I have no doubt they can. There's so much negative cultural conditioning towards mathematics by eliciting negative emotions that it's ridiculous (i.e. math is scary, math is hard, math is for smart people, etc.). I believe kids who excel at math is not due to them being smarter but due to the bypassing of the cultural conditioning. The bypassing can be due to the parents, teachers, some mentor, or their particular fascination for a math subject.
One thing that's difficult about math is all the "mathematical maturity" required, much of which is notation and... abuse of notation.
Math seems like an Arts subject, where you must learn all the cultural background, as in English Literature. I'm not sure it's necessary, given it's not logical nor self-contained, but is arbitary and historical. Then again, I'm not absolutely convinced it's not necessary...
Is it really necessary? I have so many memories of sitting in math classes and the teacher simply starts writing letters on the board and I am thinking to myself: what the hell does that symbol or letter represent? Before we even get around to the concept that the equation represents.
What you are describing is not unique to math. It is required in any discipline. In fact, it is required in any social interaction. People are constant;y building common vocabularies. The words and phrases you use at work are different than the ones you use at the neighborhood block party. Even among different friends you have different vocabularies. "Elbow" means something different to plumbers, fishers, basketball coaches, and musicians. "Panama" means something different to a very small subset of your friends. Humans are constantly building common vocabulary, and everyone maintains several different vocabularies simultaneously.
To complain about the effort to build a common vocabulary is to complain about the effort of participating in any society.
It's amazing how much this cultural conditioning can affect one's aptitude towards math. When I used to help my brother with his maths, he would oftentimes complain about not being smart enough. Shortly after, he would give up on the problem and come up with a rudimentary solution just to have an output. On these occasions, I'd prod him to solve the problem, ask him about his approach, probe his mental process, and most of the time, he'd tell me the correct steps to solve the problem. Had he not given up sooner, he would've succeeded.
I think that an interest in maths, is the fuel to help one cross from problems to solutions. With how mathematics is taught nowadays, mechanical rather than analytical, I can imagine how easy it is for kids to develop a negative sentiment towards math especially when they get dozens of rote problems every day.
I think kids see those exercise problems as arbitrary and therefore unmotivated to do them. I dunno if there's any way of making then interesting and or relevant to motivate kids to do em.
I suspect curiosity is of paramount importance, and that curiosity is not fostered in school about math.
>I think kids see those exercise problems as arbitrary and therefore unmotivated to do them.
This is exactly how I felt growing up. It wasn't until actually looking at the practical applications of math that I got it. Doing equations for the sake of it without any actual context constantly frustrated me to no end and made me hate math and physics because it made me feel incompetent and stupid, it's been a struggle getting rid of this feeling even to date.
The fact that school math is thought to be 'scary' and 'hard' is not the issue. Plenty of video games are known to be scary and hard. For instance, they contain monsters and are difficult to complete. They even have objective grades (scores).
What makes the difference is that video games aren't compulsory. So they represent true play. Which is what makes them educational, unlike a school curriculum, however much it is tweaked. Well, until something more fun than video games arrives on the scene...
Hard and scary carry differently meanings in those contexts though, surely?
Video games are safe e.g you can always try again, you can do incrementally well.
Video games are scary in an exhilarating way, you blood will pump and your pupils will dilate but at the end you can put it down, turn the lights on and watch funny cat videos to recover.
Math (at school) is hard because you get one shot. Finite lives. After enough retries society says "nope, you're done". It's hard because you can't test yourself when ever you want and your study is not directly related to your tests.
Math is scary like anxious scary. Like stay up at night dreading your exam or don't sleep for a weekend to finish your report scary. You can't watch cat videos and hope it goes away because it won't. You can't buy a new math or try it again in a few years.
Nicely illustrated. As you imply, the scariness is a property of schooling rather than of mathematics itself. And where game play is scary, it's at a level chosen by and controlled by the player, for optimal learning I suspect.
Not only that, but with "GPA" provides a running historical average that serves as a sign that:
"No matter how good the student is now, they sucked at one time and they can never escape this eventuality".
Imagine a game that's tough, full of oft-used rules, failure potentialities everywhere, limited lives, and punishes failure/learning cycle. That's School.
I love learning new areas and new techniques. And when Im teaching myself, I can fail and I go back a step or 2. Not the end of the world.
If I do similar in a class, then I set myself further back with a slowly growing chance of doing badly (D or F). And in College, then its another 1-4 credit hours to pay for the privilege. They not only fail me, but demand more money.
> I must have attempted to implement a lisp many times before I got it "right"
How did you know it wasn't right? Did you run into a problem trying to implement requirements which you knew all along and accepted? (As in, you started coding without planning for some of these requirements ahead of time, then got stuck?)
Or did you learn about and adopt new requirements which weren't easy to retrofit into the existing work in progress?
Good question. I guess by "right" here I mean an implementation that I was happy with. Not nessesarily standards compliant or complete.
In my early attempts it was things like parsing which tripped me up. I'd implement something that broke the bracketed lists up into a list of tokens and then get stuck nesting those lists into an ast. Abandon an implementation for a while and then try again at the problem, often with a fresh implementation.
Then is was evaluating expressions, when to evaluate the arguments and exploring different approaches (fexprs).
I'd do these iterations across months and with different languages. First python then JavaScript, Ruby, C.
My favourite was the JavaScript implementation. Lisp code is literate and I bootstrapped most of the built in functions by exposing the environment as a cons tree (and working with cons trees internally in the interpreter).
Video games also don't have social promotion or enforced curriculum "progression". If you die in the first level of a video game, you play it repeatedly until you legitimately merit entering the second level.
If you get a mediocre grade on a math test, even a failure, we shove you into the next subject in the curriculum. When you start failing that because new math builds on mastery of old math, we tell you you're not a math person and should probably major in Literature.
Then we go complain to the other math teachers that students are lazy and apathetic, and ask each-other how it is that math PhD students manage to spontaneously generate from such piles of shit.
I'm not disagreeing with you. 'Scary' and 'hard' becomes an issue when that's all that's being said about math. Video games are 'scary' and 'hard' but... they're also fascinating and fun. The popular math narrative is incomplete unlike video games. Why is that? Dr. Eugenia Cheng says it much better and has a good answer as to why [1].
There are some aspects of doing math that isn't fun, because it is grinding. Learning the time tables for example - you just have to do lots of exercises to get used to multiplying. No way around that.
Just like in sports, where you must also do a lot of basic practise to get good.
Making the practise like a game can be done to done degree, but ultimately, a kid is going to see through that after a while, since it gets boring.
Thats the kind of BS math they shouldn't bother teaching. Most of math is about concepts, and if 5 year olds can grapple with super-basic calculus how about teaching 10 year olds proof-based math and concepts (closer to college level) than memorization?
Its kind of dumb how arithmetic is seen as a precursor to math when its really not. You could do an ENTIRE college-level course in linear algebra or calculus or graph theory or even combinatorics perfectly well even if you have no idea what 11*15 is off the top of your head and have to look it up every time.
Its just a ridiculously backwards approach to teaching. We don't teach CS by testing people on memorizing syntax and API calls - they are rightly relegated to "google it when you need it, eventually you'll remember it". Of course algorithms are more interesting and building things is more fun so we focus on that.
Current way of teaching math is just a historical accident IMO. No real reason it couldn't be conceptual and interesting from the start if there wasn't so much inertia around current systems.
Of course understanding fourier transforms isn't reliant on my ability to do 6x7 - but arithmetic as a lifeskill is actually valuable. It's like saying that grammar and spelling aren't important because spellcheck will sort it out.
Splitting out arithmetic from "Maths" doesn't seem terribly sensible to me. Times tables aren't really problematic, and they do provide opportunities to teach valuable concepts (5x6 = 6x5).
No, you're straying away from the original point which was: there's no need to make rote memorization of arithmetic tricks and multiplication tables a hard prerequisite to everything else about math. Nobody claimed that arithmetic is absolutely useless
I think it is undeniable that both a certain level of working memory and of basic algebra is necessary to understand mathematics at the next level. Someone who could actually understand an "entire college-level course in linear algebra or calculus or graph theory or even combinatorics perfectly well", yet be unable to remember a basic multiplication table, or figure out how to calculate 11*15, would have a very interesting pathology.
It is never hard blocked on arithmetic though (in the UK at least). A bunch of simple numerical solving will be harder without being able to do multiplication in your head - but certainly someone could get through without.
I'm honestly dubious of anyone who claims that the reason people hate maths is because of times tables.
Well a lot of people who dislike math do so because they think its all about rote calculation. Not just times tables, even algebra can be seen as a completely syntactic procedure - memorizing a bunch of rules. Another example - if you are taught 10 different "templates" of problems and solutions in calculus class and memorize which equations to apply where, you can solve a lot of problems without ever building any intuition.
If someone thinks of math that way, I can understand why it would be boring to them. Thats what I mean by blocked on arithmetic. If every thing is taught as a cookie-cutter process to follow its not surprising that people associate math with blindly following processes.
> If every thing is taught as a cookie-cutter process to follow its not surprising that people associate math with blindly following processes.
A hammer and a chisel are tools. You can use them to shave a bit of wood off the door, or you can use them to carve a beautiful sculpture. You can do neither without knowing how to use them.
I agree that too much of maths teaching is "here are 100 quadratics, go solve them" - which is obnoxious and dull. But "blindly following process" (matrix multiplication, change of variables/base, sum to infinity etc) happened throughout my degree and is a component of higher maths as well. Heck, most of the proofs I remember use "tricks" that any mathematician is expected to just know.
I sure as hell don't think children are taught maths well. I remember being taught how to take the derivative of x^2 without the teacher bothering to show where that came from or what it really meant. Heck - children are given the quadratic formula to memorise and basically told "it's magic, learn it" instead of showing them how you can trivially create it by completing the square.
But mathematics does involves a ton of following some process or other to get the problem into a format you can do something with. Probably by following a different process that you've done before. I'm not sold that you can simply eliminate that aspect.
Seems like we agree. I am only saying current math is too focused on the mechanical processes, so much that many people never even see the other (more important) conceptual side. Times tables don't need to be taught in class - there is nothing to teach. They just need to be memorized by students.
> Thats the kind of BS math they shouldn't bother teaching. Most of math is about concepts, and if 5 year olds can grapple with super-basic calculus how about teaching 10 year olds proof-based math and concepts (closer to college level) than memorization?
The times tables are BS math? The thing I've used IRL literally hundreds (thousands?) of times more than I've ever used anything from, say, calculus? The thing that yes I could use a calculator for but damn would it get annoying because I use it so often? That's what we should stop teaching?
Math things I use a lot in actual real life: basic arithmetic (drilled skills/knowledge!), basic algebra, factions/percentages (converting one to the other, and scaling other numbers by them—the basic arithmetic stuff is also vital here), and generally how triangles work. That's... almost all of it, actually. Some high-school-level stats stuff from time to time, I guess.
If you don't know up to 9x9, then doing basic algebraic problems become vary cumbersome. Factoring and simplification becomes a chore and one can't get into a good flow if you are typing in 2x5 into a calculator.
On the other hand, I can't imagine anywhere that would expect one to know 11x15 off the top of ones head.
>Learning the time tables for example - you just have to do lots of exercises to get used to multiplying. No way around that.
Learning how to perform mental arithmetic with the aid of pencil and paper was a way to escape from the grind of farm work. One could get a respectable job as a clerk. However with the advent of electronic cash registers and smartphones a way around this later form of grind has also been found. So no need to make it compulsory. Which isn't to say that people don't choose to practice specific things they personally consider useful. Even gamers do that.
Can't understand modern world without understanding statistic, can't do that without algebra and basic calculus, can't do that without being at least adequate at mental multiplication (for calculating parameters in equations and understanding how they change without interrupting your train of thought every 3 seconds).
So yeah, I think you still need to grind these multiplication tables, even if you have perfect version of Wolphram Alpha on your smartphone. It won't tell you what question to ask.
Have you noticed that most adults don't know what nine times seven is? If you ask them on the street they'll mostly either get it wrong or tell you quite rudely to go away. This is despite (and because of) a decade of grind at school. Or have you noticed that people are unable to hold a conversation in a foreign language despite years of 'study' in a classroom? Whereas an enthusiastic individual can pick up basic conversation for themselves in a matter of months.
This is because coercion in education is: (1) painful, (2) prophetic, (3) it doesn't work!
What happens in real life is that if somebody needs to know something because they are keen to do something else then they will learn it efficiently, 'just in time' and precisely to the extent needed. Assuming their creativity is relatively intact...
Not really (but then I never asked random people to multiply for me :) ).
But I can kinda understand why, you almost never have to multiply 9*7 by hand. If you're doing algebra or any symbolic calculations you rarely use digits other than 0-4.
In real world Benford's law + usual rounding greatly reduces the need to multiply by 7 or 9 (or 8 for that matter).
> Learning the time tables for example - you just have to do lots of exercises to get used to multiplying. No way around that.
The way around is to just not do that.
Students get huge amounts of practice with basic multiplication if they are assigned more interesting problems/projects that use multiplication as one step, and will inevitably memorize all of the same basic facts, assuming they actually do the work.
Memorizing multiplication facts for their own sake is a soul-sucking waste of time that turns kids off from math for good.
Similarly, the way to get good at spelling is by reading engrossing books for hours every day and writing about subjects of personal interest. Studying lists of spelling words in school is beyond pointless.
These are not skills like sports or playing musical instruments, where there is a particular physical technique involving fine muscle coordination, where practicing incorrectly will establish wrong movements. Instead we’re talking about simple verbal/linguistic recall of previously seen information.
Maybe they could present kids with timetable After showing them more variate and interesting concepts, so that at that point they have both the spark of interest and the maturity to understand multiplication is important and the attitude at learning that makes it not a torture.
> Plenty of video games are known to be scary and hard
Those statements seem related because of the ambiguous definition of "scary". In relation to math, "scary" causes anxiety, whereas in relation to video games, "scary" causes excitement.
As soon as you grasp the different meanings, you can clearly see that even though both statements are about the same word they are still orthogonal.
Yes, it's complicated. Yet I don't think I committed an equivocation. As I said, what makes the difference is compulsion/coercion, which causes us to interpret scariness in one context as bad (e.g. falling off a ladder), and scariness in another as being good and part of the fun (e.g. skydiving).
videogames also encourage a learning process based around experimentation and failure, both of which were highly discouraged during the course of my math education. effective learning is a process of repeated failure & incremental improvement that stands at odds with the present incentives in the american education system.
Its the same for programming. Most of programming isn't rocket science (unless it is :P) and anyone with the ability to apply basic logic can figure out what most programs do. However the language, the text itself, gets in the way, and people throw up their hands immediately once they see something unfamiliar.
The difference between literacy and illiteracy in this case, IMO, is almost an emotional one - its the ability to say to yourself, "ok, I may not know what this keyword means, but I know what is in front of me is not magic, that it can be understood, and I can understand it with some research and patience"
It helps a lot for ones future if someone is able to be taught this skill early on.
Right. My sister is always having a hard time with math and it seems like the teachers are always focusing on solving equations rather than explaining the reasoning behind it.
It's pretty simple when you realize it's (mostly) all about lines.
This is the biggest contributor. Teachers focus on making sure they follow precise steps whenever trying to work through a problem. While that's helpful for certain things, if the student can explain why they are doing what they are doing then they won't be able to apply what they've learned to solving new problems.
It's definitely not an easy problem to solve and I certainly don't envy elementary/middle school teachers.
The negative impact of a bad teacher is overwhelming. It really reflects our species' dependence on a (recently often-recorded) oral history, even in the sciences.
In my opinion, the biggest problem is the focus on learning math subjects as dependencies for learning the next math subjects. This problem is exasperated by tying a subject to semester or school year.
If I'm not supposed to learn Calculus until I learn Algebra, that must mean I can't learn Calculus until I learn Algebra.
If my Algebra class lasts for a year, that means it must take an entire year to learn Algebra.
If I am supposed to learn Algebra first, then I must not be able to sufficiently wrap my head around anything related to Calculus.
That means that whenever I hear about "limits", "d/dx", I should immediately lose attention; after all, those are Calculus-related things, and my underdeveloped brain will not be able to comprehend such subjects for another year, and I shouldn't even try.
Of course, none of those are true. If I wanted, I could
* learn at my own pace
* learn in the wrong order
* wrap my head around limits, d/dx, L'Hopital's rule, etc. before understanding factoring, trigonometric functions, etc.
In fact, any of these would have put me far ahead in my education.
"Traditional" education requires a student to be lazy, or hold [him/her]self behind academically. To make matters worse, it requires that student to be a specific, quantified amount of lazy/held back.
For me personally, I struggled to keep up with the obscene amount of expected rote homework, while also feeling miles ahead of lectures, even though the homework and lectures were from the same class.
When I started learning about programming on my own, I suddenly had the freedom to do all of the things I mentioned above. Programming, as a subject, is especially applicable: There are abstractions, paradigms, languages, etc. You can jump into any aspect headfirst, and still get your bearings.
I think that negative cultural conditioning is an interesting point. The same thing can be said towards statistics. However, once your own personal lens is shifted to show practical application it makes the basics easier to understand.
They bypass conditioning because they are smarter and are good at it. Who do you think are the people who espouse anti-intellectual drivel like "math is scary" or "math is uncool"? Hint: not the smart people.
There's plenty of research on the "entity theory" bias, mostly applied to gender. I.e. if you tell girls that they're bad at math, on average, they will act as if they are.
Unfortunately, framing math as hard or scary only serves to reinforce stereotypes and bar people who do not have the keys to defuse that framing from advancing. It's not the rich male kids who are told math is hard and scary.
Does the "if you tell girls they're bad at maths ..." work differently with boys?
In UK girls get the better results on average. In part I think this might be because they're taught as they are generally quieter and less desirous of rough-and-tumble (in primary school, by the predominantly female teachers, say) that girls are more studious. Generalising further: boys are thus modelled as either boring swots or good for manual labour (they don't sit still, which for some reason seems to make many female teachers think means they're not intelligent).
[I'm pressing the point probably a bit much I know].
> There's plenty of research on the "entity theory" bias, mostly applied to gender.
Junk science if there ever was any. The effect sizes for these treatments are tiny, if they even replicate it all. It seems like every day now there's a new rebuttal of stereotype threat, implicit association tests, priming, etc.
>Nonetheless, the evidence overwhelmingly supports the tripartite pattern: (1)Although errors, biases, and self-fulfilling prophecies in person perception, are occasionally powerful, on average, they tend to be weak, fragile and
fleeting; (2) Perceptions of individuals and groups tend to be at least moderately, and often highly accurate; and (3) Conclusions based on the research on error, bias, and self-fulfilling prophecies routinely overstate their power and pervasiveness, and consistently ignore evidence of accuracy, agreement, and rationality in social perception.
Namely point 1, which points out it is valid science. Point 3 is the criticism that it is vastly overstated, and personally I would add it is used as a political tool far more than it should be. But the science isn't junk.
Do you now of any research on the potential downsides of the opposite of this? I have a pre-teen son whose confidence with-- and enjoyment in-- maths outstrips his ability (although this might change in either direction). He's not cocky about it; he is more like "Damn. Only got a C. Oh well, next time..."
I can't see him hurting himself with maths, but if he decided he's gonna fix something electrical in a few years time, well that could be very bad indeed.
Sometimes it is smart to embrace this. Find a grade school student struggling with math, tell them it is easy, and they'll think themselves a failure. Instead, I find it better to tell them that math can be easy or hard, and it depends upon how one is taught. Let the student know the failure to understand math isn't due to something inherent in them, but something that can be changed. (I am assuming a student who tries or has tried in the past, if they don't try at all, that takes a different approach).
Imagine repeatedly telling those who truly struggle with mathematics that math is soothing, math is easy, even dumb people should be able to do math, etc.
I would agree with this. If only by a personal anecdote: I got through calc 1 & 2 by the end of 8th grade. This was though an online class while at a smaller rural school in Montana. Being 'smarter' than my classmates had little to do with it, instead the immense bureaucracy of public school my parents fought made it happen.
Anyone commenting on this (whether saying 'yes' or 'no'), please first read the article. What it considers 'calculus', none of us here would consider calculus. Because very few 5 year olds are capable of understanding even something as basic as a line plot (or '2d graph' or however you want to call it). They simple do not have the neural matter to comprehend fairly advanced (from an evolutionary point of view) abstractions like the relationship between a value on an x and a y axis.
And of course someone is going to say 'but I did it' or 'my children do it' - sure. My 6 year old has a very rudimentary understanding of the relationship between area and volume, and speed and velocity (although I'm not even sure how much of it is real understanding or just parrotting; I'm not a very good teacher, I've learned by now, not so much in how I explain things, but more in reading feedback from students). But it's not applicable to the population at large. I've tried explaining to larger groups and at that age, understanding why 'half past 6' is called that, is already quite a feat. Let alone understanding that there's two 'half past 6's' in a day, and why that is.
I agree with that. You can hand limits as a tool to someone already comfortable with the concepts and present them as a way to quantify their intuitive understanding.
But without limits, you cannot actually do much with the concepts. Whether you call limits a concept or a tool, they are essential to the practice of calculus, and, historically, to its emergence (and not trivially so, either.)
Yeah but you can do well on the AP calculus exam without understanding limits, if you can calculate derivatives and solve some word problems related to derivatives.
Hence, the gap between "5 year old calculus" and "high school calculus" is smaller than the gap between "high school calculus" and "actual calculus"
> They simple do not have the neural matter to comprehend fairly advanced ... abstractions like the relationship between a value on an x and a y axis.
I don't think that's true. Its wholly dependent on what they've been focusing on thus far. To use the dismissive "they simply" is a misrepresentation of the plethora of candidates. Did you mean "most"? Also its not a question of brain matter but the current development of that matter towards these sorts of ideas.
Its also a completely different subject to emotion control which you brought up somewhere else. This is about understanding, not control.
In order to bring our arguments to a common frame of reference, I think that I can reword my point as follows: to do the sort of abstract reasoning that is required for what is commonly (so not like the OP, who redefines calculus as 'abstract(ish) concepts that will be useful in real math later on') considered 'calculus', one needs a part of the brain that is not just 'grown' by training, but that actually needs to grow to a certain baseline before it can function at all - after which it can be further developed by training/practice.
So in other words, some brain functions just do not physically 'exist' (the matter needed for those functions) at very early ages. You can't take any 2 year old, or 3 or 5 year old, and practice until they get it. You have to wait until that part grows naturally. And growing (or maybe just 'activating') that part earlier doesn't say anything about overall intelligence, either.
And yes, it's different from emotion control, but not dissimilar - the mechanisms is the same.
All of this, of course, for the general population. I can't deny there are child prodigies who understand these things years before their peers; they exist, it's a simple observation. My point is: some things you cannot train in children. You have to wait for the brain to get there; and children not being there yet is not (necessarily) because of lack of training.
Hopefully this clears up my position, and contrasts it with yours. If it does in the way I think it does, it becomes a matter of the state of the art in neurology and neural development. My information comes from having read some popular literature (books, not the 'parenting' section of Marie Claire, but still...) on child development; not a very authoritative position, I'll freely admit. I never got the impression anywhere that this position is controversial or even just not universally accepted, but if it is, then our (apparent) disagreement can be reframed in these terms. (to preempt - at that point I don't have anything to add except the tried and proven 'trawl google scholar until I find something that I can bend into supporting my position, to make it appear that I have Science(TM) on my side although I don't really know anything about it'; but maybe I'm getting ahead of myself here :) )
oh that's fine. :)
I'll disagree on outlook though. I believe that at least neural-net-wise a brain develops according to what one practices the most. If that's reasoning then its reasoning.
I think its a bit frivolous to just assume that abstract concepts are beyond children of that age, I think if anything its more a failing of our means of explaining them correctly.
Of course its ultimately possible when the three year old I chat to claims to understand that they are lying to me or are just seeing their response as a hoop to jump through, however I would posit that the vast majority of the required components are formed by this point. The remainder just being a process of dispelling the various foggy bits through practice and exploration.
TL;DR; I fancy the brain to be more a thing that just improves over time. Even if incorrect I certainly doubt that "reasoning" or "abstraction" are just modules that are either IN or OUT of the developing brain. I feel like these sorts of distinctions are unproductive and too coarse to make.
Like I get that there are bits that are somewhat nonfunctional. She struggles to remember but she can if she really tries or events had impact, she can process-of-elimination to find the location of one of our tribe, she can understand that a word has double or triple meaning and double checks with us when we give her an old label with new meaning.
I feel like there is plenty of stuff going on there that could be fashioned into "reasoning" and "abstraction", hence my issue with your assertion.
True; but do you have anything in mind that is less abstract than a graph and still expresses the continuous (or not, depending on what you're looking at...) nature of a relationship between two variables? Because I don't know of any way that is easier and more intuitive to understand than that. But maybe I'm just projecting based on my own preferences.
I think you have to understand how two things can change "with" each other before you can understand how to read a chart. Personally I think seeing something vary with time or distance is more immediately understandable than reading a chart.
Reading a chart yes, but that's because (IMO) reading is harder than making one. For example, no matter how much someone would explain, I could never understand the relationship between an angle and a sine wave, until I had a teacher made me draw a circle and an x/y axis alongside each other and project the line in the circle onto that axis. I still don't think one can really explain this without resorting to describing what happens visually. But again, maybe that's just me.
Most books on child development has (a) chapter(s) on neuroplasticity and neural development, and how that relates to actual skills. I've only read one and skimmed a few, the one I read was a Dutch language one called 'Oei ik groei' so that's probably not much use to you. If you start reading any book on this I'm sure you'll find plenty on it, and they'll probably point you in the direction of dozens if not hundreds of others.
But if you're referring specifically to the 'evolutionary point of view' part in my sentence, I don't even remember what I meant by that - it doesn't make much sense, re-reading my sentence. I guess I just meant that brains develop in a way that makes them, early on, incapable of doing some things that adults can. Similar to how until the frontal lobe is fully grown (18/20 years old), children/youth can't control their impulses the same way someone with a grown, healthy frontal lobe can. So it's useless punishing children for some things because they literally can't change some behaviors; it's like beating a quadriplegic until he walks because 'he should just try harder'. (This is obviously a very simplistic and probably in many ways wrong description; I'm just using this as an analogy for what I was getting at).
As someone who struggled with math early on, but then persevered through an Engineering degree, I agree with this 100%.
Math is often presented to kids as a hard challenge to overcome. Instead, math is actually just a language that helps people talk about particular subject matter faster than they would otherwise. You don't need to solve equations in order to understand how math concepts work, and for a lot of people that would already be a 10x improvement.
Shout out to - https://jumpmath.org - I interned for them in college and John Mighton made a great series of books to help kids learn math. They are not quite as playful or advanced as this article suggests, but rather simple concepts explained in simple terms, then repeated in different ways. From memory, it was especially useful for kids who thought they were bad at math, and struggled to catch up to their classmates.
Jason Roberts has been doing an awesome job rolling out a high level math program in the Pasadena school district. 6th, 7th, 8th graders learning Calculus. They even have an anual competition at CalTech called Solve.
My favorite app for kids is Dragonbox Algebra. It starts by teaching simple algebra concepts with cute shapes. A couple hours later your 5 yr old is solving complicated equations by dragging and dropping. Just amazing.
Seconded, and actually all the "We want to know" games had a lot of success with my kids. They loved the games, and they're really games first for them. I guess they'll only realize the math content in a few years ;)
Personally I'm most impressed with "Elements" [1], for geometry (from Euclid of course). I find the way they gamified geometry problems really impressive. It's very intuitive, and worked wonder even before the recommended age with both my kids. Even if you don't have kids yet, I would recommend to anyone interested in serious games to spend the ~5 bucks it costs and have a look at it.
This is my favorite book these days for exercising an aging mind : "A Moscow Math Circle: Week-by-Week Problem Sets" [1]
Is there something similar for kids i.e. a fun activity book that exercises your mind and helps you develop a love for mathematics ? (Not looking for generic puzzle books)
This reminds me of a topic that comes up a lot at my house: late homework.
What should teachers do about late homework? Should you get a zero if you don't turn it in? Should you get 50% off for anything late? Should there be a limit on how late it can be?
If a student turns in 10 missing assignment in the last week of the semester, then gets a 96% on the semester exam, what grade do they deserve?
What is the point of a grade anyway? There are so many different meanings built into a grade. A grade can teach discipline and responsibility. It can teach respect. It can teach cheating. It can teach to do just enough to scrape by. It can teach that the important thing is to be higher than others. It can represent competency in a subject. It can represent value to the income of the football team.
Education is about so many things that it's hard to have a useful blanket discussion.
The correct approach depends upon the intention. If you are trying to improve people's understanding, you take one approach. If you are trying to vet competency for potential employers, you take a different approach. If you are trying to build a minimum base on which society can be anchored, you take yet another approach.
Personally, I think math should start at teaching kids useful stuff that 100% of them are going to use such as; taxes, credit products/home loans, budgeting, investments etc. I'm sure calculus is useful and great, but the reality is only a small percentage of kids are going to use it so it doesn't make any sense to force it on EVERYONE.
I can't agree with this more. I hated math as a kid, I could learn the concepts very quickly but was prone to silly errors. So I'd fail many tests and I thought I was bad at math and shouldn't bother with it. Later doing an Engineering degree I found out math was this amazing tool that could help you model systems and do amazing things. The practice I got while solving interesting problems helped me learn to stop making silly errors. It wasn't that I was bad at math it was that I wasn't interested enough to focus. I just couldn't figure out why this wasn't the way math was taught at high school?
This article misses the point of early math education.
We don't teach elementary school kids to be future math researchers, or even engineers. We teach them things that everyone should know : reading, writing, basic science, a common cultural basis, and counting.
Arithmetic is important in everyday life : calculating change, knowing what a 10% discount means (percentages are more tricky than most people think), converting currency, measuring surfaces, etc... Calculus, not so much.
In France, they experimented with "maths modernes", which was an attempt to help kids with more advanced concept at an early age : starting with set theory and bases, delaying basic arithmetic. It failed miserably, it produced kids that were unable to do everyday life operations. We are now back to a more down-to-earth approach, with an emphasis on approximations and mental calculations.
I can second this, with a minor caveat. My daughter (third grade) loves these books, and we get a lot of good problem-solving into the curriculum as a result. The caveat, in my opinion, is that they don't provide enough cyclic review in their default configuration.
Each chapter includes 80-100 problems, divided in to usually between 4-8 sections. The problems are great, but once a section is complete they're weak on later refreshes. I've been working around this by doing even-numbered problems the first time through a section, then half of the odds a few days later when we're a couple of sections downstream, then selecting randomly from all of the unfinished problems in the entire curriculum for just a couple of extra "old stuff" problems each day throughout the year. We also supplement with a number of other great resources, if you're looking to implement a more problem- and exploration-oriented math curriculum:
1. Kitchen Table Math is great for selecting concepts to lead number talks with (for building number sense - this is the first part of our day)
2. Saxon has excellent spaced-repetition exercises for shoring up the calculation side of things, and giving the student some easy wins for confidence building (we typically use Saxon's material as a warmup before Beast Academy)
3. Thinking Mathematically (the one by J. Mason and L. Burton) has a unique and useful mental process for attacking hard problems when you're not handed a nice formula to plug things into. Once a week, we work through a hard problem using the method in this book.
4. I haven't worked it in yet, but Arthur Benjamin's "Secrets of Mental Math" has a lot of stuff in it that will solidly connect arithmetic and algebraic thinking later.
So if we block third party tracking, the Atlantic doesn’t let you see the article?
Why are we still allowing the Atlantic on HN? I understand paywalling, but they actually block content if you refuse to let them track you. Seeing an ad and consent to track are two very different things.
My Chinese-speaking Indian mother would have agreed with this article, though she long ago learned that "you can't fight city hall" -- in this case you can't change the curriculum. She taught me many fun things despite the constraints of school.
This is reminiscent of Star Trek TNG "When the Bough Breaks", where a parent of the 23rd century scolds his 8-year old child for not wanting to study calculus.
And if calculus is presented as a fun group activity, like suggested in the article, I don't see why it would hinder the kids growth. It's the other way, really.
It looks like we have a misconception of the verb "to learn". Conveying some very general intuition of the subject from adult to a kid is not learning, actually. Learning assumes conscious effort to comprehend relatively complex things, constantly overcoming pain of uncertainty, most mathematicians will agree with me here. This is not what is usually expected from a kid. Don't do that.
Imagine a toy truck that, as you drive it back and forth, plots the first and second derivatives of its position with respect to time on a little screen on the side.
I'm not even sure.. arithmetic really needs learned as much anymore... as a dev I rarely use my times table... if I need to calculate something I'll pull up a console and go into a repl and just ask php or ruby to calculate it for me...
I know HOW the calculation works and why, but I don't need to know and remember times tables. I like the idea of thinking outside the box on how to teach math and science, as I feel modern schools were built with factories and the industrial complex in mind, not technology, arts, and sciences.
if I need to calculate something I'll pull up a console and go into a repl and just ask php or ruby to calculate it for me
I'm guessing you don't spend 100% of your time in front of a computer, and even if you do, in the time it takes you to do that, someone who can actually do mental arithmetic will have approximately calculated that problem and several more, because they can stay in the "flow" of thinking about the bigger problem. As someone who can, it's astounding how many developers out there can't --- pulling out the calculator for computations as simple as 12+51 or 6x128.
It reminds me of the "developers don't need to learn how to type quickly" argument --- yes, you can be productive without, but you'll quickly find yourself to be in a handicap in contrast to others who can write, rewrite, and mentally estimate and compute the results in their head several times faster, saving much time in compiling/running/debugging/etc.
Being able to perform mental arithmetic even approximately is a very helpful skill - buy 6 items at £12.80 and get charged £110? Easily catchable with a quick "6x10+25%" mental approximation.
(I'm fairly sure there's a famous book about this kind of thing but for some reason my brain is stuck on "How To Solve It" by Polya which I don't think is correct.)
Start watching Countdown, the UK numbers and letters game. You will need your times tables for the math part. There is even a comedy version (same games with added silliness) to keep the interest of 'younger' audiences. Youtube search : 8 out of 10 cats does countdown. It's fun and funny.
As a dev, I use my multiplication tables pretty constantly, especially for quick estimates using approximate numbers. I probably didn't need to know them as early as I did...
The how is what is supposed to be being taught to children at the moment and why the math classes are shifting away from times tables and such. Rote memorization doesn't produce as good of a result as understanding how the calculations work on more intuitive level. I can't find it at the moment since I'm in the metro with poor connection but there was an article about how geometry was taught in such a way to teach kids to distrust their own common sense and instinct by relying on Byzantine rules from the getgo instead of letting kids naturally explore geometry and get a feel for shapes, angles, etc. (with the rules coming later)
The move away from rote can have an unexpected draw back. Fast mental estimation requires rote learning , otherwise you draw a blank. over-reliance on tooling like calculators have made mental skills like that decrease over the years.
