Very poor take. The author clearly has very limited experience with raising kids. Most kids won't do difficult things if you don't push them. Playing music, learning to spell correctly, doing mathematics, and so on. A very small minority of kids will do all of that easily and for the fun, but you can't rely on it. If you don't push your kid to do their 20 minutes of piano every day, they will half-ass it and will stop after 1 year and conclude they are not good at music. Same for sport. Same for reading books. Same for maths. And you know what? It's your fault. You chose to be lazy and complacent and didn't push them because it's hard to be a good parent. And now you expect me to validate your laziness? Nah.
The really important thing is to teach kids to find their joy.
At the end of my second year of piano lessons, my teacher took me into her living room, and we listened to Glen Miller records for most of the hour. And then we had a cup of tea, and she told me, "this is the music that I love. I play piano because I love that music, and I want to be able to play it myself. What kind of music do you like?" I didn't really have an answer. So she told me that we should stop doing lessons, but once I found music that I loved, she'd be happy to teach me how to play it.
In my early teens, I discovered Miles Davis. Once I had found my passion, all the hard work became play. I actually ended up learning to play jazz guitar, not piano. Even the heavy lifting was pure joy, because it had purpose and meaning.
I didn't become great at mathematics until I discovered the joy in mathematics (another brilliant teacher handed me a stack of old math contests, and said here, you might find these fun. I placed 4th among 20,000 students).
I didn't learn to write well until I discovered the joy in writings. (An absolutely brilliant English teacher who made us assign ourselves our own grades, but broke his promise in the end by upgrading all my papers to A+'s).
And I gave my kids the room to find their joy as well.
But kids are going to have setbacks, they will reach a plateau in their craft (music, painting, art, sport, ...). You need also as a good parent to help your kids go through, to not give up, because even joy to do is not always enough. This is the hard part.
From my modest experience of being a ski/snowboard instructor and trying to raise 3 boys (now 12, 16 and 18).
his whole point was that once he found joy in it, he could excel. I'm confident there's loads of things I could be good at - I'm only good at the things that I enjoy putting effort into.
Yeah, but something vital happened: you learned the basics of music theory and how to sight read music - both prerequisites to jazz guitar (and something that most guitarists don’t know). Learning piano is a great way to step into other musical instruments.
My parents pushed me hard to do piano when I was around 10-12. After a year that went pretty well I was starting to get lazy and put very little work and investment into preparing for the next lesson. They still had me play piano a full year until they eventually gave up and bitterly told me what a waste my resignation felt to them.
20 years later, I got back to playing piano, and I can't thank my parent enough for having me to continue playing in my teenage years. Because it only took me a few month to be able to play pretty advanced piano sheets compared to some of my relatives who are struggling with the basics starting it in their adulthood.
Same for maths. I feel that a lot of people like the author of this blog post are being extremely misdirected thinking math can and should be taught in a fun or amusing manner every time.
Sure, a lot of topics in Maths can be made more digestible by "gameification" to help younglings develop an intuition.
But a very big part of Maths actually requires you to sit down and painstakingly crunch down the numbers/equations, memorize and learn when to apply the correct methods to solve some problems. And even though this part can feel fun and engaging after a while, you can't expect children to exhibit such interest right of the bat without having them first struggle with the classics.
Kids don't know better. Your role as a parent is to navigate along the fine line of forcing your kid to get good exposure to the (boring) activities we adults value and letting him enjoy what he enjoys.
Only in doing that will your kid open up to the world and grow up into a functional human being.
"20 years later, I got back to playing piano, and I can't thank my parent enough for having me to continue playing in my teenage years."
One of the tragedies of being young is that few have the insight to realize that the 'boring' stuff parents and teachers are forcing us to learn will actually benefit us and that eventually we'll be very thankful that they did.
My parents nagged me all the time about studying and even though I did my fair share of it I never fully appreciated how important it was until much later.
It's a strange phenomenon, one cognitively understands the reasons but one is isolated from the reality so one is somewhat distant from it. For example, one can get upset watching war footage on TV but being there is on another level altogether (soldiers often do not talk of their experiences because they know those at home will never fully understand).
In the same way, wisdom gained through experience is almost impossible to impart to a younger generation who has no actual experience.
> One of the tragedies of being young is that few have the insight to realize that the 'boring' stuff parents and teachers are forcing us to learn will actually benefit us and that eventually we'll be very thankful that they did.
I'm 40. I don't know, perhaps I'm still young.
I did not appreciate having to learn the boring parts. Learning things for the next exam so as to forget them in two weeks... I didn't see the point then and still don't.
I managed to get by with the minimum possible, fluked my CS education, then had a career earning an order of magnitude more than the average salary. Shrug.
Maybe I'm missing something else because of my lack of education? I don't know...
I upvoted all of the above posts because - all of them share some correct arguments.
* Training is hard.
* Using your training e.g. a bicycle race is fun.
* Training is easier, if you actually know why you’re doing it and recognize some progress.
My parents forced me to play piano, right up until I told them that I'll destroy our piano if they don't lay off, and any consequences they could think of would not stop me (I was normally an obedient child, but enough was enough).
That got their attention.
30 years later I picked up classical guitar and loved it! Do I thank my parents for forcing the piano on me? Hell no.
Like I commented in another post, piano gave you the foundation for learning classical guitar (and appreciating that genre of music). Very few guitar players can even recognize note names on a staff. You’re not going to get far with classical guitar without it.
If the kid isn't enjoying the piano lessons, will forcing them to do it for 20 minutes every day really be beneficial? Sure, they will now be able to do something - something that they absolutely hate... (also, why is it always piano that parents try to force on children?)
That’s such a depressing way to see things. I’m sure most people do something they don’t utterly despise, is only because they select for their local optimum.
If you live in the bubble where you experience this, congratulation you live a wonderfully privileged life, never interact with anyone or are totally oblivious to the experiences of all the people you interact with on a daily basis.
But also, many people choose to do something they hate so they earn more money. They could be just as privileged and choose not to, just so they can compete with the Joneses and consume more...
I think this is the crux. Nobody likes to fail, kids included. And their attention span is wonky too so they may not see much value in learning from failure/s since there are so many other attractive things asking for their attention and they would rather do them.
Insane you’re getting downvoted, this 100% and then some. There’s a clear difference in outcomes between kids taught self-discipline and those who are raised standard Anglo-American ADHD style once they become adults.
Different point of view: do you consider hunting in the wilderness to be difficult?
I do, it requires being still in miserable conditions for a long time, being cold, wet, mosquitos, and then usually still no success, but frustration.
But to my knowledge, no savage kid is in need of being forced to learn it.
"children sense your true passions and naturally want to join in"
And that is my experience as well. But if you stop childrens curiosity out of limited time and patience "Be quite now!" - stop them from helping, because they are not a help in the beginning and you are faster on your own - then of course they won't just start enthusiastically some years later doing with motivation whatever it is, you define as their arbitary target now.
> But to my knowledge, no savage kid is in need of being forced to learn it
Uhh then your knowledge is very limited because that is rather well documented. Also, why are you saying "savage" like an 18th century racist? Is that in fashion again?
Oh, I am obviously a racist, by glorifying indigineous teaching methods.
But otherwise can you show, where this is documented? The natives tribes where I have some knowledge, don't force their kids to learn in the sense that is talked about here. No need to - the whole culture is about becoming a good hunter (for male individuals). So indeed lots of peer pressure, but no individual forcing.
The really important part of this is that kids mimic what they see adults they like and respect doing. If their role models spend 6+ hours in front of the television every night, that’s what they’ll do. If their role models are playing music or sport, that’s what they’ll want to do.
Yes, but one of the problems with our civilization is that we typically do the important stuff out of our children's sight, and then come home tired and try to relax. So they do not naturally get a correct idea of what we do.
Oh yeah let's turn otherwise fun hobbies into a forced chore, that will surely be great for the kid. Forcing kids to learn to spell when they have learning difficulties (eg dyslexia) doesn't usually go well either it's just causing suffering for the kid.
The problem in this discussion is that people here seem to miss that both an excessively authoritarian parenting style is bad but also going full liberal and just letting them run wild is not the solution. Sometimes children need guidance an a gentle push.
Even as a adult I sometimes need to get pushed. I sometimes take guided courses so I don't skip over the hard but important parts of learning a new thing.
Just don't push your children too hard or you do more harm than good. Accept that they are not you and have different interests and needs. Like make them practice an instrument but give them a choice which one. And if after a few years they still hate it, well you tried. Maybe it is not for them.
I think the most important part is to start early. Make your kids interested in math, music, art, and sport, before they start school. Doesn't have to be anything sophisticated, simple addition and puzzles will do for math, etc. Then you have something you can later build on.
There are also ways to make things funny, including math. Most people say that they hate math, but then they do Sudoku. So, try to make more math like this. Not all math can be transformed to funny puzzles, but after a few the kids will get positive associations with the subject, and will be more willing to learn more.
Doesn’t seem to be an issue for Asian families or ones coming the from former Soviet block as well as Jewish ones. These groups as adults tend to outperform others. There’s a reason for that: early childhood discipline and consistency being built into their cultures.
> My kid will do everything as long as it is interesting
Difficult things aren't interesting locally, you have to practice boring things in order to do interesting things in difficult subjects. Some kids do practice boring things if you just ask them, but most do not.
Sample size 3 here and they are all adults and all STEM grads.
You have to push them, but push them right. That's a combination of coercion and encouragement and helping them avoid procrastination. There are hills to climb and they need helping over them to where the good stuff is.
I remember my eldest crying over ratios at the dining table. Then algebra at the kitchen table. Then crying again at real analysis in the pub with me. She graduated with a first in the end.
The important part is to start as early as possible and absolutely not trust the school/teacher or kindergarden staff. They are badly programmed to reinforce kids in what comes easy to them and stop encouraging them after less than a handful of attempts.
If you have to restart later, no matter at which point, even up into 'the kids' 20s ( ultra late bloomers, slackers, kids disgusted by most people for reason Z, drug- or "condition X"-induced deadbeats, repressed kids with and without ADHD, failed or successful attempts by psycho-social environments ) understand three things:
1) you are not pushing, even if you are, you are demanding sth for the sake of your child AND yourself. YOU WANT THIS first and foremost. It's not a bad thing, fuck what the little fucker wants.
It's imperative for the kid to know that YOU WANT THIS no matter the obstacles. You want to see the process and result. It's a form of accountability, I guess. Kids pushing back is some dumb implicit way to check how important THEY and THE THING really are to you _or someone else_ (that counts for the ugly stuff, too). It's part of our evolutionary, hard-coded OODA loop.
2) just start at the very beginning, so that it's easy, almost effortless. The kid will be annoyed on most of the difficulty increases, it always depends on the sub-topic so don't back down. Even 20 year olds will catch up with their successful piers within some time. Neuro-genesis is awesome. Most 'grown up' stuff is child's play and a matter of baseline-human character anyway.
3) your stress level is what matters. Stay cool, be equanimous, serene, check your posture, voice, tone, the discussion won't last 5 min and will be worth it.
> Most kids won't do difficult things if you don't push them.
I guess it's true for adult humans, and other creatures as well. Instead of pushing, however, you should consider using other motivational methods (a simple prize for accomplishing something that you want from your kids works very well). Pushing can cause alienation and hate, which could affect their entire adult life.
The world needs less Asian kids being forced to play piano since a very early age and it needs more kids (Asian and not only) that are left to explore the world.
Without that exploration the kids won't make the world their world, at best they'd only make it a bad approximation of what we, our (older) generation, best think that that world should be.
Except they are now, wealthy Chinese are one of the most rapidly growing segment of tourists. Wanna guess what the wealthy successful ones learned as kids? That’s right: piano and math.
I prefer to live among educated people, thank you. I prefer my peers to go through forced history lessons, forced math lessons so they don’t tank my government, and biology lessons so they don’t tank the health system. Yes they won’t be able to determine their gender, but that will give me grandkids, thank you very much.
Same goes for piano or sports. Yes we need to pull people upwards, otherwise we’ll all become fat americans.
I kind of get the both sides of the argument. It kinda feels wrong on one hand. Lots of people agree. The other hand, it kinda, just works. Great. Everyone's hit with down-trend TFR anyway so maybe not it.
> at best they'd only make it a bad approximation of what we, our (older) generation, best think that that world should be.
Also, this part in GP doesn't feel exactly right to me. The problem doesn't seem to be in education, but rather lack of systematic resistance in current systems of society against humans weaponizing the system as tools to hamper progress of humanity as means to win minor inner struggles which is stupid. But the world doesn't seem to be moving in a wrong direction, only slowly.
Asian kids in 80s dreamed of bunches of permanent artificial space habitats running on fusion reactors. Still do. We've only gotten ground based fission reactors and space motor homes since then. But at least we are moving in that direction, just slower than at the ideal rate.
China's just done a humanoid robot marathon event. The winner completed the race. They're definitely in the future. US is, in a state not in line with site guideline to describe. And the latter is supposed to be more correct state than the other? How is that possible?
What do you mean by "science" in this context? For example, both biology and anthropology are sciences, but biology can tell us that people evolved from apes, while anthropology can tell us that a specific tribe believes that they were created by a flying serpent. Both of them are sciences, both of them can talk about the same topic (the origin of humans), but they take a completely different perspective on the topic.
Is the "transgender science" you talk about more like biology, i.e. describing how things are, or more like anthropology, i.e. describing what some (sub)cultures believe? Those are not the same things.
Because "forced history lessons" doesn't help said kids understand what history really is about, it just helps them accumulate facts, if that.
I'd say the same thing applies to math, where one can't really start understanding math until said kid is already an adolescent (unless they're a young Euler or something), so it always baffles me when I see parents filling their young kids with (fancy) arithmetics, most probably making said kids future therapy patients, all the while lauding themselves (the parents do, that is) that they're teaching their kids "maths".
Related, one of the best maths teachers I've had (this was back in high-school, in the mid-90s) was very quick to point out that we should forget almost all "maths" we had learned in elementary school, and the he very soon started to explain to us the definition of the real numbers. Or maybe this is just an Eastern-European thing, who knows? Maybe further West they do confuse arithmetics with maths until the Uni' years.
You mention Eastern Europe, are you by chance familiar with the Hejný method of math education in Czechia? Because that introduces some "math beyond mere arithmetics" concepts to the elementary school education.
Sometimes, it is possible to create a less abstract version of a more abstract thing, and thus introduce the seeds of the concept to children much younger. For example, "solve the equation 2x+1=7" is abstract, but "Peter decided to use a # symbol for a specific number, and he didn't tell us which one, but we found in his notes that # + # + 1 = 7; can you figure out which number is # ?" is simple to understand for a very young child, even if the child can only solve it by trial and error.
You’ve got it backwards: the future therapy patients are kids who are not taught discipline and persistence. Those who aren’t struggle as adults as the real world is harsh on vibe based living. Also, all those “useless facts” eventually build up on each other. They are prerequisites of knowledge and mastery.
Not if access to those things are limited while providing opportunities for other things that people enjoy.
You don't replace enjoyable things with unenjoyable things and expect the child to become a well-adjusted adult. You give them alternative enjoyable things.
Managing a child's burgeoning dopamine regulation system is a primary function of a parent. Abdicating that function for quick fixes is a form of neglect, in my opinion, just like feeding kids sugary cereals.
It is not just me who "provides the world" to my children. Also their classmates, etc. And the internet: even when someone uses it to achieve a purpose, there are various ads and algorithms that try to turn you towards something else.
A friend of mine taught remedial math at UW to incoming freshmen. She would write:
x + 2 = 5
on the blackboard and ask a student "what is the value of x?" The student would see the x, and immediately respond with x means algebra, algebra is hard, I cannot do algebra.
So she started writing:
_ + 2 = 5
and ask the student to fill in the blank. "Oh, it's 3!"
The semantic meaning of a blank is much better understood to everyone than an arbitrary letter like 'x'.
People just want to know why it's x and not something else or how a letter can have value. They might even think how can 24 + 2 = 5? They just want something to grab onto and nobody is really teaching the concept of a symbol in a math class.
I don’t recall the exact age, but when I was doing math in primary school (somewhere around age 9/10) we were absolutely using symbols - “Paul has two apples, and the basket can hold 10 apples. How many more apples can Paul put in the basket” is the same as 2 + x = 10
We did these sorts of problems for a long time, with addition/multiplication/fractions, and even when we started doing actual algebra the problems were introduced the same way “let’s look at a problem we’ve solved already, and write it in a different way”.
This becomes even more true in higher level maths where programming language style functions would make everything vastly more clear, and easily typeable, than the traditional Greek symbols. sum(x+3, 1, 4) is just so much more clear (and consistent when generalized across other operations) and practically as concise as the mathematical way of expressing that which I cannot even type. Multiple variables would be a bit dirtier, but still much cleaner than the formal expression.
Interestingly mathematical symbols in the past also regularly evolved. Then at some point we just stopped doing that and get stuck in a time which is arguably no longer especially appropriate. So we're left with rather inconsistent symbols, oft reused in different contexts, and optimized for written communication.
The formal language of math is intensely optimized for rapidly communicating with yourself 90 seconds in the future, when doing a proof or calculation, turning paper into working memory. It does seem silly to use the same language for communicating with others across unkniwn but deep chasms of context. Its remarkable that it works at all
For the purposes of education, it is important to keep in mind that "optimized for performance of a highly trained person" and "optimized for understanding of a complete beginner" are two different things.
I often see people make the mistake of trying to teach inappropriately abstract things to small children, because that's what the pros do, and we want the little kids become pros as soon as possible. Problem is, trying to skip the fundamentals is only harmful in long term.
First kids need to learn what all that stuff means, and then we can proceed to teach them the shortcuts.
The strangest part about mathematics culture is that there is a culture of vibing the notation.
Nobody in school ever tells you that there are glossaries on Wikipedia that tell you the meaning of the symbols. You're supposed to figure it out yourself using vibes.
The way mathematic notation is taught is inherently unstructured. You're expected to just get it.
its silly. itd be like introducing first year programming students to advanced maps/filters/anonymous function syntax, instead of the easier to understand for loops and if/else statements. math's "no true scottsman" approach to teaching only hurts itself in the long run.
I'm not sure if it would be easier to explain a map / filter to a first year student vs implementing the patten manually using a for loop and if statement...
Seems like a pretty easy example to make practically, for map have a collection of things, say balls or black. Pick up each one and do a thing to them, paint them blue for example.
For filter do the same except have two different colour balls, if they are yellow they get thrown away, of they are blue they get put in a bucket.
A for loop doing exactly the same you would need to explain the topic at hand, as well as explain iterating an index etc...
Explaining loops is independent of the concepts of collections though. It's also more general, since map/filter/reduce use some kind of loops under the hood anyway, the fact that probably shouldn't be ignored in education process. Unless of course you go with pure functional recursive iterator, but good luck explaining that one.
Maps and filters also require understanding of higher order functions and the very idea of passing function around as a value. I would argue that implementing map/filter with a loop and then demonstrating how this pattern is generalized as .map()/.filter() functions is better and more accessible
It's hard to debate that mathematical notation has a lot of room for improvement. High level algebra is very cryptic and often looks like an arcane incantation rather than something comprehensible for an unknowing person.
That said, as a person who moderately enjoyed math in high school and university, this functional notation would make me hate math infinitely more. It's would look like Lisp, which, at high level, looks just as cryptic as algebra. The sheer amount of braces and mistakes that would be made when reading and writing them is nauseating.
Infix notation, for all its flaws, provides important visual aid for understanding the structure of the expression (the sum of two fractions looks very different from fraction of two sums for example). Whereas with functional notation it's like working on linear textual representation of abstract syntax tree. Trust me, nobody wants to read, write or transform one by hand
Thanks for this comment. My secret shame as a programmer is that I haven't really learnt much maths, stopped at 16 in school. Writing out the sum function like you did makes perfect sense to me immediately.
What I should really do is create myself a cheat sheet of symbols to code...
The problem is, in many MANY MANY schools, teachers are more like social workers that have to compensate for utter horrifics outside of school. You got a ton of children so poor they didn't have breakfast which means their first (and all too often: only) meal will be the school-provided lunch (Covid showed that - a bunch of schools were open at least for lunches). You got children that are literally homeless and living with their parents in some car on a Walmart parking lot. You got children whose parents are in and out of jail. You got children living with their siblings in way too small, pest and mold ridden "apartments". You got children whose parents don't have money to pay for basic school supplies. You got children who are dealing with mental, physical and sexual abuse. You got children where the parents are constantly on drugs or seeking for drugs. You got children with a drug dependency on their own - if they're lucky it's just tobacco or weed, if not it's opioids. You got children with parents or siblings with serious mental or physical health issues. Or you got children with their own mental and physical health issues, or if you want it worse, children with these issues but without access to any kind of treatment. You got children that are being weaponized in nasty divorces. You got children that are being weaponized by street gangs. You got children committing crimes from petty theft to dealing drugs just to survive. You got children that have to literally work (and states like FL pushing to have more working children). You got children having their own children already (either from sexual abuse, from under-education about their own bodies, or intentionally because they fell for some stupid challenge/dare). You got children dealing with bullying, you got some who actually are bullies because they have no other way of dealing with their emotions or getting lunch money. You got children with parents with about zero interest in them. You got children who worry that they'll come home and find out their parents got snatched and disappeared by ICE. You got children who worry that ICE will storm their classroom and deport them. You got children who worry they might not survive the school day because someone will shoot at them. You got children who are constantly on the move because their parents' employment/deployment requires absolute mobility. You got children who are LGBT and have to deal with ever increasing hate against them (and LGBT youth already had significantly higher suicide rates than before the GQP made it a culture war issue).
The US doesn't have any kind of system to help these children but schools and libraries, both are horribly underfunded (there's some school districts where teachers gotta take up second jobs because the government can only afford paying them for 4 days a week), and all too often teachers have to pay with their own money for students' school supplies.
And on top of dealing with these kind of nightmares, they actually have to try and teach these children something - even if the children in question aren't anywhere near a headspace where they can actually learn.
One of my school math teacher had the same approach in another way: We were expected to use greek letters, not latin ones.
Same reasoning: It showed us kiddos that the letter was insignificant compared to the concept expressed by the letter.
So my take would be: Your friend taught the students for the first time what they were actually doing while handling equations with "a letter in it". That is no problem of algebra in itself. It just means their previous teachers sucked.
There is a game called dragonbox algebra which I'm currently working through with my son and is an absolutely fantastic approach to this problem. Sadly its now part of a horrendous subscription service and is hard to access. I find it really sad that we've had computers for decades and there are so few good maths games like this.
There are two sides to this. The system or method might be bad but also a determined person can go all the way and perform at a decent level if they put in enough time.
Even if the system was better the person still has to be able to motivate themselves and put in the time.
I’ve always found that an indictment of math education — and spent many, many hours discussing it.
When teaching addition, workbooks commonly use a box, eg, “[ ] + 2 = 5” — and first graders have no conceptual problem with this. Somehow, we lose people by the time we’re trying to formalize the same concept in algebra. There’s been many times I’ve written a box around letters in a problem and asked students “what’s in the box labeled x?”
I agree. I think the actual problem is that the student is trying to comprehend what it means for anything to have mathematical value other than explicit numbers.
Numbers and letters are taught together, but not as symbols. Letters are taught with sounds and numbers are taught with counting. The notion of a symbol isn't really emphasized much.
I would explain it more like after
[ ] + 2 = 5
what happens if you need more than one box for a complicated problem? Teach the idea that saying box #3 is equivalent to assigning an arbitrary letter for whatever reason you want, but that people more familiar with math prefer letters because they stand in for words that describe what the number is for. You might want to use 'c' for the number of cats you're trying to figure out.
In a room of five animals two are dogs. How many cats?
a = 5, c = ?, d = 2
a = c + d
so... 5 = c + 2
what is c?
Light bulb goes off: "You can do that?" Yes, you can do whatever you want and it's not all about carrying the one or whatever other rote teaching they've been given. They can get creative and be engaged, and then you let them know that actually there are some conventions people like to use for what they're trying to do. They might even believe they've invented a new idea. At least they're having fun.
most likely this very unfortunate misnomer started with fortran, where it was deemed lucrative to point out "how much programs look like mathematic formulas!".
not only is this overloading a symbol (equality) with a completely different meaning (assignment), it is also a poor choice typographically, as it represents a directional operation with a directionless symbol.
using an arrow for assignment is much better.
it's also worth pointing out that unlike most others, logic programming languages (e.g. prolog) have actual variables, not references to mutable or immutable memory cells.
I tell my kid that math is a language. You learn to speak it, just like you learn to speak any other language, slowly, by listening, understanding, speaking, intuitively recognizing patterns, rules and exceptions. When you start to become fluent you translate problems into math and solve them. At school they keep trying to make them memorize useful phrases, like a tourist that goes to Paris and learns how to say "where's the bathroom", "hello", "would you like to sleep with me", "thank you", "goodbye", etc.
If you are a physicist or an economist, you may be using mathematics as a language in the sense that you are using a mathematical description to convey an understanding of the natural world or the economy to your colleagues. But if you are a mathematician, you are interested in the mathematical objects for their own sake.
There is also a difference between the purpose of learning language and learning math. The goal of learning language is (often) to be fluent in it. In other words, the goal is to reach a level of proficiency which would allow you to not have to think about language and focus on the content of the conversation instead. On the other hand, the goal of learning mathematics is usually to be able to solve mathematical problems. Being able to do math without "thinking about it" is not usually a requirement.
"At school they keep trying to make them memorize useful phrases, like a tourist that goes to Paris…."
Like learning dozens of trig identities without any explanation about why one would need them. As I've mentioned elsewhere learning math for the sake of it isn't enough. For most of us math has to have relevance, and for that we have to link it to things in the real world.
Know what, brother? I tell you that studying the humanities in high school is more important than mathematics — mathematics is too sharp an instrument, no good for kids.
Stephan Banach quoted by Steinhaus in Through a reporter’s eyes, Roman Kaluza, 1995
Compare learning math to learning to bicycle. There is some some sweat and struggle that needs to be put in, before one "gets it". After this it can become enjoyable. I encouraged my daughter with practice exercises from a young age, but tried to avoid making it a drudge. She built up confidence and did well with it. She is also very hands on creative. She decided to study engineering and is working towards her PhD.
Those aren't nearly comparable. Riding a bike is one simple skill and as long as you're not racing that's enough for most people. Meanwhile learning maths is a years-long effort at best. I learned how to ride a bike within an hour by myself when I finally had a good reason to learn it. I can't say the same about maths.
Bike is fabulous self-correcting vehicle in most operation conditions. The trick really is just to learn to trust it when it is moving. And then what to do when it stops.
Math is layers upon layers upon layers. And then it also branches. Never really had willpower to learn it myself alone.
I'm damed sure I'd be much worse at math if I'd not been pushed in a formal environment such as a school classroom.
I liked math—especially calculus as it made sense to me—but parts became a drudgery when I could see no reason for studying them.
Right, there's always the kid in class who excels at math like a mini Euler and gets bored because the rest can't keep up but the majority of us aren't like that—doing Bessel functions and Fourier stuff as abstract mathematics without any seeming purpose can seem pointless and our only interest in them was to pass exams. (Teaching may be better these days but my textbooks never discussed the value of learning these aspects of mathematics.)
Later whilst studying elec eng/electronics it became very obvious to me how important these aspects of mathematics were. If I'd been given some practical examples of why this math was useful then I'd have been much more enthusiastic.
Same goes for the history of mathematics, I'm old enough to have had a small textbook full of log and trig tables yet if someone had asked me at highschool who John Napier was I wouldn't have had a clue. In hindsight, that was terrible.
Mathematics is often taught as if the student was going to become a mathematician à la Hardy or Ramanujan and I'm firmly of the belief this is not the best approach for the average student let alone those with few math skills.
Mathematics ought to be taught with the real world in mind for ease of understanding. For example, it's dead easy to represent AC power as a sine wave and from there use that mathematical fact to solve power problems. (Perhaps maths and physics texts should be written in tandem and synced to show relevance.)
Teachers need to take time to explain that math isn't just abstract concepts but that it's very relevant to everyday life and that tying up mathematical functions to things in the real world is actually interesting and enjoyable.
> Kids are born explorers. They naturally want to discover new things, including math.
That's true only until their senses are not shut off and attention is not fixated on screens. Exploration happens only when you have unused attention, sensory capabilities and need for a bit of hard work and risk-taking. Curiosity is less of a biological feature. It is a product of the need and the available resources (senses etc). All of these are missing now.
There is no need or motivation. And there are no available resources (senses, attention). There is no justification for exploration and hard thinking.
As someone who just finished school, I’m trying to figure out how to get genuinely interested in mathematics. I’ve never been particularly strong at it, yet I’m planning to enter a university program that demands a high level of math. The problem is, it’s hard to motivate myself to study math for its own sake. For example, I loved learning programming because it’s hands‑on—I can build something and immediately see the results. In everyday life, though, I rarely need more than basic arithmetic or simple sin/cos/tan trigonometry.
How do you develop a lasting interest in math when it doesn’t feel immediately useful?
Make it practical! Graphics programming involves linear algebra. Databases involve relational algebra. Machine learning involves requires calculus. You’ll naturally encounter hands-on tasks with tangible goals that involve learning new math.
Don't study it for usefulness, study it for beauty. Look for amazing insights.
Yes, you need some practical math as well. I did engineering, there's a lot of inelegant stuff there.
But that stuff actually tends to be right next to some very interesting things.
Here are three things you can find out.
First, there's more than one kind of infinity. You can't make a map from natural numbers like 1, 2, 3 etc to real numbers like e, 0.632268, sqrt(2) etc. Look for Cantor diagonalization.
Second, a random walk like a heads vs tails comes back to zero almost certainly. It also does so in two dimensions, like walking randomly in Manhattan. In three dimensions, it does not, and so for higher dimensions. Look for Polya.
Third. There is a way for you and me to communicate secretly, despite everyone in HN being able to see our entire exchange. Look for Diffie Helmann.
These days, there's a whole industry of people doing math videos with interesting stuff.
It's easier to appreciate math when you are disinterested in the results or applications, because the nature of academic topics near the core grouping of math/philosophy/empiricism is that they are discovered with a lot of meandering at first, and then sometime down the line they become repurposed into a direct application that can be learned by rote. School tends to instruct in some of the most directly applicable stuff first - the "three R"s" plus some civics and training aligned with national goals. And that means that school predominantly teaches associations between math and rote methods, to the disgruntlement of many mathematicians. The "meandering" part is left to self-selected professionals, so it doesn't get explored to much depth.
So I think a good motive for math study is really in games and puzzles, where the questions posed aren't about win/lose or right/wrong, but about exploring the scenario further and clarifying the constraints or finding an interesting new framing. Martin Gardner wrote a long-running column and a few books in this vein which are still highly regarded decades later.
One of my undergrad degrees is in math. As you study it, you learn to identify your assumptions (axioms), find or build interesting abstractions, prove properties about them (theorems), and then map all sorts of other things into those abstractions by figuring out that they're really the same thing. It's even more interesting when you start to find things that are different or question things you always took for granted.
Math gives you the ability to leverage the very structure and relationships of pure abstraction. It's quite the super power.
None of the specific things you learn studying math will be nearly as useful as the ability to think mathematically.
> For example, I loved learning programming because it’s hands‑on—I can build something and immediately see the results. In everyday life, though, I rarely need more than basic arithmetic or simple sin/cos/tan trigonometry.
Consider doing something that actually needs it. You like computer programming - consider making a game engine. It might be easier to learn when you can actually see that it is useful.
Keep in mind though that math is a lot of things. People obsess over calculus but that is just one type. Math is just as much the different types of symmetry in wall paper patterns as it is finding the derrivative. Don't be afraid to try different areas. If you dont know where to start, consider picking up "A Concise Introduction to Pure Mathematics" by liebeck which introduces a bunch of different math concepts and see if any feel more interesting to you.
I didn't particularly find (at the time) calculus, multivariable calculus, physics, etc. interesting as I didn't find the applications interesting at the time. I find these subjects representative of what you traditionally learn at school.
When I entered uni I discovered my passion for discrete math, algebra (groups, rings, fields, etc.), number theory, cryptography, theory of computation, etc. as they have a lot of application in CS.
That's really what did it for me - and also I had great uni lecturers. I wish they would have taught the subjects I like in highschool - the difficulty level is about the same.
N=1 datapoint here.
I studied physics in university and before I started I was not aware that physics is basically just math where the results sometimes relate to reality. The pure math courses I took were the most difficult and in the beginning I loathed them, because it felt so unattainable to get any intuition, let alone real proper comprehension for all the concepts they threw at us. For a long time I felt like I was just hanging on by threads and especially if I compared myself to those who had some innate interest in math or generally some really good intuition on the abstract concepts (or even prior knowledge) it was really demotivating. But I also felt like I had no choice but to continue and as time went on the I grew fond of it. And the feeling of being overwhelmed changed - that is to say I still was completely lost every time a new topic was breached and I could not understand even half of the proofs in class - but I did not feel so defeated about it. And I grew to like the feeling of actually completing the work sheets they gave us every week. The process of solving them was often excruciating but if you did the sense of accomplishment is real. I think for most people higher math is really difficult and that is part of why it is interesting. Another aspect I had to accept over time is that even though you can state a mathematical fact or conjecture in just a hand full of symbols or a plain sentence it does not mean that truly understand it, its implications or how you got there can be understood the same way that other prose can be. Sometimes you have to stare at, contemplate and scribble around one equation for days until you understand whats up.
If there was any advice I would give, then it's probably similar advice on how to stop procrastinating on anything that is difficult. Establish a routine first - find a spot that you will only use for studying this (like a spot in a library), start small, divide and conquer, accept that you will not understand most things easily, reward yourself for the small wins along the way, find an accountability partner or someone to study with if that's your thing, make a regular schedule with regular times where this is what you do - consistency is key, even if its just for 5 minutes, stack it onto other habits, see yourself as a scholar of math - it is what you do, lean into the discomfort, as enduring that is a valuable skill in itself.
If you love programming, there's quite a lot of programming where math is vital. Graphics, optimisation problems, cryptography, neural networks, figuring out if a hash works, projecting if an algorithm will scale...
The tricky bit is often that you need to learn some of the math before you can see how it's useful, but if you need stronger motivation, you might try diving into a slightly math heavy programming problem and learn the math as you go
I'm going to share my anecdote, because it may help, but everyone is different and your mileage may vary.
I'm a MechE by classical training (professionally I actually work doing software/network stuff, don't ask, DNS (screams internally)), so here's where it stood out for me:
Internalize what this simple example represents, think about why that's mathematically interesting, and start looking for where it applies elsewhere. You too could be roped into doing systems engineering at scales you didn't think people haven't already figured out.
University is not a good place for learning mathematics as most of your math instructors there will be very good at math and very bad at teaching people that are not already very good at math.
For whatever reason, many University programs use high level math classes as a filter to weed out 1st year students from that program. If university instructors had a genuine passion, and ability, for teaching high level math then they wouldn't accept that as an outcome.
That is unfortunately true; not only in the US, but all around the world. The particulars do depend on the instructor, and many if not most instructors try to be motivational, but the syllabus is perfectly clear: "this is a weed out class". And when it comes to test time, the syllabus wins.
The only thing I disagree with in your comment is about the instructors: they want to be employed, and they have to accept the syllabus and testing standards. It is not about passion and ability to teach (most, especially younger ones, are full of those); it is about meeting the departmental requirements.
Except maybe not calculus. I remember my calc class kind of being terrible because it was a weed out for other majors. Every math class that wasnt required though was great.
Probability/Statistics is a good excuse to learn mathematics, because paying a little attention one finds lots of day-to-day situations where is possible to apply it. For example, see the secretary problem[1].
I wish I couldve excluded everything past basic algebra and hopped right to statistics at a young age - I *loved* everything about the practicality of it, how it explained tangible relationships and illuminated the world. Algebra and calculus were so un-engaging I had those teachers calling me everything but a stupid child.
Perhaps think of them as solving logical puzzles. It's fun. Even though not always related to everyday tasks.
For me, it began many years ago when reading about Hilbert's hotel paradox. Turns out our laymen's understanding about infinity isn't as really refined.
I write mobile apps for living and indeed these stuffs are irrelevant for my work.
as someone who loved maths first but then do programming for a living, it's because solving puzzles is fun. I get the same dopamine hit whether it's a math problem, a coding task, or a video game level. but I think forcing yourself to like something is not the correct approach; you either like it naturally or you tolerate it for some other goals
Have you ever watched a video of a highly skilled tetris player? Where they fill the screen most of the way to the top and then suddenly they just combo the whole thing down and everything wraps up cleanly, and then they start fresh.
The feeling of "oh yeah, that was nice watching that mess turn into something clean and squared away" is where I get a lot of my joy from math.
But also, there are uses to math that you might be able to play with through every day, but you've never thought of those scenarios in a mathematical way.
I was walking today, and on the street there is a right angle turn. The inner portion of the turn is just a square right angle, but the outside of the turn is a radius. I started wondering to myself, if I want to be on the outside of the turn going into and exiting the turn, what would be different ways I could walk this, and what would the distance differences be.
Crossing directly across, to the inner corner and crossing directly across to the outer side again, would be 2w (for the width of the road w). Following the edge of the radius would (assuming perfectly circular), be 1/4 of a circle, so 1/42piw = 1/2 pi * w. The shortest route is a straight line, which would make a right triangle, so w^2 + w^2 = c^2, 2w^2 = c^2, sqrt(2) w = c
So crossing twice is 2w, following the edge is 1/2piw, and shortest path is sqrt(2)*w. Not super applicable, and I didn't need to do math to figure it out, but I was walking and bored, so I found joy in it. The fact that they all boil down to having w as a factor means I could figure out a nice ratio between all of them. And then I needed to mentally figure out what 1/2 pi was. 3.14/2 = 1.57... And I know that sqrt(2) is roughly 1.41 ish.
So now I know that crossing twice has a cost of 2, following the edge is 1.57, and direct line is 1.41. Following the edge is vaguely close enough to the ideal path to warrant not walking into the street to optimize the route, 1.57 / 1.41 is about ~110%. Whereas by defintion, a cost of 2 is going to be sqrt(2) times sqrt(2), so ~141% more than shortest path.
A few things to note here. First off, I'm aware that not everyone finds the same joy in doing simple mental math and thinking about problems mathematically even when there is no need to do it, but trying to think of things more minor trivial things mathematically may cause you to at least appreciate it more, which can grow into joy. And second, I wasn't doing any complicated math in my head. I just thought to myself "is it faster to cut to the inside corner and then cut back out... of course not, right?" and I was able to answer that definitively to myself. Did it matter? Was the answer probably obvious anyway? Probably, but I was able to _prove_ that. And I value facts. Finding joy in the simple things lets you build up more of a familiarity and view it more as a problem solving tool than a tedious thing to rote memorize.
A great way to build up math familiarity and see how other people find joy in mathematics would be to watch Numberphile videos on YouTube[0]. It's a bunch of mathematicians sharing things they find interesting about math. Some times are REAL hard to grasp, but some are just very interesting puzzles[1]. The puzzles don't always have clear immediate usefulness, but can often be described as "a mathematician wanted to know an answer, so they did some math to find out and prove something to themself."
Sorry, end of spiel.
tl;dr - find the joy in the simple things and use math as a tool to answer (even simple) questions to help highlight the usefulness.
The key element here is nurturing curiosity.
Since then i and my 10yr old have been sitting through a virtual math circle led by Aylean McDonald on parallel.org.uk an organization run by Simon Singh
It's not about forcing your kids to "do math", but to excel at important skills far before the benefits of being good at that skill matter.
The amount of homework/study per day that maximizes math scores on tests is significant, 1+ hours/school day by the time they're in middle school, with it helping even more for those who are starting out poor at math[0]. You'll note the referenced study doesn't even max out progress for any group - meaning most could have studied more and improved more.
I don't know any kids that voluntarily did an hour or more of solely math study per day. I know plenty that were forced, and ended up loving math or other technical fields as adults.
As a parent of young kids, obviously I haven't gone through high school yet - but I don't think many children who reach their potential in math, english, music etc will have no pressure from their parents.
> As a parent of young kids, obviously I haven't gone through high school yet - but I don't think many children who reach their potential in math, english, music etc will have no pressure from their parents.
Well it depends. I had no pressure from my parents to learn about programming but still got really good at it. Could I have gotten even better had I been pressured to "practice"? Perhaps. But then I also wouldn't like it for the reasons I do (I like making stuff, but not solving riddles) and it would feel like the dad sport situation.
I also played the piano for 6 years, starting out because I liked it. My parents didn't suggest it, but a few years in they were pushing me to continue even when I didn't like it anymore. Finished the first level of music school (6 years where I live) and haven't touched it since. Just to clarify, they weren't using any directly abusive tactics to keep me going, but they did put a lot of pressure onto it.
There's a lot of nuance to all of this and I don't completely disagree that we should occasionally pressure our kids to push their limits. What we often fail to acknowledge is that kids easily change their minds after a while. Just because they liked something at a certain point doesn't mean they still do. The easiest way to get a kid to dislike something is to make it a chore. Additionally, I think we need to ask ourselves whether it's more important to us to have a kid that's average scoring but has a (mostly) stress free upbringing, or one that excels but is stressed out by the time they hit high school. Kids absorb stress differently than we (adults) do.
2. A love of subversives and cheating the system. Basically, the guys writing leetcode cheating software are saints in my book. All subversions of the attempt to turn society into a meritocracy (a term which was originally supposed to be a slur/negative connotation - https://en.m.wikipedia.org/wiki/The_Rise_of_the_Meritocracy) is extremely good.
3. An advanced knowledge of TI basic, so I could cheat hard on every single school assignment I could get away with. AP Chemistry? I’ve got a symbolic stoichometry solver app! Calculus? CAS system in the palm of my hand!
Play stupid games with children, win stupid prizes. Maybe don’t force them to work like little slaves in their early life, and they won’t strike back at your society systems.
I had similar conclusions, but the other way around: absolutely no guidance. Fortunately by the time I programmed and sold the basic ti math exam solvers to by classmates for 2 euros a pop I had everything memorized.
Nothing like cheating the system to know the system
You realize your experience can't be generalized to anybody else except for those who were abused in the same way you were? It also isn't what people in these comments are suggesting should be done.
Your experience sounds awful but surely there is a reasonable middle ground between forcing a kid to do any math and forcing them to do it in 4+ hour sessions.
Raising kids is hard. Sad. And what do I know about it? Regardless, parents do need to get involved in their children's education. For instance, they should help their children prepare for entry exams into secondary school. This shouldn't take their child 4 hours a day. Maybe 10 minutes on some days, and 1 hour on others.
I think this isn't quite the same thing, never the less: I have dyscalculia on extreme mode (/dyslexia/autism), and I was forced to do math in the 90s in the UK, rote style. I don't know if they didn't know about dyscalculia, didn't care, or whatever, but holy hell I wish I'd never been forced to do that, it's still today a fairly painful memory and I'm in my 40s now. If you're gonna force kids into math, at least make sure they're not unable to process it correctly.
When I was having trouble learning multiplication my father made up a payment system. He made flash cards and I got a payment for every one I mastered (I had to get it right some number of times, not just once). I ended up with maybe $25 or $50 which was a lot for a kid in the 1970s.
My mother tried to give me $5 for every book of the Bible I read. I never took her up on it even though I knew about the basically freebies like Jude. I wasn't opposed to it, but it felt like –on the one hand– I didn't want to half-ass it and read a few books –and on the other– I really didn't want to read the entire Bible. So I guess that a completionist attitude prevented me from getting $30!
Oddly enough I found a great 'trick' for this. Kids hate doing math tests, but turn it into a competition and game and suddenly they love it. Print out a bunch of remedial problems, perhaps 50. And then give them 1 or 2 minutes to do as many as they can. It's just a contest to improve against your own scores over time, with prizes for the kids who score the highest after a month or whatever.
It's still literally just a math test/quiz, but somehow the context changes everything and even kids who really aren't into math were loving it, and also improving rapidly because the repetition helps instill intuition.
I imagine the part of a test people dislike is failing, and the consequences from failing. Framing it as a game without those emotional stakes fixes that.
If the teaching environment was set up to encourage learning rather than punish not having learnt yet we might not need these tricks, but that culture is slow to change
Something I've been thinking a lot about is "stealth edutainment" games.
When I was a kid, I remember "edutainment" games that were basically like normal computer games, except every so often a homework problem pops up.
I think that doesn't work super well. Better is a game which has you learning naturally, in order to play the game more effectively. For example, I've been enjoying the computer game Slay the Spire recently, and there is a great deal of mental math which is inherent to the game. If I had a kid, I think I might give them that game as a method to motivate them to learn arithmetic.
30 minutes of play per 3 days is such a brutal reality to acknowledge. One of the most wonderful experiences in all of life so drastically limited by the society we’ve constructed.
> One of the most wonderful experiences in all of life so drastically limited by the society we’ve constructed.
I could understand if someone was forced to work two full-time jobs (as my grandfather was), but I find it much harder to blame ‘society’ when so many of these situations are self-imposed.
It’s possible that I’m jaded from hearing a subset of parents complain about not having enough time with their kids but then get stuck scrolling their phone while kids want to play. I also know some parents who insist on having a spotlessly clean house every day and then complain that there is enough time to spend with their kids.
I’ve gravitated toward peer parents who have similar priorities in life which has indirectly made me happier. Seeing all of the parents in my friend circles prioritize spending time with their kids and being honest with themselves about their priorities has been unexpectedly helpful for my own sanity.
Again, nothing against parents who are really forced to allocate time elsewhere, but I’m tired of seeing self-inflicted problems of prioritization and time management be externalized as blaming society.
In some ways yes, but men have always been the ones to go hunt/farm for long hours and provide for the family, leaving the children home under the care of the mother/village for days or weeks at a time.
I would go so far as to say modern society actually enables us to be more involved in our children’s lives, especially those for whom remote work and home schooling are options.
ChatGPT makes it so easy to build a lesson/workbook for something your kid is interested in. I've used it to build workbooks on special relativity, tsolkovsky's rocket equation (including euler integration to build a scratch program), triangulation, logic gates, probabilities of simple dice games, etc. My pro-tip is to tell the LLM to format the document in LaTeX, so you get beautiful math typesetting.
You don't even have to get through the workbook. Get to a part that they need to understand better and make a detailed workbook on that part (for example, triangulation -> solving a system of linear equations).
For elementary school age kids, maybe even middle school, try getting them started with the app "Euclidea".
They won't think of it as math. It's gamified geometric constructions. Starts simple, "how do you bisect an angle" with a compass and a straight edge. It goes to a very high level that will challenge anyone.
If you formulate the warnings just right, you wouldn’t need to “force” it, as kids will be willing to look both sides themselves. They like to be alive.
And then your kids and their same generation would be replaced by their peer kids from hard working boys and girls from India and China. Unfortunately curiosity only works with brilliant minds. Normal minds plus curiosity is useless.
I think the advice is good for younger children. The author is using 14+11 as an example. Very engaged parents can have a tendency to overdo it, so it's probably a good reminder.
As kids get older, they need to learn how to struggle and overcome struggles. (I would still caution against "forcing" math.) But yes, you need to start engaging hard work and determination.
Btw, the two are not mutually exclusive. Young children should be praised for struggling at things so they begin learning that skill, too.
My daughter and loads of kids watched number blocks from around two or three up, I think it made quite a big difference- she's far ahead of where I was now, years later.
Kids do not understand the concept of "importance". At least no kid I've met. That part of their brain doesn't work. They'll trade effort for privileges or toys tho, and are little mimicry machines so they follow you if you use it.
yeah, I recoiled when the author of the post says "no bribing" - bribery is one of the most useful tools a parent has. I guess you could call it "incentive" or something, but really, it's quid pro quo.
Honestly it's so close to how the world works I can't believe 1. Avoiding loss of privilege and 2. Gaining new things as reward isn't the top two parenting tips.
But probably zeroth, most important, is modelling good behavior. Kids are mirrors.
Forcing is kind of hopeless. So is logic, and reasoning.
How children learn (they can't rely on a fully formed prefrontal cortex like adults) is very different than how adults learn (no fully developed prefrontal cortex until 25-26), learning about this can help a lot.
Learning more about the Reggio Emelia approach might help parents curious about this, it has been quite surprising how much is possible naturally. One of the best things to do is to relentlessly read to and with your kids.
Showing kids the math in every day things, especially things they already love is a helpful way of making it approachable, or at least aware.
Also, linking a topic to their interest's radar, encouraging curiosity, play in general, and letting them potentially discover it can go a long away.
When they've got something they want, teaching math and savings is a great thing. Understanding life is a lot harder without knowing a basic bit of math, and can be made a bit easier when doing it younger.
I had a math teacher that once made it clear, some stuff can just click, others is just about doing a lot of examples to learn the patterns. Doing math is very different than being creative with being comfortable to find it.
Today, I'd probably setup a good prompt to find a way for the child to share their mine to discover how they like to learn, and how they might like to learn faster and easier by taking some shortcuts through math directly or on navigating an ontology/taxonomy perspective.
When I was 8, I went to the library in our town a lot. My parents went there sometime to return their books. At some point I just stayed there when they would go home. First I was in the children/teenager section and soon in the general library, where I would read about programming and computers. I learned C by age of nine.
The "undeveloped PFC" argument is shallow, unspecific and usually just used to infantilize younger people. It may be useful if the child is under 6 years old, but at the time someone is 17 or older, it becomes essentially useless.
My learning process was always, and still is fueled by curiosity.
I'm fortunate enough that my daughter has an admirable interest (and talent) for Math since very early age. She even won a medal at a renowned nationwide Math competition when she was in Grade 5... competing in the Grade 10 category.
Maybe find an application of the subject that they might find interesting. I suppose if you can't find anything that interests them, then it's much harder to teach it.
For instance, perspective drawing might provide a nice application of 3D projective space, its subspaces, and perspectivities between those subspaces. Some of the theory of conic sections might be relevant too.
Computer graphics provides a nice application of coordinate geometry. This covers elementary algebra, Pythagoras's theorem, etc.
Even eating pizza can provide an application of differential geometry.
I tell my kids they can have letter cookies if they pick a word that starts with the letter, and can have 5 treats if they ask for 4 but know what "plus one means" or can have 4 if they recite "2 plus 2 equals ... ".
They're 3, so I don't expect that to scale, but I'm hoping it's normal reward-for-knowledge by the time we get report cards.
Forcing is kind of hopeless. So is logic, and reasoning.
How children learn (rely on the prefrontal cortex of their adults) is very different than how adults learn (no fully developed prefrontal cortex until 25-26), learning about this can help a lot.
Learning more about the Reggio Emelia approach might help parents curious about this, it has been quite surprising how much is possible naturally. One of the best things to do is to relentlessly read to and with your kids.
Also, linking a topic to their interest's radar, encouraging curiosity, play in general, and letting them potentially discover it can go a long away.
When they've got something they want, teaching math and savings is a great thing. Understanding life is a lot harder without knowing a basic bit of math, and can be made a bit easier when doing it younger.
I had a math teacher that once made it clear, some stuff can just click, others is just about doing a lot of examples to learn the patterns. Doing math is very different than being creative with being comfortable to find it.
Today, I'd probably setup a good prompt to find a way for the child to share their mine to discover how they like to learn, and how they might like to learn faster and easier by taking some shortcuts through math directly or on navigating an ontology/taxonomy perspective.
Another article where someone thinks being a parent means they understand all children. I have 4 kids, and 2 of them definitely would never do math, even basic math, for fun, ever. Their brains lack whatever pathways most people utilize to learn math, so I now have a 15 year old who has to work nearly as hard at arithmetic as she did when she first learned it. No amount of drilling, change of curriculum, buckling down or backing off has had any impact. She has absolutely no interest in math. But the kid reads faster than I do, which is not slow at all.
The only thing I know about kids after having adopted 4 of them is that none of them are alike. The only time when you can really train them to do anything consistently is when they are babies. As a result all 4 have great sleep habits :)
I read it (as a non-child), and a lot of my certainties about what young brains are and are not capable of got joyfully exploded. I'm not linking it to you proscriptively, or with a specific suggestion or riposte in mind whatsoever - you just might be interested in it.
School is very successful at convincing kids (and former kids) of two things: firstly, that the academic subjects they purport to teach are actually delineated by the school textbooks and curricula. And secondly, that the reaction people have to specific subjects within these school structures are the actual unchangeable nature of the person's relationship with the subject.
I hope one day our societies move past these two egregious and immeasurably damaging beliefs.
I haven’t read the book, but we have 100% had the “you don’t have to graduate if you don’t want to” talk with this one haha. She doesn’t want to drop out, but definitely isn’t interested in college. We want to keep that door open for her if we can, so we just remind her that staying in school requires doing some things she doesn’t like.
But yeah, at 15 it gets a little hairy. You have a kid who wants to be an adult, but in a lot of ways they are not prepared to make adult decisions still. Eventually she will have to make them, ready or not. But we have a few years left to help her, so the focus becomes how to best do that.
> School is very successful at convincing kids (and former kids) of two things:
Well, and thirdly, that your worth as a person is determined by your results in graded examinations, and by extension, your salary or some other numerical rating decided by someone else.
I as a child, a teen, and a young adult thought I hated math, I got bad grades and it bored me. I dropped out of school. I later went to college and took remedial algebra twice.
Math in school was purposeless and rigid, a rote procedure to be followed by command because that's what kids have to do.
Now, I have grown older, and my curiosity drove me to learn because I wanted to make things, machines and software and probabilistic strategies. Things that necessitate math. If you can't rotate a vector, your guy walks faster diagonally. If you can't think mathematically and you want to lift a 2 jointed robot arm that weighs several tons, you're going to tip it over, and possibly die in the process. You can do it without trig but you can't do it without thinking about math.
Once I found purpose, I began to appreciate the beauty of the more elegant solutions. I kind of fell in love with math as an adult. Now I watch numberphile with my kids and make complicated machinery and software at work.
I think a lot more people love math than realize it, because they're conflating math itself and what school calls math, which is worksheets and demands, not beauty and creation.
With my kid in elementary school, I can see how math instruction is generally terrible: teachers rarely have any enthusiasm for teaching it. I only had one great math teacher (combining enthusiasm, skill and hard work) and I've been through special math programs (in high school and uni).
Again, it is a question of incentives: someone with enthusiasm for math would likely go with a higher paying job requiring higher level math.
Still, despite the crappy teachers, I was better than most to persevere at it until high school where I had the great teacher.
But this does not scale and we are losing kids to bad teachers: how can we fix this?
Yes. And it's the same when the kids come from the same parents too. We have one kid that's willing to go very deep on math. The only does what can be figured out in 3 seconds or less. Same genetic parents, same school system.
The original concept in the article of exploration is great. Some kids want to explore math, some science, some music, and some Starcraft.
I knew some who were bad at math. Asian immigrant test scores on math are ~1/2-1 standard deviation higher than white Americans. That’s noticeable comparing groups of people but still leaves a lot of Asian immigrants who are not good at math.
There is no royal road. If all your kids are biologically yours and you and all your family are good at math and you marry someone from a similar family, you can stack the deck maybe 95/5 in favor or your kid being good at math? But that option is already off the table if you lack that talent. And there are other things you should probably prioritize first!
I don’t personally see how one person’s experience with children other than my own has any connection to my own children. That was the point I was attempting to make, though. Just because you have anecdotes to share doesn’t mean you’ve stumbled upon some universal truth. They can be helpful to share but NOT if used to dismiss other people’s experience.
Something we don't pay enough attention to is that while calculators have solved everyday math to the point we downplay it as a required skill, people are not pulling out their calculator at the grocery store to make better purchase decisions, even though we all have one in our pocket now.
So we handwave the importance of being able to do everyday math in our heads, while also not taking advantage of the tool that's a substitute for it. We're less educated but also less effective than we would be if we'd never invented automated calculation and were forced to be sharp about it.
Is there a name for this phenomenon?
And what's it going to look like a decade after AI has caused people to stop using their brain for general thinking like it's stopped them from doing math?
(I'm sure you, the reader, are very good at math and are an exception to this still-apt generalization.)
I think most people don't care about optimizing an extra $10 out of their weekly grocery run.
Probably a better example is figuring out the cost of a loan. Just multiply the amortized monthly payment by the term and compare that to the loan amount. If the difference makes you balk, then go ahead and walk.
How many people even realize that loan interest is a significant cost and would bother to do that? Or know how to do that? Most people just try to minimize monthly payments to something they can bear and sign the paperwork.
> I think most people don't care about optimizing an extra $10 out of their weekly grocery run
I remember when this kind of "optimization" was done regularly by a great many shoppers on budgets. Back in the day some stores even used to put calculators on the shopping carts.
People used to know how to budget. Apparently the average American is affluent enough to not need to be able to do this any more. I worry that the atrophy of these kinds of practical skills will cause much pain for a great many people at some point down the road.
But the reality is that's usually almost a false trade - I'm not buying one item in the grocery store I'm buying easily a dozen or more. The best way to do this would be toss the online inventory into a solver to calculate "best value" for me, but in reality it would be a waste of time because if they're out of something, or the quality looks suspect, then that blows that calculation completely. And then am I going to do this for every single item, where every minute in the store is multiplying through the rest of my day? How much is the time shopping trading off against extremely sparse leisure time?
And then there's intangibles - something being slightly cheaper doesn't necessarily mean I'm making a good trade off by buying it for my overall quality of life.
In Australia at least this whole problem was perfectly adequately solved by mandating bulk price labeling on all items in the supermarket. Products in comparable categories have per volume/weight prices listed alongside item prices.
> In Australia at least this whole problem was perfectly adequately solved by mandating bulk price labeling on all items in the supermarket. Products in comparable categories have per volume/weight prices listed alongside item prices.
Same in Europe. Mandatory labeling per volume/weight, pre-discount, after-VAT in addition to the discounted item prices. Then you glance at the shelves and make up your mind.
Learning to get to a best price per unit is a pretty useful skill that could make a lot of difference for a lot of future adults, just like financial literacy.
Some things aren't optional, and if they are seen as such, it's going to force the child to learn later in life what they couldn't earlier on.
Very poor take. The author clearly has very limited experience with raising kids. Most kids won't do difficult things if you don't push them. Playing music, learning to spell correctly, doing mathematics, and so on. A very small minority of kids will do all of that easily and for the fun, but you can't rely on it. If you don't push your kid to do their 20 minutes of piano every day, they will half-ass it and will stop after 1 year and conclude they are not good at music. Same for sport. Same for reading books. Same for maths. And you know what? It's your fault. You chose to be lazy and complacent and didn't push them because it's hard to be a good parent. And now you expect me to validate your laziness? Nah.
The really important thing is to teach kids to find their joy.
At the end of my second year of piano lessons, my teacher took me into her living room, and we listened to Glen Miller records for most of the hour. And then we had a cup of tea, and she told me, "this is the music that I love. I play piano because I love that music, and I want to be able to play it myself. What kind of music do you like?" I didn't really have an answer. So she told me that we should stop doing lessons, but once I found music that I loved, she'd be happy to teach me how to play it.
In my early teens, I discovered Miles Davis. Once I had found my passion, all the hard work became play. I actually ended up learning to play jazz guitar, not piano. Even the heavy lifting was pure joy, because it had purpose and meaning.
I didn't become great at mathematics until I discovered the joy in mathematics (another brilliant teacher handed me a stack of old math contests, and said here, you might find these fun. I placed 4th among 20,000 students).
I didn't learn to write well until I discovered the joy in writings. (An absolutely brilliant English teacher who made us assign ourselves our own grades, but broke his promise in the end by upgrading all my papers to A+'s).
And I gave my kids the room to find their joy as well.
But kids are going to have setbacks, they will reach a plateau in their craft (music, painting, art, sport, ...). You need also as a good parent to help your kids go through, to not give up, because even joy to do is not always enough. This is the hard part.
From my modest experience of being a ski/snowboard instructor and trying to raise 3 boys (now 12, 16 and 18).
You're using yourself as a refuting example, but at the end you disqualify yourself by being 4th out of 20000 in some undisclosed math ranking
his whole point was that once he found joy in it, he could excel. I'm confident there's loads of things I could be good at - I'm only good at the things that I enjoy putting effort into.
Yeah, but something vital happened: you learned the basics of music theory and how to sight read music - both prerequisites to jazz guitar (and something that most guitarists don’t know). Learning piano is a great way to step into other musical instruments.
That was exactly my sentiment.
My parents pushed me hard to do piano when I was around 10-12. After a year that went pretty well I was starting to get lazy and put very little work and investment into preparing for the next lesson. They still had me play piano a full year until they eventually gave up and bitterly told me what a waste my resignation felt to them.
20 years later, I got back to playing piano, and I can't thank my parent enough for having me to continue playing in my teenage years. Because it only took me a few month to be able to play pretty advanced piano sheets compared to some of my relatives who are struggling with the basics starting it in their adulthood.
Same for maths. I feel that a lot of people like the author of this blog post are being extremely misdirected thinking math can and should be taught in a fun or amusing manner every time.
Sure, a lot of topics in Maths can be made more digestible by "gameification" to help younglings develop an intuition. But a very big part of Maths actually requires you to sit down and painstakingly crunch down the numbers/equations, memorize and learn when to apply the correct methods to solve some problems. And even though this part can feel fun and engaging after a while, you can't expect children to exhibit such interest right of the bat without having them first struggle with the classics.
Kids don't know better. Your role as a parent is to navigate along the fine line of forcing your kid to get good exposure to the (boring) activities we adults value and letting him enjoy what he enjoys. Only in doing that will your kid open up to the world and grow up into a functional human being.
"20 years later, I got back to playing piano, and I can't thank my parent enough for having me to continue playing in my teenage years."
One of the tragedies of being young is that few have the insight to realize that the 'boring' stuff parents and teachers are forcing us to learn will actually benefit us and that eventually we'll be very thankful that they did.
My parents nagged me all the time about studying and even though I did my fair share of it I never fully appreciated how important it was until much later.
It's a strange phenomenon, one cognitively understands the reasons but one is isolated from the reality so one is somewhat distant from it. For example, one can get upset watching war footage on TV but being there is on another level altogether (soldiers often do not talk of their experiences because they know those at home will never fully understand).
In the same way, wisdom gained through experience is almost impossible to impart to a younger generation who has no actual experience.
> One of the tragedies of being young is that few have the insight to realize that the 'boring' stuff parents and teachers are forcing us to learn will actually benefit us and that eventually we'll be very thankful that they did.
I'm 40. I don't know, perhaps I'm still young.
I did not appreciate having to learn the boring parts. Learning things for the next exam so as to forget them in two weeks... I didn't see the point then and still don't.
I managed to get by with the minimum possible, fluked my CS education, then had a career earning an order of magnitude more than the average salary. Shrug.
Maybe I'm missing something else because of my lack of education? I don't know...
"…perhaps I'm still young."
Possibly so, wisdom often take years to gel and often only after life events force its notions to the fore.
I upvoted all of the above posts because - all of them share some correct arguments.
My parents forced me to play piano, right up until I told them that I'll destroy our piano if they don't lay off, and any consequences they could think of would not stop me (I was normally an obedient child, but enough was enough).
That got their attention.
30 years later I picked up classical guitar and loved it! Do I thank my parents for forcing the piano on me? Hell no.
Like I commented in another post, piano gave you the foundation for learning classical guitar (and appreciating that genre of music). Very few guitar players can even recognize note names on a staff. You’re not going to get far with classical guitar without it.
Your experience is the antithesis of mine, I wonder why people are so different.
If the kid isn't enjoying the piano lessons, will forcing them to do it for 20 minutes every day really be beneficial? Sure, they will now be able to do something - something that they absolutely hate... (also, why is it always piano that parents try to force on children?)
> they will now be able to do something - something that they absolutely hate
A valuable life skill if you want to ever have a job or get paid.
That’s such a depressing way to see things. I’m sure most people do something they don’t utterly despise, is only because they select for their local optimum.
> I’m sure most people
If you live in the bubble where you experience this, congratulation you live a wonderfully privileged life, never interact with anyone or are totally oblivious to the experiences of all the people you interact with on a daily basis.
You're of course right, it's a privilege.
But also, many people choose to do something they hate so they earn more money. They could be just as privileged and choose not to, just so they can compete with the Joneses and consume more...
Don't know about 'most' but I think many do exactly that in order to pay the bills.
Yep, 6 years of being forced to play the violin.
Sure, once I was playing it, I was fine, but I cannot explain to you the sheer dread I felt opening up the case.
Have not played in decades, despite all those lessons and concerts and orchestra sessions.
They don't hate it. They dislike being bad at it, they dislike working hard at things, and they like video games and scrolling on their phones.
And yes practicing will result in them getting better.
I think this is the crux. Nobody likes to fail, kids included. And their attention span is wonky too so they may not see much value in learning from failure/s since there are so many other attractive things asking for their attention and they would rather do them.
Insane you’re getting downvoted, this 100% and then some. There’s a clear difference in outcomes between kids taught self-discipline and those who are raised standard Anglo-American ADHD style once they become adults.
Different point of view: do you consider hunting in the wilderness to be difficult?
I do, it requires being still in miserable conditions for a long time, being cold, wet, mosquitos, and then usually still no success, but frustration.
But to my knowledge, no savage kid is in need of being forced to learn it.
"children sense your true passions and naturally want to join in"
And that is my experience as well. But if you stop childrens curiosity out of limited time and patience "Be quite now!" - stop them from helping, because they are not a help in the beginning and you are faster on your own - then of course they won't just start enthusiastically some years later doing with motivation whatever it is, you define as their arbitary target now.
Kids tend to want to partake of their own initiative in activities that are 1) physical, and 2) that they see adults themselves do.
Hunting in wilderness is a good example; so are sports, cooking, crafts, etc.
But unfortunately not all important activities that kids need to learn to become well adjusted adults in our modern societies fit those 2 criteria.
Point 2) can be hacked to an extent by modeling the behavior yourself - eg kids who see adults read books are more likely to want to read themselves.
Or staring to smartphone. Then, suddenly, we are surprised why our kids do same.
There is a huge difference between pushing your kids to overcome their current limits and forcing them to do something they do not enjoy at all.
There is indeed a difference between giving a slight push and "forcing" which is what TFA is talking about.
> But to my knowledge, no savage kid is in need of being forced to learn it
Uhh then your knowledge is very limited because that is rather well documented. Also, why are you saying "savage" like an 18th century racist? Is that in fashion again?
Oh, I am obviously a racist, by glorifying indigineous teaching methods.
But otherwise can you show, where this is documented? The natives tribes where I have some knowledge, don't force their kids to learn in the sense that is talked about here. No need to - the whole culture is about becoming a good hunter (for male individuals). So indeed lots of peer pressure, but no individual forcing.
What appears to be out of fashion is policing other people's language.
Do you do things because someone force you or beacuse you have self motivation?
I don’t have kids.
The really important part of this is that kids mimic what they see adults they like and respect doing. If their role models spend 6+ hours in front of the television every night, that’s what they’ll do. If their role models are playing music or sport, that’s what they’ll want to do.
Yes, but one of the problems with our civilization is that we typically do the important stuff out of our children's sight, and then come home tired and try to relax. So they do not naturally get a correct idea of what we do.
Yes. Due to an unfortunate event, I'm a single father. My life improved dramatically when I started doing all the chores while the kid was awake:
- We do the chores together.
- Yes it takes ages, but I need to kill the time anyway.
- Then she falls asleep and I can relax.
This is literal heaven, compared to when I played with her the whole day and then did the chores when she slept...
My take is Maths, Science and English push. Everything else let them decide what they like. Do parents push kids at every damn subject?
I think everything else like, drawing, singing, gardening, exercise, meditation could all use a bit more pushing...
Oh yeah let's turn otherwise fun hobbies into a forced chore, that will surely be great for the kid. Forcing kids to learn to spell when they have learning difficulties (eg dyslexia) doesn't usually go well either it's just causing suffering for the kid.
The problem in this discussion is that people here seem to miss that both an excessively authoritarian parenting style is bad but also going full liberal and just letting them run wild is not the solution. Sometimes children need guidance an a gentle push.
Even as a adult I sometimes need to get pushed. I sometimes take guided courses so I don't skip over the hard but important parts of learning a new thing.
Just don't push your children too hard or you do more harm than good. Accept that they are not you and have different interests and needs. Like make them practice an instrument but give them a choice which one. And if after a few years they still hate it, well you tried. Maybe it is not for them.
The problem is that it’s extremely difficult for any activity like math, music, or drawing to compete with Minecraft and YouTube shorts.
Kinda yes, but there are some solutions.
I think the most important part is to start early. Make your kids interested in math, music, art, and sport, before they start school. Doesn't have to be anything sophisticated, simple addition and puzzles will do for math, etc. Then you have something you can later build on.
There are also ways to make things funny, including math. Most people say that they hate math, but then they do Sudoku. So, try to make more math like this. Not all math can be transformed to funny puzzles, but after a few the kids will get positive associations with the subject, and will be more willing to learn more.
Doesn’t seem to be an issue for Asian families or ones coming the from former Soviet block as well as Jewish ones. These groups as adults tend to outperform others. There’s a reason for that: early childhood discipline and consistency being built into their cultures.
> The author clearly has very limited experience with raising kids. Most kids won't do difficult things if you don't push them.
Such a bold claim coming from some who does not share their credentials on the matter.
My kid will do everything as long as it is interesting. My sample of 1 contradicts your claim, but neither of us are experts on the matter.
> My kid will do everything as long as it is interesting
Difficult things aren't interesting locally, you have to practice boring things in order to do interesting things in difficult subjects. Some kids do practice boring things if you just ask them, but most do not.
Sample size 3 here and they are all adults and all STEM grads.
You have to push them, but push them right. That's a combination of coercion and encouragement and helping them avoid procrastination. There are hills to climb and they need helping over them to where the good stuff is.
I remember my eldest crying over ratios at the dining table. Then algebra at the kitchen table. Then crying again at real analysis in the pub with me. She graduated with a first in the end.
The important part is to start as early as possible and absolutely not trust the school/teacher or kindergarden staff. They are badly programmed to reinforce kids in what comes easy to them and stop encouraging them after less than a handful of attempts.
If you have to restart later, no matter at which point, even up into 'the kids' 20s ( ultra late bloomers, slackers, kids disgusted by most people for reason Z, drug- or "condition X"-induced deadbeats, repressed kids with and without ADHD, failed or successful attempts by psycho-social environments ) understand three things:
1) you are not pushing, even if you are, you are demanding sth for the sake of your child AND yourself. YOU WANT THIS first and foremost. It's not a bad thing, fuck what the little fucker wants.
It's imperative for the kid to know that YOU WANT THIS no matter the obstacles. You want to see the process and result. It's a form of accountability, I guess. Kids pushing back is some dumb implicit way to check how important THEY and THE THING really are to you _or someone else_ (that counts for the ugly stuff, too). It's part of our evolutionary, hard-coded OODA loop.
2) just start at the very beginning, so that it's easy, almost effortless. The kid will be annoyed on most of the difficulty increases, it always depends on the sub-topic so don't back down. Even 20 year olds will catch up with their successful piers within some time. Neuro-genesis is awesome. Most 'grown up' stuff is child's play and a matter of baseline-human character anyway.
3) your stress level is what matters. Stay cool, be equanimous, serene, check your posture, voice, tone, the discussion won't last 5 min and will be worth it.
Absolutely force your kids to do math.
> Most kids won't do difficult things if you don't push them.
I guess it's true for adult humans, and other creatures as well. Instead of pushing, however, you should consider using other motivational methods (a simple prize for accomplishing something that you want from your kids works very well). Pushing can cause alienation and hate, which could affect their entire adult life.
It’s amazing how far the promise of a sticker or special treat will go for young ones.
The world needs less Asian kids being forced to play piano since a very early age and it needs more kids (Asian and not only) that are left to explore the world.
Without that exploration the kids won't make the world their world, at best they'd only make it a bad approximation of what we, our (older) generation, best think that that world should be.
Except they are now, wealthy Chinese are one of the most rapidly growing segment of tourists. Wanna guess what the wealthy successful ones learned as kids? That’s right: piano and math.
Any proof of why it is best?
I prefer to live among educated people, thank you. I prefer my peers to go through forced history lessons, forced math lessons so they don’t tank my government, and biology lessons so they don’t tank the health system. Yes they won’t be able to determine their gender, but that will give me grandkids, thank you very much.
Same goes for piano or sports. Yes we need to pull people upwards, otherwise we’ll all become fat americans.
I kind of get the both sides of the argument. It kinda feels wrong on one hand. Lots of people agree. The other hand, it kinda, just works. Great. Everyone's hit with down-trend TFR anyway so maybe not it.
> at best they'd only make it a bad approximation of what we, our (older) generation, best think that that world should be.
Also, this part in GP doesn't feel exactly right to me. The problem doesn't seem to be in education, but rather lack of systematic resistance in current systems of society against humans weaponizing the system as tools to hamper progress of humanity as means to win minor inner struggles which is stupid. But the world doesn't seem to be moving in a wrong direction, only slowly.
Asian kids in 80s dreamed of bunches of permanent artificial space habitats running on fusion reactors. Still do. We've only gotten ground based fission reactors and space motor homes since then. But at least we are moving in that direction, just slower than at the ideal rate.
China's just done a humanoid robot marathon event. The winner completed the race. They're definitely in the future. US is, in a state not in line with site guideline to describe. And the latter is supposed to be more correct state than the other? How is that possible?
> Yes they won’t be able to determine their gender, but that will give me grandkids, thank you very much.
All that praise for education and yet you've fallen for the tabloid-fueled conspiracy theory that transgender science is a hoax
What do you mean by "science" in this context? For example, both biology and anthropology are sciences, but biology can tell us that people evolved from apes, while anthropology can tell us that a specific tribe believes that they were created by a flying serpent. Both of them are sciences, both of them can talk about the same topic (the origin of humans), but they take a completely different perspective on the topic.
Is the "transgender science" you talk about more like biology, i.e. describing how things are, or more like anthropology, i.e. describing what some (sub)cultures believe? Those are not the same things.
Because "forced history lessons" doesn't help said kids understand what history really is about, it just helps them accumulate facts, if that.
I'd say the same thing applies to math, where one can't really start understanding math until said kid is already an adolescent (unless they're a young Euler or something), so it always baffles me when I see parents filling their young kids with (fancy) arithmetics, most probably making said kids future therapy patients, all the while lauding themselves (the parents do, that is) that they're teaching their kids "maths".
Related, one of the best maths teachers I've had (this was back in high-school, in the mid-90s) was very quick to point out that we should forget almost all "maths" we had learned in elementary school, and the he very soon started to explain to us the definition of the real numbers. Or maybe this is just an Eastern-European thing, who knows? Maybe further West they do confuse arithmetics with maths until the Uni' years.
You mention Eastern Europe, are you by chance familiar with the Hejný method of math education in Czechia? Because that introduces some "math beyond mere arithmetics" concepts to the elementary school education.
Sometimes, it is possible to create a less abstract version of a more abstract thing, and thus introduce the seeds of the concept to children much younger. For example, "solve the equation 2x+1=7" is abstract, but "Peter decided to use a # symbol for a specific number, and he didn't tell us which one, but we found in his notes that # + # + 1 = 7; can you figure out which number is # ?" is simple to understand for a very young child, even if the child can only solve it by trial and error.
You’ve got it backwards: the future therapy patients are kids who are not taught discipline and persistence. Those who aren’t struggle as adults as the real world is harsh on vibe based living. Also, all those “useless facts” eventually build up on each other. They are prerequisites of knowledge and mastery.
Kids left to their own devices don't explore the world. They play video games and scroll on social media.
Not if access to those things are limited while providing opportunities for other things that people enjoy.
You don't replace enjoyable things with unenjoyable things and expect the child to become a well-adjusted adult. You give them alternative enjoyable things.
Managing a child's burgeoning dopamine regulation system is a primary function of a parent. Abdicating that function for quick fixes is a form of neglect, in my opinion, just like feeding kids sugary cereals.
Any patch of woods seem to cast an irresistible call to explore on kids.
If that is the world you are providing, that is the world they will explore.
It is not just me who "provides the world" to my children. Also their classmates, etc. And the internet: even when someone uses it to achieve a purpose, there are various ads and algorithms that try to turn you towards something else.
It appears that no sane parents will allow their kids to extensively explore the world that the modern society is currently providing.
So now you have a kid that’s really good at playing piano but absolutely hates doing so. Mission accomplished?
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> Without realizing it, he was doing algebra.
A friend of mine taught remedial math at UW to incoming freshmen. She would write:
on the blackboard and ask a student "what is the value of x?" The student would see the x, and immediately respond with x means algebra, algebra is hard, I cannot do algebra.So she started writing:
and ask the student to fill in the blank. "Oh, it's 3!"The semantic meaning of a blank is much better understood to everyone than an arbitrary letter like 'x'.
People just want to know why it's x and not something else or how a letter can have value. They might even think how can 24 + 2 = 5? They just want something to grab onto and nobody is really teaching the concept of a symbol in a math class.
I don’t recall the exact age, but when I was doing math in primary school (somewhere around age 9/10) we were absolutely using symbols - “Paul has two apples, and the basket can hold 10 apples. How many more apples can Paul put in the basket” is the same as 2 + x = 10
We did these sorts of problems for a long time, with addition/multiplication/fractions, and even when we started doing actual algebra the problems were introduced the same way “let’s look at a problem we’ve solved already, and write it in a different way”.
This becomes even more true in higher level maths where programming language style functions would make everything vastly more clear, and easily typeable, than the traditional Greek symbols. sum(x+3, 1, 4) is just so much more clear (and consistent when generalized across other operations) and practically as concise as the mathematical way of expressing that which I cannot even type. Multiple variables would be a bit dirtier, but still much cleaner than the formal expression.
Interestingly mathematical symbols in the past also regularly evolved. Then at some point we just stopped doing that and get stuck in a time which is arguably no longer especially appropriate. So we're left with rather inconsistent symbols, oft reused in different contexts, and optimized for written communication.
The formal language of math is intensely optimized for rapidly communicating with yourself 90 seconds in the future, when doing a proof or calculation, turning paper into working memory. It does seem silly to use the same language for communicating with others across unkniwn but deep chasms of context. Its remarkable that it works at all
For the purposes of education, it is important to keep in mind that "optimized for performance of a highly trained person" and "optimized for understanding of a complete beginner" are two different things.
I often see people make the mistake of trying to teach inappropriately abstract things to small children, because that's what the pros do, and we want the little kids become pros as soon as possible. Problem is, trying to skip the fundamentals is only harmful in long term.
First kids need to learn what all that stuff means, and then we can proceed to teach them the shortcuts.
The strangest part about mathematics culture is that there is a culture of vibing the notation.
Nobody in school ever tells you that there are glossaries on Wikipedia that tell you the meaning of the symbols. You're supposed to figure it out yourself using vibes.
The way mathematic notation is taught is inherently unstructured. You're expected to just get it.
its silly. itd be like introducing first year programming students to advanced maps/filters/anonymous function syntax, instead of the easier to understand for loops and if/else statements. math's "no true scottsman" approach to teaching only hurts itself in the long run.
I'm not sure if it would be easier to explain a map / filter to a first year student vs implementing the patten manually using a for loop and if statement...
Seems like a pretty easy example to make practically, for map have a collection of things, say balls or black. Pick up each one and do a thing to them, paint them blue for example.
For filter do the same except have two different colour balls, if they are yellow they get thrown away, of they are blue they get put in a bucket.
A for loop doing exactly the same you would need to explain the topic at hand, as well as explain iterating an index etc...
Explaining loops is independent of the concepts of collections though. It's also more general, since map/filter/reduce use some kind of loops under the hood anyway, the fact that probably shouldn't be ignored in education process. Unless of course you go with pure functional recursive iterator, but good luck explaining that one.
Maps and filters also require understanding of higher order functions and the very idea of passing function around as a value. I would argue that implementing map/filter with a loop and then demonstrating how this pattern is generalized as .map()/.filter() functions is better and more accessible
It's hard to debate that mathematical notation has a lot of room for improvement. High level algebra is very cryptic and often looks like an arcane incantation rather than something comprehensible for an unknowing person.
That said, as a person who moderately enjoyed math in high school and university, this functional notation would make me hate math infinitely more. It's would look like Lisp, which, at high level, looks just as cryptic as algebra. The sheer amount of braces and mistakes that would be made when reading and writing them is nauseating.
Infix notation, for all its flaws, provides important visual aid for understanding the structure of the expression (the sum of two fractions looks very different from fraction of two sums for example). Whereas with functional notation it's like working on linear textual representation of abstract syntax tree. Trust me, nobody wants to read, write or transform one by hand
Might I introduce you to our lord and saviour APL? Infix notation and was originally created to be exactly that, a better math notation.
https://aplwiki.com/wiki/Comparison_with_traditional_mathema...
APL »» Linear A (Well almost!)
;-)
The notation as it is works very well. It looks unfamiliar to you because you aren't familiar with it.
Thanks for this comment. My secret shame as a programmer is that I haven't really learnt much maths, stopped at 16 in school. Writing out the sum function like you did makes perfect sense to me immediately.
What I should really do is create myself a cheat sheet of symbols to code...
Just read this:
https://en.m.wikipedia.org/wiki/Glossary_of_mathematical_sym...
Thanks for this :)
> nobody is really teaching the concept of a symbol in a math class.
This was what? 5th grade?
What kind of crap teachers never taught that
> What kind of crap teachers never taught that
It's rarely the fault of the teachers.
The problem is, in many MANY MANY schools, teachers are more like social workers that have to compensate for utter horrifics outside of school. You got a ton of children so poor they didn't have breakfast which means their first (and all too often: only) meal will be the school-provided lunch (Covid showed that - a bunch of schools were open at least for lunches). You got children that are literally homeless and living with their parents in some car on a Walmart parking lot. You got children whose parents are in and out of jail. You got children living with their siblings in way too small, pest and mold ridden "apartments". You got children whose parents don't have money to pay for basic school supplies. You got children who are dealing with mental, physical and sexual abuse. You got children where the parents are constantly on drugs or seeking for drugs. You got children with a drug dependency on their own - if they're lucky it's just tobacco or weed, if not it's opioids. You got children with parents or siblings with serious mental or physical health issues. Or you got children with their own mental and physical health issues, or if you want it worse, children with these issues but without access to any kind of treatment. You got children that are being weaponized in nasty divorces. You got children that are being weaponized by street gangs. You got children committing crimes from petty theft to dealing drugs just to survive. You got children that have to literally work (and states like FL pushing to have more working children). You got children having their own children already (either from sexual abuse, from under-education about their own bodies, or intentionally because they fell for some stupid challenge/dare). You got children dealing with bullying, you got some who actually are bullies because they have no other way of dealing with their emotions or getting lunch money. You got children with parents with about zero interest in them. You got children who worry that they'll come home and find out their parents got snatched and disappeared by ICE. You got children who worry that ICE will storm their classroom and deport them. You got children who worry they might not survive the school day because someone will shoot at them. You got children who are constantly on the move because their parents' employment/deployment requires absolute mobility. You got children who are LGBT and have to deal with ever increasing hate against them (and LGBT youth already had significantly higher suicide rates than before the GQP made it a culture war issue).
The US doesn't have any kind of system to help these children but schools and libraries, both are horribly underfunded (there's some school districts where teachers gotta take up second jobs because the government can only afford paying them for 4 days a week), and all too often teachers have to pay with their own money for students' school supplies.
And on top of dealing with these kind of nightmares, they actually have to try and teach these children something - even if the children in question aren't anywhere near a headspace where they can actually learn.
thank Descartes
I thank, therefore I am...
As we are sharing anecdotes:
One of my school math teacher had the same approach in another way: We were expected to use greek letters, not latin ones.
Same reasoning: It showed us kiddos that the letter was insignificant compared to the concept expressed by the letter.
So my take would be: Your friend taught the students for the first time what they were actually doing while handling equations with "a letter in it". That is no problem of algebra in itself. It just means their previous teachers sucked.
There is a game called dragonbox algebra which I'm currently working through with my son and is an absolutely fantastic approach to this problem. Sadly its now part of a horrendous subscription service and is hard to access. I find it really sad that we've had computers for decades and there are so few good maths games like this.
There are two sides to this. The system or method might be bad but also a determined person can go all the way and perform at a decent level if they put in enough time.
Even if the system was better the person still has to be able to motivate themselves and put in the time.
I’ve always found that an indictment of math education — and spent many, many hours discussing it.
When teaching addition, workbooks commonly use a box, eg, “[ ] + 2 = 5” — and first graders have no conceptual problem with this. Somehow, we lose people by the time we’re trying to formalize the same concept in algebra. There’s been many times I’ve written a box around letters in a problem and asked students “what’s in the box labeled x?”
Pedagogy is hard.
Go from "[ ] + 2 = 5" to writing it "box + 2 = 5--what is box?". Then "b + 2 = 5--what is b?" then "x + 2 = 5--what is x?".
I agree. I think the actual problem is that the student is trying to comprehend what it means for anything to have mathematical value other than explicit numbers.
Numbers and letters are taught together, but not as symbols. Letters are taught with sounds and numbers are taught with counting. The notion of a symbol isn't really emphasized much.
I would explain it more like after
[ ] + 2 = 5
what happens if you need more than one box for a complicated problem? Teach the idea that saying box #3 is equivalent to assigning an arbitrary letter for whatever reason you want, but that people more familiar with math prefer letters because they stand in for words that describe what the number is for. You might want to use 'c' for the number of cats you're trying to figure out.
In a room of five animals two are dogs. How many cats?
a = 5, c = ?, d = 2
a = c + d
so... 5 = c + 2
what is c?
Light bulb goes off: "You can do that?" Yes, you can do whatever you want and it's not all about carrying the one or whatever other rote teaching they've been given. They can get creative and be engaged, and then you let them know that actually there are some conventions people like to use for what they're trying to do. They might even believe they've invented a new idea. At least they're having fun.
That is what math books already do.
back when we was new in programming it was similarly difficult to grok
X = X + 1
once we got it, it was a like new world!
most likely this very unfortunate misnomer started with fortran, where it was deemed lucrative to point out "how much programs look like mathematic formulas!".
not only is this overloading a symbol (equality) with a completely different meaning (assignment), it is also a poor choice typographically, as it represents a directional operation with a directionless symbol.
using an arrow for assignment is much better.
it's also worth pointing out that unlike most others, logic programming languages (e.g. prolog) have actual variables, not references to mutable or immutable memory cells.
I tell my kid that math is a language. You learn to speak it, just like you learn to speak any other language, slowly, by listening, understanding, speaking, intuitively recognizing patterns, rules and exceptions. When you start to become fluent you translate problems into math and solve them. At school they keep trying to make them memorize useful phrases, like a tourist that goes to Paris and learns how to say "where's the bathroom", "hello", "would you like to sleep with me", "thank you", "goodbye", etc.
> and learns how to say "where's the bathroom", "hello", "would you like to sleep with me", "thank you", "goodbye"
Quite a story condensed into those five phrases.
https://en.wikipedia.org/wiki/Lady_Marmalade
For sale, condom, never worn
> math is a language
I think there are some differences
If you are a physicist or an economist, you may be using mathematics as a language in the sense that you are using a mathematical description to convey an understanding of the natural world or the economy to your colleagues. But if you are a mathematician, you are interested in the mathematical objects for their own sake.
There is also a difference between the purpose of learning language and learning math. The goal of learning language is (often) to be fluent in it. In other words, the goal is to reach a level of proficiency which would allow you to not have to think about language and focus on the content of the conversation instead. On the other hand, the goal of learning mathematics is usually to be able to solve mathematical problems. Being able to do math without "thinking about it" is not usually a requirement.
"At school they keep trying to make them memorize useful phrases, like a tourist that goes to Paris…."
Like learning dozens of trig identities without any explanation about why one would need them. As I've mentioned elsewhere learning math for the sake of it isn't enough. For most of us math has to have relevance, and for that we have to link it to things in the real world.
Know what, brother? I tell you that studying the humanities in high school is more important than mathematics — mathematics is too sharp an instrument, no good for kids.
Stephan Banach quoted by Steinhaus in Through a reporter’s eyes, Roman Kaluza, 1995
Compare learning math to learning to bicycle. There is some some sweat and struggle that needs to be put in, before one "gets it". After this it can become enjoyable. I encouraged my daughter with practice exercises from a young age, but tried to avoid making it a drudge. She built up confidence and did well with it. She is also very hands on creative. She decided to study engineering and is working towards her PhD.
Those aren't nearly comparable. Riding a bike is one simple skill and as long as you're not racing that's enough for most people. Meanwhile learning maths is a years-long effort at best. I learned how to ride a bike within an hour by myself when I finally had a good reason to learn it. I can't say the same about maths.
Bike is fabulous self-correcting vehicle in most operation conditions. The trick really is just to learn to trust it when it is moving. And then what to do when it stops.
Math is layers upon layers upon layers. And then it also branches. Never really had willpower to learn it myself alone.
Learning math is equivalent to learning to cycling if you had to learn cycling from scratch with every bicycle.
I'm damed sure I'd be much worse at math if I'd not been pushed in a formal environment such as a school classroom.
I liked math—especially calculus as it made sense to me—but parts became a drudgery when I could see no reason for studying them.
Right, there's always the kid in class who excels at math like a mini Euler and gets bored because the rest can't keep up but the majority of us aren't like that—doing Bessel functions and Fourier stuff as abstract mathematics without any seeming purpose can seem pointless and our only interest in them was to pass exams. (Teaching may be better these days but my textbooks never discussed the value of learning these aspects of mathematics.)
Later whilst studying elec eng/electronics it became very obvious to me how important these aspects of mathematics were. If I'd been given some practical examples of why this math was useful then I'd have been much more enthusiastic.
Same goes for the history of mathematics, I'm old enough to have had a small textbook full of log and trig tables yet if someone had asked me at highschool who John Napier was I wouldn't have had a clue. In hindsight, that was terrible.
Mathematics is often taught as if the student was going to become a mathematician à la Hardy or Ramanujan and I'm firmly of the belief this is not the best approach for the average student let alone those with few math skills.
Mathematics ought to be taught with the real world in mind for ease of understanding. For example, it's dead easy to represent AC power as a sine wave and from there use that mathematical fact to solve power problems. (Perhaps maths and physics texts should be written in tandem and synced to show relevance.)
Teachers need to take time to explain that math isn't just abstract concepts but that it's very relevant to everyday life and that tying up mathematical functions to things in the real world is actually interesting and enjoyable.
This article has all the tells of being AI generated with random bolding and constant emdashes.
> Kids are born explorers. They naturally want to discover new things, including math.
That's true only until their senses are not shut off and attention is not fixated on screens. Exploration happens only when you have unused attention, sensory capabilities and need for a bit of hard work and risk-taking. Curiosity is less of a biological feature. It is a product of the need and the available resources (senses etc). All of these are missing now.
There is no need or motivation. And there are no available resources (senses, attention). There is no justification for exploration and hard thinking.
As someone who just finished school, I’m trying to figure out how to get genuinely interested in mathematics. I’ve never been particularly strong at it, yet I’m planning to enter a university program that demands a high level of math. The problem is, it’s hard to motivate myself to study math for its own sake. For example, I loved learning programming because it’s hands‑on—I can build something and immediately see the results. In everyday life, though, I rarely need more than basic arithmetic or simple sin/cos/tan trigonometry.
How do you develop a lasting interest in math when it doesn’t feel immediately useful?
Make it practical! Graphics programming involves linear algebra. Databases involve relational algebra. Machine learning involves requires calculus. You’ll naturally encounter hands-on tasks with tangible goals that involve learning new math.
Don't study it for usefulness, study it for beauty. Look for amazing insights.
Yes, you need some practical math as well. I did engineering, there's a lot of inelegant stuff there.
But that stuff actually tends to be right next to some very interesting things.
Here are three things you can find out.
First, there's more than one kind of infinity. You can't make a map from natural numbers like 1, 2, 3 etc to real numbers like e, 0.632268, sqrt(2) etc. Look for Cantor diagonalization.
Second, a random walk like a heads vs tails comes back to zero almost certainly. It also does so in two dimensions, like walking randomly in Manhattan. In three dimensions, it does not, and so for higher dimensions. Look for Polya.
Third. There is a way for you and me to communicate secretly, despite everyone in HN being able to see our entire exchange. Look for Diffie Helmann.
These days, there's a whole industry of people doing math videos with interesting stuff.
It's easier to appreciate math when you are disinterested in the results or applications, because the nature of academic topics near the core grouping of math/philosophy/empiricism is that they are discovered with a lot of meandering at first, and then sometime down the line they become repurposed into a direct application that can be learned by rote. School tends to instruct in some of the most directly applicable stuff first - the "three R"s" plus some civics and training aligned with national goals. And that means that school predominantly teaches associations between math and rote methods, to the disgruntlement of many mathematicians. The "meandering" part is left to self-selected professionals, so it doesn't get explored to much depth.
So I think a good motive for math study is really in games and puzzles, where the questions posed aren't about win/lose or right/wrong, but about exploring the scenario further and clarifying the constraints or finding an interesting new framing. Martin Gardner wrote a long-running column and a few books in this vein which are still highly regarded decades later.
One of my undergrad degrees is in math. As you study it, you learn to identify your assumptions (axioms), find or build interesting abstractions, prove properties about them (theorems), and then map all sorts of other things into those abstractions by figuring out that they're really the same thing. It's even more interesting when you start to find things that are different or question things you always took for granted.
Math gives you the ability to leverage the very structure and relationships of pure abstraction. It's quite the super power.
None of the specific things you learn studying math will be nearly as useful as the ability to think mathematically.
> For example, I loved learning programming because it’s hands‑on—I can build something and immediately see the results. In everyday life, though, I rarely need more than basic arithmetic or simple sin/cos/tan trigonometry.
Consider doing something that actually needs it. You like computer programming - consider making a game engine. It might be easier to learn when you can actually see that it is useful.
Keep in mind though that math is a lot of things. People obsess over calculus but that is just one type. Math is just as much the different types of symmetry in wall paper patterns as it is finding the derrivative. Don't be afraid to try different areas. If you dont know where to start, consider picking up "A Concise Introduction to Pure Mathematics" by liebeck which introduces a bunch of different math concepts and see if any feel more interesting to you.
Find math that interests you!
I didn't particularly find (at the time) calculus, multivariable calculus, physics, etc. interesting as I didn't find the applications interesting at the time. I find these subjects representative of what you traditionally learn at school.
When I entered uni I discovered my passion for discrete math, algebra (groups, rings, fields, etc.), number theory, cryptography, theory of computation, etc. as they have a lot of application in CS.
That's really what did it for me - and also I had great uni lecturers. I wish they would have taught the subjects I like in highschool - the difficulty level is about the same.
N=1 datapoint here. I studied physics in university and before I started I was not aware that physics is basically just math where the results sometimes relate to reality. The pure math courses I took were the most difficult and in the beginning I loathed them, because it felt so unattainable to get any intuition, let alone real proper comprehension for all the concepts they threw at us. For a long time I felt like I was just hanging on by threads and especially if I compared myself to those who had some innate interest in math or generally some really good intuition on the abstract concepts (or even prior knowledge) it was really demotivating. But I also felt like I had no choice but to continue and as time went on the I grew fond of it. And the feeling of being overwhelmed changed - that is to say I still was completely lost every time a new topic was breached and I could not understand even half of the proofs in class - but I did not feel so defeated about it. And I grew to like the feeling of actually completing the work sheets they gave us every week. The process of solving them was often excruciating but if you did the sense of accomplishment is real. I think for most people higher math is really difficult and that is part of why it is interesting. Another aspect I had to accept over time is that even though you can state a mathematical fact or conjecture in just a hand full of symbols or a plain sentence it does not mean that truly understand it, its implications or how you got there can be understood the same way that other prose can be. Sometimes you have to stare at, contemplate and scribble around one equation for days until you understand whats up.
If there was any advice I would give, then it's probably similar advice on how to stop procrastinating on anything that is difficult. Establish a routine first - find a spot that you will only use for studying this (like a spot in a library), start small, divide and conquer, accept that you will not understand most things easily, reward yourself for the small wins along the way, find an accountability partner or someone to study with if that's your thing, make a regular schedule with regular times where this is what you do - consistency is key, even if its just for 5 minutes, stack it onto other habits, see yourself as a scholar of math - it is what you do, lean into the discomfort, as enduring that is a valuable skill in itself.
If you love programming, there's quite a lot of programming where math is vital. Graphics, optimisation problems, cryptography, neural networks, figuring out if a hash works, projecting if an algorithm will scale...
The tricky bit is often that you need to learn some of the math before you can see how it's useful, but if you need stronger motivation, you might try diving into a slightly math heavy programming problem and learn the math as you go
I'm going to share my anecdote, because it may help, but everyone is different and your mileage may vary.
I'm a MechE by classical training (professionally I actually work doing software/network stuff, don't ask, DNS (screams internally)), so here's where it stood out for me:
https://en.wikipedia.org/wiki/Hydraulic_analogy
Internalize what this simple example represents, think about why that's mathematically interesting, and start looking for where it applies elsewhere. You too could be roped into doing systems engineering at scales you didn't think people haven't already figured out.
University is not a good place for learning mathematics as most of your math instructors there will be very good at math and very bad at teaching people that are not already very good at math.
Sorry, no. Universities are great places to learn math. You’re misrepresenting the genuine passion for teaching that many university instructors have.
Sorry, no.
For whatever reason, many University programs use high level math classes as a filter to weed out 1st year students from that program. If university instructors had a genuine passion, and ability, for teaching high level math then they wouldn't accept that as an outcome.
That is unfortunately true; not only in the US, but all around the world. The particulars do depend on the instructor, and many if not most instructors try to be motivational, but the syllabus is perfectly clear: "this is a weed out class". And when it comes to test time, the syllabus wins.
The only thing I disagree with in your comment is about the instructors: they want to be employed, and they have to accept the syllabus and testing standards. It is not about passion and ability to teach (most, especially younger ones, are full of those); it is about meeting the departmental requirements.
Except maybe not calculus. I remember my calc class kind of being terrible because it was a weed out for other majors. Every math class that wasnt required though was great.
Probability/Statistics is a good excuse to learn mathematics, because paying a little attention one finds lots of day-to-day situations where is possible to apply it. For example, see the secretary problem[1].
[1] https://en.m.wikipedia.org/wiki/Secretary_problem
I wish I couldve excluded everything past basic algebra and hopped right to statistics at a young age - I *loved* everything about the practicality of it, how it explained tangible relationships and illuminated the world. Algebra and calculus were so un-engaging I had those teachers calling me everything but a stupid child.
Haigh's Mathematics in Everyday Life [0] provides modern examples.
[0] https://plus.maths.org/content/john-haigh
Perhaps think of them as solving logical puzzles. It's fun. Even though not always related to everyday tasks.
For me, it began many years ago when reading about Hilbert's hotel paradox. Turns out our laymen's understanding about infinity isn't as really refined.
I write mobile apps for living and indeed these stuffs are irrelevant for my work.
as someone who loved maths first but then do programming for a living, it's because solving puzzles is fun. I get the same dopamine hit whether it's a math problem, a coding task, or a video game level. but I think forcing yourself to like something is not the correct approach; you either like it naturally or you tolerate it for some other goals
https://betterexplained.com might prove helpul?
Get a good teacher. They make it fun, or interesting.
Have you ever watched a video of a highly skilled tetris player? Where they fill the screen most of the way to the top and then suddenly they just combo the whole thing down and everything wraps up cleanly, and then they start fresh.
The feeling of "oh yeah, that was nice watching that mess turn into something clean and squared away" is where I get a lot of my joy from math.
But also, there are uses to math that you might be able to play with through every day, but you've never thought of those scenarios in a mathematical way.
I was walking today, and on the street there is a right angle turn. The inner portion of the turn is just a square right angle, but the outside of the turn is a radius. I started wondering to myself, if I want to be on the outside of the turn going into and exiting the turn, what would be different ways I could walk this, and what would the distance differences be.
Crossing directly across, to the inner corner and crossing directly across to the outer side again, would be 2w (for the width of the road w). Following the edge of the radius would (assuming perfectly circular), be 1/4 of a circle, so 1/42piw = 1/2 pi * w. The shortest route is a straight line, which would make a right triangle, so w^2 + w^2 = c^2, 2w^2 = c^2, sqrt(2) w = c
So crossing twice is 2w, following the edge is 1/2piw, and shortest path is sqrt(2)*w. Not super applicable, and I didn't need to do math to figure it out, but I was walking and bored, so I found joy in it. The fact that they all boil down to having w as a factor means I could figure out a nice ratio between all of them. And then I needed to mentally figure out what 1/2 pi was. 3.14/2 = 1.57... And I know that sqrt(2) is roughly 1.41 ish.
So now I know that crossing twice has a cost of 2, following the edge is 1.57, and direct line is 1.41. Following the edge is vaguely close enough to the ideal path to warrant not walking into the street to optimize the route, 1.57 / 1.41 is about ~110%. Whereas by defintion, a cost of 2 is going to be sqrt(2) times sqrt(2), so ~141% more than shortest path.
A few things to note here. First off, I'm aware that not everyone finds the same joy in doing simple mental math and thinking about problems mathematically even when there is no need to do it, but trying to think of things more minor trivial things mathematically may cause you to at least appreciate it more, which can grow into joy. And second, I wasn't doing any complicated math in my head. I just thought to myself "is it faster to cut to the inside corner and then cut back out... of course not, right?" and I was able to answer that definitively to myself. Did it matter? Was the answer probably obvious anyway? Probably, but I was able to _prove_ that. And I value facts. Finding joy in the simple things lets you build up more of a familiarity and view it more as a problem solving tool than a tedious thing to rote memorize.
A great way to build up math familiarity and see how other people find joy in mathematics would be to watch Numberphile videos on YouTube[0]. It's a bunch of mathematicians sharing things they find interesting about math. Some times are REAL hard to grasp, but some are just very interesting puzzles[1]. The puzzles don't always have clear immediate usefulness, but can often be described as "a mathematician wanted to know an answer, so they did some math to find out and prove something to themself."
Sorry, end of spiel.
tl;dr - find the joy in the simple things and use math as a tool to answer (even simple) questions to help highlight the usefulness.
0: https://www.youtube.com/channel/UCoxcjq-8xIDTYp3uz647V5A 1: https://youtu.be/ONdgXYEBihA
I adored this post right up until:
> I have an internal KPI: if in the last three days I haven’t spent at least 30 minutes playing with my kid, there’s something seriously wrong
I think I'm interpreting this ungenerously, because my knee-jerk reaction was to wonder about who is handling the other 12+ waking hours a day.
I read this as remembering to set aside time specifically for play and not just for day-to-day parenting and discipline
About a year ago I came across the concept of ‘math circles’, here on HN. It was this longish but very interesting article: https://www.thepsmiths.com/p/review-math-from-three-to-seven...
The key element here is nurturing curiosity. Since then i and my 10yr old have been sitting through a virtual math circle led by Aylean McDonald on parallel.org.uk an organization run by Simon Singh
It's not about forcing your kids to "do math", but to excel at important skills far before the benefits of being good at that skill matter.
The amount of homework/study per day that maximizes math scores on tests is significant, 1+ hours/school day by the time they're in middle school, with it helping even more for those who are starting out poor at math[0]. You'll note the referenced study doesn't even max out progress for any group - meaning most could have studied more and improved more.
I don't know any kids that voluntarily did an hour or more of solely math study per day. I know plenty that were forced, and ended up loving math or other technical fields as adults.
As a parent of young kids, obviously I haven't gone through high school yet - but I don't think many children who reach their potential in math, english, music etc will have no pressure from their parents.
[0]: https://pmc.ncbi.nlm.nih.gov/articles/PMC8025066/
> As a parent of young kids, obviously I haven't gone through high school yet - but I don't think many children who reach their potential in math, english, music etc will have no pressure from their parents.
Well it depends. I had no pressure from my parents to learn about programming but still got really good at it. Could I have gotten even better had I been pressured to "practice"? Perhaps. But then I also wouldn't like it for the reasons I do (I like making stuff, but not solving riddles) and it would feel like the dad sport situation.
I also played the piano for 6 years, starting out because I liked it. My parents didn't suggest it, but a few years in they were pushing me to continue even when I didn't like it anymore. Finished the first level of music school (6 years where I live) and haven't touched it since. Just to clarify, they weren't using any directly abusive tactics to keep me going, but they did put a lot of pressure onto it.
There's a lot of nuance to all of this and I don't completely disagree that we should occasionally pressure our kids to push their limits. What we often fail to acknowledge is that kids easily change their minds after a while. Just because they liked something at a certain point doesn't mean they still do. The easiest way to get a kid to dislike something is to make it a chore. Additionally, I think we need to ask ourselves whether it's more important to us to have a kid that's average scoring but has a (mostly) stress free upbringing, or one that excels but is stressed out by the time they hit high school. Kids absorb stress differently than we (adults) do.
Me being forced to do tons of horrible math by my abusive grandfather at a young age for literally 4+ hours at a time gave me a few things.
1. A true hatred of work, make work, and a strong desire to defend laziness as a concept (note that Bertrand Russel agrees hard with me here!) -https://en.m.wikipedia.org/wiki/In_Praise_of_Idleness_and_Ot...
2. A love of subversives and cheating the system. Basically, the guys writing leetcode cheating software are saints in my book. All subversions of the attempt to turn society into a meritocracy (a term which was originally supposed to be a slur/negative connotation - https://en.m.wikipedia.org/wiki/The_Rise_of_the_Meritocracy) is extremely good.
3. An advanced knowledge of TI basic, so I could cheat hard on every single school assignment I could get away with. AP Chemistry? I’ve got a symbolic stoichometry solver app! Calculus? CAS system in the palm of my hand!
Play stupid games with children, win stupid prizes. Maybe don’t force them to work like little slaves in their early life, and they won’t strike back at your society systems.
Maybe it wasn't the math, but the abuse.
I had similar conclusions, but the other way around: absolutely no guidance. Fortunately by the time I programmed and sold the basic ti math exam solvers to by classmates for 2 euros a pop I had everything memorized.
Nothing like cheating the system to know the system
You realize your experience can't be generalized to anybody else except for those who were abused in the same way you were? It also isn't what people in these comments are suggesting should be done.
Your experience sounds awful but surely there is a reasonable middle ground between forcing a kid to do any math and forcing them to do it in 4+ hour sessions.
Raising kids is hard. Sad. And what do I know about it? Regardless, parents do need to get involved in their children's education. For instance, they should help their children prepare for entry exams into secondary school. This shouldn't take their child 4 hours a day. Maybe 10 minutes on some days, and 1 hour on others.
I think this isn't quite the same thing, never the less: I have dyscalculia on extreme mode (/dyslexia/autism), and I was forced to do math in the 90s in the UK, rote style. I don't know if they didn't know about dyscalculia, didn't care, or whatever, but holy hell I wish I'd never been forced to do that, it's still today a fairly painful memory and I'm in my 40s now. If you're gonna force kids into math, at least make sure they're not unable to process it correctly.
When I was having trouble learning multiplication my father made up a payment system. He made flash cards and I got a payment for every one I mastered (I had to get it right some number of times, not just once). I ended up with maybe $25 or $50 which was a lot for a kid in the 1970s.
My mother tried to give me $5 for every book of the Bible I read. I never took her up on it even though I knew about the basically freebies like Jude. I wasn't opposed to it, but it felt like –on the one hand– I didn't want to half-ass it and read a few books –and on the other– I really didn't want to read the entire Bible. So I guess that a completionist attitude prevented me from getting $30!
Oddly enough I found a great 'trick' for this. Kids hate doing math tests, but turn it into a competition and game and suddenly they love it. Print out a bunch of remedial problems, perhaps 50. And then give them 1 or 2 minutes to do as many as they can. It's just a contest to improve against your own scores over time, with prizes for the kids who score the highest after a month or whatever.
It's still literally just a math test/quiz, but somehow the context changes everything and even kids who really aren't into math were loving it, and also improving rapidly because the repetition helps instill intuition.
There’s a fine line between chores and games.
Imagine having to move a round thing around some other people to get that thing into a square frame. Then, imagine that you can only use your feet!
clever!
I imagine the part of a test people dislike is failing, and the consequences from failing. Framing it as a game without those emotional stakes fixes that.
If the teaching environment was set up to encourage learning rather than punish not having learnt yet we might not need these tricks, but that culture is slow to change
Something I've been thinking a lot about is "stealth edutainment" games.
When I was a kid, I remember "edutainment" games that were basically like normal computer games, except every so often a homework problem pops up.
I think that doesn't work super well. Better is a game which has you learning naturally, in order to play the game more effectively. For example, I've been enjoying the computer game Slay the Spire recently, and there is a great deal of mental math which is inherent to the game. If I had a kid, I think I might give them that game as a method to motivate them to learn arithmetic.
30 minutes of play per 3 days is such a brutal reality to acknowledge. One of the most wonderful experiences in all of life so drastically limited by the society we’ve constructed.
> One of the most wonderful experiences in all of life so drastically limited by the society we’ve constructed.
I could understand if someone was forced to work two full-time jobs (as my grandfather was), but I find it much harder to blame ‘society’ when so many of these situations are self-imposed.
It’s possible that I’m jaded from hearing a subset of parents complain about not having enough time with their kids but then get stuck scrolling their phone while kids want to play. I also know some parents who insist on having a spotlessly clean house every day and then complain that there is enough time to spend with their kids.
I’ve gravitated toward peer parents who have similar priorities in life which has indirectly made me happier. Seeing all of the parents in my friend circles prioritize spending time with their kids and being honest with themselves about their priorities has been unexpectedly helpful for my own sanity.
Again, nothing against parents who are really forced to allocate time elsewhere, but I’m tired of seeing self-inflicted problems of prioritization and time management be externalized as blaming society.
In some ways yes, but men have always been the ones to go hunt/farm for long hours and provide for the family, leaving the children home under the care of the mother/village for days or weeks at a time.
I would go so far as to say modern society actually enables us to be more involved in our children’s lives, especially those for whom remote work and home schooling are options.
ChatGPT makes it so easy to build a lesson/workbook for something your kid is interested in. I've used it to build workbooks on special relativity, tsolkovsky's rocket equation (including euler integration to build a scratch program), triangulation, logic gates, probabilities of simple dice games, etc. My pro-tip is to tell the LLM to format the document in LaTeX, so you get beautiful math typesetting.
You don't even have to get through the workbook. Get to a part that they need to understand better and make a detailed workbook on that part (for example, triangulation -> solving a system of linear equations).
Where can one learn more about this? I want to get some activities for my kids this summer…
Me too.
For elementary school age kids, maybe even middle school, try getting them started with the app "Euclidea".
They won't think of it as math. It's gamified geometric constructions. Starts simple, "how do you bisect an angle" with a compass and a straight edge. It goes to a very high level that will challenge anyone.
Is there any knowledge that is recommended to be forced to kids?
Probably safety behaviors, e.g,, “let’s look both ways before we cross the street, regardless of whether you want to”
If you formulate the warnings just right, you wouldn’t need to “force” it, as kids will be willing to look both sides themselves. They like to be alive.
Related to what the article mentions, about playing cards, I tried to get my kids into doing basic arithmetic by playing "scopa"[0] with them.
Turns out, the one who didn't like math didn't like the game either, and the one who did like math liked the game too.
So I'm not totally convinced you can just "trick" kids into liking maths, tho for sure it's a way to get them to exercise.
[0] Scopa: https://en.m.wikipedia.org/wiki/Scopa
Learning is pain, knowing is pleasure...
And then your kids and their same generation would be replaced by their peer kids from hard working boys and girls from India and China. Unfortunately curiosity only works with brilliant minds. Normal minds plus curiosity is useless.
I think the advice is good for younger children. The author is using 14+11 as an example. Very engaged parents can have a tendency to overdo it, so it's probably a good reminder.
As kids get older, they need to learn how to struggle and overcome struggles. (I would still caution against "forcing" math.) But yes, you need to start engaging hard work and determination.
Btw, the two are not mutually exclusive. Young children should be praised for struggling at things so they begin learning that skill, too.
> Repetition is key
Even with a "normal" mind. Train consistently to gain excellence!
My daughter and loads of kids watched number blocks from around two or three up, I think it made quite a big difference- she's far ahead of where I was now, years later.
This is all fun and games until your kid has school assignments with deadlines
In the Netherlands we didn't get homework until 12 and lots of play time. Curious how that stacks up to other places.
Have them play a game like math maze 2!
They will force themselves to play... and do math in the process.
Just spent 10 minutes playing it, looks pretty fun!
Perhaps make them aware how important it is with examples from nature? https://duckduckgo.com/?t=lm&q=fibonacci+in+nature+examples&...
Kids do not understand the concept of "importance". At least no kid I've met. That part of their brain doesn't work. They'll trade effort for privileges or toys tho, and are little mimicry machines so they follow you if you use it.
yeah, I recoiled when the author of the post says "no bribing" - bribery is one of the most useful tools a parent has. I guess you could call it "incentive" or something, but really, it's quid pro quo.
Honestly it's so close to how the world works I can't believe 1. Avoiding loss of privilege and 2. Gaining new things as reward isn't the top two parenting tips.
But probably zeroth, most important, is modelling good behavior. Kids are mirrors.
Something occurring in nature doesn’t necessarily make it important to their lives.
Teach kids to do math by have them make mods for their favourite games.
Forcing is kind of hopeless. So is logic, and reasoning.
How children learn (they can't rely on a fully formed prefrontal cortex like adults) is very different than how adults learn (no fully developed prefrontal cortex until 25-26), learning about this can help a lot.
Learning more about the Reggio Emelia approach might help parents curious about this, it has been quite surprising how much is possible naturally. One of the best things to do is to relentlessly read to and with your kids.
Showing kids the math in every day things, especially things they already love is a helpful way of making it approachable, or at least aware.
Also, linking a topic to their interest's radar, encouraging curiosity, play in general, and letting them potentially discover it can go a long away.
When they've got something they want, teaching math and savings is a great thing. Understanding life is a lot harder without knowing a basic bit of math, and can be made a bit easier when doing it younger.
I had a math teacher that once made it clear, some stuff can just click, others is just about doing a lot of examples to learn the patterns. Doing math is very different than being creative with being comfortable to find it.
Today, I'd probably setup a good prompt to find a way for the child to share their mine to discover how they like to learn, and how they might like to learn faster and easier by taking some shortcuts through math directly or on navigating an ontology/taxonomy perspective.
When I was 8, I went to the library in our town a lot. My parents went there sometime to return their books. At some point I just stayed there when they would go home. First I was in the children/teenager section and soon in the general library, where I would read about programming and computers. I learned C by age of nine.
The "undeveloped PFC" argument is shallow, unspecific and usually just used to infantilize younger people. It may be useful if the child is under 6 years old, but at the time someone is 17 or older, it becomes essentially useless.
My learning process was always, and still is fueled by curiosity.
Surprised no one here has mentioned Kumon. Hated it but it works
I'm fortunate enough that my daughter has an admirable interest (and talent) for Math since very early age. She even won a medal at a renowned nationwide Math competition when she was in Grade 5... competing in the Grade 10 category.
That kid's name? Alberta Einstein.
math circles are good for this. i’d suggest it if there is one nearby
Maybe find an application of the subject that they might find interesting. I suppose if you can't find anything that interests them, then it's much harder to teach it.
For instance, perspective drawing might provide a nice application of 3D projective space, its subspaces, and perspectivities between those subspaces. Some of the theory of conic sections might be relevant too.
Computer graphics provides a nice application of coordinate geometry. This covers elementary algebra, Pythagoras's theorem, etc.
Even eating pizza can provide an application of differential geometry.
I tell my kids they can have letter cookies if they pick a word that starts with the letter, and can have 5 treats if they ask for 4 but know what "plus one means" or can have 4 if they recite "2 plus 2 equals ... ".
They're 3, so I don't expect that to scale, but I'm hoping it's normal reward-for-knowledge by the time we get report cards.
Something that might work for getting your kids interested in modular arithmetic: The Chicken McNugget Theorem.
Forcing is kind of hopeless. So is logic, and reasoning.
How children learn (rely on the prefrontal cortex of their adults) is very different than how adults learn (no fully developed prefrontal cortex until 25-26), learning about this can help a lot.
Learning more about the Reggio Emelia approach might help parents curious about this, it has been quite surprising how much is possible naturally. One of the best things to do is to relentlessly read to and with your kids.
Also, linking a topic to their interest's radar, encouraging curiosity, play in general, and letting them potentially discover it can go a long away.
When they've got something they want, teaching math and savings is a great thing. Understanding life is a lot harder without knowing a basic bit of math, and can be made a bit easier when doing it younger.
I had a math teacher that once made it clear, some stuff can just click, others is just about doing a lot of examples to learn the patterns. Doing math is very different than being creative with being comfortable to find it.
Today, I'd probably setup a good prompt to find a way for the child to share their mine to discover how they like to learn, and how they might like to learn faster and easier by taking some shortcuts through math directly or on navigating an ontology/taxonomy perspective.
Another article where someone thinks being a parent means they understand all children. I have 4 kids, and 2 of them definitely would never do math, even basic math, for fun, ever. Their brains lack whatever pathways most people utilize to learn math, so I now have a 15 year old who has to work nearly as hard at arithmetic as she did when she first learned it. No amount of drilling, change of curriculum, buckling down or backing off has had any impact. She has absolutely no interest in math. But the kid reads faster than I do, which is not slow at all.
The only thing I know about kids after having adopted 4 of them is that none of them are alike. The only time when you can really train them to do anything consistently is when they are babies. As a result all 4 have great sleep habits :)
Have you come across https://en.wikipedia.org/wiki/The_Teenage_Liberation_Handboo...
I read it (as a non-child), and a lot of my certainties about what young brains are and are not capable of got joyfully exploded. I'm not linking it to you proscriptively, or with a specific suggestion or riposte in mind whatsoever - you just might be interested in it.
School is very successful at convincing kids (and former kids) of two things: firstly, that the academic subjects they purport to teach are actually delineated by the school textbooks and curricula. And secondly, that the reaction people have to specific subjects within these school structures are the actual unchangeable nature of the person's relationship with the subject.
I hope one day our societies move past these two egregious and immeasurably damaging beliefs.
I haven’t read the book, but we have 100% had the “you don’t have to graduate if you don’t want to” talk with this one haha. She doesn’t want to drop out, but definitely isn’t interested in college. We want to keep that door open for her if we can, so we just remind her that staying in school requires doing some things she doesn’t like.
But yeah, at 15 it gets a little hairy. You have a kid who wants to be an adult, but in a lot of ways they are not prepared to make adult decisions still. Eventually she will have to make them, ready or not. But we have a few years left to help her, so the focus becomes how to best do that.
> School is very successful at convincing kids (and former kids) of two things:
Well, and thirdly, that your worth as a person is determined by your results in graded examinations, and by extension, your salary or some other numerical rating decided by someone else.
I as a child, a teen, and a young adult thought I hated math, I got bad grades and it bored me. I dropped out of school. I later went to college and took remedial algebra twice.
Math in school was purposeless and rigid, a rote procedure to be followed by command because that's what kids have to do.
Now, I have grown older, and my curiosity drove me to learn because I wanted to make things, machines and software and probabilistic strategies. Things that necessitate math. If you can't rotate a vector, your guy walks faster diagonally. If you can't think mathematically and you want to lift a 2 jointed robot arm that weighs several tons, you're going to tip it over, and possibly die in the process. You can do it without trig but you can't do it without thinking about math.
Once I found purpose, I began to appreciate the beauty of the more elegant solutions. I kind of fell in love with math as an adult. Now I watch numberphile with my kids and make complicated machinery and software at work.
I think a lot more people love math than realize it, because they're conflating math itself and what school calls math, which is worksheets and demands, not beauty and creation.
With my kid in elementary school, I can see how math instruction is generally terrible: teachers rarely have any enthusiasm for teaching it. I only had one great math teacher (combining enthusiasm, skill and hard work) and I've been through special math programs (in high school and uni).
Again, it is a question of incentives: someone with enthusiasm for math would likely go with a higher paying job requiring higher level math.
Still, despite the crappy teachers, I was better than most to persevere at it until high school where I had the great teacher.
But this does not scale and we are losing kids to bad teachers: how can we fix this?
Best advice I ever received is: You have to parent the kid you have - not the kid you want
I have 5 and can say that this is the way.
This 100%
Yes. And it's the same when the kids come from the same parents too. We have one kid that's willing to go very deep on math. The only does what can be figured out in 3 seconds or less. Same genetic parents, same school system.
The original concept in the article of exploration is great. Some kids want to explore math, some science, some music, and some Starcraft.
> The only time when you can really train them to do anything consistently is when they are babies.
This is really stupid.
If you can’t recognize a little tongue-in-cheek humor then maybe you’re the stupid one ;)
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I knew some who were bad at math. Asian immigrant test scores on math are ~1/2-1 standard deviation higher than white Americans. That’s noticeable comparing groups of people but still leaves a lot of Asian immigrants who are not good at math.
There is no royal road. If all your kids are biologically yours and you and all your family are good at math and you marry someone from a similar family, you can stack the deck maybe 95/5 in favor or your kid being good at math? But that option is already off the table if you lack that talent. And there are other things you should probably prioritize first!
I don’t personally see how one person’s experience with children other than my own has any connection to my own children. That was the point I was attempting to make, though. Just because you have anecdotes to share doesn’t mean you’ve stumbled upon some universal truth. They can be helpful to share but NOT if used to dismiss other people’s experience.
Something we don't pay enough attention to is that while calculators have solved everyday math to the point we downplay it as a required skill, people are not pulling out their calculator at the grocery store to make better purchase decisions, even though we all have one in our pocket now.
So we handwave the importance of being able to do everyday math in our heads, while also not taking advantage of the tool that's a substitute for it. We're less educated but also less effective than we would be if we'd never invented automated calculation and were forced to be sharp about it.
Is there a name for this phenomenon?
And what's it going to look like a decade after AI has caused people to stop using their brain for general thinking like it's stopped them from doing math?
(I'm sure you, the reader, are very good at math and are an exception to this still-apt generalization.)
I think most people don't care about optimizing an extra $10 out of their weekly grocery run.
Probably a better example is figuring out the cost of a loan. Just multiply the amortized monthly payment by the term and compare that to the loan amount. If the difference makes you balk, then go ahead and walk.
How many people even realize that loan interest is a significant cost and would bother to do that? Or know how to do that? Most people just try to minimize monthly payments to something they can bear and sign the paperwork.
> I think most people don't care about optimizing an extra $10 out of their weekly grocery run
I remember when this kind of "optimization" was done regularly by a great many shoppers on budgets. Back in the day some stores even used to put calculators on the shopping carts.
People used to know how to budget. Apparently the average American is affluent enough to not need to be able to do this any more. I worry that the atrophy of these kinds of practical skills will cause much pain for a great many people at some point down the road.
But the reality is that's usually almost a false trade - I'm not buying one item in the grocery store I'm buying easily a dozen or more. The best way to do this would be toss the online inventory into a solver to calculate "best value" for me, but in reality it would be a waste of time because if they're out of something, or the quality looks suspect, then that blows that calculation completely. And then am I going to do this for every single item, where every minute in the store is multiplying through the rest of my day? How much is the time shopping trading off against extremely sparse leisure time?
And then there's intangibles - something being slightly cheaper doesn't necessarily mean I'm making a good trade off by buying it for my overall quality of life.
In Australia at least this whole problem was perfectly adequately solved by mandating bulk price labeling on all items in the supermarket. Products in comparable categories have per volume/weight prices listed alongside item prices.
> In Australia at least this whole problem was perfectly adequately solved by mandating bulk price labeling on all items in the supermarket. Products in comparable categories have per volume/weight prices listed alongside item prices.
Same in Europe. Mandatory labeling per volume/weight, pre-discount, after-VAT in addition to the discounted item prices. Then you glance at the shelves and make up your mind.
Learning to get to a best price per unit is a pretty useful skill that could make a lot of difference for a lot of future adults, just like financial literacy.
Some things aren't optional, and if they are seen as such, it's going to force the child to learn later in life what they couldn't earlier on.