I would try
"set of objects"
and "set of sets"
Thanks God I teach Calculus because otherwise some smart student would come 
up with "set of sets of sets" to which I wouldn't be able to say anything 
smart (other than continuation).

Kazimierz

At 01:01 PM 10/7/07, Murray Eisenberg wrote:
>Of course I draw attention to that distinction.  It makes no difference to 
>many of the students!
>
>Bruno Nachtergaele wrote:
>>I believe it helps if you explictly draw their attention to the fact that
>>{ } is the empty set
>>but
>>{{ }} and {\emptyset} aren't.
>>Bruno
>>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
>>>
>>>  { {Ø}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} }
>>>
>>>That, despite examples to the contrary in class, and despite the comment 
>>>in the text, restated and dramatically emphasized at length in class, 
>>>that just as a lion in a cage is not the same thing as a lion uncaged, 
>>>so {Ø} is not the same thing as Ø.
>>>
>>>I don't see them nearly as often confound 1 with {1}, for example.
>>>
>>>It's unclear to me how much research into how/why students at this level 
>>>are misinterpreting {Ø} will help lead to eradicating the 
>>>misconception/mislearning.  This is not to knock theory as a basis for 
>>>action, but to wonder what theory can overcome general linguistic 
>>>insensitivity.  The relevant research might involve much earlier stages 
>>>of mental and linguistic development.
>>>
>>>But the more pressing question is what activities at the proof-course 
>>>level might disincline students from making this mistake.
>>>
>>>Charles Wells wrote:
>>>>Students commonly think that the notation "{Ø}" denotes the empty set. 
>>>>Many secondary school teachers think this, too.
>>>>
>>>>Mistakes in reading math notation occur because the reader's 
>>>>understanding of the notation system is different from the author's. 
>>>>The most common bits of the symbolic language of math have fairly 
>>>>standard interpretations that most mathematicians agree on most of the 
>>>>time. Students develop their own non-standard interpretation for many 
>>>>reasons, including especially cognitive dissonance from ordinary usage 
>>>>and ambiguous statements by teachers.
>>>>
>>>>I believe (from teaching experience) that when a student sees "{1, 2, 
>>>>3, 5}" they think, "That is the set 1, 2, 3 and 5". The (incorrect) 
>>>>rule they follow is that the curly braces mean that what is inside them 
>>>>is a set. So clearly "{Ø}" is the empty set because the symbol for the 
>>>>empty set is inside the braces.
>>>>
>>>>However, "1, 2, 3 and 5" is not a set, it is the names of four 
>>>>integers. A set is not its elements. It is a single mathematical object 
>>>>that is different from its elements but determined exactly by what its 
>>>>elements are. The correct understanding of set notation is that what is 
>>>>inside the braces is an expression that tells you what the elements of 
>>>>the set are. This expression may be a list, as in "{1, 2, 3, 5}", or it 
>>>>may be a statement in setbuilder format, as in "{x x > 1}". According 
>>>>to this rule, "{Ø}" denotes the singleton set whose only element is 
>>>>the empty set.
>>>>
>>>>This posting is based on the belief that that mathematical notation has 
>>>>a standard, (mostly) agreed-on interpretation. I made this attitude 
>>>>explicit in the second paragraph. Teachers rarely make it explicit; 
>>>>they merely assume it if they think about it at all.
>>>>
>>>>The student's interpretation is a natural one. (Proof: So many of them 
>>>>make that interpretation!) Did the teacher tell the student that math 
>>>>notation has a standard interpretation and that this is not always what 
>>>>an otherwise literate person would expect? Did the teacher explain the 
>>>>specific and rather subtle rule about set notation that I described two 
>>>>paragraphs above? If not, the student does not deserve to be ridiculed 
>>>>for making this mistake.
>>>>
>>>>Many people who get advanced degrees in math understood the correct 
>>>>rule for set notation when they first learned it, without having to be 
>>>>told. Being good at abstract math requires that kind of talent, which 
>>>>is linguistic as well as mathematical. Most students in abstract math 
>>>>classes are not going to get an advanced degree in math and don't have 
>>>>that talent. They need to be taught things explicitly that the hotshots 
>>>>knew without being told. If all math teachers had this attitude there 
>>>>would be fewer people who hate math.
>>>>
>>>>PS: My claim about how students think that leads them to believe that 
>>>>"{Ø}" denotes the empty set is a testable claim. There are many 
>>>>reports in the math ed literature from investigators who have been able 
>>>>to get students to talk about what they understand, for example, while 
>>>>working a word problem, but I don't know of any reports about my 
>>>>assertion about "{Ø}" .  I would be glad to hear about any research in 
>>>>this area.
>>>
>>>--
>>>Murray Eisenberg                     [log in to unmask]
>>>Mathematics & Statistics Dept.
>>>Lederle Graduate Research Tower      phone 413 549-1020 (H)
>>>University of Massachusetts                413 545-2859 (W)
>>>710 North Pleasant Street            fax   413 545-1801
>>>Amherst, MA 01003-9305
>
>--
>Murray Eisenberg                     [log in to unmask]
>Mathematics & Statistics Dept.
>Lederle Graduate Research Tower      phone 413 549-1020 (H)
>University of Massachusetts                413 545-2859 (W)
>710 North Pleasant Street            fax   413 545-1801
>Amherst, MA 01003-9305
>