[Maths-Education] concentric and congruent circles

[John Bibby]

Dear Kate

Greetings and thanks for your thoughts, which are partly about terminology
and partly about pedagogy.

As far as 'correct' terminology is concerned (i.e. the end-result which we
would like students to use in public discourse) I feel we should go for
'nested' definitions rather than definitions that exclude each other.

So primes, integers, reals, complex etc is a nested sequence and similarly
with trapez-thingies and paralelograms etc.

Thus concentric circles can be congruent ...  and v.v.

You say "I've been quite puzzled ... it might be about identity: if two
circles precisely overlap (i.e. are congruent and concentric), can we say
that two distinct circles exist? "

Mathematicians would be wise to steer well away from discussions about
'existence': in what sense does anything we describe 'exist'? Also, if two
things suddenly cease to exist when they are congruent, which one
disappears? And if we say that 2+3=5: does '5' exist AS WELL AS '2+3'. I
would say "Anything exists (matematically) once we have defined it". Even
the null set exists. (More generally - away from mathematics - "nothingness
exists" - wasn't 'le neant' the Sartrian term for this?

Now where pedagogy is concerned, things may be different - and pedagogy in
Malaysia may be different from that in Milton Keynes (and if so, we can
probably learn from examining the differences).

I hope that some of this helps rather than confuses.

Best regards

JOHN BIBBY




On 28 January 2010 20:51, Kate Mackrell <katemackrell at sympatico.ca> wrote:

>
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> I'm in the middle of quite an interesting experience regarding the
> definition of these mathematical terms, which I thought might be worth
> sharing.
>
> I'm developing some files for an Asian educational software company, and in
> the course of this development have been discovering that many terms have
> slightly different definitions in different places.  One example is
> "adjacent angles".  To me, these are angles on a straight line, that  add up
> to 180.  To this company, these are angles with one common arm and a common
> vertex and no specific sum.  Another example is of course the
> trapezoid/trapezium - does it have only one or at least one pair of parallel
> sides?  I've been quite happy to go along with the the company's terminology
> (I rather prefer their definition of adjacent angles), but we've now hit a
> sticking point on concentric and congruent circles.
>
> To me (and Wolfram, and lots of other authoritative online dictionaries of
> mathematics), two circles are concentric if they have the same centre.  The
> radii of the circles are immaterial - hence two congruent circles may or may
> not be concentric.  To the people I'm working for, concentric circles
> necessarily have different radii.
>
> I've been quite puzzled by this, but am finally beginning to understand
> that it might be about identity: if two circles precisely overlap (i.e. are
> congruent and concentric), can we say that two distinct circles exist?
>  Experience with dynamic geometry software gives me the automatic answer of
> "of course" - I can drag two separate objects about the screen, make them
> coincide, and then drag them apart again.  I can also construct the two
> circles to be congruent - but to have quite different properties.  The
> centre of one circle could be free to move anywhere, while the centre of the
> other circle might be constrained to a line.  The radius of one circle might
> change by dragging on the circle - and this could determine the radius of
> the other circle, but not vice-versa.  I've never thought to question the
> Euclidean definition of a circle until now - but, in a dynamic geometry
> environment, it seems to me that it's simply not sufficient, as it does not
> give me any way of distinguishing between two congruent circles that happen
> to coincide.
>
> On the other hand, the company wants to still call two coinciding circles
> "congruent", which implies that there are still two circles in a
> relationship with each other - are they just being inconsistent?
>
> I also find the question interesting of when to consider it legitimate for
> a mathematical term to have a flexibly  determined meaning (such as
> trapezoid/trapezium or adjacent angles) and when to not consider it
> legitimate.  "Concurrent" has not got any of the mathematical inevitability
> of "-" x "-" = "+".  I shouldn't be concerned about its definition being
> changed.  But I find that I am very unwilling to change.
>
> Thoughts, anyone?
>
> Kate Mackrell
>
>
>


-- 
Best wishes for 2010

JOHN BIBBY
                    (NB: this is my only current email address now.)