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[Clansgian AT wmconnect.com]

  Most people today change careers several times in their lives.
You never know what you will need to know for a certain job.

First let me point out the focus of all this which will put a lot of our 
respective emphasis in perspective.  Back in the mid 60's President Johnson's 
campaign for education pasted it's slogan on billboards and every sort of 
public 
place, I remember especially seeing them on those advertising strips on 
public 
buses.  "To get a good job, get a good education."   The country seems to 
have 
swallowed it.  Now of days you can hardly get anyone to talk about education 
without it being tied to career.  This is unfortunate.  And it is definately 
not the focus of this clan.  We need some money, not much, but some.  As the 
children establish themselves independently, they will need money too.  We 
work 
for that without selling out souls to it.  That is, career is not life, it is 
not who they are.  We have actively nipped in the bud the old bit about "What 
do you want to be when you grow up..."   Be?  They are now what they will be, 
themselves.  What they do for money is NOT what they are, although that 
notion 
serves the industrial/government/consumerist paradigm and is promoted by 
government schooling.

So you see (may see) that the notion that one needs to probe deeply into each 
artificially partitioned branch of math on the outside chance that they may 
need that particular type of math described in those particular terms and the 
ability to communicate that math in the agreed parlance because of some 
future 
possible maybe career choice is a most foolish waste of resources.

And that's what it is.  Because the second thing I'd point out in this is 
that the various "branches" of math are all artificial constructs, a paradigm 
for 
people to be able to communicate on common ground.  A really deep 
understanding of math works to keep those fences down.  Describing statics 
and dynamics, 
vehicular motion, integration, differentiation, etc. isn't the math itself, 
it's only one system of describing and communicating it.  Describing it and 
segmenting it doesn't make it more advanced.  It seems as if because you can 
describe my son's calculations as arithmatic or algebra, you are willing to 
ascribe 
to him a knowledge of same.  But since what else he knows and can deal with 
does not fit the pigeon hole of calculus, spherical trig, bodie plots etc. it 
somehow doesn't exist or isn't 'advanced' or he doesn't know it.

A person comfortable and familiar with math as it exists in the real natural 
world can easily drape the coat of nomenclature on it any time they choose.  
The opposite is not true.  

> What occupations deal with spherical triangles and 
> would likely use trig?  Astronomy, oceanography, meteorology, 
> cartography, navigation, audiology (important for musicians too.), just 
> to name a few.

Again the career thing.  Having creditials and having worked as a speech 
therapist, I'm all to familiar with audiology.  Here's the impression the 
formal 
education left with me.  Sound waves are sinusoidal.  The compression and 
rarefaction of air is the function of the sine of the angle within the cycle. 
 Do 
you know that in a class of some 20 graduating audiologists and speech 
patholotists, every one of them (except me, of course) thought that the 
schematic of 
the wavy line was what a sound wave looked like.  They thought sound was 
traveling through the air as little squiggles that looked like that.  
Yesterday on 
the launch ride the nine year old observed that when waves of the wake 
encountered the flotsam on the water, the flotsam didn't actually move.  She 
concluded 
that the water wasn't moving either but rather undulating in a regular and 
predictable pattern.  She had grasped the concept of sinusoidal wave motion, 
although she didn't use those terms ...... yet.  Now which, my nine year old 
or 
the audiology graduates, best understands spherical trig?  The distance from 
her concept to using the standard blather to describe it is short.  The 
distance 
from the blather to her understaning is vast.

> 
> That sounds like a fun day, James.  Pretty soon, your kids will be the 
> captain of their own ship, and while they will be able to read the 
> instruments and rig the sails, will they truly be able to navigate? Or 
> will it be a hit and miss proposition.  Did they use an astrolabe or do 
> they know how to use one?

How many government school products know how to use an astrolabe or even know 
what one is?  You miss the point, I think.  Schooling conditioning whispers 
in your ear that if you don't have formal training in something like 
navigation, it is far, far from you.  With his interest in historical 
shipping, picking 
up the use of navigation is within easy grasp should he ever decide he wants 
it.  There are no gaps and holes in the unschooled mind because nothing is 
far 
from them.

However let's constrast that to the schooling notion that if you don't 
specifically learn something, it leaves a gap in your knowledge.  Wow!  
Government 
school people and even dedicated schoolers at  home are without hope!  Look 
at 
all the gaps!  Even someone who is IN the field is hopelessly riddled with 
gaps.  For example, we might assume Ray has a very good knowledge of music 
since 
that's his stock and trade.  But if I posed this question, (just off the top 
of your head, Ray, no fair looking it up):  "An instrument is strung with a 
wire string.  It is tuned higher and higher until at exactly F#' two octaves 
above middle C it breaks from the tension.  I replace it with a string of 
exactly 
the same material but twice the diameter.  At what pitch does it break now?"  
If a musician doesn't know that off the top of their head, does that 
constitute a "gap" in their knowledge?

A survey done some months ago asked highschoolers what was the principle 
ingredient in French fries and 78% didn't know.  When shown a picture of a 
potato 
as a follow up, more than 20% didn't know what it was.  On this farmstead all 
an sundry can tell you the life cycle of a potato, that it need acidic 
ground, 
more than a usual share of phosphorus, that it is slightly frost hardy, and 
they can identify the main cultivars by the shape of their leaves.  The 
possibility of in depth knowledge is only exists outside the classroom.  

> 
> I guess my point is that I know your kids are learning, but as a parent, 
> I would worry that they would be overconfident in what they already know 
> and since they don't presently see the need to learn something more 
> advanced, they don't.  

First, it isn't more advanced.  Couching it in a jargon or parlance only 
gives it the appearance of being more advanced.  Also, as is part of the 
schooling 
paradigm, if the person does not pigeon-hole it and use the proper and 
approved jargon, it is deemed that they "don't know it."  Second, their 
'overconfidence' is in their ability to learn it and handle it.  Only the 
schooler would 
have overconfidence in what they already know, not the unchooler.

> 
> Ask him a question for me...I have been wanting to know what the 
> Vikings/Norsemen used for navigation.  Obviously, they used the stars, 
> but that is rather general information.  Did they use sand clocks?
> > 

He says they were coast huggers.  They mainly navigated by following ocean 
currents as if they were rivers, by monitoring the water temperature and bits 
of 
vegetation floating on it.  That's why, says he, that they didn't take off 
diagonally across the Atlantic but rather explored the European coasts and 
the 
North Atlantic where they had predictable currents to follow.