To be fair, "functional analysis is infinite dimensional linear algebra" is a saying for a reason, that's an intuition a good professor should have given you. Mine gave me it so well, I still see no difference between R^n and a Hilbert space! (on the dangers of intuitions)

This reminds me of a short lived stretch in French mathematical education, the "Bourbaki years". For those unaware, Nicolas Bourbaki was "a dozen mathematicians in a trench coat" who, in the early 20th, reformed mathematics from the ground up, their work is notoriously opaque (to some, perhaps a revelation to others), if rigorous. It served as inspiration for ambitious mathematics education reforms in the 60s and 70s.

Some of the things my father had to contend with:

- naive set theory and non-10 basis arithmetic in primary school (age 7)

- set theory in early middle school (ages 10/11), maps over sets, so I imagine things like injectivity and bijections. "[...] a different approach of arithmetic, computations often replaced by a more abstract theoretical approach": I wonder how that went in the classroom

- general Algebra introduced at 15yo, groups, rings, fields, vector spaces; 16yo mostly linear algebra (vector spaces, linear mappings etc; in true French fashion I'll bet they didn't see many 2x2 or 3x3 matrices that year)

I'm quoting Wikipedia "Mathématiques modernes" here, my father only told me of the last point himself. Though I do have his notebooks from early higher ed, they had general topology before metric space topology, for example, in year 1 or 2 (probably 1, because how can you do anything before you know topology...). It was all like this, a theory that builds on itself and you can't skip any steps. A finite dimensional space is a particular case of an infinite dimensional one, so you start with the latter. Backwards from how things were constructed and are understood by people.

The fact these reforms survived about 10 years and have been completely reverted is testament that this approach probably doesn't work.

> To be fair, "functional analysis is infinite dimensional linear algebra" is a saying for a reason, that's an intuition a good professor should have given you. Mine gave me it so well, I still see no difference between R^n and a Hilbert space! (on the dangers of intuitions)

That's the thing: not only he didn't share this intuition, he actively prohibited people from bringing it up in the class because it can lead to mistakes.

The program was to start with most generic cases (Banach spaces, metric spaces) and prove whatever is provable there, then continue adding assumptions one by one and proving stronger and stronger theorems. I think we've reached Hilbert spaces by the end of the year and that was a gotcha moment for many students (wait, these vectors were functions the whole time?), but it was too late. Everyone failed miserably at proving or understanding all the preceding theorems because without the intuition it turns into a game of symbols with no structure or hope. The only recourse were those bootleg analogies from finite spaces and Fourier analysis that a few students knew or came up with.

The "New Math" you mention above gives a good frame of reference. A beautiful curriculum that's almost impossible to understand unless you already learned math the normal way -- a formally incorrect and inconsistent but reliable way.

I also made similar mistake myself when I tried to teach the C language to 12 y.o kids without saying the dreaded "just write it like that, you'll understand later". That experiment failed completely but I only understood why two years later when I myself was subjected to the functional analysis course. Or I think the complete realization came to me even later, with the C language experiment and the functional analysis story becoming pieces of the same puzzle.

A small note on Bourbaki:

Alan Sokal's famous and brilliant hoax Transgressing the Boundaries: Toward a Transformative Hermeneutics of Quantum Gravity had a hilarious footnote that read (my highlighting):

> Miller (1977/78, especially pp. 24-25). This article has become quite influential in film theory: see e.g. Jameson (1982, 27-28) and the references cited there. As Strathausen (1994, 69) indicates, Miller's article is tough going for the reader not well versed in the mathematics of set theory. But it is well worth the effort. For a gentle introduction to set theory, see Bourbaki (1970).

I'm torn on this one, because there's no way I can accept "lies-to-children are for their own good", but on the other hand I'd hate to be overburdened with information about Peanut maximums or the Ophthalmic supergroup when all I wanted to know was how to do division or what kind of thing a mushroom is.

Once you accept that a model of reality is always somewhat incorrect you kind of stop seeing this as "lies". Reality is infinitely complex, a model however is finite and therefore understandable.

We value models for their simplicity, for their ability to disregard secondary and tertiary details to allow comprehension and prediction of primary effects.

Then the only question is: which level of the simplicity/correctness trade-off is appropriate in given circumstances? When teaching kids or even a non-professional adults, the appropriate level is often very heavy on the "simplicity" side. Adding more complexity results in less overall understanding, so a net-negative effect in the end.

I think the problem comes in when you have adults teaching kids a simplified model who don't actually understand that is what they are doing.

It is not always conveyed well that they are learning a simplified model, or using an analogy. That's when kids feel lied to.

There are very many pithy sayings to this effect, most famously "All models are wrong, but some are useful" (perhaps by George Box, but I can't find this exact phrase attributed to him)

Even Von Neumann said "truth is much too complicated to allow anything but approximations."

It's not hard. "It's sort of like that, but the details are complicated and you'll learn them later if you want" is a perfectly fine answer.

A big difference between a lie and a simplification is whether the other party knows it's a simplification. Communicating that is independent of specific simplification you use, and it's sometimes as easy as saying "is mostly like" / "sorta" instead of "is".

I was learned enough in school to know when teachers were “teaching” something oversimplified. They uniformly acknowledged that when asked, until one didn’t. That one didn’t care about the nuance and insisted I give the “expected” answer when asked and ruined my attitude in school for years (not the only event, but it didn’t help). Obviously I’ve never forgotten her.

One thing I adore about this community is the broad acceptance of “I don’t know” and “it depends” as the starting point for answers.

To bring that back into the tech/business world:

My personal rule for being an expert/consultant is to be very willing to say "I don't know, but I can find that out for you", and define that as my real expertise: know how to find out things.

Technology stacks, programming languages, all of those things come and go. The skill to pick up whatever is needed is long-term better.

It wouldn't be so nice if it was oversimplified and incorrect to the extent that the child realizes it doesn't add up.

Had this experience more than once, but the one that comes to mind:

In high school chemistry, the teacher was reviewing what we should remember from lower school about atoms & molecules (single bonds, double bonds, slots available for single atoms of different elements). I asked something like "If the model we were using treats the bonding sites on every atom of every element as equal, but says that some molecules are either rarer than others or not known to occur in nature, doesn't that mean the model is incomplete or incorrect?". She instantly shook her head with annoyance and said sternly "No.".

From my limited experience (being a child and raising one child) I would say that the problem is more that we (as adults) pretend that the simplified thing is the whole picture.

Every time someone told me: "this is a simplified version to give you a basic understanding and later you will be able to learn more accurate versions" that was tremendously helpful and sometimes it sparked my curiosity and motivated me to look stuff up myself.

I mean we should get ahead of the realisation that it doesn't add up.

When I moved to the US in 3rd grade, I was marked wrong for putting negative numbers as answers on a math test. The correct answer was "You can't subtract a larger number from a smaller number".

I had a similar experience, but was fortunate that the teacher took me aside and explained that I was right but they were teaching a simplified version. In hindsight that was really a helpful approach.

I got into an argument with my science teacher in 4th or 5th grade about whether rivers could flow north, because on a globe, north is up.

I suspect that the best approach would be not to "get ahead", but rather see this realisation develop naturally and help it along the way. It's hard to get critical thinking if everything is already criticized before you even come.

This can work with your kids though but I don't know if it can be effectively scaled to a school class. Maybe on a few dedicated play/discovery sessions, not on a regular basis.

Doing this would create a nice opportunity to teach the right and wrong ways to move the needle in scientific discourse (e.g. don't start by holding a press conference about cold fusion).

Graduation could mean an overthrowing of the known-wrong models. Congratulations, here are the next known-wrong models. Now prove these wrong.

Thankfully at very young ages where this is required, that level of critical thinking is very far off

I love your point that avoiding oversimplifications leads to disaster. It explains why I struggle to have a purist mindset while also unable to explain why things go poorly sometimes.

What's your point then?

All minds alike, big and small, benefit from simplification. I was replying to a comment that specifically referenced "small minds", that's all.