>From Anne again - who is very busy but trying to keep up with the argument: There is pedantry on all sides of course - there always will be. For me the question is 'is it important to learn about labelling?' My answer is 'yes' because labelling records distinctions and enables us to make and keep track of distinctions. So if 14 year olds (or even 4 year olds) are doing a task which generates distinctions, or which requires distinctions, introduce labelling as the way we handle these. Martin Hughes has done lovely experiments with very young children to show that if they NEED to keep track of specific quantities they will invent symbols for numbers. The same applies to 14 year olds and very recently I observed a lesson in which this happened - there was a need to label vertices in corresponding order, so they did! And by the way these were not high fliers, they were students whose prior achievement was below average for their age. Oh yes, and they were being taught by a newish teacher who has a PhD in Mathematical Physics and has not yet been fully indoctrinated into stories about what children can or cannot do at certain ages - her commitment was to their mathematics.
From: Post-calculus mathematics education [mailto:MATHEDU@JISCMAIL.AC.UK] On Behalf Of Gary Davis Sent: 15 October 2009 00:15 To: MATHEDU@JISCMAIL.AC.UK Subject: Re: Labels
a few questions (I am trying to be constructive, not deliberately provocative, or smart-arse):
* Have you worked extensively, or much at all, with 14 year olds to know, in a principled, evidential way, what they can and cannot reason and think about? * Are you familiar with the van Hiele's ideas on development in geometric thinking? * How does dynamic software - GSP or Cabri - factor in student's learning about geometry?
Where a lot of difficulty arose in Stephen Hegedus's excellent PME workshops on Symbolic Cognition was that many participants did not recognize the developmental aspect of the topic. Geometric understanding, likewise, has a large developmental component.
I'm asking the questions above because I am trying to figure out how you know what you say, such as: "at that age a triangle should be a figure with three vertices and three straight sides, and no labeling." - the reasons in other words, that you believe this. The standard way, for example, to draw a circle in GSP is to specify a center. That center does not have to have an arbitrary label such as "A", but the user has specified it, and the software has labelled it, and knows its role. Similarly, a common way to construct a triangle in GSP is for a user to specify (and so intrinsically label) 3 points, and then join them in pairs using a segment construction. If, for example, one of the specified points were constrained to lie on a specified circle, but the others were not, the labeling would be critical in a dynamic understanding of the construction. This does not seem like pedantry, and on this issue I think I'm in agreement with Tony. But more discussion would be greatly appreciated.
Martin C. Tangora wrote:
OK, let me try again.
I withdraw my previous punch line "What is there to argue about?"
James King finally raised the issue of age-appropriateness.
I would *argue* that the appropriate definition of "isosceles"
for average 14-year-olds is "having two equal sides."
If I seem to be begging the question of what a triangle is,
yes, it's deliberate. But at that age a triangle should be
a figure with three vertices and three straight sides,
and no labeling. The necessity of a savvy teacher
keeping in mind that there are, say, six labeled triangles
that all map (by the forgetful functor) to one and the same
(unlabeled) triangle, is arguable. But it would not be
age-appropriate to insist that the class learn to label everything.
That would be new-math pedantry, and no, Tony Gardiner,
I'm not uncomfortable with those ideas, I'm opposed to them.
By new-math pedantry I mean trying to teach 14-year-olds
about categories and functors, and forgetful functors,
instead of teaching them geometry.
If you have, say, an isosceles right triangle ABC,
with right angle at B, and you tell the students that
ABC is isosceles, but BCA is not, you are going
to be teaching, not mathematics, but the hatred of mathematics
(and of mathematicians).
Of course you might want to make this
entire business of labeling a major part of your course.
That is an option, but I wouldn't choose it,
and I would oppose putting it into the general curriculum.
When T.G. writes
Good teaching sometimes obliges us to smudge mathematical
distinctions. (Whoever writes this is no "new-math pedant"!)
I agree with him, in that age-appropriate class material
is not the same as graduate-school precision.
Axiomatic set theory may be lurking in our minds,
but we should not try to impose it on 14-year-olds.
Michael McConnell's anecdote about the formal definition
of "work" was very interesting to me, because I had
almost that same presentation in high-school chemistry.
We had a very weird chemistry teacher, and his lesson
on this topic began with what might sound like a good idea:
asking the students to offer their ideas toward
a definition of work. He was not a gifted teacher,
and this did not go well; it was, naturally, a game of
"Figure out what the teacher wants," and we all failed.
Finally he just said, Well, I'll tell you what work is:
Work is Force Times Distance. Confusion and rancor!
While I'm on the subject of lessons, I was puzzled
when Gary Davis wrote to me
Our students can learn a great deal about being more precise,
and much deeper, in their mathematical thought
if they are involved in framing definitions.
You can, of course, as can we all, declare by fiat
that such and such is the definition.
But does this not take away from a student
the struggle for clarity that went into this process?
Why would you not want a student to develop
a similar strength and ability to reflect
on what a definition might capture?
since I had just more or less sketched a lesson plan
in exactly that spirit:
Thus I would define a rectangle as a quadrilateral with (enough) right angles,
and then show some examples, leading up to the example of a square.
The class might object that that isn't a rectangle because it's a square;
and I would ask them to look at the definition of rectangle.
We should be able to come to agreement that the square
satisfies the previously given definition of rectangle.
Then we could all decide whether in our class the word rectangle
should be redefined to exclude squares.
Someone did make the point that it is demonstrably better
if we can get our students to agree to the "nested sets" policy
where squares are rectangles are parallelograms, etc.
because we don't have to prove so many theorems:
if we prove that the diagonals of a parallelogram bisect each other,
and rectangles and squares aren't parallelograms,
then we need to prove two more theorems
(using the same identical proof again and again).
Yet I balk at the case of trapezoids. A trapezoid is
a quadrilateral with exactly two parallel sides, right?
That was drilled into me, and being profoundly conservative,
I buy it. That way, the altitude of a trapezoid is determined,
leading to the area formula, and we don't trip over our own feet
trying to apply that formula to a rectangle, where
there is an ambiguity about which altitude.
Tony Gardiner also says
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.
I don't think this is very closely analogous to the issue
of labeling triangles; a circle (the curve) has no distinguished points,
and there is one and only one center. We can all think of
ways to define a circle without reference to a center
(e.g. plane curve of constant curvature) but they are,
again, not age-appropriate. (at least none that I can think of)
But, T.G., would you please give an example of one of
the simplest problems that students can't do
because they don't know that "'the center is part of the circle'"?