As a hungarian, I'm pretty surprised by this article, since the quality of hungarian education is getting worse and worse, thanks to the "reforms", which basically means they are spending less and less on education on all level. And however in the past, math education was world class, now the "old school" teachers are retiring and other than a handful of elite schools (Fazekas, Eötvös, Radnóti etc) most of the schools are way below the european average.
> since the quality of hungarian education is getting worse and worse, thanks to the "reforms", which basically means they are spending less and less on education on all level
I always wondered what good metrics are to measure the quality of education. Since you say “worse and worse” can you share any insight to that? I mean you could potentially refer to the OECD world rankings but somehow I don’t trust those lists, since they “feel” quite mechanistic.
I'm not the OP, but I'm from another Eastern Europe country where education level falls fast and dramatically. Few key indicators of that:
0. I was attending an elite gymnasium and we deliberately used old "hard science" (math, physics, chemistry) textbooks from 70'-80', because they were much more advance level than modern ones. Most of math/physics taught during first/second year at university now, was taught in 11-12th grade back then.
1. Anecdotally, but all teachers complain that every year students are getting less motivated, performing poorer, having shorter attention spans. With local "no children left behind" equivalent it's enough just to attend some percentage of classes to get passable grades.
2. Every few years some kind of "reform" is performed to reduce the difficulty of final exams to maintain failure rates at bay and keep statistics nice.
3. Combined points 1 with 2 this leads to universities full of students that have neither motivation nor skills to perform there. Since such students make up majority in most less popular programs, the difficulty level is also reduced to adapt.
4. Degrading higher level education quality reflects on its prestige as potential employees no longer trust universities to produce prospective candidates. This closes the loop back to point 1 as current pupils deem education "worthless".
5. Fun fact: entrance exams to non-prestigious universities from 70'-80' were (maybe still are) widely used as assignments in national level competitions when I was at school (late nineties to early 00'). To paraphrase that -- a skill level that once was expected from anybody who wanted to be accepted in university now puts you at the very top.
Instinctively, I tend to factor in a certains bias for "kids these days." Seems a lot of us have a tendency to think educational standards and/or kids' level of knowledge is poorer than before. That said, in the 80s/90s I had a lot of Russian school friends in my neighborhood. Even for primary school age, they had special "russian math" classes taught by a neighborhood parent in the afternoons. They basically followed the old 70s-80s curriculum that you guys are talking about. So basically, I'm convinced it's not just the codger instinct. They were 3-4 years ahead of the 3rd & 4th graders.
These days, I have a few eastern european (polish, slovak & lithuanian) friends with schoolkids of their own. They had the same view of the modern/western curriculum as my childhood friends' parents had. It's substandard. Two years ago one couple moved back to lithuania. She reckons the modern standard for math is as bad as it is here, these days.
To hazard a guess, I think that the old/soviet-ish system was very rigorous but not very enjoyable for the kids, especially below average kids. For example, it was very sequential and intolerant of a bad week. But, it probably produced a much better educated top half. Also, I think the old educational systems of the later soviet era got a lot of criticism for being too focused on math & science, with not enough creativity, humanities and such. Soviet era "liberal arts" were ideologically impacted, and I think people developed a tendency to just stay away.
But much of Eastern Europe has a strong tradition of arts and humanities, and a cultural tendency to promote it. From a purely traditional-cultural perspective, many Lithuanians would like as much art and philosophy in their kids education as French parents, the European high watermark for teenage philosophy classes. I think there was pressure to get these subjects into the system, with due respect.
Anyway, there's plenty of proofs-of-concept that math & science education could be a lot better. We're nowhere near the ideal pedagogical system.
You are right. Soviet education system was aimed to teach only basics to the average pupils. They were not expected to continue studying after 8-9th grade and went to vocational schools or stayed in army (it was mandatory back then at the age of 16). Studying at university was only for the very best (or those whose parents were respected Party members), so the whole system was designed that 9th grade and up was for children with above average skill-set and abilities.
Art was virtually non-existent back then and only used as propaganda tool, so it was mostly orientated towards learning various techniques, not creating original ideas.
Of course, it was different times and different world back then, it would not be optimal to just copy-paste earlier system, but I believe, there's much to learn (or remember) from it.
> Anecdotally, but all teachers complain that every year students are getting less motivated, performing poorer, having shorter attention spans.
Most of these points smack of rosy retrospective bias. Kids have great attention spans if content is delivered to them in a way they can engage in, for instance, interactive computer games. Teachers are just out of touch with kids, and really they always have been. It's why kids almost always like younger teachers more.
> Most of these points smack of rosy retrospective bias.
Well, accusations of bias can go both ways. Maybe it feels bad to admit that there is a serious problem with no obvious solution in sight; and pretending that we don't see anything is how we bury our collective heads in the sand.
Instead of accusing each other of biases, let's discuss evidence. As was mentioned in a few comments in this thread, high-school textbooks in multiple countries are gradually dumbed down, so much that 20 or more years old textbooks are now considered a material for gifted kids. (I can confirm this.) Your turn.
> Kids have great attention spans if content is delivered to them in a way they can engage in, for instance, interactive computer games.
If the only problem is that humans are losing the ability to learn without playing computer games, perhaps we could fix it by making all the necessary games. But we better start making them really fast, because there is a lot of knowledge to cover.
And while we are at this type of solution, we could also fix problems with nutrition by genetically engineering a broccoli that will taste like heroin. Situation is not that bad if kids are still willing to pay attention to addictive things.
> As was mentioned in a few comments in this thread, high-school textbooks in multiple countries are gradually dumbed down, so much that 20 or more years old textbooks are now considered a material for gifted kids. (I can confirm this.) Your turn.
And what do you think that proves exactly? Did you prove that the kids using the old textbooks actually absorbed the more advanced material? Did you prove that outcomes using the old material are better than with the new material?
Perhaps the new textbooks simply distilled the relevant material that the vast majority of kids actually grasp, without all the unnecessary detail that was just skipped over. I can think of plenty of different scenarios to explain the evidence that's been listed here, and only one of those explanations are "dumber kids and/or dumbed down education".
The exact same arguments have been trotted out about the dumbing down of liberal college education, where in the 1900s, every college degree meant exposure to poetry, art, history, philosophy and more. Modern college education is then portrayed as poor substitute, completely ignoring the fact that our body of knowledge is at least 10,000x larger than it was in the 1900s, and a direct comparison is frankly laughable.
> If the only problem is that humans are losing the ability to learn without playing computer games, perhaps we could fix it by making all the necessary games.
Talk about missing the point. As evidenced by my use of "for instance", that was merely an example. Even among adults, interactive systems are clearly more engaging, and given the environmental factors that shape modern kids, you obviously are already too old to grasp their thinking process if you can't understand that different environmental factors entails different learning processes.
Which just proves my point that adults are and always will be out of touch with the kids of their day.
"And what do you think that proves exactly? Did you prove that the kids using the old textbooks actually absorbed the more advanced material?"
Well, yes, at least some pupils did. See my comment about degrading exams difficulty and problems in contemporary national competitions. Math problems are fundamental, so it's a good indicator of skills, despite changing times. Science is boring at most times, you can't gamify everything.
> Most of math/physics taught during first/second year at university now, was taught in 11-12th grade back then.
(Foreword: Hungarian here) I didn't know that this happened in other Easter European countries as well. We used to have the same thing, that people in 11-12th grade learned (around 70'-80') the first-second year university curriculum.
But by the time I got into secondary and university education the system had changed and we adopted our system to the crappy Western-European credit and whatnot system (Bologna process anybody?).
This had the obvious drawback that the first semester of university still assumed the same knowledge in 11-12th grade from me, which I completely lacked by the time I arrived to starting my undergraduate years. In turn the first 1-2 years sucked big time in uni (for me at least).
I guess this also answers the previous commenter's question how did it get worse and worse.
I don't believe the Bologna process has any impact on degrading education, at best it's a chance for people who want to cut on education, but it doesn't imply it.
> we deliberately used old "hard science" (math, physics, chemistry) textbooks from 70'-80', because they were much more advance level than modern ones.
The same is generally true in the USA. I’d recommend going back to the 40s–60s for many undergraduate-level textbooks. (Of course, there are also many wonderful recent books, but you have to hunt for them.)
A further example is the case of the earlier mentioned KöMaL.
It faced closure this year after the government reformed the funding of journals targeting exceptional students. In the new system it was destined to receive only 1/3 of its operational budget from the government: only 16K EUR instead the necessary 48K EUR / year.
Possibly it was an oversight. After public outcry and some donation from common people to keep the journal running eventually the agency in charge promised to provide dedicated funds for the 124 year old journal.
Probably this case is more indicative of the Hungarian Approach today than the one in the article. The KöMaL helped countless talents throughout the modern history of Hungary (excluding wartime but including the communist block years) to learn and master mathematics. It is the benchmark of the best high school students. Yet it almost got sacked just over ca. 32k EUR / year (while money flows almost freely in certain non-educational directions). This is more indicative of the Newest Hungarian Approach where continuous structural reforms, forced and urged changes in teaching methods (many times just months before exams), increased bureaucracy and cutting down of even elementary independency and financial self control - with retorsion in sight for "renegades" - of the teachers makes the moral very low in the education system and the outcome unreliable, unsustainable.
Another Eastern European here.
Unfortunately, the same trend of degrading education is happening in my country of origin, Bulgaria.
Another metric, particularly for math education is the representation of the country at International Mathematical Olympiad.
Bulgaria used to score almost always in top 10, even being number 1 in 2003, beating China, USA and Russia, until the beginning of the new Millennium. Now can barely make it in the top 20.
Most of the articles I would be able to dig up are in Hungarian. Like the thread op said, PISA scores are generally used around here and they are not nice. Like he or she said, education affairs are "not great" nowdays. The government increases centralisation, forcefully decreases teacher authority and freedom, the wages are still very low (even though multiple raises had been promised), the things you need to learn are way too much and not particularly useful et cetera.
"forcefully decreases teacher authority and freedom"
"the wages are still very low (even though multiple raises had been promised)"
Just curious (I don't know anything about actually living in Communist times/system) - isn't that what it was like under Soviet [rule|influence]? What's different this time?
Many of my friends, especially the older ones say that it's pretty similar to the socialist era around here :D I don't know myself, been a kid around the shift of the regimes.
I would not say that they are doing it on all levels on purpose. (They are just not involving people who actually know how education works.) But it is visible that they are preferring skilled workers to satisfy the requirements of EU by providing cheap workforce who can work in factories or doing other manual works. Don't misunderstand me, I deeply oppose the current education politics. I am just saying that I think I understand why they do it and why they might think that Hungary can profit from it.
I would like to point out two of my favorite high school competitions in Hungary.
One is KöMaL [1]. It's a monthly journal, one has to send back solutions to the problems. The competition is during the whole school year. It has problems from math, physics and computer science, these are separate contests. I did the "P" (theoretical physics) competition. Sometimes I took a look for the "B" math problems and I could never solve a single "A" math problem, those are freaking insane.
The second competition is the Eötvös Physics Competition [2]. Unfortunately the problems are not translated to English. This is a single round competition for the whole country. There are three physics problems fitting on a piece of A4 paper (single sided). The students can use anything (any number of books, calculator), the competition is 5 hours long. All high school and first year university students participate in the same contest. It's designed to filter out the very best physics students in the whole country (typically only one or two students can fully solve all three problems).
Russia (Soviet Union actually) had somewhat similar mail in programs that IMO we're very useful to stimulate those with high interest in specific fields beyond what a school would do.
IMO stimulating and developing top end is something sorely missing in US education, which is mostly focusing on helping those behind to catch up. That is useful, but pushing 1-2% of best students as far as possible is just as important for the society long term. My 2c.
Similarly in Slovakia (and Czechoslovakia historically). The highest quality education on the elementary- and high-school level happened outside the school system.
So if anyone wants to reproduce this strategy, they should give up on schools completely, and create a parallel system of education. Make it fun, allow competition (with no penalties for not being good enough; you just don't get the medal), and keep it voluntary.
I took my first eight years of math in Hungary; admittedly, some time ago. Though I'm not sure if things have changed since then, or were different at higher levels, the way the article describes it very much reflected my experience. Starting out, there was a balanced mix of rote learning the basics (as it was done in other subjects), then moving towards creativity and gradual, independent rediscovery, and it was done so in a way that didn't feel stifling if you somehow knew to do it a different way.
Ultimately it's a small country; there was a math bee you could compete in at a local level, and then your district would send the best representatives to the countrywide event. It was a prestigious event, and pretty stressful, but ultimately fun. The questions asked at the national competition were always really oddball and obscure and required both creativity and judicious use of everything you've learned.
Come think of it, the culture of the competitions in various subjects made school really fun.
Some other aspects of my primary school education in Hungary were not so stellar. In other subjects, there was very much a focus on facts in isolation, without really understanding or delving into context, notably in History. Literature I also found limiting, as much emphasis was placed on poetry analysis, which I found to be subjective; nonetheless, diverging from commonly accepted analyses was did not result in a good mark. When I came to the US, I found an emphasis on critical thinking in the Humanities, which was a breath of fresh air.
But in math and science, the quality and method of instruction in Hungary was top-notch.
It's difficult to generalize, but by pretty much any metric you choose Hungary produced a massively disproportionate share of the outstanding mathematicians of the twentieth century.
I am curious if it's because they are hungarian, or because they are jewish. Jews pay particular attention to science and arts, from what I gather from my jewish friends. Nobel laureates, for example, have disproportionate amount of jewish among them (compared to jewish population as a percentage of world population).
You don't have to be curious for more than five minutes. Several of those mentioned are Jewish, but most of them are not. There are not disproportionately more Jewish Hungarian mathematicians, compared to the proportion of say, Jewish people in Budapest before WWII, or the proportion of Jewish people in the Austro-Hungarian Empire.
Looking only at the first half of the twentieth century, there at least as many world-ranking mathematicians from Hungary as from Poland, a country with a strong mathematical tradition, a much larger population, and a massively larger Jewish population (around a quarter of the Jewish people in the world in the 1930s).
It's worth adding that comparing the proportion of Jewish Nobel prize winners to Jewish people in the world might not be a fair comparison. Much of this can probably be explained by the strong correlation between being European (including Russian/Soviet) or American, and being a Nobel prize winner.
You don’t have to be very curious to find out that about 22% of Nobel Prize winners have been Jewish [1], which leaves your controversial Western population at under 100 million even stretching the worldwide Jewish population to 20 million.
Except for Bolyai, all mentioned above are indeed Jewish. Also all mentioned in your other comment (Fejér etc.) except Bollobás. Also Wigner, mentioned in another comment.
I stand corrected. I wrongly believed that many of these including Fejer and von Neumann were not Jewish.
I think the comparison with other populations such as Poland still seems to point out that something about Hungary in the 20th century was special, other than the presence of many Jewish people.
Ashkenazi Jews have an average IQ of 108-112 compared to a European average of ~100. This shifted mean means they have a much higher percentage of very smart people (~6 times per capita rate of >140iq which is the norm for Nobel prize winners)
This over representation of high IQ is similar to the historical over representation in Nobel prizes for hard sciences.
You can't make conclusions about the number >140 based on a shift of the mean. IQ is scaled so that the general population has a normal distribution. That doesn't necessarily mean some subpopulation has a normal distribution. Even if that was the case, the subpopulation distribution would still have a different standard deviation. (It would partly depend on the level of assortative mating within that population.)
What an odd statement. Nobody said it's because they're Hungarian. It's because of the educational system at the time, which affected everyone in said system, regardless of ethnicity or "average IQ".
With the exception of Bolyai, those are all Jewish. AFAIK Hungary did push above its weight for some time, but that was thanks to its Jewish population. Ironically Hungarians are extremely antisemitic, although nearly all known Hungarians were/are Jewish. Good riddance!
Interesting list since they received quite heterogeneous and many times mixed education, home vs. institution, Hungarian vs. foreign, also their adult achievements were many times reached/mastered in other than the educating country.
As long as the measure is the production of little children inside one of the historic borders of Hungary (or maybe elsewhere by at least one Hungarian-born parent) who later contributed to mathematics notably, this or that way, here and there, then Hungary is really good at math!
Here's a footnote about Hungarian mathematicians in the 20th century from Prime Obsession. Amazing book by the way, highly recommended.
George Pólya (1887−1985). Look at those dates—another immortal. Pólya was Hungarian. Even more striking than the rise of the Germans in the early nineteenth century was the rise of Hungarians in the early twentieth. While the German states (excluding Austria and Switzerland) in 1800 had about 24 million people, the Hungarian-speaking population of Hungary was around 8.7 million in 1900, and I believe never rose above 10 million. This small and obscure nation produced an astonishing proportion of the world’s finest mathematicians: Bollobás, Erdélyi, Erdős, Fejér, Haar, Kerékjártó, two Kőnigs, Kürschák, Lakatos, Radó, Rényi, two Rieszes, Szász, Szegő, Szokefalvi-Nagy, Turán, von Neumann, and I have probably missed a few. There is a modest literature attempting to explain this phenomenon. Pólya himself thought that the major factor was Fejér (1880−1959), an inspiring teacher and gifted administrator, who attracted and encouraged mathematical talent. A high proportion of the great Hungarian mathematicians (including Fejér) were Jewish—or, like Pólya’s parents, “social” converts to Christianity, of originally Jewish stock.
I would say it's always been a bit of a self-affirming, self-fulfilling trope among Hungarians that Hungarians are good at math and science. Some of it is founded in history, and some of it is (make-believe) national mythos.
But that's just local color; I don't actually know if it's true at all, or any time in the recent past.
This question is not just worth understanding in itself but it would be essential to ask at many levels, as science education is becoming the key to the survival of humanity. As a Hungarian who left the country 20 years ago I have been fascinated by this surprising success of the country, partly because I did not personally get a good science education there. Actually, it was very unbalanced, with most of it barely mediocre, while some of it absolutely brilliant. The successful people came from 2-3 high schools (or "gymnasiums", for ages 12-18) in Budapest in the beginning of the 20th century, and whatever success Hungary had in the natural sciences was mostly a result of the work and heritage of that generation. There is a very good description of this in Norman MacRae's book on John von Neumann, worth reading.
A quick summary of the "reasons":
1. Boom: at that time (from 1867 till WWI) Budapest was booming, more attractive to immigrants than New York. Actually, most of the city you can see today was built in those times.
2. Culture: as a result of the boom and the wave of immigrants it became a very liberal and open minded place (though this did not apply to the feudal class at all - their kids typically became soldiers or playboys)
3. Motivation: in a feudal society studying and intellectual eminence was the way to go unless you were born an aristocrat. Parents, students, teachers were willing to put time and money necessary to make their kids excel. This may not sound unusual with all those helicopter parents you see nowadays, but actually this was huge. Imagine growing up in a family where you knew - and your whole family knew - that your only chance of making it is to be the best at math your abilities allow.
4. Education: I don't know where to start, so I can only give examples. Imagine you are a 12 year old child and your teacher borrows you his favourite papers on quantum physics and asks your opinion on them. Then you give a smart comment and your teacher contacts the relevant professor at the university to have a tea with you at your house next Sunday.
5. Language: most of these kids learned Latin and Greek, and in before their teenage years also spoke at least German fluently.
6. The "marble table": Stanislaw Ulam (in his autobiography, another amazing read) and also MacRae tells about the most important ingredient, the marble table at the café (the easy-to-erase whiteboard of the time). In Central Europe mathematicians met at cafeterias, discussed all day (often meaning 12 hour days at a cafe!), challenged each other, and did rarely work in isolation, not worrying too much about "who thought of it first". They happily took young kids in, 15 year olds sipping juice and 50 year old Banach drinking something much stronger (may not have been Banach, but you'll read the book!). It was such a well-known "way of doing math" that the IAS in Princeton was officially established to re-create this culture and pull the typical American professors out of their ivory towers.
It is an elitist system, I know, and does not solve mass education challenges. But this small elite circle had an impact on almost everyone in the country's education system. Even if you weren't a Wigner, Teller or Neumann, you spent 6 years in this environment and possibly became a great teacher, similar to the one who taught half of these people, the great László Rácz [1] and taught in this fashion.
Also, a similar great science education happened in Japan at some point (50's, I think ), but I only read this as a side note in the book on Neumann. Anyway, it is possible to do this again and with two small kids I'm very interested to know how.
> with two small kids I'm very interested to know how.
My few perhaps-non-obvious bits:
If you find yourself saying "really, really (physically) small", or "N times smaller than a human hair", the approach at the top of this[1] might work better.
Estimation is now taught down to K... but oh my is it done badly. PBSKids: "estimate how many jellybeans are in the jar: look at the jar; then write down a number!" (really). But it's been a few years since I looked, so maybe there's something competent now. My suggestion is to emphasize bounding, as it supports better conversations. "How many people are reading HN today? Is it more like 1, 10, 100, 1000, 10k, 100k? Who can suggest a bound? Oh, I'm reading HN! So a hard lower bound of 1! Does everyone buy that? ...".
Toddler-initiated predictable conversations by pointing out selected elements of the environment. Eg, "a map!" and "rust!". "What does it show? Where are we? Where are we going?". "What's the thing rusting? How's the oxygen getting in? What's the rust doing? What did we have for breakfast, that we're oxidizing now?". The map one works best with urban mass transit and museums and such. Rust is everywhere.
> Anyway, it is possible to do this again and with two small kids I'm very interested to know how.
For math, I recommend Hejny method; I have great experience with it (as a student) https://www.h-mat.cz/en/hejny-methodhttps://www.youtube.com/watch?v=xm0xsBjdMe4
It is now used experimentally in Czech schools, but the textbooks are translated to other languages, too. You can buy the textbook with the teacher's manual and try it at home. Short version is that this method makes kids discover math on their own (with a lot of nudging, of course), so they can enjoy and remember it better. Works both for gifted and average children, but they will progress at different speeds.
>Also, a similar great science education happened in Japan at some point (50's, I think )
Interesting comment overall, and also the part I excerpted above. I had read Akio Morita's (Sony co-founder) book, Made in Japan, some years ago. Wonder if that (what you said about science education in Japan) was part of the reason for Sony's success.
I wonder if the review is missing some aspects of "concentrate on one subject"?
On one subject, a mentor can be a master, and master instructor. And such instruction is crucial for developing deep expert understanding. But in the future, with better content, and improved learning infrastructure, one might imagine these becoming available for more than one subject.
Both the concentration on one subject, and that subject being chess, contribute to the effort having nice properties. Like non-superficiality. An integrated and deeply organized body of knowledge and skills. Developing transferable knowledge (within the subject at least), rich feedback, and reflective building of mastery. But it's the properties that matter. Chess, or another one subject, might be taught in a way that fails to have the nice properties. And as learning infrastructure improves, the nice properties may become available without the "one subject" or "and it's chess" restrictions.
There's a long history of exceptional masters teaching their field/craft to their children from a young age, who in turn become exceptional masters. The challenge is to scale that.
The straightforward "a gaggle of masters in everyone's pocket" is AI-complete implausible. But the art of the next decade is crafting human-computation hybrid systems. Things like eye tracking, and big data, provide opportunities that no master mentor has ever had. We just need to get around to building on them.
I believe history and literature are sensitive subjects politically, hence the official ways of teaching it are somewhat limiting (intentionally). However, most teachers I had contact with/heard of are exceptionally good, so I guess it comes down to luck.
I would to see this approach applied more in the sciences as well.
I my college syntax (linguistics) class, most of the teaching was done by the teacher giving us a (carefully curated) set of data to explain as homework.
Class time was spent making sure everyone got to the same answer and understood the arguments behind it. When there are multiple reasonable answers, they also made sure that students understand the other ones and why we rejected them (sometimes the reason is as simple as 'we need to pick an answer, lets just vote'; other times it is 'both are reasonable explanations, the field went with B, read Chomsky 1987 if you want to know why.'). We would then generally talk about problems we still have (especially if the data was English, and students could come up with new data that didn't work as nicely).
"Class time was spent making sure everyone got to the same answer and understood the arguments behind it."
How many students were in your class?
With the typical undergraduate class sizes that I've TA'ed at (about 30 students or so), there's just not enough time during the class to go over everything with each person individually and explain everything to them if they don't get it. Even if there was time, if someone is slow the rest of the class will be bored to death waiting for you to explain something to them (possibly over and over until they get it). Some of the more advanced students will be bored by any explanation, while the most detailed explanations with many examples are necessary for other students.
There are office hours and lab time, and some students make the most of that, but some don't. I actually spent a lot of my time helping out students who were lagging behind and needed a lot of attention, and wound up kind of neglecting the talented ones who didn't need any help, but who I think would have gotten more out of the class if we were able to engage them more, challenge them more, and cater to their interests more rather than trying to cater to the lowest common denominator.
I'm not sure what the right answer here is in classes where students vary widely in skills, interests, and abilities.
25. It also does not typically require much individual attention; generally presenting a cleaned up version of the arguement is enough. If multiple students have a problem, they tend to have the same problem, so the work scales sublinearly. Of course, larger class sizes will always make teaching more difficult.
> After the student investigation, the teacher highlights important ideas embedded in a concrete problem, and summarizes and generalizes their findings. In particular, the teacher’s summary makes sense and is meaningful, because students have had the experience of playing around with these ideas on their own before coming together to formalize them as a class.
It's important for students to get their hands on examples and play with the ideas we're trying to present on their own. One issue I have is that this takes so much time. If you're introducing concepts that the students don't have good intuition about, you have to go so slowly. Even then it feels like some students can't follow.
I think it's beneficial when it's possible to get students to engage with the material on a daily basis as reading or homework. Hard to do when they're expected to take 4 classes (college) or 6-8 (high school) and dedicate study time to each of them.
When you use this approach, you generally cover less material, but cover it better. For most classes, this seems like a good tradeoff.
4 classes isn't really that much. You can essentially replace the time students would normally spend "studying" with them spending that time working on the problems. Further, in my experience, this type of work is far more engaging than traditional studying, so it is easier for students to spend time on it; and the time they do spend tends to be more thoughtfull.
Moreover, as people practice, they can get generically better at meta-skills such as (in no particular order) skimming, close reading, constructing mental outlines of arguments, drawing diagrams and pictures, inventing examples to match given specifications, testing concrete examples against abstract statements, generalizing specific relationships discovered in the examples to abstract laws and proving them generically, discarding or modifying abstractions to better match a wider collection of examples, working backwards, discarding dead-end problem-solving strategies while mining them for partial useful results, breaking problems down into manageable pieces, deductive logic, generating hypotheses likely to be interesting, abandoning whole problems which turn out to be too hard and switching to something else, cross-applying solution strategies from one field to another, simplifying complex arguments by discarding or separating extraneous or duplicated steps, writing up explanations in clear and coherent way for various audiences, researching past work in an efficient way, diving into new fields with completely alien notation and terminology and quickly taking the lay of the land, and so on.
These skills are more generically useful than recipes or collections of facts about any particular subject, because they serve as multipliers for quite generic learning and problem-solving efficiency.
One big problem with trying to teach a single specific course in a very problem-centered / student-driven / socratic style is that often students have not been sufficiently trained in any of these meta skills, and as a result move at much slower pace than their potential because they waste a lot of time on inefficient problem-solving methods, get side-tracked, throw away useful work, explain things poorly, don’t get around to crystallizing their useful results, and so on. The more courses get taught in this style, the faster each subsequent course can move as students figure out how to learn effectively.
You could try Polya’s How to Solve It, and follow up with Schoenfeld’s Mathematical Problem Solving (which cites a lot of other material). These are rather focused on problem solving per se, and don’t really discuss larger-scale strategies for research or learning. You can try just diving into child development and education literature. Not sure what out there really tries to be comprehensive in summarizing the best techniques for teaching and learning arbitrary meta-skills.
I think this approach is better suited to lower level schooling, where kids already spend several years learning and practicing a small number of concepts. Instead of making them do countless calculations and mechanical problems, invest some of that time in abilities that will actually stick with them and be useful in the future.
Let's make this very clear: this is not a typical Hungarian approach. This is what Fazekas and to an extent a few more similar specialized high schools do. The typical Hungarian approach is frontal instruction with no respect to the learning speed differences.
Source: Personal experience. I am a Fazekas alumni and have a Hungarian maths teachers masters as well. I am bankrolling a very small reform school in Hungary so I am in contact with current Hungarian teachers every day and also I am obviously very interested in what's going on so I read a lot.
A friend of mine enjoyed her first years in school in the Pannonian basin; I am not sure, whether they did something special - but it was enough to get her into a selective German high-school specialised in maths and sciences later in her life. Always admired her for the experience, as she repeatedly speak of it if it has been fun and games.
My high school did something similar [0]. We never had math textbooks, only a book of problems. Each night we had 10 problems we had to solve. When we showed up in class the next day, each student would present a problem on the board and discuss their solution.
It worked well for the really disciplined, rigorous kids who were super interested in math and already had a solid background in it. But for someone like me, who never quite did all the homework, it became a game of getting to class first so I could present the one problem I did last night.
There were 2 documentaries produced by famed documentarian Frederick Wiseman called High School and High School 2. In one, a group of average students are followed through a typical high school in a middle class predominantly white area. In the second one, a group of students are followed through an experimental high school in a predominantly impoverished area with mostly a Latino population. The 'experimental' nature of the school was that every single class, every last one, was completely restructured to center around one thing: critical thinking. Teaching by fiat ('this is how it is because I say so and I am the authority') was banned. Every bit of teaching was through asking questions and having them answered, students challenging teachers on an equal intellectual playing field (unequal in specific knowledge of course, but equal in capacity to reason and challenge assumptions).
The experimental school produced the highest proportion of students to go on to receive college degrees (not just attend college, but finish) every seen in the country. The results were absolutely amazing and tremendously good. But... it's harder for teachers to teach that way. They can't plan ahead. They have to know their subjects inside and out, not just read off of a lesson plan. If a student asks a question that the teacher can't answer, the teacher has to admit it and try to figure it out with them, which many teachers are not emotionally mature enough to participate in alongside an adolescent. It gives a great deal of power and agency to adolescents, and our society is obsessed with stripping adolescents of every iota of control over their own life and denigrating them as much as possible. So widespread adoption of such schooling cannot gain much support at all.
The documentaries also did a good job showing how the "traditional" schooling methods broke the 'spirit' of adolescents and sucked the love of learning and figuring things out right out of them, turning them into disinterested husks of human beings, while the experimental school left them as vibrantly full of a love of life and learning as they entered it. Such things are of course difficult to measure and generally distrusted by our "pleasure is a sure sign of hidden dangers" mentality.
I'm not "bad" at math per se, but would there be much benefit in going from the ground up learning in the Hungarian form?
I guess you could blow through the first 7~10 years of schooling in less than a year of dedicated study, but teaching yourself just up to before college level would take you a few years, right?
Is there any online courses that have this content?
I grew up and studied math in Hungary and graduated from Fazekas, the school mentioned in the atricle, and its "special math" high school class. I also spent a year in high school in the US so I have some basis for comparison.
My experience was pretty much exactly as described in the article as well and I am forever grateful for the school and my teachers for giving me a foundation I could build on later in life. I'd like to point out a few things though, others have already touched upon some of these:
First, similar to why it's hard to replicate the Silicon Valley startup model elsewhere (or at least why it takes such a long time), the issue is somewhat similar here. The method of teaching is just one part of the equation. It was a whole "ecosystem" of fantastically knowledgable and respected teachers who could anywhere else be university professors or researchers, publications (such as KoMaL), camps, competitions and other extracurricular activities aimed at elementary and high school students, and a culture of math and science being interesting and fun vs. the stereotypical hard and boring.
The way of teaching feels a little bit more of a consequence of this culture rather than the source. You can pick your own preferred origin story for how that culture emerged: a booming industrial economy in the age of the Austro-Hungarian empire, a multi-ethnic, liberal (in the original sense of the word and given the context) country, the migration and resulting concentration of Jews in Budapest, or a series of exceptional teachers and mentors. The highly visible world-wide successes in the first half of the 20th century then later provided an on-going narrative that benefited the national ethos and hence made plenty of funding available in the second half (conveniently forgetting the austro-hungarian, the economic boom, the liberal or the jewish part of the story). But in any case the culture and the support system is (or at least was) there and that one is very hard to replicate at a broad scale, although certainly easier within specific communities. As someone mentioned, even in Hungary it is not broadly present and is limited to a certain set of top schools.
Also, there is a flip side to this story. Personally, I really liked math before going to high school and completely lost interest by the time I finished. Partly because a lot of kids around me were much better so I felt like a failure, partly because I was more interested in computers and programming and also in finance, all of which was looked down upon. The prestige of winning a programming competition was nowhere near the same as placing well in math or physics. Working as a developer on the side was considered a distraction. I think this was for the better for me personally. A decent number of my classmates got burnt out and had severe depressions due to the pressure. You were almost expected to win a gold medal at the International Olympiad and eventually become a world-wide math celebrity. I can't shake the feeling that a lot of them "peaked" at the end of high school, although perhaps that's partly the result of the rapidly declining university system.
Again, I'm very grateful for what I got but it's more in terms creative thinking and problem solving than specific math skills. In fact, I got to learn other subjects, such as history and literature through the same method which I now realize is very unusual in a country where those subjects are usually heavily biased toward insitilling a national identity as opposed to fostering independent thinking.
For some reason I never knew that Hungarian Maths teaching is so outstanding. My high school teacher who came from Romania to Hungary always told us that he learned topics much earlier than we did. For example he learned equitations in grade 5 and we learn them in grade 7. So I figured that Romania must be better at teaching Maths.
Having gone through all levels of Maths in Hungary, from elementary school until BSc of university, I want to point out that this method sounds good as long as the teacher is able to keep the attention of the class. In my school, a few teachers were unable to do that and it was a disaster. Children were playing, talking and doing nothing in classes. Even preventing others from learning the material.
Fortunately, I have never had these kind of teachers. However, if someone did they were doomed at university, because the expectations were too high for them. So I think it is quite a big disadvantage in Hungary. If you miss out in high school, because you were busy being a rebellious teen, there is a good chance that you never make it. You only realize it after you started university, because it is pretty easy to get into science courses of top universities in Hungary and very hard to actually graduate.
As a hungarian, I'm pretty surprised by this article, since the quality of hungarian education is getting worse and worse, thanks to the "reforms", which basically means they are spending less and less on education on all level. And however in the past, math education was world class, now the "old school" teachers are retiring and other than a handful of elite schools (Fazekas, Eötvös, Radnóti etc) most of the schools are way below the european average.
(source: https://www.compareyourcountry.org/pisa/country/hun )
Also usually only one or two hungarian university manage to get to any of the "best 500 universities in the world" list.
> since the quality of hungarian education is getting worse and worse, thanks to the "reforms", which basically means they are spending less and less on education on all level
I always wondered what good metrics are to measure the quality of education. Since you say “worse and worse” can you share any insight to that? I mean you could potentially refer to the OECD world rankings but somehow I don’t trust those lists, since they “feel” quite mechanistic.
I'm not the OP, but I'm from another Eastern Europe country where education level falls fast and dramatically. Few key indicators of that:
0. I was attending an elite gymnasium and we deliberately used old "hard science" (math, physics, chemistry) textbooks from 70'-80', because they were much more advance level than modern ones. Most of math/physics taught during first/second year at university now, was taught in 11-12th grade back then.
1. Anecdotally, but all teachers complain that every year students are getting less motivated, performing poorer, having shorter attention spans. With local "no children left behind" equivalent it's enough just to attend some percentage of classes to get passable grades.
2. Every few years some kind of "reform" is performed to reduce the difficulty of final exams to maintain failure rates at bay and keep statistics nice.
3. Combined points 1 with 2 this leads to universities full of students that have neither motivation nor skills to perform there. Since such students make up majority in most less popular programs, the difficulty level is also reduced to adapt.
4. Degrading higher level education quality reflects on its prestige as potential employees no longer trust universities to produce prospective candidates. This closes the loop back to point 1 as current pupils deem education "worthless".
5. Fun fact: entrance exams to non-prestigious universities from 70'-80' were (maybe still are) widely used as assignments in national level competitions when I was at school (late nineties to early 00'). To paraphrase that -- a skill level that once was expected from anybody who wanted to be accepted in university now puts you at the very top.
Curious.
Instinctively, I tend to factor in a certains bias for "kids these days." Seems a lot of us have a tendency to think educational standards and/or kids' level of knowledge is poorer than before. That said, in the 80s/90s I had a lot of Russian school friends in my neighborhood. Even for primary school age, they had special "russian math" classes taught by a neighborhood parent in the afternoons. They basically followed the old 70s-80s curriculum that you guys are talking about. So basically, I'm convinced it's not just the codger instinct. They were 3-4 years ahead of the 3rd & 4th graders.
These days, I have a few eastern european (polish, slovak & lithuanian) friends with schoolkids of their own. They had the same view of the modern/western curriculum as my childhood friends' parents had. It's substandard. Two years ago one couple moved back to lithuania. She reckons the modern standard for math is as bad as it is here, these days.
To hazard a guess, I think that the old/soviet-ish system was very rigorous but not very enjoyable for the kids, especially below average kids. For example, it was very sequential and intolerant of a bad week. But, it probably produced a much better educated top half. Also, I think the old educational systems of the later soviet era got a lot of criticism for being too focused on math & science, with not enough creativity, humanities and such. Soviet era "liberal arts" were ideologically impacted, and I think people developed a tendency to just stay away.
But much of Eastern Europe has a strong tradition of arts and humanities, and a cultural tendency to promote it. From a purely traditional-cultural perspective, many Lithuanians would like as much art and philosophy in their kids education as French parents, the European high watermark for teenage philosophy classes. I think there was pressure to get these subjects into the system, with due respect.
Anyway, there's plenty of proofs-of-concept that math & science education could be a lot better. We're nowhere near the ideal pedagogical system.
You are right. Soviet education system was aimed to teach only basics to the average pupils. They were not expected to continue studying after 8-9th grade and went to vocational schools or stayed in army (it was mandatory back then at the age of 16). Studying at university was only for the very best (or those whose parents were respected Party members), so the whole system was designed that 9th grade and up was for children with above average skill-set and abilities. Art was virtually non-existent back then and only used as propaganda tool, so it was mostly orientated towards learning various techniques, not creating original ideas. Of course, it was different times and different world back then, it would not be optimal to just copy-paste earlier system, but I believe, there's much to learn (or remember) from it.
> Anecdotally, but all teachers complain that every year students are getting less motivated, performing poorer, having shorter attention spans.
Most of these points smack of rosy retrospective bias. Kids have great attention spans if content is delivered to them in a way they can engage in, for instance, interactive computer games. Teachers are just out of touch with kids, and really they always have been. It's why kids almost always like younger teachers more.
> Most of these points smack of rosy retrospective bias.
Well, accusations of bias can go both ways. Maybe it feels bad to admit that there is a serious problem with no obvious solution in sight; and pretending that we don't see anything is how we bury our collective heads in the sand.
Instead of accusing each other of biases, let's discuss evidence. As was mentioned in a few comments in this thread, high-school textbooks in multiple countries are gradually dumbed down, so much that 20 or more years old textbooks are now considered a material for gifted kids. (I can confirm this.) Your turn.
> Kids have great attention spans if content is delivered to them in a way they can engage in, for instance, interactive computer games.
If the only problem is that humans are losing the ability to learn without playing computer games, perhaps we could fix it by making all the necessary games. But we better start making them really fast, because there is a lot of knowledge to cover.
And while we are at this type of solution, we could also fix problems with nutrition by genetically engineering a broccoli that will taste like heroin. Situation is not that bad if kids are still willing to pay attention to addictive things.
> As was mentioned in a few comments in this thread, high-school textbooks in multiple countries are gradually dumbed down, so much that 20 or more years old textbooks are now considered a material for gifted kids. (I can confirm this.) Your turn.
And what do you think that proves exactly? Did you prove that the kids using the old textbooks actually absorbed the more advanced material? Did you prove that outcomes using the old material are better than with the new material?
Perhaps the new textbooks simply distilled the relevant material that the vast majority of kids actually grasp, without all the unnecessary detail that was just skipped over. I can think of plenty of different scenarios to explain the evidence that's been listed here, and only one of those explanations are "dumber kids and/or dumbed down education".
The exact same arguments have been trotted out about the dumbing down of liberal college education, where in the 1900s, every college degree meant exposure to poetry, art, history, philosophy and more. Modern college education is then portrayed as poor substitute, completely ignoring the fact that our body of knowledge is at least 10,000x larger than it was in the 1900s, and a direct comparison is frankly laughable.
> If the only problem is that humans are losing the ability to learn without playing computer games, perhaps we could fix it by making all the necessary games.
Talk about missing the point. As evidenced by my use of "for instance", that was merely an example. Even among adults, interactive systems are clearly more engaging, and given the environmental factors that shape modern kids, you obviously are already too old to grasp their thinking process if you can't understand that different environmental factors entails different learning processes.
Which just proves my point that adults are and always will be out of touch with the kids of their day.
"And what do you think that proves exactly? Did you prove that the kids using the old textbooks actually absorbed the more advanced material?"
Well, yes, at least some pupils did. See my comment about degrading exams difficulty and problems in contemporary national competitions. Math problems are fundamental, so it's a good indicator of skills, despite changing times. Science is boring at most times, you can't gamify everything.
I recommend you read http://toomandre.com/travel/sweden05/WP-SWEDEN-NEW.pdf
> Most of math/physics taught during first/second year at university now, was taught in 11-12th grade back then.
(Foreword: Hungarian here) I didn't know that this happened in other Easter European countries as well. We used to have the same thing, that people in 11-12th grade learned (around 70'-80') the first-second year university curriculum.
But by the time I got into secondary and university education the system had changed and we adopted our system to the crappy Western-European credit and whatnot system (Bologna process anybody?).
This had the obvious drawback that the first semester of university still assumed the same knowledge in 11-12th grade from me, which I completely lacked by the time I arrived to starting my undergraduate years. In turn the first 1-2 years sucked big time in uni (for me at least).
I guess this also answers the previous commenter's question how did it get worse and worse.
I don't believe the Bologna process has any impact on degrading education, at best it's a chance for people who want to cut on education, but it doesn't imply it.
> we deliberately used old "hard science" (math, physics, chemistry) textbooks from 70'-80', because they were much more advance level than modern ones.
The same is generally true in the USA. I’d recommend going back to the 40s–60s for many undergraduate-level textbooks. (Of course, there are also many wonderful recent books, but you have to hunt for them.)
A further example is the case of the earlier mentioned KöMaL. It faced closure this year after the government reformed the funding of journals targeting exceptional students. In the new system it was destined to receive only 1/3 of its operational budget from the government: only 16K EUR instead the necessary 48K EUR / year. Possibly it was an oversight. After public outcry and some donation from common people to keep the journal running eventually the agency in charge promised to provide dedicated funds for the 124 year old journal. Probably this case is more indicative of the Hungarian Approach today than the one in the article. The KöMaL helped countless talents throughout the modern history of Hungary (excluding wartime but including the communist block years) to learn and master mathematics. It is the benchmark of the best high school students. Yet it almost got sacked just over ca. 32k EUR / year (while money flows almost freely in certain non-educational directions). This is more indicative of the Newest Hungarian Approach where continuous structural reforms, forced and urged changes in teaching methods (many times just months before exams), increased bureaucracy and cutting down of even elementary independency and financial self control - with retorsion in sight for "renegades" - of the teachers makes the moral very low in the education system and the outcome unreliable, unsustainable.
Another Eastern European here. Unfortunately, the same trend of degrading education is happening in my country of origin, Bulgaria.
Another metric, particularly for math education is the representation of the country at International Mathematical Olympiad.
Bulgaria used to score almost always in top 10, even being number 1 in 2003, beating China, USA and Russia, until the beginning of the new Millennium. Now can barely make it in the top 20.
https://www.imo-official.org/results.aspx
Most of the articles I would be able to dig up are in Hungarian. Like the thread op said, PISA scores are generally used around here and they are not nice. Like he or she said, education affairs are "not great" nowdays. The government increases centralisation, forcefully decreases teacher authority and freedom, the wages are still very low (even though multiple raises had been promised), the things you need to learn are way too much and not particularly useful et cetera.
"The government increases centralisation"
"forcefully decreases teacher authority and freedom"
"the wages are still very low (even though multiple raises had been promised)"
Just curious (I don't know anything about actually living in Communist times/system) - isn't that what it was like under Soviet [rule|influence]? What's different this time?
Many of my friends, especially the older ones say that it's pretty similar to the socialist era around here :D I don't know myself, been a kid around the shift of the regimes.
I would not say that they are doing it on all levels on purpose. (They are just not involving people who actually know how education works.) But it is visible that they are preferring skilled workers to satisfy the requirements of EU by providing cheap workforce who can work in factories or doing other manual works. Don't misunderstand me, I deeply oppose the current education politics. I am just saying that I think I understand why they do it and why they might think that Hungary can profit from it.
I would like to point out two of my favorite high school competitions in Hungary.
One is KöMaL [1]. It's a monthly journal, one has to send back solutions to the problems. The competition is during the whole school year. It has problems from math, physics and computer science, these are separate contests. I did the "P" (theoretical physics) competition. Sometimes I took a look for the "B" math problems and I could never solve a single "A" math problem, those are freaking insane.
[1] https://www.komal.hu/info/miazakomal.e.shtml
The second competition is the Eötvös Physics Competition [2]. Unfortunately the problems are not translated to English. This is a single round competition for the whole country. There are three physics problems fitting on a piece of A4 paper (single sided). The students can use anything (any number of books, calculator), the competition is 5 hours long. All high school and first year university students participate in the same contest. It's designed to filter out the very best physics students in the whole country (typically only one or two students can fully solve all three problems).
[2] http://eik.bme.hu/~vanko/fizika/eotvos.htm
Russia (Soviet Union actually) had somewhat similar mail in programs that IMO we're very useful to stimulate those with high interest in specific fields beyond what a school would do.
IMO stimulating and developing top end is something sorely missing in US education, which is mostly focusing on helping those behind to catch up. That is useful, but pushing 1-2% of best students as far as possible is just as important for the society long term. My 2c.
And mail in may be better than extra classes or similar, as it may avoid various "tall poppy" issues.
And these days it can even be done electronically.
Similarly in Slovakia (and Czechoslovakia historically). The highest quality education on the elementary- and high-school level happened outside the school system.
So if anyone wants to reproduce this strategy, they should give up on schools completely, and create a parallel system of education. Make it fun, allow competition (with no penalties for not being good enough; you just don't get the medal), and keep it voluntary.
For the remaining 99% of population... probably something like Hejny method: https://www.h-mat.cz/en/hejny-method https://www.youtube.com/watch?v=xm0xsBjdMe4
I took my first eight years of math in Hungary; admittedly, some time ago. Though I'm not sure if things have changed since then, or were different at higher levels, the way the article describes it very much reflected my experience. Starting out, there was a balanced mix of rote learning the basics (as it was done in other subjects), then moving towards creativity and gradual, independent rediscovery, and it was done so in a way that didn't feel stifling if you somehow knew to do it a different way.
Ultimately it's a small country; there was a math bee you could compete in at a local level, and then your district would send the best representatives to the countrywide event. It was a prestigious event, and pretty stressful, but ultimately fun. The questions asked at the national competition were always really oddball and obscure and required both creativity and judicious use of everything you've learned.
Come think of it, the culture of the competitions in various subjects made school really fun.
Some other aspects of my primary school education in Hungary were not so stellar. In other subjects, there was very much a focus on facts in isolation, without really understanding or delving into context, notably in History. Literature I also found limiting, as much emphasis was placed on poetry analysis, which I found to be subjective; nonetheless, diverging from commonly accepted analyses was did not result in a good mark. When I came to the US, I found an emphasis on critical thinking in the Humanities, which was a breath of fresh air.
But in math and science, the quality and method of instruction in Hungary was top-notch.
Why is Hungarian math so forward looking but not the other subjects? Is Hungary especially good at math?
It's difficult to generalize, but by pretty much any metric you choose Hungary produced a massively disproportionate share of the outstanding mathematicians of the twentieth century.
can you give couple examples that are most prominent?
János Bolyai [1], Theodore von Kármán [2], Leo Szilard [3], Dennis Gabor [4], Edward Teller [5], Paul Erdős [6], John von Neumann [7].
[1] https://en.wikipedia.org/wiki/J%C3%A1nos_Bolyai [2] https://en.wikipedia.org/wiki/Theodore_von_Kármán [3] https://en.wikipedia.org/wiki/Leo_Szilard [4] https://en.wikipedia.org/wiki/Dennis_Gabor [5] https://en.wikipedia.org/wiki/Edward_Teller [6] https://en.wikipedia.org/wiki/Paul_Erd%C5%91s [7] https://en.wikipedia.org/wiki/John_von_Neumann
I am curious if it's because they are hungarian, or because they are jewish. Jews pay particular attention to science and arts, from what I gather from my jewish friends. Nobel laureates, for example, have disproportionate amount of jewish among them (compared to jewish population as a percentage of world population).
You don't have to be curious for more than five minutes. Several of those mentioned are Jewish, but most of them are not. There are not disproportionately more Jewish Hungarian mathematicians, compared to the proportion of say, Jewish people in Budapest before WWII, or the proportion of Jewish people in the Austro-Hungarian Empire. Looking only at the first half of the twentieth century, there at least as many world-ranking mathematicians from Hungary as from Poland, a country with a strong mathematical tradition, a much larger population, and a massively larger Jewish population (around a quarter of the Jewish people in the world in the 1930s).
It's worth adding that comparing the proportion of Jewish Nobel prize winners to Jewish people in the world might not be a fair comparison. Much of this can probably be explained by the strong correlation between being European (including Russian/Soviet) or American, and being a Nobel prize winner.
You don’t have to be very curious to find out that about 22% of Nobel Prize winners have been Jewish [1], which leaves your controversial Western population at under 100 million even stretching the worldwide Jewish population to 20 million.
[1] https://en.m.wikipedia.org/wiki/List_of_Jewish_Nobel_laureat...
Except for Bolyai, all mentioned above are indeed Jewish. Also all mentioned in your other comment (Fejér etc.) except Bollobás. Also Wigner, mentioned in another comment.
I stand corrected. I wrongly believed that many of these including Fejer and von Neumann were not Jewish.
I think the comparison with other populations such as Poland still seems to point out that something about Hungary in the 20th century was special, other than the presence of many Jewish people.
Ashkenazi Jews have an average IQ of 108-112 compared to a European average of ~100. This shifted mean means they have a much higher percentage of very smart people (~6 times per capita rate of >140iq which is the norm for Nobel prize winners)
This over representation of high IQ is similar to the historical over representation in Nobel prizes for hard sciences.
You can't make conclusions about the number >140 based on a shift of the mean. IQ is scaled so that the general population has a normal distribution. That doesn't necessarily mean some subpopulation has a normal distribution. Even if that was the case, the subpopulation distribution would still have a different standard deviation. (It would partly depend on the level of assortative mating within that population.)
Thanks for expanding on the assumptions I was lazy and just used a ~ to indicate it was approximate.
You're in luck! Here's a looong in-depth consideration of exactly that: http://slatestarcodex.com/2017/05/26/the-atomic-bomb-conside...
Thanks. This is pretty interesting read.
What an odd statement. Nobody said it's because they're Hungarian. It's because of the educational system at the time, which affected everyone in said system, regardless of ethnicity or "average IQ".
With the exception of Bolyai, those are all Jewish. AFAIK Hungary did push above its weight for some time, but that was thanks to its Jewish population. Ironically Hungarians are extremely antisemitic, although nearly all known Hungarians were/are Jewish. Good riddance!
> Hungarians are extremely antisemitic
One should not make irresponsible comments like this. Yes, I'm Hungarian but claims like these are just plain stupid.
> Hungarians are extremely antisemitic
Some might be, doesn't mean all of them are. So [citation needed].
Also Fejér, Turán, Rényi, F. Riesz and M. Riesz, Bollobás
Interesting list since they received quite heterogeneous and many times mixed education, home vs. institution, Hungarian vs. foreign, also their adult achievements were many times reached/mastered in other than the educating country. As long as the measure is the production of little children inside one of the historic borders of Hungary (or maybe elsewhere by at least one Hungarian-born parent) who later contributed to mathematics notably, this or that way, here and there, then Hungary is really good at math!
I would add Eugene Wigner, also Nobel laureate.
Here's a footnote about Hungarian mathematicians in the 20th century from Prime Obsession. Amazing book by the way, highly recommended.
George Pólya (1887−1985). Look at those dates—another immortal. Pólya was Hungarian. Even more striking than the rise of the Germans in the early nineteenth century was the rise of Hungarians in the early twentieth. While the German states (excluding Austria and Switzerland) in 1800 had about 24 million people, the Hungarian-speaking population of Hungary was around 8.7 million in 1900, and I believe never rose above 10 million. This small and obscure nation produced an astonishing proportion of the world’s finest mathematicians: Bollobás, Erdélyi, Erdős, Fejér, Haar, Kerékjártó, two Kőnigs, Kürschák, Lakatos, Radó, Rényi, two Rieszes, Szász, Szegő, Szokefalvi-Nagy, Turán, von Neumann, and I have probably missed a few. There is a modest literature attempting to explain this phenomenon. Pólya himself thought that the major factor was Fejér (1880−1959), an inspiring teacher and gifted administrator, who attracted and encouraged mathematical talent. A high proportion of the great Hungarian mathematicians (including Fejér) were Jewish—or, like Pólya’s parents, “social” converts to Christianity, of originally Jewish stock.
Would Paul Erdős be the most famous? There's even a Wikipedia subcategory: https://en.wikipedia.org/wiki/Category:20th-century_Hungaria....
http://www.jurisich-koszeg.sulinet.hu/files/projekt/science/...
The first 5 pages have names that are attached to lots and lots of theorems (disproportionately in combinatorics).
George Olah passed away in March.
Recently Laszlo Babai, Endre Szemeredi, Mario Szegedy
In undergraduate studies the names Denes Konig, Tibor Gallai also come up discussing graphs.
I would say it's always been a bit of a self-affirming, self-fulfilling trope among Hungarians that Hungarians are good at math and science. Some of it is founded in history, and some of it is (make-believe) national mythos.
But that's just local color; I don't actually know if it's true at all, or any time in the recent past.
This question is not just worth understanding in itself but it would be essential to ask at many levels, as science education is becoming the key to the survival of humanity. As a Hungarian who left the country 20 years ago I have been fascinated by this surprising success of the country, partly because I did not personally get a good science education there. Actually, it was very unbalanced, with most of it barely mediocre, while some of it absolutely brilliant. The successful people came from 2-3 high schools (or "gymnasiums", for ages 12-18) in Budapest in the beginning of the 20th century, and whatever success Hungary had in the natural sciences was mostly a result of the work and heritage of that generation. There is a very good description of this in Norman MacRae's book on John von Neumann, worth reading.
A quick summary of the "reasons":
1. Boom: at that time (from 1867 till WWI) Budapest was booming, more attractive to immigrants than New York. Actually, most of the city you can see today was built in those times.
2. Culture: as a result of the boom and the wave of immigrants it became a very liberal and open minded place (though this did not apply to the feudal class at all - their kids typically became soldiers or playboys)
3. Motivation: in a feudal society studying and intellectual eminence was the way to go unless you were born an aristocrat. Parents, students, teachers were willing to put time and money necessary to make their kids excel. This may not sound unusual with all those helicopter parents you see nowadays, but actually this was huge. Imagine growing up in a family where you knew - and your whole family knew - that your only chance of making it is to be the best at math your abilities allow.
4. Education: I don't know where to start, so I can only give examples. Imagine you are a 12 year old child and your teacher borrows you his favourite papers on quantum physics and asks your opinion on them. Then you give a smart comment and your teacher contacts the relevant professor at the university to have a tea with you at your house next Sunday.
5. Language: most of these kids learned Latin and Greek, and in before their teenage years also spoke at least German fluently.
6. The "marble table": Stanislaw Ulam (in his autobiography, another amazing read) and also MacRae tells about the most important ingredient, the marble table at the café (the easy-to-erase whiteboard of the time). In Central Europe mathematicians met at cafeterias, discussed all day (often meaning 12 hour days at a cafe!), challenged each other, and did rarely work in isolation, not worrying too much about "who thought of it first". They happily took young kids in, 15 year olds sipping juice and 50 year old Banach drinking something much stronger (may not have been Banach, but you'll read the book!). It was such a well-known "way of doing math" that the IAS in Princeton was officially established to re-create this culture and pull the typical American professors out of their ivory towers.
It is an elitist system, I know, and does not solve mass education challenges. But this small elite circle had an impact on almost everyone in the country's education system. Even if you weren't a Wigner, Teller or Neumann, you spent 6 years in this environment and possibly became a great teacher, similar to the one who taught half of these people, the great László Rácz [1] and taught in this fashion.
Also, a similar great science education happened in Japan at some point (50's, I think ), but I only read this as a side note in the book on Neumann. Anyway, it is possible to do this again and with two small kids I'm very interested to know how.
[1] https://en.wikipedia.org/wiki/L%C3%A1szl%C3%B3_R%C3%A1tz
> with two small kids I'm very interested to know how.
My few perhaps-non-obvious bits:
If you find yourself saying "really, really (physically) small", or "N times smaller than a human hair", the approach at the top of this[1] might work better.
Estimation is now taught down to K... but oh my is it done badly. PBSKids: "estimate how many jellybeans are in the jar: look at the jar; then write down a number!" (really). But it's been a few years since I looked, so maybe there's something competent now. My suggestion is to emphasize bounding, as it supports better conversations. "How many people are reading HN today? Is it more like 1, 10, 100, 1000, 10k, 100k? Who can suggest a bound? Oh, I'm reading HN! So a hard lower bound of 1! Does everyone buy that? ...".
Toddler-initiated predictable conversations by pointing out selected elements of the environment. Eg, "a map!" and "rust!". "What does it show? Where are we? Where are we going?". "What's the thing rusting? How's the oxygen getting in? What's the rust doing? What did we have for breakfast, that we're oxidizing now?". The map one works best with urban mass transit and museums and such. Rust is everywhere.
[1] How to remember the sizes of small things, top of http://www.clarifyscience.info/part/Atoms
> Anyway, it is possible to do this again and with two small kids I'm very interested to know how.
For math, I recommend Hejny method; I have great experience with it (as a student) https://www.h-mat.cz/en/hejny-method https://www.youtube.com/watch?v=xm0xsBjdMe4 It is now used experimentally in Czech schools, but the textbooks are translated to other languages, too. You can buy the textbook with the teacher's manual and try it at home. Short version is that this method makes kids discover math on their own (with a lot of nudging, of course), so they can enjoy and remember it better. Works both for gifted and average children, but they will progress at different speeds.
>Also, a similar great science education happened in Japan at some point (50's, I think )
Interesting comment overall, and also the part I excerpted above. I had read Akio Morita's (Sony co-founder) book, Made in Japan, some years ago. Wonder if that (what you said about science education in Japan) was part of the reason for Sony's success.
https://en.wikipedia.org/wiki/Akio_Morita
https://en.wikipedia.org/wiki/Made_in_Japan_(biography)
>Anyway, it is possible to do this again and with two small kids I'm very interested to know how.
Here's a context-relevant place to start: http://slatestarcodex.com/2017/07/31/book-review-raise-a-gen...
I wonder if the review is missing some aspects of "concentrate on one subject"?
On one subject, a mentor can be a master, and master instructor. And such instruction is crucial for developing deep expert understanding. But in the future, with better content, and improved learning infrastructure, one might imagine these becoming available for more than one subject.
Both the concentration on one subject, and that subject being chess, contribute to the effort having nice properties. Like non-superficiality. An integrated and deeply organized body of knowledge and skills. Developing transferable knowledge (within the subject at least), rich feedback, and reflective building of mastery. But it's the properties that matter. Chess, or another one subject, might be taught in a way that fails to have the nice properties. And as learning infrastructure improves, the nice properties may become available without the "one subject" or "and it's chess" restrictions.
There's a long history of exceptional masters teaching their field/craft to their children from a young age, who in turn become exceptional masters. The challenge is to scale that.
The straightforward "a gaggle of masters in everyone's pocket" is AI-complete implausible. But the art of the next decade is crafting human-computation hybrid systems. Things like eye tracking, and big data, provide opportunities that no master mentor has ever had. We just need to get around to building on them.
I believe history and literature are sensitive subjects politically, hence the official ways of teaching it are somewhat limiting (intentionally). However, most teachers I had contact with/heard of are exceptionally good, so I guess it comes down to luck.
I would to see this approach applied more in the sciences as well.
I my college syntax (linguistics) class, most of the teaching was done by the teacher giving us a (carefully curated) set of data to explain as homework.
Class time was spent making sure everyone got to the same answer and understood the arguments behind it. When there are multiple reasonable answers, they also made sure that students understand the other ones and why we rejected them (sometimes the reason is as simple as 'we need to pick an answer, lets just vote'; other times it is 'both are reasonable explanations, the field went with B, read Chomsky 1987 if you want to know why.'). We would then generally talk about problems we still have (especially if the data was English, and students could come up with new data that didn't work as nicely).
"Class time was spent making sure everyone got to the same answer and understood the arguments behind it."
How many students were in your class?
With the typical undergraduate class sizes that I've TA'ed at (about 30 students or so), there's just not enough time during the class to go over everything with each person individually and explain everything to them if they don't get it. Even if there was time, if someone is slow the rest of the class will be bored to death waiting for you to explain something to them (possibly over and over until they get it). Some of the more advanced students will be bored by any explanation, while the most detailed explanations with many examples are necessary for other students.
There are office hours and lab time, and some students make the most of that, but some don't. I actually spent a lot of my time helping out students who were lagging behind and needed a lot of attention, and wound up kind of neglecting the talented ones who didn't need any help, but who I think would have gotten more out of the class if we were able to engage them more, challenge them more, and cater to their interests more rather than trying to cater to the lowest common denominator.
I'm not sure what the right answer here is in classes where students vary widely in skills, interests, and abilities.
25. It also does not typically require much individual attention; generally presenting a cleaned up version of the arguement is enough. If multiple students have a problem, they tend to have the same problem, so the work scales sublinearly. Of course, larger class sizes will always make teaching more difficult.
> After the student investigation, the teacher highlights important ideas embedded in a concrete problem, and summarizes and generalizes their findings. In particular, the teacher’s summary makes sense and is meaningful, because students have had the experience of playing around with these ideas on their own before coming together to formalize them as a class.
It's important for students to get their hands on examples and play with the ideas we're trying to present on their own. One issue I have is that this takes so much time. If you're introducing concepts that the students don't have good intuition about, you have to go so slowly. Even then it feels like some students can't follow.
I think it's beneficial when it's possible to get students to engage with the material on a daily basis as reading or homework. Hard to do when they're expected to take 4 classes (college) or 6-8 (high school) and dedicate study time to each of them.
When you use this approach, you generally cover less material, but cover it better. For most classes, this seems like a good tradeoff.
4 classes isn't really that much. You can essentially replace the time students would normally spend "studying" with them spending that time working on the problems. Further, in my experience, this type of work is far more engaging than traditional studying, so it is easier for students to spend time on it; and the time they do spend tends to be more thoughtfull.
Moreover, as people practice, they can get generically better at meta-skills such as (in no particular order) skimming, close reading, constructing mental outlines of arguments, drawing diagrams and pictures, inventing examples to match given specifications, testing concrete examples against abstract statements, generalizing specific relationships discovered in the examples to abstract laws and proving them generically, discarding or modifying abstractions to better match a wider collection of examples, working backwards, discarding dead-end problem-solving strategies while mining them for partial useful results, breaking problems down into manageable pieces, deductive logic, generating hypotheses likely to be interesting, abandoning whole problems which turn out to be too hard and switching to something else, cross-applying solution strategies from one field to another, simplifying complex arguments by discarding or separating extraneous or duplicated steps, writing up explanations in clear and coherent way for various audiences, researching past work in an efficient way, diving into new fields with completely alien notation and terminology and quickly taking the lay of the land, and so on.
These skills are more generically useful than recipes or collections of facts about any particular subject, because they serve as multipliers for quite generic learning and problem-solving efficiency.
One big problem with trying to teach a single specific course in a very problem-centered / student-driven / socratic style is that often students have not been sufficiently trained in any of these meta skills, and as a result move at much slower pace than their potential because they waste a lot of time on inefficient problem-solving methods, get side-tracked, throw away useful work, explain things poorly, don’t get around to crystallizing their useful results, and so on. The more courses get taught in this style, the faster each subsequent course can move as students figure out how to learn effectively.
That's a nice list of meta-skills. Do you have any further reading to recommend on identifying/teaching meta-skills?
You could try Polya’s How to Solve It, and follow up with Schoenfeld’s Mathematical Problem Solving (which cites a lot of other material). These are rather focused on problem solving per se, and don’t really discuss larger-scale strategies for research or learning. You can try just diving into child development and education literature. Not sure what out there really tries to be comprehensive in summarizing the best techniques for teaching and learning arbitrary meta-skills.
I think this approach is better suited to lower level schooling, where kids already spend several years learning and practicing a small number of concepts. Instead of making them do countless calculations and mechanical problems, invest some of that time in abilities that will actually stick with them and be useful in the future.
Math is much more than what I was taught in school. I discovered that as a child when stumbling upon this Hungarian math book at the local library:
https://www.amazon.com/Playing-Infinity-Dover-Books-Mathemat...
Hey, that looks awesome! Thanks for recommending it. I put it on my bucket list.
Let's make this very clear: this is not a typical Hungarian approach. This is what Fazekas and to an extent a few more similar specialized high schools do. The typical Hungarian approach is frontal instruction with no respect to the learning speed differences.
Source: Personal experience. I am a Fazekas alumni and have a Hungarian maths teachers masters as well. I am bankrolling a very small reform school in Hungary so I am in contact with current Hungarian teachers every day and also I am obviously very interested in what's going on so I read a lot.
A friend of mine enjoyed her first years in school in the Pannonian basin; I am not sure, whether they did something special - but it was enough to get her into a selective German high-school specialised in maths and sciences later in her life. Always admired her for the experience, as she repeatedly speak of it if it has been fun and games.
My high school did something similar [0]. We never had math textbooks, only a book of problems. Each night we had 10 problems we had to solve. When we showed up in class the next day, each student would present a problem on the board and discuss their solution.
It worked well for the really disciplined, rigorous kids who were super interested in math and already had a solid background in it. But for someone like me, who never quite did all the homework, it became a game of getting to class first so I could present the one problem I did last night.
[0] https://en.wikipedia.org/wiki/Harkness_table
There were 2 documentaries produced by famed documentarian Frederick Wiseman called High School and High School 2. In one, a group of average students are followed through a typical high school in a middle class predominantly white area. In the second one, a group of students are followed through an experimental high school in a predominantly impoverished area with mostly a Latino population. The 'experimental' nature of the school was that every single class, every last one, was completely restructured to center around one thing: critical thinking. Teaching by fiat ('this is how it is because I say so and I am the authority') was banned. Every bit of teaching was through asking questions and having them answered, students challenging teachers on an equal intellectual playing field (unequal in specific knowledge of course, but equal in capacity to reason and challenge assumptions).
The experimental school produced the highest proportion of students to go on to receive college degrees (not just attend college, but finish) every seen in the country. The results were absolutely amazing and tremendously good. But... it's harder for teachers to teach that way. They can't plan ahead. They have to know their subjects inside and out, not just read off of a lesson plan. If a student asks a question that the teacher can't answer, the teacher has to admit it and try to figure it out with them, which many teachers are not emotionally mature enough to participate in alongside an adolescent. It gives a great deal of power and agency to adolescents, and our society is obsessed with stripping adolescents of every iota of control over their own life and denigrating them as much as possible. So widespread adoption of such schooling cannot gain much support at all.
The documentaries also did a good job showing how the "traditional" schooling methods broke the 'spirit' of adolescents and sucked the love of learning and figuring things out right out of them, turning them into disinterested husks of human beings, while the experimental school left them as vibrantly full of a love of life and learning as they entered it. Such things are of course difficult to measure and generally distrusted by our "pleasure is a sure sign of hidden dangers" mentality.
I'm not "bad" at math per se, but would there be much benefit in going from the ground up learning in the Hungarian form?
I guess you could blow through the first 7~10 years of schooling in less than a year of dedicated study, but teaching yourself just up to before college level would take you a few years, right?
Is there any online courses that have this content?
Reading George Polya's books might be a way of learning this technique. At least they explain very well the method of thinking about math.
I grew up and studied math in Hungary and graduated from Fazekas, the school mentioned in the atricle, and its "special math" high school class. I also spent a year in high school in the US so I have some basis for comparison.
My experience was pretty much exactly as described in the article as well and I am forever grateful for the school and my teachers for giving me a foundation I could build on later in life. I'd like to point out a few things though, others have already touched upon some of these:
First, similar to why it's hard to replicate the Silicon Valley startup model elsewhere (or at least why it takes such a long time), the issue is somewhat similar here. The method of teaching is just one part of the equation. It was a whole "ecosystem" of fantastically knowledgable and respected teachers who could anywhere else be university professors or researchers, publications (such as KoMaL), camps, competitions and other extracurricular activities aimed at elementary and high school students, and a culture of math and science being interesting and fun vs. the stereotypical hard and boring.
The way of teaching feels a little bit more of a consequence of this culture rather than the source. You can pick your own preferred origin story for how that culture emerged: a booming industrial economy in the age of the Austro-Hungarian empire, a multi-ethnic, liberal (in the original sense of the word and given the context) country, the migration and resulting concentration of Jews in Budapest, or a series of exceptional teachers and mentors. The highly visible world-wide successes in the first half of the 20th century then later provided an on-going narrative that benefited the national ethos and hence made plenty of funding available in the second half (conveniently forgetting the austro-hungarian, the economic boom, the liberal or the jewish part of the story). But in any case the culture and the support system is (or at least was) there and that one is very hard to replicate at a broad scale, although certainly easier within specific communities. As someone mentioned, even in Hungary it is not broadly present and is limited to a certain set of top schools.
Also, there is a flip side to this story. Personally, I really liked math before going to high school and completely lost interest by the time I finished. Partly because a lot of kids around me were much better so I felt like a failure, partly because I was more interested in computers and programming and also in finance, all of which was looked down upon. The prestige of winning a programming competition was nowhere near the same as placing well in math or physics. Working as a developer on the side was considered a distraction. I think this was for the better for me personally. A decent number of my classmates got burnt out and had severe depressions due to the pressure. You were almost expected to win a gold medal at the International Olympiad and eventually become a world-wide math celebrity. I can't shake the feeling that a lot of them "peaked" at the end of high school, although perhaps that's partly the result of the rapidly declining university system.
Again, I'm very grateful for what I got but it's more in terms creative thinking and problem solving than specific math skills. In fact, I got to learn other subjects, such as history and literature through the same method which I now realize is very unusual in a country where those subjects are usually heavily biased toward insitilling a national identity as opposed to fostering independent thinking.
For some reason I never knew that Hungarian Maths teaching is so outstanding. My high school teacher who came from Romania to Hungary always told us that he learned topics much earlier than we did. For example he learned equitations in grade 5 and we learn them in grade 7. So I figured that Romania must be better at teaching Maths.
Having gone through all levels of Maths in Hungary, from elementary school until BSc of university, I want to point out that this method sounds good as long as the teacher is able to keep the attention of the class. In my school, a few teachers were unable to do that and it was a disaster. Children were playing, talking and doing nothing in classes. Even preventing others from learning the material. Fortunately, I have never had these kind of teachers. However, if someone did they were doomed at university, because the expectations were too high for them. So I think it is quite a big disadvantage in Hungary. If you miss out in high school, because you were busy being a rebellious teen, there is a good chance that you never make it. You only realize it after you started university, because it is pretty easy to get into science courses of top universities in Hungary and very hard to actually graduate.