>There's no need to drag Grassman algebras or any multidimensional
>concepts into it.
>
I guess my point is that vector elements seem to become necessary
(as primitives) to support spatial extension -- in higher math.
The real numbers aren't given this power on their own -- only=20
get to "scale" the vectors. So the number line itself, according
to such algebras, has no definition minus the underlying vector
concept.
>The progression from length to position (in 1 dimension of course) is
>not too extreme.
.... but I agree with you. It's the formalizers who might have a
problem with the number line in the absence of any formal vector
concept. For an ordinary shirt sleaves math teacher, it really
makes no difference, since numbers and lengths arise from a=20
common historical root (geometric explorations), independently
of any formal concept of a vector space, or even "real numbers"
(came much later in time).
Nevertheless, I'd probably tell kids earlier, rather than later,
that vector spaces are ahead, and this number line thing is=20
going to be revisited with a different toolkit of concepts,
if they hang in long enough to get to that part.
>It's not completely necessary distinguish too assiduously the three
>normal uses of the - symbol:
>
>(1) an integral part of a number symbol as in -3, sometimes written
>elevated
>(2) a shorthand for "additive inverse of" anything, including a
>negative number, as in -(-3)
>(3) subtraction
>
I agree here as well. I'm happy enough to see '-' as an operator
across the board. -3 is a unary operator applied to a number. =20
The only reason we want the whole expression to be 'a number'=20
is we have no simpler way of writing it. It's a 'terminal'=20
expression. Likewise we wouldn't usually say 1/2 is a 'division=20
problem' -- unless we're posing it in the context of converting=20
to decimal or base two or some such.
>There is no commonly use algebraic structure (algebra, field, monoid,
>ring) for which this can cause confusion. Since subtraction is
>nonassociative, symbols like a-b-c are ambiguous and should be
>avoided, notwithstanding the normal custom towards
>left-associativity.
a - b - c =3D a + (-b) + (-c)=20
Usually this operation is defined to be associative, no? (but
maybe not commutative -- Abelian groups and rings only). It's=20
multiplication where associativity goes out the window a lot=20
of the time (matrices, quaternions and the like).
Reference: http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/vector.html
>For ordinary life, 1 dimensional distances are more relevant than
>vector positions since people need to confine themselves to fixed
>roadways, and not take to the air.
>
Well, roadways are left/right and up/down affairs as well.
My personal view is that in the age of the computer (where=20
a lot of the drudgery can be moved off to silicon), we=20
should be doing a _lot_ more to give students access to
volume, not keep all the math so flat. I'd do polyhedra=20
before I went to the relative abstraction of plane=20
geometry, left to my own devices. I call this approach
"Beyond Flatland" and write about it a lot at my website.
>In short, negative numbers are highly relevant to everyday life, not
>limited to their use as components of three-dimensional vectors.
And I completely agree. However I go back to my argument that
the math curriculum is maybe not all that well defined, as it
claims to be a formal, rigorous approach to the subject, and
yet makes all these tacit agreements and compromises in the=20
name of getting itself transmitted to the next generation. =20
My response is not that we should tighten it up more, necessarily,
but that we should accept that math is one of the humanities, a
liberal art, and depends on context (e.g. everyday life) as much
as the next discipline at many critical junctures. This, to me,
would be a more honest approach than pretending it all rests on
some "foundation" too complex to be shared with beginners (e.g.
set theory or some such claptrap). I'm from the Wittgenstein
camp, more than Bertrand Russell's.
Kirby
----------------------------
e-mail the k12.ed.math moderator at kem-moderator@thinkspot.net
submissions: post to k12.ed.math or e-mail to k12math@sd28.bc.ca
newsgroup website: http://www.thinkspot.net/k12math/
newsgroup charter: http://www.thinkspot.net/k12math/charter.html
****************************************************************************
* To post to the list: email mathedcc@archives.math.utk.edu *
* To unsubscribe, send mail to: majordomo@archives.math.utk.edu *
* In the mail message, enter ONLY the words: unsubscribe mathedcc *
* Words in the Subject: line are NOT processed! *
* Archives at http://archives.math.utk.edu/hypermail/mathedcc/ *
****************************************************************************