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Five-Year-Olds Can Learn Calculus (2014)

206 points by guiambros 9y ago

jonnybgood 9y ago
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.
- brogrammernot 9y ago
  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.
- stephengillie 9y ago
  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.
- jackmott 9y ago
  congratulations on believing a thing without rigorous evidence?
- d-sc 9y ago
  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.
- jpamata 9y ago
  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.
  - chii 9y ago
    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.
    - fspear 9y ago
      >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.
- roceasta 9y ago
  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...
  - chii 9y ago
    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.
    - Micoloth 9y ago
      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. I'm just making things up tho
    - zaptheimpaler 9y ago
      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.
- pishpash 9y ago
  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.
  - erispoe 9y ago
    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.
    - ehecatl 9y ago
      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.
    - pbhjpbhj 9y ago
      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].
    - ctlby 9y ago
      > 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. Just from today: http://www.tandfonline.com/doi/abs/10.1080/09515089.2017.1359245?journalCode=cphp20 http://www.tandfonline.com/doi/abs/10.1080/09515089.2017.135... http://internal.psychology.illinois.edu/~acimpian/reprints/jussim_BBS.pdf http://internal.psychology.illinois.edu/~acimpian/reprints/j...
- hyperpallium 9y ago
  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...
  - nogbit 9y ago
    It is necessary but could be improved. If all we had was roman numerals we would still be in the dark ages.
    - eberkund 9y ago
      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.
      - lingua 9y ago
        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.
- jamesrcole 9y ago
  > cultural conditioning I suspect that mathematics being poorly taught is also a major contributor.
  - g00gler 9y ago
    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.
  - rockostrich 9y ago
    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.
- 5040 9y ago
  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.
- headcanon 9y ago
  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.
- thomastjeffery 9y ago
  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.
gremlinsinc 9y ago
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.
- csydas 9y ago
  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)
  - chii 9y ago
    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.
- khedoros1 9y ago
  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...
- kagamine 9y ago
  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. Here is the 2017 final of the 'proper, serious' Countdown, also on YT https://www.youtube.com/watch?v=Q-VdiufVWrw https://www.youtube.com/watch?v=Q-VdiufVWrw
deleted 9y ago
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emersonrsantos 9y ago
Poor child.
stephengillie 9y ago
uoaei posted the link in a comment yesterday: https://news.ycombinator.com/item?id=15164429 https://news.ycombinator.com/item?id=15164429
hannofcart 9y ago
Please DO NOT show this to Indian/Chinese parents. Think of the children!
- stephengillie 9y ago
  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.
  - cbanek 9y ago
    _Everyone_ needs an understanding of basic calculus, whether they like it or not!
- gumby 9y ago
  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.
xupybd 9y ago
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?
- DenisM 9y ago
  The best way to learn math is to study physics. Or engineering, as in your case.
  - smcl 9y ago
    Maybe we can generalise - the best way to learn an abstract concept is to learn how applies to a discipline you find interesting.
    - kagamine 9y ago
      Sounds about right, I'm always reminded of this gif that explains why and how to calculate the circumference of a circle : https://kaiserscience.files.wordpress.com/2015/09/when-radius-is-1-called-unit-circle-then-circumference-is-2cf80.gif https://kaiserscience.files.wordpress.com/2015/09/when-radiu... and suddenly a cryptic formula from maths class is made clear through the visual display of a wheel turning.
jv22222 9y ago
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. Info and video of kids in action here: https://www.mathacademy.us/solve/ https://www.mathacademy.us/solve/ Direct link to video: https://vimeo.com/224720563 https://vimeo.com/224720563
- novalis78 9y ago
  Very inspiring - wish more schools would offer programs like this.
- dugmartin 9y ago
  Justin - when are you guys going to put out another Techzing episode?
  - jv22222 9y ago
    Jason is doing show notes as we speak!
Flemlord 9y ago
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. http://dragonbox.com/products/algebra-5 http://dragonbox.com/products/algebra-5
- yaantc 9y ago
  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. [1] https://play.google.com/store/apps/details?id=com.wewanttoknow.Euclid https://play.google.com/store/apps/details?id=com.wewanttokn...
briandear 9y ago
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.
kirillzubovsky 9y ago
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 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.
roel_v 9y ago
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.
- throwawayjava 9y ago
  TBF "what calculus is" -- i.e., the level of rigor at which topics are taught -- varies a lot even among college courses called "calculus"... 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.
  - mannykannot 9y ago
    Up to a point, but does it not all grow out of the concept of limits?
    - RobertoG 9y ago
      I would say that the fundamental idea to understand is the concept of changing ratios. Limits seems to me to be more a tool in order to calculate the ratios than a fundamental concept.
      - 9y ago
inertial 9y ago
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) [1] https://www.amazon.com/Moscow-Math-Circle-Week-Week/dp/0821868748/ https://www.amazon.com/Moscow-Math-Circle-Week-Week/dp/08218...
WalterBright 9y ago
When I finally learned calculus I was very surprised at how straightforward basic integration and differentiation are.
synicalx 9y ago
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.
- emmelaich 9y ago
  Yep, my Dad was a educational psychologist who dealt with kids who had problems with learning. 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.
unboxed_type 9y ago
Please let Five-Year-Olds play their toys instead of learning calculus, so they can grow up healthy people.
- monsieurbanana 9y ago
  Completely baseless comment. 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.
  - userbinator 9y ago
    There's also nothing stopping toys from being calculus-related either, although at the moment I can't think of any examples of such.
    - jonsen 9y ago
      A toy calculator? :p
    - pamqzl 9y ago
      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. Hmm, that's not the worst idea...
      - unboxed_type 9y ago
        Nice idea for a toy! But learning an internal machinery of that process isn't that fun if you are not inclined for that kind of activity.
    - deleted 9y ago
      [deleted]
  - unboxed_type 9y ago
    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.
Girlang 9y ago
Yes! Throw them off a high building and ask how fast will they be going when they pass the 30th, 20th, and 10th floor.
tmaly 9y ago
It would be entirely possible to teach advanced math topics to younger using just pictures. 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.
- tradotto 9y ago
  How about a video? https://www.youtube.com/watch?v=U_ZHsk0-eF0 https://www.youtube.com/watch?v=U_ZHsk0-eF0
jasongrout 9y ago
One of the best curriculums I've seen for children (grades 2-5) learning math is the Beast Academy series: https://beastacademy.com/ https://beastacademy.com/ They have challenging problems that emphasize basic and advanced concepts in a variety of ways, often posed as games or puzzles.
- aethertap 9y ago
  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.
  - barry-cotter 9y ago
    Are you home schooling? If not your daughter must be bored out of her mind at school.
deleted 9y ago
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baldfat 9y ago
We teach mathematics backwards and in the wrong order and we push students through things they have not mastered. "A Mathematician’s Lament" by Paul Lockhart https://www.maa.org/external_archive/devlin/LockhartsLament.pdf https://www.maa.org/external_archive/devlin/LockhartsLament.... BOOK published in 2009 https://www.amazon.com/Mathematicians-Lament-School-Fascinating-Imaginative-ebook/dp/B003VPWWFW/ref=pd_cp_kstore_3?tag=bisafetynet2-20 https://www.amazon.com/Mathematicians-Lament-School-Fascinat...
GuB-42 9y ago
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.
lingua 9y ago
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.
thomastjeffery 9y ago
The way math is presented to children is like so: 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.

206 points by guiambros 9y ago

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.

brogrammernot 9y ago
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.
stephengillie 9y ago
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.
jackmott 9y ago
congratulations on believing a thing without rigorous evidence?
d-sc 9y ago
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.
jpamata 9y ago
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.
- chii 9y ago
  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.
  - fspear 9y ago
    >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.
roceasta 9y ago
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...
- chii 9y ago
  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.
  - Micoloth 9y ago
    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. I'm just making things up tho
  - zaptheimpaler 9y ago
    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.
pishpash 9y ago
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.
- erispoe 9y ago
  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.
  - ehecatl 9y ago
    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.
  - pbhjpbhj 9y ago
    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].
  - ctlby 9y ago
    > 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. Just from today: http://www.tandfonline.com/doi/abs/10.1080/09515089.2017.1359245?journalCode=cphp20 http://www.tandfonline.com/doi/abs/10.1080/09515089.2017.135... http://internal.psychology.illinois.edu/~acimpian/reprints/jussim_BBS.pdf http://internal.psychology.illinois.edu/~acimpian/reprints/j...
hyperpallium 9y ago
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...
- nogbit 9y ago
  It is necessary but could be improved. If all we had was roman numerals we would still be in the dark ages.
  - eberkund 9y ago
    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.
    - lingua 9y ago
      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.
jamesrcole 9y ago
> cultural conditioning I suspect that mathematics being poorly taught is also a major contributor.
- g00gler 9y ago
  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.
- rockostrich 9y ago
  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.
5040 9y ago
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.
headcanon 9y ago
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.
thomastjeffery 9y ago
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.

gremlinsinc 9y ago

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.

csydas 9y ago
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)
- chii 9y ago
  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.
khedoros1 9y ago
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...
kagamine 9y ago
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. Here is the 2017 final of the 'proper, serious' Countdown, also on YT https://www.youtube.com/watch?v=Q-VdiufVWrw https://www.youtube.com/watch?v=Q-VdiufVWrw

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emersonrsantos 9y ago

Poor child.

stephengillie 9y ago

uoaei posted the link in a comment yesterday: https://news.ycombinator.com/item?id=15164429 https://news.ycombinator.com/item?id=15164429

hannofcart 9y ago

Please DO NOT show this to Indian/Chinese parents. Think of the children!

stephengillie 9y ago
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.
- cbanek 9y ago
  _Everyone_ needs an understanding of basic calculus, whether they like it or not!
gumby 9y ago
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.

xupybd 9y ago

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?

DenisM 9y ago
The best way to learn math is to study physics. Or engineering, as in your case.
- smcl 9y ago
  Maybe we can generalise - the best way to learn an abstract concept is to learn how applies to a discipline you find interesting.
  - kagamine 9y ago
    Sounds about right, I'm always reminded of this gif that explains why and how to calculate the circumference of a circle : https://kaiserscience.files.wordpress.com/2015/09/when-radius-is-1-called-unit-circle-then-circumference-is-2cf80.gif https://kaiserscience.files.wordpress.com/2015/09/when-radiu... and suddenly a cryptic formula from maths class is made clear through the visual display of a wheel turning.

jv22222 9y ago

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. Info and video of kids in action here: https://www.mathacademy.us/solve/ https://www.mathacademy.us/solve/ Direct link to video: https://vimeo.com/224720563 https://vimeo.com/224720563

novalis78 9y ago
Very inspiring - wish more schools would offer programs like this.
dugmartin 9y ago
Justin - when are you guys going to put out another Techzing episode?
- jv22222 9y ago
  Jason is doing show notes as we speak!

Flemlord 9y ago

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. http://dragonbox.com/products/algebra-5 http://dragonbox.com/products/algebra-5

yaantc 9y ago
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. [1] https://play.google.com/store/apps/details?id=com.wewanttoknow.Euclid https://play.google.com/store/apps/details?id=com.wewanttokn...

briandear 9y ago

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.

kirillzubovsky 9y ago

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 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.

roel_v 9y ago

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.

throwawayjava 9y ago
TBF "what calculus is" -- i.e., the level of rigor at which topics are taught -- varies a lot even among college courses called "calculus"... 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.
- mannykannot 9y ago
  Up to a point, but does it not all grow out of the concept of limits?
  - RobertoG 9y ago
    I would say that the fundamental idea to understand is the concept of changing ratios. Limits seems to me to be more a tool in order to calculate the ratios than a fundamental concept.
    - 9y ago

inertial 9y ago

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) [1] https://www.amazon.com/Moscow-Math-Circle-Week-Week/dp/0821868748/ https://www.amazon.com/Moscow-Math-Circle-Week-Week/dp/08218...

WalterBright 9y ago

When I finally learned calculus I was very surprised at how straightforward basic integration and differentiation are.

synicalx 9y ago

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.

emmelaich 9y ago
Yep, my Dad was a educational psychologist who dealt with kids who had problems with learning. 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.

unboxed_type 9y ago

Please let Five-Year-Olds play their toys instead of learning calculus, so they can grow up healthy people.

monsieurbanana 9y ago
Completely baseless comment. 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.
- userbinator 9y ago
  There's also nothing stopping toys from being calculus-related either, although at the moment I can't think of any examples of such.
  - jonsen 9y ago
    A toy calculator? :p
  - pamqzl 9y ago
    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. Hmm, that's not the worst idea...
    - unboxed_type 9y ago
      Nice idea for a toy! But learning an internal machinery of that process isn't that fun if you are not inclined for that kind of activity.
  - deleted 9y ago
    [deleted]
- unboxed_type 9y ago
  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.

Girlang 9y ago

Yes! Throw them off a high building and ask how fast will they be going when they pass the 30th, 20th, and 10th floor.

tmaly 9y ago

It would be entirely possible to teach advanced math topics to younger using just pictures. 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.

tradotto 9y ago
How about a video? https://www.youtube.com/watch?v=U_ZHsk0-eF0 https://www.youtube.com/watch?v=U_ZHsk0-eF0

jasongrout 9y ago

One of the best curriculums I've seen for children (grades 2-5) learning math is the Beast Academy series: https://beastacademy.com/ https://beastacademy.com/ They have challenging problems that emphasize basic and advanced concepts in a variety of ways, often posed as games or puzzles.

aethertap 9y ago
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.
- barry-cotter 9y ago
  Are you home schooling? If not your daughter must be bored out of her mind at school.

deleted 9y ago

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baldfat 9y ago

We teach mathematics backwards and in the wrong order and we push students through things they have not mastered. "A Mathematician’s Lament" by Paul Lockhart https://www.maa.org/external_archive/devlin/LockhartsLament.pdf https://www.maa.org/external_archive/devlin/LockhartsLament.... BOOK published in 2009 https://www.amazon.com/Mathematicians-Lament-School-Fascinating-Imaginative-ebook/dp/B003VPWWFW/ref=pd_cp_kstore_3?tag=bisafetynet2-20 https://www.amazon.com/Mathematicians-Lament-School-Fascinat...

GuB-42 9y ago

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.

lingua 9y ago

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.

thomastjeffery 9y ago

The way math is presented to children is like so: 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.