Photo Post – M.C. Escher Edition

A couple of weeks ago, I had a unique opportunity to view some of the works by artist and mathematician M.C. Escher at the National Gallery of Art. These pieces are currently not on display at the museum and our viewing was in a small group with no glass in the way.

It was a real geeky session for me and the other the math teachers in the group, even if we only got about 30 minutes. Below are a few photos of the pieces, with the rest (plus a couple of shots from elsewhere in the East building) in this gallery. 

Lineup 2

Part of the collection we were allowed to view up close and without glass. I’m sure the curators were a little nervous but no one in our group messed up anything.

Ascending and Descending

A close up of a section of one of M. C. Escher’s most recognizable works, an amazingly detailed lithograph called Ascending and Descending.

Devils and Angels

Later in his career, Escher also worked in three dimensions. In this piece, he duplicates on a sphere his original two-dimensional tessellation showing angels interspersed with devils.

Hand with Reflecting Sphere

One of several self-portraits by Escher, this one with the artist reflected in a mirrored ball.

You Don’t Need Math

Add maths

A math major who turned out to be not very good as a mathematician, looks back at his studies and nevertheless finds some lessons he learned that have “nothing to do with numbers and everything to do with life”. He offers some nuggets in this essay that will apply regardless of the subjects you studied.

1. I expect to not get the answer on the first try.

It might sound pessimistic, but I think it’s pragmatic. I was rarely discouraged, because I never expected a quick win. And if I was correct on the first stab, I was pleasantly surprised. I became well-rehearsed in failed attempts, and so much more patient as a result.

I learned that lesson very early in my mathematical studies and it was one I tried to convey to my students when I started teaching.

Accepting that things don’t always work right the first time leads directly to this.

2. I can tolerate ungodly amounts of frustration.

Writer’s block has nothing on a tough math problem, and I’ve suffered through both. Writer’s block usually boils down to you thinking you’re not good enough. With math, it feels like the universe is mocking your ineptitude.

Of course math concepts can be frustrating. But there are plenty of other fields, like writing, that have their own unique stumbling blocks. Plenty of other endeavors, academic and not, have mocked my ineptitude over my life.

But frustration is not fun for anyone. Which leads into his third lesson learned.

3. I attack problems from multiple angles

Studying math was like maintaining a toolbox. Each time I learned something new, into the big red box that newfound knowledge went. Who knew when it would be useful? Long-buried methods could be just the socket wrench I needed later on.

Again, any field of study has it’s own set of tools. And any problem worth solving requires looking at the issues from different points of view. People who are successful at anything have assembled their tools and have learned how to try different ones when confronted with a new problem.

Again, an approach we need to be teaching our students, regardless of the subject on the syllabus.

The former mathematician has a few other lessons and more to say if you care to read the whole essay. But this for me is the bottom line:

Six years into my career, I can say that being comfortable with numbers and data has been useful, but what has proved invaluable are the qualities that math imbued in me?—?patience, attention to detail, humility and persistence. That was the true reward.

So, should every student take a rigorous program of mathematics in order to gain these qualities? Of course not.

Learning to write, mastering the French horn, creating the sound design for a play, repairing an automobile, all have answers that elude solution on the first try, create frustration, and require multiple approaches to succeed.

With the right teachers, students can learn patience, attention to detail, humility, and patience by working on the skills necessary for whatever interests them.


The image Add math by Chee Meng Au Yong was posted to Flickr and is used under a Creative Commons license.

The Real Meaning of Pi

Chalkboard with Pi

Today is Pi Day. Because the 14th of March could be written as 3.14, the first three digits for the irrational number we all learned something about in elementary mathematics.

Of course, this little bit of trivia only works if you’re writing the date as we do in the US. The whole exercise falls apart in most of the rest of the world where they traditionally write the day before the month. 14.3 makes no sense.

Anyway, beyond the fluff of memorizing lots of the digits and serving actual pies to math teachers (which we do appreciate), pi is a core mathematical concept with a long history and many important applications.

In this New Yorker article from three years ago, a math professor at Cornell University briefly offers a few reasons Why Pi Matters.

So it’s fair to ask: Why do mathematicians care so much about pi? Is it some kind of weird circle fixation? Hardly. The beauty of pi, in part, is that it puts infinity within reach. Even young children get this. The digits of pi never end and never show a pattern. They go on forever, seemingly at random—except that they can’t possibly be random, because they embody the order inherent in a perfect circle. This tension between order and randomness is one of the most tantalizing aspects of pi.

A little knowledge makes for a better Pi Day.


The image is from the header of the New Yorker article.

The Mathematical Obstacle Course

In a Medium post, a “research mathematician turned educator” discusses how extremely talented students are often disillusioned by high powered mathematics competitions like the International Math Olympiad.

Of course, extremely few high school students will ever be involved in this kind of “cheap competition that brutalises the subject into a performance act”, and this piece is of very limited interest to even most math teachers.

However, this observation accurately describe the high school math experience for most students.

School maths is engineered as a relentless competition, where students are ranked and judged according to the narrowest measures of aptitude. The spoils go to those who can mercilessly commit facts and procedures to memory (irrespective, and often at the expense, of understanding), and recall them in the arbitrary confines of exams.

In most high schools, the math curriculum imposed on students is a complex obstacle course aimed directly at Calculus, a class few of them need or will ever use.

Ending Math Phobia

An engineering professor at Dartmouth College wants you to stop telling kids you’re bad at math. Excellent suggestion.

Why do smart people enjoy saying that they are bad at math? Few people would consider proudly announcing that they are bad at writing or reading. Our country’s communal math hatred may seem rather innocuous, but a more critical factor is at stake: we are passing on from generation to generation the phobia for mathematics and with that are priming our children for mathematical anxiety. As a result, too many of us have lost the ability to examine a real-world problem, translate it into numbers, solve the problem and interpret the solution. [my emphasis]

Of course, we don’t really teach math that way, so it’s not at all surprising that most people don’t leave school with that ability. They never learned her process in the first place.

Maybe if we taught math as a tool for problem solving, instead of asking students to memorize a long series of meaningless rules, we would wind up with fewer math phobic adults.

Just a thought.