Applied math in daily life, such as making change, also involves a lot of memorization, it's just that the person doesn't realize they're committing various formulas and equalities to memory.
The problem with proficiency in, e.g., making change, is that it doesn't carry over to higher-order math, logic, and reasoning. Arithmetic as taught in elementary school is attempting to achieve two things at once: proficiency in applied arithmetic, and foundational number theory (e.g. commutativity).
My mother was a waitress and emphasized skills like making change. While she never articulated the rules, I ended up developing many of the mental arithmetic techniques that (I later discovered) Isaac Asimov discussed in Quick and Easy Math: https://archive.org/details/QuickAndEasyMath-English-IsaacAs... But I never developed an appreciation for number theory until it was too late--i.e. after high school. I did get into philosophy and logic during high school, but the connection (theoretical and applied) between the two didn't click until later.
Sadly (or not?), proficiency in mental arithmetic has become much less common even among waiters, clerks, etc, at least where they don't deal in cash directly and without the aid of a register. And professional mathematicians have always humble-bragged about their impoverished mental arithmetic skills. So maybe we should drop the pretense that we're attempting to teach applied mathematics in the early years and admit the purpose is to lay theoretical foundation for higher-order math, applied and theoretical.
I only learned this in adulthood, too. I had to discover and memorize those shortcuts. Now I know that some of the "brilliant" math students I went to high school with had simply learned these skills the rest of us didn't.
The problem with proficiency in, e.g., making change, is that it doesn't carry over to higher-order math, logic, and reasoning. Arithmetic as taught in elementary school is attempting to achieve two things at once: proficiency in applied arithmetic, and foundational number theory (e.g. commutativity).
My mother was a waitress and emphasized skills like making change. While she never articulated the rules, I ended up developing many of the mental arithmetic techniques that (I later discovered) Isaac Asimov discussed in Quick and Easy Math: https://archive.org/details/QuickAndEasyMath-English-IsaacAs... But I never developed an appreciation for number theory until it was too late--i.e. after high school. I did get into philosophy and logic during high school, but the connection (theoretical and applied) between the two didn't click until later.
Sadly (or not?), proficiency in mental arithmetic has become much less common even among waiters, clerks, etc, at least where they don't deal in cash directly and without the aid of a register. And professional mathematicians have always humble-bragged about their impoverished mental arithmetic skills. So maybe we should drop the pretense that we're attempting to teach applied mathematics in the early years and admit the purpose is to lay theoretical foundation for higher-order math, applied and theoretical.