Saturday, September 10, 2011

Chapter 5: On the Game of Freeze and Multivariate Extrema (or, Why Asperger's Kids Suck at Conversation)

Warning: this post contains both scientific and mathematical content which may be boring for some readers. Read at your own risk.

During rehearsal for the play the other day, our director wanted us to get used to the feel of acting "in the round" - that is, acting when the audience circles the stage, rather than sitting on one side. To get used to the feel, she had us play an improv game called "Freeze". The game starts with two people acting out a random scene. At any point in their scene, another person may shout out "Freeze!" and, predictably, the current actors freeze wherever they are. The person who froze them must take one of their spots, and then take the scene in an entirely new direction. For example, the scene could be two people walking, and as one of them leans over to pick up a quarter, I shout "Freeze!" - take the quarter person's spot - and now the scene is us working out, and I'm trying to stretch and touch my toes. The other person must adapt to the new scene as quickly as possible.

So I'm in the middle of a scene where I'm a psychic, giving a customer messages from his dead grandma, when someone shouts "Freeze!" At this point my hands are on my temples and my partner is looking very sad. The girl who shouted takes my partner's spot, and begins the scene - "Come on, it's just the Springfield High Dance!"

I pause briefly to try and interpret the scene. I didn't quite register the name of the dance yet - my first reaction was that she was talking about some kind of dance routine, like the Macarena, which apparently I don't want to do.

"Do you just not want to dance with me?" she says, waiting for my reaction. So I figure, I obviously don't want to do this strange dance - perhaps it's one of those like Soulja Boy that I just hate - so I say:

"Just... the thought of it..." - I make use of my hands, conveniently placed at my temple - "Ugh!"

There is a brief silence. Then all the girls in the room go "Oh!" in a sad, pitying way. My new partner has a look of hurt on her face. I know we're just acting, but I get the feeling I have misinterpreted something. My reaction was not at all expected by the audience, or my partner.

Then I figure it all out. The Springfield High Dance is an event, not a routine, which she wants me to take her to. And evidently I can't stand the thought of taking her to an event, judging by my own reaction. I quickly try to readjust, find a way to get the scene back on track, but apparently the scene is perfect for someone else to shout "Freeze!" and take my spot. I go to my seat, feeling bad - though I knew it was just a scene.

Let's skip a little further into the day. I'm in my Calculus III class at State, and we're working on finding the extrema of multivariate functions. To translate, take a function of two variables f(x,y) - in other words, it takes two input values and gives one output value - and find the points where there are maxima or minima to the function. For example, take the function f(x,y) = x^2 + y^2. This function has a minimum at x=0 and y=0, because it is the lowest value of the function in its immediate neighborhood (f(0,0) = 0). Given a random function, how do you find the maxima and minima?

Every multivariate function has what is called a gradient, which is a measure of the rate of change of a function as its many variables change. It turns out that when there is a maximum or minimum at a point, the gradient is zero there. So we just have to find out the equation for the gradient, and find where it equals zero, and we have all the maxima and minima - that is, in a perfect world. Because we still need to find a way to tell which is which, and there are other points - called saddle points - where the gradient is zero, but there is no maximum or minimum. So now we also have to distinguish between saddle points and extrema.

So as the professor is explaining this, I'm playing a game I often do in Calculus class - I try and find the solution before he finishes deriving it on the board. Using some intuition, I remember that we used second derivatives in Calculus I and II to find the extrema of single-variable functions, so maybe I needed some kind of second gradient to find the extrema of multivariate functions. I began examining some way to do this and, unfortunately, did not finish before the professor. However, I did notice that his method looked nothing like mine. I did not see how he had derived it, so I just scribbled it down for future reference, and planned to perhaps derive it on my own time. I still had to finish my method, however.

Eventually I worked out the second gradient concept, but it seemed to have reached a dead end. Over the course of the next day, however, I gradually saw where I had gone wrong, fixed it, and derived my new method for finding extrema. It worked, and it still looked nothing like the professor's solution. His solution had been short and simple, while mine was complicated. His involved a few terms derived from the function itself, while mine involved a few terms derived from the function as well as a few extraneous variables that seemed to have no effect on the whole, but I didn't know how to remove. Intuition told me that if they didn't contribute to the final result, they could be pulled out somehow, and so I continued to stare at it until I saw I could turn the equation into a quadratic and pull out the extraneous terms that way. My result was exactly the professor's method. Relieved that I had finally solved the problem my own way, I set my notebook aside and moved onto other things.


You may be wondering what those two stories had to do with each other. You may also be wondering what the heck half the stuff I said in the second story was. I don't fault you if you only skimmed over it; I only ask you to consider why it was so much more difficult to read the second part that the first. I personally find the second story more interesting, but I realize that it may seem monotonous, boring and opaque to others.

The two stories highlight a key aspect of Asperger's, and that is our apparent genius to non-Aspergian people. How is it that we can understand things like what I mentioned above, with the ease with which we understand them? I will answer that question with another question: how is it that you non-Aspergian people can pick up on social cues so well? How can you look at a person's face and body language and know exactly what to say and what not to say, before even a word is uttered?

I have a theory about how the mind works. I think that our minds are just an organized collection of concepts - the ideas from which more thoughts are built from. Car is a concept. Sam is a concept. Concept is a concept. Some concepts are connected to other concepts, and when one is triggered, it triggers all those concepts it is connected to with proportionate strength. For example, if I say President of the United States, what pops into your head? Perhaps President Obama. Perhaps the Secret Service. Perhaps a feeling of anger, or a feeling of joy. Whatever appears in your head is connected to the concept of President of the United States.

Perhaps these concepts can't be indefinitely connected, however. I think that our minds have the ability to build strong connections between certain concepts with ease in some areas, and more difficulty in others. Think of it like this - your brain has a "budget" of connections it can make, and so it has to allocate them. Will it put a large number of connections in social interaction? If so, you're likely to be very good at social communication, intuitively navigating your way through conversations. Alternatively, your brain might put a large number of connections in the section of your brain devoted to math. Then you'll be able to intuit your way through complex math problems, while struggling to understand the social world. You may have difficulty in a game like "Freeze" where intuiting your situation from a few social cues is key. Or maybe your brain won't allocate most of your connections to society or math, and instead to something else, like music. And so on.

Of course, just because we have difficulty in something due to not having the right number of connections in a certain area doesn't mean we shouldn't work on it. I decided a long time ago that it would benefit me to work on my weaknesses, and so I have given it my best try to make friends and be social. I've learned a lot through practice, and made a few mistakes. My friends have been very patient to stick with me through all my failures. One key thing I've learned about living a social life with Asperger's is to not say too much. I can keep control over what I say if I speak less. When I really get talking, however, is when I start to say stupid things. I hope I haven't offended anyone by being abnormally quiet around them. It's nothing personal; I just can't think of anything worth saying at the time. It's only later when I think, "Oh! I could have said that!" I think that's a better thought than "Oh! Why did I say that!"

So, if you have Asperger's, take the time to work on your weaknesses. Find things that you don't intuitively understand and try to learn through practice. Continue to pursue the things you do well, and get good at the things you don't do well. It will only make you stronger. People without Asperger's - I give you the same advice. Try to appreciate new things. Especially math. Seriously, I've heard people talk about math like it's some sort of religion only the elite priests can understand. That's simply not the case.

I hope I haven't bored you with this awfully long post. I think it's important for everyone to realize that for all our differences, we can be quite similar on the inside. I think we all have, for the most part, the same resources - just how our brains make use of them is different.


  1. I skimmed the second story. :-)

  2. I skimmed the second one as well...but only because my Calculus 1 brain had no idea what was going on. I guess that's kind of your point, though, isn't it?

    "It's nothing personal; I just can't think of anything worth saying at the time." <-- That I related to SO much. People think I'm shy, but I really just have no idea what to say.

  3. I skimmed. But only because he explained it to me in depth earlier. However, we did discuss whether "maxima" or "maximum" is correct. That of course leads one into the world of local and global maximum values. And people might skim that too...

  4. I'm impressed by how you solved the math problem. It's like you took the same path as the very first person who ever solved a problem like that, which I think would be thrilling. Who knows, maybe you took an altogether unique path and arrived at the same point. Even more impressive. :) If I were in your situation, as soon as I arrived at the answer in a different way than the professor I would have went to him and said "check this out!", ha, and asked him if he had any ideas for getting rid of the extra variables. But you just kept working on your own until you figured it out!

  5. I read this blog entry (at the invitation of Jan Morrison) as it so happens right after reading this article from today's NYTimes:

    And it's also been my own experience, as a math professor. Thanks for saying it well.

    I enjoyed your second story. I wish more teaching were discovery-oriented. If I were your prof, I'd ask you to present your solution to the class. Of course at a large University, that's improbable. Calculus classes at Messiah College are smaller.