Julia Robinson! I recommend her for a high school audience for a few reasons: Mathematical reasons: She is best known for her work towards the solution of Hilbert's 10th Problem, regarding an ...

I have no evidence to back this up, nor do I know how I could obtain such a thing. But I strongly believe this is a vocabulary issue, and I would like to see the term "series" phased out of usage in ...

This one can be presented to students at any level, really, although the way to explain "repeat to infinity" will certainly change for your audience. It can be used to teach them that weird things ...

I find it helpful to introduce the negation of conditional claims simultaneously. For one, this better helps them to understand the "false implies false" case; but also, this helps them understand how ...

Since $\varphi$ is rather close to the conversion rate between miles and kilometers, one can use the Fibonacci numbers to convert: if $f_n$ is the distance in miles, then $f_{n+1}$ is (roughly) the ...

What are some examples of math history that can be mentioned in calculus classes, either to liven things up or to provide additional perspective / insight on the material being learned? You mention ...

The Curry Paradox is a classic. This animation resolves it:

Proving DeMorgan's Laws for $n$ sets. I like this example because it requires the $n=2$ case in the induction step. It's common to have students prove that $\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k\... View answer 19 votes If$\gcd(a,b)=1$, there exists a multiplicative inverse for$a$modulo$b$. (Otherwise, look at the$b-1$multiples of$a$, namely$a,2a,3a,\dots,(b-1)a$. They must fall into congruence classes that ... View answer 15 votes Since dtldarek's answer addresses well the issues of fairness to students, I'll mention another consideration. When writing exam questions, I try to make sure each question has a certain intent, that ... View answer Accepted answer 15 votes There are a few reasonable approaches, and they vary mostly in (a) intrusiveness towards the students and (b) effectiveness Walk intently in the direction of the student, cough, and stare at them for ... View answer Accepted answer 14 votes I doubt this would be a good idea. The only exception might be a very recently-developed field, so that they're written in modern style and notation. For most texts and papers much older than the mid ... View answer Accepted answer 14 votes Yes! I have used these a lot in an "intro to proofs" course. Typically, each weekly homework assignment has at least one problem of this variety, and I've written many like this for assignments and a ... View answer 13 votes (I have not done this exact presentation, so I cannot vouch for its efficacy. But I have used the main idea before, and it seems to help some students, and is at least a bit of fun. Also, this is ... View answer 12 votes I study pursuit-evasion games on graphs, so I will recommend using the cops & robbers game as a way to introduce graph theoretic terminology, concepts, and examples. It should also keep the tone ... View answer 12 votes This one can be shown to any student that understands the distance formula, and has a willingness to think about$n$-dimensional space for$n>3$. (It is, more specifically, an indication that "... View answer 12 votes I think it helps to make it abundantly clear whether or not you would expect an average student in your course to come up with such a "stroke of genius". If you're presenting something that might ... View answer 12 votes William Dunham's Journey Through Genius is, ultimately, about a bunch of facts, but it's written very well and can be inspiring to a budding math student. How to Lie with Statistics is just a classic ... View answer 10 votes I always provide the following example whenever a student assumes what they want to prove: Suppose 0=1. Then 1=0 must be true. Then we can add both equations to deduce that 1=1. This is a true ... View answer 9 votes I second Sue Van Hattum's suggestion that you should not be so concerned with how large the$n$is where the pattern eventually fails. I'll go one step further and recommend an example where that$n$... View answer 9 votes Perhaps you can make the topic "applicable" by comparing it to some popular social media posts of the form, "How many triangles are in this figure? Only geniuses get this right!" Take a$5\times 1$... View answer 9 votes I don't believe students subconsciously misread the notation as an exponent with a word in front. I think they are not sure at all how to read the notation! When using logarithms (whether in a basic, ... View answer 9 votes I believe this depends on how clearly the counterexample is stated. Consider this claim:$f:\mathbb{R}\to\mathbb{R}$is continuous$\implies f\$ is differentiable. Imagine a student, let's call him ...

I like this question very much. But I think the best approach is via a plethora of examples meant to demonstrate the variety of uses of equivalence classes. I doubt there is a singular example that ...

I'll share my experiences, but definitely want to hear from others, as well. I've assigned these "research project" papers in Calc II. The actual math is mostly guided, with some open-ended questions ...

How Does One Do Mathematical Research? (Or Maybe How Not To), by Lee Lady Mathematics as a creative art, by Paul Halmos I Want to Be a Mathematician: an Automathography, by Paul Halmos Also, this ...

I taught two sections of a course last semester, one MWF (50 mins each) and one TuTh (75 mins each). It was not great. The main problem is not even splitting material across days. It's an issue of ...

I'm not sure how well this suggestion would fit into your specific course set up, but I want to share something that has been helpful for me in other contexts: I like to set a general theme/...