We don't live in the UK but our kids watch Numberblocks.
Our youngest started rattling off all kinds of number stuff which I know for sure she hasn't yet encountered in school.
Me: Wow ... how do you know that?
Her: Numberblocks!
Me: Umm ... OK!
> Numberwang theme tune
I mean, sure. I do though. I still blow minds when I put my iPhone calculator in scientific mode. Math education is important for many reasons. But teaching it as a practical survival skill using no tools does a disservice to the student. Either it is useful as a problem solving exercise or it is a practical skill that should take advantage of tools. "Just memorize this stuff" isn't useful beyond teaching people to memorize arbitrary things, which backfires into hating learning.
Making a subject "fun" is alright, but making it entertaining (IME) makes for more productive engagement.
By age 6 to 7 they're expected to understand that addition and multiplication are commutative, while subtraction and division are not.
My experience, ultimately, was much less ... 'high-quality', let's say. When I left the Montessori school (by 3rd grade), I learned practically no math from then until after high school. First, in normal 'elementary' school (US), multiplication was still being covered in 6th grade. Then, suddenly (from my perspective), letters were being brought into the picture in 7th or 8th grade. So, in my arc, math started to not make sense, at all.
From my perspective, we had spent multiple years on multiplication and long division, which I already understood very well by the end of 2nd grade ... so, there was the period where I basically didn't learn anything, where it seemed like we'd reached the end of math or something. Or, perhaps, like there were some sort of subtleties remaining in multiplication and division. It just gave me a chance to be bored with all of it, boredom correlates heavily with mistakes with kids with attention issues (IMO), this fed into some sort of doubts about my understanding of everything etc., and then, suddenly, there was new material again starting in 7th grade. Material that was 'mechanical', and that didn't seem to have explanations I could understand.
Ultimately, I struggled along with that garbage through high school, then, after, took a course where we actually did PROOFS. Basic number theory stuff - modular arithmetic, etc. Bam, suddenly, the subject started to make sense.
Typing this out actually makes me slightly angry. I'm not sure I previously connected it all together - why I had so much trouble with math for some years ... how this 'arc' was pretty much perfectly engineered to make math a problem, for me. In any case, schooling through high school can be a really low quality experience at times - for some students, subjects, etc. The math curricula, methods of teaching, and progression I was exposed to, worked together, in some sense, to make the subject a problem for me. To do almost the opposite of what was intended - to pretty well impede learning. There's no one factor in that story I can point to and say 'here, fix this' ... no one involved in the story was actively attempting to do anything other than what they thought was best or what they were required to do, but, the net result was honestly worse - I now believe (and believed some years ago, even without quite this analysis) - than if I'd just been given some selection of math material to pick from and been allowed some sort of semi-self directed coursework.
Even better, though, if I'd simply had that course with proofs / basic number theory in, say, 8th grade ... guh, would have avoided so much pain, I'm pretty sure...
Teacher: “Well, there’s 100 million and one”
Child: “I was pretty close then!”
Twenty-four is the highest number! That's it. Let it go.
Or put another way, "what's ∞ + 1" basically invites the non-answer "that's not a well-formed question" whereas "what's ω + 1" gives you a whole intellectual thread to pull on.
It makes sense, since those are a lot less useful than the subjects that are taught, but something like number theory is incredibly approachable to a middle school student. And it can show students that math can be a lot less about memorization and a lot more about creative thinking w.r.t. proofs.
...and that the intellectual thread you are pulling on is a (more) artificial notion, constructed by set theorists for the sake of set theorists, not for the sake of counting or measuring in any real sense.
Or maybe put another way, taking the idea that infinity is just infinity makes a lot of sense when you’re primarily considering non-infinite numbers. When you’re primarily considering the concept of infinity and what you can do with it mathematically though, using systems that let you describe infinity with more nuance makes a lot of sense.
Or is it "Two Times Infinity"? (Hint: It isn't, because "Two Times Infinity" = "Infinity", while "Infinity Times Two" = "Infinity + Infinity". Not sure every kid knows that.)
A number is represented in set theory as a set that contains all of the numbers before it. 0, 1, 2 is {}, {{}}, {{} {{}}}...
SO! If you start with a finite "a" and increment it infinite times, you still have infinity; you haven't broken out.
But if you start with Infinity, then adding anything to it gives you {Infinity}, {Infinity {Infinity}}, etc...
Transfinite addition is not commutative!
I am in no way a mathematician. My question about the definition of addition as it relates to set theory is just that; a question.
But your question actually hints at my most profound takeaway from that whole book. I think what you're saying is right, AND that foundations-of-mathematics folks spent a long intense period searching for different set theory axioms that did NOT lead to transfinite numbers. But anything anyone could come up with that included "the axiom of infinity" led to transfinites leaking in.
Which begs the question of how to think about these things. Are they "real"? Are they an oddball side effect that we shouldn't take seriously?
I think you've arrowed right to the philosophical heart of all of this.
I think we often end up at the end of logical thought processes back at the original question - how can we observe and describe a system that we are inherently a part of?
In that context the first quantity that you refer to above is nonsensical because you can't "increment infinitely many times".
Secondly, I'm not sure your construction is correct, since your Infinity+1 set cannot be a singleton (it must contain all the numbers less than Infinity).
For the first point, I went through the book long enough ago that I can't rebuild the proof here, but iirc the more rigorous idea is that you can construct a bijection between 1+ω and ω given the recipe I had above for how to represent numbers as sets, but you can't do it for ω+1, which is bijective with ω∪{ω}. The axiom of infinity declares that ω itself is a set, opening the door for transfinite numbers.
Better?
(Full disclosure: have three children and plenty of STEM in the family)
I'm not sure that's the _default_ answer, of course one might easily get that answer if at least one parent has a STEM background.
Schools don't teach about infinity to young children. A pity, really.
A countably infinite set could be 'counted' (i.e., you could sit around labeling elements using the 'natural' or 'counting' numbers) in the sense that we might count candy. The issue for a human being is that you'd run out of time but not elements to count, at least, proceeding in the sense one might count the candy - a piece at a time. Of course, you can, instead, simply provide a 'bijection' (between the natural numbers and the set you wish to prove is countably infinite), and in a sense, you are done.
The subject of infinity and infinite sets can be kind of subtle, and for years the best mathematicians made many mistakes and had many difficulties handling these concepts in ways that didn't cause potentially serious problems (absurdities, paradoxes, etc.). I think that with the development of things like Zermelo-Fraenkel set theory, Gödel's incompleteness theorems, etc., things became a lot clearer. It's a lot easier, with all of the groundwork laid by people who worked on these, to get a good sense of what is possible and what isn't - what gets you into trouble and what doesn't. But, boy, did it twist the minds of the people trying to work it out at the time. In part, this is because it was less clear, without development in these areas, what math even is and what its limits are ... what its relationship to the structure of the universe, say, even is (something along those lines, in my opinion / experience).
(Q: Do you have kids?)
Our experience is that pretty much everything parents tell young children could potentially cause confusion later.
In no particular order: Father Christmas aka Santa Claus, The Tooth Fairy, Where Babies Come From... it's a long list, our eldest is 13 and we're not done yet.
Ha! Certainly a fair and good point.
I would propose that there is a spectrum when it comes to the 'damage', as a term that comes to mind right now, (likely to be) caused by various kinds potentially confusing information.
Given differences in the way different people understand, well, pretty much anything, I'd propose that it might best be thought of as some set of statistical distributions. Using this kind of framework*, we might be able to reasonably improve thinking about what these distributions might look like, how we might tailor the information we provide and how much work we put into trying to avoid introducing possibilities for confusion, etc. Further, I suggest 'set' as we might benefit from 'parameterizing' (thinking about distinct distributions) in terms of traits - autism, ADHD, anxiety, etc.
In my mind, and based on my experiences, I would (in part, thinking terms of the model I'm proposing here) be much more wary of asserting potentially incorrect information in the realm of math and some of the more 'abstract' subjects that people tend to have more trouble in the first place. A concept like 'Santa Claus' isn't something that a child may need to be able to use as a basis for building serious skills on, say. Of course, 'Santa Claus' can be helpful for building imagination, ability with storytelling, developing narratives, etc. ... but the fundamental information regarding some specific entity 'Santa Claus', is not really problematic, in terms of the perspective I'm trying put forward here. On the other hand, statements that are 'too strong' (or 'too weak' possibly) or using terms in ways that aren't standard in mathematical discourse ... these sorts of things can make it feel like the ground is really slipping away as you try to learn other bits about a subject that, again, for many people is ... nebulous ... it's not (so) visual, tactile, ... it's very strange in many ways, early on.
That's the best I can do, right now, in response, I think.
You raise a good point, for sure. And I'm sure there are entire books, there are papers out there in the literature, etc. Personally, I can HIGHLY recommend books like Polya's "How to Solve It" ... as a starting point regarding 'math pedagogy'. That book is a gem, IMO, and gives some real insight into how to think and problem solving in general. And, it's a good gateway to many more resources and research into these areas.
As with everything human and 'complex', there's really no 'optimum' or chance of finding any such thing, I think. Avoiding the worst impacts ... essentially, in terms of opportunities and establishing bases etc., that's doing pretty well - raising children / 'new humans' is hard.
* Which is a way I've been trained to think, sorry if it's not a great model for you - kind of best I can think of off the top of my head and with limited time this moment
Q: What proportion of children study maths long enough to understand derivatives?
Do you think 90% of attendees of Gymnasium (which I don’t think is the majority) understand derivatives? My friend’s wife who attended Gymnasium and got reasonably good grades most certainly did not, but she is my only example of a non-mathematician Gymnasium graduate, so I’m quite willing to be convinced she is an outlier.
why misquote someone and claim their idea is hard to understand?
Sure, they lack rigor, and often will just get the sketch of the idea.
And it's a lot more work to think about how to put things in the terms that a kid will understand given their knowledge so far.
But this idea? "Infinity plus one?!@" --- this is a conversation elementary school kids have on their own. Pulling it a little closer to a sane footing in ordinal analysis is not hard. Half of six year olds can handle it.
On the other hand, there's not a lot of obvious utility to teaching a six year old this particular concept early. On the gripping hand, there is a cost to keeping kids in a bubble where you don't talk about any big ideas (of whatever sort-- mathematical, philosophical, historical, linguistic) at all, or excessively dilute them to the point where they're meaningless.
Explain everything like you're talking to a fifth grader. If you can't, you don't understand your problem fully.
He spend much of his professorship agonizing about how to fit all of physics into a freshman lecture. When he couldn't, he knew we needed to think more about that area.
"Bill, despite your emphatic comments, I know for a fact that counting into the ordinals is something that children can easily learn. I have two young children (ages 4 and 9), who are happy to discuss ℵα for small ordinals α---although my daughter's pronunciation sounds more like Olive0, Olive1---and my son can count up to small countably infinite ordinals. The pattern below ωω is not difficult to grasp. Below ω2, it is rather like counting to 100, since the numbers have the form ω⋅n+k, essentially two digits"
In case anyone is curious, this person has experience teaching children mathematics. For example, on his blog, we have
http://jdh.hamkins.org/math-for-six-year-olds/ http://jdh.hamkins.org/math-for-seven-year-olds-graph-colori... http://jdh.hamkins.org/math-for-eight-year-olds/ http://jdh.hamkins.org/math-for-nine-year-olds-fold-punch-cu...
The most recent post in his category "Math for Kids" is in fact teaching how to count ordinals up to omega-squared: http://jdh.hamkins.org/counting-to-infinity-poster/
Just as a FYI, there are plenty of countries in Europe where many 6 year-olds are still in kindergarten not at school, as a result they most likely have not have properly started learning numbers or reading and writing.
https://www.statista.com/chart/13378/when-do-children-start-...
Things people say on HN :)
I do not think he means he would use symbols to explain to children, but that the notion of counting natural numbers that children have easily generalises to counting transfinite numbers.
I don't know why would it be hard for people that haven't been familiarized with a similar but different concept?