The other kids were pretty fascinated with my slightly different long division methods, but the teachers were just obsessed with making me write an `x` instead of a `·` when writing out my multiplication, and trying to make me change the methods I'd learned previously.
I stopped finding maths interesting at that point (age 11) and it breaks my heart to this day. What I knew was enough to get me a couple of 'best-in-school + gold' medals in that Year 8 maths challenge, but it was all Bs and Cs at a-level. I did rediscover a genuine interest in mathematics again at uni as part of the foundations of AI course (CS degree), but that was short-lived and frustrating, as I knew I should have been better.
But later on, I swam like a fish in water in geometry, trig, logic and loved algebra and moved from the last of my year to one of the top. I was doing integration etc a good two years before it was being taught in the higher classes. These skills are good for programming and computers.
There was a limit; basically theorems, multi dimensional stuff and unreal? math, primes, pure math etc were completely out of my scope. Strange I guess, perhaps these are connected with being good at mental arithmetic?
My point is that I also think it's bizarre on the British obsession with mental maths early on, when for many algebra might be better and easier.
Once you pretend you have mastered the basics, there is a sudden, strange focus on advanced Calculus heuristics and obscure linear algebra techniques, effectively making most people hate these subjects. The way you are taught the stuff is basically a crime against Mathematics since it drives talented people away from STEM careers.
Or maybe it's vanity. But I will say it's depressing to see all the cashiers (in the US) that can't seem to do any kind of math in their head, not even to minimize change (which seems totally ubiquitous in many other parts of the world; in fact, some cultures are aggressive about optimizing your change - China comes to mind).
I too was much better at applied maths and stats. It was fairly easy for me. But my brain just shuts down doing pure math.
To me mathematics is a thing of beauty I just find I havent found the right handle to grasp it with.
I ask myself what would be the order in which one could most easily absorb the material. Does it really start with rote learning?
What are you trying to learn and why?
Math starts with axioms, which are some statements that everything else is built from. Axioms, as far as I understand, can't be derived. These are the most basic building blocks. Then through logic and deductions other machinery is built up. There is a certain amount that does need to be internalized or memorized, but that is the same for learning the alphabet.
In math classes, at least at the university level in Canada, knowing the statement of definitions and theorems covered exactly is somewhat important. If it's a proof based course this is more important, but the theorem will tell you where it's applicable, so knowing that helps.
Math builds. It took me until a third year complex analysis course to build up enough courage to ask a prof how he solves problems. Basically he said he has a hierarchy of theorems in a given field. When he sees a problem he will go through each of them starting with the easiest to apply and check the assumptions. If it applies, then he will use it and get the result. If not, move on to the next one. He goes and looks up the exact statements and how to go through with the calculations and checks. If you forget something, look it up.
Eventually after solving enough problems you'll build up some intuition and muscle memory for the problems.
As to how to progress, I am not sure. Some mathematicians say you kind of just pick things up and then connect the threads later on. It's not always possible to march through in a linear manner because there isn't an absolute ordering in what to learn.
There should be an appendix in the back of a textbook that reviews or outlines what's required for the book. There will be a few questions that rely on outside knowledge, but one can still learn the majority of the concepts without going to far afield for review.
Start with what you want and fill in the rest of the gaps by working backwards.
I wouldn't worry about memorizing the times tables (unless maybe you're interested in number theory?) if you're trying to learn math. Know algebra, functions, and graphing as a minimum, then start to learn what you want... I would just use a calculator or pencil and paper to multiply things out on homework. If this is for personal benefit, then don't hesitate to get some numerical methods involved to help learning the material.
Hmm.. In my experience this would have been discrete math at best which is normal in CS classes. You'd have to have taken an engineering elective or math elective to get linear algebra.
Today, with AI being so hot, I'd bet that the programs include math for engineers e.g. matrices and linear algebra. Maybe an intro course in stats and probability.
The thing we called “linear algebra” involved linear maps and vector spaces and bases and dual spaces and no grids of numbers.
Students doing physics seemed to get reasonable physics maths (e.g. vector calculus things like surface integrals and Green’s theorem) without spending lots of time revising school maths.
I don’t know what the computer scientists got but I got the impression that the university preferred applicants who were good at maths to those who were good at programming. Though maybe they needed help with asymptotics: U.K. school maths doesn’t cover limits and it is hard to define big O if you don’t know the definition of a limit.
The level of mathematics shocked me. They were still learning factorisation, something I had learned years ago. It was a breeze.
Of course, in secondary school (ages 12-18) the content eventually caught up with me. Part of me wishes I could have somehow kept studying the Bulgarian way.
What is surprising even still is that school in Bulgaria was from 07:00 to 13:00. In the UK it's 07:00 to 15:00. You had more time to do homework in Bulgaria and more time to be a child.
There's also the Julia Robinson Math Festival that's supposed to be good but I don't have first-hand experience.
Ideally, a Math Circle would could be good for some kids if your area supports one.
I went to a quite high-performing school, and it wasn't unusual to have a year be 60% 'getting through the curriculum', 20% 'teaching interesting things about the subject', 20% 'exam prep', and that 20% in the middle only existed because of the extreme priviledge of both students and teachers good enough to keep up with that pace and a level of enthusiasm for the subject which allowed for teaching stuff 'not on the test'.
All the schools I know of here in the London area start around 0830 to 0900. Some operate a club for busy parents to drop off their kids early but classes don't start until the rest of the kids show up.
Eventually I went on to get a math minor in college to go with my computer science degree, but I didn't even learn the first thing about Calculus until I was 21.
Math and Science education in the USA is really really abyssal until college, and then it's sink or swim.
But truth be told it very highly depended on the quality of the math teachers. I was lucky enough to get into one of the best high-schools in my town so that we got a really good maths teacher, but the teacher "quality" was not that uniformly distributed across different schools.
At University (after the regime changed), studying hard did become more common, but it was directed at scoring a job in Canada/USA/wherever, not at escaping poverty in Romania itself...
I think those 7 years (1992 vs 1999) made a hell of a difference because of the economic "reforms" that were implemented in that time-interval and which actually saw many towns like the ones I grew up in become destitute in a matter of just a few years. Unfortunately that decade (the '90s) is not that well-studied yet in our history classes (maybe because it is too recent?), fact is I've only started to realise its true importance and its true, horrible economic and life-changing effects on our parents' generation (people who are now in the 60s and 70s) only recently. I think the same happened over almost all of the former Soviet Bloc, and imo it greatly explains some of the political tendencies we see today. But that is turning into OT, sorry for that.
That's very interesting, could you please expound on that?
Those attending “olympiads” were brainwashed into thinking that somehow they were superior and at some point we had to look down upon those whom, god forbid, chose something else but maths. Some indeed went on to have high paying office jobs in western countries, many of whom are now turning back. Apparently being trained to win olympiads doesn't yield an entrepreneurial spirit nor does it lead to riches, and after a life wasted they now just want to live a bit.
I was in school in roughly the same period as you and although beatings from teachers to pupils were very rare, they did exist and happend if the pupil was very unruly and caused trouble in class (I had one classmate who nearly assaulted a teacher during class because he didn't like the bad grade he got), but usually the teacher had the "blessing" from the pupil's parents to perform such "corrections".
Beatings were much more common in the poorer parts of the cities/country with poor performing schools and broken families (alcoholism, domestic abuse, poverty). At top schools in big cities like the one you probably went to with alumni that go to Harvard and Google, beatings were not common at all because usually the children came from mid/upper-class families. But that's very small percentage of the pupils in Romania that are in such a performant environment.
When my dad was in school in the 60's, from what he told me, receiving beatings in school from teachers was very common all around even for stupid reasons as some teachers would go on aggressive power trips if you stepped on their nerves.
Most of our olympics are currenty researchers abroad or tech leads in the country. Entrepreneurs not so much, maybe the networking types.
The hardest math problems we encountered were from the USSR olympics. There was a magazine with math problems which collected such gems.
I personally received beatings from my math teacher during middle school (5th to 8th grade), which was before '89. He would make me have the fingers up and together (think of like Italians argue) and then would hit my 5 thumbs altogether at once with a wooden stick.
It is more pronounced in universities - ask medical or economics students and you’ll uncover quite a few stories.
In primary school the older brother of a classmate has broken into a newsstand and has stolen a few porn magazines. My teacher beat him every day for a week as to serve as an example to everyone (yes he beat the little brother). He hit him around his temples so as to not leave marks and would lift him from the floor pulling by the hair. To this day i wont forget this.
Also as recently as few years ago there has been an uproar in romania as to how many parents beat their children for various reasons including poor school performance. You know, “bataia e rupta din cer”.
This happens in poor areas of romania such south or east provinces as much as in the north west.
“got jobs at X” - exactly. Excellence in romania’s education system means obedience. At the top it produces great workers. Not that working at google is not cool or getting a phd is not useful but romania needs more than that.
And while a small sample praises romania’s education system, and great maths, Romania suffers from roughly 50% functionally illiterate pupils. Just because a small sample gets good results in olympiads, frankly contests that mainly poor countries compete in, it doesnt mean the system is great. Quite the opposite.
I learned Calculus at 14, but I am now 56. We still used log tables when I was at school, calculators were only just being introduced. From what I have seen from my children, our maths education has been seriously dumbed down.
I hear this complaint form nearly every country. It seems like the system has found this "hack" where if you dumb things down enough, then scores go up across the board, giving the impression that the children are performing better, so the people in charge can meet or exceed their KPIs, and everybody's a "winner".
For example, in Romania it was very difficult to get top grades at schools a few decades ago so those grades were used as entry criterias in universities, but nowadays everyone can get top grades without a sweat by carefully rigging the system, so universities have introduced their own entry exams since when everyone has top grades then they are all worthless.
I'm also highly "skeptical of prolonged compulsory education of any quality" but, again, I think that maybe that will steer the discussion into OT territory.
On the flip side, the American capitalist economy does a remarkably good job of making productive use of people who are not academically strong (this is a notable strength vs. the Chinese, as well). And American culture & education stresses team-playing, communication skills, and trust, all of which are weaknesses of many of the Eastern-bloc programmers I've worked with.
The thing is, I don't believe that the strengths of American culture are mutually exclusive with Russian and other Eastern European strengths in mathematics and science education. In other words, we could have both. What's stopping us from ripping out just the dumbed-down math curriculum from schools and replacing it with Russian-style instruction? I guess that's what the article is about, and some families are doing just that with private enrichment classes.
What's stopping us? For starters, over a hundred years of battling desires: https://www.csun.edu/~vcmth00m/AHistory.html Not any of the sides entirely without some merit. And that is closer to America's strength, in that try as we might to enforce Universalism in some domain, we're pretty bad at it against ourselves. This is a good thing, since while the surface of possibilities does suck and we can dream it was much better, it is not entirely uniform and solid, there are yet still many cracks for the precious few (many of whom being immigrants who found their own crack just to get here) to slip through, drag some along with them, and come out to do great things.
However, the majority of people don't have this luck, and they get seriously left behind. I don't know as much about Bulgaria, but Romania has the largest percentage of functional illiteracy in the EU - almost half of Romanian high-school children can only theoretically read (they recognize the letter symbols, but can't actually read a text and understand what it meant, at the most basic level). A good percentage of people go through the mandatory K-10 education system through cheating, corruption, and basic knowledge.
Romania is very focused on national exams, one obligatory one in 8th grade and another one in 12th grade. There was a push about 10 years ago to implement some stringent anti cheating controls (cameras in each exam room, nothing fancier or more oppressive), and the pass rate plummeted from over 95% to 50% in that one year. There were entire high schools that had had straight As (10s) the year before and where no one passed the year after. This was the level of cheating and corruption.
I will also note that the stuff about having luck with your teachers is also not an exaggeration. I attended the second best high school in the country by admission grade (there is a national exam in 8th grade, and students choose their preferred high-school in a ranked vote style, and then every student is assigned to a high-school in order of exam grade + preference). This is also a high-school in the capital, and a wealthy area. I had some really good teachers in a few things, and a few really abysmal teachers in others. Even in CS, which was the high-school's specialty, I had teachers who seemed to barely know the basics (but also others who were pretty decent).
It's also important to note that there is widespread, normalized abuse in the teaching system, especially towards children with poor grades, or who are just poor. Things like yelling, demeaning, even spanking and hair cutting (for male students with longer hair, especially) are relatively common, and still considered normal in some areas (though, thankfully, fewer and fewer).
Do you know why students have to resort to cheat and corruption for passing tests and going trough the K-10 system? Is it some external factor for the students, is it just lack of good teachers, or a combination? Thanks.
For an example of the top-level mentality, the compulsory school system used to include K-12 until a few years ago. In high-school, you used to have anorganic chemistry in grades 9 and 10, and organic chemistry in grades 11 and 12. After the move to K-10, the curriculum was adjusted to have anorganic chemistry in grades 9 and 11, and organic in 10 and 12, with the cited reasoning being that you can't have students graduating out of high-school without knowing the basics of organic chemistry, can you? This, again, in a country where a good third or more of those students can actually barely read - they've been lost since around grade 2-3.
Americans will be appalled, without realizing that these are common hazards for poor, and especially black, students to face. In many cases, these tactics aren't even used for punishment, but as preemptive control measures (especially the hair-cutting).
It continues to surprise me how many parallels there are between the Eastern European and Inner City American experiences.
Unless you have a very high budget, which usually only private (or public as they're called in the uk, but they're the same thing) schools have, you can't have it both ways where both the weakest students pass and the strongest students excel.
The US system is as bifurcated as the various European nations mentioned in this thread. The elite families in the US do not send their kids to the same (frequently) underperforming schools as the middle class or poor, just as those types of families don't do that in Russia or Romania.
> The American school system isn’t designed to produce high achievers
Which system are you talking about? There is no unified American school system. Nothing remotely close to that concept exists in the US. It's not possible to generalize so broadly. There are many different education systems in the US, varying based on where you live (varying dramatically from one state or city to the next even) and or what your economic capabilities are. Your description, if we were to attempt to utilize it, applies primarily to bottom 1/2 to 2/3 of society, not the top 1/3.
An obvious example would be elite private schools in and around Washington DC. The Washington DC region simultaneously has many of the richest zip codes in the US, and vast tracts of poverty and many horrible public schools. Washington DC, broadly, presents one of the starker examples of the US bifurcation in nearly all things socioeconomic.
To illustrate the point: the only Presidential candidate from one of the two major parties, since and including the 2000 election, not to have attended a private prep school, was Hillary Clinton.
*If they don't come from the Americas
Even many of those who left for economic reasons come from richer families that are leaving because of new policies that are less friendly to the wealthy or previously privileged classes. See, for instance, white emigration from South Africa or Zimbabwe to the United States.
Also, to be clear, this is specifically an American perspective and even more specifically, about international students - labor migration from Eastern Europe to the EU is of a decidedly different character and more analogous to the comments I was making about people from the Americas in the US.
> It depends. Some of these people (their parents) left their home country because of economic reasons.
Another common reason for leaving is discrimination. Those people who leave because of discrimination are very rarely well connected or wealthy, or they usually wouldn't have to leave.
Then there are the outright asylum seekers and undocumented immigrants, who are also usually very poor and lack connections.
Most international students are not there because of discrimination they faced. Most people who are seeking asylum or undocumented in the US are from the Americas.
For international students not from the Americas, most are wealthy or well connected by the standards of their home country, although often not by American standards. It's not uncommon to hear about the banker who became a taxi driver in the US, but what you hear less of is the farmer's kid - because they never even get a chance to come.
And of course I'm generalizing! I have no doubt there are exceptions, but the fact remains that it is largely true.
So I might be comparing the best & brightest of Bulgaria, but in theory at least I'm comparing them against the best & brightest of America (though in practice I suspect it's more like the richest).
I say that as someone who has experienced both sorts of classrooms in my lifetime, which I think is a rarer experience.