Math Problems for children from 5 to 15 (2004) [pdf]

I heard about Singaporean math a while ago and looked up some youtube videos. It seemed like they showed clever ways to solve highly stylized problems, but nothing that would actually ever come up in the real world. To be fair, I only watched 3 videos, but they were all from different channels, so I assumed they were a decent sample of what Singaporean math is about.

[0] https://shop.singaporemath.com/index.php/product-category/su...

That happened to my little brother. He was reading a lot of books before he started school, but somehow un-learned it while being there. "I can't read that because we haven't learned about the letter K, yet" was something you could hear him say.

For me it's about losing the mindset that children of certain ages are incapable of doing certain things and deliberately holding them back (with nothing but the best intentions I'm sure).

I have a friend who is a published poet and prosaic genius. But she literally cannot solve 2x=4. I'm not being hyperbolic.

Pushing her in math as a child would probably have been catastrophic for her emotional well being. But limiting her in other skill sets (like literature) would be equally catastrophic in terms of wasted potential (and the well being that comes with excelling at something you care about).

I was taught math, and was a C student through high school. I relearned everything I didn't know in high school, in one semester at a community college.

In math, you know the answer, or you don't understand how to get there. Math should be pass/fail. It's different than the other subjects.

I don't believe most of my grade/middle school American teachers (mine) truely understood what they were teaching.

This seems to be uniquely American problem. Somehow in the US it's culturally acceptable and even normal to be bad at math. Elsewhere it's just another subject. You study it and get better. The expectation is that everyone in a normal school (not special needs) can learn the standard math curriculum.

My older son had trouble concentrating in the beginning (still does to some extent).

I'd say starting off with overly challenging material is probably going to be counterproductive if you're also simultaneously trying to establish a routine. My daughter started with "draw 5 beans" sort of exercises, and seeing her older brother comply with a routine it was much easier to get her to finish her studies in a timely fashion.

"Helping" too much can also be counter productive. The kid may end up expecting you to be there all the time, when really, half of the point is to develop some self sufficiency.

The feedback cycle structure may also be messed up. It may be that the kid is stuck in a vicious cycle of negative feedback (e.g. "Damn about time you finished your math! Why'd take it so long!"). WRT the anecdotes above, I doubt yelling more will yield different results.

For my kids, math oriented games provided a very different feedback cycle structure than study time (getting answers correct is literally gamified to look like rewards), and this is motivation enough for them to furiously scribble calculations on top of doodles they had carefully colored previously. It also provided a different dynamic where we can casually praise them about their in-game progress, rather than being a strictly a "boring school conversation". Another example: we started to do Monopoly game nights as a pretext to sneak in math into playtime and my son got quite into making sure people got the correct amount of change from the bank. IMHO, incorporating more positivity (real, appropriate positivity) into daily life is important.

Another more subtle and difficult to address problem is general outlook on education. I've heard, for example, my kid's teacher say things to the effect of "oof, it's monday", as if school is a chore. I've also noticed north american media also tends to portray education negatively (e.g. the nerd stereotype, ferris bueller-like tropes, etc). This is very very different from east asian culture, where the general default is that education is very important. I don't know how to fix this, other than try not to engage in negative behavior yourself.

My experience in the US is that the vast majority of math teachers, especially in primary school, don't understand math in the slightest and are abysmal at teaching math outside of rote memorization.

I knew a child that was learning about negative numbers and understood the role of primes in building the number line in 1st/2nd grade. They were learning from pure interest, but were excited by what negative numbers were about conceptually and found primes fascinating. These are the foundations of real mathematical thinking.

Seeing the student was advanced the school put the kid in a 4th grade math classes but then complained that the child didn't know the multiplication tables. Understanding multiplication tables is literally memorization, this child had never been given the task of memorizing them so couldn't possibly have memorized them. Memorizing tables says literally nothing about mathematical proficiency, whereas gaining the intuition that "subtracting a negative number is the same as adding it" requires mathematical reasoning. The teacher was unable to see this because they themselves had no notion that understanding things like inverse function in math are important tools for reasoning. The child was removed from that math class, and quickly started to see mathematics in school as uninteresting.

This is just one example, but I've ran across plenty of curious students where a math professor would be impressed but a 4th grade teacher would find them falling behind. My experience working with adults has been that most adults who think they are bad math, are more often than not ones that are getting caught up on issues with math that are good issues to have if you understand what's going on. People with a solid mathematical intuition will be confused by the rote mechanical explanations regurgitated by most elementary school teachers.

There are far more teachers that struggle at teaching math than students that struggle with learning it, but it's far easier to blame students. It's no wonder that many students grow to hate math in the US, because it feels they are being unfairly punished and they are.

You can't completely blame teachers either since the pay and respect teachers get in the US means that anyone who can do basic math will find a much better paying and rewarding job else where. In many Asian countries teachers are respected, and there is a possibility that you can attract people that understand the subject well enough to teach it.

Math must be mastered in sequence. Not learned, mastered.

For example, a child must master addition before moving on to multiplication. Not just be able to do the problems. I mean be able to do them instantly without thinking. Only then can you move forward.

Otherwise the child is hung up on a part of the lesson that’s not supposed to take any time at all. And that least to frustration. Combined with the perverted modern western idea that it’s okay to not be able to learn math, it’s a vicious cycle downward.

People who have trouble with that problem have no guts.

That sounds bad, but most of them return you to earlier states.

‘Drawing’ a transition graph until you hit a solution in your head can be done in less than a minute. On paper, it shouldn’t take more than 2.

[0]: https://youtu.be/2vdF6NASMiE

    5 - 3 = 2.
    3 - 2 = 1.

If your kid is studying 16 hours per day, bored, you're gonna get burnout.

Burnout isn't about challenge -- which is good. If you have dinner table conversations about hard topics, that isn't leading to burnout.

It's true that you forget without regular usage, but it seems the "concept" sticks around, and all you need is some refresher to be able to access it again.

The most complex was when I used some trig to calculate the angle at which I had to wrap a square column with christmas lights to ensure I covered the column from top to bottom with a single string and no excess.

For finance and stuff like that I don't even bother trying and just use calculators.

I took several years of Latin in both high school and college but outside of those academic environments I never had cause to use it and while I remember a lot of aspects of it structurally, my Latin vocabulary is almost all gone. I have at times pulled out my old textbooks just to try and see what I can do, and I can certainly work through that material a lot faster than the first time around, but I’m still needing to start at a rudimentary level to get anywhere.

1) Math teachers loves to gave out their own shortcuts, I mean they will tell us to use it every chance they gets. Then in next mathematics level, they warned that method is old and shouldn't be using it at all. Then the new teacher taught their own shortcuts. This method made it difficult to solve problems because some of the formula wasn't taught how to properly solve without shortcuts. 2) "Why? How?", lots of mathematics teachers during my education times have struggled to give out the explanation of how it get to that answer and why it is that answer. Their response is simply just nodding and "That is how I taught, so it is the answer".

It is hard for me to be able to solve mathematics because I can't conceptualize it well and struggled a lot without using technologies to help me. I do love math, I just can't enjoy math because of my past teachers have failed to educate me. And I failed myself.

I know adults from the countries that are supposed to have wonderful math education (high test scores), and they forget their math too.

I think the people who remain good at math in adulthood were the ones who developed a genuine interest in math as an end unto itself, and figured out a way to keep up with it after college.

As for being presented as "the most important thing" - well for students it is one of the most important things at that time in their lives because it opens so many career paths.

But once you are out of school and on a career path that doesn't require math (or requires just certain subset of math) it really isn't important anymore.

This is just like music. If you hope of become a professional musician mastering your instrument and music theory is pretty much the most important thing it the world for you. But if you end up becoming a programmer and don't play for 20 years - you can't pick it up and play without a lot of practice and catch up - and nobody is surprised by that.

We need to teach math a little more like we teach music.

Oftentimes math is taught as a set of rules. Do these steps in order to get the answer. Works well to pass the test with minimum effort, does not help much long term.

I quickly figured out that even if I've deeply spent time with a subject, understanding every step and derivation of some equation, if I can just quickly pop up equations (and other facts) in my head to "look" at them, it not only helps with application, but also with further understanding.

Being able to quickly recite the Taylor Series or an Inverse Fourier Transform in my head to apply in a problem beats stuff like "oh I remember understanding how it was derived, but I'd need to look it up", because all the details I otherwise once understood but did not bother memorizing might be important.

I think this is not perfect. Education is more of "progressing towards lesser and lesser lies", and changing definitions is an important part. The student might face it when he'll wonder about equation sin x = 2 , which will get to complex numbers.

Similarly, here getting a one less power of x might correspond to "divide by x". But might sometimes not - choosing that it actually does correspond to "divide by x" is a choice. Often obvious, but sometimes not - which is seen in Gelfand's explanation of why "negative multiplied by negative makes positive", or similarly, why 0^0 is 1.

Just saying that "x to one lesser power is the same divided by x" can also be seen as a convention (e.g. for some objects division can be not defined). And if it's a convention, not universal truth... then to somebody who's studying the subject this convention should be justified.

The answer is supposed to be 4mm; the only way for that to work if volume 1 is on the left is for the bookworm to gnaw its way out of the book from v.1.p.1, cross the outside of the two books without gnawing anything until it reaches the back cover of volume 2, and then gnaw its way through that cover to reach the final page of volume 2.

I don't think that's what the question has in mind. The point of being a bookworm is that you don't leave the book. So the answer would appear to require that volume 2 is shelved in front of volume 1. I don't know why that would be the case.

Does it? Did I fail at problem 2? I got:

Bottle + cork = 10

Bottle = 9cork

bottle/9 = cork

9(bottle + bottle/9) = 9(10)

9 bottle + bottle = 90

10 bottle = 90

bottle = 9

9 = 9cork

1 = cork

Similarly, in problem #2 the cork indeed costs 0.5 kopecks but in this case we're just thinking about cost conceptually, not in terms of how much money a person actually has on hand.

Reasonable assumptions, but relying on implicit knowledge nonetheless.

Since Misha is missing only one kopeck, had Masha owned any amount, the sum would have been enough to buy the book.

Therefore, Misha doesn't have any money, and the price book is what Misha is missing : 7 kopecks.

Is there some error here? I read it as "even then they did not have enough individually"

I used b-7+b-1=b to arrive at b=8... My math skills are pretty awful so I can't say if this even makes sense..

https://en.wikipedia.org/wiki/Heritage_language

https://en.wikipedia.org/wiki/Heritage_language_learning

[1] https://www.factorio.com/

(You are of by a factor of 2)

Try to draw a triangle such as the one mentioned.

I can believe, however, that this problem was on a standardized test in US at some point. This last sentence points to the answer a bit more too :)

Thus, once you've fixed the two endpoints of the hypotenuse, not all points are eligible to be the final point of the triangle. All other points in space can form a triangle with those two points, but it may not be a right triangle.

If you interpret the hypotenuse as the diameter of a circle, all -- and only -- the points on the circle, except the hypotenuse's endpoints, will form a right triangle with the hypotenuse. If the diameter's length is, as specified in the problem, 10 inches, this tells us that the circle has radius 5 inches. This is the maximum distance between the hypotenuse and the third corner of the triangle. The problem tells us that the distance from the hypotenuse to the third corner is 6 inches, which is impossible.

> (with the exception of whether you can have less than one kopeck, which the author would assume the readers knew you couldn't)

You did notice that the cork in the second problem costs half a kopeck?

>and each of them carried on walking with the same speed

I thought the "same speed" meant lady1's speed = lady2's which made no sense...

    MA = v1*x = v2*9
    MB = v1*4 = v2*x
    → v1 = 9*v2/x
    → 9*4*v2/x = v2*x
    → x = 6

The well drilled student would be able to parse both but anyone who can struggle with getting the words into their head in the right order like me could struggle.

i.e. x = 1 + x/2

the challenge is to do the necessary translation and rearrangement

> A brick weighs one pound and half a brick.

I could easily imagine a brick being balanced by a one pound weight and a half brick on the other side, and the answer was easy.

I agree that "half the brick" is trickier wording.

I also met the problem #9 at an interview, and was asked to write the solution in pseudocode, as a kind of fizz-buzz test. (A peasant must take a wolf, a goat and a cabbage across a river in a boat. However the boat is so small that he is able to take only one of the three on board with him. How should he transport all three across the river? The wolf cannot be left alone with the goat, and the goat cannot be left alone with the cabbage.)

No. This is how you encourage kids who already love math. If they're not interested (yet), this is an awesome way to turn them off.

If you do it right, you'll incorporate these into everyday life (walk down the street, see something that you can turn into a similar problem, then pose that to the kid).

Source: loved these as a kid[1], now parent who wants to encourage a healthy sense of wonder in math/sciences.

[1] Here's one for you I loved back in the day: A hen and a half lay an egg and a half in a day and a half. How many eggs do six hens lay in six days?

But, yeah, even knowing about the natural order of Pushkin on the shelf, I also gave it a thought. I was wrong and just wasted my time. It is an easy problem, without any tricks.

Eventually I concluded that the price must be between 7 and 8 kopecks, however, a kopeck is a fraction of a ruble, and Google tells me the exchange rate is currently something like 76 rubles per USD, and a kopeck is 1/100th of a ruble, so a tiny fraction of a penny, which is itself nearly worthless. Wikipedia says that hyperinflation in early Soviet Russia, inflation during the cold war, the 1998 redenomination of one new RUB ruble to 1000 RUR old rubles, and subsequent inflation in Russia all combine to mean that one kopeck in the early 90s is worth about 40,000 times less than one kopeck today. Similarly, my American son is sometimes confused why Mom and Dad pay for stuff at stores with dollar bills, but also have pennies, nickels, and dimes. Morris the Moose can buy a lemon drop for a penny, why does a small pack of lemon drops cost two dollars at the store?

The last time you could subdivide a kopeck in half into a denga was around the 1917 revolution, so if the book cost 7 kopecks and one denga, Masha could have one denga and Misha could have 6 kopecks and one denga, and they could combine to get 7 kopecks but not have enough.

The true answer is that if Masha and Misha have been collecting old kopecks forgotten between the couch cushions in their piggybanks, they'll be better off melting the coins for scrap metal, because they're not keeping up with inflation. You can barely buy a piece of paper for a ruble, much less a kopeck. Except the dengas, if they're in good condition, they should sell those to rare coin collectors for on the order of 100,000 kopecks, which is an awfully large number for a five-year-old to be dealing with.

I presume they used kopecks for a purpose and not rubles, dollars or another unit that can be subdivided. That forces the integerness of the amounts and thus the presence of a solution.

Masha needs 7 kopecks to buy 1 book

"Misha lacked one", which means to me as an native English speaker that Misha needed just one more Kopeck in order to purchase 1 book.

When it's revealed that when they pool their money that they don't have enough to purchase just 1 book, I realized that the translation is faulty and stopped looking at the rest of the problems.

https://www.wolframalpha.com/input/?i=solve+for+y%2C+%5By+%3...

I was confused by the L-shaped shading in the top-right corner of the diagram, though.