Lots of comments asking what is "Soviet/Russian Math" actually like, and how is it different.
I was lucky to get an education in three systems (Soviet Math, Romanian Math School (influenced by both French and Soviet Math school), and finally in a world top 40 university in North America.
I would summarize the Soviet/Russian Math/Physics approach like this:
- understanding of the mechanism/intuition behind the equations/methods is paramount
- teachers are astute at spotting students who memorize blindly, and will intervene to correct that
- while rigorous about notation, the mathematical representation always comes after understanding, not before
- the progression of teaching (order how material is introduced) is very well thought out
- the old soviet textbooks are generally less verbose than North American ones (less fancy), but high quality in their expression, typesetting, and ESPECIALLY (!!!) the quality of the exercises
- the Soviet Math textbook exercises are something to behold: they have funny/memorable setting (like jokes), they are short and easy to express, the numbers are chosen in such a way that the result will be a nice whole number, or pi, etc. Basically as a kid you can read one of those problems, lay down, close your eyes, and work on it in your head.
That being said, I did like some of the aspects from the so called "Western Math" (in my case Canadian university):
- teachers are more approachable, more friendly
- textbooks can be gorgeous (nice colorful plots, etc)
As an american who went through US-based curriculum in the 90s, this sounds FUCKING WONDERFUL.
Basically - it seems like the approach above treats math as a conceptual playground, where you should develop intuition and understanding.
Instead we seem to be going down the rote memorization route for most of our classes, where the goal was to apply an equation to some numbers and get the right answer, with little to no thought, and an emphasis on easy grading.
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Our physics education seems quite similar, though - and I loved that.
I have always liked math, but have never been great or very good at it. I graduated from a US public school gifted program, and a top 30 US university. The biggest problem I always had with math education is they didn’t teach the “why” nearly enough. I was lousy at memorizing formulas/methods (esp. in calculus), so I often found myself dead in the water during exams.
I used to suffer from the lack of "why" too until one day it dawned on me that there really isn't a why, or maybe, the why is always the same: for engineers doing classical physics solving problems by hand, it's convenient to define it that way. No more, no less.
When I was a math tutor in school, everyone would complain about lack of why. This was my approach to address the issue. I would describe a problem relevant to the class, say something like find the angle between two vectors, and ask them to think of a general way to do it. Usually this require some help from me but the solution was always "theirs". Then we would play with their solution: find corner cases, figure out which operations broke, develop a couple of theorems with it. Then I would show them the dot product, and a couple of its tricks (a.k.a. theorems), and they would get a good appreciation for why the dot product works the ways it does. And that's the why and the reason it stuck. It's useful way to solve a common problem. There is no magic or deep philosophical truth.
The problem is this teaching style works with a dialog to a few number of students. I am not sure how well it would work in a lecture setting. It was also not so easy on me as I would be the person who had to spot the problems in their definition, basically on the spot, and lead them to it on their own without revealing the answer. With basic things like a dot product it's pretty easy to know where to look (i.e. is their definition commutative, pretty much never) and finding issues and nudging them too it wasn't too hard. But it's likely hard to scale things like this on many metrics.
> I used to suffer from the lack of "why" too until one day it dawned on me that there really isn't a why, or maybe, the why is always the same: for engineers doing classical physics solving problems by hand, it's convenient to define it that way. No more, no less.
'Why' is shorthand for a description of the context of the operation or method being taught. My senior high-school experience was that my maths teacher just plowed through the curriculum from limits to derivatives to integrals and beyond without explaining what each was used for or why. I managed to scrape a pass in high-school calculus but never grew beyond the most rudimentary of understanding until I was taught by a university lecturer who invested time in explaining the why.
The backbone of my mental schema is narrative, so just throwing equations and processes at me does not cause the knowledge to "stick".
Teaching me that Gauss had a problem X that he tried to solve by Y, then showing me what he discovered process N, is invaluable to me because it aids my recall.
She passed her test on the first try, which was extremely rare at her school. She was taught in India, and became an American teacher as an older woman. She was an outlier. She graduated from a major public teaching program in our state of California (CSU Fullerton).
Most teachers fail their tests on the first try. This is not advanced calculus... this is elementary mathematics and reading.
The issue is that accolades in the education department do not translate to results. The state creates a monopoly on teaching via its certification process, with means education departments are guaranteed funding regardless of their results, given that once you have the credential, you're considered 'equal' to other teachers.
Despite my mother doing very well on her tests, she was fired and not granted tenure because she didn't follow her principal's teaching methodology. Even though her students were passing their own exams at higher rates and doing better (And of course my mother taught two boys who went on to make careers in science and mathematics), she was fired for not following the failing methodology.
This is what is wrong with American schooling. It's a race to the bottom and the teachers have no idea what's actual achievement because they've never achieved basic elementary schooling, much less seen it in others.
EDIT: in some ways america is doomed by its own success, in that, for those who can actually do math, reading, and writing, there are significantly more lucrative fields than teaching. Unfortunately, since teachers can't teach those skills, it's really a new aristocracy formed by parents who do know those skills passing them on to children.
So the haves spend money on education of their kids, and have-nots can't do that, relying on public schools which don't perform, and the possibilities gap is widening. Structural problem.
Yes they do. At minimum Mathematics have a why in that it is something that is arrived at logically, and provable. But when you learn mathematics, whether it be algebra or geometry and trigonometry or calculus or discrete, there is another why added, that of real life solutions. Why do we care about geometry? Well it ends up being useful in optics and physics. We like Trigonometry because it can help you build a building or bridge or shoot a cannon and take down said building or bridge. Calculus has innumerable uses in engineering and physics and finance and biology and everywhere.
I find this answer to be true, but misguided. In particular, this type of answer was the exact thing that turned me off of math for YEARS.
Maths DO have a why - We needed way to describe and model the world around us, and math was a requirement to do that.
Now - once the model and rules are put in place - Fine, you can bugger off and be as self-referential and contained as you'd like - "There is no why!"...
But lets be clear - there ABSOLUTELY is a why, and the second the world around you no longer matches the model of your maths, we start debating whether or not to throw the thing in the trash and make new rules (see set theory as the classic example...)
Programming languages do not have a why. The turing machine is an arbitrary model of computation, as is the lambda calculus, as is horn clauses. The 'why' is deciding which one is most convenient to solve a particular problem. Given that very good computer scientists come up with very different means of solving the same problem, it's clear there's no universal agreement upon which paradigm is better and that there's a lot of subjective reasoning as to why particular models are better.
Consider the natural numbers. There is no 'why' behind them. There are axioms behind them, and -- given those axioms -- there are statements about the natural numbers that can be logically reduced to the axioms, but the axioms have no why.
Moreover, it is provable the axioms have no why, because they cannot have a why, because the axioms cannot be proven except in relation to themselves. If you're so convinced the axioms have a why, please prove me and Godel wrong.
The 'why' behind natural numbers is a social one, and one of convenience. The natural numbers make it easy to solve and communicate about certain problems, but they are not the only way to solve those problem nor are they the only way to communicate about these problems.
For example, another way to deal with basic arithmetic, is to talk about numbers as sets. Now you can define certain operations on them, and completely ignore the axioms of the natural numbers. This model is way better than others for certain problems. However, you now have a new set of axioms.. and oh yeah, actually the most obvious ones are completely self-contradictory, so you'll need to choose Zermelo-Frankel or something else.
Or if you want to be even more general, you can simply talk about the lambda calculus, but good luck trying to 'prove' the lamba calculus theorems in itself, because you'll quickly hit the halting problem.
Of course you can then say... well let's get rid of that and use the typed lambda calculus, but then oh yeah you can't do anything interesting. Why are these choices made? Can the choices be justified in the systems themselves? No of course not. The idea that you can use 'logic' to derive these systems is also ridiculous because formal logic is itself a system with axiom (and a very controversial system at that).
But if you look at the lambda calculus, ZF set theory, and the natural numbers as simply models and systems that are sometimes useful, then it makes sense as to 'why'. But the 'why' exists independent of them and is not provable in them and is social and cultural in nature. It is certainly not mathematical as in order to 'do mathematics' (symbolic manipulations) you first need axioms.
Mathematics education in this country has been replaced by rote dogmatism which is why many Americans cannot handle this ambiguity.
What must be explained is that mathematics is a language and in order to communicate with other educated humans about these abstract concepts it behooves everyone to speak the same language. It is the same reason the word for 'dog' in English is taught as being spelled D-O-G. There is no why behind it. It's just the result of thousands of years of culture. Except mathematics is a more global language and more useful for different kinds of manipulations.
Again - correct but misguided in the general sense.
> It is the same reason the word for 'dog' in English is taught as being spelled D-O-G. There is no why behind it.
I agree with you completely, but I think you're guilty of speaking the wrong language in response to the question (and it would behoove you to consider it from the perspective of someone outside the field).
The question "Why" in maths almost always gets asked by someone new to the field, and they are not asking from a mathematical perspective - They are not asking you for a formal/provable "why", they're asking you what is the utility of learning this thing.
So lets go back to D-O-G. The utility is clear - I have this hairy, 4 legged animal that keeps licking me that I'd like to discuss with you. We can agree that D-O-G (or perro, or 개) refers to it.
But with math, SO MANY PEOPLE (especially those established in the field) jump right into the "Here are the rules of this system of math" without ever taking the time to talk about why someone might give a flying fuck.
It would be like me going and making up my own language and forcing you to learn it. No one else speaks it, it's got no books/literature/history, there are no works of art that reference it - it's literally the language this random person made up that serves ZERO purpose except for talking to that person.
No wonder so many kids don't like math!
Instead you need to explicitly start with the utility of math - ideally in ways that are entertaining and fun. Once a person has an application for some of the rules, they become SO MUCH MORE INTERESTING! Suddenly I care about why this rule might impact that rule over there, or why A and Z are related, or what sin/cos/tan mean.
Basically - sell me on the value proposition of your fucked up whacky language - That's what "why" is asking. Once you know those rules do something useful, it becomes a much more engaging field of study.
You are correct, but this explanation is cultural and sociological, not logical.
We must motivate math, absolutely. And to do so in my opinion starts with socratic questioning. You must convince the student that such an inquiry is even worthwhile.
One thing I'll point out is that we're not just seeing this in math. We see it in every field. More and more kids every year are insisting ( and their teachers are agreeing) that we can do away with inquiries into the English language and the humanities as well. There is a small, but continuing, effort to remove the knowledge of English masters like shakespeare and classic philosophers and treatises from the curriculum.
As a whole, American schooling fails to motivate learning of any kind. Math was the first victim, but the other subjects are also failing.
Sure, but we're in a thread talking about why some folks are choosing to send kids to "Russian" style maths teachers.
The whole discussion is from the perspective of the cultural and sociological.
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As an aside, I generally agree with you about american schooling. I think it's less a concerted effort, and more a sad reality of the fact that modern schools have essentially become federally funded child care in the US.
Indeed... As my mother was told by her principal in her inner city school for poor minority kids... "We're just here to watch them until they go to prison".
I could do calculus, in high school and college, but I wasn't very happy about it. Until I took a physics class in college which happened to be intended for the physics majors. Did the professor ever just present an equation and ask us to learn it? No! Every Single Physics Equation was explained from first principles, using calculus - a lot of integrals. Finally we were seeing a practical application, where calculus was just a tool to get you where you needed to be.
It was really helpful to me in highschool that the calculus teacher was also the physics teacher, and many students were enrolled in both classes simultaneously. You ended up learning both at the same time.
Teaching calculus through physics (and vice versa) actually makes a lot of sense.
Similarly, basic Linear Algebra and 3D graphics have great synergy. I took LinAlg and Computer Graphics the same semester in undergrad, and the first half of LinAlg led perfectly into the second half of CG. (The first half of CG was all 2D stuff, which didn't need any special math.)
I was a bit disappointed to see the differences between "algebra based physics" in high school and "Calc based physics" in college. It wasn't a whole lot more than "now rather than giving you formulas, do an integral to go from acceleration to speed, or speed to position".
It did feel good to see a practical use for the calculus I had learned, much more so than determining how much the surface area changed after adding a layer of paint 0.01" thick to a tank.
Also American, but was routed through, what was at the time, an experimental program called Integrated Math. Class was structured around word problems and the students were guided towards figuring out the formulas rather than rote memorization.
There was a ton of group work, which worked out really well for me, but if you didn't want to learn it was pretty easy to coast and let the group leader do most of the work / learning.
My British state education really did feel like "memorise just enough to jump through these arbitrary hoops which keep the government off our backs" sometimes. It's very exam-driven rather than learning for its own sake and apparently it's got even worse since schools started transitioning to academies and Gove's reforms took effect.
Normally the curriculum was split between knowledge and understanding, and problem solving. You could memorize the first part and not the second part. The problem solving sorted out who got the grades
As I was reading the summary, I was thinking "well, of course -- that makes sense. Any teaching curriculum has got to be like that so what is so special about Russian math curriculum?".
Then I read in other comments about US math curriculum and I was shocked to learn that not only they were NOT doing this, but they didn't have a better alternative. This is mind boggling. It seems almost idiotic.
My schooling was done in India. Our curriculum was quite well planned, but what is laid out here is the methodology of teaching in addition to the curriculum and textbooks. And that depended on teacher to teacher and school to school. But nonetheless, the textbooks were very well written so everyone got exposure to pretty much the same level of teaching styles, more or less.
When I came to US in college, I was surprised to learn that people didn't know basic techniques and tricks for algebraic manipulations. Of course, we were taught the basics and whys behind every trick but emphasis was also given to internalizing these tricks for quick computation by hand. This, in my opinion, is important because these tricks also become your mental models when thinking about math. And quick tricks would lead to quick thinking, roughly speaking.
It wouldn’t be an exaggeration to state that most Americans experience k-12 math as an exercise in memorization. Even relatively simple algebraic manipulation such as quadratic equations are drilled for factorization first.
Yep, it’s like teaching someone how to cook by quizzing them on recipes. I didn’t understand the purpose of calculus until grad school despite having “learned” it several times by that point.
This is significantly different style from Russian textbooks. If you want a nice one of those translated into English (albeit not the easiest to find a paper copy of), I like Piskunov’s book. https://archive.org/details/n.-piskunov-differential-and-int...
Which board was your school affiliated to, if you don't mind me asking? I went through my schooling in India too, and I had quite the opposite experience: concepts being taught with the only focus being on beating exams, rote learning prioritized instead of logical applications (in case of science), etc.
CBSE followed similar methodology as described and was good too. But only select central schools were good at teaching it - the teachers there were well qualified, and some of them are a real gem too at teaching.
A Russian mathematics professor of mine in graduate school was a student under Kolmogorov. He taught information theory and algebraic combinatorics and helped develop a number of systems for the Soviets. He was extremely challenging, but he cared less about the grades and more that you were understanding the material.
He would say at the beginning of the semester: "You must learn to build the castles in your mind." The visualization of the constructions really helps to understand how to apply the concepts in different contexts.
The Polish professor would hand out chocolate to every student before every test, so that your mind was more relaxed.
That’s very insightful - when I switched to US high school, the approach to math was baffling - everyone had programmable graphing calculators, whereas I wasn’t allowed any calculator at all, and took a lot longer to solve the most basic problems. Most questions were answered by the teacher as “it’s a formula, you follow it and memorize it” (which sounded ridiculous to me for the exact reasons above - I was taught to understand the basis first, then the applications).
I never understood why graphing calculators were required for basic calculus or math courses. Always wondered if it was part of a marketing deal with the company that manufactured them. The course is said to require a particular make and model of graphing calculator.
It is more or less exactly this. Texas Instruments invests in extensive marketing aimed at teachers, school districts, and university math departments. They get a lot of free teacher training and materials centered around TI calculators, to induce them to require all new students to buy a $100+ Texas Instruments graphing calculator every year.
I believe it is more of a standardization of knowledge when picking the tool for mathematics. TI-83 calculator is common and easily to have all the students to use the same calculator and provides various approach of how to solve it. If one student is a outlier (like using HP calculator), then it could be an issue since the instructor only have the knowledge of using TI-83 (or its variants), the instructor couldn't help that student since HP and other graphic calculators have different inputs, layouts, etc.
And not everyone is good at solving math alone without the aid of the calculator. Some people simply can't solve mathematics at all, maybe up to multiplication/division level is the best they can do. I am not good with mathematics myself and struggles with some advanced algebra. It is not the matter of trying to solve, it is matter of memorization of mathematics formula and equations that US Education drilled down so hard which is ironic when they said that calculators are forbidden to use during the exam while they are encouraging to use the calculators in class and homework... Their pedagogy is fucked and hypocritical.
Sorry you had such an experience! There really is a lot to unpack here.
As the article describes, memorization of “formulas” doesn’t actually work for my students and leaves those that it “worked” disadvantaged in future learning.
My personal guess is the “not being good at math” originates from this, at least in part. It’s a confidence killer for sure that on one had you are being forced to do something unnatural (memorize weird looking formulas) and on the other give a somewhat easy, yet demeaning, way out by saying “it’s ok, you are just not good at math”.
The calculator to study and no calculator for the test is even more ridiculous - students are put in a high pressure situation without the very tool/crutch they came to rely upon. If anything, the inverse might make more sense - homework you have a lot more time and less pressure, so try working it out on paper. Test is timed, so it’s ok to use a calculator aid, so long as you show your work.
> My personal guess is the “not being good at math” originates from this, at least in part. It’s a confidence killer for sure that on one had you are being forced to do something unnatural (memorize weird looking formulas) and on the other give a somewhat easy, yet demeaning, way out by saying “it’s ok, you are just not good at math”.
Honestly, it is more direct at the instructors instead of the mathematics itself. I am not good with math because of the instructors' pedagogy. Honestly, I do like math and enjoy doing it (Khan's Academy helps a lot!). It is their approach with mathematics is the issue. Their pedagogy are not standardized enough to have consistency with each level of mathematics. There are instructors who dismissed their student's previous instructor because their former instructor gave them the shortcut or a shorter method while the new instructor are doing the same thing. Now you have a student who have a jumbled information of mathematics and that is difficult for the student to be able to relearn a new information while they can't erase/forget the previous method.
Also I am curious why it is difficult for instructor to explain HOW and WHY that solution is the correct answer? It is like they don't want to teach the concept of mathematics which is vital for critical thinking, IMO. When I asked the instructor of this question (this is in college), their answers is "It is the way I was taught in school" and I felt that is dismissive and hand-waving away the question.
Instructors are not entirely at fault because they also received the similar education in the past as we do. The mathematics pedagogy and the curriculum need a massive restructuring and cohesive way to teach the students to ensure that the students can use the previous knowledge to the next level of mathematics for consistency without changing or influencing the students to forget everything.
Like most things there probably is more than one cause, and I suspect a pretty big one is around compound effect of instructors having poor instruction themselves and then having to admit that they themselves don’t really know the subject.
Thankfully for me, my college Calculus professor(in USA) banned calculators for Calc 1 and 2(the two required Calc classes for a computer science degree at my public university).
He was heavily avoided by many and "special" for doing so. I found it easier to forego the calculator as it allowed us to focus on methods and how/why over just moving large numbers around.
That sounds pretty awesome actually! It also probably makes it a bit harder to prepare the materials, since you need to make sure it’s reasonably computable by hand.
Perhaps, but the guy had been teaching these courses for at least 10 years so he really had it all down. I mean he taught almost completely from his brain.
Total math teacher too, his university email is so full if you email him it just bounces back.
My take on your list of advantages of soviet math/physics approach is mixed.
Fully agreed with you on the rigor regarding notation, really good progression of teaching (pacing and order in which the material is introduced), and soviet math textbook exercises being well thought-out and entertaining.
Very much disagreed on the rest, as my main personal gripe with learning math+physics in Russia was that there was zero emphasis placed on understanding of principles and logic and 100% on memorization. I spent 3 years taking physics classes there, and I learned effectively nothing, having to relearn it from scratch in the US. And that was the moment where I truly felt I understood what was going on, instead of treating physics as just another memorization exercise for a variety of random unrelated formulas. Similar with math, but to a lesser degree, because with math I was personally invested and was trying to understand the material rather than memorize, despite it hurting my grades in Russia greatly.
Overall both class and teachers didn’t like memorizers. On the other hand, it was possible to pass pretty well by memorizing. Sometimes teachers would put in random bits in tests to throw off memorizers. Sometimes it worked, sometimes it didn’t. All in all, memorizing and actual learning lived side by side.
The main difference from what West looks like, the Soviet-ish system was designed for failure. For me it looks very weird when lots and lots of people get best grades. I was raised that 10 (out of 10) is rare perfection. 9 is great. 7-8 is fine. Even 5-6 is ok if you ain’t interested in a given subject. Not everybody is super smart, not everybody is passing school with perfect grades. And that’s perfectly fine.
Unfortunately a couple decades later our education system is westernized and everybody DESERVES best grades. And the system is bending over.
>Overall both class and teachers didn’t like memorizers. On the other hand, it was possible to pass pretty well by memorizing. Sometimes teachers would put in random bits in tests to throw off memorizers. Sometimes it worked, sometimes it didn’t. All in all, memorizing and actual learning lived side by side.
I find it hard to believe, given my personal experiences and the fact that having to memorize a poem or a short story and then having to get up in front of the class to recite it word for word for a grade was an extremely common recurring homework assignment in literature classes in my Russian school (from elementary to high school).
Extra details about that type of a homework assignment for those curious: they cannot call up every student due to time constraints for each class period, so for every such assignment, only about half the students get called up (for some specific works that are "more important", they might call up everyone, but over 2 class periods; that was extremely rare though). But those assignments were so ubiquitous, you essentially got called up to the whiteboard to recite at least once every week or two.
We had those literature assignments as well. But in 12 years that happened a couple times most. Now don't get me started on literature writing assignment that are strongly advised to start in certain ways and it's best to just memorise the beginning off examples and just change few words....
Interesting to hear about an experience that's pretty much the same as mine, but rebalanced differently.
For us, writing was the same way as yours ("strongly advised to start in certain ways and it's best to just memorise the beginning off examples and just change few words"), but it only happened a few times at most, and more towards the latter years, while reciting stuff in lit classes was the norm the entire time.
And now I am starting to recall another type of assignments going until late middle school, where we had to write passages from textbooks in cursive in our "homework notebooks", mostly with small changes. Like "here is this 1000 word passage in the textbook written from the first perspective, rewrite it from the third perspective in pen". Made a typo? Well, restart, because while a couple of edits won't take much off your final score, any more than that makes a full restart a more worthy option (because it was required to use pen for those instead of a pencil; more diligent students who didn't wanna bet on getting it right on the first try, they did it first with a pencil and then traced it and erased the pencil). Another evening spent rewriting the same dull passage multiple times by hand in pen.
On the other hand, how do teach someone handwriting skills? Writing down a passage is better than random blabery IMO. I'll take the must dull passage any day over several pages of synthetic exercises.
>On the other hand, how do teach someone handwriting skills? Writing down a passage is better than random blabery IMO
I agree, however, doing it way past elementary school seems like a solid way of wasting time. When you are trying to learn writing or you are in elementary school, sure. But spending hours upon hours rewriting long passages multiple times due to random non-editable typos in late middle school felt mind-numbing and downright awful.
And of course, as you progressed in grades, less attention of graders was focused on the actual handwriting quality, with passages getting longer and more convoluted, so the handwriting tended to degrade the further you got in your elementary/middle/high school education. Not even mentioning what happened to it after high school, because by then (unless your handwriting was completely unreadable) no one cared.
And no, it didn't help in the long-term with handwriting at all. None of the adults tend to have textbook-good handwriting, it would barely even get a passing score in the best scenario (and that would be an exception). And just like in western countries, let's not even mention doctors' handwriting (but that's completely irrelevant to my point).
I've been in several provincial schools in 1990s, good and bad, and we've always had strong anti-memorizing sentiment in math classes, despite in literature classes memorizing was mandatory.
Both make perfect sense.
Now I heard it's turning more to the Western model, though.
That's why I posted my previous reply, despite me being in strong opposition of the "argue anecdata with your own anecdata" approach in discussions. But I couldn't resist replying, because all of my 9 years of experience in Russian educational system were the opposite of what OP claims in terms of memorization, and so were those of every single person I know in real life who went through that system (in other schools in Russia, not just from mine, obviously). Coworkers of mine (who went to Soviet schools a decade or two before I was even born) echoed the same sentiment in discussions on the topic, with the halfway sad "some things never change there, huh" sentiment.
Would have been less surprising if my school was doing poorly in rankings and such, but it was quite the opposite.
Of course, all my claims here are anecdata, but it just feels like something is off when the fantastic utopian description of how "soviet math" education works just runs counter to every single lived experience I had, as well as that of everyone I know in real life. Especially given that experiences of some of those people in real life I mention were separated from mine by both decades and geography (some did school in Moscow, others in smaller towns, some in soviet ukraine, etc.).
That sounds wonderful! As a product of Indian education (that prioritizes rote learning), I wish I had been subjected to such a school, maybe then I (and majority of other kids) wouldn't have fallen out of learning math like we did.
This is fascinating and I appreciate you linking it. I was mind-blown when I read page 8's claim that white supremacy manifests itself in mathematics education when "students are required to show their work in standardized, prescribed ways".
> The child of immigrants might have learned a
different way to solve a problem because that’s
how their parents were taught where they grew
up. If we just tell that student their way is the
wrong way, we risk turning them off to math for
life. If we take the opportunity to explore why
there are different ways to approach the same
problem, it can be a learning moment for the
entire class.
I certainly had this experience in school! I did many math problems mentally, using the method taught in schools today where a problem like 23 x 7 is split into ((10 * 7) * 2) + (3 * 7). As a result, showing my work was challenging, because the teachers of my time wanted us to write out the long multiplication and I didn't know writing the above expansion was an option.
Ah, yes. I was told to "show my work", but I didn't have any work to show. If they told me "For sake of illustration, show what the steps would be if you carried out the following algorithm", then perhaps I could have done that. But what they said was "show your work", and there was nothing to show.
Luckily, I had learned several years of the curriculum in advance anyway, and had contempt for what the teachers were doing, so it didn't change my opinion of math, only my opinion of school.
Part of schooling is learning and showing you can follow a specific set of instructions. As long as one was taught how to do long multiplication, I do not see the purpose of taking into account what every kids' parents taught them if the purpose of the exercise is to learn a certain method and then show that you learned that certain method by doing it.
Because in practice, that causes students to disengage from the learning process:
> If we just tell that student their way is the wrong way, we risk turning them off to math for life.
I certainly disengaged from some subjects in school due to frustration with "thou shalt" methods of teaching. I was even removed from an upper level English language class because I didn't draw the same conclusions as the teacher from the material.
was the wrong way, then that is harmful. But if the purpose of the test was to see if you can do long multiplication, and you were not able to do it because you did not want to or like to do it that way, then that is a personal problem.
The reality is that school (non university level schooling) is not purely about education or exploring the 18 million different ways something can be right or wrong. It is also an exercise in navigating one's way around other humans and their expectations and playing the game that you will have to for the rest of your life.
There is also the constraint of limited budgets and schools having to make do with perhaps not the most qualified educators. And there is certainly lots of improvement to be made, but this "racist math" stuff seems to be counter productive.
Additionally, mathematics uses standard notation and uniform rigorous standards for what is/isn't true. There is certainly some utility in teaching mental arithmetic skills, but the higher-level concepts are only accessible if one can formulate ideas in accordance with the rest of the body of research.
I had a chance to compare the systems, too but my conclusions are completely different. Teachers at the physics/math lyceum I studied at were emotionally unstable psychopaths with mood swings. They could switch from calm "kids, if you don't understand something, please, don't hesitate to ask" to the screaming at the top of their lungs of "are you a stupid imbecile? what a moron you're to ask that" just a minute later.
It was all about rote memorization. Rules, theorems, axioms. Not just the way to prove or principle behind, but the textual representation (to the teacher's liking) word-to-word. Of course, there was a division of students into multiple groups, and those with important parents had an easier time. Any of their bullshit was always graded as "A". But those actively disliked could get "F" for a perfect work. A dot at the end of the sentence is missing? "Go back to the kindergarten where you belong". Something is crossed? "What is it, a toilet paper? Go use it for wiping your ass" (they could tear it apart in front of you). So, the representation/look of the work always came first. A teacher was a lawyer, prosecutor, and a judge at the same time (just what some Americans wish to have as a state): you don't like something, go f___ yourself.
The textbooks. If you compare the best Soviet ones with some average American, you can come to the conclusion that Soviet are so good ("exercises are something to behold", "high quality in their expression", etc). But the best American books wouldn't give a single chance to the Soviet ones. In fact, even some Russian Empire textbooks so much better than Soviet (not to mention Russian), they are getting popular among parents for homeschooling (schools can't use books unauthorized by the ministry of education). The best American and pre-Soviet textbooks are born out of lectures, author's personal experience with students. Soviet authors are disinterested observers who don't care. Textbooks, apparently, were the product of central planning like process (so they were getting worse and worse, the further they went from Russian Empire epoch).
And finally, a thing that really deeply touched me at a western University. A teacher when asked a question once said "I don't know, I have to check it". Holly f... Someone who openly admitted they don't know something, no freaking way! Back at the lyceum it would be "Shut your mouth with your stupid question, we don't have time for that". Then she would find the answer and during the next class say something like "now when we have a few minutes, I'll do you a favor, here is the answer... What a moron you're not to be able to conceive this yourself".
Math was boring as hell. Worse was only literature with essays.
Except that at lyceum we did have good teachers who could admit they didn't know, and the physics teacher did anti-test, where we could pose him a (correct) problem and if he couldn't find the answer, he added +1 to quarter mark, which was a great bonus.
I was about to reply with something like this, so thank you for posting it. I don't know where the top level grandparent comment comes from, but your experience matches mine almost 1:1 (except I went to school in russia between 2001-2010, so hardly soviet anymore, and then finished up high school+college in the US). Something to keep in mind: my school in Russia wasn't just a random school, but a physics+math focused gymnasium, one of the best public schools in the city. I am shuddering to even think what it was like at less well off schools.
Everything was about rote memorization, to the level i couldn't imagine in the US. Physics? Nope, memorize everything, you don't need to understand why or how formulas relate to each other or what they mean. Math? Nope, no need to understand logic behind anything, just memorize the formulas. Even with something like programming, we had to memorize bubble sort in TurboPascal, without the algorithm ever explained at a higher/pseudocode level (it was introduced in TurboPascal right off the bat and you had to memorize it line by line, I wish I was kidding; we were graded on how line-by-line it matched the given solution, not on whether the implementation actually worked/was valid). It was pain, I barely learned anything, and was convinced that I am destined to never do well in physics, thinking "this was just not for me, i am not smart enough to get it".
Then we moved to the US, and I felt like my eyes were opened. Physics were explained from bottom up, from actual phenomena and how they worked, and formulas were just a glue connecting those phenomena and their interactions together. There was zero actual need to waste time memorizing those formulas, despite formula sheets not being allowed on the exams, because they actually explained the logic/reasoning behind it all, so it was no issue at all to simply derive those formulas on your own during the exam.
Same happened with math and programming. I actually finally understood what I was doing, rather than robotically recalling appropriate strings to put on paper from memory.
In the russian school I went to, logic didn't matter, only correct answers. For a specific example, in every single physics course I ever took in the Russia, if you got the calculation wrong for one small part of the problem (which propagated to the final answer being incorrect), but you got the rest of it right, you get 0% for that problem. While in the US high school/college, I could even omit a small part of the problem that I didn't know how to solve, put a placeholder number there, and then solve the rest of the problem correctly, and I will get points for the parts I got right. Which makes sense, because if I forgot how to solve one out of many subproblems, but know the rest perfectly, or if I miscalculated a single variable, i just lose points for that portion. It isn't "all or nothing, the only thing that matters is the final answer." I get part of the reason the "all or nothing" system was done in my russian school, it makes grading much easier. But it also introduces tons of both false positives and false negatives. You know how to solve everything, but screwed up one small calculation? No points for the problem. Oh, you got the right answer at the end and wrote a bunch of gibberish to "get" to that answer (because you just copied the answer from someone else without knowing how to solve it at all)? Well, you got the answer right, so we will give you almost all the points. Cheating was off the charts. And that is at one of the most "prestigious" and highly ranked specialized gymnasium schools.
Of course, high school in the US still had plenty of memorization, but not even close to 100% like it was in Russia for me. And once US college hit me, it became even better.
>A teacher when asked a question once said "I don't know, I have to check it". Holly f... Someone who openly admitted they don't know something, no freaking way!
This, so much. Professors/teachers behaving like humans instead of "even if i was wrong, I am marking you down because you argued with the teacher, and the teacher is always right", that was incredible for me.
P.S. sorry for the long rant, but it bottled up, especially after seeing comments proclaiming the exact opposite of what I, and every single person I know in real life who went to high school/university in Russia, have experienced. Oh, and no comment on "Russian schools in the US" (as I feel like they could indeed be great, but the "russian" part of them is imo just a marketing trick), as that's a very different beast that I have zero experience with. I was talking only about Russian schools in Russia.
I've heard from many Russian immigrant friends who insist that Soviet education was one of the few areas that was done well in USSR. Particularly, mathematics, which, other than the obligatory Marx/Lenin quotes at the top of papers, tended to avoid ideological corruption.
What's funny is that the differences in pedagogy is probably a result of that "ideological corruption". Going through the American education system, it's clear to both professors and students alike that the system was designed on the assumption that there should be winners and losers, and that meaningful education is an afterthought. My math professors complained that the way the system worked made it impossible for a student to have continuity in their learning experience, and if you measure the number of A-students in one course who get an A in immedate next course, it's random. They weren't talking about courses that have large conceptual leaps between them. They were talking about first semester calculus vs second semester calculus.
I studied in 1988-1999, and these stories may be true for Soviet elite education, but not for average Ivan. I personally studied at a school with advanced Physics and English (our teachers after graduating from uni went to Minneapolis for a year of practice). But maths had nothing close to what's described in the article or comments here. No witty simple problems, just boring theorems, then simple tasks to solve.
I did see those witty nice problems in journals and special math tasks books, but that never appeared in our lessons. That was boring like hell. Well, ...at least it wasn't dumb, we did take logarithms, derivatives etc.
But if you take an average Russian, they can't solve a simple proportion problem: say, income tax is 15%, you paid $450 of tax, how much net salary did you get? The answer is easy: 450/.15*.85 (then do it on calculator), but when I did this calculation with accountants from vocational college, they were stunned and didn't get how I did it. I'm not exaggerating a bit.
So those people were either from elites, or nostalgic.
On other courses in Soviet/Post-Soviet school.
Russian language focused mostly on orthography, punctuation and participles. Like if British school focused on spelling "coloUr" or "emphasiSe". The examples of good style were only 19th century literature, especially Tolstoi's suffocating long sentences.
Literature course is similar to what Paul Graham wrote about in his essays: old, boring and already unimportant literature, plus writing essays that must imitate literature critics. I think this was the most hated task at school, and it lasted all the way from 5th to 11th grade. Such essay writing is still obligatory till today in 2021, and I see consequences of it while teaching in a university: students write in unnatural high style, but have difficulties conveying their thoughts or selecting proper evidence (few can distinguish between facts and theories). And that's in a good university -- I'm scared to think what less smart people write. This is not a "degradation" of modern ages, it's almost unchanged since Soviet times.
History course conveyed a Communist narrative, cherry picked facts and asked you not to analyze anything but to remember dates/years. E.g. a textbook on medieval history (6th grade), a paragraph on knights and their armor started with exactly this phrase: "It was not easy for peasants to fight even one feudal lord." (then it described the armor). The entire country of Grand Duchy of Lithuania (at the time it was also called Lithuanian Russia) was omitted, except for being shown on a map. Because it was embarassing to compare that country with Russia under Ivan the Terrible (who became an icon in Stalin's age).
Geography was interesting to me, but when I got to Wikipedia in 2004 and started reading on languages and nations, I saw how much was missing from there.
Biology was a simple and rather boring literature, and the home work was to read a paragraph and be ready to retell it. Most students would simply learn them.
(Actually, with mediocre English teachers that was the case as well: read a text, called "topic" and retell -- and the teacher saw students telling the text learned by heard, but didn't care.)
So, to conclude, math in Soviet elite education was good. Other courses were probably reasonably good, because those elite schools for talented also attracted good teachers. But the average school was of much lower quality.
Friend from Bulgaria was telling me about the annual recruits for the required conscription into the army. Most were illiterate peasants. There ancestors had been illiterate since time immemorial.
Many of the Soviet-era textbooks (mainly by Mir Publishers) are now published by low-cost Indian publishers and available on Amazon.in. Just search Amazon.in for Piskunov, Irodov, Vygodsky etc.
Anecdotal, but still: What you have described might have been the case in a handful of top schools, mostly at metropolitan areas. The key difference for the "rest of us" was a high school theacher who would pick top 1-2 pupils after couple of lessons and teach at their pace for the rest of the year. This is my case and the story that was confirmed by dozens of people in my university.
This is probably a dumb question, but I'm guessing all the textbooks are in Russian?
I really would be interested in seeing this:
> - the old soviet textbooks are generally less verbose than North American ones (less fancy), but high quality in their expression, typesetting, and ESPECIALLY (!!!) the quality of the exercises
but sadly it would be no use to me if it's not in English
I also had a similar experience both on the US and Polish side. The US math exercises that I experienced through my niece were really much more focused on reading comprehension, but almost hoping that the student trips up. Maybe its because I am not a native speaker, but this seemed to be not constructive to the task at hand.
A book that appeared previously on Hackernews is Lev Tarasov's "The world is built on probability". This is not striclty a textbook, but it is translated in English in a way that preserves a lot of the feel of the original:
https://archive.org/details/TheWorldIsBuiltOnProbability
There are plenty of books in Western literature where exercises are also very good, and not only in math, but other fields too. One that comes to mind is Jon Bentley's "Programming Pearls", which has very well chosen exercises.
I went through the Soviet/Russian school system with all the advanced topics being covered in high school as others described here. My moment of zen was on my third year in one of the top tech universities in Moscow. There were quite a few students coming from Novosibirsk University who were comfortably ahead of the curve when it came to math. When they showed their transcript ("zachetka"), they had ~3000 hours of Calculus in the first few years against our ~120... As they described it, each day after the usual roster of lectures and seminars, they would spend 4 hours in class in the evening solving math problems with a teacher. While we might have spent a few hours doing homework every now and then, it was nowhere near this. Not to mention doing it in class with a teacher who would likely also give them more challenging problems and valuable feedback. The sheer amount of time and mental effort was staggering.
My personal experience is that it wasn't just hours. Those who liked math spent a lot of time on it after classes. Those who didn't, avoided extras and still did OK (they would not go to top universities, but the base would be OK, with some practice, if they decide to go to a mid-level school).
The biggest difference for me in school was a consistent, well thought out program (one for the whole country) and books. A topic would be studied once, well, and there would be enough time to master the material. Consistency across subjects, too -- if physics covered a topic in, say, the second half of the grade 7, the math needed would be covered in the math classes before then.
What I now see in the US is horrible: teachers at grade N do not know for sure what the students already know, so they repeat a lot of the background, then jump around to cover a lot of material, much of which is never mastered. Which is considered OK -- it will likely be repeated in the same haphazard fashion next year. Or not, depending on what the next teacher decides.
You’re describing an idea adjacent to “mastery” schooling; it’s the natural method people tend to use when self-taught; it’s also how some online schools, like Khan Academy, use to teach. Both of my kids earn “allowance” (extra screen time; robux; etc.) by doing Khan Academy courses. Watching how unevenly they pick up topics — there’s no rhyme-or-reason to the speed they pick up even “close” skills — makes me think there’s malice aforethought in the steady pace learning I had as a child.
The only thing mastery schooling taught me was empathy for people whose learning style doesn't fit the style used in a classroom.
It's hard to understand the the connections between and motivations for concepts when you are learning everything about a topic before moving on. I prefer the style described above, of learning the basics then moving on and circling back when more advanced aspects are needed.
Clearly different styles work better for different people.
Not to take away from the importance of a well-thought-out program, learning materials, or competent teachers, I was drawing attention to the immense time and effort that some students would invest often with an institutionalized help.
In our case, the programs were standardized, the textbooks were virtually the same throughout the union. Barred special schools and eccentric teachers, all kids were studying the same things at the same time. What differentiated the bespoken Siberians from us was that order of magnitude difference in the time put in. It challenged my sense of normalcy in many ways: people being this good at math without being apparent geniuses, universities teaching math extra 4 hours a day, realization that there is enough undergrad calculus to last 3K hours.
>teachers at grade N do not know for sure what the students already know, so they repeat a lot of the background, then jump around to cover a lot of material, much of which is never mastered.
I spent my freshman year of high school in France, and went to a French school (this is 3ème, for my fellow Frenchmen).
I recall being stunned -- in the best way possible -- when our biology teacher started his class on the first day with: "So. You've seen X last year. Now we're going to talk about related-thing-Y".
That had never happened in my US curriculum. Never.
My wife went through the French "classe préparatoire" system and describes a similar degree of raw effort that was put into her studies (and in particular, mathematics). She talks about spending 10-12 hours a day either in class or studying at home, and I have to admit the amount of knowledge in that woman's head is absolutely astounding. Her education is elite in the same way that a sports team or military unit can be elite.
There are obviously issues with pushing young adults this hard, but my overall feeling is that American schools need more of this.
As an immigrant in the US from Central Europe, I am blown away by how much time American children spend on extracurriculars vs academics (at least based on what I see around me). My extracurriculars in high school were maaaybe 5 hours a week. Some kids in the US spend 5 hours a day, every day. And don't get me started on varsity sports - that starts to resemble a job!
In hindsight, after spending nearly a decade in France (initially for graduate school), it seems very strange to me as well.
I've come away from this with the impression that American schools are the best-funded in the world, and survive by importing the best-educated from elsewhere. This is obviously a bit of a caricature, but I think it's mostly correct.
Extracurricular activities are a way for US students to stand out in terms of college admissions. When I was in high school our college advisors told us that Universities look to fill niches in each class year. Good students are relatively interchangeable, but if the University wants a Lacrosse team, an Equestrian team, some Oboe players, and stagehands for its drama department it will look for applicants that have those backgrounds already. So, if you happen to have decent grades but a background in some niche thing, you are much more likely to be selected. So in that sense, going from a B+ to an A- in terms of overall GPA isn't going to help as much as having some in-demand skill.
>Extracurricular activities are a way for US students to stand out in terms of college admissions.
You are of course correct, but I think the parent comment is implying that this is a rather unfortunate situation. I agree with him to a large extent.
My experience with the French system has left me with the sense that American schooling has to some extent cheated me out of an education. On the other hand, I look at my wife (and other "prépa" students as well) and conclude that they suffer from a certain lack of imagination and intrinsic motivation, both of which have personally benefitted me greatly, and which I attribute to something in American culture.
As my wife puts it (I'm paraphrasing, obviously): "We were never asked what we enjoyed doing; if you were a good student, you were put on the good-student-track, which was a math/science-heavy curriculum. To this day, I don't really know what I want to do; I just know what I can do, and I feel an obligation to excel at it." She's an absolute brute at math, but she doesn't like it, and I think she would have been much happier studying something like literature.
So my feeling on the matter the French educational system is one of ambivalence, overall. Nevertheless, I am convinced the US has strayed much too far in the other direction.
A bit tangential here, but our calculus/math analysis professor actually told us that the Russian and later Soviet math school drew a lot from the French, although, he might have been biased due to his apparent partiality towards Bourbaki. According to him, the university level math was taught exclusively in French during the Tsarist times, and they had some troubles coming up with Russian terminology when Soviets ordered all teaching to be done in Russian.
I can’t help but wonder what value students see for studying calculus 8 hours a day every day for more than a year.
Maybe I’m actually just subconsciously jealous, but unless you major in math, what do you even do with that? Not to mention there’s little time left to study anything else.
It's a weird article. I think this idea of 'Russian math' is very nebulous, and is probably used as marketing tool nowadays by some schools abroad. I have graduated from one of the top physics/maths high schools in Russia, and yes we had great math education there, but that was an outlier. Outside a handful of schools the math education is pretty dismal.
And even in the good math schools there are different ways of teaching maths. I.e. there is a famous system of 'sheets' where the actual teaching of formal theory is very limited, while most of learning is done by solving problems (given to you on a sheet) and then presenting solutions to the teacher. But it wasn't used in our school for example.
What definitely exists in Russia (or at least some big cities) is a system of free after-school classes, where you can go and learn how to solve olympiad-type math problems and become more interested in maths. That is definitely extremely useful to identify a few people who are talented in maths.
> What definitely exists in Russia (or at least some big cities) is a system of free after-school classe
I've been studying in a "respected" Soviet / early post-Soviet school.
These classes were super elitist and had a huge disconnect in their level from what you were taught in regular classes.
I was a straight A student and was shut off very quickly. The problems were enormous in their difficulty and teachers had zero interest in helping or educating you.
Those classes were either for 0,1% genius olympiadniks to be later recruited for Soviet science or defense, or for 0,1% elite (Soviet nomenclature) kids whose parents could afford private education.
Along with other naive/poor kids I was quite quickly reminded that I'm not welcome in the circle.
Just a reminder that Soviet system had very entrenched elites and huge discrepancy in access to education, medicine and goods.
Yeah same. Our town had a "smart kids" school and a dozen of ordinary ones. I once won the town's informatics Olympiad in my age cohort, so to prepare for the regional stage they dispatched me over to that school's auspices for training. The teacher literally ignored my presence; so in the next competition I had to use both IBM PC and Pascal for the first time in my life. Scored something like 12th out of 40, but the teacher gloated on the way back.
I seriously think the "no child left behind" type education so derided here produces better general outcome. Holding up resources for a few select students is quite the opposite of meritocracy.
Eh, I've studied in a "respected" post-Soviet school as well and can't related to this at all. Teachers were awesome and passionate, had really thorough understanding of math (and physics, and chemistry), without any hint of elitism.
It's almost as if there are good and bad math teachers (in all countries).
In China we used to have those after-hour classes too. Nowadays they charge a lot of money and the material declines in quality as well. Back then they hired some professors from the top universities to do the teaching of after-hour classes and nowadays who knows those guys are...
>What definitely exists in Russia (or at least some big cities) is a system of free after-school classes, where you can go and learn how to solve olympiad-type math problems and become more interested in maths
And for people outside of big cities there were schools-by-correspondence famously running by Moscow State University for math and MIPT for math/physics. The setup was you were mailed a small booklet every month. You had to study the material, solve problems and sent solutions back. The solutions were graded and sent back to you.
It was great on so many levels. First and the most important aspect it taught discipline and time management. The material was amazing when you were given gradual increase in complexity instead of sheer volume of simple problems or a few olympiad-level problems which can't be solved if you are not there yet.
I had that in Estonia as well, when I was in the gymnasium in 2002-03. I applied to the school for.. Exact sciences that was part of an university, for their informatics course. They sent me booklets with lessons and exercises and i sent my solutions, then received the next booklet with the results. I wasn't good at that, but it was interesting.
Somehow American Math doesn't have the same connotations. I am witnessing public schools take math out of the curriculum at elementary, middle and high schools. It is a wasted opportunity for young people to learn the beauty of math.
> I am witnessing public schools take math out of the curriculum at elementary, middle and high schools.
Is that for real? If true, this is ridiculous. Do American authorities want to introduce some kind of education for plebs?
How is such low standard for basic education is even acceptable in the modern world? It’s not only about “beauty of math” but at least basic math is required to help develop some specific cognitive and thinking skills. You can’t leave that part of children’s brain underdeveloped.
And cutting out math from schools you are deleting the future for a huge number of kids. No more future engineers and scientists and programmers etc.
Yes it's TRUE. In SF the schools district removed Algebra from the middle school curriculum. You should read The Dictators Handbook, it details why dismantling education piece by piece is a great way to stay in power. You'd expect it in a dictatorship but not in SF.
I would argue part of the problem with the American system is that trying to rush students through a rigid curriculum leaves the teachers no choice but to teach by rote memorization and cookie-cutter problems. Slowing down the curriculum is part (but not all) of the fix.
Still, whether or not this policy is good, it takes a lot of spin to turn "San Fransisco is starting algebra a year later" into "witnessing public schools take math out of the curriculum at elementary, middle and high schools", which is what you claimed.
You are both correct, but the comment you were replying to was definitely written in a way that feels misleading about "increased enrollment in higher level math classes in high school".
TL;DR from someone who read the full article: that change caused an increase in Precalculus enrollment and a decrease in AP Calculus enrollment.
Which I don't really feel good about. The quote closer to the end made me feel even worse:
>“For so long, people have held up this idea that AP Calculus is the gold standard (for college admission),” Lizzy Hull Barnes said, a district math supervisor
And those kids haven't entered college at the time of writing, and the article points out that they indeed don't know how this change will affect college admission prospects of those students. Time will tell, and hopefully my bad feeling about this change was misplaced. But given that this was written over 2 years ago (so most of those high school students have graduated by now), and we haven't had a follow-up "this change brought us some great benefits" article, I am afraid the results might have not shown any benefits of this approach.
That headline sounds a bit of an editorialized interpretation of the situation. The debate doesn't seem to be whether or not they prove their skills, it is a debate over who they prove them to, and when a change should be implemented.
Bizarre. Where I am math and English remain king and queen of getting-on-the-schedule. Being "tested subjects" from very early grades, is the main reason. They've all but eliminated recess in elementary, and have reduced hours for things like art & music, to make sure they have plenty of time for them, since school prestige and funding rely heavily on testing well just in those two subjects.
It's an open secret among educators that the brightest students learn to read at home. Early math is just practicing the same arithmetic over and over, because the logical skills for algebra just aren't present in a seven-year-old.
Art, music, and gym are actually really important in early childhood development, because they develop motor skills. Gym gives gross motor skills and major muscular development, and art and music give fine motor control. Penmanship is also important for developing fine motor control. For young children who can't yet engage in much logical reasoning, developing motor skills is appropriate and important. We tend to think in adult terms about what is the right thing for kids to learn, but small children are not adults and need to develop skills that adults take for granted.
Additionally, it's now understood that insufficient exposure to very bright light (say, sunlight) during key years is the main risk factor for nearsightedness. Cutting time outdoors during Winter, for lower-elementary kids (K-2), when school is monopolizing a good portion of daylight hours, is almost certain to increase rates of nearsightedness. Cutting recess in young grades hurts kids' eyes, permanently.
I disagree with algebra being too difficult for a 7 year old. You can learn geometric algebra at 7 with plastic blocks. It's a common way to teach math in Eastern Europe to young children afaik.
Excellent maths teachers can really bring a lot to the table with geometrical approach throughout the years.
Was occupational training ever the primary goal of public schooling -- even back in the days when most people went directly to the workforce afterwards?
You'll have to decide what the "primary" goal is, but there was certainly the sense in the past that a U.S. high school graduate would be suitable for most jobs, either blue collar (shop class used to be quite popular), white collar (academic classes, particularly English), or indeed housewife (home economy). No matter what, though, a mastery of the 3 Rs (reading, writing, and arithmetic) was considered essential to all of these (you have to know arithmetic to balance your checkbook, do taxes, measure for construction, etc.). High school wasn't necessarily a specialized occupational school, but there was an expectation you would be learning useful skills for your future.
That's what I'm getting at -- life skills are much more broad than occupational skills. Art, PE, health, sports, civics, etc. are taught in schools because they produce people with more well-rounded life skills, not because it produces better bankers or factory workers.
That's a shame. Recess and play offer some of the best learning experiences. Our schools are beginning to resemble prisons. Heads down, sit up straight, stare at computer all day.
Wait, why I have been hearing non-stop astro-turfing about "CRT in schools" in the last 6 months and I've never heard once about math being taken out? This sounds really serious, is there more I can read about Math being removed from public schools?
These are the same thing. The policy is to remove educational disparity.
Naturally force flows along the path of least resistance - rather than raising everyone up, which is hard, we are simply removing anything difficult from the curriculum, which is much easier (at the expense of actually educating anyone, a goal that nobody cares about much any more).
The easiest way to make equality is to multiply both sides by zero.
Many in teachers' unions don't want to be rated on how badly public schools are doing. It's not all the fault of teachers, granted, but incentives of the union here is not always aligned with the students. Many don't want to hear it, but it needs to be said as a factor.
There are a lot of these initiatives - like Russian math - popping up serving as test cases for how to teach subjects.
I think online programs are injecting some badly needed new energy.
>> It's a weird article. I think this idea of 'Russian math' is very nebulous
It felt a bit like a promotional piece. They really want to drill the phrase "Russian Math" into your head, presumably to promote the schools teaching it. That doesn't invalidate the notion that they teach math better than US public schools, I'm just offering a plausible explanation for the feel of the article.
Since we are piling anecdotes, I had an opposite experience -- I came to US after i had 9 grades of Soviet education in a middle-of-the-run school in a large city in southern Ukraine.
The mathematical and sciences background instilled in me in those 9 grades of Soviet school has allowed me to pretty much sail through High School(grades 10 through 12) and almost entire first year of college(as a CompSci Major at a University of California campus) without having to flex my math and science muscles.
My college had a decent-sized contingent of Bulgarian & Romanian international students. The difference in their math abilities upon entry was striking. While most of my American-born classmates struggled in discrete math, linear algebra, and vector calc, my Bulgarian friend was like "I learned this when I was 9." They were frequently tapped as TAs by the professors, because they understood the material on a level that Americans didn't.
I'm Serbian and moved to the UK when I was 10. A lot of this is a bit hazy now, but I really clearly recall my bewilderment during the first class-wide 'mental arithmetic' tests in year 5 where I thought the whole class was playing a prank on me. In Serbia I remembered doing quadratics and even touching on differentiation in the afterschool classes, but over here in the UK they were expecting me to take 20 seconds to do simple mental maths (I remember 8x3+17 being the 'hardest one').
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.
I went to a British university, CS course, at the age of 19, having completed my education in Poland. I had a very similar experience(except no one cared whether you wrote x or .) - basically at university it was like going back 5-10 years in terms of maths level. We were already doing advanced calculus in Poland in my last year at school(and being constantly told that we have to know it well or no one at university will explain it to us), and then at the UK university I went to we spent the first year just going through extremely simple algebra. It honestly felt like I was doing something wrong.
This surely depends on the university and the course. At mine (admittedly for a maths degree) the start of the course involved quickly running through the contents of the A-level FP3 maths module (which the degree did not technically require) in about two lectures as well as jumping straight into topics that weren’t very relevant to school maths (group theory and a first “set theory/welcome to proofs” course which both quickly became hard).
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.
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.
At my university in the U.K. we called the kind of linear algebra that computer scientists do “vectors and matrices.” It involved grids of numbers and maybe things like decompositions and eigenvalues/vectors. Computer scientists might get to go into topics like numerical stability or decomposition that are useful for computation. These have lots of practical uses and are relevant to topics like graph theory or Markov chains.
The thing we called “linear algebra” involved linear maps and vector spaces and bases and dual spaces and no grids of numbers.
That wasn't my experience in doing a CS degree in the 1980s - we basically did the same maths as the engineers for two years and then diverged into more discrete maths in 3rd year of a 4 year course. And this isn't even counting the pure CS mathematical components such as lambda calculus etc.
As a Brit, I was never any good at mental math at school, I still cannot do that hardest one without a calculator.
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.
In my country (and I'd say this happens in some other EU countries), you do a ton of basic algebra as a kid, then take some rushed calculus and linear algebra 1-2 years before uni.
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.
I felt the same way, but I noticed professors I admired often doing arithmetic in their head, clearly because they enjoyed doing it. So since then I make an effort to mentally calculate. And you know what? It's a more interesting hobby than you'd think. Whatever technique you adopt (or grow, in my case), it's a good, concrete, occasionally useful exercise in breaking down a problem into smaller parts, and then combining those simple results into something else.
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 still have a hard time in mental arithmetics. During my school days, this is one of the things which had instilled the "math phobia" in me.
I never really took an interest in math, until I started doing competitive programming in college. I had to literally build a new understanding of division, multiplication, modulus, etc., it's difficult to explain. But I feel really ashamed in admitting that I still suck at math, probably because I just hated it during school days and no one was there to guide me. I don't really like blaming things on others, but I am not sure what else I could've done to learn math.
I'm in exactly the same position at the age of 40. I've attempted to memorise the times table to no avail. It seems to me one of those things that you need to know cold. And then from there you can look at other things.
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?
> 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.
I think mental arithmetic is one of those things that simply requires vast amounts of practise. I’m quite slow at it, but when I’m forced to do a lot of it, say for keeping score. I quickly get faster.
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.
i remember me coming to germany with 11/12 and having to tell odd from even numbers in a test; after trigonometry and beginning of integrals in russia.
I am Bulgarian. I moved to the UK when I was 12. I had to join primary school halfway through Year 6 (the last year of primary school).
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.
UK maths teaching is fairly terrible. Not only do they start later, it's almost never taught meaningfully, instead relying on memorising rote formulae. I grew up hating maths, it wasn't until YouTube came along and the sheer amount of _good_ maths content on there that I started to 'get' the bits I just never understood as a teenager. Most of my friends will still say things like 'I just don't get algebra / I don't do maths', and I put this down to how it is taught, which isn't very well.
The main issue is the curriculum and exams emphasize this kind of rote learning (across the board, not just in maths: one stupid reason for this is many of the markers for the exams don't actually have any qualifications in the subject they're marking and need to be able to just pattern-match off a mark sheet). And because teachers and schools are judged so heavily on exam results, they are not just incentivised but basically forced to 'teach the test' as opposed to trying to get the students to actually learn anything, even when they are otherwise motivated and capable of doing so.
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'.
Ah that is lame that you had to go through that. I remember in my calc class the teacher said we would spend no time learning for the government standardized test. If you can't already pass it you have no business being in the class.
Yeah, with different tests it may be different. But in the UK the government pushes for 'more challenging tests', which means more stuff to memorise, a lot of the stuff is very specific (in science GCSE and A-levels there's specific phrases you need to use to get marks. Accurate descriptions which don't include those key phrases don't get marks), and there's a lot riding on getting near perfect marks. Someone who has a good memory and a few months to prepare and an expert on the topic with years of experience (but no specific preparation) have about the same chance with these exams.
Has it changed since the 1960s? Maths where I grew (Swindon, Wilts., ) up was pretty good and from first form (age 11, 1966) on used the SMP course where there really wasn't any memorization of rote formulae: https://en.wikipedia.org/wiki/School_Mathematics_Project
The earliest I was introduced to discrete maths like this was my first year of uni -- that course looks really interesting. I know they teach CS at high school now
Not sure about youtube, but for the right age kid, maybe late-elementary to early high school, there's a weekly "Joyful math jamboree" online. The web page for it is not updated very often, but the registration form still works.
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.
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.
I graduated high school in the USA in the past decade. I was never taught Calculus. I wasn't even very good at algebra.
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.
Romanian here, the "I learned this when I was 9" might have been a slight exaggeration so that your friend could make a point, but as an another anecdata I can say that I distinctly remember one of my first calculus classes while I was in 9th grade (I think I was 15 years old at the time) and our maths teacher trying to explain to us how real numbers are defined and, more importantly, what that definition really means. That class/hour stood with me ever since, even though 25 years have passed.
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.
Do the students bring any motivation to the classroom? Like, is there a culture of "study hard and pay attention in maths and sciences, because that's how we escape poverty" or something like that? Here in New Zealand, middle class parents put a high premium on being happy and fulfilled, and most (that I've seen across several schools) don't really prime their kids for academic success by saying "this is important, we value it, and you should too". It may be different for lower-class or upper-class kids here (I haven't spent a lot of time in those schools). Keen to know what it was like in Romania.
In Romania, from what i recall, it was usually the beatings. Didnt like math? The teacher would simply use the pointing stick to smash your hands. I was fortunate enough to have liked math but some weren’t. And so there were the beatings.
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.
You are exaggerating, I've never seen a beating in school (from a teacher to a pupil). Was in school in Bucharest from the 90's to 00's, had loads of maths and other stem classes. If you didn't keep up you'd get a bad grade, if you didn't pass the year you'd have a chance to correct it during summer school, and worst you'd repeat the year. Nothing uncommon here. And I don't share your hate against the ones competing in the olympics, I knew lots of them, later on they went into research, got jobs at Google, did PHDs at Harvard and Princeton and the likes and are now professors at top universities. And some of them are good entrepreneurs now, you'd be surprised.
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.
Depends on the teacher I guess, but we never got corporal punishment unless there were grave discipline issues like beating girls with snowballs or weaved scarves. Math? No. You just got a bad grade.
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.
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.
And there is your answer "from 90's to 00's". Beating was done in communist era, not after the fall of Iron Curtain.
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.
I graduated around 2005, and yes, beatings were common. As were bribes and sexual abuse. The sexual abuse was so much in the open that some thought it was cool to see a student girl dating a professor.
It is more pronounced in universities - ask medical or economics students and you’ll uncover quite a few stories.
Depends very much on the parents. Some families put a lot of importance on studies, and others don't. Usually those from countryside or without higher education would not care.
The most important thing is to shape your kids in such a way that they can make choices themselves.
Worst thing is when parents are pushing their children to go for the highest (their eyes) education.
A child who likes what he/she is doing, even when it is 'just' wood making becomes a much happier child.
Northern Europe is good in this. The US so so. But China and India are the worst examples. So many talented people doing things not in line with their real desires.
I grew up in Romania too (finished school in 1992), and at least during the communist regime I can't really remember a culture of studying hard to escape poverty, because that simply wasn't how you escaped poverty - it was more a mix of "choosing the right parents", the people you knew, a bit of luck, a bit of corruption and maybe some shrewdness, but Romania definitely wasn't a meritocracy at that time (that may however have changed since, at least to some degree). However, kids that studied hard and were very good in school weren't looked down upon and disrespected as they sometimes are in the West.
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 finished highschool in 1999, grew up in a mono-industrial city which had just seen its only, well, industrial place in town close down and sold for scraps, in my case (and in the case of many of my colleagues) learning maths and getting into university with free admission was really one of the few escapes available from said town and, hence, from poverty.
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.
I used to know online a Romanian who had moved to Canada sometime during high school, she made similar points about the slowness of the North American curriculum. I told her about my 11th grade AP Physics C material (which uses calculus) in America and she indicated that in Romania they forced her to learn all that stuff years ago. However she wasn't exactly helpful when I later asked if she could help me with something, she'd forgotten too much and anyway didn't have an interest to keep up or go further. I'm still left unconvinced in the superiority of other countries teaching their kids so much more so much earlier when it seems to be forgotten not long after, and of course there are the long-standing "where are the results?" counter-arguments. It's hard not to be skeptical of prolonged compulsory education of any quality.
I'm also not convinced about the superiority of any teaching system (especially now, after I've learned about and read some Ivan Illich), I was just trying to confirm OP's point that the education systems around this part of the world were indeed pushing harder when it came to some maths-related stuff earlier in a kid's "educational cycle", so to speak.
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.
I think we do see results in domains where math & science is particularly important. Eastern Europeans are consistently among the best algorithmic programmers I've worked with. My employer was co-founded by a Russian immigrant, son of a math professor. Russia beat us to space, and given the backwardness of the general eastern bloc economy they're remarkably strong in science and technology.
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.
It's a nice dream, to have a populace educated to a much more advanced level in math, but I don't think it's doable with America as it is, i.e. we can't have both. We could at least stop sliding further down, perhaps, and I'd be glad to hear about many sorts of changes in state- and nation-wide curriculum, while we're to have such things, for what I believe to be marginal improvements. At the smaller margins we have individual parents doing what they can with things like these Russian Math programs, as they have done for a long time, and I think is probably sufficient to ensure the prodigies aren't snuffed out.
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.
UK citizen here. More specifically, English - the 4 "home" UK nations have different Education systems.
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.
>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.
While this is partly true, and a point of national pride (I'm Romanian), it's also important to realize that this is the positive end of a very unequal system. If you have the good luck of having a good teacher, and enough material conditions to focus on learning, school will leave you with quite a good array of knowledge, especially in Maths and sciences.
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).
Oh wow, thanks for the post, this is really insightful.
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.
I am not an expert by any means, but I believe one of the main reasons is that there is a fixed national curriculum, that includes many different domains, with the same standards for everyone, usually withab a huge focus on knowledge accumulation and rote learning. The mentality is often centered around knowing and being able to repeat facts and formulae. The curriculum often goes into deep detail on relatively obscure subjects with no or very little context. Combined with poor teachers (both in terms of performance and financially), this leads to many, if not most, students being relatively left behind.
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.
>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)
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.
I'd be wary of forming an opinion on foreign education systems from international students, as they're generally of more privileged extraction compared to the unwashed masses who can't afford to study overseas.
Absolutely this. People seem to ignore selection effects when taking with immigrants from other countries, especially students. These people typically come from the richest/most connected families in their home country*
It depends. Some of these people (their parents) left their home country because of economic reasons. This is the primary reason for immigration from Eastern Europe to the EU and the UK. If you're in the US, yes, the visa selection program requirements make your average immigrant come from richer or more connected families. For instance quite a number of people left my country to the US using connections in neo protestant church communities - that's what I've meant by connected.
> > people typically come from the richest/most connected families in their home country
> 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.
It seems like everyone replying has ignored my caveats.
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.
I think you are generalising and exagerating a bit. As it happens, I had a colleague and an acquaitance who both worked as a taxi drivers in Chicago and knew each other. They weren't bankers, nor farmers' kids. Their parents were middle class teachers and accountants. Reality is more boring and mundane than you might imagine.
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.
Normally I'd be very wary of this, but I went to the #1 liberal arts college in the U.S. My American-born classmates included people like John Glenn's grandson, the heir to the Mead trapper-keeper company, the son of the guy who founded Kohl's department stores, the nephew of the famous Hollywood madame, etc. With the exception of the Mead guy, most of these people were functionally innumerate and never took a math course in college.
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).
Not just privileged extraction, but likely high IQ. The Russian brain teasers are to get smart students excelling. The American system is set up to get the weak student stumbling to the finish line. The American school system isn’t designed to produce high achievers which is why the high achievers are produced by the parents and not by the schools.
What you're describing isn't the US system of education.
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.
Any idea how to figure out which camp a particular school is? There are wealthy zip codes with excellent public schools, but there are also crappy private schools so the public/private distinction is not always telling. Scores also don't work because they test dumb rote learning and have become gamed. I would love to find a school where teachers are the kind who know math/other subject "intuition" behind the knowledge like they had in some Soviet schools.
> 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.
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.
Absolutely this. My father was an immigrant from an eastern block country. Just a regular family, his father was a salesman, his mother a secretary. Father was valedictorian of his high school, speaks 6 languages, ect. Scraped together enough money to study at a British grad school, finished in 1 year as valedictorian. Then he immigrated to America. So someone looking at his math achievements might think wow, they really teach those soviet immigrants math well! But it was more like the best of the best immigrate while someone more average like me would be stuck in the old country.
I think its the same case for much of the west. Here in the UK, I remember maths being not the greatest but certainly every kid got through with a passable level of competance.
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.
I think this would be a more credible concern if American teachers were equally empowered to their counterparts. To a large extent what you see out of 'foreign' students is a product of teacher's having adequate power to manage their classrooms while teaching students to standards. Meanwhile in America teacher's are shackled by their things like No Child Left Behind and a union that tolerates faculty getting physically assaulted. You don't see these things being corrected because standards and discipline have been branded as racist.
The classes where teachers are getting physically assaulted are not the same classrooms raising future IMO gold medalists. Nor are they one repeal of No Child Left behind from becoming so.
The classrooms where teachers are getting assaulted can't raise gold medalists because the teachers can't create an environment condusive to any real amount of learning.
As a Portuguese citizen, I was not exposed to Russian mathematics during my entire academic career until I entered university. Whilst slightly better than the UK, our educational system was also not brilliant when it came to maths, and we only started to do calculus close to Year 12 (the last year before university). Anyways, my one and only brush with Russian mathematics was as follows - I flunked Integral and Differential calculus on Year 1 at uni, and was getting really worried it would take me a while to do this subject. In Portugal you need to repeat a subject until you get a pass grade. Then a cousin told me I had to "do Piskunov".
In those days you didn't buy books, you'd photocopy them, so he gave me a very large photocopied manuscript of Piskunov [1] in Spanish. I had never seen anything like it. It was a bit like a game; it had very little instructions, and it started with absolutely trivial exercises, but continued on and on, relentlessly. And somehow, it got you hooked. I read the entire set of books compulsively, just to see what the next exercise would throw at me. I finished my exam really quickly and got 95% (in my rush, I made one mistake in the exam). My teacher even asked me about some of the ways in which I solved some of the exercises.
Oh yes, without a doubt. Even if you compare the level of teaching in my life time, from the 80s when I was in primary school to now it has improved dramatically. Portugal was really a developing country all the way up to the 70s and mid 80s, we have roads and infrastructure now :-) completely different place.
This is the video by the daughter of the founder of the Russian School of Math. We sent our son there over the summer, he enjoyed it and all the kids were pretty advanced in the class. California seems to want to hold everyone back in the name of equity. But that will force more people into the private school system, which is exactly what we did. I don’t trust the California government to have my children’s best interest at heart, I think they want to hamstring them in the name of equity.
Essentially large part of education falls onto the shoulder of parents, which is natural. I'm even preparing to sharpen up my Math/Physics/Electronics skills for the future. Time to re-learn those things that I mostly forgot!
> Essentially large part of education falls onto the shoulder of parents, which is natural.
Have to disagree. A parent can't be simultaneously up to speed on Math, English, a 2nd language, Biology, Science, History or any of the other subjects that a child will learn in school
School is meant to teach, parents are meant to socialise. Unfortunately that seems to have been swapped around somewhere along the line.
In the UK, by law (Education Act), parents are responsible for educating their children - this has slipped considerably, I wish it were still a central ideal : under such a regime schools should be a service that parents can use to educate their children. I'd like to, for example, use school for some things, other groups for others, and home schooling for other things (essentially Flexischooling). This is in theory an option in the UK, but it's left to individual headteachers to dictate their ideals to parents (regardless of the Edu.Act) so you have to be lucky to get a headteacher who supports your chosen pedagogy.
They absolutely can. Most adults should be able to immediately recall the learnings from elementary to high school. This is not rocket science or highly specialized knowledge.
The fact that an arbitrary american adult educated in this country cannot easily differentiate and name some works of shakespeare and provide some quotes, etc, should be o source of national shame.
I mean it's impossible to master everything but to find something that motivates the kid and keep him focused is largely parent's work. It's really difficult to ask too much from the public education system nowadays :(
They typical standard for a teacher to be certified in a subject at a given grade level, in the US, is to be proficient at least one level higher than that grade level. That ensures you really know the level you're teaching very well, and also have a good understanding of where things are going, so you can offer enrichment to students who need it, and explain motivation behind or direction of certain topics, when asked.
Oregon governor Kate Brown has signed a law that allows students to graduate without proving they can read, write or do math. The law had overwhelming Democrat support & is justified on the basis that it will benefit non-white students.
This is about suspending mandatory testing prior to graduation. They did that last year due to virtual learning, and are extending the suspension longer while things get back to normal, and while they assess whether the approach to testing they have is suitable (and in line with what other states are doing). It is not (yet) gone forever, just for a couple years. And you still have to pass courses in all of those subject orders in order to graduate.
How is your interpretation any better than the editorialized 'brown drops education requirements'. The pandemic is no excuse to lower standards. It short changes these kids. If anything, public schooling should be extended as an option for those above 18 to freely learn what they missed. This is an immense loss for these children. The last two years of High school are extremelyy important.
Not sure how exactly the lack of these skills could benefit anyone.
I understand that the lack of a school diploma is a huge drag in life and that can primarily affect people from disadvantaged backgrounds, but shouldn't they focus on improving the way they teach kids instead?
> Not sure how exactly the lack of these skills could benefit anyone.
Lower the standards across all levels -- high school graduation, college admissions, job placements. Eventually we end up with surgeons and lawyers that are illiterate. But at least it is equitable!
When I started high school in Canada, my math grades were pretty bad. Probably in the 60%s (a C letter grade). I remember staring at a quadratic equation, struggling to understand why those 3 terms drew a curve. I had no intuition for it.
The summer before 11th grade, my father decided he had enough. It was time to learn math, Soviet style. He sat me down for a few hours each morning with problems from 6th and 7th grade Russian math textbooks - which was strange to me of course because I was about to start 11th grade. One important rule was that a calculator was not allowed.
Everyday he had a list of questions ready for me that he had judiciously picked. Back in Ukraine he was a regional physics Olympiad winner, and a gold medal winner (in the Soviet Union, the top graduates from each high school were awarded a gold medal - goes to show how they valued academics I suppose). I can pull up some photos if anyone is interested.
The questions were very clever and pedagogical. You developed intuition by solving them. And you couldn't solve them if you didn't understand the underlying principles. And of course, there's the word problems. I could barely read Russian at the time, so I had to take my time, but they bridge the gap between theory and application. And without a calculator, you are forced to develop techniques for manipulating equations and numbers. You get really good at it.
I aced math and physics for the rest of high school (and later graduated with a degree in Engineering Physics).
The western education system really fails us. My dad sitting me down with those elementary Russian math textbooks and enforcing a no calculator rule was one of the best things he could have ever done for me. The Soviet mathematic curriculum was designed by some brilliant mathematicians who understood the importance of developing intuition. That importance seems to be lost here. People think that quantitative intuition doesn't matter as long as you can plug your equation into Wolfram Alpha. But when you approach math that way, you don't develop an analytical and quantitative lens.
I doubt the calculators were the problem. People going through the "AP class" route with calculators finish up all of the math (and more!) that gets covered in a french high school math class (from experience of moving to France in High School and learning zilch for 3 years).
Totally on point about the "analyticial and quantitative lenses". Multiple-choice questions and lack of "real" questions really hobble a lot of math classes.
I was lucky in middle school in particular to have classes that used textbooks with a much more indepth look at why we would do X/Y/Z than the average book (along with a system where you would work through exercises in groups of 3 or 4, so better people could help out people who were struggling more). But I had to do a hell of a lot more "work showing" in France.
The way I see it, learning to do basic math in your head is just as important if not more important then learning a procedure via a calculator. A calculator doesn't teach you anything, it just teaches you how to use a calculator.
Doing math in your head doesn't really teach you anything either though. Everyone has a calculator within hands reach these days. I don't think practicing mental math ever helped me understand principles, it just helped me memorize and learn tricks that are only helpful with doing mental arithmetic.
Calculators in calculus probably aren't a problem.
But in the US, calculators are used at almost all grade levels. My son's school allowed them while he was in elementary school, while still learning basic algebra.
As for math education in general (in the US), it's pretty terrible. The lack of practical applications of "advanced" maths is a big problem. Basic calculus didn't "click" for me until I started taking economics courses in college.
> The lack of practical applications of "advanced" maths is a big problem.
I vividly remember self-studying calculus because I absolutely wanted to know how to find the area under a polynomial curve. I knew how to find areas of normal geometric shapes, but finding the area under the curve seemed like black magic that I _had_ to learn. If schools could somehow give this to students, there would be no need for "practical applications".
Yes, exactly this. The amount of children who lost interest in a topic because of systems like this must be high. I clearly remember how I thought maths was way too easy and eventually I lost interest. Sadly I was lazy so I didn't do anything to use my skills (though one could argue it isn't the job of the child) and ended up not doing all my homework and just making up the answer on the spot when called to the blackboard, so today I'm very average which I guess is a success in the eyes of the system.
It is kinda ironic that the former Soviet Bloc, whose central ideology was built around equality, had a very streamed education system concentrated on recognizing talented individuals early, while the Anglosphere, usually renowned for its ceaseless competition, emphasizes equality at the cost of excellence now.
Well, it's only ironic because this is a bit of a simplification. Their ideology always recognized inequality of ability and comparative advantages of various individuals. The issue, at least on a theoretic level, receiving more value and power because you happened to own something (capital) instead of because you produced more or produced things that few people could produce. Which is why compensation was always unequal and often based on production.
Equality was non existent in the Soviet Union. Technically a talented researcher would earn just as much a factory worker but the researcher would have access to apparatus granted privileges like living in a better apartment, occasionally shopping in an non empty store, access to better hospitals, and so on
My second child is suffering this in a UK school - the maths is too easy, his primary school had an 'advanced' group (quotes because it wasn't really advanced, didn't go as far as I did at primary school in normal class) to push the most able kids a bit. Now high-school they're back to doing absolutely remedial basics of arithmetic.
So much wasted time in school, he's frustrated not to make progress and bored with 'maths' (truly it's lack of maths, but to the young mind that gets confused with the subject and then you lose them).
> What's happened in Canada, is that curriculum is designed to teach to the weakest student in a subject.
No, the curriculum is not watered down to meet the needs of the weakest students. Canada tends to align its curriculum to that of other western nations. On the other hand, when you're talking about math there is a bit of an issue where the background of teachers is mixed at the elementary level and students are not guaranteed to have a true specialist teacher until grade 10. That isn't to say that specialist teachers are the best teachers, but it is a bit disconcerting when a teachers college offers classes for math-phobic elementary teacher candidates (particularly since those grade levels seem to be where many children develop their attitudes towards math). It is also worth noting that the quality of teachers varies based upon region and schools, largely because teachers have a lot of choice as to where they teach.
The article also mentions that AP (and, presumably, IB) are being maintained. These programs provide a recognized curriculum and the courses are typically taken by students who want advanced classes. I also took a quick glance at the BC mathematics curriculum, which is typically offered in different tiers, and it is offered in different tiers. Science appears to take the usual tact of a general course, with specialist courses for students who want to study biology, chemistry, or physics in more depth. In other words, regardless of whatever nonsense is being spewed by the board, differentiation between interest and ability is still available.
It is also worth noting that there would be significant public push back if there was a true degradation in the curriculum. Ontario tried replacing calculus with pre-calculus about a decade ago, which the government had to reverse due to public pressure.
It depends on which catchment zone you live in, and even the schools that offer AP don't offer the same AP courses. Last I checked one offered 2 AP courses and another offered 11, so there is huge variance between the schools offering AP. These are public schools, not private schools. There are private schools that also offer AP and IB. The IB private schools cost as much in tuition for one year of high school (IB senior years is a two year program) as a Canadian university does for the 4 year degree. Some of these private schools will teach second or third year university courses to advanced high school students.
The BC math courses are offered at different levels, but even the top level math is not for students who want to move ahead or be challenged. The top level math is the bare minimum to get into a Canadian university. Some schools offer calculus 12 and many other schools don't offer it at all. I guess that's "honors" math.
The "honors" math program that has been eliminated is a program that condensed the regular curriculum. I am so confused as to how that is inequitable, but AP (which has exam costs) and IB are allowed to stay.
In some Surrey schools there are programs to allow students to spend their last year doing a trades foundation program. This isn't evenly distributed either, but is a great way to allow students to start their careers. My brothers are both in the trades, but their friends at other schools spent grade 12 in a foundation program and saved 6k in tuition.
There is even a possibility to take summer courses and spend some of your last year taking college courses or university courses in the right districts. This is for Vancouver and Burnaby students that are close to UBC and SFU, but this isn't advertised or evenly available.
My point and rant about these is that it'll be a matter of time before all of these opportunities are also taken away. If they stay, I'll be pleasantly surprised and gladly admit I'm wrong.
> It depends on which catchment zone you live in, and even the schools that offer AP don't offer the same AP courses.
I grew up in Calgary. It was possible to apply to special programs outside of your catchment area, with a choice of multiple schools for some programs. Being admitted into a public IB program comes with the expense of a monthly bus pass, not the equivalent of several years of university tuition. I would be surprised if Vancouver is any different since out-of-area students are often the means of maintaining high enough enrolment to offer special programs ranging from academics to the trades.
Something that may have been a quirk of my home city: catchment area was not a hard-and-fast rule for middle school either. There were special programs one could apply to and, failing that, approaching the school's administration directly. Granted, for something like that the family must care enough to take the initiative. That may be in short supply in some areas, but it is by no means a measure of affluence.
> My point and rant about these is that it'll be a matter of time before all of these opportunities are also taken away. If they stay, I'll be pleasantly surprised and gladly admit I'm wrong.
There is also the possibility that you'll see the opportunities taken away, then be pleasantly surprised to see them return. The education system seems to go in cycles, based upon whatever the pedagogical fashions of the day are. Then again, I doubt that we will ever see the extreme of everything being taken away. People seem to like talking about things in extremes that don't truly exist.
The IB public programs are called "district programs," which give everyone in the district the ability to apply to the programs. So you're correct about that being open to those within Vancouver.
For one program it seems that there is a roughly $1,000 cost for each level, so it's a little over $2,000 to complete the entire IB program. The other IB program seems to cost $1,000. I don't know if either of those schools waive the fees or not, but looking at other districts they say the fees are for writing the IB exams.
I can't determine if AP courses are district programs or not.
It is also worth noting that there would be significant public push back if there was a true degradation in the curriculum.
Not so sure on that one. I agree some would push back, certainly. I feel it is fewer every year, with parents not caring for anything but what a piece of paper says.
But, perhaps I am a cynic, or am reading too many such stories.
As someone thinking of raising kids in Vancouver I was really disappointed by the honours stream being removed from VSB curriculum. If the school board truly wanted equality for their students they should be looking at external factors of why kids are not performing: do they have a place at home to complete homework? Do their parents value education? Do they believe in themselves?
I do believe that there is a natural difference in intelligence, but not enough to make the difference of a student getting into the honours stream or not. A lot of the kids say “I don’t get math” or “I’m just dumb” or they don’t have a stable household or family role models of success — all of which hold them back. Naturally these external problems are much harder for school boards to tackle so they would rather chop the legs off of honours students than address the students who come from a disadvantaged background.
You can have your kids take summer courses to finish the regular curriculum early. In some cases they can then be allowed to start taking courses at UBC or SFU. It was mentioned in my Surrey school over a decade ago that there were kids in Vancouver and Burnaby doing that. I think it's a nominal fee to register in the courses while in high school. Something to explore if you're still looking at Vancouver.
> ...Everyday he had a list of questions ready for me that he had judiciously picked.
Your story is very touching, thank you for sharing it.
It also emphasizes the importance of a motivated teacher. Also I believe that such parent's involvement makes the process and the subject of learning so much worthwhile.
It's not a secret that as parents we want/need to outsource the kids into schools just to free ourselves up for what we want/need to do. Yet paradoxically we want the kids to know no less than we know ourselves.
It's just a luck if kids come across a good teacher which would help the kids demonstrate to us parents that they are worth of our attention. Kind of backwards...
> and a gold medal winner (in the Soviet Union, the top graduates from each high school were awarded a gold medal - goes to show how they valued academics I suppose).
I'm not sure where in Canada you moved to, but there are Governor General awards [1] to the top graduating students. It's bronze for top high school graduate, silver for bachelor's, and gold for higher degrees.
I finished Grade 6 in Russia (in 1995) before emigrating to Canada.
I didn't learn anything new in Math class until mid-Grade 11 [1].
[1] Except Trigonometry. But I could tell the way it was taught was completely different from the concepts I learned in the Soviet/Russian system.
It was just rote memorization of sin/cos/tan - just clever formulas for deriving the angles and edge lengths of triangles that you solved by pressing the SIN/COS/TAN buttons on your calculator, rather than the "from first principle" explanation of what these concepts meant fundamentally.
By about 1/4th of the way through I wanted to just know what 'russian math' looked like. By Halfway it was pretty clear they weren't going to tell me. I skimmed the last half, nothing stood out.
"Jane fills a bag with three types of chips. There are 3-point, 4-point and 7- point chips. Jane picks 3 chips worth 15 points. Which chips did she pick?"
Sample problem for grades 5-6 [2]:
"Pinocchio drank half a cup of black coffee. He then filled the cup back up with milk and drank one third of the mixture. Again he filled the cup to the top with milk, drank one sixth of the mixture and filled it back to the top with milk one final time before he drank the whole cup. Did he drink more coffee or milk?"
I don't like the second question. Probably the kid is supposed to see the 'trick' :he drank a full cup of coffee since he didn't refill coffee and drank all. He added 1/2, 1/3 and 1/6 of a cup of milk and drank it all, so (3+2+1)/6=1. He drank the same amount of coffee and milk.
I think it is ok as a brain teaser, but there will probably be one kid in the class to see it and all the other kids feel dumb or whatever. But I don't think it teaches you anything (maybe it does, didn't study pedagogy, and while I was good at such fun questions I preferred the more structured approach in university mathematics)
Edit: this might make the kids think you need some 'magical' insights to do math and if they don't see it they are not apt for it, while the opposite might be true.
For the other questions: 2 also seems to rely on a trick, 3 looks ok, 4 is ok, 5a looks dodgy (probably just trying out numbers), 6 looks ok
About the K1 question:these are for 3 year olds? I only met one 3 year old in my life who could read, probably I am missing something here.
Yes, but I am worried if a school uses question which can be solved with a trick that the whole instruction in class starts to revolve around tricks (which naturally will happen since the one kid who sees it will answer first).
Tricks? It's not tricks it's word problems, which is just as important as any other problem. You need to relate math to the real world and having a equation doesn't help, having stuff like Susan is taking a train at 2:32 for 50 kms going 60km/h what time did she get off helps people understand how to turn a real world problem into a math equation to solve it.
Which is a bit fiddly but hardly impossible. I think there is pedagogical value in doing the algebra accurately and I think it is annoying enough that the trick seems useful and memorable when it is pointed out. The trick is also quite broadly applicable to physics problems where there is conservation of some quantity.
Yeah I think the examples given are probably examples of hard exercises or the grades are a poor translation (maybe grade 1 has 6 year olds, for example.)
Another solution is with geometry:
Start with a 1x1x1 cube of coffee. Remove top half and replace with milk. Now remove left third of resulting combined shape (leaving a 2/3 x 1/2 x 1 cuboid of coffee). Now remove front sixth (leaving a 2/3 x 1/2 x 5/6 cuboid). Now drink it all. Now add up the volumes of the shape and write down the answer.
> I think it is ok as a brain teaser, but there will probably be one kid in the class to see it and all the other kids feel dumb or whatever. But I don't think it teaches you anything
These word problems are intended quite precisely as introduction to algebra. They show you the kinds of questions that algebra can solve in a "structural" way, which makes it way easier to grok algebra later on since motivation for the subject has been provided in such depth.
K-1 means Kindergarten and 1st grade, so 5 year olds and 6 year olds. In the US earlier grades than that are called PK-3 (pre-Kindergarten and 3 year old) and PK-4. I don't think RSM has classes for these levels.
> Edit: this might make the kids think you need some 'magical' insights to do math and if they don't see it they are not apt for it, while the opposite might be true.
That's a valid point. However, what's the opposite to having insights? Is that following routines and/or exhaustively exploring the entire problem space (which the first problem in the GP comment seems to teach)?
Teaching those might have higher pedagogical value than conditioning children to find insights (as - at least at first sight - the increase in skill in those is more directly linked to the effort the child invests in learning) However, the von Neumann story in your sibling comment suggests that some people (and so, some children in the class) will perform routines faster than the other children no matter what. Seeing a "shortcut" solution gives a chance to those who are slower at routines to arrive at a solution fast, too.
Moreover, a lot of real-world problems (in academia as well as in business - from my limited experience in both) are exercises in pattern matching and finding shortcuts rather than in an exhaustive exploration of the problem space - and helping children to collect an arsenal of tricks (and more importantly, teaching them to look for insights and patterns by giving them multiple trick-based problems over the years) prepares them to handle those real-world problems.
Yes I definitely felt dumb in my Russian math classes when someone saw a clever trick and I did not. That made me try harder to look for tricks, it's a challenge in the end.
I ended up with a Math degree very later on. So I don't see how feeling dumb harmed me.
PS: the top level professional math is 90% tricks.
#2 relies on understanding how numbers "work", i.e. on having spent time on playing with them.
#5a is a factorization exercise.
In general, Russian approach to the math includes exposing kids to the toolset as well as the theory. This particular case is solved this way, that is solved that way, etc. Keep on it for a while and it tends to develop an intuition for knowing right away which problem is solved with which approach. So, yeah, if there are shortcuts, you use them. If there aren't any, you brute-force it.
> 5a looks dodgy (probably just trying out numbers)
You can find the 4 zeros of the left side by eyeball, and use that to construct the rest of the equation group you need.
It's expected you've seen and solved this type of problem before taking the test. Not that many kids can come up with a working solution strategy on the fly.
"Trying out numbers" seems like the simpler approach to me. You have to multiply four consecutive numbers to get 1680. One of them must be 7 (since 7 divides 1680), one must be 5 or 10 for the same reason. That leaves only 3 possibilities, and you only have to check the middle one (5 x 6 x 7 x 8) since if the result too big or too small, you'll know which of the other two it is.
OK - you weren't guaranteed the solution was an integer or even real, but you should strongly suspect there's a simple answer because you didn't get generic fourth-degree polynomials in your class. Depending on the class, you might be expected to find one, two or all four solutions - but once you have the first one the rest are much easier.
Following the procedure taught in class (and I'm quite sure they didn't teach "make a guess") is safer. When you show the correct procedure and fumble say, and addition, near the end, you'll still get almost full points.
If you go off-road and get the wrong answer, it's up to how much the teacher likes you.
This doesn't line up with all the other problems, which reward having some insight as well as the concrete algebra/arithmetic skills to finish it out.
If anything, your approach sounds more like what's expected in American schools: either you have been taught a foolproof way to solve problems matching Pattern X, or you stare slack jawed at the question paper thinking "I must have been absent the day they covered this question pattern". Exactly the opposite of the kind of thinking described in the article.
Besides, as in another comment of mine on this page, there is no generic method your teacher could have taught you here that always works. (If you think so, please set the right hand side to 1681 and solve that version...). The principal method I was taught for solving cubics was to make a guess. Numerical methods came later.
Pretty sure students are supposed to switch( n % 3 ), and solve for all 3 possible values of the remainder.
> probably just trying out numbers
You can't solve a 4-degree polynomial equation by trying out numbers. It has up to 4 solutions, not guaranteed to be real, let alone rational or integer.
> You can't solve a 4-degree polynomial equation by trying out numbers
On the contrary, in practice you can't solve them any other way! Spot a simple factorization or guess a root is your best chance if you don't fancy working through the quartic formula [0], which will take you multiple pages just to write down the first step, and a numeric approximate solution is not acceptable.
Those are pretty challenging problems and they also require a bit of working out. How many of these problems are students expected to solve in, say, and hour long test?
These represent classes of problems and the test checks which classes students have covered in their studies and which they slacked on.
These two are straight-forward problems, both will be allocated some basic nominal time. You either know how to solve them or you don't. Sometimes there'd be problems that are more puzzling, because they'd combine several problem classes and will take time to unfold (this is typical in math olympics). These will be allocated more time as they require thinking and searching for a solution.
I see what you mean by classes of problems. Some of the problems from the above links I know how to do (although working out might take some time and a pen and paper) and then there are some problems where I don't know where to start. I was just never taught how to think about numbers that way even though the link says those problems are for middle school children.
From the examples I guess it is the ability to intuitively manipulate fractions? Like “oh that’s kind of like .8 of this other thing so it’s close to X” ?
I have a feeling it was more about the quality of the teachers than anything else.
I moved from “Deep” Russia itself to the periphery of the USSR (Bulgaria) when I was in first grade, and my parents had the foresight to make me repeat that grade so as to help me with learning a new language.
The quality difference was astounding. In 1992 Russia by first grade I was learning english with flash cards technique, drawing human shapes, animation, perspective, and some pretty good maths. The knowledge I gained there allowed me to learn almost nothing but the language up until about 3rd grade. And it was a school in the middle of nowhere.
I think the USSR trained some very good teachers and just sent them around everywhere, places they would not have gone themselves on their own volition kinda thing.
Oh and I remember teachers where highly respected, a thing I saw slowly degrade while the country was going through the 90s reforms.
Respect for teachers is a huge part of it. People want status as much as they want money. If culturally America venerated teachers the way it venerates soldiers we’d have fewer wars and smarter people.
> school's curriculum is based on Russian teaching traditions that emphasize reasoning and deeper understanding early on, not just memorization and practice drills. "The child should be brought to abstract level as soon as possible," she says, "meaning early introduction of algebra and geometry, not only arithmetic," and helping children figure out principles for themselves rather than spoon-feeding them.
I wish American teachers could teach the reasoning behind math.
I don't think most math teachers know themselfs though---even at some colleges.
I don't think I have even seen a math book that goes into depth on why a equation, or problem, is solved a certain way.
I would like to see most memorization in math, and most subjects nixed for good.
I have found, including myself, my early difficulties in math were due to just memorizing how to do a problem.
It wasen't until I started over (I went to a CC early. I hated high school socially, and it affected my studies. Going to a CC was the best move I made.)
I took basic math, and algebra, trig., at the community college.
It made inorganic chemistry, and physics, so easy.
I went from terrible at timed tests of regurgitating basic arithmetic facts to one of the better math students in my school once the curriculum moved on to requiring some understanding of what was going on.
I buy into having a deeper understanding, and in both work and school, it's pretty clear who has a deeper understanding and who's going through the motions.
That said, how is it that there's such range in pedagogy? So many people have studied teaching that you'd think we'd have a better idea of what works. Or maybe it's that goals are different.
Not goals - incentives. I'm from Czechia and for me math education changed dramatically for the last year of high-school. All of a sudden it was crunch time to get the best result at the standardized tests that are part of government's examination.
Up until that point the math classes were very much Soviet-style understanding-first, daily hour-long homeworks that are described in the article.
Final year was about technique memorization.
Once you're gaming a system, education quality tanks.
The dilemma with tests is that we need them to measure competency, but once you introduce them, the specific form the test takes becomes a target, itself.
I think a big part of this is society's expectations.
For example while there was bullying in my primary school - there was no bullying because you were good at math. The opposite was true - if you were bad at math it meant you're "dumb" and kids will laugh at you.
It wasn't all perfect - it was uncool to try hard (means you're dumb and have to work for it and that's boring) but it was very fashionable to instantly know every answer. So teachers had it much easier because kids had intrinsic motivation to learn math.
Another part of it was probably that the unemployment was at 20% at that point and everybody realized you have to be well educated to have a chance of good job.
The goals are different. The goal of the Russian school is to teach math. The goal of public schools is to dumb down the curriculum until everyone is equal.
The soviets were nationalists who believed the utility of the citizenry was in their contribution to their country.
You reinstill love of country, which really means love of society and others, which is another word for philanthropy, and you will quickly get this.
In America, instead of education and work being for the greater purpose of your nation and people, education is for the individual.
As usual, most people find more motivation when helping others than themselves, but the focus on ourselves in American education means it's easy to slack.
Not sure about these after-class schools, but I can compare my ex-ussr school and uni education with math or CS undergrad textbooks I read a lot in English these day. Not sure about early education though.
1. Russian math books are straight to the point, superconcrete. Hard to read in a linear fashion but very useful when student is serious about going through it ("Problems in mathematical analysis" by Demidovich is a perfect example, Mark Vygodskiy's "Elementary Mathematics Reference").
2. In most textbooks I remember nobody tries to build a dumbed down explanation of things. This might lead to the book being harder to understand without teacher's help. I remember how some American undergrad-level introductory math analysis books were trying to skip proofs, avoid certain details, giving too many intuitive explanations ().
3. Mid and late school math is pretty advanced, especially when compared to US typical level.
These days I live in UK. Kids go to school early here: 4-5 years. My daugther is 6 and is comfortable with trivial math. I've read a few secondary school textbooks and they feel quite ok.
I studied in a top school in Moldova which follows a curriculum whose foundation was defined during the Soviet union times, so could be pretty close to the Russian curriculum. Just as an idea, in grade 10-11 I was studying limits, complex numbers, derivatives, grade 12 was dedicated to integrals and solving problems with them. Math got progressively harder starting with grade 5. I remember I was filling about 2-3 notebooks of 48 pages (little squares pattern) per semester with homework and in-class class problem solving. We always had homework to do, and often it would take me hours to come up with a solution to the more difficult problems that would give me a 10 (grade A+). My nephew who finished grade 4 has no homework (Canada), but I remember my grade 4 I had so much homework for each subject (math, French, Romanian, geography, arts, etc). Heck, in grades 1-4 we even had summer homework, which were books containing exercises for various subjects, and I remember that vividly because I hated to do homework in the summer. I'm quite sure that good Russian school are also quite intense..
Can’t speak for Moldova but had what seems to be same level (specialized math school) in Russia and 10-11 was 15-17 yo (you could start at different age back then)
I don't know if it's that, but for me math "clicked" when in 2nd class of primary school in Poland we had a whole year of solving intuitive math problems. Basically linear equations of 1 or 2 variables, but that was without knowing what any algebra or even what a variable is - we started with only basic addition and multiplication.
Teacher asked us (one by one) to describe how we would arrive at the answer to problems and why that way. Sometimes it would be a contest - who guesses the answer first and that person gets to explain the process and bask in the glory of being the smartest kid ;). And then we were shown how to write that solving process as equations and practiced changing from problems to equations and vice versa.
At the end of the year most kids understood algebra.
I think the key difference is that it quickly moves from what's considered problems for kids (Ally had 3 apples, Booby had 2, etc) to abstract tasks and fundamental theorems in algebra, geometry and other areas.
The way math was taught in USSR and is still largely taught in Russia is by going as quickly as possible to calculus. I definitely studied limits and derivatives in school (around grade 8 or 9 out of 11 as I recall) and we briefly touched integrals in the last grade. There are some areas they don't really include, although in my opinion they should have, like mathematical statistics, which could be even more useful, but still.
That is a common approach recreated in some other subjects. For instances, teaching Russian includes not just basic syntax and phonetics but also just basic linguistic exercises like dissecting complex words and learning classifications, which you don't really need to talk it but they provide a deeper understanding.
Check out 'Word Problems in Russia and America' by Andrei Toom. Toom contrasts the poor state of the American math curriculum with his experiences with the Russian style, particularly its centering of 'word problems'.
If the article is accurate, one Japanese equivalent of this, known as Kumon, is almost its polar opposite -- remembering my own experiences with it as a kid, it was very much about rote memorisation and doing the same sheet of arithmetic as quickly as possible. One of the exercises Kumon made kids do at the age of 5 is to simply place shuffled magnets with the numbers 1 to 100 in order on a 10x10 board.
I can't say that this approach really turned me onto maths: quite the opposite. Past a certain level, the Kumon teachers were essentially just marking from an answer book, without any understanding of the content themselves. They had zero interest (or perhaps ability) in conveying the beauty or applications of maths to the students.
The approach that made me love maths was one where I understood the intuition and purpose behind the methods, ideally enough to develop them from the bare minimum myself.
My sister sent my niece to one, as even her 'private' middle school in Newton, MA, was just not strong enough. She did sign up my niece in every summer class she could, without overloading her. It was always my nieces' choice, and half the time she choose arts and half science. Now, my niece is able to do college entry level data science classes (she is about to become a Junior in HighSchool), and she actually likes it a lot.
American schools are too soft on science in general. I did grades 1-11 in Albania and my senior year of HighSchool in the US. Some of my schooling in Albania was done under communism, and some after communism fell in the 90s.
The Albanian school was brutal in teaching science. Biology started at 5th grade, pre-Algebra at 5, and full blown Algebra at 6, physics at 6, and Chemistry at 7. Then in highschool you did the same, but more advanced. There was no choice at all, you had to do them all. The only choice in HS was a second foreign language. The whole idea was that you have to know all the basics of ALL sciences, so in college you know what to choose and pursue. If you never tried, you will have no clue if you liked something or not. (also basic music knowledge, sheet reading, and arts was a requirement as well).
When I came to the US, I was flabergasted how behind most of the kids were in science. I took AP physics, and it became boring as it was things I had done in 8th grade. I got 800./800 on the SAT 2, physics.
The math part, I took AP calc, and it was advanced enough, especially the part B to challenge me. But this was clearly an elective that only about 30 students took it, while back home it was a requirement for all.
Unfortunately, the current movement on dumbing down math and science in the name of 'equity' is a step behind and very dispiriting. It is bound to hurt poorer but smart kids, that can't afford private tutoring and have to rely only on public schools. Extreme progressive Liberals are killing science and progress in this country, and are becoming actually regressive and backwards.
P.s. The only advantage of American teaching on science was that it relies more on experiments to teach concepts, while the Albanian one had no equipment, or lacked the basics of it. Heck, in the 90s we didn't even have glass on the windows and had to freeze all winter. Also basic electricity was lacking half of the winter.
Ps2. Most Americans have it good (condition wise), they just don't know it
Ps3. This is a good video how schooling was back then (in 88). Notice how the kids are wearing jackets inside, as there was no heating
https://youtu.be/yZD1jaKbz2g?t=251
My experience in my home country of Romania was similar to yours, however my conclusions couldn't be more different.
> American schools are too soft on science in general. [...]The Albanian school was brutal in teaching science
How on earth is brutality in teaching a good idea? What was the effect of this approach on the average Albanian or Romanian student? Sure it worked fine for a few of us, but overall the results are disastrous, just take a look at the PISA rankings.
Have you ever wondered how it's possible that the same "stupid" Americans that don't learn integrals and group theory in highschool somehow manage just fine in university and later on in life?
PISA ranking? China and Singapore are at the very top. If you think what's described is brutality, you should watch tiger moms push their children.
I've also went through what could be considered a "brutal" system, very similar to whats described, although i wouldn't use that specific term.
We had ~40 problem sets to solve every evening for homework, hours of work. I don't think anyone ever got an A+ in anything. If the student was better/stronger at something - they'd only get harder and harder questions thrown at them during oral exams until they crumble, and harder and harder sets added to their homework.
As for the "stupid" Americans? STEM classes are full of foreign kids in top US schools.
> China and Singapore are at the very top. If you think what's described is brutality, you should watch tiger moms push their children.
There are a bunch of different approaches that can lead to top scores in that ranking. notably Finland does well with a non-brutal approach.
> a "brutal" system, very similar to whats described, although i wouldn't use that specific term.
the term was used by the parent, not by me. We just happened to assess its value differently.
> they'd only get harder and harder questions thrown at them during oral exams until they crumble, and harder and harder sets added to their homework.
It depends on what you think the purpose of school should be. If you want people to understand the world they live in, understand the "why" of things, think critically and creatively etc. then problem sets and ritualistic humiliation during oral exams won't take you very far.
> Have you ever wondered how it's possible that the same "stupid" Americans that don't learn integrals and group theory in highschool somehow manage just fine in university and later on in life?
They do just fine because they aren't selected to do jobs where they need those kinds of skills, because they tend not to have them.
If they were taught the stuff at least they would have the choice.
I'm not in favour of brutalizing the kids though, I think that ruins the experience for them. But a lot of kids are more capable than we think and ought to be shown advanced subjects.
> pressure during oral exams taught to work against the clock, develop conviction and do your best to defend your position.
In some students. In others it just engendered the feeling that they're morons and that math is just "not for them". Math and science are for absolutely everybody. Not everybody will make a career out of them, but everybody can enjoy them.
> Have you ever wondered how it's possible that the same "stupid" Americans that don't learn integrals and group theory in highschool somehow manage just fine in university and later on in life?
Most people will do just fine with basic math training because it's what you normally need later on in life.
Most of my math training in the Eastern European school system has probably gone to waste. I only saw the point of advanced maths, calculus later on in technial university when I had to apply it in science. During my engineering career I only used the math I learned up to maybe ninth or tenth grade. No calculus. For materials science we also had to use maths up to tenth grade because of non linear properties of materials. However, calculus and advanced math was required to understand the physical phenomena. What went away was solving tricky integrals and series for which you had to think of an obscure substitution - the type of grinding math excercises we had to solve in high school. Last month I had to solve some RSA and ECC cryptography problems at work that could not be easily solved with existing crypto libraries. For example RSA signing checks were reimplemented using a big numbers library. That also did not require advanced maths, but rather a lot of grinding research done by reading crypto standards and technical guides.
My point was that a lot of advanced math can be taught later on. If it's forced down people's throats without them understanding why things are the way they are, all you're gonna get is ignorance and resentment.
In Romania I attended one of the "elite" maths/cs highschools. Except a hand full of people who would win medals in the imo etc. the rest of us were mostly clever, overworked automatons who learned problem solving tricks that we regurgitated on the page come exam time.
I think its not that it has to, its just more challenging and rewarding. I’ve read a lot about bulling and social problems in USA schools, but so almost none of that myself i Eastern Europe.
I think the subjects where so hard that it forced kids to cooperate, smart kids where considered a resource that potential bullies didn’t like to harm, as they would help them in the future. Additionally there was this sense of … maybe “esprit de corps” as we all were going through the same tough training.
I think that those hard problems were actually very rewarding and I saw very little of the “teenage problems” that are so prominently displayed in American media.
> I’ve read a lot about bulling and social problems in USA schools, but so almost none of that myself i Eastern Europe.
There was some bullying back when I was in school, but possibly less than in the US. However, I think this doesn't have much to do with the curriculum, but with social conditions.
In my case I was bullied in secondary school (by kids in my class) even though school was tough and I was easily part of what you would consider "valuable resources".
The result was that the bullies would copy their homework off of the kids that copied their homework from me and I still got into fights daily.
In highschool bullying disappeared completely, but that's because I went to a selective school (the equivalent of a US magnet hs) where everybody was academically oriented and also came from more stable social environments.
In less academically strong highschools there are a lot of nasty things taking place - students abusing eachother, teachers abusing students, students beating up teachers etc.
> bulling and social problems in USA schools, but so almost none of that myself i Eastern Europe
This has changed for the worse with capitalism and social inequality. We had some bullying back when I was a primary school student but it was physical bullying by higher graders towards younger students and it was mostly done outside the school. We had none of the psychological bullying that you see in the US and to a greater extent in Korea or Japan. We had roma and very poor students in our class. There was also almost no bullying during high school. The more odd kids were usually ignored and had friends of their own.
> I think that those hard problems were actually very rewarding
Well, maybe rewarding for some. I didn't like them and would rather go outside and play or ride my bike in the neighbourhood. Math for math's sake was never my thing. I only started to really like math in university when I saw its potential to solve real world problems.
> This has changed for the worse with capitalism and social inequality.
An interesting perspective which I feel might be the actual core of the problem. I grew up after the Revolution, so I can't make a before/after comparison, but (as I mentioned in a different comment) I see a tight connection between how aggressive other students were and how bad their family situation was.
> Math for math's sake was never my thing. I only started to really like math in university when I saw its potential to solve real world problems.
Perfectly legitimate. Pushing hard math early on can actually put you off the subject forever. Some of us started liking it in university, but a lot of capable students decide it's just not for them. With a different approach many might have ended up understanding and maybe even loving it.
I don't think it's really "brutal" but "challenging". So if the kid has no interest in Math he/she would regard that as brutal and for others who do have interest it's just challenging and maybe even fun. Just like people who like to crack Leetcode problems, those are challenging and fun.
> I don't think it's really "brutal" but "challenging".
Brutal was the term used by the parent, I just picked it up from there.
> if the kid has no interest in Math he/she would regard that as brutal and for others who do have interest it's just challenging and maybe even fun
Throughout my education I was in both camps at different times. The more "brutal" things got (dry, formal curriculum, tough teachers) the more it put me off. On the other hand patient and engaging teachers could get me to spend countless hours after school working on math problems.
Interests are things that develop based on intrinsic attributes but also based on environmental feedback. I suspect the latter is fast more important.
>The Albanian school was brutal in teaching science. Biology started at 5th grade, pre-Algebra at 5, and full blown Algebra at 6, physics at 6, and Chemistry at 7. Then in highschool you did the same, but more advanced. There was no choice at all, you had to do them all. The only choice in HS was a second foreign language. The whole idea was that you have to know all the basics of ALL sciences, so in college you know what to choose and pursue. If you never tried, you will have no clue if you liked something or not. (also basic music knowledge, sheet reading, and arts was a requirement as well).
I've went through similar curriculum in Croatia and I hated it - the literature they forced on us was politically correct bullshit some figurehead decided should be common culture. As a result 30% of the class read it and the rest just cheated and studied for test.
It was like this in every class - I hated cheating and studying for the test - I was lazy and it was pointless - as a result I got barely passing grades based on slightly paying attention in class. But in casual conversation I could relate way more of the basics than my peers. And I lost interest in many subjects simply because of how they were taught (study random facts because it will be on the test - no context or application). I had to relearn algebra after highschool because of how badly it was taught - and I was interested in it since I was trying to learn 3d programming and I was going to math competitions in elementary school - teachers couldn't relate any questions I asked to stuff I was interested in.
I hope my kid gets way less material to study and more opportunity to figure out what he wants to learn.
Can you share what did you do with the students that could not do algebra physics and chemistry in junior high?
I’ve always thought yours was the right way. There are some students which will do the minimal but will manage if challenged. Seems the equitable way leaves them behind.
Since it was a communist country, everyone had the same conditions (crappy), and everyone had the same fricking clothes (only two clothing factories on the whole country), and same teachers (and no private schooling allowed), and we had the same books, and no calculators.
And it is clear, if all conditions are equal, still people are not born equal. Some kids are just smarter than others, and some are just dummies, or don't care about school. Since everyone was the same ethnicity, you can't blame 'systematic x'. It was all equal. A doctor and a nurse probably lived in the same apartment, and had similar wages. (think Cuba, or North Korea today. Albania was like the North Korea of Europe).
Grades/marks were 2 - 10, 2-4 was failing, 5 was passing, 10 was excellent (and hard to get).
Grades were given in the basis of
1. homework,
2. blackboard interrogation (you had to solve and explain everything in front of the class, similar to a whiteboard coding interview),
3. Exams, flash quizzes or pre-announced.
You had three types of students:
1. Great students, and are aiming for the 10s (and usually get a 9 or a 10).
2. People that struggle to do the basic, and just want to pass the class and are aiming for a 5 or a 6.
3. Everybody else that got a 7-8.
Every exam or homework, was done with this in mind. You usually had 3 type of questions:
1. Super basic, <- If you were in class, you could do it
2. Medium, <- Some thinking is required
3. Advanced <- Usually only the really good students got these
The teachers knew, that some students would struggle, and they will let them pass if they just did the basic effort. If a student failed even that, they would have to go to 'summer class' and take additional classes and exams to pass. If you failed a class, you failed the whole year and had to repeat every other class as well.
Also, the school over time divided the students by grades. Each cohort-year, might have 4-5 classes (of 30 students). Your class was the people that you studied with, and did everything. The top students usually were placed together, and spread in the class A, and B. The rest were put on the other classes.
So, the top classes had only excellent students, or average students, but not of the failing one. The idea was that failing students will just drag down all the top students and not let them excel. A top student can make an average student better, but there was little chance to do anything with a failing one. If you were initially a failing student but started doing better, you could move up to the better classes. This is similar to the soccer relegation techniques of most soccer leagues in Europe.
But: Every class, had the same subjects, and the same load. Even the failing students had to do pre-calc. But the teachers would just be much more lenient on them. Eg. do the very basic, and they will let you pass. But if you wanted a good grade, 8 or higher, you had to work your butt off.
This is totally politically incorrect in today's environment, but even communist Albania knew better. Some people are just not smart at all, and it is better to let them just do the bare minimum and pass, meanwhile let the smarter kids do more advanced work.
T.b.h. as somebody from ost-block country, I did hate the cast-system based on the performance. In my school we had a class that selected for the worst students, and surprise, it created a class where so many people failed.
I was in class that had an even mix, some really smart kids, some not so much, and some comparable to the worst of the D-class. Like third of the of my classmates went for the sports-focused high-school, because they were decent at basketball and bad at almost everything else. But nobody was failing.
Like, A-class should have been the top students, B and C middling, and D poor. In reality, A was ordinary, B and C contained the best and the average and D was a failure.
Eh, I remember I was put in a D class. Math classes there were like "today we are going to study addition, who wants to ask this student to add two numbers?" so I volunteered and asked the guy to add 5 to 7 - I understood that the class somehow struggled with two digit numbers and wanted to see this in action. The teacher, though, realized what my plan was and said we can't do this level of complexity right now. That day I learnt something important though: that the crowd follows "the rules" without questioning them and that being a smartass isn't rewarded. We were 6-7 year olds then.
How did the A class become ordinary while B&C had a mixture of top and average? Was it that the ordinarily smart students were motivated by the sorting exams than the top smartest students?
I think part of it was that the sort happened when we were ~10yo? People from A-class were the sort of straigh-A student that then coasted on the fact learning came to them ~naturally, so maybe that is why the perceived performance declined.
Even more anecdotal thing, in my class there were more people doing extra-curicural contests, like Math-Olympiad and the like. We were even encouraged by teahers, along the lines of "No, try it even if you don't have top grades, that is the sort of thing where you need to understand what you are doing, not just ave all the right answers on the test." :D
Last hypothesis of mine is, that because we had a mix of sudents, the smarter(?) of us spend some time explaining to others (mix of goodnes of heart, being bullied and even having like a pay-for-homework manufacture?) ... and as they say, you learn best, when you explain?
But I could be completely wrong, rose-tinted glasses and long forgotten traumas and all that :D
> Since everyone was the same ethnicity, you can't blame 'systematic x'. It was all equal.
This is certainly an advantage that Albania had over Alabama. There are still people alive who remember the school system having to be desegregated under armed guard. The struggle continues: https://www.washingtonpost.com/news/morning-mix/wp/2018/08/1...
yeah, but you are missing the point completely. I gave you a real life example: Even if everyone is equal, and has the same conditions, some people are just born smarter than others, and will put more effort and do better.
Equality of conditions is a good thing to ask for, equality of outcome is stupid, as even in a murderous communist regime, was impossible to achieve as it goes against basic human nature.
That's why event though I am liberal/democrat, I don't like the today's 'progressives', as they seem completely ignorant of human nature, and have no knowledge of the history of countries that went through socialism/communism, and yet want to repeat the same destructive mistakes by trying to achieve 'equality of outcome', which is impossible.
It seems unlikely that there's just a smart/dumb setting inside of each kid.
Material conditions being equal don't mean that everything is equal. Some people have parents who value certain things and push their kids. Other people have some anxieties to get over before they can perform. Some people like being ahead of the class and put in effort to stay there.
Do you have data or anecdata on the effect of repeating an entire year? In many modern Western schools that'd be frowned upon as punitive and as likely to provoke a "well screw YOU!" response or ingrain "I must really be stupid" mindset.
Anecdotal but both my parents repeated a year (that was in the sixties in France), they went on to become a teacher teaching primary school teachers how to educate children for one and a headmistress at a relatively large primary school for the other.
Both of them looking back thought that repeating a year was a good thing for them, they went from being struggling average/below average students to top of their class and regained their confidence.
I think it really depends, in a place and time where repeating classes is more common, it maybe doesn't really destroy confidence as much whereas in areas where it's rare, then the psychological impact is much worse.
Also, one thing to note, both of them, in their experience with education, saw much better results when children were separated by level than when classes were mixed together for exactly the reason the OP mentioned. A top student can help bring an average student up but when there are both top students and very below average students, the teacher has to make the choice to either focus on the below average student or the top students and neither of those choices are good.
In France, we've had a push toward lowering the overall level and removing any elitism at schools. This has increased the amount of kids who graduate from high school with the Baccalauréat which is needed for university but has lowered the level of kids actually going to university (especially at engineering schools and elite institutions) and reduced the value of a university degree. It's a bit similar to what's happening in the US in STEM and I think it's a bad calculation for the long term competitiveness of the country.
> but has lowered the level of kids actually going to university (especially at engineering schools and elite institutions)
Come on, of course it hasn't. Entry requierements haven't changed.
The highschools providing the largest contingents of students to these elite institutions respect neither the national curriculum nor the ban of sorting students by ability without any consequences. Unsurprisingly the two most famous of these highschools are also exempt from the French purely geographical students draft and the places most politician children attend. As usual in France, the rule for all is not the rule for the elite.
In effect, the dumbing down of the national curriculum has just made a system which was already one of the most unequal in Europe even more unequal. But everything is fine. The French system only uses entrance exams and entrance exams are always fair, aren't they?
I remember being in my engineering school with prepa intégré and the teachers lamenting that we had not studied vector space in high school like students used to 15 years ago. I have a friend teaching at a math sup/math spé who complains about the lowering of the level. So yes, the level has been lowered and entry requirements have changed. I do know that in both prépa intégrés and math sup/math spé they try to compensate for the lower level out of high school by condensing all that should used to be done in high school within those two years. Of course those two most famous high school are an exception but those are such a small percentage of the total number of engineers.
All countries use entrance exams, it's not something that specific to France and of course there's an inherent unfairness to them, kids from better educated families are always going to get better results. In fact if we were talking about fairness then the disappearance of boarding schools (internats) have actually increased inequality. There was a time when a lot of kids spent the week in boarding schools, for kids from families with less focus on education (like my grandmother who kept telling my mum that she should stop reading because it'd cause her headaches), it removed them from non-optimal environment and put them in situation where they had better access to education. But I'm not sure that's desirable or optimal :)
You have to fail two classes in order to repeat the whole year. If you fail only one, you'd still could pass (but had to go to additional schooling). But if you had failed two, then there was something wrong going on with with the whole year, and it had repeated.
Usually it was kids that had some discipline problems. Perhaps ADHD, etc... but at the time, no-one cared about those aspects as school psychologist were not a thing at all.
If you failed more than one year in a row, then the teachers will just give up on you and let you just slide the next year as long as you just showed up.
My father repeated a year, bounced back from it and became a teacher later in life.
But usually it did not happen that often. When it did happen, it usually meant someone will change schools to a less demanding one (if in high school). So for example you may leave a gymnasium and go to a school for mechanics.
In elementary school it did not happen except for serious disciplinary problems. People with mental problems were able to have special programmes that tailored to their needs, so this did not result in repeating a year.
Teachers did not really like the idea of having someone repeat a year, so if you had a decent attitude you could get a passing grade but you had to put in the effort. This "effort" part is what is in my opinion biggest different, Eastern Europe schooling required you to put in a lot of work, if you were talented you didn't need to use so much time, but if you weren't natural at math, you had to put in plenty of hours to get decent grades. Problems usually were not of the sort that can't be learned through "brute force".
I've done some teaching on the side and when I was teaching a guy in a "US" school for children of diplomats, difference was that their problems were usually much more freeform and required deeper understanding but once understood, used primitive math or physics methods. Our normal schooling was different, it used advanced math or physics but the problems were often times many variants of the same problem (which allowed for brute force learning). Honestly I think that it would be the best to have both, as many of my fellow students did not really understand the material we studied but they could brute force it.
I skipped a class (went from 3rd standard to 5th) because I could clear the entrance exam during changing schools (I also topped the exam, mainly because my tutor made me work hard). In college I failed a paper because I was lazy and didn't work hard. The first incident made me happy but didn't make me think that I was very smart and the latter didn't make me think I was dumb. I knew the reasons exactly. We under-estimate how much we know ourselves.
As a parent (in Switzerland of all places) I actually consider this a very good option for kids struggling. And took the opportunity for one of our kids and it worked out great.
Edit: Switzerland has a different approach where children are segregated by their ability starting from 7th school year. So you go to Gymnasium, A level, B level, C level.
In the Netherlands, repeating a year is common. I estimate about a quarter of kids in my year had to repeat a year. If it happened twice, you had to leave school and go to a less exacting school.
I have seen less than 0.1% in my school fail like this. But yeah, it was punitive and usually did provoke a "well screw YOU!" response or ingrain "I must really be stupid" mindset.
I don't remember exactly, but it was the most advanced they offered. This was in Virginia, in 98-99. My teacher (Mr. Norris) was excellent, but it was clear that the overall course was something that I had done years before back in Albania. Since I knew a lot of things already, I was probably a bit annoying at class (having always the response), but I made some good friends in that class.
This is interesting. I studied physics at Harvard with many of the best of the best, there were a few from Romania.
I would be surprised if those people could be solving AP Physics C problems by 8th grade. 1 & 2, easily of course, but C is quite different.
If you found calculus BC hard-ish and physics easy in comparison, I think you were taking 1 & 2 as C was much harder. SAT II physics is a joke.
Virginia probably has some of the best STEM education in the entire country, at least the northern bit. Certainly better than what you can get at most schools outside of some in California, the boarding schools, and NYC magnets.
Not op, but I remember what got me into sciences was the “kids soviet encyclopedia”. Specifically the physics section.
We had “big soviet encyclopedia” and “kid” one on our bookshelf (A point of pride for my grandparents). Nobody told me that by “kids” the authors meant “teens”, so around 13yo I devoured those books whole.
Taught me structure of the atom, chemistry and lots of fascinating physical phenomena. And made most of the physics material easy to grasp up till last HS grade.
It had little maths, concentrating more on the understanding rather than the rigorous descriptions, which made the maths to describe them quite obvious, when I had to learn them in school.
Not sure if the current state of wikipedia is better or worse for that purpose- it was much better structured and paced for sure.
Why is dumbing down science teaching the fault of extreme progressive left and not, say, reduced education funding, favouring religious teaching, or even more likely, nuanced policy difficulties that don't fit nicely into a left/right divide?
Inclusive classrooms raise scores of the lower performing students. That's probably true.
It glosses over what it does to people on the other end of the spectrum.
As someone from the other end of the spectrum, growing up never having to put any effort, having few or no peers close to my level, and being constantly encouraged by the people around me to do poorly for curves etc... I can say I developed life long development/psychological problems as a result, and the idea that I or anyone in the future like me should be slowed down so everyone is more equal is deeply offensive.
I started in an average class in my year. Don't remember doing anything extra at home, or much at all, but still had an equivalent of all A's. At some point my parents realised something was wrong and moved me to the strongest class... Only then I started doing something. This is when I realised that some kids were already impossibly far ahead (we had world-level olympiad goers there).
Can't say I become world-class at anything but this helped to concentrate and work through the rest of school and uni without any problems.
I am sorry you had that experience. Mine was similar (around 1990s).
But these days, integrated classrooms (as mentioned in your article) tend to be staffed with two teachers: one with a general education background and one with special education. They work as a team and can provide differentiated learning paths. The lower teacher/student ratio can help both the gen ed and special ed students.
Because in kindergarten I was regularly asked to go hang out by myself as literally every other student in the room learned to read. When they gave me the tests they use to measure themselves I got 99th percentile scores so literally neglecting me got them the metric as doing anything for me.
Yeah, I get the general idea. I, like most HN readers likely were, was in a similar situation.
And as a certified smart guy I now ask myself and others questions like: Why is it deeply offensive to hold you back when you are literally suggesting to hold them back instead?
Because there is a distinct bias against students who understand the material who get ignored in favor of students who don't Having a bit of talent hides what you actually need which tends to get ignored because you're better at figuring things out.
I have a smart, popular nephew from a good home, who also got extra acconodations on exams for dyslexia.
You could spin that either way:
Priviliged people taking advantage of accomodations for those with problems.
Or smart people who previously got written off as disruptive troublemakers being better catered for (though maybe we'll have less entrepreneurs if we keep all the smart people in the standard education track).
In particular, the benefits of living in a society with a basic level of mathmatical (etc.) understanding seems powerful but hard to trace through.
But if we can't even talk about the trade-offs calmly and factually then its just a pointless sgouting match. The people with the academic skills should be leading the way on that.
The evidence does not suggest measurable harm for those on the other side of the spectrum from inclusive classrooms. Maybe there is some harm, but if so we haven't been able to devise a test or experiment that can find it.
I'm downvoted, but you can go look at the literature. Jump discontinuity studies for placement, testing comparison, etc. - none of them show the effect that most people on HN want them to show.
Don’t know about Albania, but in Bulgaria the teachers had very low pay, (so were / are nurses btw). I think the difference was that they were very respected.
When I managed to attend teacher parent meetings I could see the deference people had for their children’s teachers. And children also deeply respected them for the most part.
The scenes from the beginning of Breaking Bad for example are rather alien to me - I can understand them, but I’ve never encountered such things when I was a kid.
ardit33 would know and having said "A doctor and a nurse probably lived in the same apartment, and had similar wages." I question both the idea that teachers were highly paid and that there was much status to be had in general in Albania.
> But Albania can finance rigorous teaching of science
> USA spends a lot more money per pupil than any former Eastern Bloc nation. But you can spend a lot of money on inefficient solutions.
You're not wrong. Financialization of education continuing down from secondar to primary education, commodification of students—I think those are a bit better descriptions of the issue beyond "funding cuts" personally. Its incredibly easy to make things inefficient (or efficient at something else [0]) when quality/efficiency slowly fades from the conciousness of those in charge.
I'd say the more important thing is that there are large failures here, they were building before
"Extreme progressive Liberals" (which depending on how I read that I can agree with, from the left even). Saying this as someone who was a STEM kid in high school, took AP Calc at 15, etc.—I'm more mixed on the more static curriculum as described earlier, its not as important here tho
Writer Erika Hayasaki visited Cajon High. Here’s what she found:
A dozen students sat clustered at work tables inside an air-conditioned classroom, which was designed to emulate the inside of an Amazon facility. On one wall, Amazon’s giant logo grinned across a yellow and green banner. The words “CUSTOMER OBSESSION” and “DELIVER RESULTS” were painted against a corporate-style yellow backdrop. On a whiteboard, a teacher had written the words “Logistics Final Project,” and the lesson of the day was on Amazon’s “14 Leadership Principles.” Each teenager wore a company golf shirt emblazoned with the Amazon logo.
...
A public high-school classroom designed to resemble an Amazon facility, with students wearing Amazon logos on their clothing as they memorize Amazon’s leadership principles (which, it is worth noting, also include “Ownership” and “Think Big,” injunctions that hold merit for readers of this magazine when imagining how we might solve the problems exemplified by Amazon). Such a relationship between the company and public goods like a high school is part of what it means to consider Amazon as “the major working-class space of suburban and exurban socialization.”
In some places in the US it may be a funding issue, but in most places it’s not.
Take Baltimore for example. The public schools are very well-funded by any standard. And yet they have several high schools that didn’t produce a single student proficient in math.
Take New York City (which I'm familiar with, but it's likely the story is similar in many other places):
- each public school has the same budget per student. There is no "reduced education funding" for schools in district X vs district Y, or for any school, because the budget per student is equal. Each school's budget is public, and published on the web.
- favoring religious teaching: at least at the school my kids go to there's absolutely no religious teaching. I suspect this is the case for all the public schools in NYC
- nuanced policy difficulties? Not sure what you mean by this
Ok, now to answer your original question: why the fault of progressive left. As you see I skipped the word "extreme", as this is a loaded word; people who vote that way certainly do not perceive themselves as extreme.
At the recent mayoral primaries, the top candidates were the moderates Eric Adams and Kathryn Garcia (1st and 2nd place), and the progressive Maya Wiley (3rd place). As far as the education was concerned, the hot topic was the selective "specialized high schools" [1], where the admission is test-based, according to a state law passed about 50 years ago. The current mayor (Bill de Blasio, progressive) has worked tirelessly for the last 4 years or so to lobby the State legislature to get the law repealed. The progressive candidate, Maya Wiley, promised to work towards the same goal. Here's the relevant extract from her platform [2]
>> To remove barriers that separate and label children, we will: [...] Eliminate discriminatory admissions “screens.”
In other words, kill the specialized high schools program.
Eric Adams (the winner, moderate) promised to keep the program as it currently is.
In case you wonder what the admission entrance exam is like, it's just two separate tests, one for ELA and one for math. Both tests are fairly challenging. The progressives think that these tests (and especially the math one) are discriminatory in nature.
Here's a sample math question [3, p.78] "In Centerville, 45% of the population is female, and 60% of the population commutes to work daily. Of the total Centerville population, 21% are females who commute to work daily. What percentage of the total Centerville population are males who do not commute to work daily?"
This is supposed to be a "challenging" problem? It seems quite obvious to me: we simply cannot answer the question as given, because we aren't told how many in the Centerville population are classed as 'both male and female', or 'neither male nor female'.
Thank you for such a detailed comment. Maybe then I'm just missing the American context. Having lived in the UK and Australia (which have their own lion's share of education issues), to me it's obvious that students struggling should get support, and students who excel should be able to go further, and I don't think anyone imagines a system where everyone is always in one classroom with 100% the same learning.
I'm very against the idea that struggling students should just be "sacrificed" for the advanced student, however, which some of the other comments seem to be implying.
I believe you need it to figure out the total population of females who do not commute, from which you can then subtract from the total population (male and female) who do not commute to find the answer.
Also out of curiosity, from where do you draw the connection that the lackluster schooling is caused by "extreme progressive liberals"? Maybe I'm just not well versed enough in politics but it appears to jump straight to that conclusion following your personal experience.
Recent trend to dumb down math and calling 'racist'. Unfortunately it will be immigrants (like I was), that will suffer the most. Most immigrants are poor, and don't have the connections and money to go to private tutoring route and rely heavily on public education.
At least you can homeschool your kids. It takes a lot of effort but I don't think money is a big issue here unless of course you are struggling with basic income then it's best to focus on income first.
>Extreme progressive Liberals are killing science and progress in this country,
There is literally a major political party in the US that parrots science to be fake & that religion/faith in god is all that matters. Also, that "progress" (in a technological/scientific sense) is a bad thing
Last I checked, it was not the progressives that identify with this line of thinking.
The mainstream republican party does not say any of those things. You are taking fringe positions and making them emblematic of the party.
On the other hand, the democratic party openly and literally claims there is no distinction between male and female.
There are dumb policy positions all over the board. No one denies science as a whole. Everyone picks and chooses.
Consider my state of Oregon... they're literally removing the ability to do math and read as graduation requirements and the GOP minority is left asking 'why'? How can you honestly make the claim the democratic party is uniquely the party of science
> It is bound to hurt poorer but smart kids, that can't afford private tutoring and have to rely only on public schools
Why should I care about them? The internet exists. Let them learn on their own if they’re so interested. Why am I being forced to subsidize people that will just grow to resent me as a leech on society due to my “inferior mathematical ability” as you surely do?
Poverty is a far bigger problem than some Virginia schools not teaching Geometry in 8th grade (for the record - I took Algebra in 8th grade and most people in my tiny high school took it in 9th).
We should obviously care about improving education, for the good of society as a whole, for innovation, for pushing the economy forward, etc. It's dumbfounding that this would even be asked, I'm not sure anyone outside of America (i.e. no pervasive anti-intellectual culture) could conceive of a question like this
> Why am I being forced to subsidize people that will just grow to resent me as a leech on society due to my “inferior mathematical ability” as you surely do?
Why is American society so centered around appearances and perceived judgment? You'd rather damn an entire state to stagnation than risk being looked down upon by some hypothetical elitist? And math is only one of many, many subjects.. I firmly believe everyone is good at something.
> Poverty is a far bigger problem than some Virginia schools not teaching Geometry in 8th grade
And how is terrible education supposed to help with poverty? Education is a great way, arguably the primary way, that people lift themselves up.
> You'd rather damn an entire state to stagnation than risk being looked down upon by some hypothetical elitist?
Also, for the record - it's not hypothetical, you're literally an example of this hypothetical elitist as an Ivy grad (top schools only exist in order to enforce segregation against lesser people like me). I'm sure you took calculus in 8th grade while I took it in senior year but I'm not subhuman because of it.
I’m sorry, could you give some context to non-Americans? What does “taking geometry” mean? My school had geometry as a subject in grades 7-11 (with 11 being the final year), were you expected to pack it all in one? How does that work?
I'd interpret the phrase "taking geometry" in this context to refer to the quality and depth of math education available, and to a lesser degree how wealthy the school district is.
Given typical HS math curriculum (algebra/geometry up to calculus) being able to offer geometry early gives enough time to offer calculus; otherwise a school could teach it late and it won't matter because math education stops early anyway.
Also connotes to a degree college-track vs non college-track students, again due to the highest math a HS offers. It is extremely desirable to take calculus in HS to prepare for college, but if most students don't go to college, no need to take/offer calculus, no need to start the math sequence early, etc.
I went to HS in the US and I remember a split in the math curriculum starting in 10th grade (I didn't go to my HS in 9th grade so I can't say). The vast majority of kids planning on college took a combined algebra 2/trigonometry class in 10th grade; the other kids took either algebra 1, geometry, or algebra 2 (without the trig) depending on previous courses.
So the comment "taking geometry in 8th vs 9th vs 10th grade" would mean something like comparing schools for numbers of college-track students and funding (8th grade great - implies a HS with a large body of students going to college, well funded in order to offer variety of classes; 10th grade not good - implies a HS with a small body of students going to college; not well funded, math stops before calculus, etc.)
Very funny, no one will resent you as a "leech on society due to" your “inferior mathematical ability.” However, if you actually choose to literally leech on society (e.g. choose to be unemployed and uneducated) then people will. Anyway, education exists to educate. If you care about social welfare, talk to the social welfare programs.
They absolutely do - they most likely consider people like me to be subhuman entities. It’s not like I have the mental ability to get into the fancy schools or fancy companies they got into.
It is not Russian but 'Soviets math'. Ironically, Russia introduced analogy of USA's standardized right after Itina emigrated from Russia and now Russian's math is also optimized for memorizing but not for 'emphasizing reasoning and deeper understanding'
I graduated before USE, and my math classes were... mediocre. To graduate, you had to do 10 problems of average difficulty, so our textbooks didn't even have problems harder than that.
I took a look at the entrance exam at the uni I wanted to apply to and was shocked. Thankfully, I had my dad and he went through Skanavi's exercise book with me.
I look at the USE math exam every year and it's much better than my final exam (although I like gaokao more, it has more varied problems that make you combine different areas of math), but I don't know where the cutoff point for "I won't get into trouble for my students' low results" is.
That's a common criticism that mostly relies on emotions and not facts.
These standardized tests are changed little by little every year and are simply meant to a) ensure similar educational standards for smaller and remote cities b) enable kids to apply to any university in Russia.
Although some specific parents and teachers in particular school might want to focus on repetition of the same problems and tasks it doesn't mean everyone will and it certainly didn't affect me that much. In fact, having some definitive rules on how they assess an essay in Russian helped me get 100% for it the first time, since it was objective.
we are one of those parents (in Bay Area) who send our kid to RSM (Russian School of Math). We are very happy with it so far (6yo completed her first year) and, most importantly, our daughter is very happy with it too. It is not simple rule memorization and counting, even for 6yo the problems they come up with are interesting enough so that the kid wants to solve them.
What does the curriculum look like that differs significantly from standard school? Also, is your child bored in school at all? I got ahead in math early on and remember being very bored in math classes until AP classes came around.
After school programs have been a profitable enterprise in South Korea and China. I think they're still in the growth-phase in the United States with companies like Kumon, Eye Level, and Russian School of Mathematics. Competition to get kids to accel in school is high but it's also increasing the stress of these kids. Parents in South Korea, for example, have stated in numerous survey responses that they would delay having kids (among many things) because of the cost of after-school tuition.
It's a shame because education is viewed as that one normalizer which allows a child from a poor family to make it up to the top through hard work. Wealthy parents are simply gaming the system by putting kids through cram-schools and SAT programs which train you how to read between the lines and fill out Scantrons effectively.
Private tutoring is nothing new but now you're seeing tutoring becoming like a Subway's or a McDonalds franchise.
I think German math education was ok, probably not stellar, but good enough to prepare you well for engineering classes.
I went on another (short) exchange to France later in 9th grade and they were still far ahead of the German system in math at that point.
However, and this may sound stereotypical, their language education was pretty bad. While they were studying similar English literature as us, they were almost completely unable to speak English. When visiting language classes it was evident why, in Germany languages are studied interactively, in France it was only the teacher talking.
I don't know if this changed since then or it was only at this school, but it seemed like such an easy fix.
In terms of workload, I think the total was pretty similar.
I graduated from a tony American private high school, and went on to graduate from an ivy league college.
The last math I learned was basic trig in eleventh grade.
Somehow I was allowed, encouraged even, to avoid math. I never had to say the word math in college or graduate school.
As a result I do not actually know what calculus is, and while I’m sure you don’t invoke it while calculating a tip, I often struggle with that exercise.
Regrettably. I’m sure my predicament is not unique.
My favorite example is compound interest. Imagine having to solve a compound interest test to get a credit card. It seems like a basic precaution to me (like a driving license), but yet I’m not convinced all in the US can do it.
Chinese here. We followed the Russian/Soviet system of math education for quite a while. The Soviet textbooks were and are still considered superior comparing to our own (from middle school straight to university). They have a certain quality that Chinese textbooks fail to grasp.
Nowadays many schools turn to the American system (I don't know why) and the requirements for math has been dropping for a decade. The inequality of teaching resources is obvious when you compare a student from a privileged school with one from say a country-side school. The government tries to equalize things but it's very difficult to go against the top teachers and rich dads/moms.
I'm Russian and one thing I liked when I was a kid was the monthly physics & math magazine "Quantum" (Russian: "Квант", "Kvant") that was aimed at schoolchildren. Are there any such magazines in United States?
I'm from Russia, and in the US I attended GCPM - a "correspondence program" with math assignments sent to me by mail and later graded by a university professor. Was a great learning experience (the program had dedicated books and interesting math problems to explore).
It is written by a mathematician who has taught in Russia, the US and Brazil. They have a lot to say about how math is taught in the US. The paper also has a lot of sample problems.
I am from India. The school I went to (70-80s) wasn't the best, but I had an innate aptitude for math
(perhaps genetic). I was lucky to get introduced to some Russian math texts [1][2] from my 3rd/4th standards which helped me (alongwith some local texts) to get good scholarships and rankings in Maths and science
competitions (including a national level Math olympiad) and a later admission to a top level engg college in IIT
(again thanks to accessing good libraries for foreign authored books). I see the Russian books as having provided
me with a good foundation by being fun and interesting, informative, affordable, immersive and engaging and it
really fuelled my interest in math and science. Given the resurgence of the Russian school of math in the US I
wish I could be there to educate my children in that tradition. But not being there (out of choice and circumstance,
I studied in Boston for a while) I really wish something like this was available in India.
So my question is - is there some sort of a curriculum and associated training material (books, texts) available based on the Russian way
of teaching, on which to develop a training plan
locally in a country like India? Given the plethora of online school education options available today I do not
know what exact training methods are used there, but platforms exist to have a broader reach for teachers to find interested
students. I presume it would not be too difficult to setup a curriculum and training outside the
normal school one with the explicit intention of developing strong math skills based on a Russian math education
base if one wants to teach.
[1] Mainly by Mir publishers, some of which are thankfully still available online @mirtitles.org
[2] the ones I got were at throwaway prices, titles such as Yakov Perelman's Fun with Math, some
Little Mathematics Library books physics, chemistry, cybernetics etc.
Strangely enough, I'm an American who was taught Singapore math in elementary school.
Then my family moved to a nearby city where they taught standard American math and I wasn't allowed to solve math problems the Singapore way. Even though I got the correct answers, the teachers insisted I do math the "right" way, which I consistently messed up for the rest of my life.
Singapore math not only has a more reasonable pace to learning, but as someone with inattentive ADHD I found its approach to arithmetic easier for me to keep track of in my head.
Even if people have heard of Singapore math, they might not know that things like addition are done left-to-right rather than right-to-left.
I don't know about anyone else, but that is more like how I do math in my head on a day to day basis since the first step gets you closer to the answer. Just doing more addition underneath is also more visually clean than carrying numbers by writing them above the original equation. Paper is plentiful, and now I'm sure it could all be done digitally, so there's no reason to use standard American math to save space.
In my experience and opinion about studying till High school and then University here in Ontario, Canada:
- Canada seems to be obsessed to maintain high-stats. when it comes to 'literacy rate' - that is why till Grade 12, education is intentionally dumbed-down to the point any kid could just do bare minimum and still pass. Even if the kid is dumb-as-bricks, they can choose to do watered down versions of maths, physics, chemistry and still complete their High School Diploma requirements.
- However, as soon as you enroll into STEM program at University, it is on-par in terms of difficulty with their counterparts elsewhere. What was a easy-peasy style of mathematics taught in Canadian High School makes way to the old 'no-calculator and Professors don't help' style engineering calculus and maths.
This is where I found students who had studied even in 3rd World countries like Pakistan and Eritrea (I kid you not) had advantage in math and science courses throughout their degree program. Heck, it took me few tries to get into the groove but in process wasted 1000s of dollars and couple years trying to retake the courses.
The severe downside is that if your kid has above-average intelligence (as it was in my case) and if they join the Canadian education system at young age (in my case at age 13), by the time majority of these kids become adults, majority of them (as in my case) permanently loose their spark and thus get destined to think only inside the box.
I don't want to rant but another thing I notice is the leniency showed by Canadian Education system when it comes to the whole 'culture' in K-12 years. It is hands down meant to destroy bright minds / make them outcasts. The whole toxic culture of labelling those who are intelligent and/or less fashionable as nerds/geeks/dorks and nonsensical encouragement for sports and arts activities ends up alienating majority of smart kids and many just intentionally dumb themselves down to blend in with their peers.
Had the Canadian education system taken leaf from countries like Singapore/India/Pakistan/Iran/Russia/China and actually made efforts to academically grind their students and to promote discipline (with uniforms and academic competitions leading to glory) - Canada would be producing far more intelligent adults.
The current status is: Canada manages to 'import' bright and gifted scientists / university students from all corners of the World and is happy to grant them passports and claim 'Canadians are damn smart' -- what really is smart if you can take army of Canadian children and ensure they are smart-as-heck when they become adults.
I used to think that the K12 math education was really bad, but now I have a different perspective. Many public schools in China and Russia are essentially like magnet schools in the US because students need to compete for entrance. So, those schools can afford more challenging syllabus. In contrast, the US has few such magnet schools, and people of different academic capacity go into the same school. As a result, teachers can't really teach too advanced concepts or assign too hard homework. This may sound far-fetched, but you may be surprised that more than 50% of students may never be able to understand Euclidean geometry, let alone analytical geometry or set theory or probability or proof by induction, and etc. And more than 50% of STEM students in college may not truly understand calculus, let alone analysis. Case in point, roughly 50% of middle school students can't get into high school in China, even though the entrance exams are really not that hard as most of the questions are designed to examine a student's understanding of basic concepts. Therefore, we may look at the quality of the math class in the US and feel miserable, but that's only because the class is tailored to the lower 50%-ile. On the other hand, the clubs and gift programs in public schools are still of high quality -- all the more reason for not canceling such programs in the name of equity.
I am also from Eastern Europe, but did my Bachelor in the west. For a lot of people there things like limits, series etc. were almost like greek. Speaking of greek, we did some group study and there was a person there who kept saying E this E that. I was perplexed until I saw he meant Sigma as in sum.
But those who didn't know learned it and that was that. I'm not sure it's really that beneficial to force a lot on kids before they are ready to use it or know what it's for. I remember 10th/11th grade sitting and eventually realizing all this stuff we do... I will eventually need to do it to make sure e.g. a building does not crash. And that scared me to death, I did not feel prepared at all. When things are thrown at you before you're ready and can really understand it, it's sort of like you lack a connection to what is essential and what you might be able to do with it on your own. You do your little examples and tasks and solve them, but outside of that context you don't really understand it. I don't think any educational system has figured that out for the majority of kids.
Probably best to start with the Bronshtein - Semendyayev: Handbook of Mathematics. It's still widely used in a lot of slavic speaking countries. Google it, there's a pdf available.
I'll echo many of the other comments here -- seems like the Eastern Bloc's math education really out-shone the West's.
I came to Canada mid-way through high school, and breezed through with little to no effort until my 2nd year of university.
I remember math class in elementary school, how everything was explained, and if you were clever enough you could see where the teacher was going with the rest of the story, because everything inherently made sense.
This type of learning instilled in me a deep comfort with math - knowing that you can always break down a problem into a set of familiar problems, that proofs are kind of like nested Russian dolls, and that you can synthesize a solution out of first principles if you're persistent enough.
In grade 8 my teacher basically said they don't do math, gave the students a speech about how you can succeed in life without math, and then declined to teach any math except for handing out a set of math puzzle magazines one time in the middle of the year.
Also the number of people who need advanced math is so small in USA, so it’s more of a hobby than education. And people who need advanced math need more probability theory and statistics than calculus. In China, on the other hand… I might be extrapolating.
I'd be very curious to see what a primary and secondary school math program that aggressively focused on the "what will I ever use this for?" question, for most people, not just future physicists and math majors, and tailored its curriculum to that with mostly applications-based problem sets, looked like. I suspect it'd provide a lot more value to ~99% of students while never providing more than a "here's what these other branches of math you probably won't need are for, so you know what to look up or when to go learn them, on the off chance you ever do need them" survey for much of what's covered from ~8th-12th grade now, but maybe I'm wrong. I also doubt it would discourage many future math majors.
In my view math is not so much about you needing it. It's about developing abstract thinking, reasoning, logic.
It's in the same boat as sports. You will never need to run 100 yards with a ball in your hand, except for the sport itself. Yet we still do sports to train and stay in shape.
It does, but again, those skills are not valued that much in USA. I don’t like it, this is reality.
Sports classes provide nowhere enough physical activity to stay in shape though.
Ideally a country ruled by citizens require a high level of knowledge and reasoning ability. In reality people vote for aristocratic cliques in a way often determined by their birth.
So what are you suggesting? Because math is not the perfect answer, do less critical thinking? I think it's very good for the brain, necessary even. Math is not sufficient, but it's necessary. It's a pity the US schools put less and less into math, nowadays they dumb down math for equity reasons even.
I’d suggest more math as a personal advise, but I have no power outside of relatives and friends.
So I can only observe what’s happening on the national level. And that observation is that math is taught in a way that does little to promote reasoning skills and most people won’t use it. And smart people will find a way to learn, it’s easier than ever.
This got me thinking about the recent school policies for equity. Schools reduce difficulty of courses, lower standards for graduation, and cancel gift programs, all in the name of equity. I think in the long term such policy changes will hurt disadvantaged families. Family with means will simply send their kids to good tutoring schools, and the education gap among different groups of people will enlarge. It is the kids in the middle, the future backbone of our society, who will suffer most because they need strong guidance and skillful push to truly learn, yet they won't get such education because of the watered-down standards.
I'm not quite sure I understand the aversion to Russian teaching materials, other than a lingering negative reaction to communism. Gelfand's Algebra has become a real game-changing text for a lot of English-speaking students who struggle with the subject. His Trigonometry book is just as good, and helped me a lot when I needed a refresher when I became a land surveyor after being out of school for years.
The basic education concepts at ploy here are well known, just not widely deployed.
Why? IMO it's because of what I think is the sole problem with american education - parents themselves not caring if their kids actually know anything. It's actually common, extremely common, for american parents to think that the point of education is to get a piece of paper (degree) or to get a job. And a large number of the parents that think the point is actually knowing things also put no pressure on the school systems to actually provide that.
Is there evidence that knowing how to do math is actually beneficial to most people?
I think it's easy to criticize when ignoring the possibility that much of what is taught in American schools is actually just useless and very little of it is retained after schooling ends.
What I said applies to all topics. Including ones that you personally value.
As for whether math is useful or not... I'll just say I REALLY wish more of my coworkers knew their math. I work in a factory, and it's useful WAY more often than you may think.
Growing up in the US we were fed a "mythology" about Russian and Chinese schooling that claimed it was all rote memorization without actual understanding. It's been clear for a long time that this was nothing more a form of American exceptionalist propaganda. I wonder if it was ever true? I suspect not.
The tragic thing is it now appears that US STEM education (at least for non-elite schools) is closer to the rote-memorization/calculator-bot curriculums than every other school system.
Went to university in the UK, met a bunch of Russian post-docs whilst I was a researcher, and they were some of the best mathematicians I'd met.
Not unrelated, but today in the UK is A-Level results (for 18 yr olds), and the UK press and exam boards, as usual are reporting it as the 'best results ever'. This happens year on year.
There's a lot of argument about education standards getting lower in the UK. My "anecdata" is that I was the first year to sit GCSE maths exams at 16 yrs old, and we went through old O-level papers from the mid 1950s onwards (aimed at 16 year olds) to practice. Those O-level papers had advanced calculus that we didn't learn until our final year of A-levels. Those old papers were much harder.
tl;dr in the 50s/60s 16 year old Brits were taught advanced calculus. They're not now.
It depends what metric you're using for education improvement. For science based subjects, the fundamentals don't change. My narrow world view is maths and physics, and my data point of one, is that my peer group in the 1980s were taught to a lower level in maths (and physics) than in the 1950s/60s/70s, based on the content of O-level exam papers that we sat as mock exams. The fact was that we couldn't answer a percentage of the exams because we just weren't taught it.
The reverse wasn't true - we were not taught extra things that weren't in the exam, we were simply taught less.
I can't find a link, but there was talk of "remedial maths" lessons being taught at many universities in the UK to bring students up to the standards required for degrees because they're simply not taught at the same level any more. Universities on the other hand, don't have their curriculums or qualifications manipulated by the sitting government so their standards/requirements change much more slowly.
My experience with math on the Brazilian educational system was that of a shallow obsession with form. I also had great teachers and experiences but overall it was boring and left no space for creativity which at least for me are requirements for personal investment. I expected the soviet system to be heavily bureaucratic but through your PoV it looks COOL!
I wonder if there's an equivalent "hardcore" curriculum for computer science and/or software development and programming vs a softcore "US" curriculum. Would SICP be the hardcore version vs HtDP/PAPL be the softer "US" version?
Anectodal: I signed my daughter to Russian math and due to pandemic she had one semester online. The teacher was horrible and made lots of mistakes. On the other hand, some of our friends were impressed about their teachers. So, I guess it depends on the teacher a lot.
Is there something like this for adults? I've been learning mathematics on my own for a few years, but my progress has been painstakingly slow and erratic.
I would really welcome some kind of schooling aimed at people who, like me, are genuinely interested in the subject.
There’s a limited amount you can do when everyone around you is destitute. I seriously doubt future coal miners were taught sophisticated math in high school in the USSR - they took tracking and specialization to an extreme, whereas this is culturally verboten in America (for good reason).
Soviet schools weren't really specialized. They could have some classes with a heavier focus on specific subjects but all changes were minor.
So yes, coal miners learned the same stuff. Only after grade 9 kids could apply to a professional school that would teach blue-collar professions and that's the first divergence point.
P.S. In Soviet/Russian schools "class" is literally a group of kids who study all subjects together from grade 1 to 11, they are very rigid.
It's the Germans who are extreme specializers, with separate middle and high schools for vocational, professional and academic tracks.
The Union had two optional years of high school, but everyone took the first eight with no tracks or optional classes. After that you could finish school or get three years of vocational education and join the workforce.
In Australia we have a focus on STEM, but school teachers who don't understand the material.
After 2 years of philosophical maths in primary school from a special teacher, high school really killed my interest. Big shame.
A very interesting bit of trivia: Soviet school math curriculum reform was conducted in the 60s under the leadership of Andrey Kolmogorov. Some of you may be familiar with his works, especially the Kolmogorov complexity.
Basically any country that doesn't view your kid's math class as a lab to experiment with new unproven teaching techniques is fair game. Which unfortunately excludes American math.
One notable example would be the undergraduate Beginner’s Course in Topology - Geometric Chapters by Rokhlin et al. (Some “beginner” one must be, to be able to read it.)
I wonder what he older math-education in german schools is like, probably also more hard-core. After elementary school, germany has a three tracks Gymnasium, Realschule und Hauptschule. A Gymnasium prepares one for university and is still well regarded and its final tests, the Abitur, still carries a prestige. Looking back (I attended a Gymnasium), I think the quality of the education was quite high, especially in my english lessons and the social subjects. If the goal of the school was to produce good citizens capable of actively participating in a democracy and forming their own opinion, then it did it very well. Especially later, we had quite a few interesting, adult discussions with engaged teachers about various topics in history, society, art etc. So I think that the quality of my gymnasium was quite high and it really showed in subjects where capable teachers alone can make quite the difference.
But the gymnasiums should also prepare those capable for university, be more rigorous, and I think there's the problem. The division doesn't work anymore. Everybody wants their children to go to the gymnasium, because everybody has to study at the university. Even when they have no interest whatsoever in science and just want to work as a coder. Also, attending a Gymnasium can't be prestigious if everybody is doing it. I remember quite a few children struggling but getting pushed through by their parents because a Realschule is simply not an option. I was also struggling, but more because I just didn't care and there weren't really any consequences. I still passed each class. But a more rigorous math education would, I think, result in a lot of the children failing the gymnasium and a lot of angry parents who see the future of their children and their parental success in turmoil. So the Gymnasium really turned into a one-size-fits-all kind of education and I strongly suspect that especially in math (or physics) that leads to the lowest common denominator. So now I wonder what the math education was like when only few could attend the gymnasium, i suspect way more rigorous.
Is it similar in other countries?
By the way, it's totally different at the university level. The german university doesn't feel responsible for your personal success, you have to earn it. If you fail, you fail and the standard can be quite high. I also don't really see a lowering of standards in the "core" degrees, for the first big math exams I prepared myself by practicing with old ones going back into the 80s/90s. They were as difficult as the new ones. A lot of students failing out of the subjects they have chosen (can be as much as 2/3rds) just choose easier majors, for example a business-computer science combination because they have less math classes. A lot of the students that started studying with me ended up switching majors because of the more rigorous computer-science and math-classes (I think roughly 50%).
I was born in the USSR but was in New York for Jr Highschool and Highschool. Whenever my dad saw the math curriculum he assumed I was in some sort of "slow program" - he could not believe that the mainstream program is that un-challenging.
I bet there's a lot of factors to why curriculum gets watered down but I do think recent focus on how kids feel about school, self-esteem, safe spaces, etc goes against hard-core curricula.
When you learn for real (and this extends to adults as well) you are confronted with things you don't know how to do, and you have to bang against them for a while to solve them and learn. That feeling of "wait, maybe I can't do this / I don't get it" is a negative one. And it does create a situation where some percentage of kids can't hack it, so they are excluded/left behind.
I think as a culture we've been making the choice to teach/do easy things that everyone can participate in, rather than do challenging things that will force some people to grow more while leaving some behind. It's not a choice I agree with but I guess I can follow the zeitgeist and logic of it for a certain extent. It's the same as NYC, SF and other cities getting rid of the specialized schools with entrance exams. Since some people can't hack it, it's an exclusionary approach and therefore getting rid of the exams on one hand equalizes access on the other hand waters down the standards.
I do think there will be unintended but obvious consequences as we're seeing in this story: parents who know better, who want the best for their kids will invest in tutors, private schools, etc that challenge their kids when the mainstream schools do not. The effect will be that within the school system, everyone is doing easy things that everyone can do, but in outcome space there will be a larger gap between kids of parents who care/can afford something beyond public school and those who cannot. In the long run, this will cause greater inequality because there will be come percentage of kids whose parents can't give them a leg up that would have risen to the challenge in a tougher curriculum but now will not have a chance to do so.
In my eye this is unfortunate both because we're creating a less educated populace and a less confident one. The idea of being brave and confident means you take on a challenge knowing you have a good shot of eventually overcoming it. This holds at school and it holds in the work place. I am not sure how many people who never had their ass kicked (for a while) by school work and then had the experience of "getting it" will then come into a work situation and be able to "stretch" by taking on work they don't quite know how to do yet. Not the path I am chasing for my kid.
Yes, in math, in the US, maybe anywhere in the world, there is the good and the bad in programs, teachers, books, exercises, etc. So, let's see how to get around the bad:
I liked math, a lot. The SAT Math test said I had a lot of talent in math. I concentrated on math in US grades 9-12, college, and graduate school, did some math research, that later I published, got a Ph.D. in pure/applied math, and am now using some advanced and some original math as advantages in my startup.
So, I struggled through the good and bad but eventually decided that there were a lot of good math books; it was not very difficult to identify the relatively good authors and books; and the keys to learning math well were a stack of blank paper on a clipboard, a sharp pencil, a big, soft eraser, one or a few good math books in the subject being studied, a lot of good exercises, a comfortable chair, a good light, and a quiet room. That's how I learned nearly all the math I did learn; still if I want to learn some math, that is what I use.
This technique of a quiet room worked for me many times, but once was a nice surprise: The college I went to for my freshman year was selected because I could walk to it and it was cheap. The most advanced math course they would let me in was beneath what I'd already done in high school -- the high school was relatively good (MIT came recruiting; 97% of the students went on to college; one year three students went to Princeton). In my class, in 1-2-3 on the SAT Math, I was #2 and #3 went to MIT. So, I didn't want to fall behind in math so got their calculus book and started studying in a quiet room. This effort worried Mom who would find excuses for me to get up and do something else, but I still did well. For my second year of college, I went to a college with an unusually good math department and started on their second year of calculus using the same text Harvard was using. To let me start on that second year, a prof gave me a little impromptu freshman calculus oral exam. So, with the quiet room technique, I never took freshman calculus -- later taught it, applied it, etc. but never took a course in it!
In math written as theorems and proofs, for a big source of good exercises, guess the next theorem. Check your guess. Given the theorem, close the book and prove the theorem. Doing this let me get the solution to a somewhat challenging Ph.D. qualifying exam question -- I did the best in the class on the qualifying exam. This approach to exercises is good, but it is too difficult, that is, too slow, to use for all the math need to learn.
Beyond that quiet room
approach to learning, I found that to do well in graduate school, e.g., get respect from the professors, the key was, as soon as possible, do some publishable research. E.g., maybe have been pushed into an advanced course. Okay: Find some places the course and/or texts are not very clear, good, precise, complete, whatever, pick one of those, do some research to improve the situation, and publish the research. Remember: For good results, good initial problem selection can help a lot.
For calculus, yes, work through a good text and then, for a nice advantage, learn measure theory then, in particular, learn probability based on measure theory. Then in, e.g., statistics, you will have a gun while nearly everyone else has at most a knife.
But just calculus from a respected text can be powerful stuff. E.g., at
can see Einstein's special relativity done, apparently fully correctly, and where the only math used is ordinary calculus. Some 12 year old students can learn calculus plenty well enough for a lot in applications, including more advanced math.
Here is a special strategy that can work in the US: In US research universities, the math departments typically are in the school of Arts and Sciences. But such universities commonly also have engineering schools! Some of the people who give the big money like engineering more than arts and sciences! And there is the outside world! So, from contact with the outside world, maybe a job, full or part time, pick a problem where a good solution looks promising for some old/new math.
Solve the problem, and publish it in a journal with a title like Journal of Theory and Applications in .... Some journals also like to promise candidate readers that they publish not just theory but actual applications!
So, the usual criteria for publication are that the material be new, correct, and significant. Well, easily enough the solution to the new real problem can be new. Since the solution is mostly math, can pass correct. Get significant from the real problem being significant. If the math saves $10 million a month in jet fuel cost for an airline, call that significant!
For the remark, essentially, need to think before writing, I go along with that for research and challenging exercises.
Note: For challenging exercises, I found that some good research is no more difficult than some such exercises -- the exercises are good preparation for research, etc.
Generally in applying some math, will likely find some places where the old math needs some improvement, at least for the application; so, make some such improvements, and get the significance from that of the problem. That is, for picking a research problem, the Riemann hypothesis is not the only option!
For an example, I picked a problem with the Kuhn-Tucker conditions and found a solution and published it. Later I found that the famous paper in mathematical economics by Arrow, Hurwicz, and Uzawa encountered a similar problem and had no solution. My work also solves their problem. I found this research problem just from some careful, quite careful, study of the Kuhn-Tucker conditions.
So that is a way around the bad in programs, teachers, books, exercises, etc.
Some education systems put an emphasis on critical thinking, some on absorbing tons of information.
Polish education is notable for memorization. One area where it doesn't seem to hurt is medicine. It appears Polish doctors and nurses are very much appreciated when they migrate. Perhaps critical thinking is not a useful skill in medical practice?
Once upon a time I was reading an article about Polish migrants in Norway. The Norwegians observe that Poles are reluctant to send kids to Norwegian schools for a few reasons: a) the language is very different, b) Poles fear children will become rebellious, because the schools emphasize critical thinking* c) Poles fear children will become... idiots. Because they won't know too many facts. It appears despite constant complaining about pointless facts and useless information, Poles take some sick pride in the suffering. Some kind of Stockholm Syndrome.
* this reminds me of a rumor circulating about personnel of mental hospitals. Patients are often sedated not because it's good for the patients, but because it's convenient for the personnel.
No, I'm considering it. I need to travel more and compare before I make up my mind. It's unsurprising many people migrate because they're fed up with their country. Financial incentives is just another side of it, because it's demoralizing to work long and hard hours and be paid little. However, plenty of Poles migrate that can reconcile their desire for high wages with uncritical admiration for their country.
The country has taken an authoritarian turn. The aspects that annoy me personally is lack of critical thinking, lack of insight when it comes to history (it's a second state religion in practice which is two too many), low trust, bigotry, corruption, double standards, shallow XIX century understanding of patriotism. There's focus on "moral victories", heroic sacrifices and losing battles. Contempt is something very common in society - it's like everyone needs someone they can despise. Constructive criticism is very unwelcome and met with denial. Compromise is often called "rotten compromise".
Could you say that being a Pole makes you feel insecure about yourself? Or is it rather that you’re just better than most of people leaving in your country?
"Perhaps critical thinking is not a useful skill in medical practice?"
I was having a discussion about this recently, recalling the vast majority of my experiences with medical doctors who obviously just follow a cook book approach to how they practice medicine. Worse than that is how many doctors cannot correctly explain test results. Seems like although they completed a lot of schooling they did not take enough math.
I was glad to experience both methods and i can definitely conclude, that Russian/Soviet Math approach is much better! It gave me the boost to be ahead my Us and EU classmates, helped to adapt to my statistical class during my Bachelor's degree. Highly recommended for every parent to include several methodologies to their child's extra program.
We're not taught mathematics because we're not meant to think, criticize, understand epistemology, etc. We're taught to compute to make us useful and compliant, even if the computations themselves are useless.
Why do people beat themselves up for not being “as good as they should be” at math? Who cares? Find out what you naturally excel at and concentrate on that.
This notion that “American” math and sciences lag behind the world is one of the greatest lies ever told in my lifetime. And for the record, there is no “Russian” math or “Chinese” math. Mathematics is mathematics. Where the US approach differs is that they are not requiring high level of math on general public. However, if you are gifted and/or motivated, you have access to unparalleled resources.
If you count Khan Academy and MIT's OCW, sure the Anglo-sphere has unparalleled resources. But a minor point of the article is how the way Americans are taught in conventional public school is just to do the basic work. Math is treated as a hurdle to clear rather than the lifelong mental sport.
But here's a point I would have liked examined further in the article: where does one find material or instruction that would give you a competitive structural thinking needed to grapple Olympiad or Putnam-level content beyond enrichment courses? In China, the best students are scouted early by teachers and prepared by the Chinese government through special camps and their own difficult examinations. I'd assume the same thing happened/happens in Russia. In the United States, by default (and probably by design), its difficult to know on what level a student can stand in regards to everyone else until the very end. At least in a public school.
Parents and students can use proxies like how early AP subjects are offered, USA today rankings, or the success of past alumni. But, unless you're in a well-endowed private schools, magnet/exam schools, or private enrichment program like AoPS (Art of Problem solving) or Talent Identification Programs (many of which are now shutting down) there are few ways to know how well the instruction one receives as a precocious and motivated high school compare to those of the best among the nation or the world.
One could argue if you have to wait until high school to understand where you stand, it's too late. But that's my point, many students may have resources and talent but the lack the expertise to deciding where to and how begin in those crucial first years. Few states have strong "gifted" programs and the one's that do aren't really any better than "normal" instruction in China or Japan.
Those programs produced Nobel prize-winning scientists, Field prize-winning mathematicians, computational whiz kids, and CEOs of billion dollar companies, etc. I'm sure you've heard of Perelman.
But even the non-Perelmans - the "regular" students - who didn't rise to the peaks of intellectual rigor still have a stronger background in today's knowledge economy then most of their contemporaries stateside.
As far as their children are concerned, the Russians and Chinese send their best 18 year-olds to Ivies, Ivy-likes, and Oxbridge just as they send their best 18 year-olds to Moscow State and Peking. Many of them prefer an American degree for the business opportunities, better income, and the prestige of working for an American firm. A more relevant question to ask is how many would send their best 14 year-olds to the average public high school in the US, or the average comprehensive school in the UK, as full-time students for at least 3 years?
Everyone understands that Russian math means Russian approach to math. And American approach does not result in general population knowing enough math. And this is why American math is lagging behind.
The US has resources available if you know where to look, but children obviously don't, so it comes down to luck with parents (or in rarer cases a very good teacher). Perhaps the top Olympiad level types get "recruited". But speaking from experience, kids that are in the high 90 percentiles who enjoy math at a young age frequently end up bored in a typical classroom and eventually disillusioned with the subject - not provided additional resources.
By high school one can start seeking out their own resources, but motivation to do math specifically is harder to find when you've considered it a boring subject for 5+ years. And there is definitely an advantage to being introduced certain concepts at an elementary school age.
So the US approach absolutely differs in how it handles early math education for the vast majority of students, not just for the average student. I wish I learned math in the Russian style as a child, and moreover I think US math has this problem moreso than even other subjects here. Humanities classes generally have less mindless repetition and early childhood teachers could easily recommend advanced books in a way they really couldn't for math.
In many schools the same teacher will teach every subject until you hit middle school! So it is no surprise they don't have resources to give to advanced students, as a lot of them dread math themselves.
It is hard to believe any American feels the need to supplement the existing in-school lessons with after-school paid programs like Russian math or Kumon.
In the last 10 years nearly every school has adopted the Common Core curriculum, which is the product of the latest findings in educational methodology research, developed by Ed.D. luminaries like Dr. Jill Biden, and promoted by successful industrialists like Bill Gates.
I was lucky to get an education in three systems (Soviet Math, Romanian Math School (influenced by both French and Soviet Math school), and finally in a world top 40 university in North America.
I would summarize the Soviet/Russian Math/Physics approach like this:
That being said, I did like some of the aspects from the so called "Western Math" (in my case Canadian university):