I spent a lot of time up until high school forcing myself not to use a calculator, because I wanted to work on my mental skills in arithmetic, but I always make a lot of small mistakes, even now.
I have a hard enough time concentrating on 'easy' things , please don't ask me to hold numbers in my brain. I am a pretty "mathy" person but mental multiplication is not meant for the human thought process, too much is going on.
I say this half in jest, but I've met people with very good mathematical sense (especially in domains such as topology) who are incapable of multiplying 3 numbers without making a mistake. Some people have very few registers, and RAM is really slow.
I think a good part of the problem is that we learn to multiply and divide backwards. The standard US algorithm involves starting at the least significant digit and working your way up. It is much easier to do in your head if you do it from the most significant digit down. Or learn to convert hard multiplications into easy ones, for instance:
9x27 = 10x27-27 = 243
The other way to do this would be to multiply 9x7 to get 63, and then hold that in your head while you multiply 9x2 to get 18, add a zero to get 180, and then recall the 63 to add to 180.
Our insanely good fuzzy classification skills let us select and continually update a solving strategy. In computing terms: writes to long-term memory are slow, but reads are fast; short-term memory is extremely size-constrained and time-volatile, but fast for both writes and reads.
Efficient algorithms for human brains should rely as much as possible on pattern matches, long-term memory reads, and limited short-term storage—exactly the opposite of what you want to do on a computer! It’s no wonder that the polynomial-time multiplication algorithms that work fine on paper aren’t going to perform very well in our heads.
Why isn't this type of mental calculation actually taught at schools? Not every kid thinks of these "tricks". Instead, they do how the teacher showed them and do it on paper the long way. Seems like we are crippling our kids.
I heard as well that teachers are moving to a more sane method to teach multiplication.
The best part about MSB math, to me, is that at least you get a close approximation to the answer quickly. It's like successive approximations, the errors from making a mistake go down as you move along if you don't need a precise result.
For instance, what's pi * e? Good luck calculating that from the LSB... ;-)
But with MSB, you know it's about 9 because pi is on the low end of 3-4 and e is on the high end of 2-3.
So you can do
3.14 * 2.72
33 + .143 - .28*3 - something small
(idea here is that the cross term is small, three sig figs)
9 + .4ish - .9ish
8.5ish
Actual result is 8.5397.
So you can get a few sig figs by saying "eh, these digits are too small to matter" and ignoring cross multiplication of the LSBs for problems that aren't for the purpose of arithmetic.
I use this often in engineering estimation on the fly, because so many constants are only known to 1 sig-fig anyway that it's really hardly worth worrying about more accuracy. Particularly when you're building in a safety factor anyway.
Huh, so I'm not the only one who goes most-sig to least-sig when doing it in my head. Really makes things easier, oddly enough -- you end up with big numbers with lots of zeroes on the end, and add them to the smaller numbers.
In my head, what I do in the end is 9x27 = 3^2 * 3^3 = 3^5 = 243 ('everybody' knows that, because of 7^3 = 343)
And 27 x 16 ends up being (32 - 5) x 16 = 512 - 80 = 432.
And yes, I could be way faster at mental arithmetic if I just did the calculations instead of switching to cuter approaches halfway through all the time.
It's a well-known fact here that mathematicians can't do calculations. I've had professors in class ask the class to do very simple calculations for their complex examples for them. (Or they get it wrong. A lot.)
The driving factor for mathematics is laziness. I didn't even memorize the basic multiplication table (1-10) for a long time, but I remembered some shortcuts (like multiplying stuff by 9 was easy) and got by.
The multiplication table really bothers me. It's considered to be the least a child can learn in elementary school. And kids that haven't learned to do math mentally just memorise it like a poem. I really don't think this is the correct approach.
We teach multiplication as an algorithm based on breaking the problem into single-digit pieces. If you're trying to learn that algorithm, and you keep needing to figure out what all the pieces are, you won't be able to follow along with the bigger picture. It would be like trying to write an essay, but having to look up the spelling for each word in the dictionary... it would take you a lot longer, and be much more of a struggle. Instead, you memorize the pieces, and then later can focusing on understanding how all of those pieces fit together.
I feel a bit bad for not memorizing it, to be honest. Not sure if there is any other way for learning to multiply?
How do you "understand" that 6x 8 = 48?
With bigger numbers, then you can apply a system. And as I mentioned I also "cheated" a bit with the basic table. For example I would have remembered that 5 x 8 = 40 and then calculated 6 x 8 = 5 x 8 + 6 in my head, because the multiplications for 5 are easy (half the other factor x 10 +/- 5). But there might not be a cheat for every point in the table?
You can learn to better comprehend a quantity or amount. So you understand that 6 * 8 is larger than 2 * 8, and then understand that it is actually 3 times that. Another approach would be to always think of multiplication as condensed summation, so 6 * 8 = 8 + 8 + 8 + 8 + 8 + 8 = (8+8) + (8+8) + (8+8) = 32 + 16 = 48. I would let my kid write out the summation and apply an analytical approach rather than memorising it blindly. I guess you have to memorise it eventually, but a lot of kids are lost if they fail to recollect the answer.
I have a hard enough time concentrating on 'easy' things , please don't ask me to hold numbers in my brain. I am a pretty "mathy" person but mental multiplication is not meant for the human thought process, too much is going on.
That's like saying humans were never meant to run up stairs so you're taking the elevator. In any case, nobody's asking yo to hold numbers in your brain; it's OK to use paper and pencil for your calculations, as generation after generation of mathematicians did.
To be fair, I don't hold her to be the slightest bit representative, it's just that such a simple mathematical process is just... automatic. I'm not mathy at all - I've abandoned several online crypto intro courses because I can't follow at a reasonable speed when the notation comes out - but when it comes to manipulating single-digit numbers?
In her case it was significantly affected by her lack of motivation, but it was just plain odd that such a trivial thing wasn't 'automatic' for someone in a science course (the actual question was calculating V=IR). I had to simplify it with "Well, what's 2x3?", and then, 'given the answer', she could continue.
I'm enjoying your comments. Regarding the crypto: that's too bad, because a big chunk of crypto involves de minimis math, and that chunk is the most useful; crypto is as much algorithms and computer science as it is math. Consider letting us beat you over the head with crypto challenges sometime; we wrote them in ASCII text, so there's no notation. :)
I have a hard enough time concentrating on 'easy' things , please don't ask me to hold numbers in my brain. I am a pretty "mathy" person but mental multiplication is not meant for the human thought process, too much is going on.
I say this half in jest, but I've met people with very good mathematical sense (especially in domains such as topology) who are incapable of multiplying 3 numbers without making a mistake. Some people have very few registers, and RAM is really slow.