A Weblog

Today I’ve decided to post a journal together with a longer paper about games. You hear all the time that we need to inject more play into education, that we need to return to childhood, etc. But why? You don’t as frequently hear why play is useful in education. People claim things like “If learning is fun, children will learn better.” I’m not sure of the connection. I suppose that if kids are engaged in learning, then they have a better chance of actually picking something new up than if they’re not trying to learn at all. That’s like saying if you look for something you have a better chance of finding it then if you don’t look at all. Sure, I buy that. But why play? By the same argument, we could just as easily pay kids to go to school and do their homework.

Of course some people do give reasons why play is useful. In these two papers, I’m building on some insights found in a 1933 paper by Lev Vygotsky entitled Play and its role in the Mental Development of the Child. (Vygotsky, you may well know, is one of my current heroes.) I remind the reader that in play, you can find all sorts of higher-order thinking skills taking place. Imaginary play is a very natural, distilled, abstractly difficult thing to do. Yet kids seem to do it on their own anyway, and before they even step foot in a classroom. If taught effectively, I think play is a useful vehicle for transfer of skills and tons of that ever-so-hot interdisciplinary work that goes on nowadays. (Wait until I get my genetic algorithmic music up and running.)

Journal 4: Methodological Doubt, Belief, and the Structure of Play

Reflection Paper 2: Decision-making as Game: A Mode of Prediction and Solution

Peter Elbow introduced concepts of methodological doubt and belief in his book Embracing Contraries: Explorations in Learning and Teaching. They’re central to his believing game and doubting game. Traditionally, doubt has been used as the primary tool in critical thinking. This unbalanced attention really makes a lot of analysis blind to new insights that can be gleaned from a moment of pure, suspended disbelief. (My ego won’t let me pass up an opportunity to say that both games show up automatically in my coffee mug model of classroom education.)

In my first paper I remark that all games require its participants to engage in the believing game—they have to believe that the rules imposed by the game are real and that the game itself is real. There are no consequences in any game if you don’t except them. You can always pick up the ball with your hands in soccer, unless you firmly believe that you can’t. For this reason, we might frame any situation as a game.

In the second paper, I extend my ideas to show that framing a situation as a game can greatly improve your power to predict behavior and arrive at winning strategies by simply considering the acceptable moves in your game. To illustrate my point, I work through a problem of the type sometimes given in consulting or computer science job interviews. The example shows, additionally, how mathematical reasoning (which I believe is no different than plain, old, vanilla reasoning) can be used to solve a problem without once using “math.”

As always, please comment freely. I’d love to get some feedback on this stuff.

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Freshman year of college my friend Rebecca tried to explain to me the literary school of deconstruction. After some time I tried to sum up what I had heard in a phrase that (be it my own or not, and whether it be accurate or not) I have kept with me six years later.

Words have meanings, but meanings don’t have words.

Now I’m still not sure what that means, but I do know it has to be true. My friend’s grandmother, sage that she is, disagrees entirely. Meanings are the words they mean—sometimes people misuse words—but that doesn’t detract from their instrinsic definitions. But if that were true, we wouldn’t have any need for dictionaries. If words were their meanings, then words couldn’t be defined in terms of other words. That’d be silly. The other words have their own (other) meanings, after all. Imagine what a dictionary entry might look like in this alternate semantic universe:

apple, n., apple. What don’t you understand? Apple means apple.

Of course, maybe I’m taking too naive an approach. DJ’s grandmother might be onto something. How can you sufficiently define terms like ‘this’, or ‘I’, or ‘you’? This is what it is. It’s nothing else. It’s this. I am who I am. Or am I? Words, like people, take on a meaning that emerges from their use. How words are used, though, follows from larger, guiding principles. Culture helps define who we are. So, too, culture—which is really no more than a vast set of complex and subtle rules—defines what are words mean. So, words do have meaning. But only in relationship to other things (that have meaning). It’s sort of like music.

In music syncopated rhythms accent the beats which normally go unaccented. But without some concept of normal, syncopation doesn’t exist. But it does because in our music there is a structured sense of normal. And if we let loose the structure, we loose some of the meaning. Syncopation just disappears. Ironically, the tighter a straight-jacket we put on rhythm the freer we can be within its constraints: we get things like syncopation back.

In mathematics, too, Kahler manifolds are surfaces that exhibit a rich geometry. It’s thought that the physics of our universe is actually encoded on one of a special class of these surfaces known as Calabi-Yau manifolds. The thing about Kalher manifolds, though, is that their geometry is so highly structured that the surfaces are almost flat. Flat surfaces are the simplest to investigate. It turns out that these guys, by comparison, are notoriously difficult to analyze. There may be something to that—that the most useful, interesting cases often lie just on the cusp between simple and intractable—but I’m not sure what it is.

Technorati Tags: calabi-yau, apple, deconstruction, dictionary, syncopation, music, culture, identity, meaning, words, kahler, geometry, manifold, grandmother

I have a few other posts saved as drafts, and I want to get to them, especially one on technology, but I can’t pretend to have finished talking about that alternative decimal representation of the number one (0.999···) that carries with it an infinite chain of nines. To say that I’m comfortable with that representation of the number is almost as brash as claiming to have solved Zeno’s Paradox. Infinity is a funny thing. In fact, it would be more honest to say, infinities are funny things. After all there are lots of them. And there’s no reason they should all be the same—in fact, they are not.

But before we dive off the deep end, we should pause to think about what it means for finite numbers to be the same. Have you ever given much thought to statements like “5=5″? So, now, before you read on, turn to the nearest 8 year old you can find and ask her, “Does five equal five?” [That's the easy part.] Now ask her, “How do you know?” and listen for a response. [That's the medium-hard part.] If your 8 year old doesn’t sufficiently convince you that five is, indeed, equal to five, explain to her why it’s true in plain terms that no one could dispute. [That's the hard part.]

When I was taking an abstract algebra course, my professor asked the class what the number three is. We all knew it was a trick, so we waited in nervous silence, each hoping that he wouldn’t start dead-calling members of the audience. After all, college math concentrators should know what the number three is. We were studying math, and numbers, I’m told, are an integral part of mathematics. So what was the Fields medalist’s definition? He stole his answer from a six year old: the number three is three fire trucks without the fire trucks. Hold up, what? That’s a surprisinglyl useful way to think about number. I think that’s how Frege and Hume viewed number, and by some standards, they’re famous. So maybe this little kid is on to something.

Let’s pretend for a little while. You didn’t know this, but I have a pack of llamas. Every evening I feed them each one carrot for dessert at dinner time. The problem is, I can’t count. I have a bunch of carrots and a pack of llamas. How can I know if I brought out the same number of carrots as the llamas—without counting?

Well, I could feed the llamas each one carrot. If I had more carrots than llamas, then I’ll have carrots left over at the end. If there were more llamas, then I’ll run out of carrots before I finish feeding—the llamas hate that. But if the number of carrots and llamas equal, then after feeding time, each llama will have had exactly one carrot and no carrots would remain. That is, I would be able to put all the carrots in a one-to-one correspondence with the llamas. If I had more carrots than llamas, then some llamas would get the left-overs. That relationship is not one-to-one because some llamas get more than one carrot.

But we’ve got a problem, one-to-oneness is certainly necessary for there to be the same number of carrots and llamas, but it is not sufficient. If I had fewer carrots than llamas, every llama who gets a carrot gets only one. But because I run out of carrots before I run out of llamas, some llamas are left out. In order to know whether there are the same number of carrots and llamas, every llama needs to get exactly one carrot and there can’t be any carrots left over. This is tricky business.

If every llama gets at least one carrot, then we say that the pairing of carrots to llamas is onto. Ontoness is also a necessary condition, but like one-to-oneness it is not sufficient. Having more carrots than llamas leads to a pairing that is onto. Every llama gets one carrot and some get more. What we’re looking for is a matching of carrots and llamas that is both one-to-one and onto. Then for every carrot there is exactly one llama. Likewise, for every llama there is exactly one carrot. The size of the pack of llamas and the size of bunch of carrots is the same. Mathematicians like to have a standard way to talk to one another, so they call the number of elements in a group the group’s cardinality.

So, we’ve done it! We’ve found a way to determine whether two collections of things are the same size, if they have the same cardinality, without resorting to counting. In fact, we’ve secretly discovered a very powerful way of thinking. One thing we can do with our new-found friends, the one-to-one and onto functions, is define what it means to count. I don’t have the room to do it here. In fact, I don’t think we’re even going to get to infinity in this post. Instead, I’ll cop-out and refer you to some of my set theory notes. (That’s what we were doing: set theory.)

Set Theory Lesson Plans

In my notes, I’ve asked just a number of questions. I wrote these questions for a practicum for a class it took this fall. And I used them on real, high school juniors in at Codman Academy. We had a little bit more time to go over each of the questions carefully and resources that allowed dynamic, colorful diagrams—which the students largely produced for me. I was only asking questions, it was up to them to answer them. But please, don’t wait until 11th grade to feed the llamas.

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I remember when I first learned that advertisers will often use glue instead of milk in breakfast cereal commercials. The whole thing blew my mind. Initially, I felt confused. Why would they do something like that? Of course, because glue looks more like what people expect milk to look like than milk does [on camera]. Even after my personal revelation, I still felt confused. Except now my confusion came from within: why had I assumed that things on TV were what they looked like? Even now, I still feel a little uncomfortable thinking about it.

Again in the the seventh grade, another discovery left me feeling the same way: the infinitely repeating decimal 0.999··· is the same as the whole number one (1). I know that this must be true; my math teacher Mr. Heleen proved it to us. First, let’s hide the infinite string of 9s under a clean variable name, say x. Then we can distract ourselves long enough to arrive at a meaningful conclusion. Here’s what Mr. Heleen did:

10x	=	—9.999···
—x	=	—0.999···

Subtract the two lines (something that is hard to do in HTML) and you’ll get

9x

=

9,

Or, as I claimed earlier, that x = 1. Even now, I find that fact a little bit mysterious. And this is one of my central problems with algebraic methods in general. They’ll tell you that a statement is true, but they seldom lend themselves to obvious readings of just why a statement is true.

In fact, this reminds me of a frequent difference between inductive proofs and constructive proofs: inductive proofs often accompany theorems which speak only about existence—what you’re looking is out there, the proof guarantees it, but you have no idea where; constructive proofs, on the other hand, actually give you what you looking for. Constructive proofs are usually more useful than inductive proofs because they automatically satisfy existence by virtue of demonstration. (Imagine what economists would do with a constructive proof of the Brouwer fixed point theorem; and I’d understand Sard’s lemma a lot better if the proof I learned didn’t rely so heavily on induction.)

So, is it any wonder that I gravitated toward geometry over algebra in college? Geometers use inductive arguments, too, to be sure. However, problems are usually cast in ways that are about as tangible as mathematical problem can be. Some of them even have straight-forward physical interpretations. It’s not (too) hard to imagine that soap bubbles could represent minimal surfaces, for example. However, what does a Dedekind domain look like? (If you can help me visualize a Dedekind domain, I’d be very grateful. Had you helped me three years ago when I was taking algebra, I’d've been even more grateful.) Like I said, algebra is hard.

But let’s get back to our infinitely repeating decimal. Why should it be the same thing as one? Well, I suppose we should ask, what is the number one? There are lots of answers—many of them correct. In this case, one is particularly useful: the number one is a label for a point on the number line.

The decimal 0.999… is also a label. But then again it’s so much more. Both 1 and 0.999··· are directions to the points that they label. How convenient! Here’s how you read the roadmap embedded in every decimal. First you need to arbitrarily pick a point called zero. That’s up to you. Next you need to pick another point that’s a unit length away from zero. This choice is also arbitrary. In the metric system you might use centimeters. In the English system, the unit you pick might be feet. If we were measuring something large, maybe you’d choose lightyears. What you choose is really a matter of convenience.

Now the fun part comes in. The first digit d after the decimal tells you to chop up the unit length into 10 smaller pieces of equal length. This smaller distance (1/10) will become the unit you use in the next step. Then you go to walk to the d-th piece. In this case, we chop up the length 1 into 10 equal pieces and walk to the ninth piece.

In the second stage, you chop up our new unit (1/10) into 10 smaller pieces of equal length. That smaller length becomes our new unit (1/100) for the next iteration, so remember it for later. Now walk over to the piece that the second digit in the decimal tells us to go to. In the third stage, we repeat the process, always taking tinier and tinier steps. For an infinitely repeating decimal we have to take an infinite number of steps to get the point the directions describe. Eventually, the steps we take will be so small that for all practical purposes we stop. This is the idea behind a limit point.

Of course I haven’t been terribly rigorous. That’s where the algebra comes in. We already proved that 1 = 0.999··· above, but the geometry is where the understanding is, at least for me. Ideally, I would’ve had some pictures in this post—but modern technology is years behind pencil and paper. But kindergarteners can draw; more importantly they can walk. Maybe limit points aren’t especially useful in most kindergarten curricula, but I think that this shows that they probably have a fair shot at understanding the concepts. And maybe now I can put this demon 0.999··· to rest.

p.s.—Wikipedia also has an entry on 0.999··· with more pictures and deeper, more confusing jargon.

p.p.s.—Now I really need to write up something about infinity. After all, 0.999··· has an infinite number of 9s in it. What does that even mean?

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After the initial post on my robot/cartoon universe, a few of my friends and I have talked out the system. It turns out that my scheme is too restrictive in its expressiveness. Here I’ve set to free up the system.

No one has argued against the robot/cartoon dichotomy. But some have pointed out that pretends-to-be is too restrictive a connective. It only captures a very narrow (albeit common) relationship between self and self-image. Others have shown me that the connective is, perhaps, too idealistic. Pretends-to-be issues a lot of self-awareness to its referent. To balance out the relationships a little, I’ve decided to add the connective thinks it is to the mix. Thinks-it-is tries to convey whatever the opposite of self-awareness is—I’m loathe to call it self-absorption or self-deception.

Just as the split between robot and cartoon begins to blur when they are connected using a connective (like pretending-to-be), you can see that thinks-it-is is not at odds with pretending-to-be. They compliment each other through their (dual) connectives cartoon and robot. When both connectives appear in a single description, a new, complex meaning emerges from their interaction. However, the new addition complicates the taxonomy in more ways that I had first imagined. You see, pretends-to-be and thinks-it-is do not, as the mathematicians say, associate. And verbal language is not well-suited for these kinds of connectives. Let me show you what I mean.

I have a friend who is most certainly ((a cartoon who thinks it is a robot)-pretending to be a cartoon). Notice how that is not the same thing as (a cartoon who thinks it is-(a robot pretending to be a cartoon)). I’ve tried to demonstrate the difference by grouping with parentheses and hyphens (to show that the phrase wasn’t just a grammatical parenthetical). See what I mean?

Textual language handles the problem with hardly any more finesse. Parentheses and square brackets already have semi-well-defined meanings in English. The curly brace ({) is, and I’m sorry to say this, ugly in most contexts. Perhaps nested less than/greater than sign pairs would do better? My friend is a <<cartoon pretending to be a cartoon> who thinks he’s a cartoon>. Please offer up opinions and suggestions.

Technorati Tags:robot, cartoon, associate, taxonomy, connective, language, delimeter

That’s right. It’s time for another installment of “What has Josh been writing for class?” This week I responded mostly to an old article by Richard Paul—who, I think, bears a striking resemblance to Walker Texas Ranger: hold on to that.

He differentiates mainly between two types of styles of problem evaluation: weak-sense and strong-sense critical thinking. To paraphrase, perhaps unfairly, weak-sense is marred by an overly narrow subproblem formulation. It’s atomistic. First you take a big problem, chop it up into smaller problems, and then solve each of the bite-sized pieces one at a time. Paul rightly notes that oftentimes this method misses the larger problem that arrise from the interplay of the otherwise well-behaved subproblems. The mathematician in me has to note that the local-behavior-does-not-imply-global-behavior phenomenon has been a central theme in differential geometry from about its beginning. The same problem creeps up just about everywhere else you look for it. I’ve tried to talk about this before in vague terms relating to urban planning and chaos theory. Maybe I should try again sometime. But for now:

Journal 3: Weak-sense, Strong-sense, and Probabilities

I agree with Paul. Strong-sense thinking is more appropriate for lots modern problems. International conflict, curricular design, and global warming all require strong-sense critical thinking, for example. (Ordering dinner at a restaurant typically does not.) While I like Paul’s network approach to problem solving, I think the primary weakness of weak-sense thinking lies in its absolutist view of truth, not necessarily its divide-and-conquer methodology. Truth, when viewed as a certainty, is rigid and fragile. Today’s demanding social and business landscape calls for something more adaptive, fluid, and functional. (Yes, you were supposed to read that last line with an announcer’s voice.) So how do I amend his framework? With probabilities of course. Really dedicated readers will see that I’ve mostly recycled my blog entry about assumptions. But to keep things fresh, I had to add something. And you knew it would happen eventually. I couldn’t resist.

I center my discussion around a theorem from linear algebra. Gleason’s Theorem tells you exactly what the probabilistic measures on the closed subspaces of a Hilbert space are (basically they’re projection operators). And according to some, it’s central to future research in information retrieval. I use it to show the usefulness of multiple points-of-view with some scientific flare. Of course, my treatment is clumsy—but technically I’m only allowed one page per entry. How thorough could I have been? Maybe later I’ll clean this up and expand it a little. For now, it’s probably okay.

References

Paul, Richard. “Teaching critical thinking in the ’strong’ sense: A focus on self-deception, world views, and a dialectical mode of analysis.” Informal Logic Newsletter, 1982.

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Today I was reverse-stalking the links pointing to my site when I discovered that Planet 02138 uses my RSS feed to fill their content. To be honest, I’m a little touched. At first I thought it might be associated with that magazine I don’t read with the same zip code. Fortunately, it’s not. If I were to guess, it’s another service offered by the kind folks at the Berkman Center.

Anyway, here’s how the Planet explains itself:

Planet 02138 is a collection of Harvard blogs. It is a sample of opinions and ramblings by Harvard students, faculty, and alumni.

From what I saw, they nailed it head-on.

You can make your own feed reader with the software from Planet Planet. Gosh, that’s fun to say.

Trolling their blogroll inspired me to update my own. Sure, my RSS reader knows what I’m currently reading—somehow my blog got left behind, though. After all, how are you going to know what I’m [likely to be] reading?

Technorati Tags:planet, 02138, feed reader, opml, harvard, blogs, berkman center, blogroll

On Sunday, March 04, 2007, my cousin wrote the following:

Beep Beep, i turned into a robot.

Now, she didn’t know it at the time, but in doing so, she was providing some incontrovertible evidence in favor of my taxonomy of people. Sure there are a lot of classifications floating around there. We’ve all heard them: there are two kinds of people: those who think there are two kinds of people and those who don’t, for example. Then there are more comprehensive groupings. If you haven’t already, try your hand at the Meyer-Briggs-Keirsey-Jung temperament sorter—I’m an INTP, by the way—and the Which Superhero are You? quiz of MySpace fame—I’m 80% Spiderman. These extravagent typologies, for all their benefits, still require tests. There has got to be an easier way.

And so there is! In college I came up with a relatively simple though telling taxonomy of people while sitting in the dining hall. It’s nice for a few reasons: there are no tests; the results are 100% accurate; and the system is easy to learn—there are no fancy, technical definitions. In fact, there are no definitions at all. Just two primitive labels. (In modern geometry, the terms point and line are often left undefined. So at least there’s some sort of precedent for this sort of system.)

So what is my system, anyway? Well, to start off, I don’t claim that any description of a person is complete. Instead, I deal with approximate descriptors. In my system there are two, undefined building blocks: robots and cartoons.

[Note: I wish I had made one of those tree diagrams, like the ones that are used in dichotomous keys. Here's an example of the type of tree I mean.]

But that’s just the beginning. As there’s often more than meets the eye, we can have higher-order descriptors that give a more honest approximation to a person’s true character. There are six first-order personalities. We’ve already met two: the robots and the cartoons—those people who truly are pure robots or pure cartoons. A lot of people fit this description, but then there are lots of others who are hybrids. These folks might be: robots pretending to be (PTB) cartoons and cartoons pretending to be robots. For completeness’ sake, I should mention the ever abstruse robots pretending to be robots and cartoons pretending to be cartoons. These folks typically are self-aware. They’ve thought about how robots (cartoons) ideally should act, and they try to live their lives that way. Haven’t you ever met someone who was a caricature of himself? Maybe you’ve met a cartoon pretending to be a cartoon.

The great thing about this scheme is that it scales gracefully. Therefore resolution of your analysis is limited only by your level of neuroticism and time. You can take this system as far as you want: run with it. I can’t distinguish traits beyond two levels, myself. Please let me know if you ever meet someone who is unmistakably a cartoon pretending to be a cartoon pretending to be a robot pretending to be a cartoon pretending to be a robot.

Technorati Tags:jung, meyers-briggs, keirsey, typology, personality test, robot, cartoon, farey addition, stern-brocot, tree, mbti, intp, superhero, spiderman

Every few weeks, we take time to reflect on our reflections on class—a sort of mega-metacognition, you might say. This is the first reflection paper for the semester. The material builds on my journal entries and my final paper from that course on dialogue processes. The Coffee Mug Model shows up once more, but this time it’s got a little more power behind it. Take a look.

Reflection Paper 1: The Coffee Mug Model of Classroom Education

In this paper, I flesh out the idea behind a behavior space, and note that classrooms, like most other institutions are not grounded to physical space. Instead, classrooms, companies, and society itself are examples of behavior spaces—i.e., groups of actions. The language of action provides a way to communicate information, and, indeed, is more often used to transmit knowledge than verbal communication. Using these observations, I decide to center classroom instruction around a particularly useful behavior, which I call respect. Here, respect takes on a special meaning—the willingness to learn from others. Once that identification is made, I am able to show how this single behavior is especially well suited to encourage the conventional dimensions as well as progressive others around which classrooms [should] normally be designed.

Technorati Tags:behavior space, castor, pollux, minerva, dating, classroom, violence, language, communication, community, tradition, information, learning, knowledge, design, education, cct, coffee mug model, dialogue, critical thinking, respect

One of the texts we use in CCT 601: Critical Thinking is a book that came out of the Harvard Graduate School of Education group called Project Zero—yes, it’s the same one that Howard Gardner runs. The Thinking Classroom gives the educator some very concrete tools to approach some rather abstract concepts in the classroom. The format of the book is more helpful than most: two chapters cover each chunk of material. The first of the pair always introduces the concept and gives a little justification for its relevance. The second chapter illustrates the concept in practice through a handful of annotated examples. I don’t fully agree with everything they say, but I like format. That’s saying a lot.

Anyway, it’s useful to know many of my journal entries respond (in part) to this book. We also read a lot of articles, if I get the chance I’ll put references at the bottom of each of these posts.

Journal 2: Skills and Dispositions

Here I continue to investigate building learning environments from the community up. In particular, I briefly examine the differences between raw skill and dispositions actually to use those skills. I decide that there really is no difference from the standpoint of culture. Instead, I propose that the schedule (or sensitivity) of practice of a skill is built into the culture through a mechanism which I call tradition. Equipped with traditions of practice, educators can instill really abstract things like intrinsic motivation and measured risk-taking in their students simply by provided the proper community, proper culture, and proper traditions.

Let me know what you think.

P.S.—This entry is missing a graph in the right margin of the first page where it says “Performance over time.” [I drew it in by hand on the copy I submitted in class.] The graph starts out relatively flat, dips down, and then rises up above the starting level and flattens out again.

Technorati Tags:thinking classroom, education, culture, skills, disposition, intelligence, motivation, community, learning, learning environment, cct, critical thinking, traditions