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

[spike]

>... On Behalf Of Anders Sandberg
...
>
>>... Oy, good point.  My son is reacting much like scientists do when we 
> keep finding out there isn't really a simple consistent underlying 
> principle, or if so, we still don't know it.

>...Well, you could reassure you that mathematicians are on top of the game.

We do understand irrationals, transcendentals and uncomputable numbers
fairly well... Anders Sandberg

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.

The education theorists are reworking the curriculum in the US, with a
program called Common Core.  I will reserve my commentary on that for now,
and get to a special narrow focus in which I have taken interest.

One of the biggest problems we have had in education for a long time is that
it is highly dependent on a very old model of gathering children grouped by
age, then teaching them with one adult up front talking to 30 or so kids.
The problem with that is one most of us know very well: they need to shoot
for the plus or minus one sigma mean.  If you are outside that band, and
plenty of us here are way outside that group, then the traditional model of
education does little to help us, and in many ways harms us.

OK then, one aspect of Common Core is a claim that it is more effective in
reaching those outside a sigma from the mean.

I don't want to go further into that right at the moment, but will respond
if others have comments on the above.  What I really want to get to in this
thread is that we have enormous potential right in front of our eyes to help
the 3 sigma and above students once we recognize a critically important
aspect of brain development: 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.

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.

One way or the other, I hope I can convey the stunning sense of wonder I
felt when I first learned of Euler's equation, the one that still blows my
mind to this day.

Khan Academy has a really good short lecture on this topic.

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

spike