https://www.theguardian.com/world/2012/may/31/europa-french-...
So please don't think France is a backwater that just doesn't know how to do math - on the contrary, it may be the best country in the world to be born into, if you want to be a mathematician.
For what it's worth, I'm an American who honestly is pretty bad at math, but I try to keep learning, though I'm past an age where it will matter in college or higher education.
"In America, they start by teaching you how to add two numbers, then once you've mastered that you learn to subtract, then once you've got that down you learn to multiply and divide and so forth. In France they tell you: this is an arithmetic object. The operations on it are add, subtract, multiply, and divide. It's simple!
It's a good way of cranking out a bunch of Fields medalists, while everyone else has no idea what's going on."
At this point there is a specialized track (preparatory classes) to get into the most prestigious engeneering and buisness schools that will have you cram mathematics / physics / computeur science / english / a mix of litterature of philosophy in two or three years.
The student then get into the various school according to their grades on a serie of national exams.
This track is very competitive (and as such often decried) but I believe that it is a very efficient way to get the top talents to the top institutions with a maximum of learning on the way (also note that this education is free and that the very best school will even pay you).
Note that a student can also choose to get to college and ignore this track.
> A British mathematician was giving a talk in Grothendieck's seminar in Paris. He started "Let X be a variety...". This caused some talking among the students sitting in the back, who were asking each other "What's a variety?". J.-P. Serre, sitting in the front row, turns around a bit annoyed and says "Integral scheme of finite type over a field".
found it on /r/math: https://www.reddit.com/r/math/comments/7qjxbe/whats_your_fav...
I should have said the Fields Medal, as indicated in the article.
I’d much rather measure an educational system by how much it teaches the average person, than by how many Nobel prize winners it produces.
[0] https://en.wikipedia.org/wiki/Fields_Medal, second paragraph
it's interesting that some academic approaches focus on the theoretical framework and conceptual depth, with the underlying assumption that the student would "get" it once the fundamentals are understood.
The other approach, mostly in the asian schools, teach rote and speed in mechanical/mental manipulation. For example, a student would do 1000 quadratic equations, mostly by rote using substitution into the formula. That constitutes "teaching" of the quadratic equation.
But in the end, who learned more? Not sure. But i do believe that being able to manipulate numbers very quickly leads to a quality of its own, and can make understanding harder concepts easier.
It’s also clear if you’re succeeding or failing. I’ve noticed that private school parents in the US seem to hate homework. The kids get three optional problems a week and no tests. All the time that they might’ve spent thinking on their own about problems is instead spent in class, which really limits how much new stuff can be taught. The kids think they know everything because there are no grades. The kids that actually perform at grade level have private tutors giving them homework, while the rest are just playing fortnite and unable to do fractions.
What was the point then? It's a complete waste of time in that scenario.
The way I see it, doing the thing gives you ample opportunity to figure out theory on your own - repeating a menial task prompts the brain to invent shortcuts. Meanwhile I can't imagine someone learning only theory to not fail spectacularly at practical tasks - because theory doesn't capture everything; there are lots of practical, intangible things that crop up when you start doing the work. On top of that, it's easy to feel you understand something without actually understanding it.
The best way, IMO, is what philipov's father said - you have to alternate between theory and practice. It's how you can build understanding faster while continuously validating it.
Can you elaborate on this? Are you supposed to introspect while you're applying the quadratic formua 1000 times? How does being able to quickly add/subtract/multiply/divide help you better understand abstract concepts?
This should be obvious from any number of examples, from driving a car to typing as a programmer. Reducing the friction between your brain and your tools simply makes you more effective.
I studied music theory for about 5 years more than 10 years ago and remember some of it. I like to compose music and sometimes like you said I'm really losing a lot of time/energy because I don't master the basics. Yet some other times, I don't even think about music theory and let my inspiration go - and it's often better quality.
I find this effect to be true in programming as well. I find myself restricted by the frameworks I know and sometimes my mind can't think outside of it.
Tangentially, a lot of fantasy authors seem to love maps for their own sake, and provide maps of their world as illustrations. I never saw the point of these; the map never really matters to the story.
But history books provide almost the same number of maps -- usually one, before the discussion of whatever -- and it just isn't enough. Those maps really matter to understanding whatever was going on; in the real world, geography is a huge deal.
It's more about deepening the story I think and making it 'bigger' I guess or another type of story telling. You look at a fantasy map and there's typically places on the map not even mentioned in the actual story as you say, but that helps build the setting.
Look at the map for the lord of the rings for example. A good number of the places in that map are barely mentioned in the actual trilogy. Yet that map's become one of the most iconic in fantasy in general.
It's more about what the maps inspire in your imagination while you read. When some random fantasy city name gets mentioned, you can look at the map and get perspective about where it is in the world.
It's also helpful I think to keep the world consistent for authors. It gives them a frame of reference for scale and distance in their stories and likely helps keep track of all the random names and cultures and such that they make up for their stories.
Fun times and theories co-bro. Anyways.
If it were somehow possible to let kids chop wood for 10hrs, as the need arose, or whatever, and then they could go / go back to school when they were ready.
Certainly would have helped me a lot.
Ironically, her independence from ZF would make the classical Bourbaki group shiver. The same way the earlier, better known, work of Goedel did.
I don't know, if you are aware of Bourbakis spirit. There is something much more important in this context as the content-wise difference between sets and numbers. I never would argue against applying naive set definitions in a suiting way. But Bourbaki had an agenda, and this agenda is not for everyone (certainly not for me).
Well what’s wrong with that?
I got my math degree in California (at Cal) and the approach was a lot closer to what you describe French books as. Many of my courses were just a semester-long discussion of proof after proof, with breaks for exams that were more proofs.
"¿Por qué no los dos?" as they say? One can also detail how to "calculate" and "manipulate concrete data" in a rigorously correct way, involving precise concepts and perhaps some theorems. (Though an overemphasis on theorems is not necessarily good, either. Sometimes it can even signal a lack of attention to the actual "rules of the game" one is ultimately supposed to be studying, while a focus on direct manipulation would perhaps tend to avoid this. A good balance of the two styles is perhaps best.)