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May 15th, 2008, 10:16 AM   #1
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natural numbers from sets....not very natural

Hi people, first post here.
For some reason the other day, I started writing a small math paper, not academic or anything, more a simple demonstration of how maths can come about, aimed mainly at my little brother who is 8 and for my own fun/interest.

Now I like to always think of how mathematical ideas came about and the history of techniques etc.

I am familliar with math history and number theory. One thing that I don't like the idea of is Cantor's method for introducing the natural numbers through sets, where we have the empty set, or null set = 0 then the set whos only member is the empty set = 1, and so on.

I am not having a go at Cantor or sets, sets are really useful and great ideas and Cantor is clearly a brilliant mind but it doesn't seem natural, or too abstract.

Below I will post the first page of what I wrote and would love to hear feedback. none of it is new but it provides a simple view of how to introduce math through natural things. Also think an 8 year old may prefer it to sets

Quote:
Here is how to go about deducing numbers

from nature.

We start off by the simple notion of either

having NOTHING or SOMETHING.
By way of convention we will define NOTHING

to be the same as the symbol 0 and give the

sentances "the same as" and "equal to" the

symbol =.

In the same way we will give the meaning of

SOMETHING the symbol 1.
The symbols 0 and 1 have the english names

of zero and one respectively.

Now say we have "Something", like a book,

and then we have that book and another

book! We will define "and another" and

"plus" to have the symbol +.

So far we can say: A book plus another book

= something + something = 1 + 1

Adding more books to the pile in this

manner we could get, 1+1+1+1+1+1...
... means that we could repeat the process

or sequence forever.

Now we come to counting. counting is

basically asking the question, how many?
How many books do we have? We could give

the answer as 1+1+1+1+1.
However this isn't really useful and could

become too complex quickly, e.g. How many

stars in the sky?

So we shall define some new symbols to

help, along with the english word for the

symbol:

0 + 0 = 0 = zero
0+1=1 = one
1+1=2 = two
1+1+1=2+1=3 = three
1+1+1+1=3+1=4 = four
4+1=5 = five
5+1=6 = six
6+1=7 = seven
7+1=8 = eight
8+1=9 = nine

all of these symbols,

(0,1,2,3,4,5,6,7,8,9), we shall call
Numbers.

We shall also say + is an arithmatic

operation, namely addition or plus, on

these numbers.

So with what we have so far we could say we

have "six" books or 6 books, shorter than

1+1+1+1+1+1 books but we certainly don't

yet have enough numbers to count the stars!
So as I said very elementary but from it we now have what numbers are. And from here we could introduce the concept of a set, as a set of the numbers so far, however, I will leave sets till later.

I would like to continue this and i think the natural progression of subject would be: Number system (base 10), subtraction, negative numbers and the number line, multiplication and division.

I would love to hear your thoughts. Thanks
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May 15th, 2008, 10:52 AM   #2
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Sets are a good basis for mathematics (though other bases can be used instead), but they're terrible for pedagogy. I would teach numbers as counting to children (ages perhaps 1-5), and move to numbers as distance (around perhaps age 4-5: going from the discrete to the continuous). Only much later (high school or early college) would I reverse this by showing that integer problems are, generally speaking, far more difficult than real-number problems. Only *after* that would I introduce the formalist bases for all mathematics: Zermelo-Frankel set theory, Peano/Robinson arithmetic, the Axioms of Choice and Foundation, etc.
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May 16th, 2008, 02:38 PM   #3
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Eight is old enough for the approach to take into account the child's abilities. For instance, does he like mathematical problems or puzzles? Would he have the tenacity and patience to "solve" the following:

  CROSS
  ROADS +
----------
DANGER

where each letter consistently stands for a different specific digit?
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May 16th, 2008, 05:16 PM   #4
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I didn't mean it was to teach an 8 year old, I was merely saying that i liked the idea of coming up with all of maths from nature.

Obviously my brother knows what counting is and distance and crossing the road and a number of activities, if he wasn't interested in this then i wouldn't try to force him.

I said it was just fun for me to write this too and a good way to introduce maths?
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May 16th, 2008, 11:55 PM   #5
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Is he sufficiently interested (and able) to solve the problem I posted above?
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May 17th, 2008, 12:49 AM   #6
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I wasn't really writing this for him, and he probably doesn't have that much of an interest and probably wouldn't be able to solve that.

It was more just a fun little thing for me to do and i thought it would appeal to a younger audience.
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May 17th, 2008, 12:58 PM   #7
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There's nothing I can see in your approach that would make it interesting to youngsters. Also, it doesn't draw attention to the fact that, for example, five books, five coins, five sheep, etc., share something: "fiveness"!
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May 18th, 2008, 04:51 AM   #8
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Quote:
Originally Posted by skipjack
Also, it doesn't draw attention to the fact that, for example, five books, five coins, five sheep, etc., share something: "fiveness"!
This first abstraction, the most fundamental of all mathematics, has always struck me as something profound enough to merit a name.
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May 18th, 2008, 06:06 AM   #9
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Quote:
Also, it doesn't draw attention to the fact that, for example, five books, five coins, five sheep, etc., share something: "fiveness"!
To be honest, I don't understand what you mean by this.
Again I think that sounds a little too abstract, just as the set example I gave.
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May 18th, 2008, 12:47 PM   #10
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Skipjack means the object delimited by the noun phrase five markers shares the property of fiveness with the object five coins. That property is the property of having 5 member, hence the name 5-ness.

The problem with the statement "it's too abstract" is that math is abstract. Everything done in math is done on either an abstract object, or a generalized token.

I don't see how "fiveness" is any more abstract than calling a "something" a "1", and a "nothing" a "0". (pedantry follows) What something are we discussing? Can it be anything? (nevermind the abstract required to understand the concept of "anything".) Does it have to be a book? a box? What about a gallon? (and why that unit of measure?)

The something versus nothing distinction is more natural, as evidenced by the fact that certain indigenous cultures have names for only those two concepts-- certain other cultures have "none", "one", "few", and "many"-- but I wouldn't call it less abstract, and the concept of "k-ness" as a noun is a more useful abstraction than the distinction between something and nothing, as it also encompasses that distinction.

I'm inclined to agree with CRG, but I think sets should be introduced and played with a little earlier, although I think the utterly abstract formalities should be left for later in the education.
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