10 ms·

Teaching Square Roots to a Five Year Old

168 points by mkopinsky 14y ago

mturmon 14y ago
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 14y ago
  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 + 503 + 201 + 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-the-japanese-way http://lifehacker.com/5975917/quickly-multiply-big-numbers-t...
  - brazzy 14y ago
    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 14y ago
      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.
  - mturmon 14y ago
    I purposefully didn't mention the line-crossing algorithm, because I think (like other nearby commenters) that it's more of a mysterious trick than a visualization that promotes deeper understanding. Maybe as they get older, it would be fun to present the line-crossing algorithm, and then ask them if they can explain why it works. My 7-year-old would not be able to do this, but she can easily relate to the multiplication-by-area analogy and its generalizations.
femto 14y ago
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 14y ago
  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.
  - brotchie 14y ago
    Nice, I didn't realise the relationship between rotations and the matrix square root: multiplying a vector by the sqrt of the pi/2 rotation matrix rotates it by pi/4!
  - wjnc 14y ago
chris_wot 14y ago
Kids are cute when they get excited about learning something. My own kids are awesome. I hope they never lose that excitement :-)
- vsync 14y ago
  Sad to say, school will force it out of 'em. Sad to say, school is necessary to function in society.
  - learc83 14y ago
    >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 14y ago
      I don't think you fully grasped the point. Look a level deeper.
      - sp332 14y ago
        Schools are necessary for society to function?
        nmbr 14y ago
        I believe he's saying that crushing the dreams of children is required for a functional society.
        chris_wot 14y ago
        That would merely make him wrong.
invalidOrTaken 14y ago
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 14y ago
  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?
M4v3R 14y ago
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?
- PakG1 14y ago
  I learned one method in high school. I couldn't remember it, and too lazy to try to work it out, but here are some answers. http://en.wikipedia.org/wiki/Methods_of_computing_square_roots http://en.wikipedia.org/wiki/Methods_of_computing_square_roo...
  - Nav_Panel 14y ago
    I just want to point out one of the links on that page: http://en.wikipedia.org/wiki/Fast_inverse_square_root http://en.wikipedia.org/wiki/Fast_inverse_square_root Absolutely hilarious and amazingly magical method of algorithmically computing square roots. It right shifts a floating point number to essentially divide the exponent component by 2. Then it uses a weird magic number trick to get the correct result. Awesome random programming trivia.
    - ars 14y ago
      Note that this is the inverse square root, i.e. 1/sqrt(x) (which can also be written x^(-1/2) ).
- gordaco 14y ago
  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.
trvz 14y ago
"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.
- vxNsr 14y ago
  He's not wrong, though I think most people know 25^2 in general schools only teach 1-12. My mother was/is kinda a math freak (in a good way) and made sure that we knew our squares through 20 plus fun numbers like 25(625) and 45 (2025) among other things (like decimal division). She'd always decry our meager American education compared to her Soviet one, fun times.
- dantle 14y ago
  Ha ha, yes, how bad those schools must be. They are wholly dependent upon Germans to answer the tough questions, like what is the square of 25, and what's your best guess for the square root of 21. One only needs to remember a simple algorithm to compute any square by hand, and realistically a computer is always a better choice. I can't understand why you think memorizing more squares provides a better math education.
- dagw 14y ago
  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.
angrycoder 14y ago
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 14y ago
  "...We used a wooden board and pegs though..." They still do.
  - murbard2 14y ago
    But they used to too.
risratorn 14y ago
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!
quorn3000 14y ago
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 14y ago
  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 14y ago
    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 14y ago
      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.
    - tokenadult 14y ago
      To answer your question, the youngest age at which I have seen a pupil propose a proof by contradiction to me was age eight. The girl had read some Life of Fred http://www.lifeoffredmath.com/ http://www.lifeoffredmath.com/ books for children about mathematics, and had newly joined my local mathematics class. On only the second or third week of class, she came up to me after class and said, "I've discovered a proof by contradiction for the parallel postulate." As you can imagine, I found this quite amazing. (I knew her mother, and thus knew the daughter a little before she joined my class, but I would say that's rather precocious behavior even in the social circle I keep.) Her "proof," of course, was really Saccheri's flawed proof http://www.jimloy.com/geometry/saccheri.htm http://www.jimloy.com/geometry/saccheri.htm http://www-history.mcs.st-and.ac.uk/HistTopics/Non-Euclidean_geometry.html http://www-history.mcs.st-and.ac.uk/HistTopics/Non-Euclidean... that assumed the postulate to show the "impossibility" of any quadrilateral that didn't fit Euclidean geometry. But most of us have minds that begin study of mathematics with a stubbornly Euclidean set of presuppositions, so that was all right. The girl eventually advanced from my mathematics class to my colleague's more advanced class, and then did a summer at Epsilon Camp http://www.epsiloncamp.org/ http://www.epsiloncamp.org/ in that program's first year of existence.
johnmw 14y ago
What a great explanation. On a related note, if you liked that then I recommend you check out the game http://dragonboxapp.com/ http://dragonboxapp.com/ I think it is really quite genius how they have turned solving algebraic equations into a fun game for kids.
Jabbles 14y ago
How big a slice of toast do you need to use all this peanut butter?
DanLivesHere 14y ago
I wrote this! Glad to take any questions. (and why'd it come up now? I wrote it a month or so ago.)
- mkopinsky 14y ago
  Saw it somewhere online (probably Facebook) last night, and figured HN would like it.
thejteam 14y ago
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.
larrydag 14y ago
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/ http://dragonboxapp.com/
- lotharbot 14y ago
  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 14y ago
    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.
jheriko 14y ago
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 14y ago
  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 14y ago
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
mbubb 14y ago
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 14y ago
  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-Answered/dp/0571288510/ref=reg_hu-rd_add_1_dp http://www.amazon.co.uk/Questions-From-Little-People-Answere...) (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 http://www.amazon.com/Big-Questions-Little-People-Answers/dp...) 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_title_1?ie=UTF8&psc=1&smid=A3P5ROKL5A1OLE http://www.amazon.co.uk/gp/product/1593274238/ref=ox_sc_act_...) (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 http://www.amazon.com/Wonderful-Life-Elements-Periodic-Perso...) 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/1452108226/ref=pd_bxgy_b_text_y http://www.amazon.co.uk/The-Where-Why-How-Illustrate/dp/1452...) (http://www.amazon.com/The-Where-Why-How-Illustrate/dp/1452108226/ref=pd_bxgy_b_text_y http://www.amazon.com/The-Where-Why-How-Illustrate/dp/145210...)
joshuahedlund 14y ago
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.
bazillion 14y ago
I thought this was going to be about the first final fantasy games...
gonzo 14y ago
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.

168 points by mkopinsky 14y ago

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 14y ago
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 + 503 + 201 + 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-the-japanese-way http://lifehacker.com/5975917/quickly-multiply-big-numbers-t...
- brazzy 14y ago
  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 14y ago
    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.
- mturmon 14y ago
  I purposefully didn't mention the line-crossing algorithm, because I think (like other nearby commenters) that it's more of a mysterious trick than a visualization that promotes deeper understanding. Maybe as they get older, it would be fun to present the line-crossing algorithm, and then ask them if they can explain why it works. My 7-year-old would not be able to do this, but she can easily relate to the multiplication-by-area analogy and its generalizations.

femto 14y ago

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 14y ago
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.
- brotchie 14y ago
  Nice, I didn't realise the relationship between rotations and the matrix square root: multiplying a vector by the sqrt of the pi/2 rotation matrix rotates it by pi/4!
- wjnc 14y ago

chris_wot 14y ago

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

vsync 14y ago
Sad to say, school will force it out of 'em. Sad to say, school is necessary to function in society.
- learc83 14y ago
  >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 14y ago
    I don't think you fully grasped the point. Look a level deeper.
    - sp332 14y ago
      Schools are necessary for society to function?
      - nmbr 14y ago
        I believe he's saying that crushing the dreams of children is required for a functional society.
        chris_wot 14y ago
        That would merely make him wrong.

invalidOrTaken 14y ago

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 14y ago
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?

M4v3R 14y ago

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?

PakG1 14y ago
I learned one method in high school. I couldn't remember it, and too lazy to try to work it out, but here are some answers. http://en.wikipedia.org/wiki/Methods_of_computing_square_roots http://en.wikipedia.org/wiki/Methods_of_computing_square_roo...
- Nav_Panel 14y ago
  I just want to point out one of the links on that page: http://en.wikipedia.org/wiki/Fast_inverse_square_root http://en.wikipedia.org/wiki/Fast_inverse_square_root Absolutely hilarious and amazingly magical method of algorithmically computing square roots. It right shifts a floating point number to essentially divide the exponent component by 2. Then it uses a weird magic number trick to get the correct result. Awesome random programming trivia.
  - ars 14y ago
    Note that this is the inverse square root, i.e. 1/sqrt(x) (which can also be written x^(-1/2) ).
gordaco 14y ago
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.

trvz 14y ago

"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.

vxNsr 14y ago
He's not wrong, though I think most people know 25^2 in general schools only teach 1-12. My mother was/is kinda a math freak (in a good way) and made sure that we knew our squares through 20 plus fun numbers like 25(625) and 45 (2025) among other things (like decimal division). She'd always decry our meager American education compared to her Soviet one, fun times.
dantle 14y ago
Ha ha, yes, how bad those schools must be. They are wholly dependent upon Germans to answer the tough questions, like what is the square of 25, and what's your best guess for the square root of 21. One only needs to remember a simple algorithm to compute any square by hand, and realistically a computer is always a better choice. I can't understand why you think memorizing more squares provides a better math education.
dagw 14y ago
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.

angrycoder 14y ago

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 14y ago
"...We used a wooden board and pegs though..." They still do.
- murbard2 14y ago
  But they used to too.

risratorn 14y ago

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!

quorn3000 14y ago

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 14y ago
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 14y ago
  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 14y ago
    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.
  - tokenadult 14y ago
    To answer your question, the youngest age at which I have seen a pupil propose a proof by contradiction to me was age eight. The girl had read some Life of Fred http://www.lifeoffredmath.com/ http://www.lifeoffredmath.com/ books for children about mathematics, and had newly joined my local mathematics class. On only the second or third week of class, she came up to me after class and said, "I've discovered a proof by contradiction for the parallel postulate." As you can imagine, I found this quite amazing. (I knew her mother, and thus knew the daughter a little before she joined my class, but I would say that's rather precocious behavior even in the social circle I keep.) Her "proof," of course, was really Saccheri's flawed proof http://www.jimloy.com/geometry/saccheri.htm http://www.jimloy.com/geometry/saccheri.htm http://www-history.mcs.st-and.ac.uk/HistTopics/Non-Euclidean_geometry.html http://www-history.mcs.st-and.ac.uk/HistTopics/Non-Euclidean... that assumed the postulate to show the "impossibility" of any quadrilateral that didn't fit Euclidean geometry. But most of us have minds that begin study of mathematics with a stubbornly Euclidean set of presuppositions, so that was all right. The girl eventually advanced from my mathematics class to my colleague's more advanced class, and then did a summer at Epsilon Camp http://www.epsiloncamp.org/ http://www.epsiloncamp.org/ in that program's first year of existence.

johnmw 14y ago

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

Jabbles 14y ago

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

DanLivesHere 14y ago

I wrote this! Glad to take any questions. (and why'd it come up now? I wrote it a month or so ago.)

mkopinsky 14y ago
Saw it somewhere online (probably Facebook) last night, and figured HN would like it.

thejteam 14y ago

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.

larrydag 14y ago

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/ http://dragonboxapp.com/

lotharbot 14y ago
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 14y ago
  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.

jheriko 14y ago

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 14y ago
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 14y ago

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

mbubb 14y ago

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 14y ago
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-Answered/dp/0571288510/ref=reg_hu-rd_add_1_dp http://www.amazon.co.uk/Questions-From-Little-People-Answere...) (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 http://www.amazon.com/Big-Questions-Little-People-Answers/dp...) 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_title_1?ie=UTF8&psc=1&smid=A3P5ROKL5A1OLE http://www.amazon.co.uk/gp/product/1593274238/ref=ox_sc_act_...) (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 http://www.amazon.com/Wonderful-Life-Elements-Periodic-Perso...) 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/1452108226/ref=pd_bxgy_b_text_y http://www.amazon.co.uk/The-Where-Why-How-Illustrate/dp/1452...) (http://www.amazon.com/The-Where-Why-How-Illustrate/dp/1452108226/ref=pd_bxgy_b_text_y http://www.amazon.com/The-Where-Why-How-Illustrate/dp/145210...)

joshuahedlund 14y ago

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.

bazillion 14y ago

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

gonzo 14y ago

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.