# Tag Info

### Unique candidate that fails

The obvious example I immediately thought of is that, if the divergent geometric series $$1 + 2 + 4 + 8 + \dotsb = \sum_{k=0}^\infty 2^k$$ converged, it would converge to $-1$. Proof: If the series ...
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Accepted

### What is a good method for drawing a Möbius band on the blackboard?

Draw the bottom three-quarters of an oval: Flesh that out to make the bottom half of the strip: Connect one of the open ends at the top to the bottom on the other side: Now draw a straight line ...
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### What is the point of teaching variance?

Actually, your definitions are backwards: the standard deviation is the square root of the variance. In other words, one defines variance first --- it has a simpler formula, and it has simpler ...
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### How to deal with a "protest" assignment?

You need to slam him on the grade. That is what he earned. Don't be so easily manipulated by his comments on your teaching. Also I would not have sent an email apology. Just offered to meet with ...
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### Why do we teach even and odd functions?

One of the major themes of precalculus is what I call “connecting geometry to algebra”. Being able to translate between an algebraic statement like $f(x)= f(-x)$, and the geometric statement that the ...
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### How to explain that a negative number multiplied by a negative number is a positive number, and that $-(-x)=x$?

In a comment, the OP has suggested that he actually wants a practical example convincing students that the product of two negative numbers is positive. This is related to, but psychologically ...
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### Should college mathematics always be taught in such a way that real world applications are always included?

I have worked with a lot of students coming out of courses such as yours who: passed the course by blindly memorising proofs, theorems, and algorithms; learnt nothing (lasting) except solving some ...
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### What is a good reason to change calculus texts?

I would like to encourage consideration of a free textbook. The conventional textbooks are outrageously expensive. (Actually, if your department insists on one of the choices you mentioned, I'd want ...
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### Tutoring a recalcitrant/awkward/exasperating student---special needs?

How do I reach [this] kid? Let me be blunt: You probably don't. This is a person who is so intransigent that you effectively need to black-tag them. A hard lesson is that you can't save everyone. At ...
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### (How) Do American undergraduate math programs teach complex numbers?

The following is anecdotal, but based on experience as a student and instructor in American high schools in three states, as well as my undergraduate and graduate experiences. High School: In the ...
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### Getting students to actually read definitions

In my experience, students are often predisposed to "learn" by memorizing facts; that's how much of their early education worked, so that's what they're used to. When you give them a ...
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### Combative students in proofs classes

You don't say in the question what kind of school this is. It must be a four-year school rather than a community college, but there is no indication of what its admissions standards are like. If this ...
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### What are some research-level opportunities in mathematics that do not focus on proofs?

I think a more correct view is that proof is the LAST of several stages involved in researching something in math. What follows is a quickly sketched out scenario of what is often the case. Before ...
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