Mathematics, the way it is currently written, can be difficult to read. Sometimes it helps to see how people think about a topic or theorem before (or after, or during) the reading of a proper treatment or rigorous proof. The … Continue reading

## Fun with Fractions—from elementary arithmetic to the Putnam Competition—the first 1/2.

Which number is smaller, 1/6 or 1/7? The answer should be intuitively clear (without needing to find a common denominator!) and therefore it is instructive as well. Now let’s make it more exciting by taking it as far as we … Continue reading

## Reflections on the 2018 Joint Mathematics Meetings in San Diego California.

The American Mathematical Society is big; so is the Mathematical Association of America. Once a year, these giant organizations have a meeting welcoming all members of both organizations. This year there were over 6000 registered participants. That’s a lot of … Continue reading

## Proofs are not only often beautiful but also necessary

Here are a just a few examples where patterns which appear to hold, start to fail for large numbers. Of course, as humans, we have a biased view of “large” in the context of numbers. Consider the sequence, 12, 121, … Continue reading

## Math is fun–an efficient formula versus an expansive thought process

Our introductory mathematics courses are filled with finished products—not only are our proofs and procedures refined, but so too are the formulae themselves. This post gives one example of such a perfect formula along with the question: what might we … Continue reading

## For the most awesome middle school and high school teachers

Dear awesome teacher, Thank you for everything you put into good teaching. I know that’s it’s rare to see solid evidence that your extra efforts are really working. So here are two tiny examples of ways that teachers helped me, … Continue reading

## Let’s Integrate Calculus

Calculus is a full-brained subject being taught by left-brained instructors to right-brained students. I see many posts lamenting the lack of student understanding resulting from the standard way that a calculus course is taught. For example, Tevian Dray suggests: “Skip … Continue reading

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## A Parity Theorem For Drawings of Complete Graphs

This post gives a (hopefully) very accessible treatment of a beautiful mathematical fact of topological graph theory, called the “parity theorem” of Kleitman. (Two numbers have the same parity if they are either both even or both odd). (Please see … Continue reading

## Run For Third!

Imagine that it’s a tie game in the bottom of the ninth inning and there is one out. You are a base runner on first base. The outfield is playing back to prevent a double and the hitter hits a … Continue reading

## Useful generalizations, Part II

Very often in mathematics a theorem can be easier to prove if it is viewed as a consequence of a different theorem, which may be far more general than the original. Our example for this post appeared on the 2014 … Continue reading

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## Useful Generalizations, part I

Very often in mathematics we see a nice argument or proof and we realize that the same argument can prove more than what was originally intended. The purpose of this post is to do just that for two recent posts, … Continue reading

## It’s a Mean Value Theorem

“The average rate of change will be the same as the instantaneous rate of change somewhere in the interval” under reasonable assumptions, is the gist of the Mean Value Theorem. A pleasant funny review is in the slow-paced 1966 (yes, … Continue reading

## A cardinal rule of base-running is hilariously incorrect

Sometimes I watch Major League Baseball games. I’ve noticed that the following is never questioned: Unwritten Rule: With fewer than 2 outs, a base-runner on first base shall be certain that a ball hit in the air will fall in … Continue reading

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## On the value of believing that you know the answer

A beautiful inequality, the arithmetic -mean-geometric-mean inequality (AM-GM) seems reasonable but hard to prove. It states that the usual average of n positive real numbers is at least as big as the n-th root of the product. We’ll experiment with … Continue reading

## On the unreasonable efficiency of mathematical writing, part II

The writing of mathematics is very efficient. Perhaps the teaching of mathematics can afford a little more redundancy. Here is a key example: Mathematical induction-the champion of efficiency. The beautiful idea: Induction is often used (and usually taught) to prove … Continue reading

## On the unreasonable efficiency of mathematical writing

In literary writing, we are taught to provide an introduction, a body and a conclusion. We are encouraged to—inefficiently—relate parts of an essay to a central idea. In mathematics, our writing is at times too efficient. Perhaps the teacher can … Continue reading

## On Teaching Loving Mathematics

I love math—not because I think I’m good at it, but because it’s powerful, it’s supremely beautiful and it’s everywhere. Mathematicians and many others agree! However, it happens all too often that people claim not to agree. Teachers can change … Continue reading

## On the Value of Making up Your Own Question

First, we consider an example of how one might make up a question. Here is a nice fact: (1/2)+(1/3)+(1/6)=1 Let’s have some fun with this fact by making it the answer of a question. Think of it as a mystery … Continue reading