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
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