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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

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. http://www.youtube.com/watch?v=Rgw9Ik5ZGaY http://www.youtube.com/watch?v=Rgw9Ik5ZGaY Basically, to do MSB multiplication it looks like: 27x16 = 27x10 + 20x6 + 7x6 = 270+120+42 = 432 We can do accumulators much more easily from big numbers and then adjusting with small corrections than we can the reverse.

Someone 13y ago
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.
girvo 13y ago
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.
evincarofautumn 13y ago
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.
dsego 13y ago
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.
- raverbashing 13y ago
  Because the teacher a) can't b) don't care and c) shuns those who point their ignorance Yes, the way of speeding these calculations is to break it in easy steps (which might involve some simple calculations themselves)
- Gmo 13y ago
  Well, I was educated in France and I was taught those tricks.
- adestefan 13y ago
  That's how I was taught in the mid-80s at a Catholic elementary school in Pennsylvania. I believe I heard, probably on NPR, that the new Common Core standard also recommends teaching in this manner.
  - neltnerb 13y ago
    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.

Someone 13y ago

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.

girvo 13y ago

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.

evincarofautumn 13y ago

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.

dsego 13y ago

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.

raverbashing 13y ago
Because the teacher a) can't b) don't care and c) shuns those who point their ignorance Yes, the way of speeding these calculations is to break it in easy steps (which might involve some simple calculations themselves)
Gmo 13y ago
Well, I was educated in France and I was taught those tricks.
adestefan 13y ago
That's how I was taught in the mid-80s at a Catholic elementary school in Pennsylvania. I believe I heard, probably on NPR, that the new Common Core standard also recommends teaching in this manner.
- neltnerb 13y ago
  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.