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.
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.
> 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.
> 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.
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.
> 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.
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.
> 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.
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.*
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.
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.
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.
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.
People are more than their conscious minds. A neural network can compute a linear function with a small domain without having to store each case separately.
I never memorized the multiplication table, because I found it boring and unnecessary. When I had to multiply numbers, some answers just appeared automatically, while I could calculate the rest quickly enough. Over time, more and more answers would appear magically, until I no longer had to calculate at all.
Some other things I had to memorize. Those were usually lists of arbitrary names with no apparent logic behind them. And if I didn't need them often enough, they never became more than lists of random facts. For example, I often can't tell the difference between sine and cosine without recalling the memorized definitions.
Or, to give another example, Finnish language has separate words for intercardinal directions (such as northeast). Usually when I need one of them, I have to iterate over the memorized list, until I find the name for the direction I had in mind. Similarly, I had to iterate over the six locative cases in Finnish grammar whenever I needed a name for one of them.
Whether it happens consciously or unconsciously, computation takes time. So, if your theory that the brain computes the results instead of remembering them were true, it should take measurably longer to compute 9*9 than it takes to compute 2*3. I am certain that doesn't happen for me, but it could be measured for others as well.
> When I had to multiply numbers, some answers just appeared automatically, while I could calculate the rest quickly enough. Over time, more and more answers would appear magically, until I no longer had to calculate at all.
This is prefectly explained by some results becoming memorized as you see them more and more, and makes no sense if your unconscious mind were computing things. If your brain was computing these results unconsciously because it had learned the function to apply, it should have come up with results automatically for any (small) multiplication. That it didn't, and you had to consciously do the computation for some numbers, is pretty clear proof that you slowly memorized the same multiplication table, but only filled it in gradually.
Overall I'm not advocating for the importance of cramming the multiplication table. I'm just saying that people who want to do mental arithmetic, or even pen-and-paper arithmetic, can only realistically do it if and when they learn the multiplication table by heart. And, that the reason the multiplication table is taught to children is strictly to have them memorize it so that they can do arithmetic without a calculator at realistic speeds.
From my point of view, what happened with the multiplication table was practice without memorization, while the word lists were memorization without practice. Two different approaches to learning with two different outcomes.
Oh, absolutely, you learned it through practice instead of pure memorization, which overall tends to be a better strategy. But just like chess masters tend to mostly learn game positions (through practice, note trying to do rote memorization) and not some advanced logic, I am pretty sure you learned the specific results and did not learn some subconscious algorithm deeper than "lookup 9*5 in the stored table".
