My few perhaps-non-obvious bits:
If you find yourself saying "really, really (physically) small", or "N times smaller than a human hair", the approach at the top of this[1] might work better.
Estimation is now taught down to K... but oh my is it done badly. PBSKids: "estimate how many jellybeans are in the jar: look at the jar; then write down a number!" (really). But it's been a few years since I looked, so maybe there's something competent now. My suggestion is to emphasize bounding, as it supports better conversations. "How many people are reading HN today? Is it more like 1, 10, 100, 1000, 10k, 100k? Who can suggest a bound? Oh, I'm reading HN! So a hard lower bound of 1! Does everyone buy that? ...".
Toddler-initiated predictable conversations by pointing out selected elements of the environment. Eg, "a map!" and "rust!". "What does it show? Where are we? Where are we going?". "What's the thing rusting? How's the oxygen getting in? What's the rust doing? What did we have for breakfast, that we're oxidizing now?". The map one works best with urban mass transit and museums and such. Rust is everywhere.
[1] How to remember the sizes of small things, top of http://www.clarifyscience.info/part/Atoms
For math, I recommend Hejny method; I have great experience with it (as a student) https://www.h-mat.cz/en/hejny-method https://www.youtube.com/watch?v=xm0xsBjdMe4 It is now used experimentally in Czech schools, but the textbooks are translated to other languages, too. You can buy the textbook with the teacher's manual and try it at home. Short version is that this method makes kids discover math on their own (with a lot of nudging, of course), so they can enjoy and remember it better. Works both for gifted and average children, but they will progress at different speeds.
Interesting comment overall, and also the part I excerpted above. I had read Akio Morita's (Sony co-founder) book, Made in Japan, some years ago. Wonder if that (what you said about science education in Japan) was part of the reason for Sony's success.
Here's a context-relevant place to start: http://slatestarcodex.com/2017/07/31/book-review-raise-a-gen...
On one subject, a mentor can be a master, and master instructor. And such instruction is crucial for developing deep expert understanding. But in the future, with better content, and improved learning infrastructure, one might imagine these becoming available for more than one subject.
Both the concentration on one subject, and that subject being chess, contribute to the effort having nice properties. Like non-superficiality. An integrated and deeply organized body of knowledge and skills. Developing transferable knowledge (within the subject at least), rich feedback, and reflective building of mastery. But it's the properties that matter. Chess, or another one subject, might be taught in a way that fails to have the nice properties. And as learning infrastructure improves, the nice properties may become available without the "one subject" or "and it's chess" restrictions.
There's a long history of exceptional masters teaching their field/craft to their children from a young age, who in turn become exceptional masters. The challenge is to scale that.
The straightforward "a gaggle of masters in everyone's pocket" is AI-complete implausible. But the art of the next decade is crafting human-computation hybrid systems. Things like eye tracking, and big data, provide opportunities that no master mentor has ever had. We just need to get around to building on them.
A quick summary of the "reasons":
1. Boom: at that time (from 1867 till WWI) Budapest was booming, more attractive to immigrants than New York. Actually, most of the city you can see today was built in those times.
2. Culture: as a result of the boom and the wave of immigrants it became a very liberal and open minded place (though this did not apply to the feudal class at all - their kids typically became soldiers or playboys)
3. Motivation: in a feudal society studying and intellectual eminence was the way to go unless you were born an aristocrat. Parents, students, teachers were willing to put time and money necessary to make their kids excel. This may not sound unusual with all those helicopter parents you see nowadays, but actually this was huge. Imagine growing up in a family where you knew - and your whole family knew - that your only chance of making it is to be the best at math your abilities allow.
4. Education: I don't know where to start, so I can only give examples. Imagine you are a 12 year old child and your teacher borrows you his favourite papers on quantum physics and asks your opinion on them. Then you give a smart comment and your teacher contacts the relevant professor at the university to have a tea with you at your house next Sunday.
5. Language: most of these kids learned Latin and Greek, and in before their teenage years also spoke at least German fluently.
6. The "marble table": Stanislaw Ulam (in his autobiography, another amazing read) and also MacRae tells about the most important ingredient, the marble table at the café (the easy-to-erase whiteboard of the time). In Central Europe mathematicians met at cafeterias, discussed all day (often meaning 12 hour days at a cafe!), challenged each other, and did rarely work in isolation, not worrying too much about "who thought of it first". They happily took young kids in, 15 year olds sipping juice and 50 year old Banach drinking something much stronger (may not have been Banach, but you'll read the book!). It was such a well-known "way of doing math" that the IAS in Princeton was officially established to re-create this culture and pull the typical American professors out of their ivory towers.
It is an elitist system, I know, and does not solve mass education challenges. But this small elite circle had an impact on almost everyone in the country's education system. Even if you weren't a Wigner, Teller or Neumann, you spent 6 years in this environment and possibly became a great teacher, similar to the one who taught half of these people, the great László Rácz [1] and taught in this fashion.
Also, a similar great science education happened in Japan at some point (50's, I think ), but I only read this as a side note in the book on Neumann. Anyway, it is possible to do this again and with two small kids I'm very interested to know how.
[1] https://en.wikipedia.org/wiki/L%C3%A1szl%C3%B3_R%C3%A1tz