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I still don't know why long-division works. I don't think I even remember how to do it as an adult. I've always had a very intuitive grasp of math, which meant
by _bxg1 7y ago
I still don't know why long-division works. I don't think I even remember how to do it as an adult.
I've always had a very intuitive grasp of math, which meant I excelled at solving things in my head but struggled with (and resented) having to show my work in the arbitrary algorithms we were taught. I understand division despite my education.
They really should try to come up with algorithms that are more intuitively linked to a spatial comprehension of the numbers. That underlying grasp is really all that sticks with you into the real world (and it's also what makes math engaging). If students aren't getting that, they're wasting their time in my opinion.
- novok 7y agoDivision is a repeated subtraction operation, and the long division algorithm is just a nice way to lay it out. Multiplication = repeated addition Addition = a form of counting
- threatofrain 7y agoI prefer to think of subtraction as the inverse of addition, and division as the inverse of multiplication. You only need addition and multiplication if you're dealing with rationals.
- _bxg1 7y agoThat makes sense. And actually I think I could re-derive the algorithm from that explanation. Maybe they need to explain it in those terms. Or use pictures.
- dboreham 7y agoI've never thought of division as repeated subtraction: it's the inverse of multiplication, or literal division as in I have a pile of beans that I divide equally between N piles. I don't think you're wrong, just seems an odd way to think about it, and fails to account for any remainder.
- pjscott 7y agoThe remainder is what's left after you've repeatedly subtracted, and can't subtract any more before going negative. (I wish that one of my teachers had said this to me, but they were too busy getting everybody to memorize the steps.)
- alfiedotwtf 7y agoIt's weird because I was taught long division in primary school that was different to everyone else in class (different maths class). Everyone seems to do some weird subtraction for each digit, whereas I was taught to carry right (i.e almost like how carries work with addition). Way faster, and it works the same all the way to decimal places, no weird remainders. Not sure why that wasn't the standard approach.
- alfiedotwtf 7y agoCopied from comment below... Oh :) I just worked it out - I was taught to fold the subtraction and carry right in one step. I'm an idiot!
- overlordalex 7y agoFrom elsewhere in this thread I learned that this is actually called "short division"! I can distinctly remember learning long division as a kid, and I wonder why this isn't immediately taught afterwards as the logical progression
- deleted 7y ago[deleted]
- pjscott 7y agoMy parents invented a simplified division algorithm so they could teach me the four basic arithmetic operations by third grade, and it was so much easier than long division that I could never remember how to do things the conventional way. I just pretended to follow the teachers' instructions for the next few years. (All these methods had the same asymptotic complexity. Mine was slower by a small constant factor; on the other hand it actually made sense. It was just repeated subtraction!)
- ravenstine 7y agoMy mom got in trouble with the teacher for doing that kind of thing with me.
- paleotrope 7y agoYour mom didn't get in trouble she just exposed something that the people with PhDs in Education don't want exposed.
- pjscott 7y agoTeachers are a very mixed bag. Some of them are delighted to see a student learning, in whatever way that learning may happen -- and some are just annoyed that the classroom's reality fails to match up perfectly with their lesson plans. If you're a nail that sticks up, some will try to uplift you further and others will angrily try to nail you down. (And most are somewhere in between.)
- _bxg1 7y agoI repeatedly got in trouble for not showing my work in the format they wanted, whether because of intuitive leaps or because of tricks I came up with myself.
- anongraddebt 7y agoThis worked for me up until Algebra 2. Up through Algebra 1 I'd just do everything in my head, never show my work, and thus solved a lot of problems using non-standard/creative methods on the fly. Ended up hurting myself a lot. I didn't have any muscle memory for solving the trivial parts of problems as I progressed from Algebra 2 through BC Calc. This led to me choosing a non-mathy degree in undergrad. I could breeze through higher-level logic courses, but was essentially remedial for higher-level math.
- geomark 7y agoI don't understand why long division would be a mystery. It's just repeated subtraction - you are finding how many times you can subtract the divisor from the dividend, in a systematic way.
- pjscott 7y agoIt's almost never explained with such lucidity. If you find a hundred people who can perform long division, I would be surprised if twenty of them have any idea how it works.
- dboreham 7y agoYou're never going to find 100 people who can perform long division.
- pjscott 7y agoSure you can! Just look at a typical fifth- or sixth-grade classroom. Most of them can perform long division, briefly, before the test. They may even remember it for several days after the test.
- mavhc 7y agoWhat none of them can tell you is why they've been taught long division, when short division is faster. And it's because you need long division to do division in algebra
- abakker 7y agoWhat is short division? I only remember being taught long division and having a calculator. Is there another method?
- Tomte 7y agoFascinating. I didn't know, either. After looking at the English Wikipedia I found that a) what I learned in school is long division b) the notation with the dividend and the divisor separated by a parenthesis and the result on top of a horizontal bar is totally alien to me c) I've never seen short division before d) the German Wikipedia doesn't even have an article about short division
- deleted 7y ago[deleted]
- keenmaster 7y agoThat’s a good point. You know what, it’s way easier to teach how to solve for x when y + x = 3x + 3 and y = 4 than it is to teach long division. As far as I could tell, that basic algebra is also much more important than long division, so why not teach it at an earlier age? The symbology of algebra is probably great for developing minds. Just show long division for “proofs” while introducing the concept of division, and let kids use a calculator for division after that (supplemented with calculator-free drills). That way there will be more lesson time for advanced topics. Same thing with long multiplication. Show the proofs using long multiplication, then have kids do progressively harder multiplication drills where all the math is done in their head. In fact, kids might grok first principles better that way.
- LegitGandalf 7y agoSome guys in Norway made an app that teaches algebra concepts to 5 year olds using pictures, as the game levels up it slides variables and numbers in.
- keenmaster 7y agoAre you referring to Dragon Box? This looks awesome. https://apps.apple.com/us/app/dragonbox-algebra-12/id634444186 https://apps.apple.com/us/app/dragonbox-algebra-12/id6344441...
- LegitGandalf 7y agoYeah,it's amazing, I've hooked multiple friends on this app. It turned my friend's girls math fear around, which is really important given that girls are learning to fear math from teachers who are themselves afraid of math. https://news.uchicago.edu/story/female-teachers-can-transfer-fear-math-and-undermine-girls-math-performance https://news.uchicago.edu/story/female-teachers-can-transfer...
- copperx 7y agoYou linked to the iOS version. Do you have a link to the app webpage, which will surely have both Play Store and App Store links?
- buzzkillington 7y agoYou should probably read about it as an adult: https://en.wikipedia.org/wiki/Long_division https://en.wikipedia.org/wiki/Long_division Children are mostly idiots. Education is the process of painfully breaking them out of their idiocy. Bonus points if you implement a long division algorithm that works on strings in your favorite language.
- _bxg1 7y agoThing is, even now I find it useless. I can do nearly any division in my head whose digits themselves fit in my head (Miller's Law). For anything bigger, I have a pocket supercomputer with me at all times. You're right that it could be a fun programming exercise, though
- buzzkillington 7y agoIf you do code in any capacity not being able to understand long division is a huge red flag. In terms of difficulty it's somewhere between regex and being able to write a recursive function.
- _bxg1 7y agoI didn't understand why it worked (a comment here pointed out that it's just division-by-repeated-subtraction which does make sense now). But I was still able to perform it, during the time when it was relevant to my life.
- copperx 7y agoWhat's a red flag is being frustrated by long division and at the same time not being curious or resourceful enough to resolve your frustration. You blame your lack of understanding on other people. I can understand if most people shared your frustration, but not a PhD holder. Your degree basically says that you're capable of doing long, intensive research on arcane topics. What am I missing here?
- jknz 7y agoDividend is 123456 and divisor is 37. 37>1, 37>12 but 37*3<=123. Write down 3 (meant for 3000) and start again with 123456-37x3000 as the new dividend. I wish the algorithm was taught by writing the full 3000 instead of only the digit 3; this appears to be a source of major confusion and to hide the underlying additions/multiplications.
- DanBC 7y ago> They really should try to come up with algorithms that are more intuitively linked to a spatial comprehension of the numbers. Have you heard of "chunking"? This video shows old style long division (and you're right, the method is confusing). It also shows an easier method that is based on understanding numbers rather than just applying an algorithm by rote. https://www.youtube.com/watch?v=K2rBaSj2L74 https://www.youtube.com/watch?v=K2rBaSj2L74 Here's another explanation: https://www.theschoolrun.com/what-is-chunking https://www.theschoolrun.com/what-is-chunking
- cjfd 7y agoNot really something to be proud of. You basically have just admitted on a forum for programmers that you don't have very much insight in algorithms.
- layoutIfNeeded 7y agoLong division is just polynomial division with x = 10.
- nathanwh 7y agoThis comment is hilariously on the nose for how a lot of math texts read (at least in my experience) $SIMPLE_CONCEPT that you’re trying your best to understand is easy if you think of it as a specific example of $MORE_COMPLICATED_CONCEPT that you surely can’t understand without understanding $SIMPLE_CONCEPT
- jiggawatts 7y agoWait till you get into Theoretical Physics, where the same thing is being explained by people who don't even have a firm grasp of $MORE_COMPLICATED_CONCEPT, but know just enough to be dangerous with it. E.g.: I distinctly remember that many of the original articles about the Amplituhedron spent like a page explaining what a triangle was, then jumped straight into functions over higher-dimensional analogues without explaining any of the intermediate steps. Something-something-logarithm-of-a-simplex. I can grok what a simplex is, but they literally just hand-waved over the bit where they take a logarithm of it in some way. The page of explanation of what a triangle is was of zero assistance.
- jeltz 7y agoThat was quite an eye opener though when we were taught polynomial division in high school. Not necessarily for how long division works because I already understood that, but because it was an example where short division could not be used.
- username90 7y agoThis is the problem with rewarding conscientious kids, they just follow instructions without ever asking themselves why. If you were more rebellious and refused to follow steps you didn't understand then you'd get worse grades but you'd learn more.
- cptskippy 7y agoThen you never knew it? Perhaps what you were taught wasn't long division. My kids were not taught the same math that I was taught in the 80s. My 17 year old was struggling with the division algorithm they were teaching in Elementary school at the time (late 2000s). I can't remember the details but it was something I had never seen before that was a step up from throwing poop at the wall. I showed her how to do long division and she still uses it to this day. It's simple and there is a history of work to easily identify errors. The homework scene from The Incredibles 2 really hit home for me. I feel like book publishers development new methods or ways to teach the methods as a means to sell new editions of books, not because the new ways are better. This causes a rift in society because people no longer share a common understanding of a subject.
- kasey_junk 7y agoConversely I have been immediately impressed with the way my first grader is being taught math. It’s completely different than the way I was taught so I have to invest in figuring it out but it’s led them extremely quickly through multiplication & division and into algebra. I’d characterize it as “multi-algorithmic”. They come at it from a variety of algorithms, assumedly so that each student can find one that works for them. As someone who had to teach themselves a different algorithm than I was taught many painful years later this resonates. Perhaps the rift was always there and you ended up in the lucky side of it?
- thawaway1837 7y agoI remember when the common core debates were taking place, one of the memes running in parenting circles was a comparison of long division vs how common core taught division, and how much more “complicated” the common core version was. But in reality the common core way explained the actual algorithm and the reason it worked so much better, and further, the only reason it appeared more complicated was because the people who were targeted by the meme already knew long division. If you took someone completely new to both methods, they’d find the more “complicated” algorithm far easier, because a large part of its complexity was explaining why the division worked, whereas long division was about memorizing a process.
- kasey_junk 7y agoI was taught math via spatial properties and it wasn’t until I came back to it from a logical perspective that I was able to understand it. Just another anecdotal data point but my experience was completely opposite yours.
- _bxg1 7y agoTrue; part of the problem is that everyone is different.
- dannypgh 7y ago> I still don't know why long-division works I was taught long division in the 4th grade (age 9) but wasn't taught a proof of why it works until a college algebra course (age 19). For whatever it's worth a different type of long division is taught now under common core. The common core math stuff that I've seen is impressive - there is more focus on teaching things in a way where comprehension of deeper concepts should be possible. It's a shame there's so much fear of it from parents who are upset kids are learning differently than they did.
- brianwawok 7y agoYes I love the Facebook memes by parents talking about how math is so complicated now. Most American adults struggle to solve a basic algebraic equation, because it’s not used in day to day life. If a new teaching style gives us a better way to teach concepts, seems worth a try..
- tasty_freeze 7y agoI think one reason why it doesn't make sense to a lot of people is that when doing long division, it is taught with notational shortcuts that obscure what the student is really doing. Eg, say you are dividing 76543 by 5. The normal procedure is to say something like: - 5 goes into 7 once. The first quotient digit is 1. - 1 times 5 is 5, write that under the 7, and draw a short line under the five - 7 minus 5 is 2; write that under the line you just drew - the next digit is 6, so write that next to the 2 you just drew, ie, 26 - 5 goes into 26 five times; put that up in the quotient, etc That minimizing the writing, but it hides what is really happening. Really what is going on is this. How many times can you subtract 5 from 76543? One procedure would be to subtract five over and over until the remainder is less than five. The number of times you subtracted is the quotient. But in this example you'd have to subtract more than 15000 times. Instead of subtracting 5 at a time and incrementing a count, it would be faster to subtract 50 at a time and increment your quotient by 10 until the result is less than 50, and then switch to subtracting by 5s and incrementing by 1 as before. If you can see why that works, then you realize it would be even faster to subtract by 500 at a time and increment by 100, then switch to subtracting 50 at a time and increment by 10, then subtract 5 at a time and increment by 1. Etc. So rather than minimizing the amount of digit copying, it would be better at first to write everything out. In this particular case: - write the dividend: 76543 - remove groups of 50000. we can see we can subtract 50000 from this once; the leading digit of the quotient is 1 - write 50000 (150000) under the 76543 - subtract, writing 26543 (this is the current remainder) - remove groups of 5000. now we see we can subtract 25000 from this, so the next digit of the quotient is 5. - write 25000 (55000) under the 26543 - subtract, writing 1543 (current remainder) - remove groups of 500. 500 goes into that 3 times, so the next digit of the quotient is 3 - write 1500 (3500) under 1543 - subtract, writing 43 (current remainder) - remove groups of 50. it is larger than the current remainder, so the next digit of the quotient is 0. - remove groups of 5. 5 goes into 43 eight times, so the next quotient digit is 8 - write 40 (85) under the 43 - subtract. the final remainder is 3. The sequence of quotient digits was 15308. The standard way long division is taught, the child skips writing down all the digits of the remainder and it is confusing as to why this all works.