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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 continuati
by warent 3y ago
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
- alexb_ 3y agoNo it isn't. If you ask a child what comes after infinity, "Infinity + 1" is pretty much the default answer. Any kid who knows multiplication knows "Infinity + Infinity" is the same as "Infinity Times Two". The answer of "Infinity TIMES Infinity" is also popular for kids to say when they know a number bigger than their friend (who just proclaimed infinity is the largest number).
- warent 3y agoSix-year-olds know multiplication?
- jasonjayr 3y agoSome do, yes. If they have an aptitude for basic sums then pointing out that 3 x 3 is the same as 3 + 3 + 3 sets them down the right path ...
- hammyhavoc 3y agoI had it explained at a very early age as "three lots of three", and to imagine it like three boxes of three ice-creams. Treating the multiplication symbol as one would to indicate quantity in a list, thus calculating how many ice-creams there are.
- aaronharnly 3y agoFound the Brit! As an American I’d never heard the “lots of __” phrasing until I watched Numberblocks (a British show) with my kid…
- deleted 3y ago[deleted]
- scotteric 3y agoPharmaceuticals and other manufactured goods are sometimes referred to in 'lots' meaning a batch.
- logifail 3y ago> Numberblocks 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!
- technothrasher 3y agoI've never watched Numberblocks, but I do like to watch a good game of Numberwang!
- hammyhavoc 3y agoThat's Numberwang! > Numberwang theme tune
- adastra22 3y agoI think you need to spend more time around six year olds ;)
- Eisenstein 3y agoMy 6 year old nephew can do primes and I taught him to count and add in binary.
- ineedasername 3y agoI'm not sure that HN readers-- a community that will disproportionately skew towards folks educated &/or employed in STEM fields-- are indicative of other people's contact with 6-year old kids.
- 13of40 3y agoThe technique used by the Oregon public school system in the 80s went something like "Hand the child a 10x10 grid of numbers, then tell them, absent of any other context, that they must be memorized." I like your way better.
- retrac 3y agoOntario's 1990s curriculum was pretty awesome. The idea of dimension and sets were both introduced simultaneously and joined, using multiplication. Started in the 2nd grade and they just kept elaborating. Number lines and groups of items. (Tied it into geometry, too. Square numbers came up by at least 4th grade.) What is 3 x 3 but moving 3 units, 3 times in one dimension? Now, memorize these tables up to 12 x 12, you won't always have a calculator at hand.
- mulmen 3y ago> you won't always have a calculator at hand. 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 because it backfires into hating learning. Nothing about math makes it ideal for memorization and none of my math teachers spent any time on study skills.
- wpietri 3y agoNah. As somebody who ends up doing a ton of mental math, I think it's valuable. Yes, they should also learn how to use tools. But developing a feel for numbers is valuable, and I think that is much harder to do if one always relies on a calculator. (And yes, of course, this should be learned in a way that doesn't involve the kids hating it. But that's possible.)
- mulmen 3y agoPhrases like "you won't always have a calculator at hand" only serve to erode trust in the educator. It's simply not compelling, and for all practical intents and purposes is untrue. Even on backpacking trips I have a cell phone, even if it is off. If you believe mental math is useful then say that and explain the benefits. Students can smell a lie.
- hnlmorg 3y agoMy 6 year old is learning multiplication at the moment.
- Bootvis 3y agoThe mathematically curious probably do. I did.
- abofh 3y agoSome even post on HN!
- hammyhavoc 3y agoThat explains some stuff!
- deleted 3y ago[deleted]
- gorkish 3y agoThey dont usually know formal arithmetic multiplication but they well understand the concepts of repeated addition and subtraction. Most places in the world do start teaching multiplication at age 6/7.
- gus_massa 3y agoMy six-year-old likes Numberblocks https://en.wikipedia.org/wiki/Numberblocks https://en.wikipedia.org/wiki/Numberblocks https://www.google.com/search?q=Numberblocks https://www.google.com/search?q=Numberblocks . She knows a little more about multiplication than what I expected, probably 2x and 3x when x is small, (but as other sibling comments say not a general theory or how to calculate 287263 * 137167).
- ineedasername 3y agoNumber blocks is a great show, my 5yo watches and, being entertained by it, absorbs more than I could easily get him to sit still for. Then he asks me questions about what he watched and is more engaged with my answers as a result. Making a subject "fun" is alright, but making it entertaining (IME) makes for more productive engagement.
- ddispaltro 3y agoyeah they do counting by 5's counting by 2s, etc. So how many 5s in 20, they say four, yay!
- jlokier 3y agoYes. Simple multiplication and even division and fractions are part of the national curriculum at ages 5 to 6 in the UK. Which is about the age when I remember learning them decades ago too. I think we learned how to add and multiply fractions too. By age 6 to 7 they're expected to understand that addition and multiplication are commutative, while subtraction and division are not.
- apomekhanes 3y agoYes, I learned long division fairly well around that time. I was fortunate (/ disruptive) enough to be sent to a "Montessori school". Long division was definitely pushing it when I was about 5 or 6, but, honestly, given steadier instruction in math starting earlier, I suspect I could have been entirely solid on long division by that time and moving on to algebra. And, I think this is true for a reasonable proportion of children. 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...
- vmilner 3y agoChild: “Is 100 million the biggest number?” Teacher: “Well, there’s 100 million and one” Child: “I was pretty close then!”
- MalcolmDwyer 3y agohttps://m.youtube.com/watch?v=9P2ROAbQZYw https://m.youtube.com/watch?v=9P2ROAbQZYw Twenty-four is the highest number! That's it. Let it go.
- bhk 3y agoI thought that most of us learn at an early age, as a result of this kind of exchange, that "infinity" is not "the biggest number" or even a number at all, as far as the ordinary notion of "number" goes.
- JKCalhoun 3y agoMy child mind conflated infinity and God. Or maybe I was correct, I have no idea now.
- ASalazarMX 3y agoThat was the adults attributing infinite and contradictory powers to their god. Church sermons will frequently mention infinity.
- selcuka 3y agoHistorically, anything that can't be easily comprehended has been attributed to a higher power.
- thedailymail 3y agoYou're not alone! Georg Cantor was deeply concerned about the theological implications of his work on transfinite numbers, to the point that he wrote letters to Pope Leo XIII to explain why the new infinities were consistent with a God of an even higher order of infinity.
- NegativeK 3y agoNo math instruction I had ever discussed infinity with any rigor until calculus -- and even then, it was only infinity as a limit. Infinity as a concept was brushed off in the same way that the square root of negative one was brushed off until we were actually taught about it.
- NegativeLatency 3y agoIt came up a bit in some physics classes, when you can mathematically make something go to positive or negative infinity being able to remove it from the simplified calculation of something is very handy.
- anotherhue 3y agoOne of my favourite SMBCs: https://www.smbc-comics.com/comic/2014-02-16 https://www.smbc-comics.com/comic/2014-02-16
- FabHK 3y ago> Any kid who knows multiplication knows "Infinity + Infinity" is the same as "Infinity Times Two". 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.)
- colecut 3y agoThis seems to disregard the commutative property of multiplication
- kccqzy 3y agoOrdinal multiplication is not commutative.
- bigdict 3y agoWhat? Where does that follow from?
- sritchie 3y agoIt follows from the way addition is defined on top of set theory. "a + b" is implemented as "increment a (the set that represents a) b times". 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!
- RHSeeger 3y agoIs addition defined _by_ set theory, or is set theory one way of defining addition? If it's the later, then there could be other ways of defining addition that don't have the same results for infinity (because our math system doesn't really "work" for infinity, or 0, depending on the circumstances). 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.
- logifail 3y ago> If you ask a child what comes after infinity, "Infinity + 1" is pretty much the default answer (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.
- throwawaymaths 3y agoimagine my surprise when I got to college and learned that infinity + 1 was actually a number! I felt so cheated from my childhood.
- atemerev 3y agoThey are not explaining to a 6 years old, they explains to somebody who will in their turn explain it to a 6 years old, which is a different task and has to be optimized in a different way.
- gorkish 3y agoI explained basically this to my 4 year old nephew recently. He wanted to count to infinity. I asked him what is the biggest problem with counting to infinity? It's too slow. I said ok let's take bigger steps. We counted by 2's then 10's then hundreds and millions and then zillions and other ridiculous superlative numbers. It doesn't really matter because everything is still too slow. So then we said ok lets make up a number ω that is half way there, One ω, Two ω, done. He's happy. Then I told him to add one more and sent him back to play fetch with the dog.
- pacaro 3y agoI taught my kid that the way to think of infinity is that it's like hugs, there's always one more, unlike candy, which is limited and can be counted, infinity cannot be counted.
- apomekhanes 3y agoHmm, 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 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).
- singularity2001 3y agoTransfinite ordinals also known as hyperreals should really be taught in school as they make many parts of math easier: algebraic definition of derivatives (including algebraic derivative of step functions without dirac 'density') and yes: natural addition and multiplication. https://en.wikipedia.org/wiki/Hyperreal_number https://en.wikipedia.org/wiki/Hyperreal_number
- logifail 3y ago> 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?
- singularity2001 3y agoI can only speak for Germany where over 90% reach 10th grade, where derivatives are taught.
- mercutio2 3y agoHaving mechanical formulae for solving closed form equations involving the notation for derivatives… does not mean that derivatives have been understood, in my experience of tutoring not-especially-mathematically-inclined folks. 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.
- nh23423fefe 3y ago> I would focus on the principal idea: whether finite or infinite, a number is even when it can be divided into pairs. why misquote someone and claim their idea is hard to understand?
- version_five 3y agoAlso my first thought. I assume he's writing this to other people who know what transfinite ordinals are (I don't understand the explanation) and would frame it differently with an actual kid. Even in context it's a hilarious quote though, I think it's possible this was on purpose
- mlyle 3y agoI think the big assumption that kids can't get "complicated" ideas is faulty. 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.
- hinkley 3y agoRichard Feynman would be making disapproving noises. 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.
- gregschlom 3y agoIn the comments of the answer the author says they have a 4 and a 9 year old: "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"
- areyousure 3y ago> Presumably this person has no experience with 6 year olds? 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-six-year-olds/ http://jdh.hamkins.org/math-for-seven-year-olds-graph-coloring-chromatic-numbers-eulerian-paths/ 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-eight-year-olds/ http://jdh.hamkins.org/math-for-nine-year-olds-fold-punch-cut/ 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/ http://jdh.hamkins.org/counting-to-infinity-poster/
- logifail 3y ago> this person has experience teaching children mathematics 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-school-in-europe/ https://www.statista.com/chart/13378/when-do-children-start-...
- froh 3y agounless the kindergarten is playfully toying with numbers already, usually with no obligation but as an enrichment for those kids who love such activities.
- ilyt 3y agoDo we have statistics on how many pupils end up hating/loving math after that ?
- 0xBABAD00C 3y ago> In my experience with children, one of the easiest-to-grasp concepts of infinity is provided by the transfinite ordinals Things people say on HN :)
- mytailorisrich 3y agoIt's math so you can start your explanation with "Assume your 6 year old has a PhD".
- freehorse 3y agoHe has children (not sure about age right now) and discusses mathematics often with them. His tweets have had many interesting examples. 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.
- bmacho 3y agoOrdinals are hard to grasp for people that know the standard school curriculum, know about countability and uncountable sets, cardinality, and the basic properties and arithmetic of cardinality. I don't know why would it be hard for people that haven't been familiarized with a similar but different concept?