```Date: Dec 30, 2012 12:05 PM
Author: kirby urner
Subject: Re: A Point on Understanding

On Sun, Dec 30, 2012 at 12:16 AM, Paul Tanner <upprho@gmail.com> wrote:>> So my I conclude you're against this strategy of arguing pro and con a>> mathematical proposition?>> No, and in my last post>> http://mathforum.org/kb/message.jspa?messageID=7944754>> I communicated so:><< snipping repeated prose >>I don't see where I've denied any theorems in setting up a tensionbetween the deficit at each point and an argument that gets rid of thedeficit at some limit, using an epsilon / delta approach.  Rather,I've introduced several new theorems and suggested we prove them.> One of the responsibilities of [a] teacher [is] to teach his or her> students how to think such that they can avoid becoming cranks.>Lets not forget that mathematics is more than a scaffolding of provedtheorems.  It consists of conjectures (unproved) upon which otherscaffolding may be contingent, and it depends on definitions whichalter what's proved and/or provable.For example, per standard definitions, a cube with edges SQRT(2) musthave a volume of SQRT(8) because our definition of 3rd poweringincorporates the geometric hexahedron as a model of 3rd powering.In contrast, a Martian civilization might consider the regulartetrahedron their 3rd powering model.  There'd be a conversionconstant and all volume ratios would remain the same (e.g. cube totetrahedron inscribed as face diagonals = 3:1).The argument *against* teaching both Earthling and Martian mathematicsside by side would be we don't want to confuse students.  We want themto reflexively think and say "squared" and "cubed" in connection with2nd and 3rd powering, not "triangled" and "tetrahedroned".  PhDs mightexplore alternatives, but such forays into non-orthogonal (akaunorthodox) thinking would only scramble the brains of the young.The argument *for* teaching the two paradigms in close conjunction isthis approach retains more flexibility in thought and allows the roleof definition and cultural choice to remain in the foreground.  I'dconsider such multi-paradigm training more useful for studentsplanning a career in diplomacy / international relations, where rigidadherence to the "one right way" is too close to mindlessfundamentalism for comfort.  Better to scramble their brains a littlethan to encourage strait-jacketed habits of mind.So here's another case where there's legitimate debate and yet no oneis denying any proved theorems.  There's no need to fling anyaccusations that one side or the other are just crackpots.In my view, any "one size fits all" approach to math pedagogy is illadvised.  This is different from saying we should have a fast trackfor the talented and gifted and various slower tracks for those lessgood at math.  I'm saying not all who are stellar at math need thesame training, even in early childhood.  Those trying to impose the"one right way" (what topics to include or exclude, and in whatsequence) are the closest to crackpots in my vista.> This "arguing pro and con" is fine as long as theorems and proofs are> not denied - if it's only a pedagogical technique for introducing> theorems and proofs, then OK. But if the agenda is to deny> mathematical theorems and proofs, then it's not fine - it's not fine> if this "pedagogical technique" is a smokescreen for denying these> facts.>There's more to mathematics than theorems and proofs.Kirby
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