
Teaching Square Roots to a Five Year Old - mkopinsky
http://dlewis.net/blog/2013/01/31/teaching-square-roots-to-a-five-year-old/
======
invalidOrTaken
Sometime before I was eight years old (this becomes important later), my mom
taught me about negative numbers. This excited my curiosity, and I remember
walking into my mom's bathroom as she was getting ready to shower to ask her
what happened when you divided a positive number by a negative number. So yes,
my mom is a saint (she answered my question, and got in the shower).

Later, in third grade, a substitute teacher was ridiculing the work of a
student who'd had the temerity to subtract a larger number from a smaller
number. Everyone knew that was impossible. Full of indignation as only an
eight-year-old Galileo can be, I stood up (I don't know why I couldn't have
just raised my hand like a normal person) and protested that it was entirely
possible. For my trouble I shared in the ridicule, and later got a talking-to
about "teaching things when people are ready for them."

It might look like it, but I'm not leading up to a critique of the elementary
school system. My point is different entirely: math can be fun. Maybe not all
at once and with Consequences If Not Correctly Learned, and maybe "fun" in the
way programming is fun (i.e. still hard), but it's still fun.

You've got a long while before your kids have that drummed out of them. Use
it. My mom was a Dance-turned-English major, but she still answered my absurd
questions in absurd circumstances. I wish everyone could have my parents.

~~~
brotchie
I totally agree with this sentiment. My parents would always attempt to answer
any question, however absurd: What's this made of? wood. What's wood made of?
Carbon, Hydrogen, and Oxygen. What's Carbon made of? Protons and Neutrons.
What's a Proton made of? It would stop at this point because they weren't
Physicists and didn't know about Quarks and Gluons.

Following many of these experiences, I though it always appropriate to ask
probing questions, and continue to ask them until I understood a problem or
situation fully. This attitude continues to this day, though curbed in some
social situations. I'm not sure whether this persistent character trait is
correlated with my parents behaviour or caused by it.

I distinctly remember feeling quite angry during my first "adult" interactions
with many of my friend's parents. Most had equal or greater education than my
parents, yet when queried about aspect of their field that I knew they had
deep knowledge of (eg. Ships engines from a Marine Engineer, Offshore tax
havens from a Tax Accountant), they'd be very hesitant and often express "why
do you ask so many damn questions?"

In hindsight, perhaps this result was in part due to my lack of social
awareness and emotional intelligence, but I swear it's imbued their offspring
with fundamentally different characteristics. My friend's that had parents
such as these have tended to follow more qualitative pursuits (Musicians,
writers, journalists) while those with parents similar to mine ended up in
strongly quantitative fields (Engineering, Maths, Science).

Obviously my experience is a single sample from the distribution, yet I wonder
if others have observed a correlation between their own or others parent's
attitudes towards answering their children's thorny questions and life /
career attitudes? Perhaps it simply comes down to genetics?

~~~
cecilpl
My parents were like this too, and my innate thirst for knowledge led me to
persistently question people for years until I realized how much many people
seem to dislike this.

I love explaining in depth and enjoy the opportunity to explore the boundaries
of my knowledge. "Actually, I don't know! Let's find out!" comes out of my
mouth often.

But so many people seem to get irritated or annoyed when I (inadvertently)
expose their lack of knowledge. I still haven't quite figured out why, but
I've been told on more than one occasion to "Stop grilling me!" when I'm
simply curious about someone's job or how they see things.

~~~
biot
You lack the knowledge, so it's fascinating for you to learn it for the first
time. They already have the knowledge, so it likely gets increasingly tedious
the more questions you ask as reciting what they already know lacks the
fascination of discovery that you experience.

~~~
cecilpl
Then am I abnormal in that I get vicarious joy out of seeing someone else
learn something for the first time?

~~~
biot
You've probably seen/read interviews with people where the person doing the
interviewing has a list of questions already prepared and just goes through
them one by one, occasionally asking a question that was just answered as part
of a previous question or failing to dive deeper into an answer that is so
fascinating that it's a shame not to explore it more. I find those kinds of
interviews frustrating since the interviewer is putting no thought into it.

A good interviewer is someone who asks questions, then connects them with
previously learned information in order to ask a better, more insightful
question that they otherwise wouldn't have been able to ask had they not just
learned the info. It's clear that this interviewer isn't just going down a
list of things to ask. A really good interview is one where the interviewee
walks away having learned something, even though they've only answered
questions posed to them.

Asking questions is a lot like this. If you're more like the first interviewer
going through a list of disjoint questions (or questions which only probe for
further detail) then it's not surprising to me if people get annoyed. If
you're like the second interviewer asking insightful, deeper questions where
you're connecting the dots in such a way that you're asking questions that
matter and they still get annoyed then perhaps it's just them. Of course, if
it's totally new territory for you then you may need to go through seemingly
disjoint questions in order to establish a baseline of knowledge to be able to
ask more intelligent, interesting questions. You may lose people in your
attempt to bootstrap your knowledge to that level.

It's also entirely possible that they hate their job but through cognitive
dissonance they've learned to cope with it. Your questions may increase the
dissonance for the work they do in which case they'd rather avoid thinking
about it entirely.

------
femto
I think of "square root" as meaning "do half of something".

For example, A^2 means do the operation A twice, A^1 = A means do the
operation A once, A^0.5 = sqrt(A) means do half of A. What's half of A?
Something that when done twice gives A.

For example, take a number line. We have integers going from zero to infinity.
Now lets add a "negation operation", and call it -1, so we can make the
numbers from zero to negative infinity. On paper, this is equivalent to
rotating the number line by 180 degrees. Now the number line runs from
negative infinity to positive infinity. Now let's do the operation sqrt(-1),
which means "do half of a negation". On paper, this means we rotate by 90
degrees instead of 180 degrees. We now have a new number line at right angles
to the original one, and the original number line has turned into a number
plane. (Feel free, at this point, to launch into an explanation of complex
numbers, with i=sqrt(-1) meaning "move in an orthogonal direction".)

Similarly, a cube root means "do a third", and so on.

~~~
lmkg
Spoiler alert: When people talk about "group theory," this is the sort of
thing that it does. The operations of addition, multiplication, and functional
composition share certain symmetries that group theory makes formally sound.
Once you grok the relationships, and how multiplication is "the same type of
thing as" rotations, then you can see how square roots are similar to
fractional rotations.

Less abstractly... the Exponent operation lets you turn addition into
multiplication. It takes a little while to wrap your head around that, but
once you do, it's straightforward to see how square roots correspond to
fractions.

This is also concretely visible in Matrix algebra. Some matrices literally are
rotations, and the square roots of such matrices are rotations by half as
much. Matrices are nice to study in group theory, because they bridge the gap
between numeric operations and functional composition.

~~~
wjnc
Matrix algebra is one of the nicest courses to have had. Makes so many other
maths and statistics subjects more easy to grasp.

~~~
VLM
Quaternions are handy for 3d graphics too.

------
risratorn
I cannot stress how important this is, don't just shrug of the questions your
kids ask, how stupid, obvious or unexplainable the answer may be. Even if you
don't know the exact answer.

My son asked me a question some time ago "What is a tree made of?" ... "Wood"
i said, which just backfired another question "What is wood made of?" to which
I actually took the time to explain (to my best of knowledge) that wood exists
out of fibers which in turn exist out of carbon which is an atom, explaining
that materials exists out of structures built from atoms while I drew a sketch
of an atom and how it can bind to other atoms to form materials like wood,
iron, etc.

The look on his face was worth millions and it made me feel real good and
proud that my 5 year old son took interest into something that even for an
adult without proper knowledge is hard to comprehend. I'm fairly confident
that all this went way above his head but the fact that my explanation piqued
his interest and set his imagination on fire is more than worth it.

Explain all the things!

------
mbubb
That is really great. We did something a little similar with the tiles on our
floor. Geometric answers are nice as they are tangible.

I have found some beautiful books to use for reading time - but they tend to
be biology/ physics/ astrology/chemistry - rather than pure math.

Growing up the boys both drooled on, ate and generally mangled about 5 copies
of the same Animal encyclopedia. Also Roger Tory Peterson-ish "Field Guides".
Good quality illustrations; they are relatively durable and the text gives
parents the answers to thingsl like 'Where does a 3 toed sloth live?" or 'how
big does an aligator gar get?', etc, etc

No Starch press has a nice little book "The Lives of the Elements" which my
boys ( 8 and 6 yrs) devour repeatedly. We also got some of the Manga series
for fun (also O'Reilly or NoStartch).

"Big Questions for Little Minds" is a nice book. Little 1-2 page 'essays' on
"why is the sky blue" questions. They are written by experts in each area and
are fun. They also have a handful of hard vocab words for kids that age and
they are mostly written by British experts so there are some variants for
North American readers to learn.

This book is beautiful: "The Where, the Why, and the How: 75 Artists
Illustrate Wondrous Mysteries of Science"

but is more for the cool graphics than the text.

Math is a subject where parents need to improvise a bit more.

Math I am often at a loss. My kids - though they are in the same system - have
not had the same experience with learning math. My elder son grokked odd/even
numbers in kindergarten but my younger son did not get that.

~~~
DanBC
Amazon UK and US links

Big Questions From Little People Answered By Some Very Big People:
([http://www.amazon.co.uk/Questions-From-Little-People-
Answere...](http://www.amazon.co.uk/Questions-From-Little-People-
Answered/dp/0571288510/ref=reg_hu-rd_add_1_dp)) ([http://www.amazon.com/Big-
Questions-Little-People-Answers/dp...](http://www.amazon.com/Big-Questions-
Little-People-
Answers/dp/0062223224/ref=sr_1_1?s=books&ie=UTF8&qid=1362583996&sr=1-1&keywords=big+questions+from+little+people))

Wonderful Life With The Elements: The Periodic Table Personified: An Adventure
through the Periodic Table:
([http://www.amazon.co.uk/gp/product/1593274238/ref=ox_sc_act_...](http://www.amazon.co.uk/gp/product/1593274238/ref=ox_sc_act_title_1?ie=UTF8&psc=1&smid=A3P5ROKL5A1OLE))
([http://www.amazon.com/Wonderful-Life-Elements-Periodic-
Perso...](http://www.amazon.com/Wonderful-Life-Elements-Periodic-
Personified/dp/1593274238/ref=sr_1_1?s=books&ie=UTF8&qid=1362584034&sr=1-1&keywords=wonderful+life+with+the+elements))

The Where, the Why, and the How: 75 Artists Illustrate Wondrous Mysteries of
Science ([http://www.amazon.co.uk/The-Where-Why-How-
Illustrate/dp/1452...](http://www.amazon.co.uk/The-Where-Why-How-
Illustrate/dp/1452108226/ref=pd_bxgy_b_text_y)) ([http://www.amazon.com/The-
Where-Why-How-Illustrate/dp/145210...](http://www.amazon.com/The-Where-Why-
How-Illustrate/dp/1452108226/ref=pd_bxgy_b_text_y))

------
mturmon
The idea of associating multiplication with the area of a rectangle seems to
be very helpful. Commutativity falls directly out of that idea, and so does
the basic algorithm for two-digit multiplication (I guess that is
distributivity of + over *, explained by adding up areas).

Also, the idea of being able to mimic the calculator is fun for kids. They
like to be able to find an answer, and then check with the calculator (or
REPL).

~~~
NamTaf
Another great trick is using diagonal lines to calculate multiplication [1].
It's a neat little trick that really helps drive home an understanding of what
'multiplication' actually is, since it's just like the OP's grid of boxes but
turned on its side and contracted down to points. The trick is getting 45
degree lines so they line up more easily.

The advantage of lines is that you can spew out multiplication in the double-
digit by double-digit range quite quickly. 51x23 for example is super easy to
calculate - 10 100's + 17 10's + 3 1's. You're essentially just doing 50x20 +
50 _3 + 20_ 1 + 1*3, but with lines to track everything.

Show it to a kid, and watch it blow their mind.

[1] [http://lifehacker.com/5975917/quickly-multiply-big-
numbers-t...](http://lifehacker.com/5975917/quickly-multiply-big-numbers-the-
japanese-way)

~~~
brazzy
How painful then to see it called "magic" and read "I don't know how or why
this works [...] wonder why we don't teach math the way"

That doesn't sound like any kind of understanding is involved...

~~~
Sunlis
With a method like that it is very easy to accept the answer you get as magic,
because there is no understanding involved. The method shown by OP is not
practical to use for anything other than teaching concepts, but it is at least
easily expandable and actually describes multiplication. This line thing is
more of a trick than anything else.

I sincerely hope that if this is used to teach children, they also properly
explain multiplication rather than simply pulling a rabbit from a hat.

------
chris_wot
Kids are cute when they get excited about learning something. My own kids are
awesome. I hope they never lose that excitement :-)

~~~
vsync
Sad to say, school will force it out of 'em.

Sad to say, school is necessary to function in society.

~~~
learc83
>school is necessary to function in society.

No it's not. An education is necessary to function in society, it's not
necessary that it comes from school, or the type of school that forces the joy
of learning out of children.

~~~
damoncali
I don't think you fully grasped the point. Look a level deeper.

~~~
sp332
Schools are necessary for society to function?

~~~
nmbr
I believe he's saying that crushing the dreams of children is required for a
functional society.

~~~
chris_wot
That would merely make him wrong.

------
larrydag
Awesome article. I love teaching math concepts especially to kids. This is
something I will have to remember.

Here is a smartphone/tablet app that teaches kids algebra.
<http://dragonboxapp.com/>

~~~
lotharbot
I second the recommendation. My five year old nephew plays Dragon Box, and he
seems to grasp quite a few of the concepts now.

~~~
larrydag
My 9 year old finished it in a weekend. He was hooked and considered it a fun
puzzle. My only problem was trying to keep the energy going and challenge him
more.

------
angrycoder
This is a fairly old teaching method. I remember doing it over 30 years ago
when I was 6 years old at Montessori. We used a wooden board and pegs though.

~~~
bilbo0s
"...We used a wooden board and pegs though..."

They still do.

~~~
murbard2
But they used to too.

------
jheriko
I find this interesting - I too discovered the square root at 5 - but I had
little help from my parents and actually I reverse engineered it to find out
how it worked. square rooting from 0 up to 10 and scratching my head for a
while trying to work out why 0, 1, 4 and 9 were special...

Calculators are great. The first time I encountered one I learned more maths
than the next 4 years of school would teach me in mere hours...

I'm not sure how but mass education seems to have made difficult and scary
concepts like decimals and negative numbers, which are so intuitive that they
require little or no explanation when naively encountered 'in the wild'.

Everyone should let their kid play with a calculator I think. :)

I'm also strongly in favour of encouraging children to learn for themselves
through experimentation. Explaining things is difficult, and at that age - you
may not remember - but learning things is not. Not to mention that when you do
reinvent the wheel you get a truly deep understanding, rather than some rote
memorisation tricks to pass highly targetted exams.

~~~
VLM
My experience around the same age (well, maybe more like 6) was with early
BASIC home computers circa 1980, and once I learned "FOR loops" and "print"
and saw all these weird "function" things in the manual complete with handy
example code, I started randomly seeing what happened when I used functions
and got all wound up that sqr() crashed on negative numbers and almost never
was a "even round result" except for certain numbers and another thing I found
hilarious at the time was repeatedly taking the square root of ANY postive
number the computer would accept, would eventually, sooner or later, round to
exactly "1". Playing with log and antilog was another weird experience, as was
early probing of the boundaries of the scalability of factorials (so, why does
it crash on any factorial bigger than 70 or so?) I also learned that trig is
hard. I learned more about math from a TRS-80 model3 at age 6, than until I
went to university.

------
gtani
I see this a lot in my friends who've decided to raise tiger kids, so lots of
math, python programming, foreign languages, music lessons at a young age.
You'll never know how they respond. A couple turned into academic and musical
prodigies, the others are normally bright kids for whom Xbox vs. Windows
gaming is the big issue.

tl;dr time intensive, expensive, can't hurt

------
quorn3000
I've been thinking how to explain to my 8 and 6 year olds that the square root
of a prime number is irrational.

I've been over primes lots of times. I've talked about decimal fractions,
repeating decimals, and the possibility of there being numbers which are non-
repeating. I know the proof for the square root of 2 being irrational. It's
just tying it all together.

~~~
ColinWright
The easiest and most accessible way I've found to do this is as follows:

    
    
      Let's suppose that sqrt(p) = a/b
      That means that p = (a^2)/(b^2)
      That means that p * b^2 = a^2
    

Now since you've covered primes and factorization, look at how many times _p_
can turn up on each side of that equals sign. It must be an even number of
times on the right, and an odd number of times on the left.

Another way to say this is that every time you take a fraction and square it,
you never get a prime number.

And yes, I know there's a lot missing from this explanation, but the things
that are missing can then be expanded later, rather that muddying the waters
now.

~~~
alex-g
At what sort of age do you think people are ready for proof by contradiction?
I remember being taught it explicitly at secondary school (in the UK) but we
may have seen implicit uses of it earlier. I would imagine that some very
young children might find it difficult to remember the train of thought.

~~~
ColinWright
That's why flipping it to say that every rational, when squared, doesn't give
you a prime. The concept of irrartional is already tough. I've found that when
handled carefully, even quite young kids can handle at least some of it.

But you're right, proof by contradiction can be tough.

------
gonzo
My son has always had his math questions answered. His mother has a MS in both
Math and Neuroscience and a BS in CS (with a minor in History). His dad (me)
is just a dumb-ass programmer.

When he was 5 we would practice 'math' in the car as we drove to school in the
morning. Mostly addition, but then subtraction (including negative numbers).
From there we moved on to repeated addition (multiplication). Sometime in
first grade, he asked a question at the dinner table. He'd noticed that 2x2=4
and 3x3=9, but wanted to know if there were two numbers that could be
multiplied to 'make 5'.

Square roots. Did the same picture thing.

He went to school the next day, and proudly told his teacher that he knew
"square roots". We got the note back asking us to not teach him "advanced
math". The Parent-Teacher conference ensued. The teacher didn't remember what
they were.

#include <polite/conversation/about/not/discouraging/him>

sigh.

------
joshuahedlund
I love stories like these, and the stories that inevitably pop up in the
comments. My first son will arrive in less than a month and I'm excited about
upcoming opportunities like these. These stories always give me ideas that I
wonder if I ever would have thought of otherwise, and it makes me wonder how
many other simple inspirations are out there that I might not think of,
either. Is there any sort of 'hacker parenting' group/forum/resource/etc out
there for spreading stories and ideas like this? I know there's the parenting
stackexchange, but it seems more geared towards identified issues; I'm
interested in getting ideas for things I haven't even thought to identify
before the kids are too old for it to make as much of a difference.

------
DanLivesHere
I wrote this! Glad to take any questions.

(and why'd it come up now? I wrote it a month or so ago.)

~~~
mkopinsky
Saw it somewhere online (probably Facebook) last night, and figured HN would
like it.

------
M4v3R
Kind of on the topic - I know there is a way to calculate a square root
without a calculator. Of course you can do it the "brute force" way, but one
day my music teacher told be about a better way, but never explained it in
detail. Does someone know how to do this?

~~~
gordaco
I don't want to sound smug or anything, but didn't you study it in school? In
Spain we learn it in fifth or sixth grade, just after learning the concept of
square root. I thought it was basic enough to be studied at about that age in
the rest of the world.

The algorithm is a little complicated compared to other things that we study
at that age, but it's not really difficult, although most (and I mean MOST)
people, even engineers and such, forget it when they don't need it any more.
It infuriates me a little that no teacher explained to us why did it work, but
that gave me oportunity to "reverse engineer" the method a few years later,
and expand it to a generic n-root algorithm.

~~~
yakiv
> I thought it was basic enough to be studied at about that age in the rest of
> the world.

Where do you think GP went to school? Neither the post nor the profile
indicates a location.

Edit: I should say that I don't know for sure that you don't know from some
other post GP made.

~~~
gordaco
I don't know, that's why I just talked about "the rest of the world". I'm just
surprised that this is not as common as I thought it was. Cultural blindness,
I guess.

------
thejteam
Nice method.

The trick with children that age is to take what they do know, in this case
counting blocks in a square, and take it just one step further.

My wife used to teach kindergarten. The first grade teacher came up to her at
the start of her second year and asked her how the class knew how to multiply.
My wife, who is NOT a math person by any means, realized that multiplying is
just the natural progression from "skip counting", which is a skill in every
kindergarten standard.

------
trvz
"until probably 100 or 144" - ugh. That's 10 and 12. I hope he 's wrong about
that and schools aren't that bad where he lives. In Germany we've had to
memorize squares for numbers 1 to 20 and 25, which gives you the roots of 1,
4, ..., 100, 121, ..., 400, 625. For every other number beetween these, you
can estimate the root with two decimal places pretty well.

~~~
dagw
I fail to see a correlation between how good a school is and how many squares
of numbers you have to memorize. I don't remember being forced to memorize any
higher than 10 at either of the schools I went to and I'd classify them both
as 'good' in the grand scheme of things. Once you know the multiplication
table up to 10 and a couple of techniques for multiplying larger numbers there
really is no need for memorization.

------
johnmw
What a great explanation. On a related note, if you liked that then I
recommend you check out the game <http://dragonboxapp.com/> I think it is
really quite genius how they have turned solving algebraic equations into a
fun game for kids.

------
bazillion
I thought this was going to be about the first final fantasy games...

------
Jabbles
How big a slice of toast do you need to use all this peanut butter?

