"Ralston mentions that one reason the severity of teachers' bankrupt
understanding of mathematics continues to go unnoticed is that many
mathematicians do not want to be seen as teacher-bashers, as arrogant
jerks who like to belittle those who do not understand math as they
do."
I suspect this is a paraphrase.
As long as one stays on the level of generalities, Tony Ralston has a
point.
But as soon as one asks the simplest questions, it becomes clear that
there are lots of us out there (whether "arrogant little jerks" or
not I cannot say) who do not understand mathematics as Ralston does.
[This should lead to debate and consideration of the evidence, rather
than name-calling.]
For example, despite years of looking for short cuts, it now seems
clearer than ever that the process of internalising "mathematics as
she really is" has to involve the classical "hand-eye-brain"
triangle: we have to "understand" things vaguely, "see" them fuzzily,
and "do them repeatedly (by hand)" to cement understanding of
"mathematical (= mental) objects" in our brains.
As these mathematical (= mental) objects achieve a suffficeient
degree of internal robustness - we can begin (slowly and at first
very unsurely) to use them as higher mental objects - talking about
them and manipulating them in a *mental* universe, rather than
eternally grinding them out as "particular entities" by hand.
But this process is slow.
And at every stage, it seems that the *next* level of mathematics to
be internalised has to be processed in much the same way - involving
hand-eye-brain all over again.
The alternative is to produce "head knowledge" which is purely rule-
based, learned rather than grasped ("apprehended"/"comprehended"),
and hence unstable under modest perturbations.
This applies from the very outset:
* to the meanig of nubmers
* to place value
* to arithmetic (including standard written algorithms)
* etc..
Tony
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