Five-Year-Olds Can Learn Calculus (2014)
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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 suspect curiosity is of paramount importance, and that curiosity is not fostered in school about math.
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.
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...
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.
I'm just making things up tho
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.
Years later when I worked in a shop I used to cost complex print jobs in my head and co-workers though it was magic.
The way maths is taught in the UK is largely how and very rarely why.
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).
I'm honestly dubious of anyone who claims that the reason people hate maths is because of times tables.
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.
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.
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.
On the other hand, I can't imagine anywhere that would expect one to know 11x15 off the top of ones head.
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.
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.
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...
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).
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.
[1] http://www.pbs.org/video/math-is-amazing-and-we-have-to-star...
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.
You get one shot. You get little to no feed back.
That's hard and scary.
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.
Edit: misspelling that looked bad (dong=>doing)
Without the pressure I'm not scared of revisiting hard things, I'm driven to actually.
Edit: to indicate that a "simple" perceptron isn't actually that simple.
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?
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).
"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.
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.
> 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.
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.
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.
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].
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.
Just from today: http://www.tandfonline.com/doi/abs/10.1080/09515089.2017.135... http://internal.psychology.illinois.edu/~acimpian/reprints/j...
From your second link, conclusions section:
>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.
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).
Math, as taught in school, is hard.
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...
To complain about the effort to build a common vocabulary is to complain about the effort of participating in any society.
I suspect that mathematics being poorly taught is also a major contributor.
It's pretty simple when you realize it's (mostly) all about lines.
It's definitely not an easy problem to solve and I certainly don't envy elementary/middle school teachers.
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.
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.
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.
In particular, IMO the distance between AP calculus and honors calculus at an ivy is larger than the difference between this material and AP calculus.
Limits seems to me to be more a tool in order to calculate the ratios than a fundamental concept.
Hence, the gap between "5 year old calculus" and "high school calculus" is smaller than the gap between "high school calculus" and "actual calculus"
Do you have some links to research on this? Sounds interesting.
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).
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.
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 :) )
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.
That's entirely the point. You can understand calculus without understanding graphs.
It may sound crazy, but it's true. You can understand a subject without all of the dependencies academia tends to categorize Maths into.
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.
Info and video of kids in action here:
https://www.mathacademy.us/solve/
Direct link to video:
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.
[1] https://play.google.com/store/apps/details?id=com.wewanttokn...
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)
[1] https://www.amazon.com/Moscow-Math-Circle-Week-Week/dp/08218...
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.
But it was often a matter of motivation because the same kids would be able to calculate the return on a bet at the horse races in their head.
Or engineering, as in your case.
and suddenly a cryptic formula from maths class is made clear through the visual display of a wheel turning.
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.
"A Mathematician’s Lament" by Paul Lockhart
https://www.maa.org/external_archive/devlin/LockhartsLament....
BOOK published in 2009 https://www.amazon.com/Mathematicians-Lament-School-Fascinat...
One example great example is the pythagorean theorem which can be demonstrated with a series of triangles and squares.
I do not know of any current books that has done this to a degree a five year old could learn from, but there is a huge opportunity here.
The book How to Teach Your Baby Math touches on some of the fundamental differences between using visualization and symbology.
They have challenging problems that emphasize basic and advanced concepts in a variety of ways, often posed as games or puzzles.
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.
Here's a function, now practice evaluating it on this list of 200 inputs. Due tomorrow.
I would prefer:
Here's a concept. Play around with it. Tell me its limits, and I'll tell you another concept that breaks them.
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.
You can both learn calculus and play with toys.
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.
Hmm, that's not the worst idea...
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.
Here is the 2017 final of the 'proper, serious' Countdown, also on YT https://www.youtube.com/watch?v=Q-VdiufVWrw
I seem to remember one where the hostess wrote the numbers on a huge dick she'd drawn on the board. Thus giving new meaning to "Numberwang".
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.
(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.)