Dialogue and Learning Environments

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.

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Judging Authenticity

Recently, my friend Little Lamb wrote a post about how people react to identity (gender or otherwise). Now conceptions of the self have eluded me for a while, and I love reading what others have to say about the issue. Here’s a short snipet from her article—you should read the whole thing, of course—but this will do well enough to situate my post:

Of course, we do judge the authenticity of identities like these—often identity groups to which we ourselves don’t even belong—every day. We distinguish between “normal” Muslims and violent ones, women who kiss each other at parties and dykes, “real” bisexuals and gay men in denial. But every time we make judgements like these, we imply that we are better judges of authentic identity than those who live these identities. [Original emphasis]

Before I go on, I should say that I completely agree. From an observational standpoint, when someone judges the identity of another he is as a matter of fact asserting his perception of that person onto that person, perhaps against that person’s will. The question is not whether the judge is imposing his viewpoint onto another, but whether there’s any significance in the act at all. After all, in some cases it could be very useful indeed.

I grew up in a very small, white, Irish-Catholic suburb of Boston. Now it’s important that I say Boston, because already there are tremendous differences between say a Boston Irish-Catholic community and a Chicago Irish-Catholic community, and both of them, in turn, are vastly different from Irish Irish-Catholic communities. I’m not about to dismiss local variation. That said, I’m not Irish-Catholic. According to legal documentation, I’m Mexican. And as far as the law of Moses goes, I’m also Jewish. But having grown up in an otherwise homogenous environment, what being Mexican and being Jewish means to me might very well look like what being Boston Irish-Catholic looks like to you. But that’s okay. How I feel and what I know to be Mexican is largely an accident of my youth. So, whatever I think it is, it is. It’s all a matter of perspective, right? Well, maybe.

Once I went to college, I met lots of people who, like me, were Mexican, Jewish, and sometimes even Mexican and Jewish. (Now I’m going to start lumping Mexican and Hispanics into a single term. From now on, when I write Mexican you can assume I mean Hispanic. While I know this may sound clumsy and callous, it’s not. I’m Mexican after all, and who are you to tell me what it means to be Mexican—er, Hispanic?) However, unlike me, most of them grew up with other Mexicans or Jews. Consequently, they painted a very different picture when they described the Mexican experience. Still, due to legalities, I was accepted into the two groups, I think, as a matter of technicality. But the more time I spent doing “Mexican things,” the more sure of my heritage, and all the perks that come along with it, I became. I had always thought I liked spicy food because of my Hispanicidad, now there was no questioning it.

So, where does identity exist? Some might argue that identity is something that each individual chooses for himself on the inside. However, I don’t buy it. If I don’t think you’re a Mexican, then to me, you’re not a Mexican—even if you think you are. Likewise, I might think you’re a Mexican, even if you insist you’re not. The problem is that identity is not an objective fact. It lies somewhere between a speech act and something else. It may feel a little unsettlilng that you’re not in control of who you are. Identity is an emergent property of the way one person interacts with several, other people. Who you are isn’t entirely up to you, it’s up to us. Let me explain what I mean.

When I meet you for the first time, I’m going to assess the way you look, act, make me feel, etc.—I’m going to perceive you. Now, of course, I won’t get an exhaustive look at you. I probably won’t be able to guess that you’re favorite number is 11, or that you find global warming so scary that sometimes you can’t sleep at night. Everyone has to operate with incomplete knowledge. We fill in the gaps with likely probabilities based on our previous experience (some might call these probabilities assumptions) and do our best to form a belief that makes sense of the situation. Because of the way I treat you, you’re going to adjust your behavior. Your change will trigger me to adjust my beliefs and therefore behavior. Eventually, the way you act and the way I act will settle down—and voilá! What is identity other than a set of behavoirs that largely matches some (loosely if at all defined) generic shadow of behavoirs?

Humans are dynamic entities. We respond to our environment. The trick is, humans are also a part of their environment. So it’s easy to forget that other people are part of our environment, too. Before I talked about why Vygotsky thinks man is special: we use signs to store information outside of our brains. Our minds, in a very real sense, are distributed all over the world around us. It’s not so suprising, then, that each individual identity should be spread out all over a mass of other people as well.

Humans alter their environment—I write down ideas I have in a notebook I keep in my pocket, for example—so that later they can use the environment to alter our behavoir—say, like remembering what to write my next post about. What’s important to remember is that every interaction with our environment is a form of communication. Humans love gathering and piecing together clues. We impute intentionality on just about everything. So we don’t even require that the other end of the conversation come from another living entity. (Consider books, for example; if that doesn’t satisfy you, consider geologists who try to reconstruct the Earth’s past recorded in the bedrock.) And most interactions end up changing all the parties involved. (Leave no footprint after camping; reconcile after a fight to feel better; drink orange juice for energy and hydration.) The fact that we interact with other people means that we change others and are changed ourselves a little bit every day. Just like small changes slowly birthed Modern English from Old English, we, too, are not who we once were.

Few people would argue that they are exactly still their six year old selves. However, what some people might be slower to admit is that they largely have no say in who they are. Much of who we are, how we fit into society, is not up to us. It’s up to the caprice of the society we belong to, the rules of which are subtle and complex. So, let’s get back to the question of identity. It looks like it is impossible not to judge the authenticity of person’s identity. (If I agree with your perception of yourself [when it matters—fill out an online questionaire for your friend in front of your friend. You’ll see just how much of the same person the two of you see. Careful, it can get tense.] then I reinforce your conception of yourself and at the same time reinforce my assumptions about you.) That’s not the problem. The problem is not in judging, it is in how we judge. Maybe what we ought to investigate is not that we judge but the assumptions that guide our judgments.

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Backwards Compatibility

This weekend I took the Chinatown bus down to New York to visit my army special forces friends Danny and Mike and to meet their army special forces friend Zack, all up from North Carolina. Whenever Danny makes his way up the East Coast, a bunch of us convene: a few from Boston, a handful from New York, and one from DC. We’re a geographically diverse group of friends, but that doesn’t stop us. We make an effort. And that’s the problem—I think maybe we’re a bit misdirected.

Saturday, we went to a sports bar called Proof. (I immediately wondered if there was so big a math crowd in this part of the city as to sustain a bar, but I quickly realized that they probably didn’t mean a mathematical proof. Eighty proof was more the feel of the place.) The floors were the kind of dirty that black, matte surfaces always are, which set up a visual cue for the rest of the decor to follow. Everything about the bar was dark hip in that cold, uninviting way that encourages you to have fun to prove to others that you’re having fun. The neon lights that pierced through from behind the bar coupled with the bartendress’s bad dye job and caked make-up put me in the psychological dugeon of a Celine Dion concert. Proof relies on happy hour gimmicks which simultaneously feature Bud Light and a multitude of flavoured Stoli. (You could tell from the looks of the clientele that they had found their niche.) I’m not sure anyone in the bar knew why they were watching a football game. I certainly didn’t.

We established early on that no one in our party had any opinion either for or against the Colts or the Ravens. A lady bordering us said she liked the Colts, so one of us loudly cheered for her team. Lisa, Danny, and I left our uncomfortable and unsociable seats and headed to the burger joint next door just before half-time, where we almost had time enough to talk. The guy who had dragged us to Proof didn’t pay attention to the game at all. I overheard his girlfriend trying to explain to him who Adam Vinatieri is. It was no use; he wasn’t interested. By the time he realized that the three of us had left, he mobilized the rest of troops. We were going to leave. To sit down. To have dinner. Somewhere else.

Earlier that day I caught some awful brunch with my good friend Baca. We went to some up-and-coming place in Nolita called Public. Its theme: public spaces. The menu comes on a clip board. Apparently public spaces are industrial and water-stained. I had two poached eggs on garlic yogurt with kirmizi biber butter. I pressed our waitress before ordering and she admitted that “No, it’s not really butter.” She was right—it’s an oil with too much flavour for its own good. I let the hipster get the best of me. It could’ve been because I was wearing herring bone. Still, I wanted to go out to brunch. Baca merely accommodated me. Sorry, Baca.

Now here’s my point. In both instances, I hated the place we went to. But the sports bar really left me angry while, the food aside, I had a really good time at brunch. So, on the bus ride back to Boston, I started thinking why is that? And the secret is a problem in good user design.

Software designers sometimes leave in features that could be considered obsolete, needlessly complicated and confusing, or otherwise just bad in the name of backwards compatibility. The idea is that even if it’s not necessarily the best way a thing can be done, it’s a way that the user knows and therefore will expect will work, if poorly. And that’s right. The user might very well expect to see an old feature in a new realease. But expectation isn’t a good enough justification for doing something. It’s sort of like saying, “The user should be abused because the user is used to being abused.” (You might’ve heard the argument, “He may be the Devil, but at least he’s my Devil.”) People do this sort of thing all the time in all fields. Decisions made in the past are often carried well passed their realm of usefulness into the future for the sake of mindless adherence to tradition. Not to do so is like admitting your were wrong, or at least that you’re wrong now. Why do you think we won’t revise our plan for Iraq? Designers who throw in features they know to less than productive to ensure backwards compatibility have confused a means as an end.

Software is supposed to be a vehicle to help people do things that they otherwise could not have done on their own. That is, the software—like alcohol or sports bars–is supposed to be a servant, not the master. But that’s the difference between my two stories. In the first, we went to Proof because someone thought we were supposed to watch the game. Really, no one wanted to—the Pats played on Sunday, not Saturday. (We all went to dinner during the second and arguably more important half, you remember.) But brunch? I wanted to go brunch. The fact that the food sucked was only incidental. I wanted to spend time dining with Baca. And Public, bad though it was, did the trick. It provided a forum for us to catch up. That’s what I expected to do this weekend. Instead, we often got caught up doing things that are supposed to be fun rather than actually having fun.

The lesson learned is an old one: hang out with your friends at a bar, don’t hang out with a bar in front of your friends.

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The State of Grafitti: Yuppie as Mascot

About a year ago, I was at the Park Street station on my way back to Cambridge. As I waited for the train to come, I did what I always do when I’m waiting without a book: I paced the end of the platform. Rather than slowly pass my foot over the knobs of the textured, yellow safety strip,—a favorite pastime of mine—I kept to the flat brick on a well-defined route that visits the supporting columns which dwell nearest to the tunnel’s opening.

Normally I’m not struck by public graffiti, but every once in a while something unexpected crops up. This time one of my columns read: “Kill all yuppies.”

I was very excited by this message. No, I’m not in favor of killing all the yuppies. That suggestion’d put me too close at risk. There’s a very good chance, indeed, that I’m a yuppie. So, no. Please be kind to the yuppies. But here’s what’s different. Normally the graffiti that I’ve encountered are either some sort of tag—you know, a personal statement of existence and potential ownership, “Kilroy was here” or “AlL St*R” or something along those lines—or alternatively they are some commitment of love or hate (often accompanied by a slur or two). You seen them, something like “Joe is a fag” or “I love Tiffany.” Anyway, all of these examples are personally directed. They don’t extend beyond an individual. Sometimes I’ll find one that condemns a whole group of people, like my yuppies example, scrawled on a public alleyway. But those even those are gang-related or race-related. Yuppies represent something new.

Whoever wrote it got my attention because his hatred was not race-directed. It points to a larger social movement. The new segregation, if it is really new, will be intellect. And these upwardly mobile persons are central enough to earn the distinguished role of spokesperson. But what exactly are yuppies mascots of? Well, that sort of brings me to some more recent graffiti.

The Ashmont train station is undergoing some pretty hefty repairs. Officials have suspended the Mattapan High Speed Line service for a year, and the train station is hidden from plain sight by several, several ton mounds of dirt. Like most other forms of transportation in the city, the Ashmont station is going underground. It’ll take some time before things are back in order. For now, there are lots of make-shift wooden structures to take the places of the bus depot and station entrance. And that means there’s plenty of board space for community art—I mean graffiti.

The last time I was at Ashmont I noticed some of the newer pieces as I walked by one of the wooden panels. This time a website caught my eye. I haven’t seen many hypertext tags outside of the internet, but there it was: a link to someone’s myspace page. Kilroy has entered a new age and he’s updated his message. Now the statement is “I am not here, I’m here. Come find me.” It’s a revolution. Personalization on the web is at an all-time high, and movers in the field want more of it. Collaborative filtering, social navigation, blogs! They’re all in style, and they don’t look like they’re going to go away any time soon. I can’t say I mind it, either. In fact, I want to be more a part of it.

This is not the same technological revolution that your slightly older brother talked about only decades ago. No, the paradigm is different: we can read the writing on the wall. Literally. Before technology brought with it an increased level of impersonality. The assembly-line metaphor bled into everything—it’s still around, of course. Don’t worry, the transactional framework driven by the glory of mass manipulation of raw goods to form an endless supply of identical product is still very much alive. And people are still applying manufacturing-inspired methods completely out of context. And the effect is still very isolating. But lo! the very same push to maximize profit that once aimed to cut time and kill interpersonal relationships has turned a corner. Personalization is the new rage.

But will personalization help build bridges among people; won’t it keep us even more securely glued to our seats in front of our computers? I’m afraid that it can. Technologically-backed social ventures, like AOL Instant Messenger and other chat programs, have made it easier for the quiet kids to remain quiet and alone. Chat tools give the user the appearance that they’re interacting with other people. But some researchers suggest that the analogy is only that: apparent. The real satisfaction one gains from honest-to-goodness, face-to-face conversation is so much greater than its virtual manifestation that it’s almost silly to make the comparison. So, what’s going on?

The invitational nature of MySpace is different than AIM. A person’s page is like his home. Each click to that site is really a visit. That’s why it makes the news so often. Sometimes the visits aren’t just virtual. And everyone uses it: college kids, little kids, married couples. The range of demographics represented by MySpace’s users is enormous. Unlike Friendster, which originally withheld a user’s access to a stranger’s page by default, MySpace let everyone see everyone else from the get-go. Friendster was a place for people who were already friends. MySpace, I believe, was built to get people to go to and listen to new bands in concert. The idea that you’d actually meet strangers was the founding idea. Now it’s just a place find others you’d like to bone à la Craig’s List’s personals but less so. But the idea that you might meet the person attached to the website is still very much there. Isn’t that exactly what that graffiti from Ashmont Station was all about? The internet takes all the scariness out of meeting a stranger, because you don’t physically meet, and the meeting is still completely anonymous. (There’s a trade-off, though. The relationships that form are even more tenuous than those so-called and ever important “weak bonds.” Online relationships tend to be superficial and sometimes socially damaging. Like I said before, they permit the loners to find each other and stay alone. Even those of us who aren’t loners end up as loners the longer we stay online rather than outside.)

So we’ve found a cause for our mascots. Like the term itself, today’s yuppies herald the dawn of a new form of impersonalization: isolation through personalization. Technology is poised to use what it knows about you and your preferences to make a friendlier, easier experience. In the process, you get to interact with others—real or not. The interaction is deep enough to convince you that you’ve done something meaningful. You’ve made a friend or learned a new fact. (Wikipedia is a blessing and curse.) But have you really; can you rely on your friend or apply your fact?

Your iPod list has exactly the music you want to hear. And so now, people go through life not listening to each other but to themselves, plugged into a clean, white box whose world revolves around the most important person—its only person: its master is me. Time Magazine got it wrong. The person of the year is not You; it’s me. This is the society recorded in graffiti today.

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Geometry Lesson Plans

For one of my final projects, I wrote the first of three lesson plans for a high school course on plane geometry. When designing learning environments, it’s important to work around four dimensions that affect learning. They are to what extent your classroom is knowledge-centered, learner-centered, community-centered, and, of course, assessment-centered. Sadly there is no absolute consensus about what those words actually mean. And even worse, there has been considerable emphasis on learner-centered and assessment-centered environments to the near exclusion of the other two. And even worse still, many politics have tricked the general population into thinking that there is a zero-sum binary between learner- and assessment-centered classrooms. The fact of the matter is, a good instructor will make sure to provide classroom that is well balanced among all four components.

Knowledge-centered is perhaps the easiest of the four concepts to pin down. Make sure there is substance to what you’re doing. Teach something. Knowledge-centered environments require just that: knowledge. My lesson guides to geometry are filled with—you guessed it—geometry. Passing mention of concepts from real analysis and abstract algebra show up. Were I to write a fourth installment, you’d read about symmetry groups, group representations, and addition. A proper discussion about measurement would dive deep into the definition of number itself, equivalence relations, and probably prove Euclid’s so-called Common Notions. (That A=A; if A=B, then B=A; and if A=B and B=C, then A=C. Yes, students should be able to explain why self-evident facts are true, too.)

Student-centeredness takes into account what the learner already knows—or doesn’t know, or misunderstands for that matter. For this reason, my lessons are written for the instructor but led by the students. I use a list of questions that the teacher can use as a model. Taken together they form a cohesive mathematical narrative. But since the point of student-centered environments is that each classroom ought to be tailored to the individual needs of the particular students in the seats, the idea of a student-centered lesson plan that has been blindly written and mass-distributed is somewhat antithetical to its own aim. The Socratic question-and-answer method gets around that. Instructors have both the license and responsibility to dovetail the lessons in a way that best suits the students in the class.

Because of the individual nature of the plans, assessment becomes a problem. How do you figure out if the students have figured out the material if there is not one but several possible right answers? There are over 350 published proofs of the Pythagorean theorem, for example. And all of them are equally correct.

Student-directed learning has assessment built right into it. The teacher can constantly monitor student responses to gage their depth of understanding. The count of prompting questions (given by the teacher) to achieve a particular response can be used an index of mastery over the material. This sort of examination is not obvious to the students and therefore relaxes the pressure associated to more conventional means of testing. Moreover, sustained dialogue between students and the teacher promotes a collaborative, community atmosphere within classroom. Students and instructor exchange roles dynamically, which fosters all sorts of other leadership qualities and instills intrinsic motivation and proactiveness within the students. Having students talk and draw on the board takes care of three of the target components all at the same time.

So, all that you really need my plans for are the knowledge. And the notes are pretty insightful, if I do say so myself. At least have a gander at the very pretty marginal glosses. I employed some artful information mapping techniques. You’ll find that the diagrams are rather palatable. I’d be interested to know what other teachers have to say about them, how I should change them, and if I should write more.

Geometry Lesson Plans 1–3 Geometry Lesson Plans 1–3

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Eggs for Breakfast

I woke up late this morning. Having showered, I sat down to read, when I realized that I had very carelessly over-looked a crucial part of my morning routine. I had forgotten to eat breakfast. Today felt like a fried eggs kind of day. So, I left my laptop—which remains permanent affixed to the end of the kitchen table closest to the windows, just to the front and right of my slowly waking Japanese bloodgood maple—to assemble all the tools necessary to cook eggs. Naturally, I started for the frying pan. I looked for it in the obvious places: first the sink. No, it wasn’t there. Nor was it on the stove, or the cubby just above and behind the stove. Nor was it in any of the cabinets that it has been known to haunt. No, the frying pan just wasn’t around. Desperate, I called to my sister.

“Janice, do you know where the frying pan is?” I asked.

Her response was muffled, as she mumbles. Our misunderstanding was compounded by the closed door to her room, and neither of us was about to exert the energy required to open it. Minutes later we eventually came to an understanding. Sometime in the past three days my father had snuck into the apartment and thrown out the frying pan because it was scratched. Funny, the pan had been scratched for years, yet we kept it anyway. Yet today no earthly force was going to keep me from my eggs. So I resolved to brave the stores alone, despite however frenzied and therefore frightening an after-Thanksgiving mall might be.

On the drive over, I realized that I know nothing about what makes a quality frying pan. I planned to go to Williams-Sonoma, milk the clerks of their cooking ware savvy, and shuffle off to Macy’s, which, in my experience, has sold the same products at slightly lower prices.

I dodged the greeter and her catalogues at the main entrance and proceeded directly to the stainless steel pans without making direct eye contact with anyone. It took them almost no time to spot me. For one thing, the store is small. The pans are somewhat occluded from the rest of the store by a large shelf of expensive gadgets, and the closed space made me feel somewhat more comfortable. But my sustained pacing was anything but deliberate. From time to time I grabbed for a handle, though my grasp was tentative, as though personal touch finalized the purchase.

A kind lady guided me through the store’s offerings. She was shocked that I was willing to shell out more than one hundred dollars for a single pan, and, so, started me in cast iron. By the end of it, we had ventured into copper. I told her that I didn’t need that level of precision in my cooking heat distribution. We decided to stay away from non-stick surfaces. Really, I only had to decide on the exterior color. I explained that I was going to check out some other stores before I decided.

Frying pans, and similar purchases, stress me out. I’m intimidated by the permanence of my decision. Additionally, a hundred bucks is a lot of money for a part-time freelance writer and full-time education graduate student to throw down on a kitchen gadget. (I will not draw attention to the sweater I bought at JCrew to calm myself down immediately afterwards.)

According to plan, I walked out of Williams-Sonoma and into the home wares section of the Macy’s nearby. I located the same pan for the same price and promptly marched back to the friendlier store. This time I was unable to avoid the greeter, who recognized me as the boy who was looking at pans earlier. She sent someone “right over” to help me out. This time a very, perhaps too knowledgeable man helped in my assistance. His tactic: ask another bout of quick though endless barrage of high-stakes questions in order to divide-and-conquer.

He started out, “What do you expect primarily to cook in this pan?”

What? I had no idea that the typical customer base had such use-specific needs. I floundered a bit. Somehow I managed what I thought was a reasonable response.

“Omelettes,” I muttered. My helper wasn’t satisfied by my answer. He asked for clarification.

“Oh, like frittatas?”

Actually, I had large, four-egg omelettes in mind but I suppose—wait a minute. What? He just assumed I even know what a frittata is, let alone make them myself. Had it been three weeks ago, his question would have stumped me. But after a somewhat lengthy discussion with Arthur the dining hall manager and friend Karthik about the nature and classification of frittatas, I was equipped with the requisite knowledge to shop at Williams-Sonoma. Still, I sacrificed accuracy in the name of expedience and agreed, yes, that I would mostly cook frittatas in my hypothetical frying pan.

But this raises a very important point. The depths of stereotyping reach even further than I usually think. My friend Danielle lists as this quote under the heading Favorite Music on a popular online community [I hope she doesn’t mind my posting it here]:

From your glasses, I can tell you listen to that kind of music.

But stereotypes go beyond anticipated action or appearance. They can steal deep into the knowledge we assume people have. Even Danielle’s description assumes a certain knowledge base—in this case, of genre of music—but do I look like the type of person who knows what a frittata is? I’m not sure what I mean when I say I’m not. Because I’m long-winded, let’s look at a more clear-cut example, just for excessively pedagogical purposes.

Let’s say, just for instance, that we’re in immediately post-Nazi Germany—because Nazis engender a universal disgust in people, or at least people feign universal disgust because to do otherwise is socially irresponsible, and because talking about things Guantamino might make some people uneasy; either way—and that I was a grand master Nazi torturer. It’s after the war, and now I own a candy store. You, a friendly tourist and likely patron, enter my store and we strike up a conversation. Straight away, I ask you what you think of the bootikens as a torture device. Now, most people will not know what the bootikens is. [I don’t even know what it is. You can look it up yourself if you feel so inclined.] But how do you feel that I presumed you a)know what a bootikens is, b) are comfortable enough with your knowledge to discuss it in public with a stranger, and c) have an opinion of its efficacy in torture?

That’s right, I called you someone who is knowledgeable of technical torture devices; i.e., I called you a torturer. How implicit of me!

The pan, by the way, is wonderful. It conducts and retains heat very well. Be careful to use it only on a low to medium heat. And be especially careful when washing it.

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A Response (Part I)

A few posts back, reader Loki on the Run brought up several very worthwhile points in his comments. Unfortunately, it was midterm season as school and deadline season at work, and so, I didn’t have the time to write up a proper response. Hopefully, this will be a good start.

Loki wrote:

Another sad aspect of modern teaching is the notion that all students will grow to be 6 foot tall. Given that there is an approximately normal distribution of abilities, not all students are going to be able to deal with Calculus.

First off, we should be careful when we talk about abilities. It’s hard to know exactly what we’re talking about. Whenever we try to measure intelligence, we should be aware that there are at least three different things that we might actually mean. The obvious one is performance. Whatever a student actually does is all we can really ever measure. However, is that really what we mean when we speak of intelligence—what about competence and potential? These things are easy to confuse. So maybe I ought to stop and give an example of what I mean.

Take anyone who has ever tried to learn a language. Maybe you have your 1 year old nephew in mind; perhaps you tried to learn a foreign language yourself. For concreteness’ sake, let’s say you’re trying to learn Hawaiian. Now, as your teacher I want to figure just what your mastery over Hawaiian is. Therefore I give you a test. To make sure the test encompasses lots of skills, I ask you first to read a written passage on a particular, engaging topic in Hawaiian, and listen to native speaker discuss the same topic. Then I ask you to record your response on tape. Let’s say that you understood everything you heard and read, but that you have a hard time forming and expressing your own thoughts in Hawaiian. As a result, you stumble awkwardly but don’t actually communicate anything. Am I to conclude that you didn’t understand anything—that my lessons were completely lost on you? Surely, your performance suggests that you don’t speak Hawaiian any better than your friend who has no knowledge of it whatsoever. Ah, but there’s the trick: competence usually precedes performance.

There is another complication. Sometimes people extrapolate ability based solely on performance. Should we infer that because you failed your Hawaiian test that you lack the ability ever to master Hawaiian? This raises another interesting question. If student ability really does follow a normal distribution, how do we measure it? Given a good measure, we could save lots of money. Kids could be weeded out early on and pushed into ability-matched professions. We could split the alphas from the betas from the deltas from the gammas. Loki, you and Aldous Huxley would’ve made good friends, I think. Those with little potential could be spared years of needless pain and embarrassment in a school system which, by design, is destined to fail them. Except in the most extreme cases (and even then), it is difficult to gauge a person’s potential ability.

But then again, people aren’t the only things that resist easy measurement. Content, too, can evade classification. Many people point to calculus as the most advanced topic a high school student can ever hope to see—but only if he’s very smart. But why do people believe that? I doubt that calculus, whether it is hard or not, should cap any high school curriculum. (I’ve argued before that statistics would be more useful for everyone.) But I also doubt that calculus has to be hard, or even taught on its own.

Anyone who has ever ridden in a car has felt calculus. Every time a car speeds up or slows down, you feel the effects that calculus describes. Differential calculus is the study of the rate of change, and that’s something that people understand simply through living. The flip side, integration is just as natural. Anyone who has ever noticed that a three-layer cake is thicker than a two-layer cake has used calculus. Anyone who has ever stacked coins or poker chips has a rudimentary grasp of calculus. We even require kids to integrate all the time. Sixth graders have to find the area of a rectangle. By eighth grade, they’ve moved on to the volume of prisms and other solids. And it turns out that using concepts from calculus happen to be quite effective.

I spend a lot of time talking with a math teacher at an inner-city charter school in Dorchester. These kids are typically 3-7 years behind where the curriculum would place them according to their ages. And a back-to-basics approach would have them memorizing formulae blindly, because, as is typically thought, loading them up with advanced concepts would only confuse the matter. Yet empirically, we’ve found that just the opposite appears to be the case. When area is presented as the summation of infinitely thin widths across a given length, kids get it. In fact, when they come to volume, they generalize. A volume, they understand, is built out of infinitely thin cross sections. If the base remains constant, they get it. And there’s transfer!

If kids learn that the area of a rectangular solid is the area of the base times the height, they’re good to go, so long as the shape is a rectangular solid. But if asked to find the volume of a heart-shaped pan whose base and height measurements are given, they don’t know what to do. But my kids from the inner-city know what to do. They look for the perceptual invariants: is the pan made up of the same cross-section throughout? Yes. Do I know the area of the base? Sure do. Do I know the height? Yeah. No problem. They build the volume up. This is exactly how the Riemannian integral works. Kids who are well behind according to the curriculum are using concepts that are considered too advanced for most people. Yet they do it, and they can apply it out of context.

There are other reasons to introduce so-called advanced topics at a young age. Not only are many of these subjects accessible to younger audiences, their unfamiliarity helps to level the playing field. Kids learn things all the time outside of class. And the standard math curriculum no exception. Often students get a taste of some area of math before they meet it formally in school. If you change up the topics, kids who have already had adverse experiences with one math are less likely to noticed dressed up in another area’s clothing. Because of this leveling effect, Project SEED, an inner-city initiative with more than 40 years of history, throws its eigthth graders into differential calculus in order to give the kids a facile understanding of fractions. You’d be surprised to learn these same kids were doing analytic geometry as third graders. And these kids, according to many reports, lie in the lowest quartile of ability. They shouldn’t be able to add, let along understand and do calculus. So the question can’t be about ability. Or if it is, maybe it’s about how we measure ability. Or maybe it’s about how we grade mathematical content. I don’t propose to know myself.

What I’m driving at is that intelligence isn’t an all of nothing venture. And so, it’s probably impossible to quantify it with a single number, so it’s equally impossible to make sense of statements which claim that there is any sort of distribution of ability. I’m not saying that there is not a distribution of performance. We can measure performance (there’s more to say about that, of course). The trick, then, is to recognize when students have done something wonderful, like my kids who use concepts from calculus to find volume.

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The Innate Differences between Women and Math (Part 2)

Recap from Last Time: People use a set of relationships to help make decisions all the time called an ambient filter; some people might call the same set common sense. Stereotypes are a part of common sense.

Something’s not quite settling about the foundations I’ve detailed in the last post. It looks like the only thing we could say about women using ambient filters is that society conditions women to be bad at math (either by depriving them of the ability to hold tenured positions due to sexism, providing hostile working and learning environments, etc). Ah, but that ignores the nature of human existence. Like our filters, which can add or drop a relationship any time, our environment is not fixed.

This might sound a little Marxist to you; it should: Vygotsky (who got it from Engels who was inspired by Marx) loved the idea that man can shape his environment in order to shape himself. Whoa. Let’s pause a moment to digest the educational implications of that statement.

I’m told that in olden times, a person might tie a red or white string on his finger in order to remind himself to do something. Apparently, this was before they had paper and pencil and could write notes. Regardless of the specifics of the method, the general process and effect are the same: make something on the outside to trigger a response on the inside. This the the all-powerful idea of the sign. And if you dig deep enough, you can say all sorts of interesting things about social (as well as societal) effects on learning. Marx said the use of the tool makes us characteristically human; Vygotsky argues in favor of the sign. (Personally, I like the sign better.)

I know, I know, we’re moving slowly. So I’ll speed it up.

Now back to math: who were the principal investigators of mathematics since very early on? Men. And who developed the system of notation and verbal description we commonly use today? Men. And is it very likely that those who study a field of knowledge (which, by the way, may be entirely blind to the natural inclinations of its investigators) are going to devise a method of symbology that makes sense to them? Yes. And is it very likely that these representations of knowledge are going to make sense to its authors precisely because these representations automatically exploit their personal frameworks for understanding? Yes. (That is, would anyone ever record something that he understands in a way that cannot understand? No—at least not on the community-level.) Ah, then would you grant me that if there are biological differences between the way men and women think, doesn’t it make sense that because men have dominated math forever that the language of mathematics as we know it will necessarily be kinder to the male intellect than to its female counterpart? Sure it does.

So what have we learned through our very heavy-handed Socratic dialogue? It is very possible that while real mathematical knowledge doesn’t care what gender a person is, the representations we use today (in the symbols, language, and presentation at large) are biased in favor of men. Weirdly enough, that means there are innate difference between math and women. Exposition of mathematics has changed very little in the past century. The curriculum and its implementation exist primarily for historical reasons. The way people form common sense about math, therefore, hasn’t changed much, either. The trick, if what I say is correct and its effects are large, is to recast the relationships we use to describe math, and the methods by which we establish them, in a way that is meaningful to a larger audience. Of course, uprooting blatantly sexist myths about the role of women in math and science couldn’t hurt, either.

But here’s the really interesting part: we’ve shown that common sense doesn’t exist exclusively within the mind. Instead, we can leave it on the outside, in what we say, write, draw, make, build—in anything, even tangible things!—and that a throrough treatment of creative problem solving (and thought more generally) has to take into consideration the external consciousness we store in everyday objects.

(Yes, Lauren, I know. Historians have long recognized this fact. Ulrich studies teapots, I get it. Archaeologists, too. Sure. But is there anything new under the sun?)

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The Innate Differences between Women and Math (Part 1)

Leverett had its annual Sophomore Dinner (the follow-up to the universally dreaded Sophomore Outing) on Monday night. Being a member of the tutor staff, I was there to form the cohesive bonds required to form a healthy, responsive House community. Next time, I think they should serve more wine. After a fairly riveting bout of Two Truths and Lie, the dinner ended. The cool tutors met in the back of the room to catch up and gossip. Among them was my friend and not cousin Lauren, a PhD candidate in the American Civilizations Department. She does most of her stuff in women’s religious groups during the Progressive Area, but for an upcoming conference in New Hampshire on history and pop culture she’s got to stretch a little bit out of her comfort zone. Religion, it seems, doesn’t count for popular culture. I suggested that she write about the mentalists, Harry Houdini, and Robert Barret Browning. Everybody knows that magic rules. Isn’t Job everyone’s favorite character on Arrested Development, after all? I rest my case. Lauren, stubborn in her religiosity, has decided to retell one of the oldest stories from the Catholic church, this time with a Progressive Area twist: the age-old tale of the clergy pederast.

Now it should be noted that both men and women took small boys during the fifty years straddling the dawn of the twentieth century. The girls, it seems, were left out. Perhaps we think so only because we lack historical evidence demonstrating otherwise; but maybe it’s because there’s an innate difference between boys and girls that makes one more attractive to clergy than the other. I have to be careful here. This is a serious topic. And serious topics require reverence. Readers, try not to infer my personal beliefs from what I say. I can already hear several of you groaning in agony. I don’t hate women, or even children.

So, when Lauren and I convened (with the other cool tutors), I asked her how her mentalists were treating her. She insisted that she’s not writing about mentalists—which I told again told her was a poor choice—but about gender issues. So I told her that I, too, had been thinking about gender issues for one of my classes. In my Introduction to Creative Thought class, I’ve structured my weekly assignments around some serious efforts to establish a satisfactory, background-independent framework for creative problem solving. (You can see my general relativist training seeping into the vocabulary and aims, can’t you?) Of course, there are social inhibitors and enhancers. And it’s hard to incorporate society objectively into a working definition. And thus, in a very roundabout way, I explained that gender issues are very important to me, too.

Without telling you too much about my hair-brained problem solving schema, I will say that society influences just about every decision we make. Even when we’re alone, we’re not. Post-modernists love this idea. Even when you think you’re alone, the experiences culled from daily life shape the little voice in your head, opening the flood gates for society and everything that associated with it to come rushing in and drown you, the individual, out of your own mind, even without any direct, external presence. Sure, I’m being a little melodramatic. Exaggeration can be dangerous, but here it’s well worth sitting down and inspecting which thoughts are really, truly, exclusively your own. Go ahead, I dare you.

The idea is that whenever anyone approaches a problem, any problem, he makes some decisions about which relationships will be useful in finding a solution. (Yes, sticklers, I know that problem identification is not well-defined. To those of you who care, I appeal to any appropriate variant of the very robust berry picking model for information retrieval.) For example, when writing a sonnet, I might include several relationships between words I use and the number of syllables they include in my relationship set. Chances are I wouldn’t have to rely on the relationship “Wings help birds fly.” The way we choose which elements to include in a problem’s relationship set I call a filter. Filters are important because people collect what seems to be an uncountable number of relationships as they go about their daily routines. It’d be computationally impossible to consider all of them all the time. Indeed we pick up rules so often that it’s easy to do so without giving them due attention.

On the Cosby Show, Claire asked husband Cliff the following:

A parent and child were driving along one night, when, unfortunately, another car hit theirs. Only the child survived the immediate wreckage but was in critical condition. At the hospital, the attendant ER doctor gasped to see the boy on the stretcher. “I can’t operate on this boy; he is my son,” the doctor exclaimed. But how can that be?

Cliff, stuck in his ways, forgot that women can be doctors, too. When approaching the problem, he secretly used the relationship “Doctors are men.” And so the filter that’s almost always on—unless we consciously recognize and change it—I call the ambient filter. Some people might call it common sense. Where am I going with this? Ah, gender roles comprise part of our ambient filters.

This post is starting to get a little long, and I know, being a reader myself, that it’s hard slough through overtly boring entries. To read about how filters relate to why women don’t do math, continue on to the next post.

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Hurting Children

After graduation, there’s that lurking temptation to do the unthinkable: to sell your soul and jump into finance. Now I’m not hating on any of you who did this. Business is an important, even necessary part of society. So we need people to do it. The work is hard; the hours are long; but I hear the pay is pretty good. And actually, I think that my job is from a social health perspective far worse. You see, I’m in the education field.

People who go into the high-paced financial markets, well, they really can do very limited damage. Right out of school, few of us are in a position to ruin countries economically or otherwise. They keep the harm to themselves. High levels of stress combined with few hours to sleep leave the worker mentally and physically drained. Then, in those few hours they do have to themselves, many seek refuge in drugs or alcohol. Not all do, of course. But even those who do don’t really leave a lasting gash on society. Ah, but then there are those like me. The quiet, horrible types who try to help out others.

At least in business, there isn’t any real pretension to altruism. In education, that’s all we claim to do. Invest in the children today to save the world of tomorrow, and the like. However, it’s seldom that easy. Oftentimes, people deign to do charitable acts which tend more to harm than to help. Remember that obnoxious girl who tried to order her food at Boca Grande in Spanish? It took her fourteen times as long as everyone else and made everyone in the restaurant (except, possibly, the girl—she didn’t stop, after all) feel uncomfortable. That sort of thing happens a lot in education, but the effects are more permanent. Try as we might, people like to simplify complicated processes because, well, that’s human nature.

I freelance for a publishing company in the math textbook division. Right now I write tests for an accompanying middle school textbook series. And let me tell you, while it’s hard to write a good math textbook problem, it’s very easy to write a bad one. Many states, and indeed the country at large, have pushed for more so-called real-world math. These over-contextualized problems do wonders to confuse and hinder understanding. The research shows how bad they are, but people seem to love them. Or, rather, they love to make their children do them. No one actually loves to do them. That’s why many parents won’t help their children do their math homework. (And whoa, what a message that sends the kids: math is unimportant; it’s okay not to be good at math; do it now and soon it’ll be over. Why don’t we accept a similar level of ignorance in other fields? It’s embarrassing not to be a “reading person” but perfectly fine not to be a “math person.”)

Motivated by the enthusiasm and reward real-world problems brought Agatha Christie (to be honest: I don’t know anything about Agatha Christie aside from this quotation, which pops-up in math education reading from time to time. In fact, up until recently I thought she was Angela Lansbury), I rely on her words. They float around in my head and guide my writing:

I continued to do arithmetic with my father, passing proudly through fractions to decimals. I eventually arrived at the point where so many cows ate so much grass, and tanks filled with water in so many hours I found it quite enthralling.

And so I try to sneak in problems that use only thinly veiled real-world examples, but are secretly robust, real math problems. I’d include some examples, but I’ve signed a non-disclosure contract.

Some of my problems don’t have numbers at all, and even ask students to draw and label their answers. Of course, for every problem I come up with that I think is mathematically constructive, I submit six or seven others that I think are damaging. And here’s the problem: I actively hurt children. I help to spread and reinforce American mathphobia, one problem at a time. Because of me (and those like me), children learn to believe that math is boring, calculation according to some magic set of standards that devious, smart, and totally absent people make up. Still, it’s nice to know that I’m fighting back the cancer of classically construed middle school math, albeit not by much.

And the textbook series that I’m writing for isn’t extremely terrible. The authors sprinkle in short and extended response questions among the rote drill calculations. Some of the questions are open-ended. And they’re big on listing the standards each problem uses. Yet the text introduces the meat of each standard through by example, leaving the student to abstract and generalize rules on his own. (This is quite generally a dangerous practice.) Obvious over-contextualization aside, these margin notes do encourage basic metacognitive reasoning. In a small, roundabout way, they ask the studenst to think about what they’re thinking about. More practically, the kids (and their parents) know up-front what material they’re accountable for. And they get to see that these problems weren’t made up completely at random. Someone thought about them. So the cost of the materials is justified, right? Yes, I think it’s a political ploy. A good one, though.

And this is the most frustrating part about it. The standards trick people into thinking that there is some golden set of content and skills that a person should have in order to be considered mathematically competent or numerically literate or whatever fashionable buzzword you can come up with. The fact of the matter is, there isn’t. Math isn’t about what you know, it’s about how to organize what you know. I don’t know much graph theory; does that mean I’m innumerate? No way. I can do more geometry than plenty of professional graphy theory mathematicians, I’m sure. They know what they like; I know what I like. The crazy thing is, I know how to reason the same way as the graph theorists. The take home: the mathematical content of a textbook is really a vehicle for the abstract reasoning behind it all. For this reason, curricula can really be a lot more flexible than they are. Now don’t get me wrong. I’m not going to say that kids shouldn’t learn arithmetic. I will argue that maybe they should learn it another way. Even when we publish fancy standards in our books but forget to change the way we approach those standards, we really haven’t done anything. Kids have been learning how to add in just about the same way for over a century. Meanwhile there’s been lots of ground-breaking research done on how people learn, think, and understand over the course of the last one hundred years. Why do we so willingly ignore it?

But I do have a curriculum, and I use it. Meanwhile, I can only do so much to take into account the kids who’ll be using my books. We’re never going to meet. I don’t know anything about them, except, possibly their average age and vague geographic location. It’s important to have a good sense of what they know, how they understand it, and how they learned it. Projecting two years into the future about strangers is hard stuff. I have to write blind to my reader.

Whatever its impact, I’m very lucky to have the opportunity to work on textbooks. With some careful thought and hard work, maybe I can make a small contribution for the better in middle school education (before running back into academia to play for the rest of my life).

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