Joel David Hamkins, a mathematician and logician, writes about set theory, infinite chess, logic puzzles, and teaching math to children.
HN commenters link his logic puzzles like Cheryl's birthday variants and discuss his posts on math education and transfinite ideas.
> Far better to show them the most effective technique, let them practice it, and move on.
This can work for the base levels of what most folks think as mathematics, namely arithmetic and algebra. But it starts falling apart as students get to more advanced algebra and mathematical topics. Memorizing a particular algorithm does not lend to understanding mathematics. In particular, the parallels between algebra (and some arithmetic) and geometry are non-obvious and not easily conveyed via a series of exercises and then jumping to the next topic. See the visualization examples of `18x5` in the linked article. Letting students see the correlation between the numbers and the objects (squares and rectangles) greatly helps their understanding of the subject. Discussing it reinforces this experience and gets them to see it beyond the exercises they've been doing, abstracting the concept beyond a handful of cases and to the general case.
> Riding bikes and mathematics were never evolutionarily selected for, so we have to learn them by, well, learning them.
There are many ways to learn and to teach. Teacher-tell, student-regurgitate is one method. And it works for some (many) subjects, at least in getting facts and figures into students heads. Teacher-tell, student-explain gets the students past thinking about facts and thinking about whys and hows of the world. (You can expand tell beyond spoken word to shown, demonstrated, etc.) Teachers can take it further by starting on a…
My favourite trivia about "On Computable Numbers" is that Alan Turing got the definition of computable reals wrong! The way he defines them - namely that a computable real is a Turing machine that generates the sequence of digits of the real - has severe issues, because addition and other basic operations are incomputable due to subtle diagonalisation arguments (see https://jdh.hamkins.org/alan-turing-on-computable-numbers/ for instance).
It's interesting that Turing didn't realise this, as the whole paper revolves around diagonalisation arguments in a few places.
The "modern" take is to define a computable real to be a program that takes an epsilon as input and returns a rational number within that epsilon from the real being represented. Or something similar - in this case it is very simple to define addition, for instance, as you have control over these bounds.
One more fact - equality is incomputable for both of these representations! There's no program that can uniformly decide whether two computable reals are the same, no matter how you cook up the definition. This one maybe isn't too surprising - in some sense no matter how many digits or bounds you check between two reals you can never be sure there isn't a smaller gap between them you haven't gotten to yet.
I like Cheryl's transfinite logic puzzle, which takes logic puzzles to a ridiculous extreme.
http://jdh.hamkins.org/transfinite-epistemic-logic-puzzle-challenge/
The puzzle is inspired by Cantor's transfinite ordinal numbers, which are pretty cool. You don't need to know anything about transfinite numbers to solve the puzzle, but they are interesting in their own right. The simplified idea is suppose you count 1, 2, 3, ... up to infinity (ω). Everyone knows adding 1 to infinity doesn't get you anywhere, but suppose you can do it and get ω+1, ω+2, ... The limit is 2ω. But then you can go 3ω, 4ω, ... Which obviously :-) gets you to ω*ω = ω^2. Then you can keep going with bigger and bigger infinities. See https://en.wikipedia.org/wiki/Ordinal_number
My favorite version of this is a real mind fuck.
http://jdh.hamkins.org/cheryls-rational-gifts/
Two people are told rational numbers of a particular form. They want to know whose number is larger:
> Albert I don’t know.
> Bernard Neither do I.
> Albert Indeed, I still do not know.
> Bernard And still neither do I.
> Cheryl Well, it is no use to continue that way! I can tell you that no matter how long you continue that back-and-forth, you shall not come to know who has the larger number.
> Albert What interesting new information! But alas, I still do not know whose number is larger.
They keep saying “No matter how much we do this, we still won’t know” and eventually they know. It’s great and I recommend reading it.
> 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-seven-year-olds-graph-coloring-chromatic-numbers-eulerian-paths/
http://jdh.hamkins.org/math-for-eight-year-olds/
http://jdh.hamkins.org/math-for-nine-year-olds-fold-punch-cut/
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/
Someone (I had thought it was MJD, but uncle GOOG is not pulling it up on his blog) once told a story about how software estimation was a bit like transfinite recursion on ordinal numbers: sometimes you have an estimate in ℕ, and it's possible to say "I'll be done in 5 days, give or take", but sometimes your estimate is above omega, and the best you can do is tell someone who asks "Right now, I don't know how long it'll take, but come back in 5 days (give or take) and then I can tell you".
(and it wasn't JDH, either: closest I get there is https://jdh.hamkins.org/transfinite-nim/ )