The math exams of my life
andreinc.net
andreinc.net
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.
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...)
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.
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.
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.
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.
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.
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
Of course, you might say they were Jewish, but they are the product of the Hungarian School of Mathematics.
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)
- 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.
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.
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”.
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.
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".
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
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.
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.
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.
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.
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
My kid might do this someday, that's why I'm asking.
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...
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)
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).
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√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.
Romania, Bulgaria, Hungary and Poland had 7.5x the slots than China, and only produced 1.23 times the gold medals. Not that impressive.
(Sure, I could Google this, but I'm guessing this question may lead to some interesting conversation :) )
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.
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.
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.
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 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.
> 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.
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.
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.
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.
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.
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).
I'm sure there is some amount of unseriousness polluting the test results. It's unclear to me how much.
„Students in Romania scored less than the OECD average in mathematics, reading and science.“
https://www.oecd.org/publication/pisa-2022-results/country-n...
4-6 hours?
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.