A lot of that stuff is about internalizing rules. By doing times tables up to 10x10, you have just enough memorized to understand patterns of multiplication, how it behaves and how to manipulate it. By working out long hand you understand the patterns of positional notation and some mechanisms to manipulate it. Division long hand also gives you an opportunity to experience how division is the inversion of multiplication, as long hand division is literally running long hand multiplication backwards (it’s trickier cause you are more likely to run into a fraction tho) - and the concept of modulo is also incredibly useful for daily life.
Even these things are building foundations for later math. Times tables and long hand division aren’t the only way to teach math, but they are a way that works. It’s not directly relevant, sure, and hasn’t been for almost 100 years. But it’s a foundation and has importance in that way
A lot of that stuff is about internalizing rules. By doing times tables up to 10x10, you have just enough memorized to understand patterns of multiplication, how it behaves and how to manipulate it. By working out long hand you understand the patterns of positional notation and some mechanisms to manipulate it. Division long hand also gives you an opportunity to experience how division is the inversion of multiplication, as long hand division is literally running long hand multiplication backwards (it’s trickier cause you are more likely to run into a fraction tho) - and the concept of modulo is also incredibly useful for daily life.
Even these things are building foundations for later math. Times tables and long hand division aren’t the only way to teach math, but they are a way that works. It’s not directly relevant, sure, and hasn’t been for almost 100 years. But it’s a foundation and has importance in that way