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 😛