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.
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?
I dropped my kids off at pre-school the other day. There is a little box full of sea shells, and my boy picked one up and fingered the little lumps and bumps on it.
I thought, damn, I have no idea how those little bumps on the shell are made. So I went on the internet and read as much as I could and found some nice links. Then I did a show and tell with the family. There wasn't that much interest, but later in the week I heard from his teacher that my wife had been telling the other kids how shells made, and what lives in them. My boys learned how to draw spirals.
That's my story.
I like the questions that kids ask, like what is an atom, what are protons, quarks etc. Then we get to say "No one really knows" and talk about the biggest machine in the world (LHC) (it may no be the biggest - but the story has to be good) and lab coats and scientists (like Dora the explora - that one's a bit of a stretch.) I think kids like to hear that we all question stuff and some people try to answer those questions.
I love it when my kids say 'I think so-and-so' and you can tell them that other scientists thought that and did an experiment to see if they were right. That son, is a valid and interesting question.
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.
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.
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.
I doubt they're annoyed by their "lack of knowledge" being exposed, it seems more likely to me that they're annoyed because they're trying to do something more important (to themselves/their boss/whoever) than answer your persistent questions..
It would be great if people were taught to say "I don't know" (in other words, to tell the truth).
It's such a liberating thing to be able to say.
I tutor kids, and sometimes they ask stuff I can't answer. I tell them I don't know but will find out, then move heaven and earth to find out. A week later, when I tell them, the look of surprise on their faces can be funny: an adult actually admitted to not knowing, and then kept a promise to find out!
There are often campaigns to "get children/young people interested in science". Except you don't have to teach people how to be interested in science, instead people/society teach people to not be interested in science. Children sometimes ask "why" constantly. Some people tell children to "not ask questions" or "just because". Then when they're 15 we wonder why they've stopped asking why. Grrr.
I went to the science museum (don't know what is it called exactly) in Barcelona and there they have demonstrations of many laws of physics, such as if you are spinning and extend your hands, you will slow down and vice versa.
There were descriptions for each of the demonstration and yet I saw parents with kids who just went from one to the next, didn't bother to understand or read about the demonstration and in general were completely useless in teaching their kids.
Our times are sad. Do you see any encouragement toward science in the Simpson's? Read any Jules Verne book, you'll see why in these times most kids dreamt about science. Now even Lego became figurines you have to collect, in my time Lego was a big box of colorful squares used to build anything that crossed one's mind.
Read any Jules Verne book, you'll see why in these times most kids dreamt about science.
Well the past wasn't brilliant either. Remember in those times, only boys should dream about science. Women were obviously incapable of doing science & maths, so there's no point encouraging them, there's probably something wrong with a woman who wants to do men's work like science.</sarcasm>
In the old days:
"Some people tell children to "not ask questions" or "just because"."
In the wikipedia/google era:
"That's not important enough for our limited disk space / griefers thought it would be funny to trash it so they feel better when they make others feel sad"
I think you'll see an interesting societal shift as the current generation grows up. Its not a positive one. Going from "stop asking questions", "there's no answer" without necessarily judging the questions themselves to "your questions are wrong and your ideas suck and you're wrong to even wonder about this topic" is even more negative.
The 'fun' of maths isn't the same fun as you get from, say, playing another game of some casual computer game where you're simply switching off and doing something relaxing. It's fun in the sense that it's a challenge and provides immense satisfaction when you work out problems with it.
I've noticed that I get a rarely-rivaled sense of satisfaction when I see different bodies of knowledge that I've covered come together. For example, if my physics from school/uni help me solve a problem I'm facing at work, it gives a really good, satisfying feeling of mastery over the topics at hand. Deriving a solution without guidance, simply from your own existing knowledge, is a wonderful feeling.
The problem with this is that unless you embrace that 'inquisitive for inquisitive's sake' attitude, you don't easily justify fully exploring subjects like maths to the point of gaining that capacity. It's the same as programming, just as you touched on. Someone could be a moderate programmer if they learn what they need to spit out basic code to achieve an end goal, but you won't feel the same satisfaction of mastery as if you embrace the art and program things 'just because'.
The good thing is kids seem to be born with this attitude innately - just look at any young child explore the world around them. It's only really through social conditioning that it seems to be beaten out of them, as you say. Through good role models and leadership (e.g.: from teachers, parents) providing adequate material to feed that curiousity, it can be avoided.
It's fantastic that this kid's parents are willing to do what they did, just as your mother was willing to feed your curiousity. Understanding how significant it can be to the development of a child throughout their life is, I think, the key to fully appreciating the role that teachers (formal and informal) have in our society and why they should be valued far more than they often are at present.
As a corollary to this, perhaps one of the most heartbreaking moments for a kid whose parents welcome their questions and feed them information is when you finally discover they don't actually know everything and that you've exhausted their ability to teach about a certain topic. I remember reaching that point several times and it always upset me, all the way from first playing around with computers and dad not having all the answers, to finally getting to maths beyond what either of my parents were taught towards the end of my high schooling.
It's upsetting, but I guess it's also gratifying if you can step back from the moment and realise that it means you're also becoming just as wise as those who you've always looked up to. That's powerful.
I have three children and they frequently will ask me question that either I never knew, or have long since forgotten. I take these times to demonstrate another lesson -- how to learn.
When I was a child it would end up with a trip to the encyclopedia (we were a Collier's family) and if that did not satisfy our needs, a visit to the library. A few times, we have had to visit the library, generally so my kids can check out a book or two on a specific interest, as the internet/wikipedia/kahn academy now have so much content there are few things an interested mind cannot learn.
I think it is very important for children to learn how to learn and how to admit when you don't know something.
i don't have any kids of my own, but have had plenty of opportunities to babysit. what i always do if kids ask a question i don't know is tell them i don't know, but lets go look it up. Then i go on my phone or on a computer to wikipedia and look it up with the kids right there. Even if the kids aren't able to read yet, there are pictures and you can explain it to them as you learn it yourself. I'm hoping i might spark their curiosity this way, and also once they're a little older the desire to go and research something that they don't know but are curious about.
Haha! I had the same story about substracting "impossible" numbers (I distinctly remember that it was 6-9) when I was like 5 or 6. Fortunately I had a very good teacher and she dismissed it politely, saying something like that was a little advanced and would be explained another year. We still talk and laugh about the anecdote whenever I met her on the street.
Math IS fun, but it's damn hard to teach, and that's why math teachers end up being hated more often than the rest. Learning math is much more about finding your way to the "eureka" moments than letting someone do it for you, but it's very satisfying once you get the hang of it.
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.
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.
Reminds me of category theory, which I tried to understand a little of yesterday, based on a discussion of Haskell. I don't know enough of either to know what the difference is between the two.
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!
I don't like this explanation at all. This fuzzy thinking works out because you knew it to work out based on your much less fuzzy understanding of the complex numbers. Similarly fuzzy thinking in a new problem domain won't work out 90% of the time.
That's another interesting way to do it. When I read the blog post and saw the two legs of what I assumed would be a right pythag triple triangle forming, I thought I was about to see the classic Pythagorean theory and its geometric proof whipped out, which might be a little too much all at the same time for a kid (or, maybe not?). Hard to describe in text but its pretty much what you get with a 3x3, 4x4, and 5x5 gridded squares connected to each other giving a 3,4,5 right triangle.
The problem is this rapidly devolves into "very nice anecdote but why can't you use any random lengths" and "why does this only work in 2 dimensions" and next thing you know its gettin pretty deep which is a bit much to start with WRT whats a sqrt.
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.
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.
great pictures are such a valuable thing to have in a book for kids. even if the text is hard, if the pictures are fascinating and amazing, it helps inspire curiosity and wonder, and hopefully encourages kids to learn more, both now and as they get older.
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).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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
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
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.
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.
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?
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.
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.
I'm sorry, that wasn't at all my intention. In any case, I wouldn't be mocking him, but the educational system he studied under (which definitely wouldn't be his fault; to be honest, I've also suffered a quite bad educational system). And I didn't answer him because someone else had done so already in another comment, linking to wikipedia, so I didn't want to be redundant.
We also learned it in, I think, 4th grade or perhaps 5th. This was in the US. And yes, I've also forgotten it. I don't think I ever used the method outside of that lesson.
I guess it's this: if you have a guess G at the square root of N, compute N/G. If G is too large, N/G will be too small and vice versa (well, as long as your initial guess is larger than zero)
Also, G' = (G+N/G)/2 will be a better guess than G.
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.
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.
"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.
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.
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.
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.
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.
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.