Exactly. Math has historically relied on rote memory for most mental math. Kids would have to fill out their times tables, addition tables, etc until they memorized them. I still remember getting pop quizzes in elementary school that looked like this:
You only had two minutes to fill out the entire thing, which meant you only had 1.2 seconds per answer. You didn’t have time to actually calculate them. The point was that you were expected to have them memorized ahead of time instead of calculating each one.
But rote memory is laughably bad at actually teaching concepts. You may know that 12x5 is 60, but you don’t have any understanding on why, or other ways to do that same calculation without rote memory. And rote memory is only decently reliable up to ~12x12. Anything past that, and it becomes too much info to track; kids simply start forgetting answers.
The kids who were good at math (and I mean actually good at math, not just good at memorizing things) quickly devised methods to do this shit in our heads easily. Keeping track of multiple numbers in your head gets confusing. So “line them all up, add straight down, and carry 1’s” sort of falls apart if you’re doing it in your head. Especially if you’re trying to keep track of more than three or four numbers at a time.
Essentially, 127+248+30 is the same as 105+250+50, but the latter is much easier to parse in your head. But yeah, the parents (who primarily relied on rote memory) didn’t understand why the new method would be more effective, because they didn’t understand the concepts surrounding the math.
i used to hate the times table but i definitely think it’s essential to mental math. even if you vaguely remember it it will help. like knowing 42 shows up somewhere in the 7x and 6x may help you remember 6x7. or if you remember a neighbor you can just add or subtract the number once. for example if you don’t remember 7x6 it definitely helps to know each neighbor (both of which are easier to me since one is a 5x and one is a square number)… so either you think about 7x5 which is 35 so you can add another 7 to it or 6x6 which is 36 so you can add another 6 to it.
Oh I agree. My point wasn’t to say that rote memory is useless. I simply wanted to point out that it’s bad at teaching concepts. By teaching the concepts first, students are better prepared for later (more complicated) math courses. Anyone can memorize that 8x8 is 64, but understanding how to arrive at that answer is just as important.
I’m all for the multiple paths to solutions, but they aren’t even doing times tables these days. We drill it a little at home, but he struggled with just getting it memorized. I don’t know why they don’t drill a little. Honestly, they seem to have the kids sitting on the computer doing adaptive math most of the time.
I think it’s good to have a good set of these tables memorized and then based off those you can bounce your tricks. Eg if you know 5x12 by heart, you get 5x24 by intuition. Or even if you know 24/2 for that matter. I use simple examples but this could scale to less memorable numbers too.
It’s really helpful for quadratic factoring, too, since knowing at a glance that –56 is ±7 × ±8 keeps your working memory free to actually focus on the mathematical skills/concepts/problem.
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 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.
I only sort of agree. I still think that by forcing you to do that, by making you practice, makes the calculations “muscle memory” in that you aren’t memorizing the answers but can do the calculations faster and faster each time.
Sure. Some people could memorize them. But others will learn to calculate quickly.
It frees you up to do more complicated arithmetic. Geometry would be too slow if you didn’t innately understand 3*60=180. Which you don’t get without 3*6=18
Exactly. Math has historically relied on rote memory for most mental math. Kids would have to fill out their times tables, addition tables, etc until they memorized them. I still remember getting pop quizzes in elementary school that looked like this:
You only had two minutes to fill out the entire thing, which meant you only had 1.2 seconds per answer. You didn’t have time to actually calculate them. The point was that you were expected to have them memorized ahead of time instead of calculating each one.
But rote memory is laughably bad at actually teaching concepts. You may know that 12x5 is 60, but you don’t have any understanding on why, or other ways to do that same calculation without rote memory. And rote memory is only decently reliable up to ~12x12. Anything past that, and it becomes too much info to track; kids simply start forgetting answers.
The kids who were good at math (and I mean actually good at math, not just good at memorizing things) quickly devised methods to do this shit in our heads easily. Keeping track of multiple numbers in your head gets confusing. So “line them all up, add straight down, and carry 1’s” sort of falls apart if you’re doing it in your head. Especially if you’re trying to keep track of more than three or four numbers at a time.
Essentially, 127+248+30 is the same as 105+250+50, but the latter is much easier to parse in your head. But yeah, the parents (who primarily relied on rote memory) didn’t understand why the new method would be more effective, because they didn’t understand the concepts surrounding the math.
i used to hate the times table but i definitely think it’s essential to mental math. even if you vaguely remember it it will help. like knowing 42 shows up somewhere in the 7x and 6x may help you remember 6x7. or if you remember a neighbor you can just add or subtract the number once. for example if you don’t remember 7x6 it definitely helps to know each neighbor (both of which are easier to me since one is a 5x and one is a square number)… so either you think about 7x5 which is 35 so you can add another 7 to it or 6x6 which is 36 so you can add another 6 to it.
Oh I agree. My point wasn’t to say that rote memory is useless. I simply wanted to point out that it’s bad at teaching concepts. By teaching the concepts first, students are better prepared for later (more complicated) math courses. Anyone can memorize that 8x8 is 64, but understanding how to arrive at that answer is just as important.
I’m all for the multiple paths to solutions, but they aren’t even doing times tables these days. We drill it a little at home, but he struggled with just getting it memorized. I don’t know why they don’t drill a little. Honestly, they seem to have the kids sitting on the computer doing adaptive math most of the time.
yeah obviously, but i thought they were supposed to teach the times table after multiplication in general anyway
I think it’s good to have a good set of these tables memorized and then based off those you can bounce your tricks. Eg if you know 5x12 by heart, you get 5x24 by intuition. Or even if you know 24/2 for that matter. I use simple examples but this could scale to less memorable numbers too.
x5 has its own trick, for me it’s ÷2x10 (or x10÷2, whichever feels more intuitive) so 5x24 => 24/2 => 12x10 = 120
It’s really helpful for quadratic factoring, too, since knowing at a glance that –56 is ±7 × ±8 keeps your working memory free to actually focus on the mathematical skills/concepts/problem.
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 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 •
I only sort of agree. I still think that by forcing you to do that, by making you practice, makes the calculations “muscle memory” in that you aren’t memorizing the answers but can do the calculations faster and faster each time.
Sure. Some people could memorize them. But others will learn to calculate quickly.
It frees you up to do more complicated arithmetic. Geometry would be too slow if you didn’t innately understand
3*60=180. Which you don’t get without3*6=18