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Posted:
Oct 14, 2009 8:01 AM
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>> anyone who insists that
>> * given a labelled triangle ABC
>> * ABC and BCA are the same triangle
>> should never be let near a geometry class.
> Here he is making this fundamental error, of not distinguishing
> between a triangle and a labelled triangle.
Perhaps it is not I who have failed on this score. (Nothing is
gained and much is lost by labelling things we find uncomfortable
"new-math pedantry" or "descriptivist poppycock".)
This remarkable exchange shows that
* though psycho-logically and developmentally the visual image of
* "triangle" is prior to the mathematical notion of "labelled
* triangle"
* logically the notion of "triangle" is secondary.
This is absolutely standard in mathematics.
(Aside: It also explains why it is dangerous to say glibly
"mathematics is all around us". There is something all around us
that is related to mathematics, or even that gives rise to
mathematics. But mathematics is an artificial activity that is quite
different from whatever it is that is "all around us".)
We can *suppress* the labelling.
We can *imagine* the labelling whilst recognising that the particular
labels are arbitrary (like the x and the f in our favourite
definition of a function) - and hence we can pretend that the
labelling is irrelevant.
We can talk informally amongst ourselves about "triangles" without
explicitly mentioning the labelling.
However, as we have proved incontrovertibly here, the conversation
will only be fruitful if we all know that the triangles we are
talking about carry a hidden "shadow-labelling" (with arbitrary, but
for the moment unknown labels).
Informal words like "triangle" (unlabelled) are like pronouns.
They only make sense if one remembers (informally) that they need
some kind of quantification for what is the left unspecified.
Students fail to solve the simplest problems about circles because
they have been taught by teachers who failed to insist that
* the centre is part of the circle*
and so should be marked, and will probably be needed.
Many conversations use pronouns in a way that is either cavalier or
deliberately deceptive - so that different speakers (e.g. husband and
wife; or employer and union) can sometimes avoid disagreeing openly
because they interpret the same pronoun in different ways.
This is OK as long as the conversation does not really matter.
As soon as it does (as in mathematics, or in serious domestic or
industrial disagreements), this kind of fudge is a recipe for
disaster.
Pronouns are very useful (as is the word triangle); but they are
useful for precise communication only as long as all parties
understand what they are actually referrring to (namely a labelled -
ordered triangle) and agreeing not to be more specific than is needed
at the time.
Good teaching sometimes obliges us to smudge mathematical
distinctions. (Whoever writes this is no "new-math pedant"!)
At other times one teacher or one system may choose to smudge, while
other teachers or systems choose not to.
(This is perhaps most transparent in plane geometry where different
systems make very different choices - say in ensuring that the
notation helps to make important distinctions between some or all of
* line AB (doubly infinite)
* line segment AB (e.g. covered by a bar)
* length of line segment |AB| [bar], or m(AB[bar])
* ray AB [covered by arrow] (half infinite starting at A and passing
through B).
In England all of these are denoted by "AB", and one can observe the
result (in contrast to many European countries).
The advantage of labelling is that it allows one to make precise,
unambiguious statements and to clarify meanings.
Any attempt to capture SAS congruence without labelling the two
triangles reduces mathematics to the level of word play.
("If AB = A'B'. <ABC = <A'B'C', BC= B'C', then triangles ABC and
A'B'C' are congruent" is completely clear, provided one has the gut
knowledge - which all grade 9 kids should have - that the labels are
what we might call "free", so are not themsleves part of the
definition. Noone gains - and the students all lose out - if we try
to replace this with verbiage. We all probably use infomral
shorthand such as "two sides and the included angle", but to get
large numbers of students to enjoy applying the SAS criterion
successfully, it is almost essential to begin by requiring them to
lay out their congruence so that corresponding vertices are aligned
(vertically):
A B C
A' B' C'
Only in this way can one begin to mechanise the process Anne Watson
so rightly highlighted: the process of getting students to "see"
those features of a situation that the mathematics represents so
subtley and so economically.
Tony
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