The math exams of my life(andreinc.net)
andreinc.net
The math exams of my life
https://www.andreinc.net/2024/01/09/the-most-important-math-exams-of-my-life
20 comments
Yes, my math grad student time ws an excellent preparation for coding interviews. In addition to the oral exams, I found TA'ing was also an excellent preparation. Students show up with lots of questions, which you get to solve on your toes at a board...
Classes and competitions also generally have an allowed 'bag of tricks' from which you build your solutions. White-board coding interviews are the same. If you want to do well, you get to know what the 'atomic' operations are in your bag of tricks, how they combine into useful tools, and how to spot when they might be useful in the 'wild.'
Restating/reworking a problem into a form where some tricks are more obviously applicable is also a useful skill. These are general problem solving skills, which are pretty independent of what's actually in the bag of tricks... Draw a picture, try a concrete example, identify the pen-ultimate step, etc.
When I give whiteboard interviews, I have a favorite question with about four reasonable stopping points. My general approach is to work with the candidate until they get stuck, and then see how handle being stuck and getting un-stuck. So, even if you have trouble with the 'easiest' part of the problem, I can still get a good read on the general problem solving skills. (And if you sail through to the hardest level solution without getting stuck, you're probably worth hiring anyway...)
Classes and competitions also generally have an allowed 'bag of tricks' from which you build your solutions. White-board coding interviews are the same. If you want to do well, you get to know what the 'atomic' operations are in your bag of tricks, how they combine into useful tools, and how to spot when they might be useful in the 'wild.'
Restating/reworking a problem into a form where some tricks are more obviously applicable is also a useful skill. These are general problem solving skills, which are pretty independent of what's actually in the bag of tricks... Draw a picture, try a concrete example, identify the pen-ultimate step, etc.
When I give whiteboard interviews, I have a favorite question with about four reasonable stopping points. My general approach is to work with the candidate until they get stuck, and then see how handle being stuck and getting un-stuck. So, even if you have trouble with the 'easiest' part of the problem, I can still get a good read on the general problem solving skills. (And if you sail through to the hardest level solution without getting stuck, you're probably worth hiring anyway...)
Your experience matches my own with TA'ing being a fertile training ground for solving problems from a variety of perspectives on your toes. I also appreciate your resultant interview approach with which I've also found great success.
The point for me that is most revealing is the adaptation to the sticking point, as the questions are engineered to zig, when a candidate zags more than from solving any particular software problem.
The point for me that is most revealing is the adaptation to the sticking point, as the questions are engineered to zig, when a candidate zags more than from solving any particular software problem.
This sounds amazing, like the many scenes from science-themed movies that feature two scientists, often an elder and a younger. The younger is scrawling out the solution to some near-intractable problem on the blackboard, while the elder guides and goads, perhaps pointing out a flaw in the proof or an overlooked detail. I believe Good Will Hunting and Theory of Everything had scenes like this.
Regrettably I've squandered my youth and it's too late for my chance at the blackboard, solving something really hard under the eye of a judging but patient mentor. So now I code webshit for a living. Congrats on that doctorate, man.
Regrettably I've squandered my youth and it's too late for my chance at the blackboard, solving something really hard under the eye of a judging but patient mentor. So now I code webshit for a living. Congrats on that doctorate, man.
This is similar to oral math exams in Danish high school.
The main differences being that the teacher submits the curriculum, and the student draws 1 topic on the day of the exam and is given something like 20 minutes to prepare between drawing and presentation. All submitted topics are known to the student, through the submitted curriculum, which has all been taught throughout the year(s).
Grade is dependent on how well you do the proof related to the topic and how well you manage to reference related topics in the curriculum.
The main differences being that the teacher submits the curriculum, and the student draws 1 topic on the day of the exam and is given something like 20 minutes to prepare between drawing and presentation. All submitted topics are known to the student, through the submitted curriculum, which has all been taught throughout the year(s).
Grade is dependent on how well you do the proof related to the topic and how well you manage to reference related topics in the curriculum.
Why not a written part too? I heard it's both usually.
I didn't experience this, but I was at dinner with someone who had recently emigrated from Russia, so I decided to ask, "What is it about the education systems in formerly Soviet domains that created such a strong passion for computing?" He answered in two parts:
1. Scholars who might've considered studying literature and philisophy might have a hard time competing on the global stage, as the Soviet state didn't take kindly to the idea of promoting anything that could be perceived as anti-Soviet ideals, even if it's for the sake of an academic exercise. Not that the Soviet Union was alone in this practice, but this practice in particular affected their academic community to the extent that many who might've considered literature or philosophy changed their minds.
2. Trade restrictions between the 50s and 60s with large portions of the West created a large demand for semiconductor products on behalf of the state, as the USSR understood the strategic importance of this technology early on. While trade restrictions were gradually relaxed in the decades leading up to Perestroika, the domestic industry for computer products had been established, similar to China's own semiconductor industry and Deng Xiaoping's economic reforms which opened the country to global trade.
This is mostly just the verbal account of one person followed by my own personal research, so this is by no means an authoritative take. If there are others with more knowledge (acquired through research or lived experience) I'd love to hear it, as my knowledge of the history of computing has a Soviet-sized hole in it.
1. Scholars who might've considered studying literature and philisophy might have a hard time competing on the global stage, as the Soviet state didn't take kindly to the idea of promoting anything that could be perceived as anti-Soviet ideals, even if it's for the sake of an academic exercise. Not that the Soviet Union was alone in this practice, but this practice in particular affected their academic community to the extent that many who might've considered literature or philosophy changed their minds.
2. Trade restrictions between the 50s and 60s with large portions of the West created a large demand for semiconductor products on behalf of the state, as the USSR understood the strategic importance of this technology early on. While trade restrictions were gradually relaxed in the decades leading up to Perestroika, the domestic industry for computer products had been established, similar to China's own semiconductor industry and Deng Xiaoping's economic reforms which opened the country to global trade.
This is mostly just the verbal account of one person followed by my own personal research, so this is by no means an authoritative take. If there are others with more knowledge (acquired through research or lived experience) I'd love to hear it, as my knowledge of the history of computing has a Soviet-sized hole in it.
Eastern Europe in general has a strong math & science culture. It operates on a lot of levels and is hard to summarize, but it predates the 50’s for sure.
How so? There were no Eulers in Eastern Europe except for Russia, which had a prestigious science academy. No Poland, Hungary, Bulgaria, Romania, etc. had that.
Well, for Hungary, you have John von Neumann and Eugene Wigner, who both finished an Elite School in Budapest: https://en.wikipedia.org/wiki/Fasori_Gimn%C3%A1zium. And you also have Erdos.
Of course, you might say they were Jewish, but they are the product of the Hungarian School of Mathematics.
Of course, you might say they were Jewish, but they are the product of the Hungarian School of Mathematics.
Does Rubik count as well? Not a mathmatician, but everybody knows him for a reason.
Ulam, Banach? Other people who made the atomic bomb? Often polish-jewish
https://en.m.wikipedia.org/wiki/List_of_Polish_mathematician... (on a side note, the table should be reworked with sortable dates and names - last name perhaps first but Im not sure how to do it)
https://en.m.wikipedia.org/wiki/List_of_Polish_mathematician... (on a side note, the table should be reworked with sortable dates and names - last name perhaps first but Im not sure how to do it)
There were no Eulers anywhere except Switzerland (later Russia), though I guess there was a Gauss in Germany, so I think that’s a slightly poor example. A pretty random list of polish mathematicians you may have heard of:
- Copernicus
- Marie Curie
- Banach and Tarski
I think it’s worth noting that the history of the modern borders and what happened inside them is more complicated than that of (say) Switzerland so, for example, someone may have come from what is now considered Poland but be seen as a Prussian or Russian scientist based on where the political entity that was around when they were working.
I think it’s reasonable to say that the math & science culture goes back at least as far as when socialist-style education systems were being set up.
- Copernicus
- Marie Curie
- Banach and Tarski
I think it’s worth noting that the history of the modern borders and what happened inside them is more complicated than that of (say) Switzerland so, for example, someone may have come from what is now considered Poland but be seen as a Prussian or Russian scientist based on where the political entity that was around when they were working.
I think it’s reasonable to say that the math & science culture goes back at least as far as when socialist-style education systems were being set up.
In America it's the opposite direction post-Apollo program . Outside of Silicon Valley at least.
> If there are others with more knowledge (acquired through research or lived experience) I'd love to hear it, as my knowledge of the history of computing has a Soviet-sized hole in it.
The famous book Mathematics, its Content, Methods, and Meaning by Alexandrov, Kolmogorov, et al. has two chapters on computing, which is interesting both to take a sneak peak into Soviet-era techniques, as well as to understand the importance Kolmogorov and friends gave to the topic.
The famous book Mathematics, its Content, Methods, and Meaning by Alexandrov, Kolmogorov, et al. has two chapters on computing, which is interesting both to take a sneak peak into Soviet-era techniques, as well as to understand the importance Kolmogorov and friends gave to the topic.
What educated industrial society didn't have a passion for computing?
I guess my language in the original comment didn't really convey the full context of the discussion. My use of the phrase "passion" was referring to what appeared to be the observed success of these countries in the fields of mathematics, statistics, and computer science I thought were disproportionate to other measures of human developent. For example, the OECD data indicates that Hungary invests a lesser proportion of its budget into education compared to other countries mentioned in the article, or even other OECD countries[1]. However, Hungary is clearly established itself as a creator of serious mathematical talent based on the IMO stats shared by the author of the post (even thought the math skills used to win gold at the IMO are very different from what's needed to advance the field as a whole[2]).
At a very superficial level, the question is akin to, "Why does Argentina have such a successful national soccer team, when there are other countries with a strong cultural link to soccer and much more capable of pouring money into building a good team?". There is clearly some nuance I was missing when diving into this question but, I don't know what I don't know. I started by asking the question to learn more.
1: https://www.oecd-ilibrary.org/sites/7688ce27-en/index.html?i... 2: https://x.com/dwarkesh_sp/status/1712221099344458092
At a very superficial level, the question is akin to, "Why does Argentina have such a successful national soccer team, when there are other countries with a strong cultural link to soccer and much more capable of pouring money into building a good team?". There is clearly some nuance I was missing when diving into this question but, I don't know what I don't know. I started by asking the question to learn more.
1: https://www.oecd-ilibrary.org/sites/7688ce27-en/index.html?i... 2: https://x.com/dwarkesh_sp/status/1712221099344458092
> Between a=5√2 and b=2√13 which one is bigger?
No need to memorize square roots here. Simply square both sides and it’s clear that 50 < 52.
If x^2 < y^2 then x < y (ignoring negative numbers…).
More generally, these types of “math tricks” are about solving for the answer. Not solving for the parts of the question. It’s a great lessom for life, especially working in software.
No need to memorize square roots here. Simply square both sides and it’s clear that 50 < 52.
If x^2 < y^2 then x < y (ignoring negative numbers…).
More generally, these types of “math tricks” are about solving for the answer. Not solving for the parts of the question. It’s a great lessom for life, especially working in software.
Author here, it's probably what I did, but i am sure i also had to memorize quite a few radicals, and also multiplying numbers up to 40X40 + other tricks to multiply fast.
Memorized non-integer radicals outside of 2, 3, and 5? Those at least come up in geometry.
Any idea where I can get more English questions like the ones you’ve listed? They’re quite enjoyable! Though some of your translations are a bit off:
> A cube has one of its sides 2cm. The total area of the cube is ...
Pretty sure you mean “surface area”.
Any idea where I can get more English questions like the ones you’ve listed? They’re quite enjoyable! Though some of your translations are a bit off:
> A cube has one of its sides 2cm. The total area of the cube is ...
Pretty sure you mean “surface area”.
The sources are scarce at least for everything that's older than 2010, but I did started to collect exercises like this from various forgotten places.
I plan to put them into a github repo this month.
Sorry for the translations. I will try to check them again, and maybe to some corrections.
I plan to put them into a github repo this month.
Sorry for the translations. I will try to check them again, and maybe to some corrections.
Don't be sorry! I enjoyed your post, and your English is very good.
Here's another tiny little nitpick if you want to improve your translation a bit more: In English we don't say "inequation", we say "inequality".
Here's another tiny little nitpick if you want to improve your translation a bit more: In English we don't say "inequation", we say "inequality".
When you get the repo up, can you tell me? I'm very interested in a summary like that.
Here is a raw dump for the Baccalaureate Exam:
https://github.com/nomemory/romanian-math
I will reply to you here.
One of my grandmothers, born about 100 years ago in Romania (she’s not with us anymore), told me many times she was denied from pursuing math education because of then Nazi policies of “numerus nulus” i.e. the exclusion of jews from public universities.
There’s a fascinating wikipedia page on the topic of racial academic exclusions - https://en.m.wikipedia.org/wiki/Numerus_clausus
And related to math exams, this paper of problems that looked easy but were quite difficult (and which were given to jews in the USSR) is also quite interesting — https://arxiv.org/abs/1110.1556
There’s a fascinating wikipedia page on the topic of racial academic exclusions - https://en.m.wikipedia.org/wiki/Numerus_clausus
And related to math exams, this paper of problems that looked easy but were quite difficult (and which were given to jews in the USSR) is also quite interesting — https://arxiv.org/abs/1110.1556
Quite a lot Jewish people were influential in mathematics in Romania.
https://en.m.wikipedia.org/wiki/Leon_Birnbaum
https://en.m.wikipedia.org/wiki/David_Emmanuel_(mathematicia...
https://en.m.wikipedia.org/wiki/David_Emmanuel_(mathematicia...
https://en.m.wikipedia.org/wiki/Meinhard_E._Mayer
https://en.m.wikipedia.org/wiki/Isaac_Jacob_Schoenberg
And many others.
My grandfather who become a math teacher eventually had a Jewish math teacher. So was my grandmother's brother, he had lots of Jewish math teachers.
Unfortunately things become bad during ww2, but before that Jews were overly represented into teaching math.
https://en.m.wikipedia.org/wiki/Leon_Birnbaum
https://en.m.wikipedia.org/wiki/David_Emmanuel_(mathematicia...
https://en.m.wikipedia.org/wiki/David_Emmanuel_(mathematicia...
https://en.m.wikipedia.org/wiki/Meinhard_E._Mayer
https://en.m.wikipedia.org/wiki/Isaac_Jacob_Schoenberg
And many others.
My grandfather who become a math teacher eventually had a Jewish math teacher. So was my grandmother's brother, he had lots of Jewish math teachers.
Unfortunately things become bad during ww2, but before that Jews were overly represented into teaching math.
Thanks for this. My grandmother came of university age in the 40s, so this all makes sense.
Disproportionately, not "overly".
The fact that I could figure out the writer was a fellow countryman just by reading the title of the HN post alone speaks volumes about this country’s obsession with math.
I went through something very similar to the author (went to the best high school and yadda yadda), I then went on to get dragged through even more math in Comp Sci. I was fairly good and I loved the way of thinking math makes you discover, but I hated the grind and my grades suffered as as result, started dropping below a 9 at the end of HS and barely got passing grades in Uni.
IMO this system is thoroughly fucked due to how disconnected it is from teaching students something that’s actually relevant and applicable in every day life.
I went through something very similar to the author (went to the best high school and yadda yadda), I then went on to get dragged through even more math in Comp Sci. I was fairly good and I loved the way of thinking math makes you discover, but I hated the grind and my grades suffered as as result, started dropping below a 9 at the end of HS and barely got passing grades in Uni.
IMO this system is thoroughly fucked due to how disconnected it is from teaching students something that’s actually relevant and applicable in every day life.
It depends on the maths teacher on how good or bad one can be at maths (I used to be bad at maths before HS because of a teacher, I was actually good at maths in HS also because of a teacher), even though I agree with you that we have an obsession with it, and especially about those Olympics that are of no use (I always like to point to people that no matter how many "gold" medals we got at those olympics we've never had a Fields medalist until now).
What's not told to students, or not instilled into their heads, is that, first, maths as taught in uni has no connection to how maths is taught in HS, and that's a good thing (there are a few exceptions that confirm the rule, meaning HS math teachers that "approach" the philosophy of how maths is taught at uni but, again, they're very few and far between), and second, the first two years of uni are some of the most important years in one's education when it comes to his/her future professional life. Later on during uni you're sort of specialising, but during those first two years you should construct the theoretical base for your future career in any field related to maths (which covers a lot of today's tech).
You mentioned teaching students stuff that is not relevant, and of course that many of my colleagues at uni (Comp Sci at Bucharest Politehnica) at the end of the '90s - the start of the 2000s had the same opinion, that's why the class that was teaching Object Orientated programming applied to Java was almost always full (or as full as a Politehnica class could have been back then), while the Matematici Speciale I + II class (I think the Americans call it Algebra I and II, not sure, never been there) was attended at some point only by 15-20 students out of a total of 100-120. It turns out that nowadays matrices and manipulation of matrices (which those classes covered quite in detail) is a lot, lot more important compared to writing some stupid videos games in Java applets (which I think was the subject of a class in like 2002-2003, something like that).
Also, because this is an American forum and we're talking about how maths is studied in Romania, I'll always remember our first year Calculus teacher at Politehnica (mr. Răbâncă, a legend in his own right if you search for his name on the early Romanian web) who was bad-mouthing the American Green Visa Lottery system in between talking about some infinite series or what have you. Apparently he had won one such visa sometime in the mid-'90s, had actually gone to the States only to find out that he was unemployable in his field of study there, or at least at the beginning, to quote him: "I got there and they asked me to mop the floors. Of course I came back". That was a big loss for the States, a big win for us.
What's not told to students, or not instilled into their heads, is that, first, maths as taught in uni has no connection to how maths is taught in HS, and that's a good thing (there are a few exceptions that confirm the rule, meaning HS math teachers that "approach" the philosophy of how maths is taught at uni but, again, they're very few and far between), and second, the first two years of uni are some of the most important years in one's education when it comes to his/her future professional life. Later on during uni you're sort of specialising, but during those first two years you should construct the theoretical base for your future career in any field related to maths (which covers a lot of today's tech).
You mentioned teaching students stuff that is not relevant, and of course that many of my colleagues at uni (Comp Sci at Bucharest Politehnica) at the end of the '90s - the start of the 2000s had the same opinion, that's why the class that was teaching Object Orientated programming applied to Java was almost always full (or as full as a Politehnica class could have been back then), while the Matematici Speciale I + II class (I think the Americans call it Algebra I and II, not sure, never been there) was attended at some point only by 15-20 students out of a total of 100-120. It turns out that nowadays matrices and manipulation of matrices (which those classes covered quite in detail) is a lot, lot more important compared to writing some stupid videos games in Java applets (which I think was the subject of a class in like 2002-2003, something like that).
Also, because this is an American forum and we're talking about how maths is studied in Romania, I'll always remember our first year Calculus teacher at Politehnica (mr. Răbâncă, a legend in his own right if you search for his name on the early Romanian web) who was bad-mouthing the American Green Visa Lottery system in between talking about some infinite series or what have you. Apparently he had won one such visa sometime in the mid-'90s, had actually gone to the States only to find out that he was unemployable in his field of study there, or at least at the beginning, to quote him: "I got there and they asked me to mop the floors. Of course I came back". That was a big loss for the States, a big win for us.
I went to an international school, so I somehow got impressions of the different math systems of various countries through friends from around the world.
My impression was that South Koreans were the most intensely drilled kids. They would show up and panic about being left behind due to the pedestrian nature of IB higher math. Like they would literally show up, see that we are doing some integral, and be like "oh am I in the wrong room? "
Scandinavian countries and North America seemed to be the gentle touch. Particularly Americans seemed to show up as if they'd never seen algebra by the time we got to the last couple of years of high school. I remember seeing my cousin's math homework from the US and it seemed too easy for his age. Likewise with a Danish cousin.
The British system seemed soft at the time. I sat the GCSE and A levels a year early, thought I'd done badly with the syllabus being a bit different, but the marking was so lenient I got the top mark anyway. My kid is in that system now and he's able able to answer questions for exams that happen in five years, so maybe it's just that the exam is for a very broad range of abilities and he's had good teaching.
Eastern Europe was on a par with the top countries. I was actually on a math team where we'd do contests with other international schools. Most years, the winning team was Koreans and East Asians. The year we won it, my two teammates were Polish. You could tell they were well drilled from their previous education. I still talk to them, great friends.
I never knew how big a deal IMO was. I know a couple of Iranian gold medallists, and they seemed to have special classes to prepare, lasting a significant amount of time, with a later field's medallist teaching them. I know a guy who did the whole Chinese system, also hugely impressive and with a lot of work. By contrast, a kid from my school got into our national team with barely any coaching before the big event.
My impression was that South Koreans were the most intensely drilled kids. They would show up and panic about being left behind due to the pedestrian nature of IB higher math. Like they would literally show up, see that we are doing some integral, and be like "oh am I in the wrong room? "
Scandinavian countries and North America seemed to be the gentle touch. Particularly Americans seemed to show up as if they'd never seen algebra by the time we got to the last couple of years of high school. I remember seeing my cousin's math homework from the US and it seemed too easy for his age. Likewise with a Danish cousin.
The British system seemed soft at the time. I sat the GCSE and A levels a year early, thought I'd done badly with the syllabus being a bit different, but the marking was so lenient I got the top mark anyway. My kid is in that system now and he's able able to answer questions for exams that happen in five years, so maybe it's just that the exam is for a very broad range of abilities and he's had good teaching.
Eastern Europe was on a par with the top countries. I was actually on a math team where we'd do contests with other international schools. Most years, the winning team was Koreans and East Asians. The year we won it, my two teammates were Polish. You could tell they were well drilled from their previous education. I still talk to them, great friends.
I never knew how big a deal IMO was. I know a couple of Iranian gold medallists, and they seemed to have special classes to prepare, lasting a significant amount of time, with a later field's medallist teaching them. I know a guy who did the whole Chinese system, also hugely impressive and with a lot of work. By contrast, a kid from my school got into our national team with barely any coaching before the big event.
Compared to when I was at school, 25-30 years ago, the current UK maths syllabus is much harder. It still focuses far too much on "how" rather than "why", though.
the UK A level maths exam does cover a dramatic range of ability - there are a fair few gimme questions and some ambitious ones towards the back of the exam. There are also specialties, of which you usually only have to pick one (iirc decision maths, mechanics, or statistics. Then if you take further maths you do the remaining ones plus some more "why" stuff with formal proofs etc)
In general, having taken maths and further maths, after moving to denmark and studying in university, I found myself a lot further ahead than my peers for years (and subsequently slacked off and had to relearn the discipline aha). I also found that I had quite an advantage at doing things in my head/with blackboard, wheras danish students wanted to go to matlab/computer immediately. I think i'm glad that I learned it this way, although I often will use matlab, I do find it has helped me catch mistakes or things that just dont feel right
In uni in denmark they do have strange oral maths exams though sometimes. That was interesting. Personally I quite liked it, although I know some of my peers absolubtely hated it, and spent days feeling nauseous at the prospect of facing a panel of professors
In general, having taken maths and further maths, after moving to denmark and studying in university, I found myself a lot further ahead than my peers for years (and subsequently slacked off and had to relearn the discipline aha). I also found that I had quite an advantage at doing things in my head/with blackboard, wheras danish students wanted to go to matlab/computer immediately. I think i'm glad that I learned it this way, although I often will use matlab, I do find it has helped me catch mistakes or things that just dont feel right
In uni in denmark they do have strange oral maths exams though sometimes. That was interesting. Personally I quite liked it, although I know some of my peers absolubtely hated it, and spent days feeling nauseous at the prospect of facing a panel of professors
So you did high school in the UK and uni in DK? How did they convert the grades?
My kid might do this someday, that's why I'm asking.
My kid might do this someday, that's why I'm asking.
For me they actually didn't, there are two admissions tracks, quota 1 and quota 2. The first is much more based on your "gpa" type thing, which is converted by the educational institution you apply for. The danish grades system is strange, a 12 point scale, with the only possible grades being 0 (a fail), 2 (barely passed), 4, 7, 10, and 12 (excellent grade).
Personally my A levels results weren't great so I applied via quota 2, where it is more about the personal statement you submit and other external achievements. I had done work in open source and won some programming competitions which meant I got in relatively easily. Amusingly enough they did want me to submit proof of my English proficiency which was hard because I had never done an exam in it but it was easy enough with a phone call to resolve that (presumably partly because they could hear me conducting the phone call in english aha)
here you can find the guidelines they use to convert foreign exams https://ufm.dk/uddannelse/anerkendelse-og-dokumentation/find...
Personally my A levels results weren't great so I applied via quota 2, where it is more about the personal statement you submit and other external achievements. I had done work in open source and won some programming competitions which meant I got in relatively easily. Amusingly enough they did want me to submit proof of my English proficiency which was hard because I had never done an exam in it but it was easy enough with a phone call to resolve that (presumably partly because they could hear me conducting the phone call in english aha)
here you can find the guidelines they use to convert foreign exams https://ufm.dk/uddannelse/anerkendelse-og-dokumentation/find...
What degree did you do that you needed english proficiency?
Robotics batchelors (and subsequently masters), offered in english (although one or two of the first year classes were in danish it wasn't too onerous to get around). A few years ago the government had an agenda to reduce foreign students so axed all (most?) of the english language batchelors, so it probably isnt available anymore.
> Back in school, I had to remember a few radicals to use them as needed. √13 was one of them. I have forgotten them by now. We also had to learn the algorithm to compute any √ as needed
That’s very interesting. In Poland we also had problems of this type, but we were taught to solve them by simply bringing everything under the square root and comparing the numbers. Definitely resulted in much bigger numbers but meant less memorization x)
That’s very interesting. In Poland we also had problems of this type, but we were taught to solve them by simply bringing everything under the square root and comparing the numbers. Definitely resulted in much bigger numbers but meant less memorization x)
The comment for that particular problem also caught my eye. There's absolutely no need to remember radicals to solve it; just square both sides and compare.
I'm about seven or eight years older compared to the poster (am also Romanian) and most certainly we didn't need to memorise √13, this is actually the first time I hear about it.
I think (and now that I've checked I'm 100% sure) that we knew by heart the first two decimals of square root of 2, meaning 1.41, and possibly Euler's number (2.71) too, even though I'm not so sure about that anymore (this was 25+ years ago).
I think (and now that I've checked I'm 100% sure) that we knew by heart the first two decimals of square root of 2, meaning 1.41, and possibly Euler's number (2.71) too, even though I'm not so sure about that anymore (this was 25+ years ago).
That problem can be solved with estimation of the roots on a number line. It's not the most precise way of calculating it but when visualized you can clearly see that the root of 13 will have a fractional part above .5, while the root of 2 will have a fraction below .5.
Did you actually solve it?
5 times <1.5 is <7.5, and 2 times >3.5 = >7 so "more or less than .5" is insufficient for this comparison.
5 times <1.5 is <7.5, and 2 times >3.5 = >7 so "more or less than .5" is insufficient for this comparison.
You can solve it, however, by knowing that
P.S: 0.45^2 = .2025 and 1.45 = 2.125 is an amusing coincidence.
5√2 < 5 * 1.45 = 7.2
2√13 > 2√12.96 = 2 * 3.6 = 7.2
P.S: 0.45^2 = .2025 and 1.45 = 2.125 is an amusing coincidence.
> If you take Russia and Ukraine aside, Romania, Bulgaria, Hungary, and Poland have less than 80 million citizens together, but have accounted for 261 Gold Medals through the years, more than China or USA, the World’s two Superpowers. Of course, most of those students (and the ones that don’t qualify to the International but are way above the world’s average) leave Romania (or Poland, Hungary, and Bulgaria) to more prosperous, more developed countries.
Romania, Bulgaria, Hungary and Poland had 7.5x the slots than China, and only produced 1.23 times the gold medals. Not that impressive.
Romania, Bulgaria, Hungary and Poland had 7.5x the slots than China, and only produced 1.23 times the gold medals. Not that impressive.
Yeah, that was a very weird comparison. Not only do those countries send 6 people each, those four started participating in 1959 while US started in 1974 and China in 1985. Though arguably, those first olympiads featured fewer countries, (with presumably stronger competition) so they had to compete against each other for medals.
Another factor might be the number of distinct participants. I believe the Chinese enforce a one-participation limit, though I don't know if that would make it more or less likely to get gold.
Another factor might be the number of distinct participants. I believe the Chinese enforce a one-participation limit, though I don't know if that would make it more or less likely to get gold.
Slots aren't meaningful. There is very little random variation in performance on the event. More slots per capita just means more room for less capable people to compete and lose.
of course they are. The Chinese team selection tests are notoriously more difficult than the IMO, and they enforce a one-time participation limit. If they were to send 24 people, not only would they probably be returning with ~20 golds, they'd be taking away the golds of the other nations (as only the top 1/12 get gold)
Yes, China having more slots would mean unclear factor of more gold medals for China. (Although China has only earned 6 gold medals in half of its years, and 5 or 6 gold medals in only 24/34 years, despite having 1/6 the world population so proportionally should claim ">8" of the ~50 gold medals each year, so China does appear to be under performing the average country (per capita) in half the years.)
But those 4 smaller countries having 1/7 slots wouldn't mean 1/7 gold medals, since they only send 5-10 gold medalists each year in those slots.
Romania averages over 1 gold medal per year. China, with 70x the population of Romania, should easily be getting 6 gold medals per year almost every year and about 70 theoretical golds of their deep bench got tested, if it has comparable per capita performance to Romania.
Of course all these stats are kind of silly because of sampling limitations.
But those 4 smaller countries having 1/7 slots wouldn't mean 1/7 gold medals, since they only send 5-10 gold medalists each year in those slots.
Romania averages over 1 gold medal per year. China, with 70x the population of Romania, should easily be getting 6 gold medals per year almost every year and about 70 theoretical golds of their deep bench got tested, if it has comparable per capita performance to Romania.
Of course all these stats are kind of silly because of sampling limitations.
On most years China wins six golds, so more slots would definitely lead to more golds.
Genuine question: Why did they have more slots?
(Sure, I could Google this, but I'm guessing this question may lead to some interesting conversation :) )
(Sure, I could Google this, but I'm guessing this question may lead to some interesting conversation :) )
Because they are 4 countries, and china is only 1. Also they started IMO, and china first participated 26 years later.
Having grown up in Norway, our school system was (still is?) much more focused on lifting all boats, and making most pupils pretty decent and well-rounded at most subjects, rather than focusing resources on the most apt ones. But that's just how things are in general - our culture has always been focused on what's best for the masses.
That said, I can't remember our math AP classes being any difficult. Whenever I read about HS students in other countries taking proof-heavy classes, I have zero recollection of that.
We had a couple of Russian kids, and one Polish, in our HS class. At least the Russian kids told us that the math classes were much harder back in Russia, but that's about it.
EDIT: To expand - we have two AP math classes in HS, and two AP physics classes. To qualify for STEM studies at Uni., you usually need two math AP + one physics AP.
You get some extra "competition points" if you take more AP classes, some points for age, some for military service, etc.
But that's about it. No entrance exams, no standardized tests, etc. The whole country gets the same exams, and for first-time applicants it comes down to your GPA + points from AP classes. For people after that, it's the same but all the different age points, military points, etc.
That said, I can't remember our math AP classes being any difficult. Whenever I read about HS students in other countries taking proof-heavy classes, I have zero recollection of that.
We had a couple of Russian kids, and one Polish, in our HS class. At least the Russian kids told us that the math classes were much harder back in Russia, but that's about it.
EDIT: To expand - we have two AP math classes in HS, and two AP physics classes. To qualify for STEM studies at Uni., you usually need two math AP + one physics AP.
You get some extra "competition points" if you take more AP classes, some points for age, some for military service, etc.
But that's about it. No entrance exams, no standardized tests, etc. The whole country gets the same exams, and for first-time applicants it comes down to your GPA + points from AP classes. For people after that, it's the same but all the different age points, military points, etc.
That sounds similar to how my school in Australia worked.
I wanted to study Engineering at university so was required to have a certain amount of advanced Mathematics courses as well as physics.
We had two Math Streams, the advanced stream and regular stream. I was in the advanced stream but I had friends who were in the normal stream I remember feeling jealous because the normal stream got to do lots of excursions and other "fun" activities. For example my friend's entire math class played ten pin bowling at local bowling ally (ostensibly so they could compare scores and learn about median and distributions). Their class also got to visit the horse racing track to learn about odds and probability.
I think these were ham-fisted attempts to make mathematics "fun". Meanwhile my advanced class got to enjoy none of this and were stuck in the classroom all the time. At the time it felt more like a punishment then a reward.
I wanted to study Engineering at university so was required to have a certain amount of advanced Mathematics courses as well as physics.
We had two Math Streams, the advanced stream and regular stream. I was in the advanced stream but I had friends who were in the normal stream I remember feeling jealous because the normal stream got to do lots of excursions and other "fun" activities. For example my friend's entire math class played ten pin bowling at local bowling ally (ostensibly so they could compare scores and learn about median and distributions). Their class also got to visit the horse racing track to learn about odds and probability.
I think these were ham-fisted attempts to make mathematics "fun". Meanwhile my advanced class got to enjoy none of this and were stuck in the classroom all the time. At the time it felt more like a punishment then a reward.
I hate the term "brain drain".
I have a very similar background. The elite schools he talks about after passing those exams early. Hundreds of exercises, day after day, for a decade on end, classes, after school programs. Eventually a PhD.
I'm a human being, not some statistic to bolster the GDP of some corrupt country that doesn't give a damn about me. It's not "brain drain" it's just building a better life for yourself. Other countries want to give me a better life, to give my kids a better life, where they don't need to deal with corrupt cops that shake us down, doctors that demand bribes to provide the most basic care, wars, etc.
> Brain drain is a sad reality few try to counter.
My extended family was abused, expropriated, and at times just killed for centuries by the political system of these countries while not letting us leave. I'm so glad that my generation is the last one to have any ties to that place. Everyone in my family got educated and left.
Brain drain is the best thing that's ever happened to us. It's not something to counter.
I have a very similar background. The elite schools he talks about after passing those exams early. Hundreds of exercises, day after day, for a decade on end, classes, after school programs. Eventually a PhD.
I'm a human being, not some statistic to bolster the GDP of some corrupt country that doesn't give a damn about me. It's not "brain drain" it's just building a better life for yourself. Other countries want to give me a better life, to give my kids a better life, where they don't need to deal with corrupt cops that shake us down, doctors that demand bribes to provide the most basic care, wars, etc.
> Brain drain is a sad reality few try to counter.
My extended family was abused, expropriated, and at times just killed for centuries by the political system of these countries while not letting us leave. I'm so glad that my generation is the last one to have any ties to that place. Everyone in my family got educated and left.
Brain drain is the best thing that's ever happened to us. It's not something to counter.
> My extended family was abused, expropriated, and at times just killed for centuries by the political system of these countries while not letting us leave. I'm so glad that my generation is the last one to have any ties to that place. Everyone got educated and left.
> Brain drain is the best thing that's ever happened to us. It's not something to counter.
When people say countering brain drain, they typically mean stopping the things that lead to it, not forcing people to stay in miserable conditions. So, in this instance, stopping the abusing, expropriating, and killing.
> Brain drain is the best thing that's ever happened to us. It's not something to counter.
When people say countering brain drain, they typically mean stopping the things that lead to it, not forcing people to stay in miserable conditions. So, in this instance, stopping the abusing, expropriating, and killing.
The point of "brain drain" is not to accuse those who leave of anything. The point is that, judging from the perspective of the country itself, or from that of the people who stay, having all your brightest minds leave is devastating.
And, as others have mentioned, forcing people to stay against their will is not what anyone wants as a cure to brain drain. The point is exactly to try to create the appropriate conditions to make people not want to leave, so they can help repair and improve the damage of many generations past.
And, as others have mentioned, forcing people to stay against their will is not what anyone wants as a cure to brain drain. The point is exactly to try to create the appropriate conditions to make people not want to leave, so they can help repair and improve the damage of many generations past.
I can't agree more. I'm from China, and did Math Olympiad before as well and ended up in US. And most of good ones are here as well, except those would works well with that system.
Everyone left? No one innocent lives there anymore?
Not one member of my extended family. No. Thank goodness.
Is concentrating resources on the gifted more productive in national outcomes than raising the overall bar? Would the resources devoted to the gifted be better spread out to the whole population? I think it depends on the quality of normal schools. If it is poor, it is easier to lift and protect the gifted, who are most receptive to education, from the rest.
The gist I got was that they were sweeping everyone below the 85% mark under the carpet and then MK Ultraing the top 15% to be stepford math children.
Mihai was absolutely brilliant but better to emulate in intellect than interpersonal skills. ;)
Thanks for sharing your article - I enjoyed reading it! The number of stunningly capable CS colleagues I know who came out of Romania is impressive.
Also, small typo: in the translation of problem 1 from the third exam, the first case of f(x) says m <= 1 and the second case says x > 1. The first case should say x <= 1 as it does in the original problem.
Thanks for sharing your article - I enjoyed reading it! The number of stunningly capable CS colleagues I know who came out of Romania is impressive.
Also, small typo: in the translation of problem 1 from the third exam, the first case of f(x) says m <= 1 and the second case says x > 1. The first case should say x <= 1 as it does in the original problem.
Makes me sad. None of these questions can ignite a passion for maths or connect maths to the real world.
Those are quite interesting. The test for 13 year olds looks pretty easy for 13 year olds; the high school one looks challenging for that cohort and the undergraduate admissions test is harder than any test I was given at that age.
To get into MIT in the 80s the only formal test was the SAT plus you wrote a few essays. My high school didn't do "AP" subjects nor compute GPA nor class rank, just ABCF twice a year, and had essentially no electives. So basically they just looked at my SAT scores I guess. I could not have passed that Romanian test at that age.
To get into MIT in the 80s the only formal test was the SAT plus you wrote a few essays. My high school didn't do "AP" subjects nor compute GPA nor class rank, just ABCF twice a year, and had essentially no electives. So basically they just looked at my SAT scores I guess. I could not have passed that Romanian test at that age.
but 1500+ SAT scores were rare back then, compared to today. Yet MIT is much more competitive. This means additional criteria are needed beyond SATs.
If a school only accepts students capable of attaining high grades, in what sense is that school good? That's no proving the school teaches well, it's proof that they're trying to avoid having to teach.
Being selective means they can start further, go faster, and expect motivation.
Even if the value is coming entirely from your fellow students, that has no effect on deciding which school you want to go to. You want to be in that environment, and also get the status from the exclusivity.
Even if the value is coming entirely from your fellow students, that has no effect on deciding which school you want to go to. You want to be in that environment, and also get the status from the exclusivity.
I've studied in not-so-elite math/physics/CS high school in Russia. It has a focus on full coverage of school program so alumni would crush the exams to get into universities they want. The entry exams were hard because in the first year they cover the entire program of specialties without cutting the corners (i.e. van der Waals equation, nearly all known to science properties of triangles and 4-side polygons, methods to ease brute force a polynom of Nth degree) with extensive homework. This is an impossible task for anyone who isn't already good because there's so much to learn and only 24 hours in a day.
The actual quality of the education at these schools varies. I went to one of the ones mentioned, and had a mix of good and bad teachers (including an awful math teacher one year, and an awful informatics teacher in another year - two of the supposedly most important subjects for that HS).
However, having an entire class that actually is at the bar for which the curriculum is designed, and having an entire class of students who generally want to learn, does allow teachers to actually spend most of their time teaching the curriculum as intended. We had no problem being up to date with the Bacalaureat matter, for example. No one in the entire HS flunked their exam either.
The general state of high schools in the country is terrible though. Teachers don't care to know the curriculum, they don't have time to go through it if they want anyone in the class to actually be listening, many poor kids don't have time or energy for school among their many other chores (especially in the more rural areas of the country, children are a big part of house and field work) etc. Many children who nominally finish high school probably haven't even heard of a quarter of the questions they'd get on the Bacalaureat.
It's also important to note that there is a massive cheating culture in Romanian schools, all the way up and down. There was some push ~10 years ago to stop cheating at the national exams, installing cameras and such, and the Bacalaureat results dropped from ~80% pass rates to ~40% nation wide in that year; there was one case of an HS where the previous year 90+% of the students had gotten a 10/10 on their exam, and after the anti-cheating measures the following year, not one child got the 5/10 required to pass.
The Capacitate/National Tests exams that you take at 14 were even worse, with all sorts of advanced questions that were either cheated through or at the very best rotely memorized. In my time, we were required to do literary analysis on one of these exams, writing a 2 page essay on things like "the characterization of <female heroine in major novel>". After some reform, today you get questions like "here is a newspaper article about a meteor. How large is the meteor? What is it made of?" and still many many children don't pass - because those are the actual results when you take cheating out.
Not to mention, Romania consistently ranks last in the EU at PISA tests. And around 50%+ of the country is functionally illiterate (that is, they technically know the letters and words, but can't actually read an article and tell you what it means).
However, having an entire class that actually is at the bar for which the curriculum is designed, and having an entire class of students who generally want to learn, does allow teachers to actually spend most of their time teaching the curriculum as intended. We had no problem being up to date with the Bacalaureat matter, for example. No one in the entire HS flunked their exam either.
The general state of high schools in the country is terrible though. Teachers don't care to know the curriculum, they don't have time to go through it if they want anyone in the class to actually be listening, many poor kids don't have time or energy for school among their many other chores (especially in the more rural areas of the country, children are a big part of house and field work) etc. Many children who nominally finish high school probably haven't even heard of a quarter of the questions they'd get on the Bacalaureat.
It's also important to note that there is a massive cheating culture in Romanian schools, all the way up and down. There was some push ~10 years ago to stop cheating at the national exams, installing cameras and such, and the Bacalaureat results dropped from ~80% pass rates to ~40% nation wide in that year; there was one case of an HS where the previous year 90+% of the students had gotten a 10/10 on their exam, and after the anti-cheating measures the following year, not one child got the 5/10 required to pass.
The Capacitate/National Tests exams that you take at 14 were even worse, with all sorts of advanced questions that were either cheated through or at the very best rotely memorized. In my time, we were required to do literary analysis on one of these exams, writing a 2 page essay on things like "the characterization of <female heroine in major novel>". After some reform, today you get questions like "here is a newspaper article about a meteor. How large is the meteor? What is it made of?" and still many many children don't pass - because those are the actual results when you take cheating out.
Not to mention, Romania consistently ranks last in the EU at PISA tests. And around 50%+ of the country is functionally illiterate (that is, they technically know the letters and words, but can't actually read an article and tell you what it means).
Your conclusion assumes the PISA test actually reflects students' abilities. This may not be the case. My daughter when attending a Romanian school told me that students know the PISA results play no part in their performance rating and many of her colleagues treated the exam either as a joke maybe even supplying deliberately dumb answers or at best as an irrelevance - 'shove anything down and finish with it'. Her experience was that the teachers were, let us say, not helpful, in countering this attitude. So, incorporate the PISA results into students' records and see if the test results change.
The functional illiteracy finding is not based on the PISA test results. It is separately measured, and it correlates to the PISA scores.
I'm sure there is some amount of unseriousness polluting the test results. It's unclear to me how much.
I'm sure there is some amount of unseriousness polluting the test results. It's unclear to me how much.
A number countries have stats that account for the socio-economic background of the kids. In the UK it's called progress-8. It looks how how good a school is vs what would be expected from the mix of students they have.
One company basically asked me this question in a phone interview. To disentangle the student's quality from the education's. I did not work there.
Good students make a school good. A school with high entrance standards provide high quality peers to learn with and from.
The PISA results of Romania paint a different picture:
„Students in Romania scored less than the OECD average in mathematics, reading and science.“
https://www.oecd.org/publication/pisa-2022-results/country-n...
„Students in Romania scored less than the OECD average in mathematics, reading and science.“
https://www.oecd.org/publication/pisa-2022-results/country-n...
The topics and problems presented, particularly the Bacalaureat, are very similar to older Greek national exams problems for high school students (up until the late 2000s I think). Nowadays the mathematics problems for the Greek national exams are much easier. However, we still insist on rigor and proofs (e.g. proofs of continuity, differentiability etc.).
> you make the kids take 4-6 of Math a week (or more),
4-6 hours?
4-6 hours?
Yes, 4-6 math classes, each roughly 1h. The Romanian school day is divided in 50 minute classes + 10 minute breaks in principle (the teachers have the ultimate say for when to take a break, so if you have 2 Math classes one after the other, the teacher may choose to merge them).
exams /s
"As a fun fact, the current mayor of Bucharest, Nicusor Dan is a two-time Gold Medalist at the International Mathematics Olympiad."
The famous reform mayor of Bogota, Antanas Mockus, also started in math!
https://en.wikipedia.org/wiki/Antanas_Mockus
And prime minister of Singapore was a senior wrangler at Cambridge: https://en.wikipedia.org/wiki/Lee_Hsien_Loong : "His college tutor, Denis Marrian, later described Lee as "the brightest mathematician he had admitted to the college". Béla Bollobás said that Lee "would have been a world-class research mathematician", but his father did not realise this and persuaded Lee to leave the field."
Do we have any mayors coding in C, Java or Python?
That's the kind of person we need more of in government!
Is it really still this hardcore math meritocracy in Romania?
In Poland it became gradually less hardcore as 80s baby boom passed through universities and at the same time private universities started to appear everywhere. Now it's pretty laid back comparatively.
In Poland it became gradually less hardcore as 80s baby boom passed through universities and at the same time private universities started to appear everywhere. Now it's pretty laid back comparatively.
I was expected to solve the problems at the board. They didn't necessarily expect me to solve them immediately, but they wanted to hear me think out loud and communicate my ideas. There were around 5-10 questions total over the course of I think a few hours, ranging in complexity from trivia (i.e. memorized facts) to computations to proofs. Some of them I found solutions for and some of them I didn't, and they assigned me some to solve later as "homework" (though they did pass me without it being conditioned on the homework).
In hindsight, it was very similar to white-board coding interviews, but I didn't know anything about coding interviews at the time. I can't really think of any other place such a format would be appropriate as a math exam. When else would a paper math exam not be sufficient.