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The multiplication table doesn't have patterns, or it only has a few. You really do need to remember all of the 100 results. I know what 7*8 is, and I know the

The multiplication table doesn't have patterns, or it only has a few. You really do need to remember all of the 100 results. I know what 7*8 is, and I know the rules for exponents, so I can compute 7e10*8e11. But I can't "deduce" what 7*8 is by any rule, it's just a fact I remember. I have certainly not added 7 to itself 8 times in decades.

LouisSayers 2y ago
> I can't "deduce" what 7*8 is by any rule But you can break this into a different problem knowing that 2^3 = 8, and doing 7*2*2*2. This isn't as fast but is in a way more useful because while 7*8 is fairly easy to remember you're not going to remember 17*8 etc but you can problem solve it fairly quick. There are other ways of seeing the multiplication table as well. For example 9 times something can be thought of as 9*x = 10*x-x. I never learnt these, but simply realised over time that there are different approaches to doing calculations.
- Dylan16807 2y ago
  > But you can break this into a different problem knowing that 2^3 = 8, and doing 7*2*2*2. Doing that multiplication all the way through is super slow. When they said "can't" they meant in an effective sense, since they did mention repeated addition as an option. And that's not an effective way to get there. > There are other ways of seeing the multiplication table as well. For example 9 times something can be thought of as 9*x = 10*x-x. Yes, you can do that one. But that's just about the only fast trick there is.
  - wizzwizz4 2y ago
    Multiplication and division by 2 and 5 correspond to each other. Multiplication by 11 is shift-and-add. There are many such tricks.
    - Dylan16807 2y ago
      The first one doesn't help with remembering times tables. The second one is outside the 1-10 times table we were talking about.
      - wizzwizz4 2y ago
        6 is 2×3, so 6×5 = 30. Likewise, 17 = 2×8 + 1, so 5×17 = 85.
  - lelanthran 2y ago
    > Yes, you can do that one. But that's just about the only fast trick there is. I dunno about that. For division, anyway, there's a bunch of fast tricks that give you a decent approximation (i.e. decent precision, maybe to the nearest integer) Someone recently was surprised that I worked out the VAT (Value Added Tax, 15%) on a very large number in a few seconds. It's because its 10% of the number plus `(10% of the number)/2`. It's easy to get 10% of any number. It's easy to halve any number. It's a fast trick because there's two easy operations. There's a bunch of similar "tricks": 1%, 10%, 25% and 50% are fast to calculate in your head (at most 2 easy operations, like `(half of half of N)`). Then you either add or subtract N. Or you multiply N by 2. At most three easy operations gives you 1%, 2%, 4%, 5%, 10%, 11%, 12%, 14%, 15%, 20%, 21%, 24%, etc To someone who doesn't know how you are getting the answer it might seem like you are a human calculator because you get so many of those quickly, and they don't see the ones you don't do in 3 easy operations (say, 13%, which is 10% + 1% + (1% * 2)). IOW, it looks like a very impressive trick, but it isn't.
    - Dylan16807 2y ago
      > For division, anyway Did you not see the neighbor comment where I made it extra clear I'm talking about times tables in particular? There are tons of math tricks! But the multiplication table of the numbers between 1 and 10 is mostly rote.
- tsimionescu 2y ago
  I completely disagree. First of all, at the time children learn the multiplication table, they definitely don't know the concept of exponentiation. Secondly, 7*2*2*2 is not some immediately obvious shortcut. Also, learning multiplication with numbers higher than 10 still relies on knowing the multiplication table. 17*8 is 7*8=56, hold the 5, 1*8 + 5 = 13, so 136.
  - LouisSayers 2y ago
    > Also, learning multiplication with numbers higher than 10 still relies on knowing the multiplication table You've actually just proved my point - you used a method of breaking down the problem into a different problem and then solving it rather than simply memorising. If you give the same question to multiple people there will be numerous ways different people use to go about solving it. As an example, I might solve this by doing 20*8 = 160 3*8 = 24 160 - 24 = 136 Or 10*8 = 80 7*8 = 56 80+56 = 136 And I might apply different tools like the one I originally mentioned within these calculations. I know that 80+20 is 100 and so "borrow" 20 from 56, so that I can easily add 100 and 36 together. These ways of calculating happen in your mind very quickly if this is how you get used to calculating.
    - tsimionescu 2y ago
      Sure, but all of those work for numbers higher than 10, and all assume you know the multiplication table by heart. The multiplication table (the result of multiplying every number between 1 and 10 with each other) is something you have to memorize. You can get away with memorizing only some of these results and computing the others based on them, but it's basically impossible to do any more complex arithmetic if you don't know most of it by rote memorization.
  - lelanthran 2y ago
    > Also, learning multiplication with numbers higher than 10 still relies on knowing the multiplication table. 17*8 is 7*8=56, hold the 5, 1*8 + 5 = 13, so 136. Interestingly, I do those less efficiently: 17 * 8 = (10 * 8) + (7 * 8) = (80) + (56) = (80) + (50 + 6) = 130 + 6 I think the reason I do it this way is because I get an approximation sooner when the numbers are very large i.e. I get the most significant digit first, and can stop calculating when I get the precision I require.*
jltsiren 2y ago
Of course there is a pattern: (n+1)x = nx+x. Your brain can learn it just fine, and then it can multiply numbers without having to burden your slow inefficient symbolic reasoning machinery with rules and facts.
- tsimionescu 2y ago
  How is that pattern useful for replacing memorization of the multiplication table? 7*8 = 6*8 + 8 - fine. I still need to memorize what 6*8 is, or go through the extraordinarily slow process of expanding this recursively: definitely not an option in school.
  - jltsiren 2y ago
    It's a linear function your brain can learn. Your brain, not the conscious you. A lot of learning is about bypassing the slow inefficient consciousness that thinks it's in charge. In sports and other physical activities, you don't memorize the right moves. You practice them until you can do them automatically. The same approach also works with cognitive activities.
    - tsimionescu 2y ago
      If that were how people learned the multiplication table, you would see that people take longer to come up with the result of 9*9 than it takes them to compute 3*5. I have never seen anyone work this way, and so I believe it's far more likely people just remember the results in a table. "9*9=81" is simply a fact you have stored, not a computation you perform, even subconsciously. Edit: I should also note that it's pretty well known people learn arithmetic as symbol manipulation and not some higher order reasoning. The reason this is pretty well established is that historically, the switch from Roman numerals to Arabic numerals led to a huge flurry of arithmetic activity, because it was so much easier to do arithmetic with the new symbols. If people had learned by subconsciously calculating the underlying linear functions and not through symbolic manipulation, the switch would have been entirely irrelevant. Yet for most mathematicians in Europe at the time, doing 27*3 was much easier than doing XXVII*III.

LouisSayers 2y ago

> I can't "deduce" what 7*8 is by any rule But you can break this into a different problem knowing that 2^3 = 8, and doing 7*2*2*2. This isn't as fast but is in a way more useful because while 7*8 is fairly easy to remember you're not going to remember 17*8 etc but you can problem solve it fairly quick. There are other ways of seeing the multiplication table as well. For example 9 times something can be thought of as 9*x = 10*x-x. I never learnt these, but simply realised over time that there are different approaches to doing calculations.

Dylan16807 2y ago
> But you can break this into a different problem knowing that 2^3 = 8, and doing 7*2*2*2. Doing that multiplication all the way through is super slow. When they said "can't" they meant in an effective sense, since they did mention repeated addition as an option. And that's not an effective way to get there. > There are other ways of seeing the multiplication table as well. For example 9 times something can be thought of as 9*x = 10*x-x. Yes, you can do that one. But that's just about the only fast trick there is.
- wizzwizz4 2y ago
  Multiplication and division by 2 and 5 correspond to each other. Multiplication by 11 is shift-and-add. There are many such tricks.
  - Dylan16807 2y ago
    The first one doesn't help with remembering times tables. The second one is outside the 1-10 times table we were talking about.
    - wizzwizz4 2y ago
      6 is 2×3, so 6×5 = 30. Likewise, 17 = 2×8 + 1, so 5×17 = 85.
- lelanthran 2y ago
  > Yes, you can do that one. But that's just about the only fast trick there is. I dunno about that. For division, anyway, there's a bunch of fast tricks that give you a decent approximation (i.e. decent precision, maybe to the nearest integer) Someone recently was surprised that I worked out the VAT (Value Added Tax, 15%) on a very large number in a few seconds. It's because its 10% of the number plus `(10% of the number)/2`. It's easy to get 10% of any number. It's easy to halve any number. It's a fast trick because there's two easy operations. There's a bunch of similar "tricks": 1%, 10%, 25% and 50% are fast to calculate in your head (at most 2 easy operations, like `(half of half of N)`). Then you either add or subtract N. Or you multiply N by 2. At most three easy operations gives you 1%, 2%, 4%, 5%, 10%, 11%, 12%, 14%, 15%, 20%, 21%, 24%, etc To someone who doesn't know how you are getting the answer it might seem like you are a human calculator because you get so many of those quickly, and they don't see the ones you don't do in 3 easy operations (say, 13%, which is 10% + 1% + (1% * 2)). IOW, it looks like a very impressive trick, but it isn't.
  - Dylan16807 2y ago
    > For division, anyway Did you not see the neighbor comment where I made it extra clear I'm talking about times tables in particular? There are tons of math tricks! But the multiplication table of the numbers between 1 and 10 is mostly rote.
tsimionescu 2y ago
I completely disagree. First of all, at the time children learn the multiplication table, they definitely don't know the concept of exponentiation. Secondly, 7*2*2*2 is not some immediately obvious shortcut. Also, learning multiplication with numbers higher than 10 still relies on knowing the multiplication table. 17*8 is 7*8=56, hold the 5, 1*8 + 5 = 13, so 136.
- LouisSayers 2y ago
  > Also, learning multiplication with numbers higher than 10 still relies on knowing the multiplication table You've actually just proved my point - you used a method of breaking down the problem into a different problem and then solving it rather than simply memorising. If you give the same question to multiple people there will be numerous ways different people use to go about solving it. As an example, I might solve this by doing 20*8 = 160 3*8 = 24 160 - 24 = 136 Or 10*8 = 80 7*8 = 56 80+56 = 136 And I might apply different tools like the one I originally mentioned within these calculations. I know that 80+20 is 100 and so "borrow" 20 from 56, so that I can easily add 100 and 36 together. These ways of calculating happen in your mind very quickly if this is how you get used to calculating.
  - tsimionescu 2y ago
    Sure, but all of those work for numbers higher than 10, and all assume you know the multiplication table by heart. The multiplication table (the result of multiplying every number between 1 and 10 with each other) is something you have to memorize. You can get away with memorizing only some of these results and computing the others based on them, but it's basically impossible to do any more complex arithmetic if you don't know most of it by rote memorization.
- lelanthran 2y ago
  > Also, learning multiplication with numbers higher than 10 still relies on knowing the multiplication table. 17*8 is 7*8=56, hold the 5, 1*8 + 5 = 13, so 136. Interestingly, I do those less efficiently: 17 * 8 = (10 * 8) + (7 * 8) = (80) + (56) = (80) + (50 + 6) = 130 + 6 I think the reason I do it this way is because I get an approximation sooner when the numbers are very large i.e. I get the most significant digit first, and can stop calculating when I get the precision I require.*

jltsiren 2y ago

Of course there is a pattern: (n+1)x = nx+x. Your brain can learn it just fine, and then it can multiply numbers without having to burden your slow inefficient symbolic reasoning machinery with rules and facts.

tsimionescu 2y ago
How is that pattern useful for replacing memorization of the multiplication table? 7*8 = 6*8 + 8 - fine. I still need to memorize what 6*8 is, or go through the extraordinarily slow process of expanding this recursively: definitely not an option in school.
- jltsiren 2y ago
  It's a linear function your brain can learn. Your brain, not the conscious you. A lot of learning is about bypassing the slow inefficient consciousness that thinks it's in charge. In sports and other physical activities, you don't memorize the right moves. You practice them until you can do them automatically. The same approach also works with cognitive activities.
  - tsimionescu 2y ago
    If that were how people learned the multiplication table, you would see that people take longer to come up with the result of 9*9 than it takes them to compute 3*5. I have never seen anyone work this way, and so I believe it's far more likely people just remember the results in a table. "9*9=81" is simply a fact you have stored, not a computation you perform, even subconsciously. Edit: I should also note that it's pretty well known people learn arithmetic as symbol manipulation and not some higher order reasoning. The reason this is pretty well established is that historically, the switch from Roman numerals to Arabic numerals led to a huge flurry of arithmetic activity, because it was so much easier to do arithmetic with the new symbols. If people had learned by subconsciously calculating the underlying linear functions and not through symbolic manipulation, the switch would have been entirely irrelevant. Yet for most mathematicians in Europe at the time, doing 27*3 was much easier than doing XXVII*III.