I guess that greatly depends on your teacher. However, I will say that “doing the numbers” and understanding are pretty strongly correlated in math. BTW the same goes for English literature where reading more books greatly increases your understanding.
it’s a different kind of understanding though. also, vocabulary in school is always presented in context, while mathematics usually isn’t, save for contrived examples, because you can’t gradually introduce stuff the same as with language.
like, i never got an intuition for division. i have to brute-force it every time. during school i would ask for help and nobody else seemed to get it either.
Edit:
what i wanted to say wasn’t entirely clear, so let’s try again:
doing the numbers is only useful when you are working towards understanding. at least when i was in school, after an intro to multiplication, the table for e.g. 7 was presented “without comment”: we were to fill it in while timed, and if we did it quickly enough we were considered to have “learned” it, and got to advance to 8.
I think your example with the multiplication tables is a great one. It is important for students to have a understanding of what multiplication is both as a building block of more complex math, and because multiplication is one of the most practical skills we learn in school. Having said that, rote learning of multiplication tables is also a useful skill. By learning the multiplication tables you free up cognitive resources when learning something more complex.
i don’t know about that, i would prefer to build an intuition. i know people who simply have the entire thing memorized and “look up” the answer when prompted. which of course completely breaks down if you introduce an operand higher than 12.
You need both. Take 1718. Your understanding of multiplication should tell you that this equals 1010+107+108+8*7. Now your rote learning will allow you to calculate this quickly as 100+70+80+56=306.
and no, my rote learning has not prepared me for that. nothing like that was ever presented to me. i went from multiplication tables to factorisation and never mentally connected the two. as a result i can’t do factorisation in my head at all, despite doing 80% of a master’s in engineering.
Yeah your education failed you. What I am effectively doing is “factoring” 17*18 into (10+7)(10+8), before working out the parentheses, but it’s easier because you only work with numbers and not with x’s. A nice in-between step towards algebra.
yeah that’s what’s so interesting. like obviously i can see the steps with actual numbers but replace with unknowns and it’s 50-50 whether i would be able to do it. and since i work with optimisations every day, i have had to reconstruct this stuff from first principles without a theoretical understanding so i need to go the long way around every time.
I guess that greatly depends on your teacher. However, I will say that “doing the numbers” and understanding are pretty strongly correlated in math. BTW the same goes for English literature where reading more books greatly increases your understanding.
it’s a different kind of understanding though. also, vocabulary in school is always presented in context, while mathematics usually isn’t, save for contrived examples, because you can’t gradually introduce stuff the same as with language.
like, i never got an intuition for division. i have to brute-force it every time. during school i would ask for help and nobody else seemed to get it either.
Edit:
what i wanted to say wasn’t entirely clear, so let’s try again:
doing the numbers is only useful when you are working towards understanding. at least when i was in school, after an intro to multiplication, the table for e.g. 7 was presented “without comment”: we were to fill it in while timed, and if we did it quickly enough we were considered to have “learned” it, and got to advance to 8.
I think your example with the multiplication tables is a great one. It is important for students to have a understanding of what multiplication is both as a building block of more complex math, and because multiplication is one of the most practical skills we learn in school. Having said that, rote learning of multiplication tables is also a useful skill. By learning the multiplication tables you free up cognitive resources when learning something more complex.
i don’t know about that, i would prefer to build an intuition. i know people who simply have the entire thing memorized and “look up” the answer when prompted. which of course completely breaks down if you introduce an operand higher than 12.
You need both. Take 1718. Your understanding of multiplication should tell you that this equals 1010+107+108+8*7. Now your rote learning will allow you to calculate this quickly as 100+70+80+56=306.
you’ll need to escape the asterisks:
\*and no, my rote learning has not prepared me for that. nothing like that was ever presented to me. i went from multiplication tables to factorisation and never mentally connected the two. as a result i can’t do factorisation in my head at all, despite doing 80% of a master’s in engineering.
Yeah your education failed you. What I am effectively doing is “factoring” 17*18 into (10+7)(10+8), before working out the parentheses, but it’s easier because you only work with numbers and not with x’s. A nice in-between step towards algebra.
yeah that’s what’s so interesting. like obviously i can see the steps with actual numbers but replace with unknowns and it’s 50-50 whether i would be able to do it. and since i work with optimisations every day, i have had to reconstruct this stuff from first principles without a theoretical understanding so i need to go the long way around every time.