[ExI] teaching of math and science, was: RE: discovery of irrational numbers

[Anders Sandberg]

On 2013-12-19 17:37, spike wrote:
> Anders here's an idea, on which I invite your comment.  Since you have met
> my son recently, your outlook on this will be most valuable.
...
> if we stay right on the leading edge of a
> student's ability to learn, being right there with the new concepts right as
> the brain's synaptic development is ready to comprehend it, a
> high-performing student can progress at an astonishing rate.

Yes. One useful trick is to introduce a new domain, let the student play 
around for a short while with the things in it (vectors, electrical 
components, word classes, Roman emperors) to get a sense of what can be 
done inside the domain, and then the instructor gives useful information 
about how to solve problems or figure out things in the domain. Too 
little toying around and the student will just parrot, too much toying 
around and they will have to rediscover stuff that took bright people a 
few centuries. Too much instruction and the creativity is gone. 
Balancing these factors are the skill of a true educator.


> For instance, one of the ideas I had was a novel way to introduce the notion
> of imaginary numbers.  I have long objected to that name, real and
> imaginary.  It makes it sound too much like imaginary numbers are just a
> plaything or don't have any basis in reality.  That name imaginary was an
> unfortunate choice.  Imaginary numbers are real!
>
> So here's my idea.  When I introduce the concept, instead of calling them
> imaginary numbers, I call them vertical numbers.  I show my son the complex
> plane (oy, complex numbers, another bad choice of names) then show him that
> all the numbers he has worked with so far are horizontal numbers, and
> everything which contains square root of negative 1 are vertical numbers.
> All numbers which have both are ambi numbers, or ambinums.  You can't count
> with vertical numbers or ambinums, but there are plenty of reasons we need
> them and use them.  Then after he gets comfortable with vertical numbers and
> ambinums, then I introduce the standard terminology and attempt an
> explanation for why they have those names.

I wonder if vectors might be even better as a start. Explain that 
composite numbers are really useful, show how they can be added and 
subtracted, including multiplied by normal numbers to make them longer 
or shorter. And these things are easy to represent as ambi numbers.

Then you can start discussing what the "right" way of multiplying them 
are. Maybe show that component-wise multiplication is pretty pointless, 
that the dot product is neat but not really like normal multiplication - 
you want to get something that is the same type of thing as the two 
things you multiply. The cross product is kind of the right thing, but 
there is no "one" that leaves cross products invariant. Meanwhile, if 
one defines multiplication in the complex way one gets neat things in 2D 
at least - easy ways of doing rotations, and an explanation of the 
square root of -1.

At this point vectors and complex numbers come apart, and it is a good 
point for explaining that different mathematical structures (algebras) 
are good for different things. Vectors are great for space, electricity, 
force and similar stuff. Complex numbers complete the usual numbers in 
lovely ways.

When he asks about 3D complex number you will have succeeded. But 
explaining why there is no division algebra in 3D but one in 4D is a bit 
trickier...

> Anders, have you any ideas on how to use our current info tech to help the 3
> sigmas?

Look for systems like Minecraft, Celestia or interactive physics 
programs that let you set up your own objects and contraptions and then 
let them loose. Sometimes suggest challenges to make.



-- 
Anders Sandberg
Future of Humanity Institute
Oxford University