### Reality Check

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There is such a great gap between what we assume that the students know and can understand, and what they actually know and are able to understand. When we teach calculus for example, we take it for granted that they know high-school algebra and precalculus inside out. True, in order to enroll in Calc2, they were supposed to know Calc1, which means that that they were supposed to know precalculus etc. But they don't! Even those that get an A have a very shaky understanding of very basic high-school algebra (that is much more important than calculus). 95 percent of even the A students get an A because they do the symbol-manipulations correctly, and know how to reproduce the solutions in the review problems. Those who are B and C students, are not as good fakers. This is like the Turing Test, have a calculus student simulate understanding, but very few actually understand what is going on.

And why should they? Calculus, and even high-school algebra, are very abstract. For the prof., and even TA, it is all ``obvious'', so most of the instructors can't even understand how it is "not possible to understand". The few professors [who are ] aware of students' lack-of-understanding [..] are still bound by the demanding, largely obsolete, curriculum, teaching students to do, mostly by rote, things that computer algebra systems can do so much better today.

There is such a great gap between what we assume that the students know and can understand, and what they actually know and are able to understand. When we teach calculus for example, we take it for granted that they know high-school algebra and precalculus inside out. True, in order to enroll in Calc2, they were supposed to know Calc1, which means that that they were supposed to know precalculus etc. But they don't! Even those that get an A have a very shaky understanding of very basic high-school algebra (that is much more important than calculus). 95 percent of even the A students get an A because they do the symbol-manipulations correctly, and know how to reproduce the solutions in the review problems. Those who are B and C students, are not as good fakers. This is like the Turing Test, have a calculus student simulate understanding, but very few actually understand what is going on.

And why should they? Calculus, and even high-school algebra, are very abstract. For the prof., and even TA, it is all ``obvious'', so most of the instructors can't even understand how it is "not possible to understand". The few professors [who are ] aware of students' lack-of-understanding [..] are still bound by the demanding, largely obsolete, curriculum, teaching students to do, mostly by rote, things that computer algebra systems can do so much better today.

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