Mathematics Educators Stack Exchange
Mathematics Educators Stack Exchange

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Reference on study habits; cramming vs studying a bit every day

I am teaching Linear Algebra this semester, with pre-recorded classes and weekly meetings, and the students have a few short and simple exercises to solve every week, which count a bit (20%) to their final course grade. This was designed in order to…
Luiz Cordeiro
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Playful problems in mathematics

This answer describes an analogy between finite state machines and mazes. This allows for some playful exercises, like Draw a representation of a word accepted by the following automaton... which sums up to "solve this maze" in more "respectable"…
dtldarek
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What is the right notation to use in multivariable chain rules?

The following "chain rule" is in my multivariable calculus course: If $f$ depends on $x$ and $y$, but $x$ and $y$ depend on $t$, then $\frac{df}{dt} = \frac{\partial f}{\partial x} \frac{d x}{d t} + \frac{\partial f}{\partial y} \frac{d x}{d…
Chris Cunningham
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Transitioning proof based math courses online

I'd love to learn from anyone's recent experiences teaching online proof based math courses, especially those that have a large group of students who will be working asynchronously. My usual proof based course includes some lecture, with a lot of…
Mathprof
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Multivariable limits

Multivariable limits are harder than their one-variable counterparts, and textbooks examples usually focus on limits that don't exist when approaching from different straight lines. This gives the false impression that testing straight lines is "the…
Mark Fantini
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Ideas for explaining 4D and higher dimensions

I introduced the hypercube (to undergraduate students in the U.S.) in the context of generalizations of the Platonic solids, explained its structure, showed it rotating. I mentioned Alicia Stott, who discovered the $6$ regular polytopes in…
Joseph O'Rourke
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2D drawings of 3D objects in printed school textbooks: orthogonal or perspective?

There is a tradition in the use of orthogonal projections to represent 3D objects in printed school math textbooks. On the other hand, perspective projections represent better the way as we "see" real objects with our eyes. Take a look to…
Humberto José Bortolossi
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Confusing verbal descriptions of function transformations

While teaching Function Transformations, I found the verbal descriptions of stretch and squeeze really confusing. So for $y = f(x)$, $y = 2f(x)$ is said to stretch $f(x)$ vertically by a factor of $2$; $y = \frac{1}{2}f(x)$ is said to squeeze…
reflectionalist
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A more natural motivation for the appearance of generalized eigenvectors in linear system with repeated eigenvalue

When I teach constant coefficient linear differential equations, the usual guess of an exponential can be motivated because it is "approximately" a fixed point for the differentiation operator. The aesthetic here being that mathematicians look at…
Jon Bannon
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Terminology for parts of limit notation

When we talk about: $$\lim_{x\to{c}}f(x)=L.$$ Is there a formal name for the number "$c$"? I know that the notation means "$L$ is the limit of $f(x)$ as $x$ approaches $c$". It just would be nice to be able to refer to "$c$" separately without…
Ari
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Is short division taught these days and if not, why not?

tl;dr I'm interested in opinions on short division. Below I discuss my experience dealing with young children and long division versus short division. For those that don't know of it, wikiHow has a nice explanation on short division. Recently I've…
Simply Beautiful Art
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An intuitive explanation of l'Hôpital's rule for ∞/∞

L'Hôpital's rule for the indeterminate form $\frac00$ at finite points can be given a nice intuitive explanation in terms of local linear approximations. See for instance this textbook or this one. And for limits as x approaches $\pm\infty$, we…
Mike Shulman
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Algorithmic thinking problems

In Norway we will have a new national mathematics curriculum for all ages including high school beginning august 2020. A fundamental change is the new focus on so called algorithmic thinking. In practice this means that we are going to add the use…
Jostein Trondal
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Symmetry in polar functions - how to explain

In the precalculus curriculum I am teaching (using Stewart's book Precalculus: Mathematics for Calculus, 7th ed.), we do a bit of polar graphing, which includes discussion of symmetry on polar graphs. We teach the students to test for symmetry in…
G Tony Jacobs
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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?

I am a teaching assistant for an intro programming course. One assignment asked for the averages of a certain ratio, but most students, rather than returning $$\frac{\text{sum of all ratios}}{\text{total ratios}},$$ gave $$ \frac{\text{sum of the…
JohnnyApplesauce
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