user52817
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Why is it possible to teach real numbers before even rigorously defining them?
59 votes

Expansion of mathematical knowledge does not unfold in Bourbaki progression. This is true at the level of both societal and individual knowledge. Just as the invention and significant applications of ...

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How to respond to “solve this equation” in a basic algebra class
27 votes

Ask your pedantic colleagues to reconcile their formatting expectations with the context "solve $F=m\cdot a$ for $a$". Would they prefer to see just $\frac{F}{m}$, or maybe this expression jazzed up ...

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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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How to explain that the sums of numerators over sums of denominators isn't the same as the mean of ratios?
18 votes

One observation is that (sum of numerators) divided by (sum of denominators) is not well defined. For example, let's work with the two ratios $a=\frac01$ and $b=\frac11$. The ratio of the sum of ...

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Is there a more telling name for "Calculus 2"?
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18 votes

In my job, I evaluate university math courses for transfer equivalency on a regular basis. In the US, "Calculus 1" typically refers to single variable differential calculus up to the fundamental ...

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Why do we teach even and odd functions?
14 votes

Learning to think about functions abstractly should be one goal in precalculus, and function symmetry helps. Also suppose we carefully protected a student from knowing anything about function symmetry....

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Why do we teach the Rational Root Theorem? (high school algebra 2)
12 votes

I think an important learning outcome of this part of high school Algebra 2 (theory of polynomials) is developing mathematical capacity to juggle several disparate abstract tools all at once, similar ...

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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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Why the fear of polynomial long division?
11 votes

I like to include synthetic division as a topic in a college algebra or precalculus course. It is an opportunity to take a 20 minute digression to talk about Horner's method, which is used in several ...

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Should $\varphi$ be monotone in the integration by substitution?
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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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Ideas for explaining 4D and higher dimensions
10 votes

I highly recommend "Flatland The Movie." Your institution should be able to purchase it. You can find a free trailer on the internet. When I was young, I read the book "Flatland: A Romance of Many ...

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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 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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I want a "true" proof by contradiction of an implication P => Q
9 votes

I think you are overlooking the fact that proof by contradiction must invoke the tautology $(P\ \hbox{or}\ \neg P)$, called the law of excluded middle. To prove $P\Rightarrow Q$ by contradiction, we ...

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Why should or shouldn't we teach functions to 15 year olds?
9 votes

Some abstraction is perfect for children. Mitsumasa Anno was a Japanese writer of children's books and he had many innovative approaches to introduction functions. This is from Anno's Math Games II. ...

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

It might be fun to have your students pretend that the only functions they know are sums of monomials $cx^n$ where $n\in{\bf Z}$, and in particular, play like they know nothing about the function $f(x)...

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How to make students comfortable with the use of axiom of choice in analysis
9 votes

As others who answered have pointed out, this issue is not the countable choice involved in defining the sequence that makes this challenging for learners. Rather, the difficulty is the semantic ...

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

There is a model of how people progress towards abstract reasoning through the subject of geometry called the Van Hiele model. The model describes five levels: visualization, analysis, abstraction, ...

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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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Fear of 3-dimensions
8 votes

After teaching Calculus 3 for many semesters, your question certainly resonates. I think it is empowering to coach students in how to sketch surfaces in 3D. Their tendency from curves in 2D is to ...

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An intuitive explanation of l'Hôpital's rule for ∞/∞
8 votes

Your approach is in fact geometrical, if we see all this happening on the Riemann sphere, i.e., the one-point compactification of ${\bf C}={\bf R}^2$. Briefly, we have the usual coordinate $z$ on ${\...

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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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How should I answer questions about the purpose of learning math?
7 votes

This is among the oldest questions about learning that has been asked. When a young person started learning geometry with Euclid and asked him why he should learn geometry, Euclid replied, "Give ...

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Does anyone use the cubic formula these days?
7 votes

It seems unlikely that the Cardano formula has even been of serious analytic use, i.e., used to approximate roots of a cubic. At least since the inception of calculus, Newton's method can be used to ...

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Why do you need to distinguish between apparently identical objects in probability?
7 votes

This is a very good question. The issue comes up frequently. I explain this using a toy model: throw two regular six-sided die. What is the probability that the sum is 3? With some physical modeling, ...

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Easy examples of correspondence between global and local, as preparation for Gauss's theorem and Stokes's theorem
7 votes

One idea that comes to mind is the fact that a connected graph has a Eulerian circuit if and only if every vertex has even degree. Another idea that comes to mind is just the Fundamental Theorem of ...

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How does one explain that transformations 'inside' a function operate in the opposite direction than intuition suggests?
7 votes

I think a key tool here is tables of values. My approach has always been to give students functions like $f(x)=x^2$ and $g(x)=f(x-1)$ and ask them to fill out tables of values for some range of $x$, ...

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Who actually uses $\mathbf i$, $\mathbf j$, $\mathbf k$ for the standard unit vectors?
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7 votes

Search arxiv.org for "unit vectors i, j, k" and you will find examples of research papers where the notation is used. Many are from physics communities investigating phenomena in three-dimensional ...

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L'Hopital's Rule: Why do we need it?
7 votes

When teaching calculus I like to include L'Hospital's Rule because it exemplifies how the subject is an impressive arsenal of calculational methods, and because it is easy to explain why this rule is ...

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