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.

Hey, Alexa. Explain Your Algorithms.

AI Cover

Lately I seem to reading a lot about artificial intelligence. Between all the self-driving car projects and many, many predictions about robots coming for our jobs (and our children), the topic is rather hard to avoid. The topic is interesting but also somewhat scary since we’re talking about creating machines that attempt to replicate, and even improve upon, the human decision making process.

One of the better assessments of why we need to be cautious about allowing artificially-intelligent systems to take over from human judgement comes from MIT’s Technology Review, whose Senior Editor for AI says “no one really knows how the most advanced algorithms do what they do”.

If a person makes a decision, it’s possible (theoretically) to learn how they arrived at that choice by simply asking them. Of course it’s not as easy with children, but most adults are able to offer some kind of logical process explaining their actions. Even if that process is flawed and they arrive at the wrong conclusion.

It’s not so easy with machines.

There’s already an argument that being able to interrogate an AI system about how it reached its conclusions is a fundamental legal right. Starting in the summer of 2018, the European Union may require that companies be able to give users an explanation for decisions that automated systems reach. This might be impossible, even for systems that seem relatively simple on the surface, such as the apps and websites that use deep learning to serve ads or recommend songs. The computers that run those services have programmed themselves, and they have done it in ways we cannot understand. Even the engineers who build these apps cannot fully explain their behavior.

We’ve never before built machines that operate in ways their creators don’t understand. How well can we expect to communicate—and get along with—intelligent machines that could be unpredictable and inscrutable?

However, far below the level of television-ready robots and whatever Elon Musk is up to now, AI is a topic that educators should probably be watching.

More and more edtech companies are developing (and your school administrators are buying) “personalized” learning systems that include complex algorithms. These applications may fall short of being intelligent but will still collect huge amounts of data from students and then make “decisions” about the course of their educational life. 

It’s unlikely the salespeople can offer any clear explanation of how the systems work. Even the engineers who wrote the software may not have a good understanding of the whole package. Or know if there are errors in the code that could result in incorrect results.

And it’s not like you can ask the computer to explain itself.


The image is the logo from Steven Spielberg’s 2001 film AI: Artificial Intelligence, a somewhat mediocre example of the genre. And a movie would have been far different if Stanley Kubrick had lived to direct it.

Understanding Blockchain. Sorta. Maybe.

You’ve probably heard of Bitcoin, the “virtual” currency that was all over the news when it shot up in value in January and then dropped off just as quickly in February.

You may have heard about Blockchain, the technology behind Bitcoin and literally thousands of other virtual currencies. A concept that everybody says is going to change everything, but can’t exactly say how or why.

Do you understand any of it? I think I have a basic grasp of the blockchain concept but don’t hold me to that.

On his weekly dose of premium cable sanity, Last Week Tonight, John Oliver does an ok job of explaining all the pieces of the technology. But he does a great job of putting it all in context (“…if you choose to invest in the cryptocurrency space, just know that you’re not investing, you’re gambling.”).

The video is worth 25 minutes of your time. If nothing else, skip to the very funny closing that includes Keegan Michael Key.

However, even if you have no interest in the Bitcoin and cryptocurrency part of this topic, educators should at least have at least a fundamental understanding of the blockchain concept.

There is a lot of chatter about the technology having a big impact on education1 and edtech vendors are already finding ways to weave at least the word “blockchain” into their products.

When the marketing guys shows up at your door, one of you in the discussion needs to have a basic understanding of what you’re talking about.


HBO always posts John Oliver’s main segment to the Last Week Tonight YouTube channel. It’s worth subscribing to.

1. Audrey Watters’ guide on the topic, The Blockchain for Education: An Introduction, is two years old but still the best explainer on the topic I’ve read.

The European Approach to Protecting Your Data

Almost four years ago, the highest court in the European Union (EU)1 ruled that citizens of member countries had a “right to be forgotten”. Of course, that ruling left some holes and more than a few questions. But it did trigger some increasingly public conversations around the general topic of privacy and personal data.

That discussion, paired with some massive data breeches at high profile companies, led the EU Parliament to create a new set of laws2 dealing with data security and privacy. Those rules, the General Data Protection Regulations (GDPR), will become effective in the EU beginning in May.

In general, the GDPR sets strict guidelines for the kind of data that can be collected from individuals by companies and organizations, and how that data can be used. That data includes anything that can be used to specifically identify a person (including social media posts, location info, photographs, etc.), as well as not so obviously personal information like race, religion, and politics.

GDPR also requires companies to obtain more specific consent from the user as well as explaining more clearly how their data will be used. Specifically excluded is vague language like “Improving users’ experience”, “marketing purposes”, or “future research”. Companies must also make it easy for users to withdraw their consent and are then required to delete the material they’ve collected. 

So what has any of this got to do with those of us not living in Europe? Plenty.

While the regulations are specific to the member countries of the EU, most of what I’ve read about them suggest that all of us in the US, and elsewhere in the world, will likely be affected by them.

The law applies to any company or organization that does business in the EU member countries and collects personal data from their citizens. That includes many based in the US, familiar names like Facebook, Google, Microsoft, Apple, and more. Since most multinational corporations shuffle information around the world, it’s very likely that they will need to adapt their data handling practices everywhere, not just in Europe.

Plus the law also also provides for some pretty hefty penalties for misusing or failure to secure the data, including fines of up to €20 million or 4% of “global turnover”, whichever is larger. To put that in some perspective €20m (about $24m US at the moment) is pocket change for Facebook. 4% of their total income is not.

I know, all of this is pretty geeky stuff.

However, it’s also important if you’re concerned about the data most companies are already collecting about you and others. If you’re interested in more details of the GDPR in basic, non-legal language, check out this rough guide to GDPR and/or this short summary directed at US corporations.

Of course, the EU laws are not perfect. There will likely be much confusion when they take effect, and when the first law suits follow not long after. It will be interesting to see whether the big data collectors will be forced to change their behavior. Or will they just find new ways to continue their current practices? After all, our information is the foundation of their massive profits.

Beyond that, there’s also the larger question of whether the US should implement similar laws? It’s not likely to happen in this political climate, with political “leaders” who claim that the “free market” will protect us all. But maybe some outside pressure on US-based companies may effect some need change.


The map is from the BBC, showing the current configuration of the European Union. Of course, their home country, the United Kingdom, is in the process of a very contentious “Brexit” from the EU, so that map could change in 2019. In more than one way if the people of Scotland and Northern Ireland make some hard decisions.

1. Very tangential side note: I love that the official anthem of the EU is based on Beethoven’s “Ode to Joy”. Certainly more uplifting music than the militaristic tones of most national anthems.

2. In some of what I’ve read, experts says that GDPR isn’t so much “new” law as it is a clarification of many different data and privacy regulations that are already on the books, combined with court rulings. Either way, GDPR is likely going to change the way companies do business in the EU, and possibly elsewhere.

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.