Math Problems for children from 5 to 15 (2004) [pdf](imaginary.org)
imaginary.org
Math Problems for children from 5 to 15 (2004) [pdf]
https://www.imaginary.org/sites/default/files/taskbook_arnold_en_0.pdf
347 comments
> I'd recommend taking a look at Singapore Math[0] to get an idea of what kids in those age ranges are actually capable of doing, provided that adults shed preconceptions that children ought to be "sheltered from hard scary stuff", and instead encourage them.
So much this. I've managed to make learning fun for my three year old son. All too often we turn some daily scenario into a fun exercise and a well-meaning family member will exclaim "he can't possibly know that!". I assume they intend to shield him from the inevitable failure they believe I'm setting him up for by asking these questions, but he usually figures it out. And when he doesn't, he still gets a kick out of understanding it when we go through it together.
I repeatedly ask them to not make these comments. The more often he hears them say these things, the more liable he is to start believing them himself and say "I can't figure this out because I'm only x years old".
I believe that if kids were allowed to be challenged and excel at the things they show an interest in and predisposition to, the scholastic standard would be much higher. Instead of adults deciding what children of certain ages are "supposed to" be able to do and not to do.
My biggest fear at the moment is for his excitement at learning being crushed when he starts school.
So much this. I've managed to make learning fun for my three year old son. All too often we turn some daily scenario into a fun exercise and a well-meaning family member will exclaim "he can't possibly know that!". I assume they intend to shield him from the inevitable failure they believe I'm setting him up for by asking these questions, but he usually figures it out. And when he doesn't, he still gets a kick out of understanding it when we go through it together.
I repeatedly ask them to not make these comments. The more often he hears them say these things, the more liable he is to start believing them himself and say "I can't figure this out because I'm only x years old".
I believe that if kids were allowed to be challenged and excel at the things they show an interest in and predisposition to, the scholastic standard would be much higher. Instead of adults deciding what children of certain ages are "supposed to" be able to do and not to do.
My biggest fear at the moment is for his excitement at learning being crushed when he starts school.
> My biggest fear at the moment is for his excitement at learning being crushed when he starts school.
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.
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.
Maybe try explaining to him that school and other institutions tend to optimize for the average person, so unless his goal is being average, he'll need to take responsibility.
My personal experience is very very different, and is why I "quip these are too hard."
I explain below, but since Singapore was mentioned, I need to ask a cultural question first:
What do parents and teachers from outside the US do when a child DOESN'T understand math? How is math taught so that kids don't end up crying, getting sick, or hating themselves when faced with math problems? Or is there selection bias: it does happen, but "those kids" are left behind, and never seen by the rest of the world?
My personal experience with kids and math:
I'm in the US, and have one child who repeatedly could not complete math work during school, and would bawl and protest how stupid they were when given math homework at home. Completing it would take hours. Far behind, they could not multiply at age 8 or do fractions at 11. Tutors couldn't cover in an hour what other students finish in 10 minutes. Yet doctors indicated there's no learning disability.
So this leads to my perspective: a prodigious child may certainly be capable of these and enjoy the challenge. But for some others, this may succeed in making them feel worse about themselves, because it's yet another example math they don't understand.
I explain below, but since Singapore was mentioned, I need to ask a cultural question first:
What do parents and teachers from outside the US do when a child DOESN'T understand math? How is math taught so that kids don't end up crying, getting sick, or hating themselves when faced with math problems? Or is there selection bias: it does happen, but "those kids" are left behind, and never seen by the rest of the world?
My personal experience with kids and math:
I'm in the US, and have one child who repeatedly could not complete math work during school, and would bawl and protest how stupid they were when given math homework at home. Completing it would take hours. Far behind, they could not multiply at age 8 or do fractions at 11. Tutors couldn't cover in an hour what other students finish in 10 minutes. Yet doctors indicated there's no learning disability.
So this leads to my perspective: a prodigious child may certainly be capable of these and enjoy the challenge. But for some others, this may succeed in making them feel worse about themselves, because it's yet another example math they don't understand.
I appreciate this point. But I don't think it's about setting extremely high standards across the board and leaving those who can't keep up behind.
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).
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 agree. There is a lot of Be Like Me in the thread. The arguments need to be exposed to more diverse psychological scrutiny than is available on HN.
We can care for students, emphasize their gifts, provide math education when it is desired, AND achieve good educational outcomes. These aren't mutually exclusive...
We can care for students, emphasize their gifts, provide math education when it is desired, AND achieve good educational outcomes. These aren't mutually exclusive...
Go over the basics with him again, and again. I bet the teachers rushed him through fractions/percentages.
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.
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.
I feel like middle school is the worst age range to be teaching/learning pre-Algebra and Algebra. My experience was that teachers just expected students to either teach themselves or blindly follow and repeat the steps one by one. Geometry and trigonometry seem like they would be a better fit for that age range.
You may not believe it, but parents outside the US do not do anything special when a child doesn't understand math.
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.
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.
Bad at math is OK (and being too good at math means you are weird) attitude is (maybe was?) very prevalent in UK.
But why? Where does that come from? Why is it different in Eastern Europe? My experience was that it was just another subject; those that were more diligent were as good at it as at other subjects (many were girls, as girls tended to be more diligent in general). Many were bad at it but they were bad at most subjects.
I would guess the 'being bad at maths is OK' or 'maths is uniquely hard' is a self-fulfilling prophecy, meaning many students don't try as hard as with other subjects.
I would guess the 'being bad at maths is OK' or 'maths is uniquely hard' is a self-fulfilling prophecy, meaning many students don't try as hard as with other subjects.
My wife frequents chinese parent forums, and according to parents there, study time in a lot of households involve children staring at the ceiling and parents yelling... a lot. The thing, though, is I never hear about the situation improving by yelling more.
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 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.
> I'm in the US, and have one child who repeatedly could not complete math work during school, and would bawl and protest how stupid they were when given math homework at home.
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.
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.
Developing some fluency in basic arithmetic is a necessary step in correctly and quickly solving more complex problems. It is regrettably the school didn't take the obvious step: task the kid to memorize the multiplication table. A week later, the kid is ready to roll.
Learning to play a musical instrument is similar: there is some degree of practice and 'rote memorization' required to level up.
Learning to play a musical instrument is similar: there is some degree of practice and 'rote memorization' required to level up.
[deleted]
> I'm in the US, and have one child who repeatedly could not complete math work during school, and would bawl and protest how stupid they were when given math homework at home. Completing it would take hours. Far behind, they could not multiply at age 8 or do fractions at 11. Tutors couldn't cover in an hour what other students finish in 10 minutes. Yet doctors indicated there's no learning disability.
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.
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.
This is a solid point and I'd like to color that with my step-mom's experience tutoring an adult (lower twenties) in math. After running into frustration starting with basic algebra and working backwards through some basic arithmetic, it was realized this student lacked some very basic number sense. Step-mom took the student outside and started with counting exercises. "How many tires are on this street right now?" It started off with guessing. After some work with basic number sense and counting, they were able to puck back up, go through arithmetic, and then back to algebra relatively quickly. The student went on to pass their college algebra class.
I don't think anyone could have said it better.
I mean other subjects you can skip topics and still be ok, but with math you are screwed.
I can't speak for "the world," but in Eastern Europe, everyone is expected to struggle and fail often in school. If your 11-year-old is the only one in class struggling and failing, as is typical (US aims for >90% correct), you've got a mental health crisis. If your 11-year-old were surrounded by kids all struggling and failing, just at different levels, it'd be normal.
Get rid of the preconception/expectation that all kids learn at the same rate. So instead of grouping kids by age, group them by skill level. It’s odd to assume that just because a child is N years old that they should all be expected to be at a certain skill level.
That’s just called tracking and it’s already present in many countries.
Yes, I’m aware. But the context of my comment is about the US where this is not common.
Math tracking isn't common in the US...?
Math tracking exists but whether it exists and to what level varies by school district. It’s probably not the same as what you’re thinking. The variance in levels is also limited (based on what I’ve seen) and, in my opinion, carries a stigma because of how it’s presented (ex: honors track is literally named in a way to say “these kids are better”). My comment is more about normalizing the idea of separating by level such that kids don’t feel dumb for being in a “slower” track and to also have the variance span more than a single year/grade level.
How I thought about tracking was allowing kids to take math courses which had different trajectories leading to different "capstones".
If your child is taking Math 6 in 6th grade, that might be considered "normal". An accelerated course would be called Math 6/7, after which is Math 7/8. Then instead of Algebra 2 you could take Algebra 2/Trig.
So largely devoid of terms like "honors" or "slow" or "fast", but whatever you call it, kids understand what's happening — they are jumping ahead, going with the median, or falling behind.
As a relevant tangent, CA is considering proposals under the framework of Equitable Math to de-prioritize algebra as the capstone for middle school and calculus as the capstone of high school. Under the new framework, children will take classes with the same trajectory up to the last year in high school where they choose their own capstone, such as data science or calculus.
If your child is taking Math 6 in 6th grade, that might be considered "normal". An accelerated course would be called Math 6/7, after which is Math 7/8. Then instead of Algebra 2 you could take Algebra 2/Trig.
So largely devoid of terms like "honors" or "slow" or "fast", but whatever you call it, kids understand what's happening — they are jumping ahead, going with the median, or falling behind.
As a relevant tangent, CA is considering proposals under the framework of Equitable Math to de-prioritize algebra as the capstone for middle school and calculus as the capstone of high school. Under the new framework, children will take classes with the same trajectory up to the last year in high school where they choose their own capstone, such as data science or calculus.
You have to sit down and teach him the math and sit with him while he practices.
Then being behind is completely your fault.
I'll tell you my experience as someone who went to school in India. I believe that kids there are taught to cram, and I get that you need to understand , but I believe a lot of math is also remembering stuff ( multiplication/division, trignometry, algebra). And kids just practice an insane amount of problems, that it just becomes second nature.
I’m inclined to agree having seen this exact thing happen myself. On the flip side, the same reasoning is why most gifted programs have been cancelled in western countries. I think a good education system should cater for both.
Singapore Math & Prodigy are good recommendations. I'd also add IXL, Beast Academy, AOPS & RSM to the mix.
Our public school here in Indiana was training middle schoolers for the Math Bowl statewide competition. I spoke to one of the teachers at the school and volunteered to help. She handed me a bunch of math problems. I quickly hacked up a web app to help the students train. Imagine my shock & surprise when a month later, our humble public school team took home the first prize[1], in a tournament that had some 300+ schools, many of which had private coaches. Congressmen from Indianapolis drove down to our little town to hand over the trophy & plaques!
Since then, I do a weekly zoom session with those middle schoolers, sort of a Summer Math program. We work through AMC 8/10 problems & finish up with a friendly competition on the web app so I can track their progress.
I believe competition math can be a lot of fun if taught well.
[1] https://twitter.com/Hoosier47906/status/1400221783173775369
Our public school here in Indiana was training middle schoolers for the Math Bowl statewide competition. I spoke to one of the teachers at the school and volunteered to help. She handed me a bunch of math problems. I quickly hacked up a web app to help the students train. Imagine my shock & surprise when a month later, our humble public school team took home the first prize[1], in a tournament that had some 300+ schools, many of which had private coaches. Congressmen from Indianapolis drove down to our little town to hand over the trophy & plaques!
Since then, I do a weekly zoom session with those middle schoolers, sort of a Summer Math program. We work through AMC 8/10 problems & finish up with a friendly competition on the web app so I can track their progress.
I believe competition math can be a lot of fun if taught well.
[1] https://twitter.com/Hoosier47906/status/1400221783173775369
I couldn't agree more. It's always interesting seeing people shelter their kids from thinking that might become frustrating.
I forgot who said it but there was a quote like: "don't be afraid to push your brain, you won't break it!" Of course someone will reply to this talking about burnout which is real but a common sense approach to introducing cognitively difficult topics to kids is very different from that.
A couple of weeks ago one of mine (4yo) asked what the 'D' in 2D and 3D is since I had mentioned it to her while watching a movie. I took out a small ruler and went over to a corner of the TV unit. We then put the ruler along one edge and I explained how that's one measurement or 1D and how we can make a dot with a marker anywhere along that line. We then repeated for the next perpendicular edge and said that it's now two measurements or 2D. You can see where this is going. After 3D we talked about how we can now put a point anywhere in this imaginary cube.
She responded by asking how 4 rulers would look!
Kids are incredible and we frequently underestimate them.
(edited to add the age in there since it's relevant)
I forgot who said it but there was a quote like: "don't be afraid to push your brain, you won't break it!" Of course someone will reply to this talking about burnout which is real but a common sense approach to introducing cognitively difficult topics to kids is very different from that.
A couple of weeks ago one of mine (4yo) asked what the 'D' in 2D and 3D is since I had mentioned it to her while watching a movie. I took out a small ruler and went over to a corner of the TV unit. We then put the ruler along one edge and I explained how that's one measurement or 1D and how we can make a dot with a marker anywhere along that line. We then repeated for the next perpendicular edge and said that it's now two measurements or 2D. You can see where this is going. After 3D we talked about how we can now put a point anywhere in this imaginary cube.
She responded by asking how 4 rulers would look!
Kids are incredible and we frequently underestimate them.
(edited to add the age in there since it's relevant)
I don't think people understand burnout.
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.
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.
I agree with your general point, but if you want to give math problems that inspire creative, rigorous thought, you need to make them very clear, with as little ambiguity and assumed knowledge as possible. These problems don't do that. Examples:
Kopecks being indivisible, there being only one book at a specific price in the first problem, books being on a shelf in a specific order (13).
>>A brick weighs one pound and half the brick. How many pounds does the brick weigh?
As a native speaker, that doesn't even make sense, and I don't know a natural way to express it that doesn't do most of the work of the problem. (Another comment indicates it means "a brick's weight is equal to half of a brick plus one pound".)
Kopecks being indivisible, there being only one book at a specific price in the first problem, books being on a shelf in a specific order (13).
>>A brick weighs one pound and half the brick. How many pounds does the brick weigh?
As a native speaker, that doesn't even make sense, and I don't know a natural way to express it that doesn't do most of the work of the problem. (Another comment indicates it means "a brick's weight is equal to half of a brick plus one pound".)
Well, it sounds awkward in Russian as well as English but it works.
Sure, but the actual document states that
The book is addressed to school and university students, teachers, parents – to everybody who considers the thinking culture an essential part of the personality development.
Not sure where the 5yold number came from.
The book is addressed to school and university students, teachers, parents – to everybody who considers the thinking culture an essential part of the personality development.
Not sure where the 5yold number came from.
[deleted]
Which singaporemath? There are several.
Yeah, would be interested to see an example other than the link provided, which seems to be to purchase workbooks. Are there any good free resources (or free to try/evaluate) online?
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.
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.
We bought the Intensive Practice series[0], and paced study time at a couple of pages a day (which takes about 20-30 mins), semi-supervised (i.e. kids mostly work on their own, but if they don't understand something, we help clarify). It does take some hand holding at the beginning though.
[0] https://shop.singaporemath.com/index.php/product-category/su...
[0] https://shop.singaporemath.com/index.php/product-category/su...
Interesting. So was this a supplement to math learned at school or was it the primary math?
I was expecting you to name dimensions or one if the other programs they offer.
I was expecting you to name dimensions or one if the other programs they offer.
It isn't related to school curriculum (our kids go to US public school). It's supplemental in the sense that they get to practice more math exercises than a kid that only relies on school/common core curriculum.
Most school math is pathetically slow. IIRC, they don’t learn algebra till 8th or 9th grade and even then spend an entire year on it.
Curious if you ever looked into Beast Academy and compared it with Singapore math?
My kid is doing beast academy which looks good so am curious how they compare.
My kid is doing beast academy which looks good so am curious how they compare.
These seem pretty challenging for a 5 year old. I am pretty sure if I was interviewing senior engineers this would stump them - and I would get walk outs: https://i.imgur.com/lRfEQOs.png (from his book).
Guess they didn’t watch Die Hard ;) [0]
[0]: https://youtu.be/2vdF6NASMiE
[0]: https://youtu.be/2vdF6NASMiE
You'd get walkouts? Any person with normal intelligence and average facility with math realizes within 3 minutes that (2*3)-5 = 1.
People who have trouble with that problem have no guts.
People who have trouble with that problem have no guts.
random1538 didn't say what kind of engineers they were hiring, but if it was software engineers I'd be confused by it too. I know the solution to the problem, but only because I've seen it before and thus know the method to solve it.
If they were hiring cooks / chemists / anyone that is expected to have experience with measuring liquid volumes, it might make sense to assume they have the experience needed to solve this problem.
If they were hiring cooks / chemists / anyone that is expected to have experience with measuring liquid volumes, it might make sense to assume they have the experience needed to solve this problem.
[deleted]
This requires realizing that you can stop pouring. It isn't as in your face obvious as you're making it out to be.
The solution became immediately obvious to me after reading your comment. This leads me to believe that the question isn’t testing for math ability nor intelligence. It just requires you to realize the trick. Maybe it tests for creativity / lateral thinking ability?
It is not a trick so much as thinking what kinds of operations can be done with these vessels and water. Not necessarily a good interview question, but may be a good discussion with a young student.
The search tree for that isn’t very wide. At every step, there are at most 6 things you can do: fill vessel A/B from the tap, empty vessel A/B in the sink, fill up vessel A/B from vessel B/A.
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.
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.
[deleted]
According to a famous blog post, many "senior engineers" also can't do FizzBuzz either.
5 - 3 = 2.
3 - 2 = 1.Where are you finding these guys?
> 1 dog year is about 7 human years
This is the average; the correspondence is not linear...
This is the average; the correspondence is not linear...
Probably true, but the point was that the daughter was capable of quick multiplication, the age of the dog was..decidedly a minor detail.
Classic HN
Parent comment belongs in the hall of fame right next to the classic "Dropbox is trivial—it’s just an FTP server under version control":
https://news.ycombinator.com/item?id=8863
https://news.ycombinator.com/item?id=8863
I see your "classic HN" and counter with a "I'm one of today's lucky 10000!".
I was taught the 7 years rule as a kid and only learned as an adult that it's an average, not true for all dog breeds.
I was taught the 7 years rule as a kid and only learned as an adult that it's an average, not true for all dog breeds.
It's also not really true for young dogs. A 1-year-old dog of most breeds is reproductively mature, whereas a 7-year-old human is not.
On a related subject, here is a recent experience and I dont know how to deal with it.
I have been helping my kids with their homework during the pandemic, I thought it would be easy since I got very good grade 25+ years ago. And then when I sat down doing it. I couldn't remember a thing. Not a single thing. All of a sudden, apart from basic algebra, all of the maths were gone. Zip, Zero. I couldn't remember how sin cos tan works any more. It was like a few years of memory in my brain went missing. For some people it may be funny and have a laugh about it. For me it was shocking, quite horrifying and depressing.
I am thinking if I should relearn all those maths again. If so how do you go about it? Most of my friends aren't any good at maths so they thought not remembering any thing was not a problem.
But for some strange reason all the basic for Physics, Chemistry and Biology were still there. At least half of it. It was just maths. I dont know if anyone else have similar experience.
I have been helping my kids with their homework during the pandemic, I thought it would be easy since I got very good grade 25+ years ago. And then when I sat down doing it. I couldn't remember a thing. Not a single thing. All of a sudden, apart from basic algebra, all of the maths were gone. Zip, Zero. I couldn't remember how sin cos tan works any more. It was like a few years of memory in my brain went missing. For some people it may be funny and have a laugh about it. For me it was shocking, quite horrifying and depressing.
I am thinking if I should relearn all those maths again. If so how do you go about it? Most of my friends aren't any good at maths so they thought not remembering any thing was not a problem.
But for some strange reason all the basic for Physics, Chemistry and Biology were still there. At least half of it. It was just maths. I dont know if anyone else have similar experience.
I decided to start a CS undergrad degree 15 years after finishing my first undergrad degree.
The university had a math placement test. I didn’t remember almost at math, but spent about 3 weeks going through the placement test review materials for 30 min to an hour a day. Got almost perfect score on the placement test.
I did retake calculus 1 and 2 by my own choice since I wanted to know it quite well, and much of that seemed completely unfamiliar.
So it’s much much easier to learn a topic the second time around even if it’s forgotten. To get up to speed on it, you could use the placement test materials—collegeboard has some standard tests and materials to review for those tests, or your local university might have review materials for an in-house test.
I will say, I completed calc 2 a year ago now, and I already feel it slipping away again due to disuse. Now I’m onto new math topics I never took the first time around, like linear algebra and higher levels of calculus.
The university had a math placement test. I didn’t remember almost at math, but spent about 3 weeks going through the placement test review materials for 30 min to an hour a day. Got almost perfect score on the placement test.
I did retake calculus 1 and 2 by my own choice since I wanted to know it quite well, and much of that seemed completely unfamiliar.
So it’s much much easier to learn a topic the second time around even if it’s forgotten. To get up to speed on it, you could use the placement test materials—collegeboard has some standard tests and materials to review for those tests, or your local university might have review materials for an in-house test.
I will say, I completed calc 2 a year ago now, and I already feel it slipping away again due to disuse. Now I’m onto new math topics I never took the first time around, like linear algebra and higher levels of calculus.
I have a sneaking suspicion that there's something fundamentally wrong with how we approach math in school, given that:
1) It's presented as the most important thing in the world, pretty much, and
2) I've forgotten most of it past the first semester of algebra 1 in high school but that's mostly because... it wasn't important, at all, for me. And I think that's overwhelmingly the typical experience.
Honesty, I struggle to even talk fluently about early grade school math. "You can flip around the terms in a multiplication problem and the result's the same, because of the... uh... transitive property? Maybe? I think that's the name?"
Meanwhile, aside from when I'm trying to help my kids with math, life goes on just fine.
1) It's presented as the most important thing in the world, pretty much, and
2) I've forgotten most of it past the first semester of algebra 1 in high school but that's mostly because... it wasn't important, at all, for me. And I think that's overwhelmingly the typical experience.
Honesty, I struggle to even talk fluently about early grade school math. "You can flip around the terms in a multiplication problem and the result's the same, because of the... uh... transitive property? Maybe? I think that's the name?"
Meanwhile, aside from when I'm trying to help my kids with math, life goes on just fine.
Everybody says it's important, but for the wrong reasons. It's treated like a contest, to get "ahead," get high test scores, get into a desired college, and hopefully major in STEM. Then it can be safely forgotten.
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.
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.
I have a hard time to conceptualize mathematics because of the teaching methods and how they presents the information.
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.
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.
Math is a skill, just like playing an instrument. Just like an instrument, if you don't practice regularly you lose the skill. People have no problem accepting this when it comes to a musical instrument, but for some some reason our schools seem to teach people that math doesn't require ongoing practice.
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.
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.
But should it open up so many career paths? We treat math at the same importance as it was in 70 years ago before computers and whatnot.
Should someone genuinely bad at math or disinterested in math be precluded from a CS degree and the opportunities it provides?
Should someone genuinely bad at math or disinterested in math be precluded from a CS degree and the opportunities it provides?
I find it very similar to primary education language classes. Unless you use it as an adult after school, you’re not going to retain the knowledge for very long. And most people aren’t going to be using either set of skills in their adult lives after school.
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.
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.
Nice thing about language classes is that being bad at a language (or just not being interested) doesn't preclude many career paths. Math on the other hand is a clear gate, which doesn't make sense since you can literally forget and still do well in your career (as the parent poster mentioned).
> Nice thing about language classes is that being bad at a language (or just not being interested) doesn't preclude many career paths.
It does outside the English-speaking world. In many non-English-speaking countries—including Japan, where I live—English education is similar to mathematics education: All children have to study it and ability at school English is treated as an indicator of overall academic ability, but many children struggle with it and by adulthood most people have forgotten most of what they learned.
In Japan, school English education is also affected by problems similar to those mentioned in other comments on this page, including English teachers who themselves are not skilled at the language, educational policies that require that all children study the same material at the same age, and, sometimes, an overemphasis on rote memorization and teaching-to-the-test.
There’s a huge industry in Japan serving adults who have forgotten most of their school English—or didn’t learn much in the first place—and who now want to get better at it in order to advance their careers.
It does outside the English-speaking world. In many non-English-speaking countries—including Japan, where I live—English education is similar to mathematics education: All children have to study it and ability at school English is treated as an indicator of overall academic ability, but many children struggle with it and by adulthood most people have forgotten most of what they learned.
In Japan, school English education is also affected by problems similar to those mentioned in other comments on this page, including English teachers who themselves are not skilled at the language, educational policies that require that all children study the same material at the same age, and, sometimes, an overemphasis on rote memorization and teaching-to-the-test.
There’s a huge industry in Japan serving adults who have forgotten most of their school English—or didn’t learn much in the first place—and who now want to get better at it in order to advance their careers.
If you can forget math, it means that you memorized it. I don't think one can ununderstand math.
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.
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.
It's definitely possible—common, even—to forget things you didn't learn by memorization.
I use math often, but most of the time it's basic math. Simple things like ratios when trying to calculate per-unit costs in a grocery store when two things are displayed with different units, or converting between Fahrenheit and Celsius. Basic multiplication for tip calculation.
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.
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.
Oh, yeah, to be clear I use math (well, I apply mathematical algorithms and formulas) many times a day. But the ROI for my time spent on formal math eduction peaks somewhere around 3rd grade and declines fast after that.
I (genuinely) wonder how much that is attributable to having no actual use for other math, vs
1. not having been taught math early enough for it to be second nature
2. not having been taught useful every day applications of the math so as to keep practicing it
I've also forgotten quite a bit of math, but I also frequently encounter scenarios where I acknowledge that having a better handle on it would be advantageous to myself or others. For example, a better understanding of statistics and probability would certainly help political discourse in our society.
1. not having been taught math early enough for it to be second nature
2. not having been taught useful every day applications of the math so as to keep practicing it
I've also forgotten quite a bit of math, but I also frequently encounter scenarios where I acknowledge that having a better handle on it would be advantageous to myself or others. For example, a better understanding of statistics and probability would certainly help political discourse in our society.
>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.
that doesn't seem trivial at all.. wonder how that's done.
that doesn't seem trivial at all.. wonder how that's done.
The length l of the Christmas lights is the hypothenuse of a rectangular triangle of height h, the height of the column. So, if the slope angle is α, we have sin(α) = h/l, or α = arcsin(h/l).
Soundness check: that doesn’t have a solution if h > l. Looks good.
Soundness check: that doesn’t have a solution if h > l. Looks good.
does this assume the xmas tree is shaped like a column or a cone?
edit: ah, i re-read the original problem and it does mention column. i thought it was a xmas tree that was being wrapped.
edit: ah, i re-read the original problem and it does mention column. i thought it was a xmas tree that was being wrapped.
That would be harder, yes. Reading https://en.wikipedia.org/wiki/Conical_spiral#Slope, you want a logarithmic spiral (you need a constant angle to make the problem make sense)
Luckily, arc length isn’t too gnarly for those (same Wikipedia page), but you still have one equation with two variables.
I would have to think hard about whether those give you a unique solution.
I also doubt that spiral would give you uniform coverage of the cone (and that probably, is the real requirement, not constant angles), but again, I would have to do some thinking.
Luckily, arc length isn’t too gnarly for those (same Wikipedia page), but you still have one equation with two variables.
I would have to think hard about whether those give you a unique solution.
I also doubt that spiral would give you uniform coverage of the cone (and that probably, is the real requirement, not constant angles), but again, I would have to do some thinking.
oh, interesting variation for uniform coverage! that is indeed what i'd want for the tree. in building a road around a cone, a constant angle would be more desirable.
Suppose you’ve got a 16 foot strand of lights and an 8 foot column. If you unwrap the column in your mind, you can see you’ve got a right triangle with a hypotenuse of 16 and vertical leg of 8. What’s the angle that the hypotenuse makes with the floor? It’s the angle whose sine is opposite/hypotenuse = 8/16 = 1/2. That’s 30 degrees. So wrap the lights around the column at a 30 degree angle and it’ll be close (with a bit of slop thanks to rounding corners on the column).
my xmas isn't shaped like a column, it's a cone.
edit: ahh, the original question was for a column. i misread it and thought it was for a xmas tree.
edit: ahh, the original question was for a column. i misread it and thought it was for a xmas tree.
If you unwrap a cone you get a circular sector. Similar idea.
Years ago I had to correct a Bridge design I was tasked with writing a program to draw out the complex curved shape.
The engineer had used 2d instead of the 3d formulae :-)
The engineer had used 2d instead of the 3d formulae :-)
I have been working through the Art of Problem Solving Volume 1. I was a competent, though by no means excellent, maths student 20 years ago. AOPS was exactly the refresher needed to find those neurons again. Everything came back. However, had I jumped right into Trigonometry, I too would have been feeling like part of my mind was erased.
The math will come back, but you need to sit down and give yourself a structured program and, most importantly, time to actually do some exercises.
The math will come back, but you need to sit down and give yourself a structured program and, most importantly, time to actually do some exercises.
This is why math teaching pedagogy is important. I'm a fan of first principals and pattern finding for learning math (see Mathematician's Lament by Lockhart [0]).
Most kids in the US are historically taught memorization tricks. You have a kid who can't recall if x^1 = 0 or 1 or if it was x^0 = 1 or 0. They can't _remember_ some fact like a needle in a haystack of thoughts. However, the student who understands that x^3 = x * x * x and x^2 = x * x, will quickly know that x^1 must be x, and if each step is "divide by x", then x^0 must be 1.
I'm curious where the current math education trends will take us on this path, but I do like that they seem to focus more on understanding rather than rote memorization.
For sin, cos, and tan, they are much more re-discoverable if you are familiar with the unit circle's basics.
[0]: https://www.maa.org/sites/default/files/pdf/devlin/Lockharts...
Most kids in the US are historically taught memorization tricks. You have a kid who can't recall if x^1 = 0 or 1 or if it was x^0 = 1 or 0. They can't _remember_ some fact like a needle in a haystack of thoughts. However, the student who understands that x^3 = x * x * x and x^2 = x * x, will quickly know that x^1 must be x, and if each step is "divide by x", then x^0 must be 1.
I'm curious where the current math education trends will take us on this path, but I do like that they seem to focus more on understanding rather than rote memorization.
For sin, cos, and tan, they are much more re-discoverable if you are familiar with the unit circle's basics.
[0]: https://www.maa.org/sites/default/files/pdf/devlin/Lockharts...
Yes, but in my experience, it helps to thoroughly understand (down to first principles if you want to), and then memorize anyway.
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 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.
A person taught his son about sine and cosine. He himself got introduced to them as ratios of side lengths in a right triangle, but he didn't like the idea of changing definition when angles become more than 90 degrees, so he defined those functions as abscissa and ordinate of a point on a circle of unit radius, centered at origin.
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.
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.
x⁰ is generally a matter of definition and not a fact reasonably accessed from deeper underlying fundamentals. It just so happens that the definition fits this story that you have for reasons of convenience. Also, you know, 0⁰.
Knowledge atrophy is real. I've even talked to math PhDs who have forgotten areas of math they have definitely learned and excelled at but hadn't been using actively.
But I think your brain still subconsciously possesses knowledge of these supposed forgotten math skills. This is the reason why relearning these concepts will take way less time than learning them the first time. So I think just don't be afraid to relearn it.
But I think your brain still subconsciously possesses knowledge of these supposed forgotten math skills. This is the reason why relearning these concepts will take way less time than learning them the first time. So I think just don't be afraid to relearn it.
Lots of good courses on Coursera and edX. Khan Academy is good too. I particularly recommend the A-level prep sequence from Imperial College London on edX
But if you really want to maintain and maybe even further develop your math skills after getting back up to speed, I think the best long term strategy is to do personal creative and/or commercial projects in domains that interest you and that make heavy use of math. E.g. low level 3D graphics programming, etc
But if you really want to maintain and maybe even further develop your math skills after getting back up to speed, I think the best long term strategy is to do personal creative and/or commercial projects in domains that interest you and that make heavy use of math. E.g. low level 3D graphics programming, etc
Same - I was a straight A student, loved solving math problems, but now I don’t remember a thing.
I think it’s just how our brain works - it gets rid of knowledge that we don’t use any longer. Muscle memory like swimming or riding bicycle stays, but seems like language and math skills don’t retain unless they are being practiced.
I don't think so. The feeling described here is familiar to me with certain areas of maths, ones that I definitely knew and have then forgotten seemingly entirely, but when I had to get back into them it was nowhere near having to relearn them.
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.
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 information isn't erased - it's just that the retrieval synapses haven't been reinforced. It is relatively easy to do that.
Yes, and I believe that still existing but somewhat inaccessible information isn't just what was learned on the surface, but also includes the hard-earned intuition that was formed on the topic.
When I help kids with math homework I usually skim their textbook to see how they learned how to do it. This both refreshes my own memory and also makes sure that I am teaching it the same way they learned it (I can show them other methods after they master the way the teacher wants them to do it).
If you don't use something you are at risk of not retaining it at all.
About 10 years after I got my masters degree I browsed through some notes made during my studies. I was very surprised to find out that it's not that I don't remember some things, I didn't remember if I ever learned them.
Not sure why Physics, Chemistry and Biology stuck with you. I'm sure I don't remember 90% of history, geography, literature and many, many things.
What stuck for me are things that I was learning myself anyways. Math, physics, chemistry, a bit of biology. Same way I retained a bit of electronics even though school never attempted to teach me that. The rest went to hell and I don't regret a single thing forgotten from primary school and high school.
Curriculum for such young humans is aimed at keeping little buggers from annoying their parent for x hours a day, not for usability and future retention.
Kids don't even need decades to forget this stuff. I vividly remember coming back to school after summer break and knowing I forgot everything I learned last year and feeling safe because I'll most likely have no use for that information this year or later (except for math because it's the only thing in school that can be learned only on the foundation of simpler math that you need to learn earlier and retain).
About 10 years after I got my masters degree I browsed through some notes made during my studies. I was very surprised to find out that it's not that I don't remember some things, I didn't remember if I ever learned them.
Not sure why Physics, Chemistry and Biology stuck with you. I'm sure I don't remember 90% of history, geography, literature and many, many things.
What stuck for me are things that I was learning myself anyways. Math, physics, chemistry, a bit of biology. Same way I retained a bit of electronics even though school never attempted to teach me that. The rest went to hell and I don't regret a single thing forgotten from primary school and high school.
Curriculum for such young humans is aimed at keeping little buggers from annoying their parent for x hours a day, not for usability and future retention.
Kids don't even need decades to forget this stuff. I vividly remember coming back to school after summer break and knowing I forgot everything I learned last year and feeling safe because I'll most likely have no use for that information this year or later (except for math because it's the only thing in school that can be learned only on the foundation of simpler math that you need to learn earlier and retain).
I realized a few years after school I had mostly forgotten elementary calculus.
I still had my textbooks (Apostol volumes I and II) and re-read them. Things were again right with the universe--I could do elementary calculus.
A few years later, I realized I had again forgotten it. I decided for variety to buy Spivak's "Calculus" and read that instead of reading Apostol for a third time. Yet again, I could do elementary calculus.
The next time I realized I had forgotten elementary calculus, I re-read Apostol again (although just Volume I). To try to make it stick, I did every exercise in the book.
I of course have since forgotten elementary calculus. I'm not sure if doing all the exercises made it last longer or if I forget it as quickly as I usually do.
The next time I decide to relearn elementary calculus, I think I shall first make sure I have a long supply of problems covering the entire subject, and then after I finish the textbook I'll do a few random problems a week so that I have to actually use the stuff.
I still had my textbooks (Apostol volumes I and II) and re-read them. Things were again right with the universe--I could do elementary calculus.
A few years later, I realized I had again forgotten it. I decided for variety to buy Spivak's "Calculus" and read that instead of reading Apostol for a third time. Yet again, I could do elementary calculus.
The next time I realized I had forgotten elementary calculus, I re-read Apostol again (although just Volume I). To try to make it stick, I did every exercise in the book.
I of course have since forgotten elementary calculus. I'm not sure if doing all the exercises made it last longer or if I forget it as quickly as I usually do.
The next time I decide to relearn elementary calculus, I think I shall first make sure I have a long supply of problems covering the entire subject, and then after I finish the textbook I'll do a few random problems a week so that I have to actually use the stuff.
I've been helping one of my kids with some online high school classes, and we just read through the course material together and work the problems. Despite my having taken those subjects before, all of the material is new to me. I have no memory of learning that stuff in high school, despite graduating with AP classes.
It's nice because I get to learn new things, so I'd recommend that rather than teaching yourself before teaching your child that instead you just learn the material together. If you point out the stuff that is confusing to you and how you find the answer then your kids can learn that process as well.
And I can never remember the formulas for sin, cos, and tangent either so I just keep a graphics book handy.
It's nice because I get to learn new things, so I'd recommend that rather than teaching yourself before teaching your child that instead you just learn the material together. If you point out the stuff that is confusing to you and how you find the answer then your kids can learn that process as well.
And I can never remember the formulas for sin, cos, and tangent either so I just keep a graphics book handy.
This is like riding a bike isn't it? First few steps are a bit shaky but then you're back pretty soon.
Also keep in mind modern media has an explanation for everything online, there's not much below graduate level that isn't explained in several ways by several people.
Also keep in mind modern media has an explanation for everything online, there's not much below graduate level that isn't explained in several ways by several people.
Same issue/question. I was a pro until I stopped actively using it 10+ years ago and now, well, my math is embarrassing compared to teenage me.
I'm pretty sure the only way to pull that knowledge back into "actively useable" would be to start studying a la college again. I imagine it'd be a lot easier since we would be revisiting it instead of learning for the first time.
Hard to get excited about studying math relative to my other priorities :\
I'm pretty sure the only way to pull that knowledge back into "actively useable" would be to start studying a la college again. I imagine it'd be a lot easier since we would be revisiting it instead of learning for the first time.
Hard to get excited about studying math relative to my other priorities :\
I learned math way better as a math teacher than I did as a student because I had to figure out how to explain it - which meant I had to learn it first. Open up your child's math textbook and read the section they're working on, get to where you understand it yourself, then teach them. The textbooks do teach the material, and as an adult I found them to be easy to understand and sufficient explanations.
I had similar experience, but different outcome. I also had forgotten many formulas, but was able to derive everything from basic algebra. Quadratic formula, sine and cosine of sum of angles, derivatives, etc.
Some of those things took much longer than necessary, but I made it a point to not look anything up on principle. How can I explain something if I can't do it myself?
Some of those things took much longer than necessary, but I made it a point to not look anything up on principle. How can I explain something if I can't do it myself?
If you look up the definitions of sin, arcsin, logarithms, etc, does it mostly come back to you? Or do you feel like you need to completely relearn? I’m wondering if in your case all you need is to take a little time for a math refresher.
sin/cos for me were quite common since I'm fond of geography and geometry. So, even though they weren't needed at all, I had areas to apply them.
I never needed any math like log/exp at work, but somehow remembered it, probably because I used to do some fast estimations of things, for instance, "how big a pool of water you need to store energy to heat a house in winter", or "how fast will energy dissipate from the pool".
And that was probably thanks our school physics teacher, who showed that such napkin calculations were easy.
I never needed any math like log/exp at work, but somehow remembered it, probably because I used to do some fast estimations of things, for instance, "how big a pool of water you need to store energy to heat a house in winter", or "how fast will energy dissipate from the pool".
And that was probably thanks our school physics teacher, who showed that such napkin calculations were easy.
I Googled what I forgot till I found text that was interesting to read, basically.
For me, once I have found something to remind me, it all went back fast.
For me, once I have found something to remind me, it all went back fast.
I ran into this too. The process of relearning Math with my kids has made me much stronger than the first time around.
They're interesting to adults, too! Simple enough that it feels like you should be able to blurt out the answer, I'm more than twice the maximum recommended age and a professional engineer, but (at least for me) it takes some thought. The top recommended three:
> 1. Masha was seven kopecks short to buy a first reading book, and Mishalacked one kopeck. They combined their money to buy one book to share, but even then they did not have enough. How much did the book cost?
> 3. A brick weighs one pound and half the brick. How many pounds does the brick weigh?
> 13. Two volumes of Pushkin, the first and the second, are side-by-side on a bookshelf. The pages of each volume are 2 cm thick, and the cover – front and back each – is 2 mm. A bookworm has gnawed through (perpendicular to the pages) from the first page of volume 1 to the last page of volume 2. How long is the bookworm’s track?
I do take objection to the answer to question 13 - the author seems particularly set on one way of loading the bookshelves as correct.
> 1. Masha was seven kopecks short to buy a first reading book, and Mishalacked one kopeck. They combined their money to buy one book to share, but even then they did not have enough. How much did the book cost?
> 3. A brick weighs one pound and half the brick. How many pounds does the brick weigh?
> 13. Two volumes of Pushkin, the first and the second, are side-by-side on a bookshelf. The pages of each volume are 2 cm thick, and the cover – front and back each – is 2 mm. A bookworm has gnawed through (perpendicular to the pages) from the first page of volume 1 to the last page of volume 2. How long is the bookworm’s track?
I do take objection to the answer to question 13 - the author seems particularly set on one way of loading the bookshelves as correct.
> A brick weighs one pound and half the brick. How many pounds does the brick weigh?
I am a native english speaker and am having a hard time parsing this one. The only sane interpretation I can think of is that one pound + half the brick = the whole brick.
EDIT: I think the reason it is so confusing to me is because "and half the brick" sounds like (the start of) an independent thought. "A brick weighs one pound and half the brick was painted yellow".
This version is much clearer, IMO: "A brick weighs one pound plus half a brick". Maybe there is a fear that wording the problems too clearly makes the solution obvious.
I am a native english speaker and am having a hard time parsing this one. The only sane interpretation I can think of is that one pound + half the brick = the whole brick.
EDIT: I think the reason it is so confusing to me is because "and half the brick" sounds like (the start of) an independent thought. "A brick weighs one pound and half the brick was painted yellow".
This version is much clearer, IMO: "A brick weighs one pound plus half a brick". Maybe there is a fear that wording the problems too clearly makes the solution obvious.
These problems are worded to be deliberately confusing, especially #1. Is it a translation issue or are they worded because the math itself is too obvious once the wording has been deciphered?
Here’s a way to use algebra to grind out the solution to #1 with no particular insight needed.
Assume that prices and the amount of kopeks a person has are both represented by nonnegative integers. Let A be Masha’s kopeks, let I be Misha’s kopeks, and let B be the price of the book. We are given the following:
A = B - 7 (1)
I = B - 1 (2)
A + I < B (3)
Substituting (1) and (2) into (3) yields
B - 7 + B - 1 < B (4)
This simplifies to
B < 8 (5)
B = 7 satisfies (5) and, from (1) and (2), implies that A = 0 and I = 6, which together satisfy the givens (1), (2), and (3). So B = 7 is a solution. Furthermore, we cannot have B < 7 or else (1) would imply A < 0, contradicting the assumption that our variables are represented by nonnegative integers. So B = 7 is the only solution.
Assume that prices and the amount of kopeks a person has are both represented by nonnegative integers. Let A be Masha’s kopeks, let I be Misha’s kopeks, and let B be the price of the book. We are given the following:
A = B - 7 (1)
I = B - 1 (2)
A + I < B (3)
Substituting (1) and (2) into (3) yields
B - 7 + B - 1 < B (4)
This simplifies to
B < 8 (5)
B = 7 satisfies (5) and, from (1) and (2), implies that A = 0 and I = 6, which together satisfy the givens (1), (2), and (3). So B = 7 is a solution. Furthermore, we cannot have B < 7 or else (1) would imply A < 0, contradicting the assumption that our variables are represented by nonnegative integers. So B = 7 is the only solution.
It's also quite easy to do intuitively.
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.
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.
Couldn't the book cost 7.5k and one has 6.5 and the other has 0.5? Along those lines, isn't anything in the range of costing 7->8 (non-inclusive) acceptable (e.g. 0.9k and 6.9k)?
I was wondering the same thing but Kopecks are not currently subdivided.
https://en.wikipedia.org/wiki/Kopek
https://en.wikipedia.org/wiki/Kopek
The second question's solution requires half-kopeks.
These problems aren't current or modernized. There's no way a bottle with a cork in problem #2 costs 10 kopek.
current 10 kopek is worth 0.0013$
They probably come from time when 0.5 kopek was the smallest coin.
#1 problem doesn't have single non-zero solution if the smallest coin has larger or smaller.
current 10 kopek is worth 0.0013$
They probably come from time when 0.5 kopek was the smallest coin.
#1 problem doesn't have single non-zero solution if the smallest coin has larger or smaller.
That’s a nice approach. (It’s the same one given by bencollier earlier: https://news.ycombinator.com/item?id=27885681). I regard it as requiring a bit of insight, as opposed to my approach, which is more like grinding gears to reach a conclusion.
This is one of my main observations growing up with math: it's the moments of beauty and elegance that are the most exciting, but the grinding gears thing is also a necessity. They complement each other. For instance when you're just learning the basics there's a lot of these "wow what an insight" but over time you figure out that people have distilled it into a mechanical procedure, which also has some attraction to it. Something like quadratic equation turns the search for a pair of numbers that add up some one thing and multiply to another into a simple formula. You then use that mechanism to build ever more elaborate ones.
I came to the same conclusion the same way but it felt wrong due to the phrase "They combined their money to buy one book to share". Perhaps the phrase lost something in translation.
The problem says they combine their money which implies Masha has more than 0 kopeks though.
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Yes this is the flaw in the question. They should have said that the sum of their money is insufficient. Combine implies a physical action which can't happen if one of the parties has nothing.
> They combined their money to buy one book to share, but even then they did not have enough.
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..
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..
What is your objection to question #13?
I suppose the question doesn’t mention that volume 1 is on the left and volume 2 is on the right but I guess that would be assumed by any speakers of left to right languages.
I suppose the question doesn’t mention that volume 1 is on the left and volume 2 is on the right but I guess that would be assumed by any speakers of left to right languages.
> I suppose the question doesn’t mention that volume 1 is on the left and volume 2 is on the right but I guess that would be assumed by any speakers of left to right languages.
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.
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.
Diagram for the desired solution, on the shelf:
The first page of V1 is the rightmost page (shelved) of Volume 1, and the last page of V2 is the leftmost page (shelved) of Volume 2. So the bookworm ends up going only through the covers. Having volumes shelved in order from left-to-right is conventional in left-to-right languages since that's the same direction we read, and you'd want to "read" through the titles to find the volume you wanted.
V1 V2
| V1 | V2 |
+----------+----------+
|9876543210|9876543210|
The first page of V1 is the rightmost page (shelved) of Volume 1, and the last page of V2 is the leftmost page (shelved) of Volume 2. So the bookworm ends up going only through the covers. Having volumes shelved in order from left-to-right is conventional in left-to-right languages since that's the same direction we read, and you'd want to "read" through the titles to find the volume you wanted.
This is incorrect. You seem to be making the same error as the author says the editors made in the footnote at the bottom.
If volume 1 is on the left, and the worm goes from page 1 of volume 1 to the last page of volume 2, it travels 4mm in a straight line.
Page 1 of volume 1 and the last page of volume 2 will be right next to each other, if volume 1 is on the left.
If volume 1 is on the left, and the worm goes from page 1 of volume 1 to the last page of volume 2, it travels 4mm in a straight line.
Page 1 of volume 1 and the last page of volume 2 will be right next to each other, if volume 1 is on the left.
Wow, this relies on both books being in the same orientation, with front cover to the right. It assumes a lot. For perspective, I for years kept books shelved upside down because that orientation was easier for me when reading spines.
I guess it relies on the books being ordered and arranged the same way they'd be in every single bookstore and library in the world (in left-to-right ordering countries).
But it's true, maybe this is too much to assume. Most of the time when I've seen this puzzle it's shown the book spines in an image to make it clear, and many people still can't get it. Then again, that would rely on knowing whether it was using top-to-bottom or bottom-to-top book title orientation, so perhaps the only solution is for the author to spell out "the first page of volume 1 is next to the last page of volume 2."
But it's true, maybe this is too much to assume. Most of the time when I've seen this puzzle it's shown the book spines in an image to make it clear, and many people still can't get it. Then again, that would rely on knowing whether it was using top-to-bottom or bottom-to-top book title orientation, so perhaps the only solution is for the author to spell out "the first page of volume 1 is next to the last page of volume 2."
> in every single bookstore and library in the world
Not so fast. Some cultures (Japan for one, I think China and Taiwan as well?) have page-ordering right-to-left but books are generally stacked left-to-right from what I've seen (and in ether case bookstores don't order volumes differently depending on if it's native right-to-left books or foreign right-to-left ones).
Not so fast. Some cultures (Japan for one, I think China and Taiwan as well?) have page-ordering right-to-left but books are generally stacked left-to-right from what I've seen (and in ether case bookstores don't order volumes differently depending on if it's native right-to-left books or foreign right-to-left ones).
German books have the orientation of the writing on the spine flipped. I don't like storing books upside down, so it makes a mess in my mixed English and German bookshelf.
There's a language-wide order in Germany of direction the of writing on the spines of a books?
I just checked my bookshelf and my books go both ways.
I just checked my bookshelf and my books go both ways.
Ah. I just checked, and while all the English ones seem to have a consistent orientation, the German ones indeed don't. Never noticed, huh...
The point with this question is that if volume 1 is on the left and volume 2 is on the right, the first page of volume 1 is facing right and the last page of volume 2 is facing left, so the only thing between them is the two covers. Hence, the answer is 4 mm.
It also assumes that the pages aren't facing out.
could also be the case that the two volumes are empty, ie have no pages
The problem statement gives us that there are 2cm of pages in each book. So they are not empty. The confusion is in which order the books would be on the shelf, and consequently which direction the bookworm would be moving and through what.
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what book has no pages? Also the problems states:
> The pages of each volume are 2 cm thick
> The pages of each volume are 2 cm thick
The pages would be less than 20mm thick in that case.
> side-by-side on a bookshelf
Seems pretty explicit to me
Seems pretty explicit to me
Assuming volume 1 is on the left and the books have their spine facing out.
Reasonable assumptions, but relying on implicit knowledge nonetheless.
Reasonable assumptions, but relying on implicit knowledge nonetheless.
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Which volume is to the left? Thus the ambiguity.
> A bookworm has gnawed through (perpendicular to the pages) from the first page of volume 1 to the last page of volume 2.
How would a worm eat perpendicular to the pages and go from the first page of volume 1 to the last page of volume 2?
It can only be vol1->vol2
How would a worm eat perpendicular to the pages and go from the first page of volume 1 to the last page of volume 2?
It can only be vol1->vol2
> How would a worm eat perpendicular to the pages and go from the first page of volume 1 to the last page of volume 2?
Sorry, what is the question supposed to be? You're positing a contradiction between two facts:
- The worm's path is perpendicular to the pages.
- The worm's path begins at the first page of volume 1, and ends at the final page of volume 2.
What's the contradiction?
Sorry, what is the question supposed to be? You're positing a contradiction between two facts:
- The worm's path is perpendicular to the pages.
- The worm's path begins at the first page of volume 1, and ends at the final page of volume 2.
What's the contradiction?
The bookworm could have moved right to left from volume 1 to volume 2.
Assuming the spines are facing out, the books are right-way up, and volume 2 is on the left of volume 1, then the answer would be 44mm.
While the answer does assume that V1 is on the left, there's no contradiction in your statement. If V2 happened to be on the left, it would still be perfectly logical for "a worm eat perpendicular to the pages and go from the first page of volume 1 to the last page of volume 2." They would simply have to go through more pages.
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My answer to #1 is less than 8 kopecks, and Masha has less than one. There's a problem: nowadays kopecks are the minimal unit of currency. Either it means you have to think of the old Imperial money units (polushka, 1/4 of kopeck), or think of fractional amounts of money.
I assumed kopecks were pennies. If Misha needs one, and Masha doesn't have enough to give her one, than Masha must have none. So the answer follows from that.
How can they combine their money if one of them doesn't have any?
But naturally, at first you assume the numbers are natural. :)
Well, yes, otherwise you'd have an infinite number of solutions.
Seems like Mishalacked isn't great at making deals.
1. Depends on whether kopecks are divisible into a smaller monetary unit or not. If they are divisible into 100 units, I believe the answer is "anywhere between 7.00 and 7.99 kopecks".
(Problem #2 requires kopecks to be divisible.)
(Problem #2 requires kopecks to be divisible.)
I think part of the point of this brochure is to think about the problems intuitively in the context they are presented. So in the first problem it's just kids trying to buy their first book, it would be silly to think Masha had a fraction of a kopeck (assuming you understand what a kopeck is, I really think it should have been translated as cent) and that the answer could be in range [7, 8). This may be what he talks about when he says that many academics fail at these problems.
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.
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.
Indeed, but it likewise seems intuitively reasonable to think that a book costs much more than 7 cents (or 7 times whatever the atomic unit of currency is) and that over 100 times the atomic unit is more reasonable.
Before 1917, they were divisible into 4 polushkas.
They're not. It's like a penny. Sure you may have parts of cents like with gas, but for girls buying books, it's the lowest currency value.
Long ago, many currencies had more divisions than they do now.
The penny of GBP came in quarter pennies (farthings) until 1950.
Half US-cent coins were made until 1857.
The penny of GBP came in quarter pennies (farthings) until 1950.
Half US-cent coins were made until 1857.
> (Problem #2 requires kopecks to be divisible.)
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
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
I read the second statement ("the bottle itself is 9 kopecks
more expensive than the cork.") as:
Bottle = 9 + Cork
Your statement (Bottle = 9 * Cork) would be "the bottle is 9 times as expensive as the cork".
I solve it to:
Bottle = 9.5 Kopecks.
Cork = 0.5 Kopecks.
Bottle = 9 + Cork
Your statement (Bottle = 9 * Cork) would be "the bottle is 9 times as expensive as the cork".
I solve it to:
Bottle = 9.5 Kopecks.
Cork = 0.5 Kopecks.
Works even if Kopecks are divisible: say the books value is 7.5, Mash must have 0.5 (7.5 - 7) and Misha 6.5 (7.5 - 1), however now when you combine them they sum to exactly 7.5 not less than, the only way to arrive at less than is if Masha has 0. So its always exactly 7
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6.5 + 0.5 is 7.0, not 7.5, so that should be valid.
The book's cost can lie anywhere between [7, 8).
The book's cost can lie anywhere between [7, 8).
Right on the original typo, but still not convinced, I’ve rephrased to original to try to be clearer
> say the books value is 7.5, Mash must have 1.5 (7.5 - 7) and Misha 6.5 (7.5 - 1), however now when you combine them they sum to exactly 7.5 not less than
There are several problems with this:
- 7.5 - 7 is 0.5, not 1.5
- 1.5 + 6.5 is 8, not 7.5
- 0.5 + 6.5 is still less than 7.5
The problem specifies that 2x - 8 < x. There is no way to constrain this to the specific solution x = 7. Everything would work fine if the book cost -2.6 kopecks.
There are several problems with this:
- 7.5 - 7 is 0.5, not 1.5
- 1.5 + 6.5 is 8, not 7.5
- 0.5 + 6.5 is still less than 7.5
The problem specifies that 2x - 8 < x. There is no way to constrain this to the specific solution x = 7. Everything would work fine if the book cost -2.6 kopecks.
Thank you :) I’m being very clueless today
6.5 plus 0.5 is 7.0, they don't add up to exactly 7.5.
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A kopeck is like a cent (1/100th of a ruble).
For Russian-speaking readers here who may want to challenge their children, here's the (original?) open book in Russian: http://ilib.mccme.ru/pdf/VIA-taskbook.pdf
I suspect Russian-speaking readers of HN mostly have kids who can’t read/write Russian, unfortunately.
Like my daughters, for example.
Like my daughters, for example.
There are interesting Wikipedia articles on this topic: "Heritage language"
https://en.wikipedia.org/wiki/Heritage_language
https://en.wikipedia.org/wiki/Heritage_language_learning
https://en.wikipedia.org/wiki/Heritage_language
https://en.wikipedia.org/wiki/Heritage_language_learning
How actively did you try to teach them to do that?
They did go into a Russian child care where they learned basic reading/writing.
But after they went into US elementary school, we didn’t want to interfere with their English learning, so without practice they have lost reading/writing skills, but still can speak.
But after they went into US elementary school, we didn’t want to interfere with their English learning, so without practice they have lost reading/writing skills, but still can speak.
Thank you for that link! I can read Russian and that version is much clearer.
I try to make math fun in my house. A couple of things I’ve found that work:
1. Anytime there’s a “guess how many are in the jar” contest, I get my kids to use the appropriate formula for volume to see if they can guess the right answer. They usually get really close.
2. Show them how math helps them win at games. Monopoly is great for this where you can calculate the ROI for different properties on the board, how many houses are ideal, etc. you can go further with likelihood of landing on certain properties too.
It works. The hardest thing about math motivation as a kid is “where will I use this?”
Moneyball (movie) is a good one too.
1. Anytime there’s a “guess how many are in the jar” contest, I get my kids to use the appropriate formula for volume to see if they can guess the right answer. They usually get really close.
2. Show them how math helps them win at games. Monopoly is great for this where you can calculate the ROI for different properties on the board, how many houses are ideal, etc. you can go further with likelihood of landing on certain properties too.
It works. The hardest thing about math motivation as a kid is “where will I use this?”
Moneyball (movie) is a good one too.
> It works. The hardest thing about math motivation as a kid is “where will I use this?”
I struggle to motivate myself to brush up on or move farther in math as an adult, for similar reasons.
Some recreational math is kinda fun, but mostly worthless except as a pastime.
I struggle to motivate myself to brush up on or move farther in math as an adult, for similar reasons.
Some recreational math is kinda fun, but mostly worthless except as a pastime.
Play Factorio [1]
[1] https://www.factorio.com/
[1] https://www.factorio.com/
Just pick up (analog) electronics as a hobby, and it becomes relevant and necessary like nothing else. Add some light signal processing, and now you understand why you had those Algebra I and II courses in university.
Moneyball is one of my favorite movies!
> The hypotenuse of a right-angled triangle (in a standard American examination) is 10 inches, the altitude dropped onto it is 6 inches. Find the area of the triangle. American school students had been coping successfully with this problem over a decade. But then Russian school students arrived from Moscow, and none of them was able to solve it as had their American peers (giving 30 square inches as the answer). Why?
That's a good one. I know the answer but won't reveal it since it's a fun one to discover yourself.
That's a good one. I know the answer but won't reveal it since it's a fun one to discover yourself.
https://en.m.wikipedia.org/wiki/Altitude_(triangle) for reference, I guessed what the issue was but wasn't familiar with the term altitude
Exactly. In grade school the formula being taught was "base times height over 2" but no one mentioned the term altitude.
Since none of the replies so far have it correct, here's a spoiler:
https://math.stackexchange.com/questions/1594740/v-i-arnold-...
https://math.stackexchange.com/questions/1594740/v-i-arnold-...
Fair enough. I was going to wait a bit longer to provide the hint. I do like that there are non-Euclidian answers. However, I doubt non-Euclidian geometry was expected on standardized tests tests in US, if the anecdote is to be believed that this came from say SAT or ACT.
Probably, I'm too proficient in English to see the problem.
* guess #1: they read it as a Pythagoras triangle, with sides 10, 6 and 8, hence they answered 6 * 8 / 2 = 24.
* guess #2: they could not make sense of "altitude dropped onto it".
* they tried to convert units and forgot that for area, the coefficient is squared?
* guess #1: they read it as a Pythagoras triangle, with sides 10, 6 and 8, hence they answered 6 * 8 / 2 = 24.
* guess #2: they could not make sense of "altitude dropped onto it".
* they tried to convert units and forgot that for area, the coefficient is squared?
The problem is that right triangles have to obey a constraint. The angle opposite the hypotenuse is 90 degrees.
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.
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.
> which is impossible
Or, generally, is only possible in a non-Euclidean geometry (which the Russian students apparently did not know very well).
Or, generally, is only possible in a non-Euclidean geometry (which the Russian students apparently did not know very well).
> 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.
I've known that since I was a kid. What I didn't know until ~40 years later is that there is a generalization of that. If AB is a chord of a circle, and C and D are any points on the circle that are on the same side of AB, then angles ACB and ADB are the same. I have no idea how I never came across that before. It's called the Inscribed Angle Theorem, and is in Euclid.
When I read that I tried to prove it. First try was geometrically. I just could not get it. (Yes, I know that in fact it is easy...I've always sucked at geometry).
Second try was with vectors. What it is saying is that the dot product of AC and CB should be the same no matter where C is if you move C around on the same side of AB. That led to some ugly expression that would need to be constant. Mathematica said it was constant, but Mathematica doesn't show its work and I could not figure out how to show it.
Next try was with physics. Imagine that the circle is a very large circular train track, and there is a train on the track whose front is at A and back is at B. Imagine you have two cameras at the center of the circle, one pointed at A and the other, mounted directly on top of the first, pointed at B.
If the train starts moving, you'd have to turn the cameras to keep them pointing at the front and back of the train. With the cameras at the center of the circle, you'd have to turn them at the same rate. That's because from your point of view at the center of the circle, the angular velocities of any two points on the train are the same.
What the the Inscribed Angle theorem implies is that this also works if the cameras are on the circle. I.e., from the point of view of someone standing on the circular track, looking at a train moving elsewhere on the track, all parts of the train have the same angular velocity.
Dropping the train, what we have then is that the Inscribed Angle theorem is equivalent to claiming that a point moving around the circle at constant angular velocity as seen from the center of the circle also had constant (but not the same constant!) angular velocity as seen from an observer on the circle.
It was then easy to set up a point moving around a circle at constant angular velocity in polar coordinates (r = 1, θ = t), convert to Cartesian coordinates, shift the viewpoint to somewhere on the circle, go back to polar coordinates, and differentiate θ(t) with respect to t. That gave an expression that was easy to see was a constant. QED. Whew...
...and then I had another go at doing it with elementary geometry, and it turned out to be easy after all. Something you might reasonably see on a high school geometry homework assignment.
I've known that since I was a kid. What I didn't know until ~40 years later is that there is a generalization of that. If AB is a chord of a circle, and C and D are any points on the circle that are on the same side of AB, then angles ACB and ADB are the same. I have no idea how I never came across that before. It's called the Inscribed Angle Theorem, and is in Euclid.
When I read that I tried to prove it. First try was geometrically. I just could not get it. (Yes, I know that in fact it is easy...I've always sucked at geometry).
Second try was with vectors. What it is saying is that the dot product of AC and CB should be the same no matter where C is if you move C around on the same side of AB. That led to some ugly expression that would need to be constant. Mathematica said it was constant, but Mathematica doesn't show its work and I could not figure out how to show it.
Next try was with physics. Imagine that the circle is a very large circular train track, and there is a train on the track whose front is at A and back is at B. Imagine you have two cameras at the center of the circle, one pointed at A and the other, mounted directly on top of the first, pointed at B.
If the train starts moving, you'd have to turn the cameras to keep them pointing at the front and back of the train. With the cameras at the center of the circle, you'd have to turn them at the same rate. That's because from your point of view at the center of the circle, the angular velocities of any two points on the train are the same.
What the the Inscribed Angle theorem implies is that this also works if the cameras are on the circle. I.e., from the point of view of someone standing on the circular track, looking at a train moving elsewhere on the track, all parts of the train have the same angular velocity.
Dropping the train, what we have then is that the Inscribed Angle theorem is equivalent to claiming that a point moving around the circle at constant angular velocity as seen from the center of the circle also had constant (but not the same constant!) angular velocity as seen from an observer on the circle.
It was then easy to set up a point moving around a circle at constant angular velocity in polar coordinates (r = 1, θ = t), convert to Cartesian coordinates, shift the viewpoint to somewhere on the circle, go back to polar coordinates, and differentiate θ(t) with respect to t. That gave an expression that was easy to see was a constant. QED. Whew...
...and then I had another go at doing it with elementary geometry, and it turned out to be easy after all. Something you might reasonably see on a high school geometry homework assignment.
SPOILER:
Well my answer but it's a correct one.
˙ʇsǝʇ ǝɥʇ uo „ǝlƃuɐıɹʇ ɥɔns ou s,ǝɹǝɥʇ„ ɹǝʍsuɐ ou sɐʍ ǝɹǝɥʇ ǝsnɐɔǝq ʇı ǝʌlos ʇ,uplnoɔ sʇuǝpnʇs uɐıssnᴚ ssǝnƃ I
˙lɐnbǝ ǝɹɐ sǝƃpǝ ɹǝɥʇo oʍʇ uǝɥʍ s,ʇI ˙ގ ʇsoɯ ʇɐ ǝq plnoɔ ʇı uo pǝddoɹp ǝpnʇıʇlɐ ǝɥʇ 0⇂ ǝsnuǝʇodʎɥ ɥʇıʍ uǝɥʇ ǝlƃuɐıɹʇ pǝlƃuɐ-ʇɥƃıɹ ɐ s,ʇı ɟI
Well my answer but it's a correct one.
˙ʇsǝʇ ǝɥʇ uo „ǝlƃuɐıɹʇ ɥɔns ou s,ǝɹǝɥʇ„ ɹǝʍsuɐ ou sɐʍ ǝɹǝɥʇ ǝsnɐɔǝq ʇı ǝʌlos ʇ,uplnoɔ sʇuǝpnʇs uɐıssnᴚ ssǝnƃ I
˙lɐnbǝ ǝɹɐ sǝƃpǝ ɹǝɥʇo oʍʇ uǝɥʍ s,ʇI ˙ގ ʇsoɯ ʇɐ ǝq plnoɔ ʇı uo pǝddoɹp ǝpnʇıʇlɐ ǝɥʇ 0⇂ ǝsnuǝʇodʎɥ ɥʇıʍ uǝɥʇ ǝlƃuɐıɹʇ pǝlƃuɐ-ʇɥƃıɹ ɐ s,ʇı ɟI
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Well, there's many possible answers. I'll give a few:
- Russia doesn't have "inches"
- The question has English terms like "hypotenuse"
- The Russian students were younger
- There were differences in conventions, e.g. which side is down. An altitude dropped onto the hypotenuse of a right triangle could be perpendicular, or either of the two other sides.
- And so on...
A lot of these problems are designed for a conversation rather than a solution.
- Russia doesn't have "inches"
- The question has English terms like "hypotenuse"
- The Russian students were younger
- There were differences in conventions, e.g. which side is down. An altitude dropped onto the hypotenuse of a right triangle could be perpendicular, or either of the two other sides.
- And so on...
A lot of these problems are designed for a conversation rather than a solution.
Inches is not the issue. It could be any units. The joke about the American vs Russian students was a jab at the American students, not the Russian ones! Something about the Russian students seeing something that American student couldn't. I don't agree with the premise of course, just providing the extra info as a hint.
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 :)
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 :)
Funny in germany we learn the formular for just that:
Area_triangle = Base_triangle * Height_triangle / 2
/edit: fixed factor of two
Area_triangle = Base_triangle * Height_triangle / 2
/edit: fixed factor of two
That’s either a nice example of hemibel thinking (http://libertycorner.blogspot.com/2004/07/hemibel-thinking.h...), of your memory going away, or of an extremely lousy education.
(You are of by a factor of 2)
(You are of by a factor of 2)
I'm not sure sure if you made a mistake or if you are joking
The problem is exactly that the application of a formula requires some assumptions to be met, the first of which is logical consistency.
Try to draw a triangle such as the one mentioned.
Try to draw a triangle such as the one mentioned.
lol?
There's a funny story. Before PISA, Finland looked up to the German school system, which was clearly considered superior by both sides.
When PISA came out in 2000, Finland was surprised to come out on top for Europe. Germany's performance in math was abysmal -- behind the US even. People started flocking to see what Finland did, and stopped looking up to Germany.
https://nces.ed.gov/pubs2002/2002116.pdf
The German school system has since gradually improved -- it's no longer behind the US -- but it's far from world-leading.
When PISA came out in 2000, Finland was surprised to come out on top for Europe. Germany's performance in math was abysmal -- behind the US even. People started flocking to see what Finland did, and stopped looking up to Germany.
https://nces.ed.gov/pubs2002/2002116.pdf
The German school system has since gradually improved -- it's no longer behind the US -- but it's far from world-leading.
One problem I give children aged 5 – 7 is the following.
How old are you? (They answer X.)
How many years did it take for you to become X years old?
I've found that at 6 years old they start to relate their age with how many years it took to become that age. At 5 they usually can't make this connection.
How old are you? (They answer X.)
How many years did it take for you to become X years old?
I've found that at 6 years old they start to relate their age with how many years it took to become that age. At 5 they usually can't make this connection.
> 12. A tide was in today at 12 noon. What time will it be in (at the same place) tomorrow?
What is this, a maritime exam?
What is this, a maritime exam?
Yeah, this one doesn't have anything to do with math, does it?
Off the top of my head, I'd guess that tides lose about an hour per day, so the tide would be in at 1:00 pm.
Ah, I guess the point of this is that the next tide is half way between those two, and so would be just after midnight.
But to know this, you'd have to know that the tide's cycle is slightly longer than the day's cycle.
Off the top of my head, I'd guess that tides lose about an hour per day, so the tide would be in at 1:00 pm.
Ah, I guess the point of this is that the next tide is half way between those two, and so would be just after midnight.
But to know this, you'd have to know that the tide's cycle is slightly longer than the day's cycle.
Considering the moon was at its highest today at 12:00, and will be at its highest again in ~29 days at 12:00, what time will the moon be at its highest tomorrow, which is 1/29 of its cycle?
The question then turns into dividing 24 hours into 29 parts, or about 48 minutes per day, so 12:48.
Bonus points (not in math) go to whoever knows that a tide can also come when the moon is at its lowest, or half that time, or 0:24 or so.
The question then turns into dividing 24 hours into 29 parts, or about 48 minutes per day, so 12:48.
Bonus points (not in math) go to whoever knows that a tide can also come when the moon is at its lowest, or half that time, or 0:24 or so.
It wouldn't be a Russian textbook without some completely unreasonable problems.
At 1pm, considering that tides usually go with 6 hours and 15 minutes increments?
(I hope my memory is right, took the RYA exam a while ago)
(I hope my memory is right, took the RYA exam a while ago)
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But what's noon plus 12 hours 30 minutes?
Is there an answer list anywhere? It seems to me that some of these questions could have multiple answers depending on how you interprt the question. I know it is more about the thought exercise, but on some it would be nice to have the correct answer in order to learn how to find the solution.
I only worked through the first 13, but none of them seemed to me to have multiple solutions (with the exception of whether you can have less than one kopeck, which the author would assume the readers knew you couldn't).
> I only worked through the first 13
> (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?
> (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?
#18 does not have any answer. It's impossible to cover the squares as requested.
What do you mean? You just provided the answer. It's unique.
Another interesting book in this vein if these problems tickle your fancy: https://www.amazon.ca/Math-Three-Seven-Mathematical-Preschoo...
Working on math problems together with your kids is a fun way to learn how they think and reason. It has led me to have a deeper emotional connection with my kids as I learn what they struggle with in school. I have slowly learned that some times framing the problem a certain way helps them to grasp what is being taught better than brute-forcing them through exercises and homework.
Working on math problems together with your kids is a fun way to learn how they think and reason. It has led me to have a deeper emotional connection with my kids as I learn what they struggle with in school. I have slowly learned that some times framing the problem a certain way helps them to grasp what is being taught better than brute-forcing them through exercises and homework.
A lot of comments complaining that many of the problems don't have definitive answers, e.g. #12 (tides) and #13 (bookworms).
I think the point for some problems is not finding the answer so much as developing logical, analytic and critical thinking skills; to learn the value of looking at a problem from different perspectives and the necessity to sometimes think outside of the box; and to be able to find the holes and ambiguities -- sometime to even reject the question outright.
I think the point for some problems is not finding the answer so much as developing logical, analytic and critical thinking skills; to learn the value of looking at a problem from different perspectives and the necessity to sometimes think outside of the box; and to be able to find the holes and ambiguities -- sometime to even reject the question outright.
I met one of Vladimir Arnold's grandchildren at a high school math summer camp. One of the things that stood out to me was his insistence that his proofs were as simple and understandable as possible. Unlike other students, he would intentionally avoid using sophisticated techniques. He would brag that a "third grader" could understand it. His preference for simplicity and elegance was clearly inherited from his grandfather.
Not commenting about the difficulty, but some of the later questions definitely require a bit more specialised knowledge. #74-77, "uniform, dense, smooth map, open set". Basically some topology questions. Some other questions are just weird, #54- just plot a parametric equation? Also a few integrals?
I think anyone (kids included) could go pick up this stuff fine, but doesn't quite match with the blurb given (why not define some of these things in the questions and work through it that way).
I imagine then what this could be good for is if you use these questions as a bit of a teaching guide/goal. Maybe work your way through the concepts/problem with the end goal being the question, but then introducing stuff as necessary, rather than just throwing the entire question out there.
I think anyone (kids included) could go pick up this stuff fine, but doesn't quite match with the blurb given (why not define some of these things in the questions and work through it that way).
I imagine then what this could be good for is if you use these questions as a bit of a teaching guide/goal. Maybe work your way through the concepts/problem with the end goal being the question, but then introducing stuff as necessary, rather than just throwing the entire question out there.
The problem here is that most people won't use most of the math knowledge they gain (and then lose) in school. But for programmers, it's different. Math can be fairly common in programming, especially depending on what field you're working with, and you wouldn't want to lose the knowledge you gained in school, but, unfortunately, most people will lose it.
What I propose is a better way to find math knowledge. In contrast to programming, math problems are harder to find solutions for, and the information available is quite sparse and hard to find online, from my personal experiences.
When I try to tackle programming problems, most of the time I won't memorize code snippets or algorithms, instead I'll have a mental link pointing to the name of the algorithm or the specific page, that I can search up, find, and then implement.
This is the total opposite of what school tries to do. They try to force memorization, which should come naturally. A better way to do it is to let the students have the equations and the necessary information that they need, then they can fit the puzzle pieces together to solve the problem. We live in the age where everything is becoming more and more documented, and we're still forcing memorization on people.
What I propose is a better way to find math knowledge. In contrast to programming, math problems are harder to find solutions for, and the information available is quite sparse and hard to find online, from my personal experiences.
When I try to tackle programming problems, most of the time I won't memorize code snippets or algorithms, instead I'll have a mental link pointing to the name of the algorithm or the specific page, that I can search up, find, and then implement.
This is the total opposite of what school tries to do. They try to force memorization, which should come naturally. A better way to do it is to let the students have the equations and the necessary information that they need, then they can fit the puzzle pieces together to solve the problem. We live in the age where everything is becoming more and more documented, and we're still forcing memorization on people.
Anybody who has a credit card, mortgage, or retirement account needs to have a high-school level understanding of math.
Disagree - or maybe I just don't know what high-school level is.
Kind of going in the same direction, I wonder whether it would be good to have children start out learning math in the context of accomplishing concrete, real-world tasks that require mathematical problem solving, and then only gradually abstracting from this concrete starting point if an individual seems to have an aptitude for math.
Example: learning statistics in the context of gathering information about the health of people in a village.
Example: learning statistics in the context of gathering information about the health of people in a village.
You cannot learn without learning. And yes, that means some memorisation.
These problems are very simple and surprisingly fun to work with. Their purpose is to develop "thinking culture" (Arnold says this in the intro). He also writes that "the worst at solving these simple problems are Nobel and Fields prize winners" :) Now, I think that's Russian hyperbole or poetic licence, if you will, but it does illustrate an issue I run into nearly daily teaching maths to 13 - 15 year olds: Some of the kids think in ways I have a hard time understanding and often solve problems in ways which are difficult for me to grok because they are not using mathematical language to explain what they're doing. This happens especially with kids who generally perform poorly in maths. They solve problems similar to these in ways I have a lot of trouble understanding because it's like we are speaking two different languages. These problems are trivially easy if you translate them into simple equations (the ones I've done, anyway). The problem is I don't think that's what Arnold means when he says "Thinking culture", so I suspect I might be cheating ;)
Edit: Nr. 13 is wonderful :)
Edit: Nr. 13 is wonderful :)
I'm looking at the first question and wondering:
• What is the intended path the student is supposed to go through to figure this out? Guess and check? Some thing more specific?
• What would you say is the generalizable lesson for the student? How does solving this problem, which is an edge case, help you think about other problems in the future?
• What is the intended path the student is supposed to go through to figure this out? Guess and check? Some thing more specific?
• What would you say is the generalizable lesson for the student? How does solving this problem, which is an edge case, help you think about other problems in the future?
If Misha is 1p short, and combined with Masha's fund still cannot afford a book, how much money could Masha have possibly had?
Derivation of answer to the problem #16 here: https://datagenetics.com/blog/may32013/index.html
On the subject of teaching math to kids I found two books quite fun (and my daughters have been enjoying them as well).
1. Moebius Noodles
2. Avoid hard work!
Both of these take a playful approach to teach quite advanced mathematical concepts. What I particularly liked was a focus away from calculations and numbers.
The argument given in one of the books is that starting to teach math by counting and then calculation is like only starting to read books after children learned the alphabet. Children are very capable to figure out more advanced maths concepts even without being able to calculate fluently yet.
Both of these take a playful approach to teach quite advanced mathematical concepts. What I particularly liked was a focus away from calculations and numbers.
The argument given in one of the books is that starting to teach math by counting and then calculation is like only starting to read books after children learned the alphabet. Children are very capable to figure out more advanced maths concepts even without being able to calculate fluently yet.
Those books look cool, and it appears they are available for download on a name-your-price basis (literally says "Type the amount (from zero to infinity)"). What a generous and cool idea!
The struggle is real. Someone help me out here please:
>5. Two old ladies left from A to B and from B to A at dawn heading to-wards one another (along the same road). They met at noon, but did not stop, and each of them carried on walking with the same speed. The first lady came (to B) at 4pm, and the second (to A) at 9pm. What time was the dawn that day?
I think dawn was at 6am. This assumes that Old Lady A travels the full distance in 10 hours and Old Lady B takes 15 hours to travel the same distance.
I don't know how I arrived at that answer other than some sort of trial and error. I don't like it.
Any advice?
>5. Two old ladies left from A to B and from B to A at dawn heading to-wards one another (along the same road). They met at noon, but did not stop, and each of them carried on walking with the same speed. The first lady came (to B) at 4pm, and the second (to A) at 9pm. What time was the dawn that day?
I think dawn was at 6am. This assumes that Old Lady A travels the full distance in 10 hours and Old Lady B takes 15 hours to travel the same distance.
I don't know how I arrived at that answer other than some sort of trial and error. I don't like it.
Any advice?
Say the middle point is M. We know that for each person the ratio of time to go between M->A should be the same as to go between M->B since the distance is fixed only the rates are different.
First lady got to B at 4 pm so her M->B is 4. Second lady got to A at 9 pm so her M->B is 9. Since they both left in the morning, if we call the first lady's M->A time to be x, we know that the second lady's M->B has to be x as well.
Using the fact that the ratio of M->A : M->B should be the same for both of them, we can set up the equation
4/x = x/9 ==> x^2 = 36 ==> x = 6
Since getting to M was at noon and 12-6=6, dawn was at 6 am.
(For what it's worth, I did trial and error too cause I was lazy and didn't feel like thinking... then noticed the nice pattern and back justified it. :P)
First lady got to B at 4 pm so her M->B is 4. Second lady got to A at 9 pm so her M->B is 9. Since they both left in the morning, if we call the first lady's M->A time to be x, we know that the second lady's M->B has to be x as well.
Using the fact that the ratio of M->A : M->B should be the same for both of them, we can set up the equation
4/x = x/9 ==> x^2 = 36 ==> x = 6
Since getting to M was at noon and 12-6=6, dawn was at 6 am.
(For what it's worth, I did trial and error too cause I was lazy and didn't feel like thinking... then noticed the nice pattern and back justified it. :P)
The text is confusing to me.
>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...
>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...
That's exactly how I read it. It was the first obstacle to overcome. "OK, they're walking different speeds, and the statement means that neither of them changed their own unique speed over the course of their walk."
It took a bit of convincing myself, but eventually I became confident that it did not mean that they were each walking at the same rate as each other after their lunch meeting.
But even then, I struggled with figuring out how I arrived at 6AM.
I like the method of switching to the 24 hour clock (as opposed to the AM/PM clock), and setting their arrival times as 16:00 and 21:00, with their meeting at 1200, then solving it.
It took a bit of convincing myself, but eventually I became confident that it did not mean that they were each walking at the same rate as each other after their lunch meeting.
But even then, I struggled with figuring out how I arrived at 6AM.
I like the method of switching to the 24 hour clock (as opposed to the AM/PM clock), and setting their arrival times as 16:00 and 21:00, with their meeting at 1200, then solving it.
just summing it up, v{i} are ladies speeds:
MA = v1*x = v2*9
MB = v1*4 = v2*x
→ v1 = 9*v2/x
→ 9*4*v2/x = v2*x
→ x = 6After their noon rendezvous: One lady traveled some distance in 9 hours, and the other lady traveled some other distance in 4 hours. Before their noon rendezvous: you switch both the distances and the ladies. If x is the number of hours before noon that they traveled, you thus get that the ratio of 9 to x (what the first lady covered in 9 hours, the second covered in x) is that of x to 4 (what the first lady covered in x hours, the second covered in 4). Thus x is the geometric mean of 9 and 4, or in other words, x=6 and sunrise was 6 hours before noon at 6am.
Simply translate the words into equations:
(16 - t)v = d
(21 - t)w = d
(12 - t)(v + w) = d
Let C be the point where they meet in the middle and s be dawn.
It takes Lady 1 `12 - s` time to walk AC and 4 time to walk BC. It takes Lady 2 `12 - s` time to walk BC and 9 time to walk AC.
Let's find the ratio of time it takes Lady1 to walk a distance to time it takes Lady2 to walk a distance. The ratio should be the same for all distances since their speed doesn't change.
Thus, (12-s)/9 = 4/(12-s) <=> s^2 -24s + 108 = 0. The quadratic formula gives us s = 6, 18. Since 0 <= s < 12, s = 6.
It takes Lady 1 `12 - s` time to walk AC and 4 time to walk BC. It takes Lady 2 `12 - s` time to walk BC and 9 time to walk AC.
Let's find the ratio of time it takes Lady1 to walk a distance to time it takes Lady2 to walk a distance. The ratio should be the same for all distances since their speed doesn't change.
Thus, (12-s)/9 = 4/(12-s) <=> s^2 -24s + 108 = 0. The quadratic formula gives us s = 6, 18. Since 0 <= s < 12, s = 6.
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> 3. A brick weighs one pound and half the brick. How many pounds does
the brick weigh?
huh? Good luck 5 year old. This is how you get kids to hate math.
huh? Good luck 5 year old. This is how you get kids to hate math.
No. This is how you get kids to love math, if you do it right.
There's an American theory that math problems should be doable, that kids should score 90+% if they're doing well, and that struggle makes people hate things.
That's contradicted in American sports, where coaches push people really hard, boot camps, frat hazing, and cult indoctrination.
There's an Eastern European theory that math problems should be hard, interesting, involve struggle, and often too difficult to solve.
On the whole, Eastern Europeans seem to do better for turning out kids who love math.
There's an American theory that math problems should be doable, that kids should score 90+% if they're doing well, and that struggle makes people hate things.
That's contradicted in American sports, where coaches push people really hard, boot camps, frat hazing, and cult indoctrination.
There's an Eastern European theory that math problems should be hard, interesting, involve struggle, and often too difficult to solve.
On the whole, Eastern Europeans seem to do better for turning out kids who love math.
> This is how you get kids to love math, if you do it right.
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?
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?
The point is this: you hand out problems that the kids can solve after some struggle with the problem to give them confidence that they can solve the problems that come their way. That's actually a concept in German pedagogy.
The American way is to drill the kids with an algorithm and then hand them 20 more problems that are solved with the same algorithm, no insight required.
The American way is to drill the kids with an algorithm and then hand them 20 more problems that are solved with the same algorithm, no insight required.
Classic Eastern European pedagogy is different. You give kids problems many /can't/ solve a lot of the time to build resilience.
Kids in the West come out with a fear of failure. Good problems in Western pedagogy are the German style ones, where kids struggle for a bit and then get it, to "build confidence."
Kids in the (Soviet-era) East come out understanding there are problems which are easy to solve, hard to solve, and impossible to solve, and you'll face all three in school and in life. You won't know which one you've got until you've tried.
Kids in the West come out with a fear of failure. Good problems in Western pedagogy are the German style ones, where kids struggle for a bit and then get it, to "build confidence."
Kids in the (Soviet-era) East come out understanding there are problems which are easy to solve, hard to solve, and impossible to solve, and you'll face all three in school and in life. You won't know which one you've got until you've tried.
> This is how you get kids to love math, if you do it right.
...
>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).
So you agree, this is how to get kids to love math IF you do it right?
...
>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).
So you agree, this is how to get kids to love math IF you do it right?
[deleted]
This particular word problem is a clever one that can be sold a with different mental models (variable in an equation, imagining a brick being split, and probably other ways I haven't thought of). But it's a dumb problem with no reasonable analogy to real life. Who weighs things in comparison to a fraction of themselves plus a basic weight unit? I wonder if the problem is confusing not because of the wording, but because people rely on inferred context to understand language. This context is asinine, so people might think, "Clearly, that cannot be the intended meaning. I must have misread it."
Exactly. If the problem does not involve the "Aha! moment" the kids won't love it. And there's no way to have both "Aha!" and 90+% scores.
I actually got to hate math in grade 3 with such problems. Nobody explained to me that I could just write down an equasion, at least my father (physicist) could not. And all I saw in the magazines were cryptic answers like "it's plain obvious that for Jose-Rammstein conjuncture, x = 10"
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.)
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.)
I would not blame the problems here, but rather the lack of support.
No, because this is written in broken English, and is ambiguous. If you want to make kids think that math is all about "ha, gotcha! Technically this question could be interpreted to mean X" then by all means this is an excellent question. Anyone with a basic understanding of English would have phrased the question differently, unless they were a Russian with a mediocre understanding of English syntax, or were purposely trying to be clever and ambiguous. Both are probably true in this case.
But the actual math in these is fairly trivial. The entire challenge is in deciphering the wording.
> 12. A tide was in today at 12 noon. What time will it be in (at the same place) tomorrow?
These problems seem so poorly organized and motivated, I don't understand the curation in putting them together in one document, especially one that goes from ages 5 to 15.
Math problems shouldn't feel like random problems where you have to squint really hard or be really clever to see the connection to life. They should build with deliberateness into a worldview or a recognized skillset that a child can later translate into life wins.
These problems seem so poorly organized and motivated, I don't understand the curation in putting them together in one document, especially one that goes from ages 5 to 15.
Math problems shouldn't feel like random problems where you have to squint really hard or be really clever to see the connection to life. They should build with deliberateness into a worldview or a recognized skillset that a child can later translate into life wins.
I can't even read this as english.
I feel these "problems" are more about poor grammar than they are math.
I feel these "problems" are more about poor grammar than they are math.
Replacing "the brick" with "a brick," I always thought the question was fine:
> 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.
> 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.
The problem might have be created when bricks were hand made and there was no assumption that different bricks have same weight. So it was important to stress that it weighs 1 pound and half of itself.
Exactly. It would read better as "The weight of a brick is equal to one pound plus the weight of half an equivalent brick"
i.e. x = 1 + x/2
i.e. x = 1 + x/2
I could parse it fine but if I were writing this I would include more redundancy e.g. "weighs one pound and half the weight of the [maybe use our to suggest to the reader it's the same one] brick".
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.
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.
in this case, the line between grammar and math is fuzzy - english (or russian) and mathematical symbols are two different languages here which are each capable of describing the same thing.
the challenge is to do the necessary translation and rearrangement
the challenge is to do the necessary translation and rearrangement
Very classic problem but yeah maybe a bit overkill for 5 year olds
http://jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/Sei...
http://jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/Sei...
That uses the wording "plus half a brick" instead of "plus half the brick." I think the "a brick" wording could be a little clearer for kids.
"One Pound and Half the Brick" means one pound is the other half. It's interesting to visually explain it to kids so they start to see that "fraction" (half) doesn't have to mean "weird number".
2 pounds ?
brick_weight = 1pounds + 0.5brick_weight
[deleted]
Only read the first 4 problems.
1 and 4 have no answer and I doubt teaching math by utilizing
logic fallacies does any good.
First depends on potential subdivision.
Fourth depends on the fill level of the vessels.
Don't think that is the right way to teach anything.
I don't think the answer to 4 depends on the fill level of the vessels. All that matters is that the volumes of the liquids in the barrel and the glass are unchanged, so that volumes of the foreign liquids will be equal.
> Fourth depends on the fill level of the vessels.
I don't think it does.
I don't think it does.
1 has an answer
I wonder if #27 is supposed to be proven without Fermat's Little Theorem. (The question is if p is an odd prime, then 2^{p-1} = pk + 1 for some integer k). Since p does not divide k, it follows from Fermat's Little Theorem that p | (2^{p-1} - 1).
Any idea where to find the answers? These are extremely intriguing to do, even now.
A paper with interesting problems for children: http://toomandre.com/travel/sweden05/WP-SWEDEN-NEW.pdf
What I would like is a book of projects/real world problems/applied maths relevant to kids from age 5 on that could be used to motivate them better than rote memorization.
Sunshine Math is a great set of math problems for grades K - 8. I couldn't find a publisher, but PDFs of scanned copies are available online.
The problem #13, mentioned in the introduction, is said to be really tough for academicians, but I'm not! Probably I'm just plain stupid.
He didn't specify the order (or/and script, if you prefer) in which the books were placed, so it's ambiguous.
Tomes of Pushkin have a natural order ascending from left to right. It may be a cultural thing, but it would be obvious for any Russian child 5 years old or older. No tricks here.
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.
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.
The same order a multi-volume set would be displayed in any bookstore or library in a left-to-right ordering country.
First question:
> 1. Masha was seven kopecks short to buy a first reading book, and Misha lacked one kopeck. They combined their money to buy one book to share, but even then they did not have enough. How much did the book cost?
My goodness, that's an impressive 5 year old.
> 1. Masha was seven kopecks short to buy a first reading book, and Misha lacked one kopeck. They combined their money to buy one book to share, but even then they did not have enough. How much did the book cost?
My goodness, that's an impressive 5 year old.
I wonder if there's an issue in translation. How could they "combine" their money if Masha had nothing to begin with? But if we assume quantities of money can be non-integers, then the problem is underconstrained.
Kopeck - "It is usually the smallest denomination within a currency system."
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.
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.
The solution to the second question requires half-kopeks.
I looked at that problem after posting my original and was utterly confused. Would you trust yourself with an answer involving half-pennies? Seems like an error. As others pointed out, in #1, unless the kopecks are integer, the answer is underconstrained. #2 requires non-integer kopecks.
The problem is that the girls can't afford to buy a book singularly.
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.
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.
I believe your reading is correct. The problem is stating "Misha lacked one kopeck". Consider this solution:
Masha has 0 kopecks
Misha has 6 kopeks
The book costs 7 kopeks
Masha has .5 kopecks
Misha has 6.5 kopeks
The book costs 7.5 kopeks[deleted]
[deleted]
Yeah, this was confusing b/c to "be short" does imply that you have some money, otherwise you're 100% short of buying anything!
Hint: It's much easier if you don't know linear algebra.
Can someone help me with this question.
I tried wolframalpha but the question seem to make no sense.
https://www.wolframalpha.com/input/?i=solve+for+y%2C+%5By+%3...
https://www.wolframalpha.com/input/?i=solve+for+y%2C+%5By+%3...
Got it: cost is 7
https://www.wolframalpha.com/input/?i=solve+for+y%2C+%5By+%3...
https://www.wolframalpha.com/input/?i=solve+for+y%2C+%5By+%3...
You can constrain the result to whole numbers by adding "y mod 1 = 0" as an additional constraint.
https://www.wolframalpha.com/input/?i=solve+for+y%2C+%5By+%3...
https://www.wolframalpha.com/input/?i=solve+for+y%2C+%5By+%3...
neat trick, thanks.
Please don't spoil it for the rest of us. @dang
Knowing the answer is 7 isn't a spoiler. The point is to work it out. You'd have to click the URL and see the WA input to be spoiled about that.
Honestly this confused me at first read through. If I'm understanding it right, the book costs 7?
I figured the same, but that's a curious definition for "combined their money". That means that Masha has no money, and Misha has six kopecks, so they combined 0 + 6 and are still short of 7. If the price was eight kopecks, they'd have one and seven each, and would have exactly enough. If it cost nine kopecks, they'd have 2 and 8, and would have more than enough.
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.
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.
> kopeck in half into a denga was around the 1917 revolution
Really? I think I had 1/4 kopek coin somewhere when I was a kid (old foreign coin). I don't think it was this old.
Really? I think I had 1/4 kopek coin somewhere when I was a kid (old foreign coin). I don't think it was this old.
I figured it did.
Clearly you have not seen primary school math questions in Singapore's curriculum.
I printed this, will try them all, then translate to my language and try with my kids. I love this stuff.
Some of the first "problems" have multiple answers. I don't think that is how anything should be taught.
First problem depends on whether or not smaller coins exist, fourth problem depends on the actual fill volume of glass and barrel.
Those questions do not have an answer.
Didn't read further than that.
This story needs a nerd-snipe warning
> 12. A tide was in today at 12 noon. What time will it be in (at the same place) tomorrow?
Am I a moron or is this a really bad problem?
Am I a moron or is this a really bad problem?
I had to look up the answer[0].
> Due to the Moon's orbital prograde motion, it takes a particular point on the Earth (on average) 24 hours and 50.5 minutes to rotate under the Moon, so the time between high lunar tides fluctuates between 12 and 13 hours, generally being 12 hours and 26 minutes. So, if the tide has affected the place at 6:00AM, then it would generally take around 12 hours and 26 minutes for the place to be affected by Ebb i.e 6:00 AM+12 hrs+26 mins=6:26 PM.
Seems like a pretty tough question for a child?
[0]: https://www.toppr.com/ask/en-my/question/if-a-place-is-affec...
> Due to the Moon's orbital prograde motion, it takes a particular point on the Earth (on average) 24 hours and 50.5 minutes to rotate under the Moon, so the time between high lunar tides fluctuates between 12 and 13 hours, generally being 12 hours and 26 minutes. So, if the tide has affected the place at 6:00AM, then it would generally take around 12 hours and 26 minutes for the place to be affected by Ebb i.e 6:00 AM+12 hrs+26 mins=6:26 PM.
Seems like a pretty tough question for a child?
[0]: https://www.toppr.com/ask/en-my/question/if-a-place-is-affec...
I agree - I don't see how this could be answered outside of guessing without outside information.
I assume that these are meant to be universally understood. If the tests were given in fishing villages, it may have been a much easier premise since much of their world is defined by the tides.
Some of those are not even Grammatical
"A brick weighs one pound and half the brick. How many pounds does the brick weigh?"
"A brick weighs one pound and half the brick. How many pounds does the brick weigh?"
#18 is impossible. This margin is too small to provide the proof.
blueplanet200's proof can fit in a sentence or two. So sounds like you came up with an entirely different strategy. I'm curious what it is (doesn't matter if it's not as elegant).
spoiler for how to prove it:
think of colors of chessboard (black/white alternating) and what colors a domino piece will always cover.
think of colors of chessboard (black/white alternating) and what colors a domino piece will always cover.
Nice.
I was confused by the L-shaped shading in the top-right corner of the diagram, though.
I was confused by the L-shaped shading in the top-right corner of the diagram, though.
No offense, but Masha is broke af
Damn I am dumb
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rcoumet(2)
There are also great math-oriented games these days (I had some good success w/ prodigygame.com[1]).
My youngest daughter is 6 and can solve simple multiplication and division problems. Sometimes she even surprises me. Some time ago, we were introducing ourselves to a new neighbor and the convo went somewhat along these lines:
- my son: how old is your dog?
- neighbor: she's 8
- son: she's so small, is she a puppy?
- neighbor: oh no, she's grown up. 1 dog year is about 7 human years, so-
- daughter [interrupting]: oh wow, so then she is 56!
The other day, she came to me beaming to explain how she had just solved 38/2 (by doing 40/2, 2/2 and subtracting the results). Gotta say it's a joy to see a kid that enjoys math.
[0] https://www.singaporemath.com/
[1] https://www.prodigygame.com/main-en/