Speaking of multiplication. Ken Iverson teaches us to do multiplication by
using a * outer product to build a times table for the digits involved.
+-+--------+
| | 3 6 6|
+-+--------+
|3| 9 18 18|
|6|18 36 36|
|5|15 30 30|
+-+--------+
Now you sum each diagonal:
(9) (18+18) (18+36+15) (36+30) (30)
9 36 69 66 30
And just normalize as usual:
9 36 69 66 30
9 36 69 69 0
9 36 75 9 0
9 43 5 9 0
13 3 5 9 0
1 3 3 5 9 0
The multiplication table is easy and just continued practice for your
multiplication facts.
You don't need much more machinery before you have the kids doing Cannon's
order n systolic array algorithm for matrix multiply, on the gym floor, with
their bodies. This assumes that the dance teacher is coordinating with the
algorithms teacher. Of course if there isn't something relevant going on that
warrants matrix multiply then all is lost. I guess that's a job for the
motivation teacher. :-)
-David Leibs
On Jun 15, 2012, at 12:57 PM, Pascal J. Bourguignon wrote:
> David Leibs <[email protected]> writes:
>
>> I have kinda lost track of this thread so forgive me if I wander off
>> in a perpendicular direction.
>>
>> I believe that things do not have to continually get more and more
>> complex. The way out for me is to go back to the beginning and start
>> over (which is what this mailing list is all about). I constantly go
>> back to the beginnings in math and/or physics and try to re-understand
>> from first principles. Of course every time I do this I get less and
>> less further along the material continuum because the beginnings are
>> so darn interesting.
>>
>> Let me give an example from arithmetic which I learned from Ken
>> Iverson's writings years ago.
>>
>> As children we spend a lot of time practicing adding up
>> numbers. Humans are very bad at this if you measure making a silly
>> error as bad. Take for example:
>>
>> 365
>> + 366
>> ------
>>
>> this requires you to add 5 & 6, write down 1 and carry 1 to the next
>> column then add 6, 6, and that carried 1 and write down 2 and carry a
>> 1 to the next column finally add 3, 3 and the carried 1 and write down
>> 7 this gives you 721, oops, the wrong answer. In step 2 I made a
>> totally dyslexic mistake and should have written down a 3.
>>
>> Ken proposed learning to see things a bit differently and remember the
>> digits are a vector times another vector of powers. Ken would have
>> you see this as a two step problem with the digits spread out.
>>
>> 3 6 5
>> + 3 6 6
>> ------------
>>
>> Then you just add the digits. Don't think about the carries.
>>
>> 3 6 5
>> + 3 6 6
>> ------------
>> 6 12 11
>>
>> Now we normalize the by dealing with the carry part moving from right
>> to left in fine APL style. You can almost see the implied loop using
>> residue and n-residue.
>
>> 6 12 11
>> 6 13 0
>> 7 3 0
>>
>> Ken believed that this two stage technique was much easier for people
>> to get right. I adopted it for when I do addition by had and it works
>> very well for me. What would it be like if we changed the education
>> establishment and used this technique? One could argue that this sort
>> of hand adding of columns of numbers is also dated. Let's don't go
>> there I am just using this as an example of going back and looking at
>> a beginning that is hard to see because it is "just too darn
>> fundamental".
>
> It's a nice way to do additions indeed.
>
> When doing additions mentally, I tend to do them from right to left,
> predicting whether we need a carry or not by looking ahead the next
> column. Usually carries don't "carry over" more than one column, but
> even if it does, you only have to remember a single digit at a time.
>
> There are several ways to do additions :-)
>
>
> Your way works as well for substractions:
>
> 3 6 5
> - 3 7 1
> -----------
> 0 -1 4
> 0 -10 + 4 = -6
>
> 3 7 1
> - 3 6 5
> -----------
> 0 1 -4
> 10 -4 = 6
>
> and of course, it's already how we do multiplications too.
>
>
>
>> We need to reduce complexity at all levels and that includes the
>> culture we swim in.
>
> Otherwise, you can always apply the KISS principle
> (Keep It Simple Stupid).
>
>
> --
> __Pascal Bourguignon__ http://www.informatimago.com/
> A bad day in () is better than a good day in {}.
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