardinals tends to be either boringly easy or impossibly hard. Ordinals, on the other hand, you can just play around with and actually get somewhere.

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