
Algebraic and calculus concepts may be better way to introduce children to math - tokenadult
http://www.theatlantic.com/education/archive/2014/03/5-year-olds-can-learn-calculus/284124/
======
kalid
(Disclosure: I'll be working with Maria on the calculus series for kids.)

I'm a big fan of the conceptual approach. One of the largest problems I see
with math education is that we don't check if things are really clicking.

I graduated with an engineering degree from a great school, and still didn't
have an intuitive understanding for i (the imaginary number) until I was about
26.

Go find your favorite tutorial introducing imaginary numbers. Got it? Ok. It
probably defines i, talks about its properties (i = sqrt(-1)) and then gets
you cranking on polynomials.

It's the equivalent of teaching someone to read and then having them solve
crossword puzzles. It's such a contrived example! (N.B., this anguish forced
me to write a tutorial on imaginary numbers with actual, non-polynomial
applications, like rotating a shape without needing trig. See
[https://news.ycombinator.com/item?id=2712575](https://news.ycombinator.com/item?id=2712575))

Calculus needs these everyday applications and intuitions beyond "Oh, let's
pretend we're trying to calculate the trajectory of a moving particle."
They're out there: my intuition is that algebra gives a static description
(here's the cookie), while calculus describes the process that made it: here's
the steps that built the cookie. Calculus is the language of science because
we want to know how the outcome was produced, not just the final result. d/dx
velocity = acceleration means your speed is built up from a sequence of
accelerations.

~~~
lifeisstillgood
Yes, yes, yes.

I have tried (fitfully) three times to sit down and learn what calculus
actually means. I still don't get it. And I suspect that clients and families
will push it ever further away.

Yes, give me real world puzzles and applications for learning new mental tools
and I will probably love them. Make it abstract enough and I cannot see the
value in learning it.

~~~
kalid
I was the same way :). I have an intuition-first guide to calculus, here:

[http://betterexplained.com/calculus/lesson-1](http://betterexplained.com/calculus/lesson-1)

I think the essentials on how to think with calculus can be conveyed in 1
minute. Let me know if the above helps :).

~~~
ht_th
I see that your examples of teaching fast (one minute, ten minutes) all use
discrete models for continuous processes. For people who understand
continuousness mathematically, going between discrete models to continuous
models and vice versa is easy. However, there are indications that the other
way around -- students who perceive these changing situations mathematically
as intrinsically discrete -- going from discrete to continuous models might be
extreme difficult (due to in-commensurability). Of course, students can learn
procedures to make the switch from discrete to continuous models, but even
then it is possible that this procedural experience is not build on deeper
mathematical understanding. Tricky stuff, learning mathematics :-)

~~~
kalid
Great point, thanks for the comment.

I think the key missing insight for me was that a continuous process and a
discrete process can both point to the same result.

A pixellated word on a screen ("cat") conveys the same meaning as a perfectly
smooth vector of the same word. The math idea, to me, is "can a discrete
description/process" point at the same result that we get from a continuous
one?

The idea of limits, is essentially figuring out when a discrete epsilon can
still lead us to the true value of f(x) (if this trick works, we call it a
continuous function, and can use calculus with it.)

~~~
ht_th
Yes. You might be interested in Cornu, B. (1991). Limits. In D. Tall (Ed.),
Advanced Mathematical Thinking (Vol. 11, pp. 153–166). Retrieved from
[http://dx.doi.org/10.1007/0-306-47203-1_10](http://dx.doi.org/10.1007/0-306-47203-1_10)
and similar works. Volumes have been written about the limit concept, learning
the limit concept, and different approaches to the limit. It is one of the
more interesting mathematical concepts one encounters in high school, not in
the least because of the ability to use it successfully without understanding
the underlying concept and the philosophical implications of the concept.

Unfortunately, I doubt that these implications will be discussed in your
average Mathematics classroom. Actually, my experiences with mathematics in
primary, secondary and a lot of tertiary education, both as a student, teacher
and observator, leads me to believe that conceptual discussion is rare, whilst
procedural discussions (how to perform some calculation) are plenty and
abound.

------
nikhizzle
I love it. This really appeals to me because of the all the bad ideas I had to
unlearn in my 20s:

\- math is about numbers and arithmetic operations on them \- being good at
math meant you were good at arithmetic \- some people (meaning me) just didn't
have what it took to be "good" at math. Reinforced by my high school math and
physics teachers.

I hated math because I didn't understand that mathematics is a system for
representing abstract concepts and manipulating them.

Eventually on my 4th try to get calculus, I took a class from nick
fiori([http://www.yale.edu/education/about/faculty/fiori.elw062111](http://www.yale.edu/education/about/faculty/fiori.elw062111)).

His teaching methods opened my mind, and I've gone on since then to become an
ardent amateur mathematician with minor publications and career tangents in
machine learning and data science.

I can only imagine what would have happened if I had been taught math well
from an early age.

~~~
roasty
Good to hear, any math books you'd recommend to someone who's not there yet,
or remember from that class?

~~~
nikhizzle
Sorry, did not learn much from books. The only one of note is the Tom Mitchell
Machine Learning Book, which already requires a basic advanced mathematics
fluency. I also believe it is a little out of date.

I can recommend a few professors who really opened my mind to how to use math,
all at UCSC (CS grad school for me):

\- Dave Helmbold (Machine Learning) \- Kevin Ross (Operations Research) \-
Martin Abadi (Security)

------
bane
Yes yes yes.

The mechanics of basic calculus are remarkably simple. After a while it
becomes a game of sorts with a pretty simple set of rules. Once you learn to
"see" parts of the equation as symbol blocks (instead of numbers needing
computation), you can move them around freely so long as you follow the rules.
Algebra is simply a subset of the rules.

Teaching kids to think symbolically will help them in so many other fields.

I'm pretty convinced that with a little thought, you can teach basic
derivation and integration to pretty young kids. Carefully craft the problems
to avoid difficult division problems, avoid trig, let them use lookup tables
for multiplication and you _might_ be able to get kids under 10 to even do
some of this.

Then "vertically integrate" other algebraic and trigonometric concepts into
this framework, like adding new pieces to the game.

~~~
sp332
Hm, you're talking about teaching symbolic manipulation using calculus, but
for me the real ideas of calculus are integration and derivation. I think
those ideas are very valuable, and symbolic manipulation is just one way to
solve those problems.

~~~
bane
Can you provide some examples? To me, the basic mechanics I remember (for
simple equations like x^3 d/dx) were almost entirely "solvable" by moving bits
of the equation around...

1\. move the 3 in front of the x

2\. where the 3 was, subtract 1 from it and put that back where the 3 was

but if you think of it as symbols you get more like

1\. move the exponent symbol in front of the x symbol

2\. subtract 1 from that exponent and put it back where the original exponent
symbol was

For me at least, bridging the gap from the specifics of the first example to
the more generalized second approach meant that when I was given something
like

3x^(y^4) d/dx my mind was absolutely blown because I didn't know if there was
some rule I needed to know if the exponent had an exponent or something.

Going from the years of the arithmetic approach to really groking the symbolic
manipulation approach was _really_ hard for me, and looking at lots of young
kids suffering through algebra it was the same.

Learning how to recognize the symbols through the specificity of the numbers
was something I don't think I really got at all until calculus and it's
because the mechanics of the rules of simple derivatives and integration
really force you to recognize the underlying symbols...more than algebra did
(which for me just seems like a really complicated arithmetic rather than
what's actually a pretty simple symbol manipulation exercise).

It's odd, because like most technically minded kids, I was already great
working with symbols and breaking things down and building them up. I could
build huge structures out of various small assemblies with legos, write simple
software, crank out papers for English class that would guarantee an A and so
on, but understanding that the teachers no longer wanted me to be a human
tabulation device but to be a manipulator was something I never really
cottoned on to.

I know from spending a little time with nieces and nephews younger then 10
that you can teach basic derivatives in about an hour or two and have them
fairly reliably doing simple work like the above in an afternoon if you don't
worry them too much about simplification of the result or all the arithmetic.

My _gut_ says that if you start adding more rules, like what do you do when
you have something like (x^3)/(x^2) and how exponents should be subtracted,
and with a little care in the examples you show them of this rule in action,
you end up showing them all of algebra, fractions, exponents and later trig,
logs, etc. while building up and hanging all of these concepts off of
calculus.

TBH I don't know enough about early childhood education to know if this should
be used _instead_ of basic arithmetic or not, but I bet if you use shapes and
colors instead of letters and numbers and start to teach basic rules, you can
just slowly introduce numeracy later anyways when their brains are more
developed.

~~~
GeneralMayhem
No offense, but what you're proposing would be a huge step backwards. All of
those examples you've given are extremely special cases of a general concept,
where the concept is relatively easy to understand (two points make a line, we
only have one point, so take the limit) although the implementation is complex
(requires either understanding algebra or knowing lots of rules.)

If you try to teach a little kid the power rule right off the bat, they will
both be turned off of math (because it's just pushing x around for no clear
reason) and have learned absolutely nothing of the important bits of calculus.

~~~
bane
> where the concept is relatively easy to understand (two points make a line,
> we only have one point, so take the limit) although the implementation is
> complex (requires either understanding algebra or knowing lots of rules.)

The pedagogy of Mathematics education is fundamentally broken and we've lost
generations of math users because of it.

Approaching it from this sense "two points make a line, we only have one
point, so take the limit" _is_ the method today and with all respect, it's
been a terrible terrible failure...even if it is "correct".

A few kids will grok it and turn into computer scientists or physicists or
mathematicians of some sort, and the other 98% will take the bare minimum to
get their high school education, and if they go on to college see which degree
programs require math and which don't and select the B.A. degree that doesn't.

Anecdotal, but I don't know a single person who enjoyed their K-12 math
education. I know lots of people who enjoy math, and found that joy in college
or later, but found the educational experience of K-12 so abysmal and
torturous that they completely swore off even pursuing fields with heavy Math
components. There are lots of artists, musicians, writers and historians who
would probably be great Mathematicians, Physicists and Scientists were it not
for the piss poor job we do indoctrinating kids into math.

It's not just that students fail to learn mathematics, it's that the
pedagogical experience is so negative that they swear off ever even trying.
And IMHO, a very big part of that is the (to the student) endlessly pointless
jargon filled inapplicable overly rigorous and formal mess that is Math
education today.

I've sat in the audience on some very heated round tables about promoting STEM
education in my region and I've come away convinced that getting more people
into STEM is critical to long-term economic success, but students are not only
_not_ drawn into STEM, they're actively driven away from it by the pedagogical
approach of what amounts to a single class every year. Students love science
classes and science labs, they love shop class and learning engineering, they
love computers and everything else to do with STEM _except_ for the K-12
Mathematics education. And that loses them, it simply drives them away from
all of these great (and often lucrative) fields.

More importantly, I think fields which are not traditionally Math focused,
could benefit greatly from a better general Math education. Read a social
sciences academic paper and see what I mean for a quick example.

> If you try to teach a little kid the power rule right off the bat, they will
> both be turned off of math (because it's just pushing x around for no clear
> reason) and have learned absolutely nothing of the important bits of
> calculus.

I want to agree particularly with this, "it's just pushing x around for no
clear reason". The fundamental problem with learning Mathematics in K-12 is
applicability. Beyond arithmetic, students have almost zero examples of why
they should bother learning anything else. _Most_ of the population gets by
just fine without anything more complex than arithmetic and figuring out
percentages (and even then that's a stretch).

When students go to their parents to seek guidance, their parents _also_ don't
have any idea why they have to learn all this stuff. _They_ certainly don't
need it in their day jobs and can't provide a hint as to why it's important
other than the student needs to study it for the grades.

And to be honest, even if their parents _do_ use Mathematics in their work and
can provide examples, it's likely that the student's ability to relate to that
work is very limited. The reinforcement that all this time spent learning Math
is pointless is much stronger than the reinforcement that it's important or
useful. Getting a kid to crank through 20 or 30 algebra problems is _much_
harder than say, a 5 paragraph essay for English class because the entire time
they're doing this they're saying to themselves "why am I doing this? Math is
useless! At least learning to write a little has some kind of use!"

But children will play "pointless" games for hours and hours and hours - and
not even ones they're especially having fun with, just ones that hold their
interest (if you've ever watched a 9 year old vent frustration at their Xbox
you'll know what I mean). If we can turn Mathematics education into a kind of
"game", then fill in the details and formal bits as they age, they'll at least
be able to relate to it even if they don't understand the application or
relevancy.

And the truth is, once you get to Calculus and get it, it's actually pretty
fun and pretty easy. That's a high enough discipline for most STEM jobs and I
firmly believe that every K-12 student should be able to do what we call
"college level" Calculus by the 10th grade. So why not try to capture the
things that make Mathematics at that level fun and easy, and I think that's
the symbolic manipulation, even if it is hard to establish relevancy, and get
them used to doing it from a very early age. Kids can move blocks around and
stack them before they can walk, why can't the blocks be bits of equations?
And why can't moving the blocks have little game-like rules they can learn?

~~~
GeneralMayhem
>Read a social sciences academic paper and see what I mean for a quick
example.

I have actually done some graduate-level work in sociology and history, and
the papers and books I read were mostly examples of very good statistical work
and well-thought-out process analysis. You can put your STEM-master-race badge
away.

>Beyond arithmetic, students have almost zero examples of why they should
bother learning anything else.

When I say "for no reason," I don't mean "for no _day-to-day practical_
reason." Playing with abstract concepts is and should be its own reward; that
was the whole point of TFA. Mechanically memorizing how to take the
derivatives of polynomials is neither a fun abstract concept nor a boring-but-
necessary practical skill.

>If we can turn Mathematics education into a kind of "game", then fill in the
details and formal bits as they age, they'll at least be able to relate to it
even if they don't understand the application or relevancy.

"Gamification" as a cynical ploy to get kids to sit still long enough memorize
their times tables may or may not work. But even if it does, it's only gotten
them to play the game long enough to pass them to the next level; it has
deliberately shifted their interest away from the joy of learning for its own
sake. That is __not __what the article is about, and it 's not helpful in the
long run.

>Approaching it from this sense "two points make a line, we only have one
point, so take the limit" is the method today and with all respect, it's been
a terrible terrible failure...even if it is "correct".

No... no, it isn't. The approach today, for the majority of students, is to
learn the bare basics so that you can plug them into an equation and find out
what the marginal cost of widgets will be next year given a certain set of
equations. And in any case, it comes so late that kids have been taught that
"math" is something that actually _is_ boring and useless.

~~~
bane
> Playing with abstract concepts is and should be its own reward

I'm sorry, but you're just simply wrong on all points. Promoting the status
quo in math education, as you are doing, has been, is, and will continue to be
a failure that drives kids away from learning. There are now decades of
evidence of the failures of k-12 education to address this need and I find it
unbelievable that you haven't gotten the picture yet.

I'm not saying that what I'm proposing is correct, but continuing the very
poor pedagogical approach that you support is not going to solve the
educational failures that we're experiencing today. What we need are
fundamentally new approaches to Math education. You are not providing any
insight into what those approaches should be.

I'm sorry, but in a discussion to fix and change what is obviously utterly
broken in k-12 maths education, suggesting to just continue the course is not
a helpful contribution and is simply part of perpetuating the problem.

This has been recognized for so long, that it has finally percolated out of
educational establishment, which has failed to address the problem with
undereducated and unqualified teachers, student motivation, repeated failures
in curriculum development ( _Common Core_ is simply the latest joke of a
curriculum), and has reached levels as high as the White House for targeting.
You _have_ to know that the current approaches are failing if the President
has to get involved.

From the "Report to the President...K-12 education in STEM"
[http://www.whitehouse.gov/sites/default/files/microsites/ost...](http://www.whitehouse.gov/sites/default/files/microsites/ostp/pcast-
stemed-report.pdf)

"Schools often lack teachers who know how to teach science and mathematics
effectively, and who know and love their subject well enough to inspire their
students. Teachers lack adequate support, including appropriate professional
development as well as interesting and intriguing curricula."

"As a result, too many American students conclude early in their education
that STEM subjects are boring, too difficult, or unwelcoming, leaving them
ill-prepared to meet the challenges that will face their generation, their
country, and the world."

"Put together, this body of evidence suggests that grade-school children do
not think as simplistically about STEM subjects as conventional curricula
assume. They are capable of grasping both concrete examples and abstract
concepts at remarkably early ages. Conventional approaches to teaching science
and math have sometimes been shaped by misconceptions about what children
cannot learn rather than focusing on students’ innate curiosity, reasoning
skills, and intimate observations of the natural world."

[http://books.google.com/books?id=7v2gRGuxw4sC&pg=PA253&lpg=P...](http://books.google.com/books?id=7v2gRGuxw4sC&pg=PA253&lpg=PA253&dq=National+Research+Council.+\(2001\).+Adding+It+Up:+Helping+Children+Learn+Mathematics.+Washington,+DC:+National+Academies+Press&source=bl&ots=AgRbbDK-Z4&sig=kAkWvPXWg36IJ6wAynh-
tpmfRtY&hl=en&sa=X&ei=7LYVU62RFePw0wGm7oHoAQ&ved=0CCYQ6AEwADgK#v=onepage&q=National%20Research%20Council.%20\(2001\).%20Adding%20It%20Up%3A%20Helping%20Children%20Learn%20Mathematics.%20Washington%2C%20DC%3A%20National%20Academies%20Press&f=false)

"The first principle of How People Learn emphasizes both the need to build on
existing knowledge and the need to engage students' preconceptions --
particularly when they interfere with learning. In mathematics, certain
preconceptions that are often fostered early on in school settings are in fact
counterproductive. Students who believe them can easily conclude that the
study of mathematics is 'not for them' and should be avoided if at all
possible."

I differ from this report in that they continue to promote a bottom-up
approach to maths education. I think math should be taught from a top-down
approach, like nearly every other discipline. You don't learn to bake a cake
by first spending 10 years learning about chemistry, agriculture, nutrition,
animal husbandry, distillation etc. You say "I want to bake a cake" and you
start with a simple cake recipe. Then the next time you say "I want to bake a
different cake" and you use a more complicated recipe. And so on and so forth
until you don't need a recipe and are putting together your own cakes from
scratch.

The "cake" I'm proposing is calculus...and I believe, from experience teaching
basic calculus to kids under 10, that this is realizable and beneficial.

The end result is that I don't expect kids to come out of K-12 baking cakes
that look like this [http://www.askmamaz.com/wp-content/uploads/2013/01/Most-
Beau...](http://www.askmamaz.com/wp-content/uploads/2013/01/Most-Beautiful-
Wedding-Cakes-22.jpg)

when the average STEM work-a-day mathematics really requires them to do this.

[http://www.recipegirl.com/wp-content/uploads/2013/06/Red-
Whi...](http://www.recipegirl.com/wp-content/uploads/2013/06/Red-White-and-
Blue-Cheesecake-Cake-7.jpg)

Right now, your approach is producing students who don't even bother with cake
baking, and when pressed into service produce this
[http://fc05.deviantart.net/fs70/f/2012/070/1/b/ugliest_cake_...](http://fc05.deviantart.net/fs70/f/2012/070/1/b/ugliest_cake_ever_by_great_5-d4sfvj4.jpg)

This needs to change.

------
cabinpark
A big secret in mathematics is that many ideas in mathematics are rather
simple and straight forward.

Formulating and proving these ideas in a rigorous and logical manner, however,
is the difficult part.

~~~
yardie
Explains my joy in puzzles and sequences. Sometimes I don't know why the next
number in a series is. I used my intuition, arrived at the correct answer, and
now the hard part is explaining myself.

------
nilkn
This has, to some extent, been known for a long time. How many stories have
you heard about math prodigies? I think the problem, though, is in that very
sentence. Rather than considering whether this material might be more
accessible than traditionally thought, we just assert outright that the child
is a prodigy for learning calculus at a young age and then learn nothing from
the observation.

I learned calculus in sixth grade and was and am very bad at arithmetic. It
was easier for me to understand the fundamental theorem of calculus than it
was for me to do long division in my head. I don't think I'm alone. I think
I'm just among the few who, for one reason or another, actually made a serious
attempt to learn the subject at the time.

~~~
nzp
> It was easier for me to understand the fundamental theorem of calculus than
> it was for me to do long division in my head. I don't think I'm alone.

You're not. I had the same experience. I have severe problems with mundane
arithmetic, my mind just locks up when faced with concrete numbers and even
simple numerical calculations. OTOH, no problem whatsoever with geometry,
(abstract) algebra, calculus etc. I remember when I first got calculus in high
school, how natural and easy it felt, how it simply made sense and how all
around pleasant and interesting it was, and it struck me how different this
experience was to the other more "basic and mundane" areas I struggled with
during school. A few years ago I was amused when I saw an interview with an
astrophysics PhD who struggled with doing basic arithmetic. It's a source of
amusement for her now, but you can imagine the dread she had to endure in
"school". So, no, we're not alone.

What the article says when it compares what children face in school under the
label of "mathematics" to torture akin to digging a trench with a spoon sounds
very true to me.

------
mathattack
The article headline is a little sensationalist. It's very rudimentary, and
it's more about learning limits than equations. That said, games like
DragonBox highlight that kids are capable of being introduced to these
concepts well before schools get around to it.

~~~
MichaelGG
My 5-yr-old played DragonBox a bit - first for the colourful critters and
powering up her monster, but then for the problems.

It was pretty neat when I wrote down 2 + x = 7 and she was able to rearrange
the equation to solve for x. (The later stages get considerably more
complicated.) It's fantastic that kids think in terms of symbolic
manipulation. Most adults still don't get that, and think maths is just about
coming up with numbers.

~~~
mathattack
I haven't figure out how to convert the on-line intuition into pen and paper.
I saw a paper on their website. It still seems like a leap, when the kids
haven't formally been taught multiplication, division and fractions. I'm glad
that it worked in your case.

~~~
MichaelGG
Hmm, have you tried having them do the actual problems on paper? No numbers or
letters, just copy the screen onto paper:

Like write out:

    
    
      Side 1         |     Side 2
        Fly     Box  |   Mushroom
    

And have them solve into:

    
    
      Side 1         |     Side 2
        Box          |   Mushroom    Inverted Color Fly ("Dark Fly")
    

From there it's just a visual replacement to turn Box into x, Mushroom into a
7, and Fly into a 2. The game introduces "inverse color = -", so going from 2
to -2 should be a visual translation that doesn't actually require them to
understand subtraction or inverse.

After they're all done, they _might_ notice "hey, wait, I recognize 7 -2 and
it can also be written as 5!" But that's not relevant to manipulating the
equation, is it?

Sure, without arithmetic they won't be able to calculate the most simple form
(7 + -2 -> 5 or 2/4 + 3/6 -> 1) -- but that's a different skill. And people
seem to have more issues with handling the equation than actually doing the
arithmetic part.

------
yardie
Does anyone know of any resources to introduce math to a 6yo? I have apps like
Dragonbox to introduce algebra to my son. He played with it, completed it, but
now is back to the crap IAP games typical on tablets these days. I wouldn't
mind a few more like Dragonbox.

I also use flashcards to help with the rote math homework.

~~~
tokenadult
I don't ever use flashcards for mathematics learning. Not once, not ever. (I
hardly use flashcards for foreign language vocabulary either, not even for
Chinese characters, except for recognition of single characters in isolation,
because there isn't any one-to-one correspondence of words between languages.)
Working a lot of problems[1] eventually makes the math facts second-nature,
without flash cards.

I do recommend to my children and my students that they consider doing one
thing I did as a kid, namely fill out a multiplication table (the one I did in
childhood was 30 × 30) by hand by hand calculation. I did that in odd moments
during the school day and kept the table stashed in my desk between moments of
working on it. Filling in a multiplication table helps learners notice number
patterns (it made me very conscious of perfect squares, for example) and
produces a tangible accomplishment at the end.

For a child the age of your child, I enthusiastically recommend the Miquon
Math materials,[2] which are inexpensive and take a thoughtful, playful
approach informed by higher mathematics for learning much of the content of
the first several years of elementary mathematics. All you need to use with
the Miquon Math books are Cuisenaire rods,[3] which are available from various
sellers and are a fun plaything in themselves.

[1]
[http://www.epsiloncamp.org/ProblemsversusExercises.php](http://www.epsiloncamp.org/ProblemsversusExercises.php)

[2] [http://miquonmath.com/](http://miquonmath.com/)

[3]
[https://en.wikipedia.org/wiki/Cuisenaire_rods](https://en.wikipedia.org/wiki/Cuisenaire_rods)

~~~
yardie
Brilliant! thanks.

I did the multiplication as a child myself, only 20x20. I've also tasked him
with the table, 10x10.

------
3rd3
Discussion on reddit:
[http://www.reddit.com/r/math/comments/1zfizg/5yearolds_can_l...](http://www.reddit.com/r/math/comments/1zfizg/5yearolds_can_learn_calculus_why_playing_with/)

------
tokenadult
I see the article title (which I put on the submission, and one comment
commented on) was replaced by an HN moderator with a rewording of the article
subtitle. I guess that's okay.

The article was especially eye-catching for me, because it was first posted
among our mutual Facebook friends by my mentor in homeschooling mathematics,
the mother of the first United States woman to win a gold medal in the
International Mathematical Olympiad. The teacher featured in the article as
the developer of innovative teaching methods for young learners is also a
Facebook friend of mine, and we are part of a network of parents and teachers
who are curious about how to stimulate interest in mathematics among young
learners by doing things differently from the United States typical school
curriculum in mathematics. I'm grateful that my children have been exposed to
approaches like that (which I follow imperfectly in my homeschooling my
children and in teaching other learners in classroom courses for my
occupation), as that has helped my oldest son launch into the adult world with
good problem-solving skills for hacking at the startup where he works.

There is a continual tension in mathematics education between teaching topics
in their logical order as we understand mathematical topics in light of modern
higher mathematics and teaching topics in their historical order of
development (which is largely what happens in school mathematics courses).
Children need some concrete, tangible experience with counting and with shapes
to understand much about mathematics. The cool abstractions that motivate
higher mathematics may be inaccessible to children who have no experience with
tangible examples of those absractions. (See

[http://www.latimes.com/opinion/commentary/la-oe-adv-
frenkel-...](http://www.latimes.com/opinion/commentary/la-oe-adv-frenkel-why-
study-math-20140302,0,5177338.story)

an article submitted to Hacker News yesterday, for more about this.) But it
looks like we can gain in mathematics instruction by letting children play
with more sophisticated representations of mathematics than most children get
to play with. Games are great learning tools. One of favorite games for
teaching mathematics I learned about from John Holt's book _How Children Fail_
(originally published 1964, which I read in 1971 on the advice of my school's
assistant principal). The twenty questions game asks children to find a number
from 1 to 1,000,000 by asking twenty yes/no questions, which the person who
chose the unknown number must answer truthfully. It is an interesting
mathematical exercise to show that twenty questions is (barely) enough for
finding one number in one million if the twenty questions are used optimally.
I have played this game many times with the children in my classes, and the
opportunity to play this game again at the end of class is one of my strongest
incentives for the children to stay focused during a lesson on a Saturday
morning to get through the lesson content efficiently.

Playing with challenging problems appears to be the royal road for learning
mathematics.

[http://www.epsiloncamp.org/ProblemsversusExercises.php](http://www.epsiloncamp.org/ProblemsversusExercises.php)

I hope ideas like this spread through many communities in the United States
and the English-speaking world, so that more young people gain more
opportunity to learn to enjoy challenging mathematics early and often.

~~~
notthemessiah
As for math games, I can recall a few interesting examples:

Algebra for children:
[http://www.wired.com/geekdad/2012/06/dragonbox/all/](http://www.wired.com/geekdad/2012/06/dragonbox/all/)

Interactive tutorial for sequent calculus:
[http://logitext.mit.edu/logitext.fcgi/tutorial](http://logitext.mit.edu/logitext.fcgi/tutorial)

Ancient Greek Geography:
[http://sciencevsmagic.net/geo/](http://sciencevsmagic.net/geo/)

~~~
protomyth
DragonBox on iOS is pretty good at teaching algebra.
[http://dragonboxapp.com](http://dragonboxapp.com)

------
michaelfeathers
I know that when I learned calculus the hardest part was reconciling the
notion of infinitesimals with algebra. The delta/epsilon definition of a limit
was unsettling. It's a shame. Much of calculus is easy to grasp intuitively -
Riemann integration is easy to see and explain.

~~~
Pxtl
Yes, the concepts of limits and derivatives are simple and intuitive, but the
concept of the epsilon/delta proof just completely soars over students heads.
I didn't learn the epsilon/delta stuff until _after_ all that other stuff, and
I don't feel like it was the wrong order of things.

That and the prof I had never just really explained that epsilon just means
"really insanely small" which probably would've helped.

~~~
ChristianMarks
I had to learn the epsilon-delta definition before I really understood limits.
As far as insanely small goes, as you must know, the definition is equivalent
to "for every epsilon < M, there exists a delta >0, ..." where M is some
positive constant (it could be 1 or 1/10^100...). Sometimes it helps to play
with the quantifiers...

------
aaronsnoswell
Relevant anecdote: For a high school ICT project my class had to make an
educational game in Visual Basic. I made a 'math invaders' clone where you
answer simple algebra sums (e.g. 5 + x = 12) to defeat the relentless hordes
of aliens. At home my 8yo brother watched me play for about 5 minutes, then
exclaimed 'Oh I get it, the letters are like numbers that you have to find!',
then went on to play the game himself for hours. He learnt algebra about 6
years before any of his classmates through gamification, and years down the
track enjoys math more than many of his friends.

------
craigching
This looks extremely interesting, I have a 5 year old daughter that I want to
see the playful side of mathematics before she hits the typical US Maths
curriculum. One of my guiding documents has been "Lockhart's Lament" [1], but
trying to figure out how to make that a reality for her has been difficult for
someone like me with so little time! I am going to check this out tonight!

[1]
[http://en.wikipedia.org/wiki/A_Mathematician%27s_Lament](http://en.wikipedia.org/wiki/A_Mathematician%27s_Lament)

------
ef4
If you're interested in this kind of work, and especially if you're also
interested in using computers to open up new kinds of mathematical play and
learning, you should read Seymour Papert's "Mindstorms" book.

The ideas are powerful and radical. Radical enough to explain why run-of-the-
mill schools never successfully integrated awesome tools like LOGO and its
descendants. They can't deal with learning that's so child-driven and free.

------
vasundhar
There was a study done in a place quite unexpected to me now, and interesting
back then when I was in school myself.

My Math Teacher told there were some schools where a pilot was conducted
teaching calculus concepts from 5th,6th std students and were monitored till
they did their graduations and career paths.

Interestingly the people who were chosen randomly, performed equally well and
their decisions in addition to math were more logical.

This study was done in India :)

------
Myrmornis
"Revolutionizing the way math is taught" and not "memorizing multiplication
tables as individual facts rather than patterns" sound like very worthy and
important goals. At the same time, it's hard to judge this particular
initiative based on the article; it might be Hippie Math for Rich Kids Who
Will Study Humanities Subjects and Become Trustafarians by a Berkelely Yoga
Instructor"

------
NAFV_P
> _But they also need to see meaningful (to them) people doing meaningful
> things with math and enjoying the experience._

I was wondering if this was essential for the learning process of five or six
year olds.

Making maths enjoyable for young children would be the primary concern, but
showing something meaningful (not necessarily to them) being done with maths
by someone meaningful (to them)? It would work better if:

 _The kids require to see something that is relevant to_ their _interests or
needs being solved with maths by any old codger, as long as said codger can
adequately manipulate algebra._

By the way, who is going to teach these children algebra? A fair number of
teachers won't be up to the task. Reminds me of the _learn to code_ buzz-
phrase being passed around - first off you need to spend money and time on
teaching teachers how to code.

EDIT: bloomin' asterisks.

------
sudhirj
[http://siterecruit.comscore.com/sr/atlantic/broker.js](http://siterecruit.comscore.com/sr/atlantic/broker.js)
is holding up the entire site for me. Async, anyone?

------
kps

      “What is learned without play is qualitatively different. It helps
       with test taking and mundane exercises, but it does nothing for
       logical thinking and problem solving. These things are separate,
       and you can’t get here from there.”
    

This is the core of what people mean when they — legitimately — talk about
‘passion’. Did you learn computing (or any field) for fun, or alternatively
because you thought it paid well, or your parents wanted you to do it, or
whatever.

(Not to be confused with ‘passion’ as mbaspeak for ‘naïvité enough to work
ridiculous hours for peanuts’.)

------
eitally
Maria co-authored a great book aimed at making higher level math concepts fun
for schoolchildren titled Moebius Noodles. The site appears to be down now,
but here's the FB page:
[https://www.facebook.com/moebiusnoodles](https://www.facebook.com/moebiusnoodles)

I highly recommend it, as well as signing up to receive periodic emails from
Yelena/Maria with additional activities, tips and information.

------
fdej
Obligatory Star Trek reference:
[http://www.youtube.com/watch?v=ETt8GJRbqLc](http://www.youtube.com/watch?v=ETt8GJRbqLc)

------
beat
I've thought for a long time that an abstract approach might work better than
a concrete approach for introducing children to math.

I remember when my kids were in kindergarten, and their math education started
with _estimation_. My first thought was, "Great. Now they've turned _math_
into a touchy-feely all-answers-are-right-answers nonsense subject!"

~~~
gshubert17
I don't want to turn math into a touchy-feely all-answers-are-right-answers
nonsense subject, but estimation is a good skill in math, engineering, and
computing.

Two examples: Fermi problems[1] and Jon Bentley's "back of the envelope"[2]
estimates.

[1]
[http://en.wikipedia.org/wiki/Fermi_problem](http://en.wikipedia.org/wiki/Fermi_problem)
[2] [http://www.csie.fju.edu.tw/~yeh/research/papers/os-
reading-l...](http://www.csie.fju.edu.tw/~yeh/research/papers/os-reading-
list/bentley-cacm84-envelope.pdf) and
[http://www.eecs.harvard.edu/cs261/background/p176-bentley.pd...](http://www.eecs.harvard.edu/cs261/background/p176-bentley.pdf)

~~~
beat
I agree that estimation is important. But so important that we start children
with it? That smells wrong to me. Estimation is about intuitive thinking.
Mathematics is usually a child's first introduction to formal, rigorous
thought. Don't go soft on it.

------
apalmer
I definitely appreciate that math is much more than numbers and equations...

On the other hand teaching a kid math has to at some level at some point
revolve around taking some numbers and getting other numbers out...

I cant really see how one can do algebra without knowing arithematic. algebra
is fundamentally a different beast but you just cant do it without some kinda
arithmatic

~~~
lutusp
> On the other hand teaching a kid math has to at some level at some point
> revolve around taking some numbers and getting other numbers out...

Yes, but perhaps not the very first thing. To me, the first thing should be
interesting, aesthetically pleasing patterns -- images -- that have
mathematical meaning
([http://arachnoid.com/mandelbrot_set/graphics/title_image.jpg](http://arachnoid.com/mandelbrot_set/graphics/title_image.jpg)).
Then, when the student's curiosity has been aroused, we can "open the machine"
and show its inner workings.

> I cant really see how one can do algebra without knowing arithematic.

But that's easy. I'm not necessarily recommending it, but one can easily have
algebra without numbers. Remember that the most useful algebraic equations are
those in which the numerical values are irrelevant.

------
nroose
Ever since I took Discrete Math in College, I have thought it would be a great
way to introduce school kids to Math.

------
alexvr
Does anyone else find math like trig and calculus dramatically easier to
comprehend than stuff like combinatorics?

~~~
nilkn
Combinatorics has always been easier to comprehend for me, but the problems
have always been harder. That's because calculus depends on higher-level
concepts but is otherwise rote, whereas combinatorics depends on only the most
elementary concepts but the problems quickly become very complex. I think this
is why combinatorics features so much more strongly in math competitions than
does calculus.

------
chrisBob
Am I the only one that enjoyed math more earlier than now? I loved calculus
and geometry in high school and undergrad, but now in graduate school I can't
follow the physicists at all, and my eyes just glaze over.

