Teaching Maths Like Music

Music class is where we take out our staff paper, our teacher puts some notes on the board, and we copy them or transpose them into a different key.

This sounds like the worse music class ever, doesn’t it?

Paul Lockhart argues that mathematics education is like this. For someone who loves math, our current classes are awful.

For me, maths is a wonderful activity. It’s an opportunity to get creative, to come up with my own problems, to get frustrated as the solution I’m crafting is getting too ugly, to get excited as a beautiful proof emerges. It’s something I enjoy watching others do. My breath is regularly taken away by an elegant proof or by an explanation that flows perfectly.

Most of us understand how people can fall in love with music… Yet when I tell my daughter that I love maths she just shakes her head, telling me I’m an alien. Because to her maths is about taking out her staff paper and copying some notes.

I wish her math curriculum had been closer to her music curriculum. That it encouraged creative exploration, that it fueled awe and wonder, that it recognized mistakes were something to be embraced on the way to greater creativity, and not something to systematically correct.

Watch Toby Hendy summarize Lockhart’s essay on YouTube.


Deconstructing Noam Bardin’s Post

I get excited about a lot of things.

Sometimes I get excited because someone I admire agrees with me. In these cases, if I dont force myself to stop and think, I can loose my critical sense.

This happened to many of us with the recent explosive piece by Noam Bardin, Waze’s ex-CEO who recently left Google.

Thanks to Charlie Gedeon for desconstructing the article and helping me be more critical. He points out that bashing on entitled Google employees hides the fact that the lion’s share of revenues are still going to much less visible people (eg. shareholders).

There is something wrong with the dominance of big tech. It is important to cultivate a diversified innovation culture — the likes of which is found in startups. But the picture Noam Bardin paints is too simplistic.


Our Experimental Deficit

We don’t run enough experiments in the wild.

In fact, I think our drive for global equity, although laudable in its desire to increase the quality of life of everybody, is having the unfortunate side-effect of reducing the number of experiments being run in the wild.

A century or two ago every country was running their own little in-the-wild experiment, with their own mix of social, economic and cultural rules. Naturally in most countries the outcome sucked more than in the best countries, hence the drive to adopt the rules of the best country.

But if you’ve ever been in a lab you know that the worst enemy of success is the fear of failure. It’s shutting down an experiment too early because the first few datapoints suck. It’s giving up on your “hunch” that your crazy new approach is bound to work even though it hasn’t worked yet.

Unfortunately, a lot of the ideas that can have a positive impact on human lives cannot be experimented with in the lab. UBI is one of them. We can have debates as to whether or not UBI is a good idea, but these will systematically lead us to our own opinions on human nature. Some of us have a “hunch” that the experiment should eventually work, some of us think it never will.

This is typical in academic labs, and if some scientists have a really strong “hunch” that an approach should eventually work, they’ll pursue the research even though their colleagues think they are crazy. This is the dynamic that gave us Deep Learning.

But a “hunch” that UBI should eventually work cannot be tested on computers. It needs to be tested on humans interacting together at a large enough scale.

I don’t know how to solve this ethically, but I still feel the relentless march of global equity, for all the good things it’s bringing to the world, is increasing our experimental debt in a way that may not be visible but that is likely to have long term impacts. In the meantime, from the few experiments we have with UBI, it seems like it might be a pretty good idea


Falling in Love Again (With Maths)

If you like maths just a bit I’m sure you remember your first encounter with the solution to the Towers of Hanoi puzzle. If you kept doing maths you probably consider the problem too trivial now, but try to go back to the emotions you felt that very first time.

I personally vividly remember my excitement as I discovered recursion through these stacks of moving discs. I was blown away by this new mental model – splitting a problem into smaller versions of itself – and how it allowed me to structure a seemingly devilish problem in a way that made it trivial to solve.

In today’s Mathologer video, the amazing Burkard Polster brought back these emotions not only by allowing me to dive back into the Towers of Hanoi puzzle, but by taking it three steps ahead into 4 pegs and 5 pegs versions of the problem. As always, he does it with exceptional clarity and delightful animations.

Brighten up your Saturday and sharpen your mind by watching this. I bet you’ll enjoy it even if math is not your thing.


Chess is a Videogame

Pandemic? Climate crisis? No: PogChamps!

For a small fringe of the elite chess world the biggest threat to society today is the fact that chess has become a Twitch phenomenon and that popular streamers are playing it for an audience… Even though their ELO rating is not even 2000!

PogChamps 3, the amateur tournament at the heart of this crisis, is streaming on Twitch right now and some GMs are not amused. Quoting Super Grandmaster Yan Nepomniachtchi: “PogChamps 3 as popcorn stuff is displacing any real chess content and this is just terrifying.

Personally it’s this debate that I find popcorn-worthy… Such elitism is totally antithetic with the online gaming culture that is powering this chess renaissance. Young people seem to be treating Chess like any other online video game. Something you can play with friends, for fun, where you can watch pros but where you can also watch streamers who are just trying to make a good show.

I also think the phenomenon is very interesting from an AI perspective. When Deep Blue beat Kasparov in 97 it felt like a crisis in the world of chess. Today nobody cares about the fact that engines have left humans in the dust. In fact engines have turned into excellent and very democratic trainers. A membership on will buy you a deep analysis of every game you play, highlighting your good moves and showing you where you erred. You’ll get personalized lessons based on the mistakes you most frequently make. You’ll get chess puzzles calibrated for your weaknesses. It’s a clear invitation to continuously improve.

This renaissance of chess has affected my household directly: my son has become enamoured with the game. I tried paying it with him often when he was younger but he could never be bothered to learn what pinning a piece meant and why a discovered check was so dangerous. Now he’s playing the Ponziani and correcting my Sicilian defense. To see him bite into something and having all the tools needed to fuel his passion is putting a huge smile on my face.

The Internet can be pretty darn awesome.


Learning About Ellipses

You get a circle with radius r. Its area is πr². You get an ellipse with major and minor axes a and b. Its area is πab.

The circle’s circumference is 2πr. The ellipse’s circumference is π(a+b), right? No.

I’m incredibly late to the party, but I just learned that there is no closed formula for the circumference of an ellipse. This blew my mind. I really felt like one of “the lucky 10000“. I also felt like sharing it with you, in case you cared to join me.

This launched me in an exploration which culminated in this paper, that I’m reading avidly. There are so few of these “scientific papers” that are written with the reader in mind. I can only imagine what a world of academics who really cared about being understood — rather than being published — would look like.


Adding More Maths to Deep Learning

Anybody who has tried to learn Deep Learning quickly realized that it involved a lot of maths. However, despite all the equations you encountered, much of Deep Learning is poorly understood from a mathematical standpoint.

Our understanding is progressing, though, and this Quanta Magazine article does a good job of summarizing recent advances on the theoretical front.

Within the sprawling community of neural network development, there is a small group of mathematically minded researchers who are trying to build a theory of neural networks — one that would explain how they work and guarantee that if you construct a neural network in a prescribed manner, it will be able to perform certain tasks.

Boris Hanin, a mathematician at Texas A&M University, likens the situation to the development of another revolutionary technology: the steam engine. At first, steam engines weren’t good for much more than pumping water. Then they powered trains, which is maybe the level of sophistication neural networks have reached. Then scientists and mathematicians developed a theory of thermodynamics, which let them understand exactly what was going on inside engines of any kind. Eventually, that knowledge took us to the moon.


Sometimes Things Just Are Complex

When you encounter a complex system it’s very tempting to point at all its flaws and imagine you could rebuild it from the ground up in a much more simple way. I’ve entertained that thought very often myself. I’ve often been wrong, as this article by Venkatesh Rao points out.

The Authoritarian High-Modernist Recipe for Failure:

  • Look at a complex and confusing reality, such as the social dynamics of an old city
  • Fail to understand all the subtleties of how the complex reality works
  • Attribute that failure to the irrationality of what you are looking at, rather than your own limitations
  • Come up with an idealized blank-slate vision of what that reality ought to look like
  • Argue that the relative simplicity and platonic orderliness of the vision represents rationality
  • Use authoritarian power to impose that vision, by demolishing the old reality if necessary -Watch your rational Utopia fail horribly

The big mistake in this pattern of failure is projecting your subjective lack of comprehension onto the object you are looking at, as “irrationality.” We make this mistake because we are tempted by a desire for legibility.

Via Sentiers by Patrick Tanguay


Cut-and-Pasting Text from Images on Android

I just discovered the most useful hidden feature on my Android phone.

You know when Facebook won’t let you cut & paste an excerpt from a post, or when you wish you could copy the text from an Instagram image? Now you can.

Swipe up to go to the applications carousel which shows a the screen of every application currently running on your phone. Within the carousel you can swipe your finger on any of these screenshots to select text. It works even if this text is not selectable in the app, or if it’s embedded in an image.

It’s a very useful feature when you like to cut-and-paste excerpts for your posts, like I do.

I tested this on a Pixel 3 with vanilla Android 11.


About Working with Yoshua Bengio

I have the immense privilege to be collaborating with Yoshua and some of his students — shoutout to Anirudh Goyal — on what is the most exciting research I’ve ever done. I love his research direction, but most importantly I love his research philosophy:

What matters to me as a scientist is what needs to be explored in order to solve the problems. Not who’s right, who’s wrong, or who’s praying at which chapel.

Read more about the researcher who’s most influenced me in this IEEE Spectrum article.