Games: a Ludic Structure for Problem-Solving

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

Paper 2 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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Words and Meanings

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.

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On Connectives and Language: Some More Robots and Cartoons

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.

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Critical Thinking Journal/Weak-sene, Strong-sense, and Probabilities

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 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.


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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Cartoons and Robots: A Taxonomy of People

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.

The Robot/Cartoon Taxonomy of People
Level 0Robot Cartoon
Level 1Robot pretending to be a CartoonRobot pretending to be a RobotCartoon pretending to be a Cartoon Cartoon pretending to be a Robot
Level n Robot (prentending to be a {Robot or Cartoon})n Cartoon (pretending to be a {Robot or Cartoon})n

[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.

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Critical Thinking Journals/The Coffee Mug Model

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 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.

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Critical Thinking Journals/Skills and Dispositions

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 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.

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Critical Thinking Journals/Culture of Thinking

Well, last semester I kept a journal for my class CCT 602: Creative Thought. This semester I’m doing the same for CCT 601: Critical Thought. I think that what I’m writing now is more interesting. I’ve been able to build on my work from previous classes, but somehow things seem to be coming together now. To indulge my narcissism, I’ve decided to post my papers right here on my blog—that way at least my grandmother can read them.

Journal 1 The Culture of Thinking

In this entry, I try to tease out some of the more obvious components of society. In doing so, I look for applications in a learning environment context. Values pop out as a the centerpiece of attention—and whether a classroom is structure to enable the learning and use of higher-order thinking skills is really a commentary on the values of the classroom. The implication is somewhat surprising: there is no such thing as a morally neutral education. Every action in a classroom is a statement of value judgment.

In particular, I introduce a concept of central importance to my later journal: a behavior space. After all, how can you “take me to Funkytown?”

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Current Event Mad Libs

So I’ve been itching to respond to retired Miami Heat’s Tim Hardaway’s full and public disclosure of his sweeping hatred. (If you missed it, check out the article on CBS Sportsline, for example.)

I’m not going to give some holier-than-thou exegesis of social ills or anything like that. There are too many others out there who’ve already done that for me. Besides that’s not how I roll. I’m going to leave up to you, the informed reader, to decide how you feel. I just want to situate his comments so that we’re in a better position to judge it. To begin, though, here’s what the AP reported Hardaway said:

You know, I hate gay people, so I let it be known. I don’t like gay people and I don’t like to be around gay people. I’m homophobic. I don’t like it. It shouldn’t be in the world or in the United States.

The fact that Hardaway is commenting on gays is almost irrelevant. Let’s see how his words sound when we make a few substitutions.

You know, I hate black people, so I let it be known. I don’t like black people and I don’t like to be around black people. I’m blackophobic. I don’t like it. It shouldn’t be in the world or in the United States.

Now how about another go at it?

You know, I hate blue-eyed people, so I let it be known. I don’t like blue-eyed people and I don’t like to be around blue-eyed people. I’m blue-eyedophobic. I don’t like it. It shouldn’t be in the world or in the United States.

That could be funnier. Let’s fill in something outrageous.

You know, I hate iguanas and carrot cake, so I let it be known. I don’t like iguanas and carrot cake and I don’t like to be around iguanas and carrot cake. I’m iguana-and-carrot-cake-ophobic. I don’t like it. It shouldn’t be in the world or in the United States.

Now, let’s try just one more time. Then you can go back to work.

You know, I hate dignity, so I let it be known. I don’t like dignity and I don’t like to be around dignity. I’m dignophobic. I don’t like it. It shouldn’t be in the world or in the United States.

That’s it. Mull over what you’ve read, and report back on what you’ve learned from it.

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A few weeks ago my friend Michelle called me a little after one in the afternoon. The ring of the telephone woke me up and I stumbled across the room to answer her call. I looked down at the little display, saw that it was Michelle, and then put on my best telephone face to accept her “Hello, Joshy.”

Michelle called to tell me that she had accepted a new job—she was in need of a new one, believe you me. This was fantastic news. Usually Michelle can spot my false wakefulness, even over the phone, like one of those empaths from Star Trek (or something). But this time, her excitement—in conjunction with my keen theatrical abilities—distracted her from the reality of my slumber, which is exactly what I was aiming for.

I wasn’t, but the question still remains: could I have justifiably been upset at Michelle for waking me up? Well, yes but mostly no. It was one in the afternoon. Most people are awake when the sun’s up. So, Michelle was right to operate on the assumption “people are awake when the sun’s up.” The world is a crazy and complicated place. We have to live our lives despite only having access to a very small amount of knowledge about our environment. Therefore most of our judgments aren’t certain. Instead, they’re best guesses that approximate what we should do if we actually knew everything there was to know. Thankfully, we’re not totally in the dark.

People are very good at working with probabilities because lots of the events in the world have a high probability of certainty. That tree in the park you saw this morning on your way to work will probably still be there during your commute home later tonight. The position and function of the knobs and buttons on your stove are not going to switch themselves around when you’re not looking—with high probability. So it’s not surprising that people believe that there are certainties in life. And maybe if you were able to know everything about everything at every time, then the world would work according to a small set of fixed laws. Unfortunately, no one—as far as I know—has that sort of depth of knowledge and understanding. So, for practical purposes, we’re left interacting with probabilities.

Now we get into trouble when we confuse probabilities for certainties. Then we become locked into a stereotype. That’s right, I think stereotypes are simply misapplied probabilities. Several years ago some fledging stand-up comedian trying to break it big played the Conan O’Brien Show. He included two “postive stereotypes” that stuck with me. “All Jews can fly and Mexicans are made out of candy,” he claimed. Being (sort of) both Jewish and Mexican, I can say from experience that very few Jews whom I know can fly and even fewer Mexicans are made out of candy. So what makes his stereotypes wrong? Well, probabilistically his claims aren’t well supported.

Here’s another perhaps less inflammatory claim: men are taller than women. I bet a lot of you agree with that. But let’s hold up just a second and see just what the sentence is saying. There are a lot of words missing that really ought to be there. My claim doesn’t mean all men are taller than all women. If you cite your friend from college on the women’s basketball team who towers over everyone else in a crowd, you haven’t disproved anything. What I really mean to say is that on average men are taller than women; i.e., if you pull a random man and a random woman off the street and compare their heights, record the answer, and then repeat the experiment several times over, then in general, you will find that the man is taller than the woman.

So what are assumptions: they are the most probable results from a distribution of possible results that we adopt as fact based on our experience. Experience varies, so assumptions vary. The key is to remember that sometimes outlying events can happen, and we must be open to the possibility that they do. Most Mexicans aren’t made out of candy, but don’t let me fool you into believing that none of us are.

All that said, Michelle should’ve known, given her previous experience, that there was a high likelihood that I would be sleeping at one in the afternoon. Don’t forget that not all assumptions apply in all contexts. These things are conditional, after all. So, she’s only partially excused.

And while I’m on my soapbox, it’s worth pointing out that because people almost exclusively interact with probability distributions, probability and statistics really need to be given more attention in school curricula. Over emphasis on deterministic systems tricks students into believing that the world really operates on certain events. I can’t think of anything further from the truth.

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