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Comment by agumonkey | Hacker News Reader

I can only assume but culture makes things very different. I often said that French abstract algebra books stopped me from learning for 10 years (after many serious attempts). One book from Garreth Williams unlocked 80% of my issues. The shocking part was the approach. It was very operational, you calculate, you manipulate concrete data. In French books it's all theorems and concepts. Learning curve to make a Haskeller faint. And that book is not the only instance, Gilbert Strang lectures on youtube are also similar in style. It also correlates with the pragmatic engineering of US history (electricity comes to mind).

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I feel like it's worth pointing out here, for Americans who don't know better (no blame implied), that the French have one of the best mathematical pipelines in the world. They have almost as many Nobels (13 vs. 11) despite having 1/5 the population. Guardian article about them here:

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.

Do you mean the Fields Medal?

Ha! Yes I do. In fact you can tell my amateur-ness from this, that this misstatement didn't immediately jump out as wrong (though I do know there is no Nobel for math, when I stop and think about it).

I should have said the Fields Medal, as indicated in the article.

A paraphrased anecdote I heard about French math 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."

'Fields' or 'fields' or both, I laughed out loud. Mathematicians might find this quite funny.

Note that this was true at a time (some of my math teachers learned group theory in middle school) but that nowadays math education is much less abstract until the end of highschool.

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.

Tangent joke:

> 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...

To the question "what is 2 + 3" a French primary school pupil replied: "3 + 2, since addition is commutative". He did not know what the sum was equal to and could not even understand what he was asked about!

https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html

To be honest, what he's criticizing is also what is lacking in post-primary school math education. The mechanical reduction (3 + 2 -> 5) leads to a road block when you start thinking into solving equations.. because there's no single path of reduction. You need to see the formulas as material you can transform in your mind.

There is no such thing as a Nobel prize for math.

The strongest plausible interpretation of what the previous comment said is that it refers to the Fields medal, which has been described as the mathematician's Nobel Prize [0]. Assume good faith.

[0] https://en.wikipedia.org/wiki/Fields_Medal, second paragraph

I don't believe anyone here thinks France is an Academic backwater. France has had some of the most brilliant scientist/mathematicians since the 1600's.

> I feel like it's worth pointing out here, for Americans who don't know better (no blame implied), that the French have one of the best mathematical pipelines in the world. They have almost as many Nobels (13 vs. 11) despite having 1/5 the population.

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.

The two might not be exclusive. Perhaps one of the side effects of the masses having a good grip on trigonometry is producing more Nobel laureates.

I studied Engineering for a period in France at the time (undergraduate) I was very impressed at the level of rigour and abilities of my French peers. However this does not seem to link well with actual abilities as a practising Engineer. For instance French engineers are not particularily coveted.

> It was very operational, you calculate, you manipulate concrete data

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.

>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.

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?

In my experience, those rote mental muscles mean that the basics are never slowing you down when tackling a harder problem. It's comparable to having a perfectly organized toolbox vs a mess of tools in your shed - you can do the same things, but you're free to focus on the core of the current task instead of distractions in the trivial parts.

This is precisely it. I have a math-heavy job and play/write music, and the same phenomenon plays out in both situations: when reading sheet music or thinking in music theory is in your muscle memory, your entire conscious mind can focus on the actual, creative task at hand.

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.

It's very interesting because in the examples you described (music and programming at least), I sometimes feel limited by the tools I know.

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.

This might be controversial but this is why I like Python a lot. It just allows me to think about the problem without wrangling with syntax.

I really dislike dealing with large engineering systems in Python, but I fully agree when I just need to solve a self-contained problem and don't need to worry as much about it being maintainable or readable (or more accurately, don't need to worry about interacting with Python code that other people have written). I currently work a lot with both Python and C++, and I find myself occasionally trying to write C++ Pythonically and it's a huge headache (eg map/filter or list comprehensions are infinitely more readable than stuff like std::transform).

Yes, we can easily experience this by writing code in an unfamiliar programming language, or playing tennis left-handed. The theory is all there, but the muscle memory is missing.

In fact, the Soviet approach also emphasized solving problems. There are books entirely devoted to problem sets, and there are other books devoted entirely to theory. My grandfather was a Professor of Mathematics at Moscow State University before we emigrated, and as I was growing up, he would always stress that it does one no good to just know about swimming if you can't actually swim. The people who learn the most are the ones who have competency in both theory and application. To get that competency, you alternate between focused practice in one and the other, just like an athlete alternates training different muscle groups.

The rote learner learned nothing more than the ability to regurgitate a mechanical process while the academic learner learned understanding, in my view. You have to work through some examples to solidify it but that should be totally secondary to understanding the meaning and concepts. Like a physics teacher of mine put it, rote math learning is like teaching sea lions to honk horns for fish - students give the right memorized answer and get the grades without really learning anything.

Everyone can learn to solve equations quickly if they’re forced to do it 1000 times. They might forget it the day after the test, the might not know why they’re doing anything, but you can be damn certain you know what you’re doing if just for that brief window in time.

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.

> They might forget it the day after the test, the might not know why they’re doing anything, but you can be damn certain you know what you’re doing if just for that brief window in time.

What was the point then? It's a complete waste of time in that scenario.

In my experience, it's easy to think one has understanding without actually understanding. The process of applying that tentative understanding to a problem to test it is essential to figuring out what the flaws are in the tentative understanding so that real understanding can eventually be achieved. Once it gets to the point that a person can crank through problems without significant errors, then I think further problems don't do much good except as an occasional refresher.

Hence the importance of word problems, I think. They force a synthesis of abstract and concrete that really tests/enhances understanding on all levels.

Applying reasoning to a similar but more familiar domain is a good sanity check for said reasoning. Let's then apply what you said to programming. Would you say that the "academic learner" who read a lot of CS and programming books but haven't written a single program on an actual computer learned understanding, while a "rote learner" who keeps going through hundreds of small programming exercises learned to "regurgitate a mechanical process"? Doesn't pass the sniff test to me.

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.

I did say you need to do some exercises, just not an extreme amount or at the expense of theory. And these math exercises are different from programming - they amount to applying some process to different numbers over and over again while at least there would be some degree of problem solving if you wrote a bunch of genuinely different programs.

Note that I don't mind conceptual depth. But if you fail to bootstrap then you're stuck. Calculation kinda helped me connecting the notation with simpler things and bring back confidence and gradual progress

Thank's for the insight. I struggled with geography until I discovered history...again, after high school of course. I still struggle with algebra, so I look forward to checking your suggestions out. I think my problem as a kid was I was just hyper, all the time. I could cut wood outside for 10 hours but couldn't sit still for a moment.

Child labour is an untapped resource.

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.

Agreed. There is a severe lack of mindfulness and focus in modern education, and strenuous physical exercise like chopping wood can be a real mindfulness-workout.

> I struggled with geography until I discovered history

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.

My favorite sequence of maps in a book was “Black Hawk Down”. The first chapter begins with a world map, showing Somalia, and it lays the geopolitical setting for the events. The second chapter has a map of Somalia, showing where Mogadishu is, and sets the scene there. It keeps zooming in, until the final chapter shows a map so tight it has individual people (who you have since met) and which side of the street they were on.

>the map never really matters to the story.

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.

Imagine if the lord of the rings' world was like when we were still conquering and exploring all around, and then some dude, tolkien's lost brother, or something, twin, perhaps, with co-systemic minds, mapped out the unmentioned parts, and then you could just span the two things together.

Fun times and theories co-bro. Anyways.

You’re right. I’m a French student myself and our education system has always been very "Cartesian". This is also partly due to the fact that we have a special competitive exam system for entering higher education (top business and engineering schools) where you spend 2-3 years absorbing unfathomable quantities of math, economics and history/geography. The emphasis on mathematical abstraction and especially proof here is insane.

My experience was the opposite. School in America tried to make me just memorize a bunch of stuff and it didn't work at all, but using resources like Khan Academy to build understanding and working through proofs of everything was all I needed. Lots of shallow exercises did nothing for me compared to a few deep exercises/proofs and reading.

Bourbaki school of thought. Read Dieudonné - not a single( I think to remember) illustration. Kindergarten starts math education with set theory. This was the imaginated ideal.

> Kindergarten starts math education with set theory.

Well what’s wrong with that?

Emphasis on 'theory', not 'set'.

Not a bad idea at all. You can have a separate class called “Arithmetic” and that can be used to learn arguably the most boring application of mathematics: operations on numbers. Maybe distinguishing mathematics from arithmetic will make people hate the subject less. Set theory alone you can get to continuum hypothesis. I’m sure children would find the idea that some infinite sets are bigger than other infinite sets quite fascinating.

> Set theory alone you can get to continuum hypothesis.

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).

I'm the right age to have done New Math [1] in my first years at school, my memory of it is that it seemed fine but at age five I wasn't really thinking about comparing different educational styles.

[1] https://en.wikipedia.org/wiki/New_Math

In my experience with Russian style mathematical thinking, it often involves hardcore theory in pursuit of some hard-nosed applications (where every assumption and approximation in the theory is well-motivated). When the two come together, there’s an ineffable magic and wholesomeness that I’ve seldom seen elsewhere. Some of the Russian Olympiad problems are absolute beauty.

> It also correlates with the pragmatic engineering of US history (electricity comes to mind).

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.

Ah well my views were too biased then :)

> It was very operational, you calculate, you manipulate concrete data. In French books it's all theorems and concepts.

"¿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.)

I dunno, Rudin, the most popular abstract algebra textbook in English, is quite dense.

"abstract algebra", are you sure you aren't talking about analysis? It's not usually used as an introduction anyway.

Rudin's Principles of Mathematical Analysis is the top analysis book in the US too.

sorry yes... wrote that comment way late at night.

"popular"

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