user52817
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When two equivalent algebraic statements have two "different" meanings
2 votes

The two statements are equivalent, assuming care with quantifiers. In the original form $\frac{n}{m}=\sqrt{7}$ we have an equation over the real numbers. We trade this for the second form $7m^2=n^2$ ...

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Topics for a general education course
2 votes

As a department chair, I think this is the most difficult and expensive course to staff, unlike college algebra, elementary statistics, calculus, etc. I want the best faculty who can bring their own ...

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"Function" vs "Function of ...": how much does it contribute to students difficulties?
6 votes

I really appreciate your question. It gets to something I think about often when teaching calculus. You should read about Ed Dubinsky's notion of reflective abstraction, rooted in Piaget. At one ...

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Teaching Asymptotes
3 votes

Mathematically a horizontal line is asymptotic to itself. So for example the horizontal asymptote of the constant function $f(x)=3$ is the line $y=3$. This issue of whether the definition of a ...

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Why does the widespread erroneous definition of "irrational number" persist without being taught?
7 votes

The definition of an irrational number as a "number which is not rational" is not without its own difficulties. It presumes that we have a clear definition of a real number. The audience you refer to ...

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How to solve the problem of Wolfram Alpha?
8 votes

You can also use Wolfram Alpha to look up definitions of words. So by reductive reasoning, there is no need for infants to learn vocabulary because when they are old enough to use a smartphone, people ...

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Applications of MVT for Integrals, suitable for calculus 1
8 votes

I think the best "real" application of the mean value theorem for integrals is to make a rigorous proof of the fundamental theorem of calculus.

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Necessary trigonometric identities for k-12 students
3 votes

Keep in mind that the Law of Sines and Law of Cosines are not identities in the same sense that your items 1-21 are identities. Rather these two laws are geometrical facts that are particular to ...

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Why would you teach Calculus before teaching Real Analysis?
10 votes

In many respects this is what is done in European universities, where the degree is often a three year program. First year students reading maths take analysis. Of course they probably learned ...

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How can I motivate the formal definition of continuity?
7 votes

The prototypical way for a function to not be continuous is that of a jump discontinuity. Imagine a jump discontinuity on the order of a few micrometers, like the width of a hair. If you are tracing ...

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Students strictly follow the steps and notations in sample problems without understanding them
2 votes

In many respects this problem reminds me of the sort of exercises that came out of the calculus reform movement of the 1990s. The problem is "conceptual" in the sense that it cleverly does not rely on ...

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Why unlike terms cannot be simplified?
1 votes

The "high level" answer is that a polynomial algebra is a free commutative algebra generated by the indeterminates. But this would probably not be a very satisfying answer to 8th graders. A possible ...

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What good is the phrase "Taylor series"?
1 votes

Indeed we could get by without referring to Taylor series, and instead just always refer to a "power series." But then maybe we should also stop referring to Fourier series and simply refer to "...

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When higher grades are more common than lower grades
6 votes

Although I have serious issues with "outcomes based" education, its philosophy does combat the pernicious notion that grades should have a normal distribution centered on "average." It is not ...

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Why are induction proofs so challenging for students?
9 votes

The reason why learning mathematical induction is difficult is developmental. Your audience of under-prepared college students probably accept "proof by verifying the first few cases" as more ...

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Why do we conventionally treat trig functions as going anti-clockwise from the right?
2 votes

Perhaps the convention is rooted in how the big dipper rotates around the north star--in a counterclockwise direction. Also think about how the earth rotates around its axis--in a counterclockwise ...

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Given a 3 4 5 triangle, how do you know that it is a right triangle?
21 votes

The fact that there is a 3-4-5 triangle that is a right triangle is unique to the Euclidean plane. There is no such triangle in the spherical or hyperbolic planes. Since the Pythagorean theorem is ...

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Teaching about absolute values not centered at $0$
1 votes

Going beyond the good answers here that have been given, and which work for teaching in a classroom, one could instead focus on pliable strategies that work well on GRE exams. When I tutor students ...

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How do you explain why perpendicular lines have negative reciprocated slopes?
10 votes

One approach is to start with the fact that a 90-degree rotation results from a reflection across the $x$-axis followed by a reflection across the line $y=x$. (In general, reflecting across two ...

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Wiggins' question #12
1 votes

Coxeter, in his book titled Projective Geometry, describes a dictionary game called Visch (short for "viscious circle"): Point=that which has position but not magnitude Position =place occupied by ...

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Pedagogical quandary with the definition of $i$
1 votes

The choice of i versus -i is correlated with the geometric orientation of the the plane. One choice, i, correponds to counterclockwise orientation which matches the righthand rule for orientation. ...

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Should $\varphi$ be monotone in the integration by substitution?
Accepted answer
11 votes

There are two formulations for definite integrals: $$\int_{\phi(a)}^{\phi(b)} f(x)\, dx=\int_a^b f(\phi(t))\phi'(t)\, dt$$ and the one you state: $$\int_{\phi([a,b]}f(x)\,dx=\int_{[a,b]} f(\phi(t))...

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How best to explain the logarithm to the mathematically naive?
1 votes

You are close. Floor($\log_{10}(N))+1$ is the number of digits.

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Logarithm Tables - How were the values reached?
5 votes

Gauss said "You have no idea how much poetry there is in a table of logarithms." The first paragraph of this paper might get you pointed in the right direction ON THE DISTRIBUTION OF PRIMES—GAUSS’ ...

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Domain of an exponential function (US Common Core, High School Math)
1 votes

It is a poorly written question because it presumes there is some sort of universal definition of what is meant by an "exponential function." You can closely replicate the graph with $f(x)=3+e^{-1/...

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How can I convince students that Fourier series are useful?
3 votes

I often use the module "How to Tune a Radio--Trigonometric integrals explain tuning a radio." It is in Volume 3 (Applications of Calculus) of the MAA Resources for Calculus Collection (MAA Notes ...

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A parabolic arc is not semicircular. But students think so
12 votes

This might be a good opportunity to inject some history into your course. Aristotle thought that cannonball trajectories were line segments, up at an angle then straight down. Look in Chapter 5 of "...

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Simple "real world" l'Hôpital's rule problem?
5 votes

I have a colleague whose hobby is to always figure out how to a evaluate any given limit without L'Hospital's rule. Every week in the lunch room, there is yet another example. I think a very good ...

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System of Equations Generator
3 votes

If you have access to software such as Maple, Mathematica, or Matlab, then you can figure out how to have it generate an nxn matrix A with integer entries and with determinant 1. Once you have such a ...

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In introductory statistics hypothesis testing, why not always use P-values?
1 votes

If we place value in understanding the meaning of a p-value, it seems almost essential to discuss the test statistic, and hence the corresponding critical value. (The p-value is the probability that ...

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