We are never finished learning and relearning math

“Astronomer Copernicus, conversation with God” by Jan Matejko (1872). Image from Wikimedia Commons

In school, you learned that the Earth was the center of the universe. At least, you learned that people used to think that. Your science teacher then explained to you that actually, the Earth revolves around the sun instead. A man named Nicolaus Copernicus proved it during the Renaissance, and set off a scientific revolution.

If a teacher just wanted to teach you about orbits and the solar system, there would be no point in telling you that people used to be wrong about the sun and the Earth. There would be no point mentioning Copernicus, or Kepler, or Galileo…


Flour mills, McDonald’s, and cars that drive themselves

Photo by MAZOUZI ABDELADIM on Unsplash

Self-driving cars will take over the world of long-distance delivery, and it makes sense once you consider, of all things, automatic flour mills and McDonald’s. These 3 products all seem unrelated, but they all adhere to a simple principle: high volume, low labor costs. Automatic flour mills and McDonald’s became unstoppable for their respective industries. Self-driving cars, or really self-driving 18-wheeler trucks, will likely become unstoppable too.

The Mill Man

In 1785, Oliver Evans changed flour milling forever. Back in his day, flour was milled by hand. To get flour, a farmer would grow wheat on their own farm, take the wheat to…


Photo by Febrian Zakaria on Unsplash

Eventually, we will go back to eating at restaurants. Yet the modern restaurant may never be the same, and we can only wonder what restaurants of the future will look like.

Restaurants change the way we eat food, often drastically. Even putting the pandemic aside, a young man or woman in 1921 would be stunned to learn that today we have Doordash, Instagrammable food, fast food, drive-thru windows, and modern chain restaurants that long for the nostalgia of the 1950s, the 1930s, even the 1430s.

Did you know that there’s a restaurant where you can eat a balloon? People in…


A quick example of a common mistake we make with percentages

Photo by Icons8 Team on Unsplash

One reason the 2016 U.S. Presidential election was… erm, remarkable?… was that it was the first election where both major-party candidates had an unfavorability rating of over 50% (more accurately, “first election” as in the first held after pollsters started recording unfavorability).

The other day someone told me this: “It was the first election where a majority of Americans disliked both candidates.”

No. No! That’s not what this statistic means. It could be the case that a majority of Americans disliked both candidates, but it’s much more likely that most Americans disliked one candidate and had a neutral/favorable rating of…


Photo by Road Trip with Raj on Unsplash

The man who got me into reading books never once told me to read a book. When I was 18, I went to therapy through my college. The therapist I met with was the catalyst who inspired my reading habit, a habit that has since changed my life.

Growing up, I mostly kept my distance from books. You could count on one hand the books I had read in the first 18 years of my life. …


Photo by Kaleidico on Unsplash

I believe that a good book changes the way you view something forever. It doesn’t always change your mind, but it changes your view. A great book from any genre does this, from self-help to journalism to fiction. The Steve Jobs biography changes your view of Steve Jobs. A good novel, like Little Fires Everywhere, changes your view of family values, adoption, etc.

In 2020, I finished 40 books. To be honest though, if you read one great book in a year, I think you’re doing well. The exact number does not really matter. One great book that changes something…


Photo by Steve Johnson on Unsplash

When teaching linear algebra, the concept of a basis is often overlooked. My tutoring students could understand linear independence and span, but they saw the basis how you might see a UFO: confusing and foreign. And that’s not good, because the basis acts as a starting point for much of linear algebra.

We always need a starting point, a foundation to build everything else from. Words cannot be written without the foundation of an alphabet. Ancient civilizations believed the universe was formed from 4 classical elements — water, earth, fire, and air (Long ago…). …


A different take on the meaning of math

Photo by Andrew Ridley on Unsplash

I can’t believe that teachers don’t tell you this, but math is not the study of numbers. We haven’t been lied to, but it’s like we all collectively forgot what the real point of math is.

At some point in geometry class, you probably had to prove that two triangles were congruent to each other. One way is to show that they have 2 angles and 1 side in common, and then you can conclude that the two triangles are congruent. …


Using the cartography package in R to create plots without a purpose

Your analysis is supposed to reveal something new. It should reveal an insight into a trend, or an abnormal pattern. Employers for data science jobs like to see that your analysis is useful and impactful.

But that’s boring. Instead, we can make graphs that are completely, totally useless, which is what I spent the last couple of hours doing.

The cartography package in R helps you turn your data into eye-catching maps. For example, R comes with a dataset of SIDS death in North Carolina. …


Photo by Lucas Benjamin on Unsplash

Our eyes see color using only three types of cone cells — which take in red, green, and blue light — and yet from those three types we can see millions of colors. Beyond being a nice, efficient biological feature, this illustrates an important concept in linear algebra: the span. To explain span intuitively, I’ll give you an analogy to painting that I’ve used in linear algebra tutoring sessions.

(This is part two of a three-part series explaining linear algebra concepts using the analogy of painting. The first part is on linear dependence/independence, which you can read here. …

Mike Beneschan

Math Tutor | Orange County, CA | If you want to reach out beneschan@gmail.com

Get the Medium app

A button that says 'Download on the App Store', and if clicked it will lead you to the iOS App store
A button that says 'Get it on, Google Play', and if clicked it will lead you to the Google Play store