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What is something other people seem to grasp easily that eludes you? (lemmy.world)
submitted 2 days ago by to c/asklemmy@lemmy.world
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For me it's time signatures in music. I grew up playing instruments, played in several orchestras, etc. But no matter how many times someone tries to explain time signatures to me, it makes no sense. I don't understand why people place such importance on it. To me, notes have a pitch and a duration, maybe some dynamics. That's it. I don't understand why people try to needlessly complicate it. It doesn't give you anything.

My wife will hear a song and go "Oh neat, it's in 5/8." And I'm like "How tf do you know that, and why does it matter?"

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[–] 6 points 2 days ago (19 children)

division. i can do basic fractions but anything more complicated feels completely unintuitive.

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  • [–] 11 points 2 days ago* (16 children)

    Huh. It sounds as if maybe you missed a basic building block as a child.

    Once you're missing a foundational concept, everything after that is difficult. It's sadly not uncommon - maybe you were ill that week, or you had a teacher that couldn't explain in a way that worked for you, etc.

    My work is in maths and stats, my husband is a lawyer, so when he did an introductory finance course a few years ago, I tried to help him understand some formulas that involved percentages. It was an unmitigated disaster. So I called my mom (a retired math teacher) and she started by drawing a circle and saying this is a pie, we cut it into slices, etc. like you would for a young child. At first I was horrified, but she kept building on it, checking he understood at each point, until they reached a point where he wasn't going, "yes, yes, OK". She explained that piece to him and something clicked. Problem solved.

    Anyway, that was a long anecdote to tell you, you probably just need to find a good maths teacher who is willing to spend a couple of hours with you building up, until you find that gap. I could ask my mom if you like? She's genuinely a really good teacher.

    Edit to add: Alternatively, go through the Khan Institute lessons. Skim them until he starts doing something that isn't obvious. Then go back and watch the previous lesson.

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  • [–] 5 points 2 days ago (15 children)

    i'm a senior computer engineer currently specialising in signal processing algorithms on constrained systems 🙃 so i have really tried getting that intuition to stick, including the khan stuff, for years, but no. i have to double-check everything i write out because i can't visualise it as soon as there's division involved, unless it involves powers of two.

    it genuinely is as you said, it feels like i was ill for a week in middle school and suddenly i didn't understand anything. and now i don't have the energy to do studying on my off hours. thanks for the offer though, that's very sweet.

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  • [–] 3 points 1 day ago* (6 children)

    Hmm. I know the point of this thread wasn't supposed to be a bunch of strangers info-dumping your weaknesses at you, but I'd like to take a crack at this one 😉

    Problem one may just be that you overestimate how much the rest of us "intuit" or "visualize" division. It may just be that the rest of us use the tool for its outcomes rather than because it feels right, as opposed to visualizing it. Maybe it's just the language you chose to use, and it doesn't imply what I said, but it does make me very curious how you visualize addition and subtraction 😛

    Okay, that aside, division is the opposite of multiplication, that's the thing it does, but it's also how we reconstruct its algorithm. Pies and their slices are fine for fractions, but here we're talking about division. So instead I'm going to start with dealing cards. Each card will be a ©, and we'll start with 15 of them, and we have 4 players.

    ©©©©©©©©©©©©©©©

    Okay, first round, we deal out the first 4, one to each player

    © © © ©

©©©©©©©©©©©

    We have 11 remaining. Next round:

    © © © ©
© © © ©

©©©©©©©

    2 to each player so far, 7 remain. Round 3!

    © © © ©
© © © ©
© © © ©

©©©

    3 to each player in 3 rounds, and we can't do another round because we have fewer than 4 remaining. Specifically 3. So we could now answer that 15 divided by 4 is "3 remainder 3", because that's how many we get per player when there are 4 players, and then there's 3 left over.

    The reason I've constructed it this way is because we've reconstructed the primitive square intuition of multiplication, but from the other direction! I think we can see how 4×3 makes 12, and then also how we did the dividing by re-distributing the first 12 cards of our 15 into 4 columns to work out how many rows our square would have needed to be to get us to 12. We knew we were interested in 4 columns, that's our problem setup, but we didn't know how many rows that meant, so we divided them literally amongst the columns until we ran out. Then we looked at the square we'd built and said "aha, I guess it was 3 rows!"

    To put it in algebra format and play around a little:

    15 = 12 + 3
15 = 3×4 + 3
15 - 3 = 3×4
12 = 3×4
12 ÷ 4 = 3

    Just playing with division being the opposite of multiplication. Maybe that didn't help, in which case ignore it and go back to the cards thing! 😅

    Okay, so that's algorithm number 1, and it works every time. With integers... So what the heck is long division?

    Well basically it's a heuristic algorithm on top of this attempt to make squares out of numbers, where we want to use our intuition about multiplication, because it's easier, in order to do division. And basically the only reason to do that is because dividing each number by getting a set of counting stones in that quantity and divvying out a square is time consuming and we're busy people.

    So we're doing 15÷4. We bascially guess how many rows we think this will have, by our intuition about the 4s times-table. We go "okay, maybe 2? 4×2=8, 15-8=7, that's bigger than 4, so I underestimated, but not by much. So I try 3, and that gets us our 4×3+3 from before. If we'd started with a guess of 4, we'd have got 16 and known it was too high and come down to 3. It's about guessing how many rows our square would have if we divvyed things out like before, but hopefully without having to actually do that work. And we check that guess by calculating the remainder and making sure it's smaller than the number of columns in our square, because if it wasn't then we could go another row.

    Okay, now what about long long division. Like 1573÷48?

    Just like with long multiplication, we want to leverage our intuition about smaller numbers to break down bigger ones. So we're going to break down 1578 into 157×10+3. Then we divide 157 by 48. I don't know about you, but I'm not great at my 48 times tables, but 48 is near 50 and 157 is near 150, so I'm going to guess 3. 48×3 is 144, 157-144 is 13, so we did good. 157÷48=3R3, or 157=48×3+3. That's great! But it wasn't the problem we were ultimately trying to solve. 😛 So we need to put back what we took off before, meaning our running answer so far isn't 3, it's 30. And our remainer isn't 13, it's 133, because we broke 1578=157×10+3 before, so 13×10+3 is 133.

    So now we have a smaller new problem! 133÷48. Again, I'm going to go on my "about 50" intuition from before and say 2. 2×48=96, 133-96=37. So far so good! So now we add our 2 to our answer of 30 so far and get 32, and our remainder is 37. And, in this case, we're done! So 1573÷48=32R37.

    An interesting property of this process, is to check our work, you can long-multiply 48×32 and you'll notice some old friends. You'll do 2×48 and get 96, you'll do 3×48 and get 144 which due to position is actually 1440, and you'll add those together to get 1536. Then you can add 37 to that to get 1573!

    And one last step here. By generalizing our division algorithm to be based on multiplication and subtraction, we can extend this process to non-integer numbers in a way that's difficult with our "card dividing" method. We can multiply and subtract non-integer numbers already, after all. And even more than that, if we want to get rid of our tidy remainders and replace them with dirty dirty decimal places, we can continue to extend this process infinitely! Instead of stopping at remainder 37 we can extend our question to 1573.0, convert our remainder to 370, and then ask how many times 48 goes into that, get 7R34, but remember it's not actually 7, it's 0.7, so our total so far is 32.7 and our remainder is not 34 but 3.4, and if we check we can see that 48×32.7+3.4 is in fact 1573.0!

    And if we want to eliminate that remainder altogether we can just keep adding new zero deicmal places to the right, and keep working our remainder smaller and smaller until it's either perfectly zero, or we decide it's small enough that we're going to round it to zero from here. Or maybe we get repeating digits, which is a story for another time! 😉

    So, I don't know if any of that helped at all! But I enjoyed typing it, so thanks for reading! Also, my "long division" section was done conversationally rather than in the typical "frame" that long division goes in. That's because I was explaining my thought process, but the frame is doing the same thing, it's basically just there to help you keep track of where you're at in the problem, which deicmal place everything is aligned to, etc. But it's not essential to the process! There's no magic, besides algebra I guess 😛

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  • [–] 1 point 1 day ago (4 children)

    holy shit, now that's a post. very well written, easy to follow. nice.

    unfortunately, what you've described is basically what i've been doing, which is way more steps than i tend to need with the others. i don't know if it exists in other places but we had an entire module in middle school on "överslagsräkning", which basically translate to "rough calculations". it was about using iterative methods like this to land at a plausible solution to a problem.

    unfortunately, i mostly need actual solutions. as an example i'll tell you what i did today.

    today, i tried to refine an algorithm that, given a set of points (represented by a c array) and a "fractional index" (a positive non-integer value) can give result for a number between two defined points. i had previously done this through simple linear interpolation, but since we sample data from actual physical objects the points tend to represent a curve, and the common way to solve that is with a lagrange polynomial. so, i found an algorithm that, given a set of n + 1 points, evaluates the unique lagrangian of at most degree n that intersects all the points at the given index. i split the terms out to an (a + b) / c form and then i realised that i have no fucking idea what i'm doing.

    like yes, i can run the algorithm to see if the numbers line up, but i could never actually do the math. i'm not talking about the algebra part, i'm talking about the arithmetic. if i plug some numbers in and do the parts i can do, i end up with something like 0.23 / -7 and i don't even know where to start. i have no idea where that would even land. so, while i can verify that my test data works, i have no way to quickly extrapolate numbers. which in turn means i can't come up with plausible input. i'm back to using a calculator for that, which is so damn annoying.

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  • [–] 2 points 1 day ago* (3 children)

    Ah! I think I see what you mean. Well, I night. You mean you don't have a rough sense of what to expect of your answer without just computing it? I'm not 100% sure I do either 😛

    But there's some things we can maybe do. The easiest one, in this case, is that we can temporarily play with orders of magnitude and put on a factor of 100 to get 23/7 = ~3 which means when we take that 100 back off your answer is going to be roughly -0.03. And again, that answer comes from leveraging multiplication insights, to aid division. 23 looks like 21, and I know 21 is a multiple of 7.

    Any magnitude we add to our numerator is countered by any magnitude we add to our denominator. So if we had 0.23/73, we could multiply by 100 again to get 23/73, and then multiply by 10 again to get 230/73, but we can also equivalently reduce the denominator by a factor of 10, giving us 23/7 ish. We've actually moved by 3 orders of magnitude to get there, so our new "gut" answer is -0.003 ish.

    That's my first guess of what you meant. But you did say you needed "actual" solutions. So maybe you didn't mean you wanted an intuition about approximations, and you actually wanted to do the long division?

    Just in case, I'll do another worked example of what you might do on paper if you didn't want to use a calculator.

    0.23÷7. 7 goes into 0 zero times, easy, that's our ones place solved. 7 goes into 02 zero times as well, so our answer so far is 0.0 and we're moving on to hundredths. 023 is larger than 7, and as before it is recognizably near to 21, a multiple of 7. 7×3 is 21, so our answer so far is 0.03, and our remainder is 23-21=2 (which is actually 0.02, but we keep multiplying things by 10 during the algorithm so we don't have to worry about that). Okay, we move another order of magnitude down and have 20÷7, which again looks a lot like 21, but smaller so we know it's 2. 0.032 and a remainder of 20-14=6. 60÷7 is 8, so 0.0328 and 60-56=4. 40÷7 is 5, 0.03285 with remainder 40-35=5. 50÷7 is 7, so 0.032857 and remainder 50-49=1. 10÷7 is 1, 0.0328571, with 10-7=3. 30÷7 is 4, 0.03285714, remainder 30-28=2. And now we're in trouble. Because we've already seen 2. 20÷7 is 2, remainder 6 again. So it just loops now!

    -0.032857142857142857142.....

    And that's less of a division problem and more of a problem with decimal numbers rather than fractions. It's simply a limitation of the way we write things. So there isn't an exact numerical answer for this particular division, you just have to decide where is enough decimal places for your application.

    I don't know if any of that helped at all... but it's an attempt!

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  • [–] 1 point 1 day ago (2 children)

    it's a good attempt too, because i had completely forgot how to do that (mostly because we were taught something called the "chair method" for long division which tends to take up entire pages) and it reminded me of another thing: times tables.

    whet i was in school, we got these arithmetic worksheets with times tables on them once a week, (one meek it was 4×n, then 5, then 6...) and we were supposed to do them while being timed. iirc we had five minutes at the start of maths. i don't think i have to tell you how stressful that is with no intuition as well as fine motor issues. anyway, point being, i don't remember ever doing one for division. we went from pieces of pies, to the chair method, to short division problems. now, i've since learned that apparently the worksheet method isn't a good pedagogical tool, but it still feels like we skipped a step.

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  • [–] 2 points 1 day ago (1 child)

    For sure, yeah. I think that honestly division just isn't as intuitive, for people. It's weird. But you're right, maybe none of it is as intuitive as it feels and it was just the school hammering things in 😛

    Like, a classic example, it feels intuitive that 1/3 + 2/6 = 3/9. I know those are fractions and not division, but I think it goes together. And it makes sense why it doesn't work that way if you really dig into it, in this particular example all three of those fractions are actually ⅓, so it's a+a=a, which is only true for a=0. You can draw pictures of pies and stuff, but I do agree it doesn't feel quite as "natural" as addition or multiplication. There is also a giant hole in it with division by zero, which isn't the only time the math runs into and edge, but it is the first one people encounter...

    Well, it's been fun chatting about math with you anyway!

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  • [–] 1 point 1 day ago

    Oh oh, one more thing that may help the long-division process seem less mystical is to realize you can stop anytime. Like, when I did the first pass and got an answer of just 30 and a remainder of 377, we could stop there!

    48×30+133 = 1573

    That's true. So we could say the answer is 30R133. It's just that typically we want the remainder to be below 48, because we have an answer that isn't as accurate as it could be. But there's nothing magic about going down to the zero place and not extending into the decimal places like tenths and hundredths. If we only cared about the accuracy of our answer to the nearest hundred, we could find that our answer is 30ish and be happy, and just ignore the remainder from that right away. The process of long-division is constantly about refining an answer, and gives true but maybe imprecise answers all the way along.

    Viewed from that lens our answers actually went through this chain:

    1573 = 0×48+1573 (0R1573)
1573 = 30×48⁺133 (30R133)
1573 = 32×48+37 (32R37)
1573 = 32.7×48+3.4 (32.7R3.4)
etc

    Don't know if that helps demystify it further!

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  • [–] 2 points 1 day ago (7 children)

    i’m a senior computer engineer currently specialising in signal processing algorithms on constrained systems

    Ah. Hmmm. I mean, division is just multiplication turned on its head, so maybe it would help to start with how you visualise multiplication, and see whether you can extend that to work for division too? 🤷‍♀️

    If it makes you feel any better, my arithmetic is horrendous. I understand complex mathematical concepts easily, but don't ask me to add 2 big numbers. It's ridiculous 😂

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  • [–] 1 point 1 day ago* (6 children)

    you know cuisenaire rods? those counting stick things. i think in those. that means addition makes a line, multiplication makes a plane, exponentiation makes a cube. higher exponents make cubes of cubes.

    division doesn't really fit in that visual because it's a quantised system. fractional multiplication fails for the same reason.

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  • [–] 1 point 1 day ago (5 children)

    Division is taking a plane and breaking it into equal lines. I assume when you multiply you are already giving that plane bounds because anything else wouldn't give you a result. It takes a few passes to go the other way, but your facility with powers of two can help you narrow it down quickly enough. You're just trying to find out how long the box of a certain width has to be to fit a given number of blocks.

    Maybe it helps, maybe it doesn't. It's surprising how many ways people have found to achieve the same goal.

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  • [–] 2 points 1 day ago (4 children)

    yeah i understand the principle, but the issue is i have to "go through the motions" every time. i can't do a quick ballpark for division in the same way as for other arithmetics so i can't self-correct.

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  • [–] 1 point 1 day ago (3 children)

    I'm not sure if that's a familiarity issue or an intuitiveness issue. I'd say I'm very good at math, but I'm also weakest at division mentally. I think part of it is we come across the other operations more frequently in life and our brains are more tuned for them, whether that be nature or nurture. I think the only way to change that is practice.

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  • [–] 2 points 1 day ago (2 children)

    the question is where to start practising. i am absolutely terrible at most math; i can't balance an equation to save my life, i tend to get all my trig backwards, and the only thing i consistently get right for calculus is that odd-power exponentials are assymmetrical. despite that i struggled my way up to differential equations and found that annoyingly, combinatorics and boolean algebra came naturally.

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  • [–] 2 points 1 day ago (1 child)

    you are describing my experience with math almost 1:1, it's uncanny... division is unintuitive, i've re-learned calculus from scratch like four times during my school career and none of the actual math made intuitive sense, trig never clicked. i've been doing computer graphics for like 6 years now and despite trig being everywhere i still need like 30 seconds to puzzle out what sin(0) is for the thousandth time. but i aced combinatorics without even trying as soon as i heard that the funny fraction-looking thing with parentheses is called "n-choose-k", and if you give me a boolean expression with 5 levels of nested parenthesis and i can simplify it in my head in seconds.

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  • [–] 2 points 1 day ago

    i know, right?! like, a friend of mine was doing a homebrew computer project a year or so ago and had issue with some complex instruction logic in the alu or whatever, and without having done it for over a decade i just sketched up a karnaugh map and struck two terms out. i even confused myself with that one.

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