I can't help thinking there is something missing in the discussion so far, which is about how students 'see' diagrams, and how they 'read' them, because this process - which is so self-evident to those of us who do a lot of geometry - is non-trivial and bridges their perception of the shapes in the diagram to their conceptualization of the properties and relations embedded in the diagram and hence the need to, and power of, labeling.
-----Original Message----- From: Post-calculus mathematics education [mailto:MATHEDU@JISCMAIL.AC.UK] On Behalf Of Jonathan Groves Sent: 13 October 2009 09:39 To: MATHEDU@JISCMAIL.AC.UK Subject: Re: Why do we do proofs?
Tony Gardiner wrote:
> > I myself really do not know how most mathematicians > define > > "isosceles triangle" > > There is something wrong when discussion of such > matters uses words > like "I agree" or "I disagree". > > A circle can be defined in an unnatural way without > reference to its > CENTRE. > But for the beginner (and most of us remain relative > beginners for a > very long time), the definition is not of a "circle", > but of > > * "a circle with centre O and radius OA" * > (i.e. the locus of all points X [in the pre-ordained > plane] such that > OX = OA).
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.
> Similarly, we should never define "isosceles > triangle", but rather > > * "isosceles triangle with base BC" * > (i.e. triangle ABC is *isosceles with base BC* if AB > = AC). > The base is part of the definition. > > So whereas a square is a rectangle (and a > parallelogram, a > quadrilateral, and a polygon to boot), > > an equilateral triangle ABC gives rise to three (or > maybe six) > isosceles triangles. > > > This is not pure mathematical pedantry; it is needed > in Grade 9 (and > to avoid adults getting tied needlessly in knots). > > Tony
Again, students should be aware that labels are used for convenience but do not affect what the object actually is. And specifying which of the three sides of an equilateral triangle is the base does not change the triangle itself; in all cases, the same triangle is being discussed.