I want to echo some of themes Jay's themes with a reflction on Dirac and
why school maths isn't much like maths.
"I think that there is a moral to this story, namely that it is more
important to have beauty in one's equations that to have them fit
experiment. If Schrödinger had been more confident of his work, he could
have published it some months earlier, and he could have published a more
accurate equation. It seems that if one is working from the point of view
of getting beauty in one's equations, and if one has really a sound
insight, one is on a sure line of progress. If there is not complete
agreement between the results of one's work and experiment, one should not
allow oneself to be too discouraged, because the discrepancy may well be
due to minor features that are not properly taken into account and that
will get cleared up with further development of the theory."
Paul Dirac-- Scientific American, May 1963
But Subrahmanyan Chandrasekhar notes of Dirac:
Elegance in physics is as much in the eye of the beholder as it is in any
other field of human endeavor. Dirac’s formulation appeals to physicists
because, by being a little vague and ambiguous about its precise
mathematical structure, it enables them to grasp and manipulate the
physical content of the theory with a clarity and power that would be
greatly diminished if one were distracted by certain complicating but
fundamentally uninteresting mathematical technicalities. But for
mathematicians, those minor technical matters lie at the heart of the
subject. Quantum mechanics becomes ill-formulated and grotesque if it does
not properly rest on impeccable mathematical foundations.
You get similar/contrary sorts of insight from reading Feynman's
autobiographies and his discussions with Murray Gell-Mann. There is a
certain lack of purity in is thoughts ;-).... but then according to
Boltzmann "Elegance is for tailors".
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But these struggles happen in school curricula. We have a back to basics
struggle that is keen on explicitly applicable mathematics with
concetration on the solution of everyday arithmatic problems and the
memorisation of algorithmic rules for their solution. This is the "hard"
end of the applicable maths spectrum.
"Pure" mathematicians had some influence on the curriculum reform in
mathematics in the 60's and 70's by inclusion of mathematical topics
which were less obviosly applicable at the time like set theory. the
basis for this links to the ideas of having a structural understanding of
mathematics.
A curriculum development which seems to have been driven out of the UK
curriculum is a process oreinetation whereby learners are given an
opportunity to act as mathematicians (either pre or applied) and undertake
investigations. These may be investigations that involve modelling,
finding solutions to (relatively) hard problems or just playing with ideas
to see if a geeneralisable pattern (what some mathematicians like to call
equations) emerge. Few people are given opportunities to think like Dirac
in schooling, for most maths is just something there to learn.
Dirac's thoughts may also point to some issues in "crisis"
Martin Owen
Labordy Dysgu- Learning Lab
Prifysgol Cymru Bangor- University of Wales, Bangor
"How do you explain school to a higher intelligence?"
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