
Five-Year-Olds Can Learn Calculus (2014) - guiambros
https://www.theatlantic.com/education/archive/2014/03/5-year-olds-can-learn-calculus/284124/?single_page=true
======
jonnybgood
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.

~~~
roceasta
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
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.

~~~
zaptheimpaler
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.

~~~
Ntrails
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).

~~~
eecc
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

~~~
Ntrails
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.

~~~
zaptheimpaler
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.

~~~
Ntrails
> 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.

~~~
zaptheimpaler
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.

------
roel_v
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
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
Up to a point, but does it not all grow out of the concept of limits?

~~~
RobertoG
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.

~~~
mclehman
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.

~~~
mannykannot
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.)

~~~
throwawayjava
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"

------
kirillzubovsky
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.

------
jv22222
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)

~~~
dugmartin
Justin - when are you guys going to put out another Techzing episode?

~~~
jv22222
Jason is doing show notes as we speak!

------
Flemlord
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
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.wewanttokn...](https://play.google.com/store/apps/details?id=com.wewanttoknow.Euclid)

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

------
lingua
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.

------
synicalx
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
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.

------
xupybd
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
The best way to learn math is to study physics.

Or engineering, as in your case.

~~~
smcl
Maybe we can generalise - the best way to learn an abstract concept is to
learn how applies to a discipline you find interesting.

~~~
kagamine
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-
radiu...](https://kaiserscience.files.wordpress.com/2015/09/when-radius-
is-1-called-unit-circle-then-circumference-is-2cf80.gif)

and suddenly a cryptic formula from maths class is made clear through the
visual display of a wheel turning.

------
GuB-42
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.

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

------
baldfat
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....](https://www.maa.org/external_archive/devlin/LockhartsLament.pdf)

BOOK published in 2009 [https://www.amazon.com/Mathematicians-Lament-School-
Fascinat...](https://www.amazon.com/Mathematicians-Lament-School-Fascinating-
Imaginative-ebook/dp/B003VPWWFW/ref=pd_cp_kstore_3?tag=bisafetynet2-20)

------
tmaly
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
How about a video?
[https://www.youtube.com/watch?v=U_ZHsk0-eF0](https://www.youtube.com/watch?v=U_ZHsk0-eF0)

------
jasongrout
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
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
Are you home schooling? If not your daughter must be bored out of her mind at
school.

------
thomastjeffery
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.

------
stephengillie
uoaei posted the link in a comment yesterday:
[https://news.ycombinator.com/item?id=15164429](https://news.ycombinator.com/item?id=15164429)

------
briandear
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.

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

~~~
stephengillie
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
_Everyone_ needs an understanding of basic calculus, whether they like it or
not!

------
emersonrsantos
Poor child.

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

~~~
monsieurbanana
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
There's also nothing stopping toys from being calculus-related either,
although at the moment I can't think of any examples of such.

~~~
pamqzl
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
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.

------
gremlinsinc
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.

~~~
userbinator
_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.

~~~
zimpenfish
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.)

