Re: Is this math test too easy?
From: J.E. (troubled6man_at_yahoo.com)
Date: 11/27/04
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Date: 27 Nov 2004 15:12:52 -0800
Tim Brauch <RnEeMwOs.pVoEst@tbrauch.cNOoSPAMm> wrote in message news:<Xns95AEC270C8B8webmastertbrauchcom@63.240.76.16>...
> troubled6man@yahoo.com (J.E.) wrote in
> news:39d6e584.0411261009.1160f7be@posting.google.com:
>
> >> Pure mathematics uses a very ambiguous notation in which symbols mean
> >> one thing at one level or in one discussion, and something different
> >> in another domain or another discussion. I'd expect mathematics to
> >> use a notation that is completely and universally unambiguous, like a
> >> computer programming language. I don't understand how f(x) dx can
> >> represent an integral of a function if the syntax appears to say "f
> >> times x times d times x" from an algebra standpoint. Much is left
> >> unsaid in mathemtical notation, and it's very irritating, especially
> >> since mathematicians like to brag about how precise mathematics is.
> >
> > I would love to fix this. There is absolutely no reason (that I know
> > of) that we (mathematicians or those who use mathematics) shouldn't be
> > able to fix the symbols for common mathematics so that every symbol
> > has a direct meaning unabmbiously and independant from context. That
> > could even be the standard of "common mathematics" when some registers
> > a new symbol to be used for an exclusive purpose. One way is have
> > economy is to define some symbols broadly, like addition so that they
> > have a definition for a wide variety of objects (scalars, vectors,
> > functions). And the other is to simply use more symbols, which will
> > make conversations about math in ASCII more difficult, but forums like
> > Usenet can always rely on context as we do today.
>
> One problem is that in Abstract Algebra,
Wow, I should flesh out my comment better if you misread it THAT
badly. Anything that it for math people can be abstract. We could
even have a notation like circle addition or circle multiplication
that MEANS that it doesn't have a standard interpretation, but then we
don't teach anything with a circle in it in primary or secondary
school (maybe).
> you often define the operation
> "*" or "x" and sometimes it really means addition and then powers mean
> multiplication. Then again, looking at permutation groups, you need
> "*" to be defined as composition (not to mention either left or right
> composition). And thus, the main problem with standardizing notation is
> that certain properties exist beyond our notation. Even if we
> standardized a symbol for an action in an algebraic group, with every
> group, the action can be defined differently.
But let's not teach Abstract Algebra in high school then, OK? Or do
you disagree? If we do, then let's not use a+b and axb or a*b or
(a)(b) or whatever it is that we agree to be standard for normal
addition and multiplication.
> And you run into the same problem with equivalence relations and partial
> orderings and total orderings in combinatorics. You use the same
> symbol, but define the relationship to mean something specific in that
> case.
Then use a circle for "use context" or make different symbols. You
could prove a theorem about a "circle <", which I'll call o< and then
have many symbols < ~< << that satisfying the o< properties. I think
a reserved set of variables and clarity about when a definition is
abstract and when specific is good.
> And if we want to look at applied math, getting away from all of the
> abstractness, try doing matrix manipulations. if A and B are matrices,
> and you need to multiply them together in the matrix sense, you would
> write AxB (or maybe A*B or something else similar). However, if you
> wanted to multiply them together component-wise, that is
> C(i,j) = A(i,j)*B(i,j) you need something to define that.
Having a standard notation where (i,j) means {{0,i},{1,j}} is good
especially if (i,j,k) means {{0,i},{1,j},{2,k}}, where the pattern is
known and standard. If A is defined as a function A(i,j) and B is
defined as a function, then yes you could define a new function
C(i,j)=Sum_k A(i,k)*B(k,j), sometimes people use summation conventions
pretty well, we'd have to get mathematicians to agree and probably to
some studies in math education research to find out what is easiest to
teach.
> In reality, the best you can hope for is defining symbols at the
> beginning and make sure things are well-defined at the beginning.
The problem is that for pedagogical purposes we'd have to decide what
was "the beginning" and stick with it the whole time.
> Then there is the whole issue of variables and constants. Sometimes
> "i" is a constant in an equation, sometimes it is a variable.
Well, that's just sloppy. A variable could be Vi, and an individual
could be Ii. Just because people have been historically sloppy
doesn't mean you should be sloppy yourself.
> And most
> characters can be used in a similar fashion. Even matrices and sets in
> ASCII are usually denoted by capital letters.
If you reserve V and I for Variable and I for individual, then yoy can
VA VB or VC for variables are you want and IV IB or IC for individuals
all you want.
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