Author: admin

One of the problems on the 6th grade NY State math exam looks like this:

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What is the value of the expression below when r = 2 ?

          9 – 3r

A   0

B   3

C   6

D   12

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My first thought upon reading this was, more or less: “Nine minus six equals three”. So I looked at the four choices to see where the 3 was (in this case, next to B).

But then, afterward, I realized that what had gone through my mind was really more like this:

(1) Because I had just read `r = 2′, the `3r‘ looked to me just like a `6’. I know it doesn’t say `6′ — it says `3r‘ — but I saw a `6′.

(2) Because I saw something that looked to me like `9 – 6′, I just did the subtraction.

I think the take-away here was that I looked for a way to get rid of the variable r as fast as I could. In this case it happened so fast that it was done before I was consciously aware of what was going on. Something deep in my mind was apparently saying: “Danger, variable encountered! Must be eliminated!”

This reminds me of what we do all the time with natural language — something most people encounter far more often than they encounter algebraic expressions. When somebody says to us: “Wear that tie I like,” — and we happen to know that they really like the pink tie — we might actually end up thinking that they had said: “Wear the pink tie.”

In other words, we substitute the variable right away, as fast as we can, going from the abstract description (“that tie I like”) to the concrete result (“the pink tie”). Except in real life, with real objects, we make these sorts of substitutions so easily that we generally don’t even notice that we’re doing it.

Perhaps mathematical reasoning starts with something as simple as learning to repurpose the ways of thinking we all use for natural language (which is an innate ability), so we can apply those ways of thinking to numbers and variables.

Some colleagues and I have been going over the standard NY State 6th grade math tests, as part of our process at the Games for Learning Institute of trying to understand how to help kids learn this stuff. I hasten to add that learning to do well on these standardized tests is definitely not the same as learning math.

In general such tests aim at only the lowest semantic level, and doing well on them requires mastery of concepts that are only barely removed from plugging in a formula. This is certainly not the sort of generative and multi-faceted view you want kids to have if you expect them even to glimpse the bracing beauty, joyful delight and sheer heart-stopping wonder of the actual mathematical universe.

Nonetheless, I was fascinated by my own process of reading these questions and answering them while simultaneously trying to catch a glimpse of what was going on in my mind. What sorts of concepts did I bring to bear upon reading a problem? When did I solve a problem by transforming it into a different one? How much energy did I spend on solving the underlying problem itself, as opposed to understanding the wording of the problem?

One of the difficulties in addressing these questions is that for most problems the answer just seemed to come to me, all at once. Because I’m not actually still a sixth grader, I ended up going through the steps and reaching a result faster than I could catch what those steps were, so the process often seemed instantaneous (as I suspect it would be to most of the people reading this).

But with a concerted effort I was sometimes able to slow the process down and watch what was going on in my head — a little like watching a film in slow motion. Some of what I observed was very interesting, and some was downright surprising. Over the next days I will try to share a bit what I saw.

The recent bizarre diatribe by Pat Robertson, right after the recent earthquake in Haiti, is generally described merely as “thoughtless”, or “in bad taste”, or “the ramblings of an old man.” But I think there is something much more specific going on here. Read for yourself what the man said:

“Something happened a long time ago in Haiti, and people might not want to talk about it. They were under the heel of the French, you know, Napoleon III and whatever. And they got togethter and swore a pact to the Devil. They said we will serve you if you will get us free from the French. True story. And so the devil said Ok its a deal, and they kicked the French out, the Haitians revolted and got themselvers free. But ever since they have been cursed by one thing after the other, desperately poor.”

Context is everything. And part of the context here is that Haiti is the only nation in history that was ever formed as the result of a successful military revolt of black slaves against a white colonial power. This “pact with the devil” myth is an old story. By repeating it now, Robertson is fulfilling a familiar role. Jews know this role all too well from their long history with the myths of the “Blood Libel”.

In the case of the Jews, the essence of the myth is that Jews kill Christian children for some sort of bizarre secret ritual purpose. Of course belief in such myths would give indignant Christians license to deny rights to Jews, or even to kill them outright.

In the case of blacks, the essence is even more insidious. The myth Robertson is repeating essentially says “Those black people could not possibly have won a war against us white people. Since they did, it must have been the devil who actually won the war for them.”

Notice how particularly, pointedly ugly this is. He is very specifically using an occasion of great tragedy to assert a claim that blacks are inferior to whites. Having done this, he is effectively casting any aid or help for Haiti not as a hand reaching across from one nation to another, but rather as a gesture from a superior people to an inferior people.

It is difficult even to talk about something like this. The sheer horror of contemplating minds that can spin this kind of filth is right at the edge of what any sane and reasonable human being can take in.

But it is important, when this kind of naked hatred (which so often disguises itself in one sort of sanctimony or another) shows its true self out in the open, that we look at it straight on, no matter how disgusted we may be.

We owe it not just to the people of Haiti, but to ourselves, to understand the nature of hate agendas, and the ways that those agendas disguise themselves, so that we can be effective in counteracting them. Don’t kid yourself that you are safe from this sort of thing. Sooner or later, no matter who you are or where you live, the people who spread these kinds of monstrous ideas of hate will come after you and your own children.

Suppose I would like to drop into my web calendar the up-to-date price of an AMTRAK train ticket from New York to Boston on February 17 at 5pm.

That sure sounds easy enough, but it’s surprising how hard it is to implement. You can’t just point to some page on AMTRAK’s website. The only way to find the price of a ticket on-line is to enter your start/destination and time/date into a web form. Then AMTRAK then crunches that data for you and generates a page that shows you the current ticket price.

There is nothing in the address field of that results page to indicate which route or date you’d asked for. That’s all done on AMTRAK’s server — the page you end up looking at is custom made for you by that server, on the fly.

I was talking today with Murphy Stein — a colleague and Ph.D. student here at NYU — and we came to the conclusion that it might be possible to solve this problem with a macro capability. In other words, you go through your usual process of entering the data into the web form, but you tell your browser to track what you’re doing, and it stores (somewhere) all the steps you just went through. It also lets you highlight the price on the results page, and it remembers where you highlighted.

Later, your browser periodically goes on a little web crawl onto AMTRAK’s site, entering the same values that you did into AMTRAK’s on-line web form, and then “clicking” on the same “GO!” button that you’d clicked on. Your browser then looks at the place on the results page that corresponds to the number you had highlighted. Unless things go horribly wrong, there will indeed be a number there (although it won’t be the same number if prices have gone up).

Every time you look at your web calendar, you will see an up-to-date price (how up-to-date depends on how often your browser goes crawling for updates).

None of this is sure-fire. For one thing, we are relying on AMTRAK to not change their on-line query form. For another thing, we are relying on your browser being able to find the price on the updated results page, working just from the location of the price you’d originally highlighted. That can turn out to be tricky if AMTRAK hasn’t designed their page sensibly.

Also, of course, AMTRAK might become unhappy if its customers’ browsers keep revisiting its site on their own every hour or so. At some point all of those repeat cyber-visits will start to overload company web servers.

So maybe none of this is practical. But it’s a nice thing to think about — being able to create a live and up-to-date link from anywhere to anywhere else on the web.

Quite a while back I talked about how our interactions with each other are framed by our inability to read each others’ minds (in September 2008, actually).

Recent events — as you might imagine if you’ve been reading this blog steadily — have led me to wish, at least for a moment, that I could actually read the mind of another. People have the ability to put on such smooth smiling faces, and we may never learn about their inner pain until it is too late to help them.

But as I think more carefully on this thought, I realize that society as it is now constituted could not exist if we could read each others’ minds. Just about any social, legal or ethical convention you can think of would be torn apart beyond recognition if we could peek into each others’ heads.

In fact, it’s not really clear that the notion of an “individual”, as we currently understand that word, would continue to have any meaning. So much of our essential being is predicated on the inviolable privacy of our own thoughts, and upon our ability to navigate the difference between those inner thoughts and the self that we outwardly show to the world.

Even to those who are closest to us.

The more I think about it, the more it seems that if we were all to wake up tomorrow morning with the ability to read each other’s thoughts, the result would be a vision from hell.

Guess we’ll just have to muddle through without the mind reading.

In the last few days there has been a tragedy in Haiti of immense proportions. Perhaps more than a hundred thousand dead, with millions left in a state of extreme suffering. There is no way to fully take in something this huge. Intellectually we can understand the enormity of such a calamity, but on an emotional level I don’t think our human minds are built for events on this scale.

At the same time — literally at the same time — an event occurred to which I obliquely alluded in my blog post the day before yesterday. At the time I was too overwhelmed to speak of it directly. The essential facts were as follows: Somebody I knew personally, a man I esteemed highly from within my own everyday life, died suddenly, and quite unexpectedly. I had last spent time with him that very same day.

Not surprisingly (or so people tell me) as I’ve walked around New York City these last two days I have seen him everywhere. A stranger will walk into a restaurant, or out of the subway, and for a moment, out of the corner of my eye, it appears to be the man I knew. But of course it isn’t, and won’t ever be.

As many of you know, the immensity of an unexpected death can for a while overwhelm everything else. For some period of time after, everything looks just slightly off — you temporarily lose track of “normal”. You observe people discussing politics or relationships, having a stupid argument over something or other, you see the texture of everyday life, and none of it quite adds up. Part of your mind thinks “What difference does any of this make?” There’s a necessary period of readjustment, a gradual feeling of the day-to-day reemerging, of things going back to approximately where they had been. But not quite exactly where they had been.

Because the reminder stays with you that the day-by-day life we live, that every touch of a friend’s hand, that every conversation over dinner (even the stupid arguments), is infinitely precious, infinitely worth fighting to preserve.

And that brings me back to Haiti. I understand that what I have just witnessed in my own personal sphere these last two days, the sense of loss and incomprehension, is also happening about fifteen hundred miles away, but on a vastly larger scale. All of those many thousands of individual lives, colleagues, mothers and daughters, fathers and sons, beloved spouses and lifelong friends, taken in an instant. And the ones who survived are left trying to figure out what “normal” is — how you can recapture the simple innocence of being able to take a day for granted, without constantly questioning reality itself, without continually seeing the dead out of the corner of your eye.

I think the best way we can honor the suffering, in addition to giving to the rescue effort, is to try to keep in mind — however difficult it may be — that every single life lost in Haiti is as deep a tragedy as the loss of someone we know in our own life. We must try to look at what has happened straight on, rather than from the corners of our eyes. I know that this is not possible to do well, but it is necessary to try.

Stepping gingerly away, for now, from the sad events of yesterday, I turn my mind to something altogether less difficult.

I was giving a lecture the other day during which I happened to mention that I had been using Google for several years before somebody finally pointed out to me that there were ads on the right side of the page. Not only had I never clicked on any of these ads, I had not even noticed them.

After my talk, a woman came up to me and pointed out that the fact that I had not noticed the ads did not mean they had been ineffective. “Do you think,” she asked, “that your buying patterns might have been influenced by the presence of those ads in the periphery of your vision?”

So. Perhaps these ads had been working on me all the time on a subliminal level, slipping in beneath my radar, as it were. But how could we test such a thing?

I’m thinking it would be interesting to run user tests in which we modify Google search pages (through some sort of proxy server, presumably). We would insert little artificial ad lines, mixed in with the real ads over on the right side of the page. These artificial ad lines could contain, for example, references to two different colors — one color for half of the participants, the other color for the other half.

After having been exposed to one or the other of these ads over a period of time, participants would be asked to chose one color from a selection of colors. If we see a systematic preference for the color mentioned in that participant’s artificial ads, then we will know that there is indeed a subliminal effect from these ads, even if the ad is never clicked on.

After all, if people are managing to put thoughts into your head, it might be useful to know about it.

Today I am trying to deal with some tragic news.

I suspect that tomorrow I’ll find a way back to my sense of optimism, and will once again be able to talk about the many ways that the future has just started. But today isn’t a good day for me to attempt that.

So please bear with me, as I try to find a bridge between yesterday and tomorrow.

There was a day, some years ago, when you and I walked along a beach, in the sincere belief that our lives might become intertwined forever. On that day we did not speak of this, but we each knew, in our hearts, there was more at stake than a train ride from the city and back.

Then you were with him, and I with her, and the secrets we shared from that day stayed with us, unviolated, entire. Only we two have glimpsed that other life, that other future, the one never lived.

There are moments, for each of us, when we still stand in a harbor, watching the tall ships sail out to sea. They are magnificent, lovelier by far for the unknowable secrets they keep.

I have often thought of you in the time from then till now. Perhaps if the wind had blown the other way, if a certain moment had lasted but a moment longer, we would have sailed to sea together.

It was just a day, only one among many. Yet even now it stays with me.

That other life. The one we didn’t have.

Today a colleague told me “I wish I could live in the future”. My immediate response was to point out that this is de facto impossible, even if you were to possess the requisite time traveling tech. I was actually thinking, as I said this, of those immortal words of Buckaroo Bonzai: “No matter where you go, there you are.” I didn’t bother quoting Mr. Bonzai (co-inventor of the oscillation overthruster, for those of you who didn’t know) because I was pretty sure my colleague would not have gotten the reference.

My larger point was that the human brain is simply not wired to sustain a sense of novelty. Unfortunately, all new things on our event horizon become reduced to the mere normal with astonishing rapidity, and our voracious and fickle appetite for the new and different can all too quickly lead us to consume the very change we wish to enjoy. We eat the future for breakfast, by mid-morning we have indigestion, and by lunchtime we are hungry again.

One of the projects of Will Wright’s wonderful Stupid Fun Club is a robotic waiter. Unsuspecting customers in an ordinary looking diner find themselves being served by a very polite robot, an attentive cybernetic being crammed with mechanical relays, electric motors and blinking LED lights. The results, surreptitiously recorded on video, are rather interesting. After a moment, most customers simply take their unusual new waiter in stride. After all, they’ve seen robots in the movies — why not at their local restaurant?

To make things more interesting, Will and his colleagues then have the robo-waiter get the order wrong. The unfailingly polite mechanical servant brings back coffee instead of tea, or a bagel instead of biscuits. At this point all customers react in the same way. Completely putting aside the wonders of being served by a mechanical man, a marvel of the future, a harbinger of the world to come, they just get annoyed. They ordered tea, not coffee, dammit.

And so I come to my thesis. Even if you were to build a time machine, put on your silver lamé suit, set your flux capacitor to full forward thrust, and emerge two hundred years in the future, you would have only about five minutes to enjoy the sensation, more or less. During that time you might marvel at the wonders of antigravity, the graceful arc of the protective energy dome over your city, the glint of sunlight off the floating skyscrapers in the sky above, or the way your brain tickles from the seamless techno-telepathy that that appears to have rendered both TV and the internet obsolete.

But after about a minute or so, your brain’s novelty normalization filter will begin to kick in. Within three minutes everything around you will start to seem obvious, even prosaic. After five minutes you’ll once again simply be living in the ordinary present. Yes, it will be a present that contains floating cities, free infinite energy, shimmering holograms you can control with pure thought. But none of that will matter once you get used to it.

It will just be normal.