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.