Date: Dec 5, 2012 4:07 AM
Author: GS Chandy
Subject: Re: In "square root of -1", should we say "minus 1" or "negative<br> 1"?

Joe Niederberger posted (GSC's remarks and questions interspersed):
> >I mean, by the time you figure out the cars or
> un-losing poker hands, you have probably developed
> the elements (habits of mind) behind formal thinking.
> Another answer (sorry):
> Really "formal thinking" leads one down the
> "Martinez" road where one realizes that the rule
> could be different, but the usual rule as is
> preserves other nice properties like distributivity,
> identity.

Rather late in the day (I'm sure), I must confess that I do not 'get' what you mean by the 'Martinez road'. Would be most grateful for any references; Martinez' is, I'm certain, a teacher/educator who did something worthwhile - I'd like to learn something about him.
>That seems about where Peter Duveen ended
> up but long time after formal schooling.
> Simply working through the logic of some
> time-reversal scenario or other way of illustrating
> the sign rule, may in fact stand in the way of deeper
> understanding, because the person may think that
> somehow "proves" the matter.

In regard to 'deeper understanding', check through the attached document "Deep Logic". I have found, in practice, that the OPMS process that I often discuss INVARIABLY leads to a true 'deeper understanding' of the issue discussed. It is, I claim, a crystallization of the process through which we all naturally 'learn'. See attachment "How a Child Learns". I've also attached a document "What Is Modeling?", which discusses the basis on which Warfield developed his 'systems modeling'.
> It good, but its not enough by itself.
> Nor do I think "virtuous circles" (Manzur) prove it.
> Cheers,
> Joe N

Indeed, Mazur's 'virtuous circles' ("virtuous cycles", I would believe) are only a (relatively small) portion of 'deeper understanding: they are useful, but are not everything. I recall I had long ago prepared a document discussing 'vicious and virtuous cycles' in some detail (based on Warfield's systems science approach) - but I'm afraid I can't locate it right now.