Comment by vanderZwan

Yes, I agree! And also that a lot of the fun stuff is hidden behind historically opaque terminology. Although I'm also sympathetic to the fact that writing accessible explanations is a separate and hard to master skill. Once you understand something it can be really hard to step back into the mindset of not understanding it and figuring out an explanation that would make the idea "click".

I think a lot of maths is secretly a lot easier than it appears, but just missing an explanation that makes it easy to get the core idea to build upon.

For example, I've been meaning to write an explorable[0] for explaining positional notation in any integer base (so binary, hexadecimal, etc) in a way that any child who can read clocks should be able to follow. Possibly teaching multiplication along the way.

Conceptually it's quite simple: imagine a counter that looks like an analog clock, but with the digits 0 to 9 and a +1 and -1 button. We can use it to count between zero and nine, but if we add one to nine, we step back to zero. Oh no! Ok, but we can solve this by adding a second counter. Whenever the first counter does a full circle, we increase it by one. A full circle on the first counter is ten steps, so each step on the second counter represents ten steps. But what if the second counter wants to count ten steps? No problem, just add a third! And so on.

So then the natural question is... what if we have fewer digits than 0 to 9? Like 0 to 7? Oh, we get octal numbers. 0 and 1 is binary. Adding more digits using letters from the alphabet?

The core approach is just a very physical representation of base-10 positional, which hopefully it makes it easy to do the counting and follow what is happening. No "advanced" concepts like "base" or "exponentiation" needed, but those are abstractions that are easy to put on top when they get older.

I've asked around with friends who have kids - most of them learn to read clocks somewhere between four and six, and by the time they're eight they can all count to 100. So I would expect that in theory this approach would make the idea of binary and hexadecimal numbers understandable at that age already.

EDIT: funny enough the article also mentions that precisely thanks to positional notation, almost every adult can immediately answer the question "what is one billion minus one".

[0] https://explorabl.es/