set notation
- From: wellsoberlin <wellsoberlin@xxxxxxxxx>
- Date: Sat, 06 Oct 2007 19:05:11 -0700
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 whoseonly 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}" .
.
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