Brendan W. Sullivan
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What female mathematician can I introduce to my High School students?
60 votes

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 ...

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How can I teach my students the difference between a sequence and a series?
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41 votes

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 ...

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Impressive examples where a "proof by picture" goes wrong
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31 votes

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 ...

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How to teach logical implication?
31 votes

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 ...

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What interesting properties of the Fibonacci sequence can I share when introducing sequences?
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28 votes

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 ...

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Historical tidbits to liven up calculus classes
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26 votes

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 ...

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Impressive examples where a "proof by picture" goes wrong
23 votes

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

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Good, simple examples of induction?
21 votes

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\...

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Examples and applications of the pigeonhole principle
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 ...

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What are some good rules for handling student questions during exams?
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 ...

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What to do when students are not keeping their eyes on their own test?
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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 ...

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Using original texts while introducing new concepts in class
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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 ...

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"Correct the following mistake"-style questions?
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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 ...

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For calculus students, what should be the intuition or motivation behind series?
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 ...

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Content for a 40-minute lecture on graph theory for high schoolers
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 ...

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Impressive examples where a "proof by picture" goes wrong
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 "...

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Presenting a solution with a stroke of genius
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 ...

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What are some great books for exploring mathematics? (not kids' books and not textbooks)
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 ...

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Writing up a proof that assumes what is to be proven?
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 ...

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Patterns that unexpectedly fall apart at large $n$
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 $n$ ...

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Is there a simple real-world problem I can use to motivate a formula for $\displaystyle \sum_{i=1}^n i $?
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 $5\times 1$ ...

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Do students confuse $\log_ab$ and $\log a^b$?
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, ...

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Explanation of Counter-example?
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 ...

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How to motivate equivalence classes
9 votes

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 ...

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How can we help students learn to write about their mathematics?
8 votes

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 ...

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Books/(auto)biographies/references on how mathematicians study/studied (as students)?
8 votes

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 ...

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How do I adapt a MWF class into a TuTh class?
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8 votes

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 ...

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How can I give feedback that is not demotivating?
7 votes

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/...

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Drawing vs Constructing
7 votes

I would emphasize the algorithmic nature of a construction over the ad hoc nature of a sketch/drawing. Tell your students that a construction must be accompanied by a specific sequence of steps (a "...

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What prerequisites would college students need for a course based primarily on Euclid's elements?
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7 votes

I really think this depends on what you intend to cover and what you want students to learn from it. Do you hope for your students to work through the major parts of the books and develop a true ...

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