Strongly disagree that memorization isn’t important. It’s THE foundation to be able to do effectively do more advanced stuff.
Take the equation (5678 • 9876). Use long multiplication and you only rely on doing a bunch of single digit multiplications and additions. It’s so much faster to be able to instantly know each step instead of having to recalculate these “atomic” steps again and again in your head.
You generally don’t need to be able to solve multiplications involving double digits in your head. It’s nice-to-have but otherwise useless, as long as you’re able to calculate the ballpark of the result.
For example, (38•63) is roughly 2400 and I can then calculate it on paper instead of in my head.
Head calculations are just so much more error-prone than written calculations. Don’t do them if you can avoid them. There’s a reason why math students (at a university) are infamous for being unable to make the simplest calculations in their head. It takes effort that could be spent somewhere else.
I would argue that memorization is important, but what you memorize and how you arrive at that is very personal. Forcing kids to memorize very specific things, and trying to enforce memorization (as opposed to the ability to arrive at the solution) seems like a bad idea to me.
I still don’t have the 10x10 multiplication table memorized, and I took physics in high school and work as a programmer. I have a use for knowing number multiples, and have domain-specific numbers memorized (2^8=8*8=256, 256*256=65536), but what I don’t remember off the top of my head I can figure out from the things I do know, from certain tricks, and from brute force mental math juggling numbers.
And the important thing to me is, I learned what I know not because somebody told me this is how I should do things, but because I picked them up as needed, a mix of memorizing common multiplications and figuring out tricks (like multiples of 9*N for N<11 being the digits N-1 and 10-N)
I strongly disagree that memorization is important or foundational to advanced math. It definitely is useful, but you don’t need it. And the more advanced your math gets, the less valuable it becomes.
My experience is that university-level math explicitly tells you to not memorize values and formulas, but to get comfortable finding solutions directly, because then you actually learn what is going on and have methods that are universally useful.
In the real world memorization is even less useful. You will never be as fast and accurate as a calculator, or remember as many values as a precomputed table has. So why bother?
I meant basic memorization, not any advanced stuff. If you have to re-derive everything basic from scratch again and again, you will be less effective at advanced stuff.
This is not to say the basic stuff should just be memorized. Rather, it should first be understood and only then be memorized.
And definitions must be memorized, otherwise you’re screwed. For instance, try proving something is a group if you forgot the definition of a group. Yes, the definitions have reason for being the way they are (which you will likely learn) but definitions just cannot be derived from your mind during an exam.
In OP’s example with memorizing multiplication tables instead of doing them on-the-fly: This is a core skill required for so much later on. You don’t want to waste time and energy thinking about how e.g. 7•8 = 7•2•4 = 14•4 = 14•2•2 = 28•2 = 56 because that’s a quick way to lose focus. Especially if you – like me btw – have to invert a 7x7 matrix with two variables x,y put in a bunch of positions (and linear combinations of them) in an exam.
Strongly disagree that memorization isn’t important. It’s THE foundation to be able to do effectively do more advanced stuff.
Take the equation (5678 • 9876). Use long multiplication and you only rely on doing a bunch of single digit multiplications and additions. It’s so much faster to be able to instantly know each step instead of having to recalculate these “atomic” steps again and again in your head.
You generally don’t need to be able to solve multiplications involving double digits in your head. It’s nice-to-have but otherwise useless, as long as you’re able to calculate the ballpark of the result.
For example, (38•63) is roughly 2400 and I can then calculate it on paper instead of in my head.
Head calculations are just so much more error-prone than written calculations. Don’t do them if you can avoid them. There’s a reason why math students (at a university) are infamous for being unable to make the simplest calculations in their head. It takes effort that could be spent somewhere else.
I would argue that memorization is important, but what you memorize and how you arrive at that is very personal. Forcing kids to memorize very specific things, and trying to enforce memorization (as opposed to the ability to arrive at the solution) seems like a bad idea to me.
I still don’t have the 10x10 multiplication table memorized, and I took physics in high school and work as a programmer. I have a use for knowing number multiples, and have domain-specific numbers memorized (2^8=8*8=256, 256*256=65536), but what I don’t remember off the top of my head I can figure out from the things I do know, from certain tricks, and from brute force mental math juggling numbers.
And the important thing to me is, I learned what I know not because somebody told me this is how I should do things, but because I picked them up as needed, a mix of memorizing common multiplications and figuring out tricks (like multiples of 9*N for N<11 being the digits N-1 and 10-N)
I strongly disagree that memorization is important or foundational to advanced math. It definitely is useful, but you don’t need it. And the more advanced your math gets, the less valuable it becomes.
My experience is that university-level math explicitly tells you to not memorize values and formulas, but to get comfortable finding solutions directly, because then you actually learn what is going on and have methods that are universally useful.
In the real world memorization is even less useful. You will never be as fast and accurate as a calculator, or remember as many values as a precomputed table has. So why bother?
I meant basic memorization, not any advanced stuff. If you have to re-derive everything basic from scratch again and again, you will be less effective at advanced stuff.
This is not to say the basic stuff should just be memorized. Rather, it should first be understood and only then be memorized.
And definitions must be memorized, otherwise you’re screwed. For instance, try proving something is a group if you forgot the definition of a group. Yes, the definitions have reason for being the way they are (which you will likely learn) but definitions just cannot be derived from your mind during an exam.
In OP’s example with memorizing multiplication tables instead of doing them on-the-fly: This is a core skill required for so much later on. You don’t want to waste time and energy thinking about how e.g. 7•8 = 7•2•4 = 14•4 = 14•2•2 = 28•2 = 56 because that’s a quick way to lose focus. Especially if you – like me btw – have to invert a 7x7 matrix with two variables x,y put in a bunch of positions (and linear combinations of them) in an exam.
Edit: substitute unescaped *s with •