Forwarded at Ed Dubinsky's request. (This is the second person who can
read the list, apparently, but whose attempted postings are being refused!)
Murray
-------- Original Message --------
Subject: Re: Set notation
Date: Sun, 7 Oct 2007 19:01:28 -0400 (EDT)
From: Ed Dubinsky <[log in to unmask]>
To: Murray Eisenberg <[log in to unmask]>
CC: [log in to unmask]
References: <001501c8091e$9c7d8600$eeebf004@u2v8c5>
<[log in to unmask]>
Bill Fenton and I developed a course on elementary discrete mathematics
using an innovative pedagogy and supported by a textbook. It has been a
few years since I taught this, but Bill has done it more recently and may
be able to give some details. I believe that using our approach, the
difficulties mentioned in this discussion and many others can be dealt
with effectively. We have done some studies that produce data that tends
to support the effectiveness of our pedagogical strategies.
Our pedagogy is based on theoretical analyses of the mental activity that
can lead to learning the mathematical concepts that are treated. It uses
cooperative learning, students doing certain relatively simple tasks on
the computer and small-group problem solving in the classroom.
Ed
On Sun, 7 Oct 2007, Murray Eisenberg wrote:
> Ted Stanford wrote:
>> I have seen students, even graduate students, read
>> the empty set symbol as "phi" rather than as "empty set".
>
> Perhaps that's because nobody ever told them that the symbol comes from a
> letter of the Norwegian alphabet.
>>
>> One thing we might try is not using the phi-like symbol
>> in introductory classes. We could use the empy bracket
>> symbol {} exclusively at first, which would give students
>> a constant reminder of the meaning.
>>
>> If you list the power set of {1,2,3} without using the
>> phi-like symbol, then every element of the list
>> has the same overall structure - a pair of brackets
>> with some stuff in between, and maybe this
>> would make more sense to beginning students.
>
> Yes, of course: I've tried that, too.
>>
>> I'm not sure I like the "lion in the cage" explanation.
>> It may work as a memorization device (like "soh cah toa"
>> for remembering which trig function is which), but I
>> don't see how it would help students make real sense
>> out of set theory notation.
>
> What's wrong with analogies? That's one way humans make sense out of new
> ideas.
>>
>> -Ted Stanford
>>
>> -----Original Message-----
>> From: Murray Eisenberg <[log in to unmask]>
>> To: [log in to unmask] <[log in to unmask]>
>> Date: Sunday, October 07, 2007 9:31 AM
>> Subject: Re: Set notation
>>
>>
>>> 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
>>>
>>> {Ã}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}
>>>
>>> or as
>>>
>>> { {Ã}, {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 {Ã} is not the same thing as Ã.
>>>
>>> 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 {Ã} 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 "{Ã}" 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 "{Ã}"
>> 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, "{Ã}" 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
>>>> "{Ã}" 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 "{Ã}" . 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
>>
>
>
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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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