Here is another issue in basic understanding of set
theory. Students often don't understand that the elements
of a set are always distinct from each other. They don't
understand, for example, that {2,2} = {2}. There are a lot
of good reasons for them not to understand this. Some
examples:
1. Elementary school textbooks will often show a picture of a "set
of apples" which contains five identical apples (or flowers, or
cars, etc). The books are mostly concerned with cardinality, but I think
pictures like this are misleading (at least for those students
who later on need to understand precise set theory).
2. Students learn a lot these days in K12 about "data sets".
Of course, it is perfectly legal for a data set to have repeated
elements. The data set 2,2,3,4 is not the same as the data
set 2,3,4. So again, the word "set" is being used in a way
that is mathematically meaningful, and yet is at odds with
its use in set theory.
3. A popular method in middle shools for finding gcd and lcf
is to make a Venn Diagram of the prime factors of both numbers.
For the numbers 12 and 18, for example, one would put the prime
factors of 12 in one circle, the prime factors of 18 in the other, and
their common prime factors in the intersection. Of course, in
order to make this work, you have to list the prime factors with
multiplicity. So there would be a 2 and a 3 in the intersection of
the circles, while the 12 circle would have an extra 2, and the 18
circle would have an extra 3. It's a nice visualization of the comparison
of the two prime factorizations, but once again we are dealing with
"sets" which are not sets in the standard mathematical usage.
-Ted Stanford
-----Original Message-----
From: Charles Wells <[log in to unmask]>
To: [log in to unmask] <[log in to unmask]>
Date: Saturday, October 06, 2007 8:02 PM
Subject: Set notation
>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.
|