Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » Education » mathedu

Topic: Labels
Replies: 10   Last Post: Oct 14, 2009 5:17 PM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Guest

Watch Watch this User
Labels
Posted: Oct 13, 2009 9:55 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

I have to say I am with Tony and Anne on this point. Contrary to what
Jonathan writes:

"Tony, it does help to discuss labels of circles and other geometric
objects, especially if there several circles, several triangles, several
angles, etc. in the same discussion or diagram so that it is clear which
object we are referring to. However, whether a geometric figure is a
circle or not does not depend on labels (or even if it is labeled).
Students should know this fact. Labels are not parts of definitions of
geometric objects because the property of a geometric object does not
depend on its name for it; rather, we use labels for convenience or
sometimes out of necessity to make it clear which object is being
mentioned."

I think it is very important in a number of situations to have labels.
One simple example follows. One can think of many more, as Tony has
indicated. My daughter was, many years ago, in year 7 at a private girls
school of some repute in Melbourne, Australia. She came home having to
study geometry over Easter. The prescribed text's "definition" of a
polygon was "A polygon is made of straight lines." I began by throwing
some pick-up-sticks on the floor. "No, not like that!" my daughter
exclaimed, and showed me a drawing of what I would think of as a
polygon. I drew some lines, going from point to point, but not closing
up the figure. She objected and indicated the figure should close up. I
drew 4 points, labelled 1,2,3,4 (as if cyclically arranged at the
corners of a square, yet to be drawn) and proceeded to join them 1 to 3
to 4 to 2 to 1. My daughter objected that the two "diagonal' lines
crossed. We talked back and forth (definitions and refutations?) until
she and I agreed - after looking at several other texts - that, for us,
pre-requisite data to define a polygon is a set of points labelled by
positive integers, joined from first to last, in order, and with a
couple of other conditions obtaining (no three points in a line, no
lines crossing except at specified points). She was happy, she could do
the set problems without confusion and after Easter she taught her year
7 teacher something about polygons (related to the example above with 4
points).

Can one define polygons in other ways. Yes one can. Is the labelling an
intrinsic part of the definition as she and I framed it: yes it is. Do
the properties of the resulting figures depend on the labelling: yes
they do.

Tony's examples are even more compelling. And I haven't even mentioned
clarity of thought and perception as did Tony and Anne.

As Tony points out there is an essential difference between an arbitrary
after-the-fact naming of parts, and labelling that is intrinsic and
essential to the definition and to a deeper understanding.

Gay Davis





TONY GARDINER wrote:
>> whether a geometric figure is a circle or not does not depend on
>> labels (or even if it is labeled). Students should know this fact.
>> Labels are not parts of definitions of geometric objects
>>

>
> No wonder students have difficulties if those of us who teach them
> are so confused, so happy to parade our confusion in public, and so
> reluctant to rethink when it is clear that we have got something
> wrong!
>
> There is a very basic confusion here.
>
> If God were to exist, then he would probably not owe his existence to
> whatever "name" we might use. However, as the OT use of "Jahweh" -
> or "JHWH" - illustrates, we mortals may need a name in order to grasp
> the idea and to communicate with one another about it.
>
> For all I know it may be true that the *idea* of a "function" does
> not need the use of letters as names for entities/variables/elements
> of the domain/etc. - such as "f" and "x". But I defy anyone to make
> the idea of function *clear* and useable without invoking such
> labels.
>
> In exactly this sense, the definition of a circle OBLIGES anyone to
> give a name to the centre and to a radius.
>
> And the definition of an isosceles triangle includes the base (and
> apex) and so requires us to label the vertices - say ABC (at which
> point one can say clearly:
> "If in triangle ABC we have AB = AC,
> we say it is *isosceles with apex A and base BC*".
>
> Those who prefer to try circumlocutions (for circle, or for isosceles
> triangle) will fail, and will at the same time increase the level of
> totally unnecessary confusion in the world.
>
> The things that mathematics forces on us are quite different from
> "life-style choices". Noone cares whether I am for or "against"
> geometry books giving correct definitions. But anyone who insists
> that
> * given a labelled triangle ABC
> * ABC and BCA are the same triangle
> should never be let near a geometry class.
>
> I tried to say this once - and I clearly failed.
> I apologise for having now had to try to say it more clearly in case
> it was genuinely not understood.
>
> Tony
>
>




Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.