Curious to see comments quipping that some of the math problems geared towards young kids are too hard. 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.
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.
> 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.
> 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.
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'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.
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.
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.
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 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.
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).
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...
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 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.
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.
> 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.
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.
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.
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.
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.
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 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".)
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.
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.
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.
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.
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).
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.
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.
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.
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/