Lattice multiplication - the usual name for that algorithm - is just about as old as the algorithm you're familiar with. I do know people, mostly people who suffer from various levels of dyscalculia, who find it easier than the other method. However, every single state - including Texas - requires students to also learn the standard algorithm (the one you're used to).
The goal in teaching different methods of multiplication (or addition, or subtraction) is to engage the students in thinking about why the methods work rather than simply memorizing algorithms, plugging in numbers, spitting out answers. Rote memorization without understanding... well, it works really well for a few years, and then kids hit a wall, usually either in the 4th grade or, if they have good memories and pattern recognition skills, around pre-algebra. There's just too much to remember, and it's too complicated. Conceptual understanding takes longer, but it's better.
I've spent a lot of time over the past 10 years talking to people on this issue. This may surprise you to hear, but a lot of people don't know why the algorithms they memorized work. They don't understand that 18 x 22 is the same as 18 x 20 + 18 x 2... even though they do that on paper every time they multiply! And consequently they also don't understand that it's the same as 20 x 22 - 2 x 22, or 9 x 11 x 4, or... well, I could go on. The point is, they don't really grasp what they're doing. They just memorized the steps, and they want their kids to memorize the steps. If these people make errors in their calculations, they often can't identify and fix the errors either.
It's the same with partial quotients division (which your video calls "big 7"). First, the video is quite correct, that algorithm is more optimized for mental math than the long division algorithm you're familiar with - and it's really quite easy once you're used to it. It hurts your head now only because you're unfamiliar with it. I know many people - and this time not just people with learning disabilites! - who prefer to use it the majority of the time, though, again, every state requires the teaching of the standard long division algorithm. (Which is good, as it's a necessary precursor for learning to extract square roots.) The partial quotients algorithm makes it clear what you're doing when you divide and why it works. Again, this may seem shocking, but I have met adults who didn't understand that when you divide 3 into 369 what you're really doing is dividing it first into 300, then into 60, then into 9 and adding those results together. If they made an error in their calculations they couldn't find and fix it, because they didn't understand it. And the long division algorithm is complicated! Memorization without understanding means you're more likely to make mistakes.
I think I'd be better off making lines in order to never have to know how to multiply ever again.
Again, you've got it backwards. We don't teach children to make arrays so they "never have to know how to multiply". We do it so they do know how to multiply. You can teach a crow to recite "two times two is four, two times three is six", but they won't understand it. Simply memorizing algorithms isn't understanding. Drawing an array makes the relationship between the symbols and the action explicit. Of course 3 x 4 = 12, because if you make three rows of four dots each, hey, you get twelve dots. You can think of this as training wheels. When your child first learned to ride a bike, you let him have training wheels. And then when he got experienced, he didn't need them. Well, arrays, and partial quotients, and the lattice method - these are training wheels for understanding. Once you understand, then you don't need to spend as much mental resources on memorization, which frees you up to do more complex arithmetic later.
It's like... it's like the difference between phonics and the old look-say method. A four year old can memorize the appearance of hundreds of words in a fairly rapid time and read any number of books with those words in them... but once they encounter unfamiliar words, they're stuck, and sooner or later they won't be able to memorize any new words. Or you can spend 100 hours laboriously teaching the 74 basic phonograms used in English. That takes longer. They won't be "reading" books in their first week, or even their first month. But once you've done it, they don't need to memorize any more random word shapes and can decode just about any English-language text that comes their way. For decades, America has been doing look-say math - memorize the algorithms, don't ask why. And it is crippling. This is the reason there are adults who go "I don't really understand fractions" or who don't comprehend why 12 x 3 must be the same number as 6 x 6 and 9 x 4 (a little more work with arrays would've fixed that up) or who can't see why 4 must go into 368 the same number of times that 2 goes into 184. They're stuck on arithmetic when they are all of them capable of at least handling statistics, basic algebra, and geometry. And why? Because at some point, somebody told them that they should just memorize the algorithms and not worry about why they worked.
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Date: 2018-06-25 12:53 am (UTC)The goal in teaching different methods of multiplication (or addition, or subtraction) is to engage the students in thinking about why the methods work rather than simply memorizing algorithms, plugging in numbers, spitting out answers. Rote memorization without understanding... well, it works really well for a few years, and then kids hit a wall, usually either in the 4th grade or, if they have good memories and pattern recognition skills, around pre-algebra. There's just too much to remember, and it's too complicated. Conceptual understanding takes longer, but it's better.
I've spent a lot of time over the past 10 years talking to people on this issue. This may surprise you to hear, but a lot of people don't know why the algorithms they memorized work. They don't understand that 18 x 22 is the same as 18 x 20 + 18 x 2... even though they do that on paper every time they multiply! And consequently they also don't understand that it's the same as 20 x 22 - 2 x 22, or 9 x 11 x 4, or... well, I could go on. The point is, they don't really grasp what they're doing. They just memorized the steps, and they want their kids to memorize the steps. If these people make errors in their calculations, they often can't identify and fix the errors either.
It's the same with partial quotients division (which your video calls "big 7"). First, the video is quite correct, that algorithm is more optimized for mental math than the long division algorithm you're familiar with - and it's really quite easy once you're used to it. It hurts your head now only because you're unfamiliar with it. I know many people - and this time not just people with learning disabilites! - who prefer to use it the majority of the time, though, again, every state requires the teaching of the standard long division algorithm. (Which is good, as it's a necessary precursor for learning to extract square roots.) The partial quotients algorithm makes it clear what you're doing when you divide and why it works. Again, this may seem shocking, but I have met adults who didn't understand that when you divide 3 into 369 what you're really doing is dividing it first into 300, then into 60, then into 9 and adding those results together. If they made an error in their calculations they couldn't find and fix it, because they didn't understand it. And the long division algorithm is complicated! Memorization without understanding means you're more likely to make mistakes.
I think I'd be better off making lines in order to never have to know how to multiply ever again.
Again, you've got it backwards. We don't teach children to make arrays so they "never have to know how to multiply". We do it so they do know how to multiply. You can teach a crow to recite "two times two is four, two times three is six", but they won't understand it. Simply memorizing algorithms isn't understanding. Drawing an array makes the relationship between the symbols and the action explicit. Of course 3 x 4 = 12, because if you make three rows of four dots each, hey, you get twelve dots. You can think of this as training wheels. When your child first learned to ride a bike, you let him have training wheels. And then when he got experienced, he didn't need them. Well, arrays, and partial quotients, and the lattice method - these are training wheels for understanding. Once you understand, then you don't need to spend as much mental resources on memorization, which frees you up to do more complex arithmetic later.
It's like... it's like the difference between phonics and the old look-say method. A four year old can memorize the appearance of hundreds of words in a fairly rapid time and read any number of books with those words in them... but once they encounter unfamiliar words, they're stuck, and sooner or later they won't be able to memorize any new words. Or you can spend 100 hours laboriously teaching the 74 basic phonograms used in English. That takes longer. They won't be "reading" books in their first week, or even their first month. But once you've done it, they don't need to memorize any more random word shapes and can decode just about any English-language text that comes their way. For decades, America has been doing look-say math - memorize the algorithms, don't ask why. And it is crippling. This is the reason there are adults who go "I don't really understand fractions" or who don't comprehend why 12 x 3 must be the same number as 6 x 6 and 9 x 4 (a little more work with arrays would've fixed that up) or who can't see why 4 must go into 368 the same number of times that 2 goes into 184. They're stuck on arithmetic when they are all of them capable of at least handling statistics, basic algebra, and geometry. And why? Because at some point, somebody told them that they should just memorize the algorithms and not worry about why they worked.