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Re: Set notation
Posted:
Oct 19, 2007 4:39 PM



Ed Dubinsky cannot get his messages read by MATHEDU, so here is the response he wanted to make.
Bob Burn Research Fellow, Exeter University Sunnyside Barrack Road Exeter EX2 6AB 01392430028
Original Message From: Ed Dubinsky [mailto:edd@math.kent.edu] Sent: Wed 17/10/2007 13:06 To: Burn, Robert Cc: MATHEDU@jiscmail.ac.uk Subject: Re: Set notation See below for responses.
On Wed, 17 Oct 2007, Burn, Robert wrote:
> Just two thoughts. > 1. To regard the empty set as a thing, is quite a step. I dont think it > rates as a thing from the perspective of Euclid's Elements.
I agree completely.
> 2. Some autobiography might be illuminating: at what point did readers > of this list recognise the distinction between the empty set and {the > empty set}? I think for me it was at the construction of the natural numbers.
I don't know about myself, but I can tell you what works really well (that means, a high percentage of students get it): Having students write computer programs that construct sets (including the empty set) and perform actions on them such as checking their cardinality, forming unions, intersections, etc.
Ed
> Bob Burn > Research Fellow, Exeter University > Sunnyside > Barrack Road > Exeter EX2 6AB > 01392430028 > > > > Original Message > From: Postcalculus mathematics education on behalf of Smith, Alexander J. > Sent: Sun 07/10/2007 23:50 > To: MATHEDU@JISCMAIL.AC.UK > Subject: Re: Set notation > > Let us not forget the following word of Feynman. > > (My humble experience is that it is a happy event when an undergraduate mathematics major can intuitively distinguish between the empty set and the set which contains only the empty set.) > > Feynman's words: > > The power of instruction is seldom of much efficacy except in > those happy dispositions where it is almost superfluous. > > There isn't any solution to this problem of education other than > to realize that the best teaching can be done only when there > is a direct individual relationship between a student and a good > teachera situation in which the student discusses the ideas, > thinks about the things, and talks about the things. It's impossible to learn very much by simply sitting in a lecture, or even by simply doing problems that are assigned. > > > ___________________________________ > From: Postcalculus mathematics education [MATHEDU@JISCMAIL.AC.UK] On Behalf Of Ralph A. Raimi [rarm@MATH.ROCHESTER.EDU] > Sent: Sunday, October 07, 2007 5:17 PM > To: MATHEDU@JISCMAIL.AC.UK > Subject: Re: Set notation > > 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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I am going to move to a new internet access system, but the change will be a long process. For the foreseeable future, I will use two systems simultaneously and at some later date, I will drop one of them.
So, until further notice, please send all messages for me to both of the following addresses:
edd@math.kent.edu eddub@mindspring.com
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Ed Dubinsky edd@math.kent.edu eddub@mindspring.com http://www.math.kent.edu/~edd/
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