I would like to pose a question to anyone who studies a highly mathematical science. What do we mean when we say we "understand" a particular concept?
I can write an equation on a board and recite what each symbol stands for. This does not, however, constitute understanding. It simply means we can recall what word is associated with which symbol. Ultimately it seems to me that to have an "understanding" of a concept simply means we can manipulate a set of symbols according to a particular set of rules to achieve what is considered the "correct" answer.
However, when undergoing new research this cannot be the case as the generation of new ideas precedes mere symbol manipulation (or does it...?).
At the level of undergraduate teaching, however, it seems almost that being able to associate particular symbols with particular words and being able to manipulate them constitutes an "understanding" of the underlying concepts. Is this really understanding or just a sign that understanding can be easily feigned?
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