Adam
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Grading a limit problem
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50 votes

The student changed something which was indeterminate ($\infty-\infty$) into something which was not ($\infty\cdot \infty$). How does that not merit a perfect score? Changing indeterminate expressions ...

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Which product of single digits do children usually get wrong?
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46 votes

https://www.theguardian.com/news/datablog/2013/may/31/times-tables-hardest-easiest-children There are links to a dataset in the article. As far as I can tell, this isn't a formal study: But some new ...

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How should I teach logarithms to high school students?
28 votes

Coming from the perspective of someone who reteaches this material at the college level, neither the graph perspective nor the list of properties perspective really translate into a deep understanding ...

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Should figures be presented to scale?
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22 votes

As an answer so that I can paste in a picture: I think that the problem with the one on the left is that it is possibly "good enough" that someone might think that it is to scale and that ...

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What is the right notation to use in multivariable chain rules?
17 votes

What's wrong with this?: $$ \frac{df}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt} + \frac{\partial f}{\partial t} \frac{dt}{dt}$$ with $\frac{dt}{dt}...

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Counterexamples to "stable digit" theory of error estimates
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13 votes

How about $\frac{1}{3}x^3-\frac{1}{4}x+\frac{1}{12}+\epsilon$? When $\epsilon=0$, this cubic has two distinct roots: a single root at $-1$ and a double root at $\frac{1}{2}$. If we let $\epsilon>0$ ...

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Functions can be divided into odd and even components - name of theorem?
11 votes

Take the function $f(x)$ and write the functions $\frac{1}{2}(f(x)-f(-x))$ and $\frac{1}{2}(f(x)+f(-x))$. They are odd and even, respectively, and their sum is $f(x)$. Nothing fancy required so I ...

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Ten options for multiple choices questions
10 votes

Presumably, you are trying to reduce the chance that someone will get a correct answer by guessing. In that case, I think that 10 options are more than you need. Suppose that we have an $N$ question ...

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What's the best way to explain multivariable limit problems to students who are not familiar with $\epsilon-\delta$ proofs?
10 votes

You can make this $\epsilon/\delta$ proof easy enough that an interested student should find the argument believable without experience with $\epsilon/\delta$ proofs. Divide both the numerator and ...

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Fun set theory for kids
9 votes

Questions about infinity are one way to go. e.g. 'Are there more natural numbers or even natural numbers?' Intuition says there are more natural numbers ($\mathbb{N}$) than even natural numbers ($2\...

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Different ways to multiply decimals
8 votes

Combining your last two methods: $3.9*7.5 = 4*8 - 0.1*8 - 4 * 0.5 + 0.1*0.5$, which can be thought of as computing the area of the big rectangle below, cutting off the two extra strips along the edge, ...

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How can I explain why numerical integration is easy, but symbolic integration is hard?
8 votes

You should expect numerical integration to be "easier" [1] than symbolic integration because it is answering a fundamentally weaker question. That is, symbolic integration, if you can do it, gives you ...

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How would you introduce Frullani integral to students?
8 votes

I almost certainly would not introduce that integral to freshmen. I probably wouldn't introduce it to sophomores or juniors either; not without some particular application in mind. (I don't know of ...

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"Real world" examples of implicit functions
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8 votes

Many linkages have this sort of behavior and are, perhaps, relevant to a mechanical engineer. For example, you could analyse a bicycle suspension --- a type of four-bar linkage. https://en.wikipedia....

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How to teach abstract algebra for the first time?
8 votes

I think that you probably need to give them some hands-on experience before any of it will sink in. I assume that they are CS students? After you give them the definition and a few examples, you could ...

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Examples of basic non-commutative rings
8 votes

How about $\mathbb{Z}[i]$ together with conjugation, thought of as a ring element? Conjugation and multiplication by $i$ do not commute.

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Are these assumptions in statistics correct or beneficial?
7 votes

It is worth considering that, if the ages would have been recorded as integers, rather than intervals, the assumption would have still been wrong in a similar but less obvious way. That is, a 25 year ...

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Integral calculus from the modern viewpoint
7 votes

There are many techniques of integration. Some of them, like integration by parts, are important theoretically. Integration by parts shows up in the derivation of the Euler-Lagrange equations in ...

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Proportional density function question
6 votes

You are asking a lot of them, even though each little bit of the problem isn't especially difficult and was probably covered in a class that they have taken. You need to remember a bit of trig/...

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Python programming - math library that uses degrees by default
6 votes

May I suggest giving them a bit of header code? i.e. Instead of using from math import sin try import math def sin(theta): return math.sin(math.radians(theta))

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Quadratic equations using complex math but with no imaginary roots
6 votes

Per request, I am expanding my comment into an answer. Given a quadratic equation, $ax^2+bx+c=0$, the quadratic formula $$ x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$$ will only involve complex numbers ...

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Real World use of the Function $(\sin{x})^x$
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6 votes

I suspect that if $(\sin x)^x$ shows up in any physical situation, it will be highly specific and not really a natural or worthwhile thing for a calculus student to spend their time on. Perhaps ...

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How to correct a wrong mental picture of the limit?
6 votes

If your students really think that the limit is something that is realized after infinitely many steps, I think that you are doing them a disservice trying to dissuade them. Taking the limit is ...

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Why are so many online sources "wrong" about directional derivatives?
5 votes

It is worth noting that your definition of directional derivative allows for non-continuous functions to be differentiable (in all directions). It isn't enough to look at just straight-line paths for ...

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Group theory by geometry
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5 votes

Since you are doing this geometrically, I would suggest a different tact on your treatment of normal subgroups: You can think of them as stabilizers of certain extra decorations. For example, in a ...

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Why do we study Cantor Set?
5 votes

I think that you may be selling short the value of a counterexample! They are quite useful for making sure that you have not proven too much. i.e. When you have a plausible but only semi-formal ...

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Should I go over examples straight from the textbook in Calculus lectures?
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5 votes

Sometimes Stewart picks exactly the example I would have picked, even if I hadn't been prepping from his book. It is usually when the natural example is relatively simple. I wouldn't worry too much if ...

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Why are proofs written in flowery language incomprehensible?
5 votes

Honestly, both seem pretty bad and don't prove the statement in an obvious way. For example, there is confusion betwen irreps and the representations of individual elements. You can make the $G$-...

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In what curricula are "rectangles" defined so as to exclude squares?
5 votes

I just came across this discussion https://news.ycombinator.com/item?id=16605831 which reminded me of your question. The perspective isn't exactly math and isn't exactly not-math but rather, a ...

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Examples of basic non-commutative rings
5 votes

Another elementary ring to consider is the Exterior Algebra of a vector space. That has the advantage of having both commutative and non-commutative parts, and having a relationship to linear algebra ...

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