On Sun, 7 Oct 2007, Murray Eisenberg wrote:
> I just experienced this phenomenon (again!) in the first exam in our proofs
> course, where the question was to list the elements of the power set of
> {1,2,3}.
>
> Several students gave the answer as
>
> {Ø}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}
>
> or as ... but to wonder what theory can overcome general linguistic
> insensitivity. The relevant research might involve much earlier stages of
> mental and linguistic development.
An anecdote from the days of "the new math" in America, when
elementary school teachers were instructed to tell the kiddies about sets,
unions and such:
The teacher, having seemingly absorbed the idea of distinguishing
between a set and its members, and bent on transferring the lesson to her
class, asks "the set of all boys to stand up", and then, "the set of all
girls to stand up".
(Excuse me: I meant "bent on transferring the lesson to the
members of her class". The class cannot absorb a lesson any more than
the set of all boys can stand up.)
Which is to say that we (even mathematicians) are accustomed
to conflating the set with its members in daily speech, and have really no
reason to be pedantic about it until careful reasoning in mathematics
requires it of us. I see little reason to try to teach such things before
university mathematics begins to consider theorems regarding which, and
regarding whose proofs, this distinction has some application. As we all
have seen, the lesson simply won't go over, except for some few who don't
need it anyhow, not even if they aspire to careers in science or
mathematics, for they will learn it easily enough when the time comes.
Ralph A. Raimi Tel. 585 275 4429 or (home) 585 244 9368
Dept. of Mathematics, Univ.of Rochester, Rochester, NY 14627
<http://www.math.rochester.edu/people/faculty/rarm/>
"Algebra is conducive to symbolic reasoning." ....PSSM, p.345
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