A Weblog

No, I haven’t been off on vacation in the Caribbean—but I have seen the third pirates movie. It was great. But I wanted to let anyone out their who is interested that I’m going to lead an informal seminar on general relativity at UMass/Boston this summer starting June 4th. Here are the details.

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.

Technorati Tags: vygotsky, play, imagination, games, rules, prediction, decision-making, higher-order knowledge, belief, doubt, mathematical reasoning, education, learning, solution finding, transfer, cognition, thinking, critical thinking, creativity, soccer, peter elbow, epistemology, coffee mug model

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.

Technorati Tags: infinity, set theory, one-to-one, onto, injectivity, surjectivity, number, lesson plans, education, llamas, carrot, zeno, paradox, counting, cardinality, math, mathematics, frege, hume, function, map

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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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.

Technorati Tags:richard paul, critical thinking, strong-sense, weak-sense, probability, quantum information theory, logic, gleason, hilbert space, information retrieval, ir, education, learning, stereotypes, marketing, linear algebra, coordinate, dialectic, education, teaching, classroom

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

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.

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?”

Technorati Tags:tishman, perkins, jay, education, classroom, learning environment, society, value, culture of thinking, critical thinking, learning, knowledge, community, cct, behavior space

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.

Technorati Tags:assumptions, probability, stereotype, society, philosophy, average, mexicans, jews, candy, flying, conan o’brien, education, curriculum, reform, statistics

During the winter session I took a class on dialogue processes. Most people are familiar with debate. We have clubs for this sort of thing after school, after all. In the standard set up, a debate has two or more opposing sides. They bat each other over the head with facts and name-calling until one of them submits and declares a surrender. Dialogue is the opposite of debate. Instead of looking for a product (i.e., winnning), dialogue focuses on a process (i.e., learning). It’s ideal in education because it nicely ties together the sometimes competing interests of knowledge-, student-, and assessment-centered learning environments by a clever structuring of its community base.

I’ve posted the final paper I wrote for this class [late]. It’s short and only very briefly describes my “coffee mug model” for the classroom. Basically, this thing is predicated on the idea that respect is the willingness to learn from another [person or thing].

I know I’ve been in situations when I know that the person who’s taking to me is much more knowledgeable than I am, that I should pay attention to what he’s saying, but that because I don’t respect the guy, I just can’t learn from him. In the classroom, I think that learning from another person is respect, by definition. Think about it. How many times do opposing viewpoints talk right passed each other? The reason is because they’re not willing to learn from the other. Chances are paying attention to your opponent can help out your cause. Sometimes, you might find that there really isn’t any conflict at all. Instead, it’s all perceived (rather than real) conflict. Golly, communication is powerful stuff.

I still owe you guys a post about assumptions. Consider this the beginning of it.

Also, if you have the time, please come to Seven Old Ladies get lost in the loo tomorrow nigth at Blanchard’s Tavern (turn down your volume before you follow the link). For those of you who don’t know it—and be ashamed if you don’t—Blanchard’s Tavern is one of the few bars around here that tries (really, really hard) to stay honest to its 18th century foundings. They serve things like loganberry wine and Brunswick stew. (You can check out the full menu for yourself.) And they’re a steal at only $3 each.

Tomorrow’s event is going to be raucous—the volunteers who run this thing promised me. So come on down. Bring a canned good or expect to donate $1 to the local food pantry. We’ll sip on General Washington coffee and sing along to old sea shanties. And if you can’t make it tomorrow, you can show up any Saturday. Every Saturday.

Do it.

Technorati Tags:dialogue, debate, learning, classroom design, learning, education, perspective, conflict, colonial, new england, patriots, tavern, history, loo