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.
What does "Russian Math" look like?
What does "Russian Math" look like?
"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?"
[1]https://f.hubspotusercontent30.net/hubfs/981338/Blog/Element...
[2]https://f.hubspotusercontent30.net/hubfs/981338/Blog/Middle_...
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 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.
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.
A good review article on the subject: Persson, Ulf and Toom, André: Word Problems in Russian Mathematical Education, available at http://toomandre.com/my-articles/swedish/ULFENG.PDF
If you see the trick, you save some time on the calculation. If you don't you have to add some more numbers. It's not that difficult.
This trick discussion reminds me of the great anectode about von Neumann:https://news.ycombinator.com/item?id=5950755
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.
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.
#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.
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.
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.
If you go off-road and get the wrong answer, it's up to how much the teacher likes you.
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.
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.
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.
Coffee concentration before first,second,third,fourth drinking = 1,x,y,z
x = 1/2
y = x*2/3 + 0*1/3 = 1/3
z = y*5/6 + 0*1/6 = 5/18
Coffee drunk = 1/2 + 1/3 x + 1/6 y + z
= 1/2 + 1/6 + 1/18 + 5/18
= 3/6 + 1/6 + 2/6
= 1
Milk drunk = 0 + 1/3(1-x) + 1/6(1-y) + (1-z)
= 1/6 + 2/18 + 13/18
= 1/6 + 5/6
= 1
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 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.
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.
If that's the difference, why were the goals so different there, and how do we make US public school goals more like the Soviet ones?
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.
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.
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.
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.
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.
So maybe this is a US problem.
EDIT: a few example books added
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.
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.