That's interesting. I'd seen Arnold's work on teaching mathematics, and I'm very much aware of the dry formalism in Bourbaki books. However, I was not aware of the Arnold-Serre debate.
But the thing that really caught my eye here is your claim that there were visible effects of Bourbakism polluting mathematics.
Do you have any examples or specifics of that "visible effect". My question is a genuine one, not a challenge to what you are saying.
If you speak/read French, try to have a look at elementary/middle school math books from the 60's/70's in France. It was a disaster and forced a lot of kids to imagine they were bad at maths.
I had discussion about it with Cedric Villani once and he told me that was not a disaster because of the books or the material, but the teachers were not good enough.
Eg. when teaching set theory, teachers were debating what should be a perfect representation of a set (is it an ellipse or circle... etc.?)
Children were completely lost because of this.
If you don’t draw a perfect ellipse to represent a set then you had a bad mark and so on and so forth.
Sure but what good is an education method if the teachers don't understand it. I don't doubt that if you had a great teacher that would work, but considering how focused French education is on mathematics, this was one of the main reason why the social ladder broke.
> I understand, but on the same period, France grew a whole generation of brilliant mathematicians. I don't think that the two are completely unrelated.
If you could prove me that there were more brilliant mathematicians educated in France during the 60's/70's compared to other periods, I'd be interested.
Even if this is the case (which I'm not sure), at least it's pretty clear it lowered the median (whether this is good or not is another debate ;)
But how did Bourbaki influence elementary/middle school mathematics? Bourbaki mathematics is basically at the level of graduate school. Except in countries where a graduate degree is required to teach mathematics, I can't imagine how there would even be a connection between the two worlds. Is that the case in France?
Again, I'm not disputing what you are saying either, merely trying to understand what happened.
Is it still going on in French schools today? Or was it 60's/70's only?
AFAIK the idea at that time was to try to apply the Bourbaky method to teach kids elementary maths - e.g. try to explain the concept of sets before kids know how to add 2 numbers (I'm caricaturing but not that much - you had to explain to kids Peano before addition..). Someone else might have a better answer for that as I'm too young to know. Wikipedia French page on 'New Math' mentions it but not in details.
> Is it still going on in French schools today? Or was it 60's/70's only?
Unless you are in a reaally elitist high school in Paris with a really old teacher, this should'nt happen nowadays.
I'm Czech and this revolutionary method of teaching came here in the 90's, when I entered my grammar school. Probably, it had a political “east vs. west” motivation among others, shortly after the velvet revolution...
The first thing I learned in the 1st year of schools were basic concepts of set theory. We were drawing circles, ellipses and Venn diagrams (even though we didn't call them like that) filled with images of apples, plums and cherries. Teachers explained to us what an intersection, union and set differences are and we were supposed to draw items into one set, but not in another set, etc.
I recall these exercises as funny and playful. They were similar to IQ tests in the sense that the exercise is logical, slightly entertaining, but highly abstract and loosely related to the world you know.
And I think this was the main issue. The 2nd topic we learned was simple arithmetics as in standard educational systems. However, at that time, I didn't see any relationship with the concepts of set theory.
Was the system any good? Hard to say. AFAIK, it was dropped after a few years. Eventually, I obtained a PhD in computer science, so at least, the system wasn't a complete disaster for me. :-)
I think teaching sets like this totally misses the point. Finite sets are trivial (and pretty useless), so there is no use for schoolkids, it is just a waste of time.
On the other hand, arithmetic for very young kids like second-graders is probably also a waste of time. If you waited till they're older (or they express a genuine interest) I expect they'd learn it quickly enough and with less chance of learning mainly to hate math.
Perhaps it made sense in ye olde days when many students would not go on to junior high.
You mean adding and substracting numbers? Personally I think it should be taught (and is here) to first graders. Also here no kid hates math before 5th grade (overly generalizing).
My sample number might be too small (so it is probably anecdotal evidence at most), but I would say german primary schools (which is grade 1-4). To me it looks like the math hate is coming later starting somewhere in secondary school between grade 5 to 7.
Thanks. I based my 'probably' on a few things -- it's admittedly not strong:
- There was an American school about a lifetime ago that tried that strategy, and the principal claimed it worked fine.
- A general impression that before mass education it wasn't uncommon for people who got schooling to start years older than our start, with no impression that they learned arithmetic any worse. E.g. http://www.scientiasocialis.lt/pec/files/pdf/vol57/90-101.Pi... "In general the pupils began their arithmetical instruction at 10–11 and this
education prosecuted for two years."
- Piaget's picture of stages of development (in my vague understanding) suggests that arithmetic beyond a very concrete level would be developmentally unnatural for younger kids, and more natural later. Apparently Montessori schools do better on this score?
- Unschoolers sometimes reach adulthood with less understanding of math than state-schoolers, but if the average is worse, I haven't heard of it. Anecdotally they're fine.
- Hate and ignorance of math is very widespread (I've read similar claims about average French people with their substantially different school system)
- This jibes with my general experience, having gone to school, etc.
- There's not much reason to expect a claim this far from mainstream to have been carefully studied. Maybe it has been and refuted -- I just don't very much respect the status quo and so I expect there are improvements that would 'easy' except for the obstacle that it's very hard to meaningfully change the system. And this strikes me as a plausible (though unambitious) one.
When I learned set theory and relations for the first time, I was in love with the abstractiveness of mathematics. It was so intuitive and I truly felt that I've got the grasp of fundamental principles of maths. The sad thing was, we were introduced to set theory in Grade 11(17 years old), that's too late IMO, it should be taught around Grade 4.
In most countries the schoolbooks had been changed to teaching about sets and stuff in the first grade. Generally more abstract or top-down approach. Old school approach was more bottom-up.
In my country I have been still learning from old schoobooks while classes below had new schoolbooks - thats why I remember that well.
Recently I have been helping 11 year old with classes (remote learning now). And my impression is that this top-down approach is still present but the schoolbook was full of practical life examples (money issues, understanding newspaper articles with pecentages and percent points etc.)
But the thing that really caught my eye here is your claim that there were visible effects of Bourbakism polluting mathematics.
Do you have any examples or specifics of that "visible effect". My question is a genuine one, not a challenge to what you are saying.