Re: LOGO-L> Total Turtle Trip

[George Mills <mills@softronix.com>]

This is two messages in one. John Gough's and my Reply.
John, I have not had some messages go out today as well.

I copied one piece of your message up here to reply to it.

> I haven't followed all the arguing procedures and alternative procedures
> here, but it seems clear enough that we have in fact had several procedures
> which are saying and doing more or less the following:
> * take a fixed point,
> * tell the turtle to move so that it swings around that fixed point, always
> keeping the same distance from the fixed point.
> [The correct word is "conker"]

I disagree here, I claim the POLY method (at least as proposed) does not
have this property. Or to put it more accurately it does not rotate
about
the center with the given radius. The other methods posted do.

John Gough wrote:
> 
> Dear George,
> I am receiving messages from the Logo list, but am unable to send any. (A
> local network fault, I suspect.)
> 
> Could you please forward this to the list, as you seem to be able to do
> this. With apologies for presuming on you this way,
> John Gough
> 
> Date: Thu, 30 Oct 1997 09:47:26 +1100
> To: mikedoyle@cix.compulink.co.uk (Mike Doyle), logo-l@gsn.org
> From: John Gough <jugh@deakin.edu.au>
> Subject: Re: LOGO-L> Total Turtle Trip
> 
> >On Monday 27th at 21:46 Chuck Shavit <CShavit@MagicSquare.com> wrote:
> >> This last issue is related, I think, to what Mike Doyle was saying.  The
> >> relation between MCIRCLE and a mathematical circle is somewhat fuzzy.
> >> Mike's mistake, if I understand him correctly, is that he thought that
> >> MCIRCLE is the only Logo-ish way to draw circles.  But maybe I missed
> >> something in what Mike was saying.
> And Mike Doyle <mikedoyle@cix.co.uk> replied 30/10/97
> [forget the time - whose time zone are we using anyway? even "day" is
> rather moot, when we talk electronically virtually-at-once around the
> world]:
> >I think perhaps you missed the bit from /Turtle Geometry/ and /Mindstorms/
> >that defined Turtle geometry as the geometry of a movement and a turn. To
> >my mind (in 1982) this implied that the one impossible construction in
> >Turtle geometry is a circle.
> 
> I am puzzled by this. It suggests that Mike is objecting to the possible
> existence of "circles" as definable objects within a construction of
> Euclidean geometry based on spatial actions. Yet this approach to geometry
> was popular in the heady days of New Math(s), when it was known as
> Transformation geometry.
> 
> A straight line was the set of points generated by a spatial operation
> known as a translation.
> 
> A circle was defined as a set of points generated by rotating (and rotation
> was a defined spatial operation) one point around another fixed point,
> maintaining a constant distance between points. (A different way of
> introducing this compass-like definition was to discuss a set of points
> known as a 'locus', or points traced by a moving object that moved
> according to certain defined rules.)
> 
> Other spatial operations included reflections.
> 
> I am not trying to sketch an account of this transformational approach to
> geometry, only to remind those of you who may have experienced it in
> school, and to alert others to its existence. It was widespread through
> North America, and versions of it were used in the popular British
> Secondary mathematics curriculum known as SMP (School Mathematics Project:
> Cambridge University Press).
> 
> The point is that Papert's treatment of turtle geometry was clearly (well,
> clear to me) compatible with such an approach, where circles are acceptably
> defined in terms of points described by moves and turns.
> 
> If Mike rejects circles, as defined this way, then he seems to be rejecting
> circles in transformation geometry. Seems a big rejection.
> 
> >I noticed that all circle procedures in this thread are written with the
> >objective of producing a perceptual circle on the screen.
> 
> Yes, of course. What else is Logo geometry doing but creating perceptions.
> I don't see this as an objection, except when we pause to make the obvious
> point that a perception (such as the gross one of a right tetragon) may be
> a polygon, which is only approximately a "circle".
> 
> >The Logo
> >instructions in no way define the operation of describing a circle as, say
> >a child swinging a cocker (horse chestnut) on a string would.
> 
> I haven't followed all the arguing procedures and alternative procedures
> here, but it seems clear enough that we have in fact had several procedures
> which are saying and doing more or less the following:
> * take a fixed point,
> * tell the turtle to move so that it swings around that fixed point, always
> keeping the same distance from the fixed point.
> [The correct word is "conker"]
> 
> >The Logo-ish
> >way I prefer is that used in Geomland, a Euclidean geometry microworld,
> >where the compass is admitted as a virtual instrument.
> 
> It seems equally clear that a well constructed ARC-command is doing only
> what a compass approximates.
> 
> >The interesting relationship between Turtle geometry and digital
> >representation, on the one hand, and Euclidean geometry and analogue
> >representation, on the other appears to have been buried by Turtle
> >Geometers delight in crafting obscure notation.
> 
> Argument ad hominem, methinks?
> 
> >I become increasingly concerned that the influence of mathematicians on
> >the development of the use of computers in schools has been unhelpful.
> 
> This also puzzles me. I don't think Mike's argument has been made any
> clearer by his attempt to attack turtle geometry, or his sketches of
> alternative approaches.
> 
> There is far more subtlety in mathematics, as a conceptual body of
> knowledge, that we usually realise. As far as I am concerned, the
> mathematical concepts we are trying to teach children are only human
> constructions (with the possible exception of a few not quite constructed
> concepts, such as none-ness, one-ness, twoo-ness, in-ness, out-ness, and so
> on, which we have probably had built into our brain genetically -
> everything else has been mentally and culturally constructed on top of that
> biological basis). I do not believe there IS a god-given or universally
> acknowledged defintion of a "circle", to come back to Mike's example by way
> of objection. As teachers, our task is to try to make as clear and useful
> to our students as possible the cultural conceptual constructions of
> mathematics, and other subjects.
> 
> As far as I am concerned Logo polygons are NOT circles (but did anyone ever
> really claim they were? Mike has probably been attacking a straw man), but
> they are good way of attempting to build a mental picture of what a circle,
> if we could ever apprehend one, might be. As I said earlier, successive
> polygons were good enough for Archimedes, who was a straight-edge and
> compass man if ever there was one.
> 
> But maybe Mike is a Platonist, and he believes there really IS a god-given
> universally acknowledged definition of a circle. But he is going to be very
> dissatisfied with any physical attempts to SHOW circles, because all
> physical models are approximations of Plato's archetypal IDEAL circle.
> Apart from anything else, it seems clear that to the extent that Einstein
> is right about the universe, we don't live in a Euclidean universe anyway,
> so "circles" are theoretical, not real, at best.
> 
> As far as this list is concerned, while such questions promote interesting
> attempts to do things in dialects of Logo, and also prompt reflection on
> what we think we're doing it for, the discussion has a point. If we want to
> try to talk about other things beyond this maybe we reach irreconcilable
> philosophic differences.
> 
> jugh@deakin.edu.au (John Gough)
> Lecturer in Education
> http://128.184.132.3:80/sci_dev/Staff/jgough.htm
> Deakin University SDS, 221 Burwood Hwy, Burwood, Victoria 3125, Australia
> phone:   Australia, Melbourne area code 03 9244 6390
> fax:     Australia, Melbourne area code 03 9244 6734

-- 
===============================================================
George Mills
email: mills@softronix.com
http://www.softronix.com
The www page contains some very powerful educational software.
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