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#LyX 2.0 created this file. For more info see http://www.lyx.org/
\lyxformat 413
\begin_document
\begin_header
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\begin_body
\begin_layout Title
Curious Minds
\end_layout
\begin_layout Author
By: no relevance
\end_layout
\begin_layout Author
Dedicated to al l who have not been given enough chances to succeed and
to a night in 'the Kings' on Friday 11th March 1983
\end_layout
\begin_layout Author
Thanks to all of those who gave me a chance to possibly fail again.
\end_layout
\begin_layout Chapterprecis
A story of art, maths, art, physiology, science etc.
in a single collection known as knowledge.
A story of questions with no answer, a story of answers which question
current knowledge and mostly a story of hope, hope that we can really and
truly ask large questions of small issues.
\end_layout
\begin_layout Part
Basic Schooling
\end_layout
\begin_layout Standard
\end_layout
\begin_layout Standard
In school, like many others I had to learn arithmetic, how to add and subtract
and when this has been achieved then teachers go further and teach us times
tables.
There are even songs about times tables like the poems and songs used to
teach the alphabet.
Then there is English (or whatever your language is) or literature (it's
called English in Scotland).
In these classes you learn to read at an early age which is great.
Of course art and music are also taught and here teachers let us draw with
crayons or watercolour paints, sing and generally be creative.
In not one of these classes was I never shown why we do all this stuff,
as with foreign language or science, religion etc.
In School I was taught like others to repeat and the best got certificates.
I am not saying this is bad to receive certificates for regurgitating informati
on, although I am not sure it deserves al l the respect of the system.
the kids who ask why, usually received reports with could do better , often
loses attention or easily distracted .
I was similar to the latter in primary School, but was, at least initially,
relatively quiet as I was taught to not speak unless being spoken to and
sit nice and don't be bad.
It was obvious questions and in particular debate was being bad, and, yes,
I was belted often in later years in primary school!
\end_layout
\begin_layout Standard
I wonder how many others recognise this story as similar to their own or
very similar to what they encountered.
It is later in life I realised that the education of a child is of paramount
importance and perhaps mass education is not quite working as we thought
it should.
I have to be careful here and allow the reader to recognise I am 100% committed
to education, but with the caveat that teaching repeating tasks or theorems
etc.
is not, in my opinion, a complete education.
I am not impressed by people who can recite large texts or numbers (Pi
etc.) off by heart, my dicta-phone can do that, it's not that clever.
I am also unlikely to be impressed by a child remembering all the times
tables all the way up to 12 times 12 and I do not think that deserves certifica
tes and honours at early ages (actually honours do not impress me at all
either, but that's completely another matter).
I agree completely with rewarding and making kids feel special and interesting,
it's a good start but a better way is to reward creativity in all areas
and this is difficult, but then again should teaching be easy ?
\end_layout
\begin_layout Standard
Before continuing we must recognise something important.
The vast majority of people can get through life admirably and very successfull
y with the basic tools of early education (reading, writing and arithmetic),
in fact arguably in most cases this is more than adequate.
This perhaps should be an area that is hived out to an elementary level
and yes, a recondition in this area is probably important.
Also the recondition of asking questions is becoming more acceptable and
even encouraged now in schools, as I have been reliably informed so this
is progress and very welcome.
One area that has been an issue and still is though is the creative side
of children, we seem to knock that out of them in the push for this elementary
education and that's a terrible crime.
\end_layout
\begin_layout Standard
It's also worth noting that this early stage mass education for the masses
only works to an extent (a very limited extent).
I say extent as it does not seem clear that the education by batches of
ages is very clever or indeed the industrial conveyor belt approach is
at all beneficial to anyone.
In any case we can assume some good comes of it, although perhaps in a
sloppy and uncaring way in many cases.
\end_layout
\begin_layout Standard
Now imagine for a moment a situation where early schooling was teaching
more why rather than how ! By this I mean start with why ? then teach a
method (and also reiterate it's only one method) of how to achieve this.
I prefer several methods of how, but that requires a level of questioning
that is perhaps too abstract for some of today's teachers (I do not mean
this in any derogatory way), but I bet the really dedicated would also
love to learn some of the answers as well.
\end_layout
\begin_layout Standard
If I were a teacher, I would hope I would relish the days every year when
the first student of any age challenged me to a question I did not know
(never mind the answer, like what does this God look like or where did
we come from type questions).
Even better if a student showed me a different method of existing thinking,
a method I had not realised or seen before.
Now that would be magnificent.
I fear this is not the attitude I recognise as being fully accepted from
my own schooling (there are a lot of genius teachers out there though)
and surprisingly as I will outline later, this acceptance of difference
is less tolerated in later years at college.
\end_layout
\begin_layout Standard
1.1.
Some teaching ideas
\end_layout
\begin_layout Standard
OK! So what do I mean by all this gibberish ? Well it would appear that
teaching addition and subtraction are relatively easy to explain.
I have one apple and get another so I have one + one = 2 apples (more later).
If I remove one then I have 2 minus 1 = 1 apple.
There it works! That's one method for addition and subtraction that we
can all teach children and do so from an early age.
So what about the next steps multiplication and division ?
\end_layout
\begin_layout Standard
These are just as simple as addition and subtraction but we make them extremely
difficult by teaching patterns and rhymes and times table songs and lose
children's interest in an amazing subject very early in their formative
years.
For the repeaters this method of teaching we use is great, for the thinkers
and potential genius's well, we may lose them at this crucial stage, unless
you have a good recall then what we call (incorrectly at this stage) maths
becomes difficult straight away.
\end_layout
\begin_layout Standard
To compound this we have parents, the initial yardstick of how well we are
doing and source of comfort to us, the people we most want to please.
How do they t in? Well they reward us for remembering times tables and
scold us for not remembering, saying s/he is not good at maths (considering
s/he has not being doing maths, then this is quite a statement).
This is, to me, amazing! We would be as well kicking children in the face
and congratulating them for not bleeding too soon!
\end_layout
\begin_layout Standard
So how can we teach a better or at least different, why ? This is actually
easy in principle, to teach addition you could say something like, I have
2 screw-holes and
\end_layout
\begin_layout Section
OUTSIDE THE BOX
\end_layout
\begin_layout Standard
I need to screw a screw (hmm! verb == noun) into each screw-hole so I need
1 plus 1 screw.
There that's why we need to learn to add.
\end_layout
\begin_layout Standard
So continuing, say you were handed 3 screws for this job children.
Now we need to remove one screw to make 2 which is the amount we need.
So simply throw one screw away (or subtract it) and we have again 2 screws,
perfect for the job.
Many such examples teach a why ! and in this case it is very difficult
to separate the why from the how as it is relatively simple to add and
subtract.
This is an example of how using addition you can also subtract.
By adding the number of screws you remove from the pile (of 3) then you
add again the original pile (now 2) and add to the pile you removed (1).
\end_layout
\begin_layout Standard
So subtraction can be done using only addition! what about multiplying?
\end_layout
\begin_layout Standard
Well this is easy 5 times 3 (or multiplied by) is 5 + 5 (multiplied by 2
now) +5 and that's us multiplied 5 by 3, 5 + 5 + 5 now again use only addition
to get the answer 15!
\end_layout
\begin_layout Standard
So multiplication can be done using addition! what about division?
\end_layout
\begin_layout Standard
lets reverse the above example and This is a tiny (very small) bit more
difficult.
say what is 15 divided by 5? so here we start with 15 screws and remove
5 from it, there's still some left so again take screws one at a time adding
them up to 5, we still have some left so again removing one at a time and
adding to 5, then the amazing thing, we have none left from our original
bundle, and we did this 3 times, therefore the answer is 3.
15 divided by 5 can be done by removing groups of 5 till the pile is empty
and this can be done exactly 3 times, wow !.
This means an elementary school kid who can add, can also subtract, multiply
and divide, without any additional rules or mechanisms over and above simple
addition and guess what? no remembering tables yet.
We can cover remainders and fractions in a couple of minutes, there terrificall
y easy too (maybe easier than this so far).
\end_layout
\begin_layout Standard
1.2.
Outside the box
\end_layout
\begin_layout Standard
Here is an interesting exercise, count your fingers, yes 1 2 3 4 5 6 7 8
9 10, great so we can count to 10 (use one hand a count to 5 if its easier).
Now did we count to 10 or did we miss a number? this is a much more signicant
question and one that confounded and caused great consternation for many
hundreds of years so the question is very difficult indeed.
In today's mathematical world we did miss something.
Let me explain hold up 2 fingers one hand and we will subtract 2 from it
(don't jump to conclusions here!) so on one hand put down 1 finger (you
can hold up one finger on the other hand to use our subtract by addition
method, as one finger on one hand is put down, then hold up a finger on
the other hand).
Now put down another finger, so we subtracted 2 from the first hand (some
of us know because we have 2 fingers held up on the other hand and can
count to 2).
So we have how many left? when we counted our fingers there was 10 of them
(if you count thumbs as fingers, for the pedantic).
\end_layout
\begin_layout Standard
So how many now? none you say excitedly none so the answer is none (which
is accurate), but in maths today we call nothing, something, we call it
zero (or nought or nothing).
So what's the answer? is it zero? we did not have zero fingers, we started
counting at 1 (again stay with me), otherwise starting at zero would give
\end_layout
\begin_layout Standard
8
\end_layout
\begin_layout Standard
1.
BASIC SCHOOLING
\end_layout
\begin_layout Standard
us 9 fingers (try it), this is shown by simply counting backwards on your
fingers, so on one hand 10,9,8,7,6 (one hand gone) and we have still ve
fingers on the other hand, what 5 + 6 = 11 (told you not to be so hasty).
\end_layout
\begin_layout Standard
So if we start counting at 0 we have 9 fingers, or if we count back from
10, we have 11 fingers, think about this for a while, it will help you
understand some more of maths than you imagine.
Now revisit the subtracting by adding this does not occur (you add the
numbers you remove so 5, therefore 5 + 5 left = 10, were you hasty?).
So to start at zero is a bit crazy, after all zero what? fingers? cars?
blades of grass? but if zero is a number then why not start there?
\end_layout
\begin_layout Standard
So is 0 actually a number? is 2 - 2 a number or is it actually nothing,
not a number just nothing? we will come back to this (bet you can't wait).
\end_layout
\begin_layout Standard
1.3.
Far outside the box
\end_layout
\begin_layout Standard
Counting as in the previous passage is rather easy and we can see it clearly.
What about the alternatives or differing approaches, are there any? An
interesting thing is that we are taught to read left to right and do so
almost automatically.
Why then do we count right to left, it all seems so easy and natural, but
it gives us limitations almost immediately.
for instance, how can we add a number to an irrational number (a number
when written in decimal approaches infinity, or maybe does, but we cannot
get to the end of it like pi or the square root of two).
\end_layout
\begin_layout Standard
Well the good news is we can add such numbers together and quiet easily
actually.
Instead of adding for instance left to right as normal we can do it the
other way round.
\end_layout
\begin_layout Standard
1234 + 4567 =? Normally we would carry out this task right to left like
7+4=1 carry 1, then 3+6(+ carry 1) = 0 carry 1, then 2+5+1 = 8 and nally
1+4 = 5, so answer is 5801 (you will note we answer backwards although
ignore that naturally).
What if instead though the problem was 1234 .
.
.
+ 4567 .
.
.
=? (the ...
meaning, onwards towards infinity), this introduces an important issue
as we cannot go to the right hand number, because we cannot reach infinity
to find it.
Is our system broken then? well only if we stay in the box and think for
a second that the mechanical teaching we have had is all there is.
\end_layout
\begin_layout Standard
The answer is so simple that it's not even funny, importantly doing this
exercise in junior school would probably mean you failed the test as your
working would be wrong, even though the answer is correct.
Let us demonstrate this wrong method now.
\end_layout
\begin_layout Standard
In steps, first part (left to right) is 1+4=5, next part is 2+5=7, next
part is 6+3=9 and last part (the trickier part) is 7+4 = 1 and this time
we carry 1 backwards so our previous 9 becomes 10 or zero and carry backwards
again so the second thought becomes 7+1 or 8.
so the answer (read forward this time) is 5801...
and the numbers go on to infinity if we so desire.
\end_layout
\begin_layout Standard
If we then try a trickier version 9235+1986 =?in this case the answer is
9+1=0 carry 1 back.
We note here there is no back so we add an extra digit to the beginning
so for the first number we now have 10.
Then we have 9+2=1 carry 1 backwards so then we have 111 so far.
Then 3+8=1 carry 1, so we have 1121 so far then 6+5=1 carry 1 so we have
the answer 11221...
Which works, but importantly we have
\end_layout
\begin_layout Standard
1.4.
MORE QUESTIONS
\end_layout
\begin_layout Standard
9
\end_layout
\begin_layout Standard
started a system where if we find another new number in our series then
we're set up with the thinking that would allow us to continue the process,
even when the carry back carries back several times, perhaps even to the
beginning of the number.
\end_layout
\begin_layout Standard
I hope by now questions are starting to fester away and if there not, they
should be! To think we have not got to religious studies yet!
\end_layout
\begin_layout Standard
1.4.
More questions
\end_layout
\begin_layout Standard
So now! do we teach the tables and rhymes ? absolutely not, what a terrible
idea, now the teacher has to talk about all this and see what clever ideas
the kids can come up with, now they have a new skill.
Time to play about, do you want 6 minus 3 sweets or 6 divided by three
sweets ? If you cut a card till a bit falls o, how many pieces are there
(does a card have a limit of cuts in it?)? Now ask, If I write a number
on the board with this small piece of chalk and then another and then another
and so on (actually do it!) can you see the chalks getting smaller, so
how many numbers are in the chalk? If I keep speaking and don't eat or
drink how many words are inside me? will all my words run out? let the
kids discuss and talk crazy till they run out of crazy and start evaluating
the answers properly, watch for the crazier kids and encourage this (probably
at back of class with lots of energy and devilment, or at the front and
very quiet).
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Now some other subjects can be tackled the same way, boil a kettle dry and
ask the kids where all the water is? Look at the heat waves coming o a
radiator, ask the kids what it is (open the window and ask if anyone can
see the heat escaping).
Ask kids what the heat waves above a road are and they will be amazed when
you tell them the heat waves are the sky1 ! (maybe you will be to).
Then discuss the heat waves and say to a child, they can put there hand
half in the hot air and half in the colder air just at the edge of these
waves.
An adaptation of this is look at wind on the sea or loch, there is an end
to it in the plane of the water, therefore you must be able to put your
hand part in a wind and part out of it, it does end.
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Now we can explain some more phenomena, these heat waves are actually causing
some light air to rise2, therefore sucking air from a heat source and sending
it up higher.
So the air must be getting apparently pulled in from below this heat wave,
pulled from all around it and pushed upwards.
Easy !
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1This is a subject discussed by Richard P Feynman during talks on the subject
of quantum electrodynamics, where light passing through hot air can refract
(or bend, wow big debate, it's actually taking the fastest route) showing
heat haze on a road is the sky to.
I wont waste a good book, buy it (QED).
2This is also untrue.
The hot air does not necessarily rise, it stays where it is, the heat increases
the pressure if the area is conned, in this case it's not so the heat spreads
apart the air molecules and the cooler air brings them together (or allows
the mass attraction to take over), the air neither rises or falls, simply
the pressure differs.
This is another example misstated in many subjects, including physics.
The hot stuff rises, what nonsense! The heat rises to reach the lesser
pressure area, the stuff stays where it is relatively speaking, or does
it? Keep thinking! As a molecule heats it will go in all directions, rising
to a lesser pressure is the most efficient escape route in reality, therefore
heated molecules may seem to rise as they bash against their partners,
but in doing so they heat their partners also.
It's a very interesting subject for another whole book.
Basically how much stuff rises is somewhat inversely proportional to the
transfer of heat capability the stuff possesses.
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This is creating a pressure gradient in fact, this is explained now.
The pressure at the bottom is low as stuff is being pushed up so it's not
weighing on us now, it would feel lighter standing at this point as it's
as though were being lifted (ever so slightly).
Try it put a feather there and watch it rise.
Now discuss hand gliders and birds soaring on thermals.
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So then we can expand, if there is a low pressure here, then it must be
causing a higher pressure above as this is where we are pushing the stuff
to (air).
This makes sense, I hope.
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Interestingly though if we expand this thinking we must be sucking air sideways
towards the point of the heat (that radiator in the class), so we are creating
an even higher pressure at the side of the heat source.
So as we push air up and suck it in sideways the air we push up, cools
and falls down again to be sucked into the heat source again.
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This explains the beginnings of understanding worldwide weather systems,
and winds or pressure gradients as they are confusingly called.
It's a simple matter of a heat source pushing hot air up, sucking the same
(or in some places nearby) air back in from below as it cools.
This is a great thing to teach a child.
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Now you can show the child some tricks in nature.
As he or she can put their hand in the heat source part way they can also
put their hand in rain part way.
So you can have rain falling on part of you only, or on half of your house.
How so?.
Well it's simple now if we bring all this together.
If the heat source is near water it heats the water, evaporating that and
mixing it with hot air and lifting it up.
As it rises it cools, as we already know, and in doing so forms clouds
which is simply all the previously hotter water getting cooler and again
coming back closer together, thus forming larger water concentrations,
until it's mass is greater than the force exerted upon it by the heat (Brownian
motion) and it falls.
Interestingly though as the water falls in spheres the bottom part of the
sphere is closer to the earth and is pulled by gravity more, The outer
edges spread out though due to the air resistance pushing on the bottom
part of the sphere (which is stronger than the gravitational force) and
the sphere splits into smaller spheres (increasingly slower though), this
continues until they hit the ground.
These clouds become heavy as they cool and cannot hold together as steam
or vapour any more and form water, as water cannot oat in normal air, it
falls down, as rain.
The clouds have an edge and part of this divide is wet and the other is
dry.
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Now children can put all that together and realise why deserts have heat
but not much rain and seaside places tend to be wet, especially in summer
as the land heats faster than the water and sucks the sea vapour towards
beaches as it pushes the hot air up, which cools and forms clouds.
This is called a sea breeze and explains the thin clouds round the coasts
in summer (have you ever wondered why there are clouds at the seaside on
a sunny day), and also the relaxing breeze we feel in hot summer afternoons
at the beach.
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Now we can teach people about gravity! and we should be clear here, we never
teach people gravity, even our best scientists can only guess at it, in
fact Newton was embarrassed by it and Einstein theorised about it.
We have no clue what gravity is.
We can, however, explain the effects of gravity localised to our little
spot in the universe, and we do.
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1.5.
NATURE
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When kids are told of this phenomenon it's confusing and so well it should
be, we wrap a thing around it call it a force and leave it at that, but
we are so wrong! Anyway as we nd out the earth is spinning many children
will say things like if I jump up will I land in another place ? or leave
the brake of the car and let it drive around backwards to the direction
of the earth's spin (I certainly did and still wonder) using no fuel ?
We tell them gravity stops that and pulls us straight back down or the
car is stuck to the ground too hard due to gravity that it will not move
with the brake o, if on a straight or level road.
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However, we do not tell the whole picture, as the earth turns it does affect
things as a child would imagine (and we stop them thinking this, what a
shame!).
As the earth turns the air we pushed up, or air in the upper atmosphere
is moved from the location it left earth from (slightly).
Gravity has a lesser eect on water vapour (well actually its overcome with
heat rising) and the place where the air is pushed is backwards from the
point where it left the ground source (as the earth turns below it).
This is again confusingly named the Coriolis eect.
We need to let children know their original assumptions about the turning
planet are true.
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Children believe gravity is such an amazingly powerful force when they are
young, to be told later, it is the weakest of all forces known? why do
we do that?
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Another interesting factoid here is that if you jump out a 7th story window
the earth moves up to meet you with a proportional force that you fall
to meet it.
This is the force or measure of force we call momentum, which is actually
velocity multiplied by mass, relatively the velocity the earth meets you
at is proportional to the speed you meet it at.
The proportion is the mass of the earth compared to you.
Weird but true, now discuss!
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1.5.
Nature
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Nature is the most important thing to teach kids and yet it is ignored,
kids may look out a window at some grass and tree's or a bird making a
nest and what do teachers do, well your in trouble now kid! This is school
! no looking at nature.
This is the best time to teach a child, what are you looking at ? grass,
oh so what can you tell me about grass? (people depended on the spread
of grass, we would still be in Africa if it were not for grass, assuming
we did all migrate from there).
There's loads of things to learn from grass (try to destroy it by burning,
drowning etc.) or investigating birds or trees, in fact everywhere in nature.
There are more unknown organisms in a single square meter of a common garden
than the total number of known organisms identified buy humans to date,
we can observe nature for many years to come, before we can consider ourselves
even to have an infantile knowledge of the world around us.
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Now put a small plant or bush in a pot, one which will grow fairly large
in a few months, we will come back to this.
Water and feed it and explain what's happening all the time you do.
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So nature is life's textbook, always changing, never the same twice and
always fascinating, all aspects from the shape of clouds and all you can
teach about that, to a rock that seems to never alter, cosmic rays, heat,
cold, wet, dry, you bet it alters (later photons smashing into it throwing
electrons off back into photons when they hit air again, oh the list is
never ending).
Important to note here that like the
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times tables and rhymes for teaching 'times tables' it is important NOT
to teach all different names for stuff in nature but teach what things
actually are, what they are made of, how they interact with other 'things'
and importantly how there all different, even 2 birds of the same family
or breed are completely different, as people are.
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Now some more interesting observations for children, when they look at,
say a rock or better a bird that flies away.
Now ask where is the bird? is it gone? or is some of it left ? Ask each
of the children what they remember about the bird, some may say it had
wings, others, it was black, had a beak etc.
Now discuss why they can remember what it looks like and if it's really
gone what bit of it got in our head ? and if it looked at us, is a bit
of us in it's head ? is a bit of us flying and soaring through the sky
chirping and freely soaring in currents of some of that warm air ripple.
Now you may remember the bird your whole life, many years after it's not
a bird any more in the world outside your window, but is it a bird still
in your head? your head is made of matter and that birds impression takes
up a piece of that matter (perhaps) so is it alive, in you? These lessons
may start to become interesting to teachers as well now.
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What can nature not teach us? when you think of something then don't teach
kids that thing (as your very likely wrong).
Use nature always and when teaching if you cannot reference a natural issue
to depict your notion of the piece of knowledge your about to impart, simply
don't.
This is fairly easy as everything we do is defined by nature, even the
unnatural things ! It's just important to think about it (like dividing
by adding !).
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OK back to the bush we planted, have the kids look at it and especially
look at the amount of dirt in the pot.
The amount is the same as when the thing was planted, but wait the thing
is much bigger, where did all the stuff come from ? where did the mass
of stuff that's there now, leaves, possibly flowers, more stalk or bark,
lots of stuff and it all weighs something.
It never came from the dirt, the dirt is still in the pot! what happened.
Well all the stuff came from the air (vast majority, some plants have lots
of water content, but dry a tree out and it's still lots of stuff ) now
is the time to talk of chloroform and photosynthesis not in years to come,
now when the kids can actually see what happened even if they can never
understand how they will always remember what happened and what photosynthesis
does, it takes air and sunlight and turns it into carbon, same stuff as
inside us is now inside the plant.
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Now talk about kids eating less than their parents but growing faster and
when they eat the same the kid still grows, so how's that?, why don't the
parents keep growing, where's the bones and skin all coming from, again
how is may be not so important here but what actually happens is very easy
to show.
If they don't understand all the complicated maths and science behind this
they will and should understand what the outcome is, being able to prove
it now should seem less of an issue.
It's easier to prove things you can see and touch, hear feel etc.
than some absurd equation or huge Latin word from a book.
Nature is proving to be very helpful now I hope!
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1.6.
THE ZERO INFINITY THING AGAIN
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1.6.
The zero infinity thing again
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Zero, nothing, void, empty and so on, zero what ? are zero skyscrapers equal
to zero fingers, are innite3 skyscrapers equal to infinite fingers ? well
the latter is wrong you can enter a skyscraper and go into a room, can't
do that with a finger, no matter how many you have, but the former would
appear to be true! So zero being the opposite size (or dimension) of infinity
is not true ! but there are interesting effects on one another 0 ∗ ∞ =
0 and ∞ ∗ (any number except 0) = ∞ and 0/∞ = 0 and ∞/0 = U undefined
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This is interesting if you analyse it.
So infinity can be destroyed or created by zero! Really interesting is
the undefended cases, undefended! yes in the simplest equation, never mind
Reinman's zeta function based problem or Fermat's last theorem, simply
infinity and zero together are enough to confound mathematics.
Are we not all taught that in School? I wasn't!
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Perhaps you would not have been taught that the proof of 1+1=2 has never
been mathematically proven! Two mathematicians, Russel and Whitehead wrote
over 300 pages in the book principia mathematica where this proof was attempted
, but failed to do so (Godel showed the laws in what was thought to be a
proof of this as he introduced his incompleteness theorem).
Again there is a lot we are not taught, why?
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If zero is a number is it on your calculator ? is infinity then also on
your calculator and if not why not ?
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Sorry if that's now confusing but it's actually simple and expected, first
of, none of the above answers are infinity or zero, the answers tend towards
these numbers but will never get there.
So zero and infinity are destinations a traveller can journey towards but
in many cases, alas never reach.
Therefore if somebody asks how long will it take to count to infinity you
could reply I will finish tomorrow as this would be true every time you
say it (as long as you say it at least daily).
but to count to zero on the other hand can be done very quickly, but only
if we can assume 1 is a number and can be proven! oh dear this is madness
is it not.
What do we mean can we prove there is such a thing as 1.
Perhaps it's easier to just do the repeating thing and get a certificate
after all, is it not? Simple thing here is to write 1/3 as a decimal, 0.333...,
now multiply by 3 to get 0.999....
Oh oh this is not 1.
More soon! We need to consider time and space to understand there may be
a 1, but only a 1 there cannot be two 1's and we cannot really define a
1 unless we can freeze time! Yes this is true, are you clutching your arithmeti
c certificate yet and saying this is wrong, this is wrong over and over
again.
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Well actually it's not madness it very much depends on the way you try to
get to zero, for instance if you half a number every time, you will never
get there, so these
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3I remember being told as a child by a teacher, infinity is defined as such:
if you give a monkey a typewriter and infinity it will type out the complete
works of shake spear word perfect.
I immediately thought the ribbon (for younger folk the ribbon is the old
mechanical printers version of a print cartridge) would run out, and then
the paper would run out so given this infinity with infinite objects does
not work, there's many infinities in this definition, like number of trees
(therefore planets and universes), spare parts, tools for the spare parts
etc.
etc.
in fact to me this definition was one of many and possibly infinite infinities.
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real numbers or integers (or whole numbers) are very handy as you can remove
real positive integers from another positive integer one at a time till
zero appears, but again they're approximations and no such thing exists
as I will prove later (mathematics is full of notions and real numbers
one of them).
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So back to the zero thing.
For thousands of years it did not exist along with it's partner (which
we have shown to be strange bedfellows indeed) infinity.
In fact zero was seen as the void or a place where evil lived and infinity
was seen as everything, a place where god lived (any god, it does not matter
here).
There have even cases of people being killed for attempting to use infinity
or zero (especially an early Pythagorean who mused over a thing called
an irrational (think crazy) number, which we will come to later).
Just like the system of counting we use today (the base 10 system) took
thousands of years to agree on so did the concept of zero.
Therefore when teaching children to count to 10 and being delighted, think
carefully are you instilling blind faith into them at this stage, after
all what does counting to 10 actually mean, 10 what?
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1.7.
Limits and things
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Zero is a notion, infinity is a notion and the base 10 system is a notion
(base anything is).
By this I mean none of them really exist unless we introduce limits into
nature (mathematical limits are discussed later, so don't jump ahead).
The limits we use to represent numbers are pretty straightforward but missed
by many.
For instance we can say we have 2 apples, but immediately we are working
within limits without realising it.
The limit here is our notion of an apple, after all if somebody held up
an apple and orange and stated I have 2 apples, we would realise our limits
have been breached pretty quickly.
These limits govern much of modern life and for good reason actually.
So taking this further lets say a person holds up a granny smith apple
and a golden delicious apple and says I have 2 apples, again limits are
introduced.
The person is holding up 2 dierent things but it's OK in this case the
limit has not been breached, there still apples, but there still different
things, so we can have 1 of something and 1 of another thing equal to 2,
what about dividing these 2 things now - can we? If that person then increases
the limit or parameters of his things by adding more words to the description
like saying I have 2 green apples (well they may be, but..) then we impose
further limits almost without thinking and can evaluate whether this is
true.
Add some more detail, i.e.
saying I have 2 green apples each with a leaf on them and we may break
the limit, note here the leaf in question is also limited to fall within
certain parameters for our human way of thinking.
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Therefore we can use these numbers like 2 apples within limits and only
within limits.
This is true for everything in nature and therefore everything in maths
and science as these are the study of nature, even though we forget this
quickly.
So many limits can be boiled down to individual words (like apple) and
these words inserted in our basic vocabulary, thereby allowing us to ignore
the fact that many numbers and systems of calculation are in fact not real
but purely notional.
This may hurt your head a little and it should.
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So numbers represent things limited for us to easily digest, however the
whole notion of adding and dividing can be shown very quickly to be against
nature, i.e.
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1.8.
MIDDLE EARTH
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in the above example we cannot get to 1 thing again unless we maintain a
limit, but imagine for instance we do something a bit different with this,
while we are playing about anyway.
Take the apples half each of them (or more precisely cut them in 2 relatively
equal mass objects) and we have 4 halves of apples.
Now if we take a half of each apple (preferably a red and a green one)
and put it together in our hand and say to another person look I have an
apple, our limit thinking is challenged, no you don't have an apple you
have 2 halves, would be the likely answer.
Think about this 2 times 1/2 is not one any more, our limits have gotten
us into some trouble.
How do we get over this fact then? well actually the answers pretty poor,
we train ourselves and children to ignore it, like the ancients ignored
zero as evil we currently ignore this kind of thing, to the extent many
people reading this will simply not accept it or even be able to question
it.
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But wait ! we can't use precision as we cannot measure into infinitesimally
small differences to categorise them, we need limits!.
Yes we do need limits but we also need to see those limits and understand
them, otherwise we a removing some vision that could be important in understand
ing nature related to maths and science etc.
and as we already know this is vitally important.
Just realise for now the notion (yes notion) of 1 is a notion and not really
a thing that exists without limiting factors.
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1.8.
Middle Earth
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Another interesting area not covered in modern teaching is what is sometimes
referred to as the midd le earth syndrome (well I refer to it that way).
This is looking and taking care of perspective in our assumptions of nature
and all it holds.