To teach effectively, you have to know when your students have understood something and when they haven’t the most efficient way to discover that is to listen to them and to watch their faces. A teacher who only lectures, and does not encourage questions, might as well be replaced by a book or a movie. This means listening to your students and encouraging their questions.
Teachers, you will be more comfortable in your job if you try to do it well, and don’t think of your students as the enemy. As philosopher John Culkin said, “We don’t know who discovered water, but we are certain it wasn’t a fish.” That certain mathematical errors are common among students may be partly a consequence of biases that are built into our language and culture, some of which we aren’t even aware of. But both involved “thinking outside the box” par excellence - i.e., seeing past the assumptions that were inherent in our culture and our language.
Both are fairly simple, in retrospect, to anyone who has studied them. These discoveries are remarkable in that neither involved long, involved, complicated computations. Ultimately, what are the sources of errors and of misunderstanding? What kinds of biases and erroneous preconceptions do we have? Two of my favorite historic discoveries are Einstein’s discovery of relativity and Cantor’s discoveries of some of the most basic rules of infinities. (There is some overlap among these topics, so I recommend reading the whole page.) … Of related interest: Paul Cox’s web page, and the books of Bradis, Minkovskii, and Kharcheva and E. OTHER COMMON CALCULUS ERRORS, including jumping to conclusions about infinity, loss or misuse of constants of integration, loss of differentials.UNWARRANTED GENERALIZATIONS, including Euler’s square root error, x x.ERRORS IN REASONING, including going over your work, overlooking irreversibility, not checking for extraneous roots, confusing a statement with its converse, working backward, difficulties with quantifiers, erroneous methods that work, unquestioning faith in calculators.CONFUSION ABOUT NOTATION, including idiosyncratic inverses, square roots, order of operations, ambiguously written fractions, stream-of-consciousness notations.ALGEBRA ERRORS, including sign errors, everything is additive, everything is commutative, undistributed cancellations, dimensional errors.ERRORS IN COMMUNICATION, including teacher hostility or arrogance, student shyness, unclear wording, bad handwriting, not reading directions, loss of invisible parentheses, terms lost inside an ellipsis.(I would rather see new, original errors!) I caution my undergraduate students about these errors at the beginning of each semester. I am tired of seeing these same old errors over and over again. This web page describes the errors that I have seen most frequently in undergraduate mathematics, the likely causes of those errors, and their remedies.
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