State of the Discussion

First we smash all the (electronic*) calculators.
Get the logarithm table and/or the slide rule back.
That's how one teaches the basics!
In all seriousness, I work as a tutor and by now mainly for math. Before one can teach them abstract concepts, they need to get the basics right and that means calculations (preferably without electronic help). I see no justification to teach set theory** when they get still puzzled by "Is 123 divisible by 3 without residue?" or "What's the square root of 81?". If that's 19th century, then Gott erhalte Franz den Kaiser!
To me that sounds like the unfortunate discussion about why and whether kids still have to learn to write by hand in our modern age. Why learn orthography when there are spellcheckers? And what by the way is the use of kids reading fictional literature of guys long dead or ancient (pre-1990) history?
Yes, understanding should have priority over rote learning buy everyday math is still mostly basics. And I am cynical enough to say that we are failing there already. Imo math at school should concentrate on practical problem solving not Zermelo's theorem.
*abaci will be tolerated
*exception: difference between natural, integer, rational and real numbers.

For any mathematician alive today, mathematics is a subject that studies formally-defined concepts, with a focus on the establishment of truth (based on accepted axioms)
Absolutely, however teaching "formally define concepts, axioms, proofs, etc" are taught in PLANE GEOMETRY, not so much in calculation/algebra type classes.

Here's a post from Keith Devlin working through some thoughts about the tension between calculation and mathematical thinking.
https://devlinsangle.blogspot.com/2018/05/calculation-was-price-we-used-to-have.html
For any mathematician alive today, mathematics is a subject that studies formally-defined concepts, with a focus on the establishment of truth (based on accepted axioms), with various forms of calculation (numerical, algebraic, set-theoretic, logical, etc.) being tools developed and used in the pursuit of those goals. That’s the only kind of mathematics we have known.
Except, that is, when we were at school. By and large, the 19th Century revolution in mathematics did not permeate the world’s school systems, which remained firmly in the “mathematics is about calculation” mindset. The one attempt to bring the school system into the modern age (in the US, the UK, and a few other countries), was the 1960s “New Math”. Though well-intentioned, its rollout was disastrous, in large part because very few teachers understood what it was about – and hence could not teach it well. The confusion caused to parents (other than mathematician parents) was nicely encapsulated by the satirical songwriter and singer Tom Lehrer (who taught mathematics at Harvard, and did understand New Math), in his hilarious, and pointedly accurate, song New Math.
As a result of the initial chaos, the initiative was quickly dropped, and school math remained largely unchanged while real-world uses of mathematics kept steadily changing, leaving the schools increasingly separated from the way people did math in their jobs. Eventually, the separation blew up into a full-fledged divorce. That occurred in the late 1980s. The divorce was finalized on June 23, 1988. That was the date when Steve Wolfram released his mammoth software package Mathematica.[...]
Devlin is really good on matters pedagogical, and always worth the read.
I do tend to think, though, that students will have a very hard time with understanding math (or written communication) if they have not had enough experience with doing the work, and not seen enough examples to get an idea of the possible range of approaches to doing the work, etc.. Early in my teaching I tended not to give enough examples, figuring that teaching the conceptual side would lead students to sort through their own database of examples to see the underlying principles. I've since learned that most students come in having seen and understood too few examples, and having no idea of more than one approach to the tasks they have been called upon to do.
I do a lot more modeling of approaches, and evaluation of those approaches, now that I'm finally starting to figure out this whole teaching thing.

I was taught using the School Mathematics Project, which seemed OK to me. But I may not be one of the "normal people".
I suggest that being able to divide accurately with pen and paper is now almost useless, whereas being able to divide approximately in one's head is useful for avoiding fat-finger errors. That is, the underpinnings have turned out to be more important than the algorithms.

Camel notation* from computer programming would possibly be better: InternalCompustionEngine.
It is interesting that it is widely used in domain names, e.g. KaiserPermanente.org Clearly the sales and marketing folks think it will be easier to parse that way.

Michael, the scrambling thing is called Typoglycemia. In German it's Buchstabensalat (letter salad) and at a pedagocic course I had to suffer through at university as Badewanneneffekt (bathtub effect, no idea why).
https://en.wikipedia.org/wiki/Transposed_letter_effect#Internet_meme
With long words with completely random letter scrambling it's rubbish even if first and last letter are retained.

"Whole language" may have been a wrong turn if it really resulted in ignoring phonetics (I doubt that it really did in practice), but the fact is that proficient reading requires recognition of whole words.
To paraphrase a friend, "No one can get a PhD dissertation out of pushing phonics. You have to claim that something else is better." Or at least that something else is as/more valuable than recognizing the words early on. The new things are all well and good, but memorizing a few hundred words-as-a-chunk is still necessary. You can't sound out "bat" and "cat" forever; at some point it has to be automatic.
There are assorted postings -- the internet has made them more common -- that ask whether you can read "Aoccdrnig to rscheearch at Cmabrigde Uinervtisy, it deosn’t mttaer in waht oredr the ltteers in a wrod are, the olny iprmoetnt tihng is taht the frist and lsat ltteer be at the rghit pclae." Is this just an English thing? Does Romanized Japanese tolerate the same sort of misspellings for fluent readers?
New Math was the same sort of thing. It pushed a much broader view of what math was than just the algorithms. Look, long division is done the way it is because hundreds of years of experience informs us that it's the best way to get the right answers when you have to do a hundred division problems a day, day after day. New Math failed when the teachers pushed the broader view but didn't teach the mechanics.

It's just convention that one does not write railwaystation or particleaccelerator or internalcombustionengine but imo those are perceived as units.
German also has the useful convention of capitalizing nouns. Using Internalcombustionengine would at least signal it's a noun, even though it starts off with an adjective. Camel notation* from computer programming would possibly be better: InternalCompustionEngine.
The example everyone remembers from college German is Handschuh (hand shoe) as a generic glove/mitten term, Fingerhandschuh for gloves, Fausthandschuh for mittens, Panzerhandschuh for armor, etc. The German is at least consistent. In English, it's another of the English/Norman dualities. Glove is from Old English; mitten is from Norman for mitten; gauntlet is from the Norman for glove. I understand there are more types of Handschuh that correspond to some of the other uses English has piled on gauntlet, like "throw down the gauntlet" or "run the gauntlet".
I'll just go ahead and invite Hartmut to explain how wrong I am :^)
* Off and on for a half-century now I have occasionally tried to adopt camel notation when I'm writing code. It never lasts, and I always go back to underscores: source_index rather than sourceIndex in something I've been writing this week.

I'm puzzled by this. I'm not good at languages, relative to my other skills, but switching alphabets - Cyrillic, Greek, Georgian... is trivial.
It's not onerous, no, but it is a factor on at least two levels in my experience.
First off, it can create some noise when particular letters look similar to letters in the other language that are not phonetically equivalent, and that usually triggers a bit of recursion in the reading process. It's not a lot of load on the system, but it is processing power that is not being used to make sense of the meaning. Writing English using the Greek alphabet barely affects reading comprehension when deciphering the message when one is fluent in English. Combine a lack of fluency with the need to decipher and the effects compound.
Second of all, it messes with the pattern recognition that one relies upon when skimming a text. When I'm reading Swedish or Spanish, I can skim the text fairly easily and a lot of the language has enough root-equivalency to make those reading skills transfer. That sort of whole-word pattern recognition doesn't fire the same way when I am faced with another alphabet.
All of these things mess with your language in the same way that when a student is asked to write about an unfamiliar topic with its own technical vocabulary, they often end up writing language that has a greater number of grammar and spelling errors than when they are writing about familiar topics. The familiar has a much simplified processing economy.
And again, with functional and transactional language, these difficulties are much less pronounced than when dealing with more complex and nuanced subjects.
At least that is my experience, and it seems to match with my observations of how my non-native student writers interact with texts. Actual linguists would likely have a lot to say about the places where I'm wallpapering over some complex topics, or missing the boat entirely.

...the alphabet contributes somewhat to that difficulty...
I'm puzzled by this. I'm not good at languages, relative to my other skills, but switching alphabets - Cyrillic, Greek, Georgian... is trivial.
(I struggle with Hebrew, where some of the letters are just too alike.)

It occurs to me that a lot of education should be guiding people to what they are good at. I always thought I was good at languages, but I realize that I am really good at the languages that are in roman letters. Had I known, I might have opted for vietnamese instead of thai, or even earlier, dove into chinese first (I have several english native friends who have fluency in Japanese and Chinese and all of them did Chinese first and I don’t know anyone who went the opposite way).

Instant translation is fine for functional and transactional language, but it hits its limits pretty quickly as language complexity increases and becomes problematic for understanding as soon as there is an intertextual element at work. I see this a lot with my international students when they are working their way through English texts with the help of translation software. They miss a lot of the features that the authors are using to communicate - parallelisms, homophones, puns, etc.
To be fair, a lot of my native language domestic students miss those things too, but the international students have the reading skills to catch those elements in their own languages, and would notice those things if they were actually working with the original text.
One thing I can add that speaks to lj's first point. Language-wise I've studied Spanish, French, Swedish, and Ancient Greek. I can muddle through in Spanish, and would probably be able to attain fluency in any of the first three in a few months with immersion. Greek, however, never sticks particularly well, and the alphabet contributes somewhat to that difficulty. It's one more unfamiliar element (deciphering) that takes up processing power that would otherwise be used for linguistic sense-making.

In case anyone is interested in the subject (and in lieu of fraught AI summaries):
https://en.wikipedia.org/wiki/English_as_a_lingua_franca
It's entirely possible that English will become the lingua franca for international communications, but if it does, I'd expect, like Hartmut, that it continues to shed irregular constructions and colloquialisms and that native dialects will be treated as quaint variants with charming local color. I also predict that both Americans and Brits will complain bitterly that ELF is "not proper English" when that happens, and resent any standard that treats ELF as the paradigm.

Sorry if I misunderstand you, but are you saying that the use of national languages in the native countries will disappear or be reduced? I don't think that would be realistic.
Yes, that is exactly what I am saying. Not that it will necessarily be quick. But it will happen.
See, for example, the various Native American / First Nations languages. They haven't, quite, died out and there are various efforts to save one or another of them. But the reality is that native speakers are overwhelmingly old. Children may retain some fluency, in order to speak to their grandparents. But for everyday use, they speak English. And the children's children will be straight Anglophones.
We see the same phenomena in immigrants. My wife's immigrant grandparents were functional in English, but spoke Japanese with family and friends. Her parents were fluent in Japanese (from talking to parents and aunts and uncles when growing up), but generally spoke English except when talking to the older generation. My wife and her siblings? Even having lost virtually all of the Japanese I studied in college, I still speak more than they do.**
Granted, there is more inertia when it comes to languages with a big population base. So it will take longer. But modern communications mean that the next generation will be exposed to English far beyond the classroom. And anyone who interacts with the outside world, from academics to taxi drivers, will need to use it routinely. Already do, actually.
** When my wife and I first got together, we made occasional trips half the length of California to see her family. About the third trip, I got taken off to see Grandma. My future mother-in-law gave Grandma an explanation of who was this blue eyed blond, then introduced us. I remembered enough to say Hajimemashte. Grandma just lit up; from that moment, as far as she was concerned, I was in.

Re: changing systems of roman-alphabet spellings in japanese: it would be good to get Hartmut's input, since (IIRC) there was a systematic change in "official" German spelling ten? twenty? years ago.
Re: English taking over the world. There's a saying that "English is the Lingua Franca of Science".
Which I find amusing because "Lingua Franca" literally means "French", but (IIRC) is in Latin.
So there's three of the contenders for 'language to take over everything' right there in one sentence.
Why, yes, I *am* easily amused, why do you ask?

I am involved in an international organization (ICANN, if you care). I have a nagging (unspoken) embarrassment because everybody speaks English. Most of them barely distinguishable from native speakers.** Partly that turns out to because they did college or grad school in the US, or perhaps the UK or Australia.
But there I am, speaking only English. The German I learned in high school is mostly gone. The Japanese I studied in grad school is also gone. And, of course, because I grew up in California I know a few bits and pieces of Spanish. (Actually, taking a few Spanish classes is on my Really Need To Get Around To This list.) For the moment, I try to at least learn how to say thank you in the language of wherever we are meeting.
I have the suspicion that English will eventually end up as the world language. There will no doubt be long and bitter fights to preserve the national language. But they will, in eventual history, be seen as futile fighting tetreats.
These days, the world requires a common language to function. One will come: the only question is which one it will be. The Chinese will argue, as their economic power increases, for Chinese (Mandarin), but a tonal language is simply too difficult for anyone not raised in one. Spanish might be a viable option, but it lacks a serious economic power to push it. French might have a chance, except that its spelling is nothing approaching phonetic, which makes it hard to learn.
English, thanks to British and then American economic dominance (plus the fact that India, with its huge population, already uses it because its people speak 5 mutually unintelligible native languages), is already getting there. A lot of countries, not just Japan, start English lessons in grammar school. I won't claim that nothing could displace English. But it would take, at minimum, a couple of centuries of economic and cultural dominance.
** Working groups typically work in English (even if there are no native speakers involved), with a few having simultaneous translations for French and Spanish. For the three major meetings, we get translations in the 5 UN languages (English, French, Spanish, Russian, and Chinese) plus Arabic. And the language of wherever we are meeting, if it isn't one of those.

It's as if someone had deliberately did a lot of work to make it really, Really, REALLY look like a high-level conspiracy.
Of course, it doesn't take much work to make a conspiracy look like a conspiracy. And how likely is it that the incompetents involved could manage to fake one convincingly? Or even do a lot of work to try to?

Acosta, fed prosecutor that let Epstein off with a wrist slap on Federal charges, then got a Trump Sec. Labor job.
FL state charges could have happened, except "guess who? Pam Bondi" was FL atty general.
Epstein "suicided" during Trump's first term, Bondi now saying "nothing to see here" with fiddled video.
It's as if someone had deliberately did a lot of work to make it really, Really, REALLY look like a high-level conspiracy.

I know that the mainstream Dem attitude is that all this conspiracy crap is bad and should not be encouraged
I just don't get the Dems sometimes. Just because there are people who turn this into a wild conspiracy story that can be used for partisan purposes doesn't mean that there is no there there.
It was the same with the sexual assault allegations against Clinton. A more principled attitude would be beneficial.