A Mark of distinction

We’re not quite done with the seven dwarves. There is still the question of how to decide which dwarves can reach the nine button and which can only reach the eight button (the digits need to alternate in the right pattern for things to average out properly), but we can come back to that another time.

This evening I went with a friend to see the revival of the farce Boeing Boeing (in a translation by Beverley Cross of Marc Camoletti’s french original) on Broadway. I mostly went to see Mark Rylance, who is quite possibly the finest english speaking actor of his generation. He is a legend in the U.K., but virtually unknown here. I had been fortunate enough to see a number of productions that he had directed and acted in when he was the Artistic Director of the Globe Theatre in London. The sheer range of this man’s acting talent is bewildering – from Shakespearean tragedy to modern farce, he holds the audience completely in the palm of his hand at all times. Imagine what it might be like to play pick-up basketball with Michael Jordan: He might play down near your level to keep it fun, but the immense coiled strength and skill is always there, waiting to be unleashed in an instant.

At the Globe I had especially liked Rylance’s conception of Twelfth Night, the one where Viola dresses like a man, impersonating her twin brother Sebastian (whom she believes is dead). Duke Orsino, having fallen in love with the Lady Olivia, hires this comely young “man” as a go-between, whereupon Lady Olivia falls for Viola (whom she thinks is Sebastian) and Viola falls for Orsino – which is also awkward, because the Duke thinks she’s a man. Then of course the real Sebastian turns up, and everything goes pear shaped (a wonderful British phrase taught to me by a young lady from London).

Rylance went back to the custom of Shakespeare’s day, whereby all the parts are played by men. Of course in the 21st Century this makes for quite a different impression. Rylance cast himself as the Lady Olivia. Imagine if you will a man playing a woman tormented by her love for what she thinks is a young man, but who is actually a woman, who is also played by a man. Meanwhile, this second young man playing a woman playing a man is in love with the Duke, a man who is in love with the first woman being played by a man.

Got that?

Well, it worked splendidly, and the audience was able to follow every moment of it, largely because Rylance directed and played the part of Olivia with absolute conviction. In lesser hands this conceit could have been a complete mess, but Rylance’s powerful direction and performance ended up bringing out the strengths of Shakespeare’s immortal comedy of errors.

In Boeing Boeing Rylance plays a man from Wisconsin, and he slips into a flawless midwestern accent – most audience members will assume he’s from somewhere near Madison. The entire farce is a delight, a complete success, and much of the comedy rests on Rylance’s uncanny ability to channel the extreme emotions of farce, from shy sobriety to hysteria to uncontrollable lust, either switching between one and the next at the instant, or, on occasion, managing to convey all three at once.

The evening was lovely, and the audience very happy. At the end Christine Baranski, who plays a french servant eerily like Edna ‘E’ Mode (the fierce if diminutive fashion designer created, directed and played by Brad Bird in The Incredibles), flung her cigarette out to the audience, and it landed in my lap. I shall treasure it always.

From Dopey to Dopamine

Continuing on from yesterday… Yann’s sparse numbers inspired me to think about how you might picture a base system that is not an integer – say 1.7, which is between one and two, or 9.3, which is between nine and ten. Doug’s comment the other day points to work that shows how to do this for one particular non-integer – the golden ratio φ, which is (1+√5)/2. But what about all the other non-integers?

Here’s a thought: Imagine Snow White and the Seven Dwarves has been reimagined by a crazed mathematician. In this version, each of the dwarves rides in an elevator – one elevator each for Bashful, Doc, Dopey, Grumpy, Happy, Sleepy and Sneezy. Each elevator represents one digit. When Snow White calls out a seven digit number, each dwarf is supposed to press the elevator button corresponding to his digit.

Snow White makes sure her elevator operators have clean hands

But there’s a problem – the dwarves might not be able to reach all the buttons, because some of the buttons might be too high and therefore out of reach:

Suppose the taller dwarves, Doc, Grumpy and Happy, can reach the ninth floor button, but the four shorter dwarves can only reach up to the eighth floor. Also, in a curious plot twist, our crazed mathematician author decides that Snow White can slip the dwarves a magic potion to make them all taller. As the potion continually increases everyone’s height, at some point each dwarf realizes he can reach one more button. By carefully doling out the magical potion, Snow White can set her button pushing friends to start to look like a base system between base to and three, or between base nine and ten, and so forth.

If we replace the dwarves by brain synapses, we can see that as we get more synapses firing, say if the brain’s dopamine level rises, then fewer neurons will be needed to represent the same variety of brain states. In effect, the brain is shifting further from base one and nearer to base two. Of course the scale is different: Snow White had only seven dwarves, whereas our brain has about 100 trillion synapses.

What would we call this way of representing numbers? Bashful binaries? Dopey decimals? Snow White and the Seven Digits? I’m open to suggestions.

Rounding the bases

As promised, the explanation. Which will require a little detour into the research of my brilliant colleague Yann LeCun, who creates machine learning algorithms. That is, he figures out how to get computers to learn. One of his inspirations comes from the way the human brain itself learns. And that can lead to some pretty interesting questions.

At some level, a computer’s memory consists of bits – and each bit can either be one or zero. There is something analogous that goes on in our brains. Each synapse of a neuron can be thought of as a bit. In response to stimuli from the world, a neuron can either fire a signal across that synapse (one) or not (zero).

It’s not hard to see that it takes a lot more energy for a neuron to make a one than to make a zero. In fact, if all of the neurons in our brain were to fire their synapses at once, the heat generated would fry our brain from the inside, and we’d be dead in less time than it takes to say “fried brains”.

Not surprisingly, the “bits” of our neural synapses tend to spend most of their time at zero, and much less time at one. This tendency saves our brains from frying, but at a cost: it means we need a lot more neurons. I’ll explain.

Given, say, a million different things to distinguish, if you had to have a synapse for each one, then you’d need to use a million synapses. This is the base one representation we talked about yesterday – like representing the number 42 with forty two zeroes. But there are many many millions of things that humans need to distinguish, and if we worked this way, pretty soon our brains would run out of space. This is sometimes known as the “Grandmother problem”: If there had to be a special synapse that fired only when you recognized your Grandmother (as well as one dedicated synapse for each such recognition task), then your head would probably need to be as big as a house. And that would most likely upset your Grandmother.

There are two opposing influences here: Our brains need to have a lot more zeroes than ones, so they won’t fry anytime we think too hard, but they can’t have all zeroes, or you run into the Grandmother problem. What the brain actually does is a compromise – something between base one (Grandma gets her own synapse) and base two (deep fried brain from too many synapses firing all the time). The brain actually works somewhere between base one and base two: mostly zero bits, and then the occasional one bit.

Yann uses something like this property in his work. Yann’s learning algorithms represent aspects of the things to be recognized as long strings of bits. In any given string, almost all the bits are zero, and only a few are one. Like 0000100001000000000100000100. It turns out that this is a good way to make machine learning algorithms efficient: by allowing the occasional one bit, the strings can be kept reasonably short (ie: you can avoid the Grandmother problem). And by making sure most of the bits are zero, you can get away with a lot less computation, because you can concentrate all of the computing work just where the one bits are. Just as your brain avoids getting fried by not firing too many synapses at once, Yann’s algorithms avoid frying the computer’s CPU by keeping down the number of computations.

This is all very interesting, but it’s a little abstract, isn’t it? Are we really talking here about something like base 1.7 or base 1.3? Well, yes we are. But in order to explain properly, I’m going to need to recruit that army of short people in elevators. Which I will do tomorrow.

Touching all the bases

Most people these days count in base ten. And most computers count in base two. In school we learn that there are other possibilities: like base one, or three or four or five. But nobody ever talks about base 1.7, or base 3.4, do they? Why is that? At this point you might be thinking “why would anybody want to count in base 1.7?” That’s a very sensible question. Well, ok then, I’ll tell you.

Let’s take the number 42 for example. It has two digits in base ten, right? Well, in base two it has six digits – 1010102 – in base three it has four digits – 11203 – and in base four it has three digits – 2224. In base one it’s a lot bigger:


That’s 42 zeros, in case you were wondering. But suppose you wanted to use about twenty digits to represent 42? Then you might want something between base one and base two.

Does this sound crazy? Well, it will probably get crazier. Tomorrow I’ll talk about why I’m thinking about this. It’s a tale that goes from brains that overheat when they think too hard, to problems with grandmothers, to computers that learn on their own, to armies of short people in elevators.

All this and more! Stay tuned.

Make believe it came from you

Recently I was thinking about that wonderful old Ahlert and Young song I’m gonna sit right down and write myself a letter, and I wanted to make sure I had the lyrics right. So I did what most people do these days: I did an internet search. And I discovered, to my utter disbelief, that on the internet the lyrics did not exist.

Oh sure, there’s a Wikipedia page, where you can learn that Fats Waller had a huge hit with this song in 1935, and that since then it has been recorded by everyone from Frank Sinatra to Bing Crosby to Billy Williams (#3 on the charts) to Nat King Cole, Barry Manilow, Dean Martin, Scatman Crothers, and even Bill Haley (in a rock and roll version, of course). And if you type “I’m gonna sit right down and write myself a letter” into Google – with the quotes – you get about 136000 hits. But not a single one of those pages has the actual song lyrics themselves. Not one.

I was mentioning this over dinner recently to a group that consisted mainly of hip New York new media artists. They were pretty unanimous in their response of “Who cares?” Around the table there was general dismissal of my concern that great popular culture of eras past should be alive and celebrated on the internet. To me their attitude was incomprehensible. What are we here for if not to inspire those who come after, to pass the torch of what is best in our time and in times before, so that its light might illuminate the minds of the future?

Yes, I know you can purchase these recordings and transcribe the words for yourself, and I’m sure there are books with the lyrics printed in them. But we think of the internet as a vast and growing reflection of our culture; yet it has these sorts of strange and inexplicable holes in it. And it seems that many of the intellectuals of this new medium simply do not care.

Today I mentioned the strange absence of these song lyrics to my uncle Artie, who is in his seventies. He knew exactly what I was talking about, and seemed rather appalled that anyone would think of great song lyrics as disposable. Of course they were all right there in his head, fresh as the first day he heard them. And what did he do? He sat right down and wrote me out the lyrics:

I’m gonna sit right down
And write myself a letter
And make believe it came from you
I’m gonna whisper words so sweet
Gonna sweep me off my feet
Some kisses on the Bottom
I’ll be glad I got ’em
I’ll write and say I hope you’re feeling better
And close with love the way you do
I’m gonna sit right down
And write myself a letter
And make believe it came from you.

So there you have it – now he song is on the internet. Soon it will be up on Google. Then I’m gonna sit right down and search there for this lyric. And make believe it came from you.

May day

Isn’t it mysterious that so many cultures celebrate May the first? In France they all go giving each other lilies, in the Rhineland boys deliver a tree to the girl they love (I’m not making this up), in Scotland they gather on the beach naked, and in England they all just go around Morris dancing.

Of course this is also the day where worker’s rights are celebrated around the world, this being International Worker’s Day. Except of course in the U.S., where we worry that too much worker solidarity could lead poor people to get uppity. Next thing you know poor people will be demanding the right to vote in Indiana! Not to worry – it’s clear that’s not going to happen any time soon, now that our learned Supreme Court has proudly brought back the poll tax. It’s not like those pesky poor people would know how to handle something as complex as democracy anyway, right?

April may very well be the cruelest month, but personally I think of May as the most polite. After all, isn’t it the only month of the year that announces itself by asking “May I”? 😉

Emily’s poem


“And when the gyre slowly turn
And cut into the bone
The fire lit, the soul will burn
    Til nought be left but stone

The ageless one can sleep no more
The Wraith has circled thrice
That He may rise who came before
    To weave His ring of ice

What is up shall then be down
What is far be near
For He shall come to wear the crown
    And He shall rule by fear

And nought but they who know the book
Can hope to break His spell
Yet well we know a single look
    Can burn the Devil well

Sing the ancient words aloud
Write them on the air
Turn His mantle to a shroud
    His glory to despair

Oh ageless one I call ye now
I call thee from thy sleep
A single strike upon His brow
    And let the blade go deep

Oh let the gyre slowly turn
And cut into the bone
A fire lit, a soul to burn
    Til nought be left but stone”

Scenes from the novel IX

Patiently she worked over the surface of the wood, polishing it to a fine burr, until she could glide her hand smoothly over the hickory stick. When she was done with that, she screwed the top off the first oil jar. It had a nice smell – this first oil had always reminded her of butterscotch. She worked it in, all over the surface, making sure not to miss anything. Then she carefully screwed the cap back on and unscrewed the second jar. Three oils in all, in the proper order, just like Grandma used to do it. When she was done, the polished surface gleamed like fine marble. She lifted the hickory stick up in her hand, just feeling the weight of it. Then she swished it back and forth a few times. Strong, light, just a little flexible. Exactly right.

Then Emily held the stick straight upright with her right hand and placed one end gently onto her upturned left palm. Slowly, carefully, she took her right hand away. The stick seemed to sway a little, and then it held steady, pointing straight up to the ceiling. She concentrated, and into her mind came the words of the old poem, the one that Grandma had taught to Mom, and that Mom had taught to her. As she did this, the stick started to spin like a top, slowly at first and then gradually faster. Then it lifted up, and floated gently off her outstretched palm. Carefully she took her hand away, and the stick continued to spin in the air.

OK, so far so good. This was the part where she was supposed to say the poem out loud. No, wait, not yet. Not until the music. She let the rod pick up speed. It started to make a whirring sound, and then the whirring cleared and turned into a kind of musical whistling. That was it – that was what she was supposed to wait for. It was the old tune, the one Grandma used to hum. And now it was time to recite Grandma’s poem. Except it wasn’t Grandma’s poem anymore. It was Emily’s poem, because now everything depended on her.

Smiling just a little at the thought of that, she started to recite the words.


Continuing the topic from yesterday – which again comes out of lots of interesting conversations with Jan Plass and other colleagues – the question always comes up of whether the potential educational benefits of games for learning might run into the coolness problem: kids may not want to play a game that’s supposed to be “good for them”. If you give middle school kids a game that’s really fun, but by playing it they know that they are actually doing math homework, would this awkward fact make the game so uncool that it is actually no longer fun?

Yet how could things be worse than the way they are now? Standard practice today is to give kids ages eleven through thirteen lots of boring homework exercises to hone their math skills. It’s hard to argue that the current approach is the optimal way to win their hearts and minds, or to show them that math can be fun and exciting. The same material covered via well-designed game play could hardly be any worse for motivating learning than the current status quo. In fact, there is every reason to think it would be better.

But still, I am reminded of that wonderful 1928 New Yorker cartoon drawn by Carl Rose and captioned by E.B. White:

Old E.B. hit the nail right on the head, didn’t he? Kids know. Kids always know.

So how will this play out? Can games make education more relevant and fun? Or will kids merely look at games in education as a surreptitious attempt to make them eat their vegetables? On the other hand, if our educational system embraces game play properly then perhaps school itself may come to seem a lot cooler.

Anatomy of gameplay

When you play a computer game, what are you learning? And how can we describe the essential pieces of that learning process? My colleague Jan Plass, whose research studies the effective use of technology in education, and I have been having some pretty fascinating discussions and spirited debates about how a game can function as a kind of vehicle for learning. One intriguing problem is how to visualize this vehicle in a way that clearly shows how all of its parts work together. Here is one such visualization:

  1. Player’s understanding – ability to play well
  2. Game mechanic – the rules of play
  3. Aesthetic design – graphics, sound, etc.
  4. Narrative drive – the “story” moving the game forward
  5. Extrinsic rewards – points, winning, etc.
  6. Intrinsic rewards – getting better, improving skill

Everything revolves around the interplay between (1) the model in the player’s head and (2) the mechanic of game play. All of the other pieces are in support of that interaction. Graphics, sound, and so forth (3) serve to add clarity and aesthetic pleasure to the experience, but they are really scaffolding to support that central player/game-play dynamic.

Similarly, a game generally provides some sort of narrative thrust (4), such as “get three in a row”, “capture the enemy king” or “buy up all the real estate”, which helps to focus the interaction in the player’s mind by giving it context.

In addition, the player is rewarded for good play in two ways: extrinsically by earning points, winning games, being the high scorer among a group of peers, and so forth (5) and intrinsically by gaining improving in skill, confidence and ability to take on more advanced challenges (6). In many computer games this increase in skill is often rewarded by “leveling up” – a case where intrinsic reward is augmented by an extrinsic reward.

A game that does not continue to give intrinsic rewards would soon grow tiring. This is why, for example, only small children enjoy tic-tac-toe: Because the highest level of attainable skill is quickly reached by older children and adults, and then there are no more opportunities for intrinsic reward.

The entire above model only really makes sense as a support vehicle to “move” the player’s mind, through a path of maximum fun, in a way that provides continual pleasurable challenges. This tends to result in ever increasing skill, as long as the game lasts. This path of maximum engagement, neither too easy nor too difficult for the player’s current skill level, corresponds to the mental state that Mihály Csíkszentmihályi refers to as “flow”.

One can visualize this as a landscape with an optimal slope. Game narrative and extrinsic rewards move the player and the gameplay forward, while the intrinsic reward of the player’s progression in ability moves the entire structure upward along the slope:

As the entire vehicle moves, the nature of all the components within it might be changing as appropriate: the player’s mental state, the nature of the gameplay, the aesthetic components, the narrative of the game and the rewards offered, both extrinsic and intrinsic.

A good game is one that moves this vehicle along this upward terrain by evolving all these components appropriately to maintain a maximum sense of fun challenge.

It could be argued that the only difference between a good game and a good educational game is in the opportunities for knowledge transfer of the intrinsic rewards (6) out of the game and into other contexts. For example, a rewarding experience playing Worlds of Warcraft can increase a player’s abilities in the “science” of alchemy. But one could well envision an alternate version of WOW with exactly the same gameplay, yet which challenges the player to improve his/her skills in chemistry or some other skill that transfers well to the world outside of the game. That version of WOW could indeed be an effective educational game.