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UNDERSTANDING

	Fish don't understand swimming.

	J. S. Bruner (1967)
%2Toward a Theory of Instruction%1, Norton, New York
says (p. 44) "any idea or problem or body of knowledge can be
presented in a form simple enough so that any particular learner
can understand it in a recognizable form".  This seems quite
wrong; some things are genuinely difficult to understand.  At
least I find some mathematical ideas apparently beyond my
ability to understand.

	Understanding is one of the key mental qualities whose
formalization is difficult, but first we must notice an ambiguity.
Understanding a word is often taken to mean recognizing it and
using it for a single purpose in accordance with its meaning.
Thus a computer operating system may understand the word
"logout".

	The sense of "understand" that I want to understand is
exemplified by "He understands the epsilon-delta definition
of continuous function".  This definition has the advantage for
study that the mere ability to repeat it is obtained much more
easily than that ability which a mathematics instructor would
regard as understanding in a college student.  Experiments in
teaching such concise but difficult ideas should be undertaken
and some attempt made to identify the intermediate states of
understanding.

	There is a parallel between understanding and
"knowing about".  Someone who knows how automobiles work
will have to know about carburetors, but when ⊗X correctly
says that ⊗Y knows about cars, ⊗X may never have heard of
carburetors himself, although ⊗X would easily convince himself
that this knowledge is essential to understanding automobiles.
⊗X has ways of convincing himself that ⊗Y does or doesn't
understand automobiles that don't require that he understand
automobiles himself.

	We might usefully study a narrowed notion of knowing
about, namely ⊗Y knows about cars if ⊗Y can tell whether an
account of fixing a car is coherent, e.g. he should reject
an account that reported "reinflating the left front carburetor"
or "getting a milage of ten carburetors per hour" or "fixing
the carburetor by filling it with sand".  He should know that
the gas line is connected to the carburetor and that the
carburetor mixes gasoline with air.