391–398 of 398 posts
Pretty sure if I tried to explain this to my kid niece she'd just say: "I'm uncomfortable. Can I go now?"
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> Transfinite ordinals also known as hyperreals should really be taught in school as they make many parts of math easier: algebraic definition of derivatives Q: What proportion of children study maths long enough to understand derivatives?
I can only speak for Germany where over 90% reach 10th grade, where derivatives are taught.
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.
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Yes, I thought you meant otherwise. Yes, I was confused about it. Yes, I really thought that. Truly.
You truly thought I didn't realize that "3.14" is an abbreviated representation of π, or that I somehow missed years and years of using the "repeating" sign above various decimal representations, or all those "..."s, such that it was plausible I meant the obviously-wrong thing rather than the correct thing? This stuff is hammered in in US K-12 school. [EDIT] Look, I don't mean to be a dick, performative misreading an…
“Decimal numbers” is not a term routinely used by mathematicians (quite distinct from primary and secondary teachers of arithmetic who are, unfortunately, rarely mathematicians), precisely because of the confusion you, perhaps unwittingly, elicited. If you mean by this phrase all infinite series with a decimal approximation, then you’re talking about the reals. Some people thought you meant this!
Other people, also quite reasonably, interpret “the Decimal numbers” to mean all numbers that can actually be expressed with (finite) decimal notation, in which case you are talking about (a subset of) the rationals.
It is extremely important, when discussing different sets, to be clear about the difference between these two.
In my experience with children, one of the easiest-to-grasp concepts of infinity is provided by the transfinite ordinals, since it can be viewed as a continuation of the usual counting manner of children, but proceeding into the transfinite: 1,2,3,⋯,ω,ω+1,ω+2,⋯,ω+ω=ω⋅2,ω⋅2+1,⋯,ω⋅3,⋯,ω2,ω2+1,⋯,ω2+ω,⋯⋯ Presumably this person has no experience with 6 year olds? This explanation is horrendous haha
I don't know why would it be hard for people that haven't been familiarized with a similar but different concept?
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You’re thinking of isomorphic, not equal.
They meant "their cardinalities are equal". It's honestly an easy mistake to make, especially if typing on a small screen. Or especially if having a discussion where sizes of infinity are already being discussed.
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I majored in math and my biggest problem with this is that you don't get to "do" anything infinitely many times in the math that I'm used to. In discrete contexts where infinity is used, you instead can "do" something an unbounded but finite number of times. In a continuous setting you are allowed to pick an arbitrarily large (finite) number. In that context the first quantity that you refer to above is nonsensical b…
Sorry, of course you're right on "Secondly". The right construction is ω, ω∪{ω}, ω∪{ω}∪{ω∪{ω}}... 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…
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Sorry, of course you're right on "Secondly". The right construction is ω, ω∪{ω}, ω∪{ω}∪{ω∪{ω}}... 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…
Thanks, sorry for being pedantic. These sorts of constructions tend to trigger some kind of defense mechanism in me.
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Hmm, that could potentially cause confusion later. There are 'countable' and 'uncountable' forms of infinity / infinite sets. 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 on…
> Hmm, that could potentially cause confusion later [...] (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