I find the view of solving equations very helpful, when I explain injective and surjective functions: A function $f:X\to Y$ is injective if the equation $f(x)=y$ has always at most one solution, ...

At German universities, one of the first lectures in mathematics is "Analysis 1" which is a kind of "rigorous calculus" and there one always proceeds more or less like this: We start with an ...

I also tried to estimate this for my lectures in advance for some years but couldn't really figure it out. However, after a several years of teaching math I got pretty good at estimating. Here is what ...

I would be careful with the type of result for which one needs a lot of new math to digest the explanation. For example, I would avoid talking about $1 + 2+3+.. = -1/12$ because there is basically ...

Some more recent books on inverse problems that also stress applications are: Statistical and Computational Inverse Problems, Jari Kaipio and Erkki Somersalo, https://www.springer.com/de/book/...

I'd say a good approximation is often better that an exact result. This may sound counterintuitive, but as the phrase is vague anyway, here is a longer explanation what I mean: An "exact result&...

Besides applicability in topics like integration and Fourier analysis, it also connects algebra to calculus at least in the way that multiplication of even/odd functions behaves like addition even/odd ...

Drawing an analogy to the multiplication of matrices (where order does matter): If I read "$A$ multiplied by $B$" I would probably think of $AB$ first, but probably I would realize that ...

I'd say that it is not "linearly" vs. "nonlinearly" or "whole book" vs. "paragraphs" but rather you should change your focus when starting to read. When you transition to research, you should not ...

I have not tried anything similar myself, but if I had to, I would start by looking at Gilbert Strang's book "Introduction to applied mathematics" in chapter 8 "Optimization" (8.1 is "Introduction to ...

I haven't heard of deliberate practice before but what you describe sounds pretty much like what we do in math courses at the university where I am (and also at most others in Germany): We give ...

I do not know if there is an accepted definition of what a mathematician is. There are teachers of mathematics and professors of mathematics, for example, and most people agree that people of the ...

This can get quite some list… Here are a few: function: a mapping in mathematics vs. a "feature" a "purpose" and "functionality" in everyday language root: e.g. a point where a function is zero vs. ...

As far as I remember Krantz's book How to Teach Mathematics is not restricted to undergraduate courses and contains a lot of great advice. His book A Mathematician Comes of Age is not focused on ...

$\newcommand{\RR}{\mathbb{R}}$I am not really sure if I understood your question correctly, especially if we have the same understanding of what "multivariable calculus" is. If by multivariable ...

Regarding the question in the title on how to deal with late submissions: First some background. I collect hand written homework and no electronic submissions. Usually the the submission is before ...

I have two intuitions to offer: A sequence $(a_n)$ may have cluster points (these are points such that every neighborhood contains infinitely many elements of the sequence, or, more precisely, for ...

$\newcommand{\RR}{\mathbb{R}}\renewcommand{\span}{\mathrm{span}} \newcommand{\vol}{\mathrm{vol}}$ Here is my own answer: Let's assume that you know that the volume of an $n$-dimensional ...

What is called "fórmula del chicharronero" in Central Mexico (see the answer by Rodrigo Zepeda) is called "Mitternachtsformel" ("midnight formula") in middle school in some parts of Germany. This is ...

If you simplify a term by adding and subtracting something you call this a "nahrhafte Null" in German (probably translates to "nutritious null"?).

I haven't tried this in a class but it worked in one-on-one conversation with kids at basically all ages (well, ability to count is needed but that's basically it). It also has the benefit, that you ...

My first take is $$\ln(x) = \int_1^x\frac1t dt.$$ Granted, some texts introduce the natural log of the inverse of $\exp$ but other texts define $\ln$ as above and the $\exp$ as the inverse. If I ...

For multiple choice questions it is much better to ask in a slightly different way, namely Are the following numbers even? yes no 1.) O O 17 2.) O O 22 3.) O O 33 4.) O O ...

So called tensile structures in architectures are indeed minimal surfaces. Popular examples are the Olympiastadium in Munich: or the former Millenium Dome in London:

Besides mastering the material in the course one thing that the students have to learn during studies is to communicate mathematics in written form. Almost nobody comes to university and is able to ...

There are tons of reasons: Signal theory: The key phrase here is bandwidth. If you want to transmit a signal over an analog channel you could modulate it onto a carrier frequency. To ensure that you ...

I think the situation is tricky. Without knowing more about the actual problem it may well be that this proof by patchwork will be finished after a few iterations. This happens when every ...

I think one important distinction is: Is the joke about mathematics or about mathematicians? An example of the former is the "rat poison principle" from Steve Gubkin's comment or any other "funny ...

Directly referring to the phenomenon in the title: Consider $$x_n = \sum_{k=1}^n \frac{1}{k}.$$ Then for any $p$ and $n\to\infty$ $$|x_{n+p}-x_n| = \sum_{k=n+1}^{n+p}\frac1k \leq \frac{p}{n+1}\to 0.$$ ...
Actually, I don't think that it is quite right that there really "is an invisible $-1$" in front of $-x$. I would say that one defines the symbol $-x$ to be the one that fulfills $x + (-x) = 0$ and ...