I 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.