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

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.

Technorati Tags:judgment, identity, gender, race, hertigage, hispanics, emergence, interaction, society, assumptions, evolution, change, vygotsky, environment, self

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We may have spent a hundred years investigating how people learn, but the best way to learn to ride a bike is to get on one and try, and to pick yourself up when you fall off and try again. Having a parent run along behind to hold the bike up is good at first, as are trainer wheels, but eventually, you have to spend time riding the damn bike.

It seems to us that many (perhaps most) students today have been given the idea that they have no responsibility to learn and that teachers have all the responsibility for their failure to master the material. They believe in instant tratification [sic] and will not put in the time with the homework and the exercises. That is, they will not ride the bike and expect to become BMX celebrities simply by being told about angular momentum and bearings and friction.

Loki is, to some extent, right. If students don’t take responsibility for their learning, then there’s no hope. Despite the teacher’s best efforts, a kid who’s bent on shirking the material won’t learn it. The old adage “You can lead a horse to water” comes to mind. But perhaps Loki is being a little too hard on the students, on the teachers, on everyone. Still, it’s difficult to know what Loki means by responsibility. You might be suprised that incentives (such as money or the promise of a class party) are less effective at bolstering performance than really explicit directions and prompts. (Don’t believe me? I’ve got references.) So maybe we should at least hold teachers responsible for letting the students know what they’re responsible for.

That said, I’d like to acknowledge that people can do more with the help of others than they otherwise could alone. Some psychologists have studied this phenomenon formally. They’ve identified a zone of proximal development (ZPD). The ZPD is something like the teaser accompanying the end credits of a television show that gives some hint as to what will happen next time. The very existence of the ZPD shows that learning is necessarily social. Or at least, effective learning is social. I’m not going to argue that people cannot learn alone. But we’re talking about building effective classrooms. Let’s not make it harder for the students just because we can. So, to use Loki’s metaphor, it’s very useful to have training wheels and parent nearby. On this neither of us disagrees. The problem comes in when we try to decide what the parent (or teacher) ought to do.

In the model which Loki presents as standard, it seems first the teacher presents a repository of knowledge—very likely in the way of facts and procedures, e.g., the product of two negative real numbers is a positive real number, or the algorithm for multi-digit addition—then the students memorize and reproduce the facts and procedures. Teachers evaluate the degree to which students have mastered the material by way of tests. It is very likely that these tests ask the students to answer questions written in a format consistent to the teacher’s original presentation. To perform well, the students need to memorize and drill until their responses become automatic. This form of evaluation suffers from at least one critical problem: it cannot distinguish between accurate performance and thorough understanding.

The performance of a good novice and an expert can often appear the same. For example, a child who simply learns his addition tables by rote can respond as quickly and accurately as another child who has a reasonable grasp of the mechanism represented by addition. Thus when the two students move onto problems which require a “carry,” the first student will have a significantly harder time simply because he has more facts to memorize, whereas the second student will be able to generalize the rules of addition to accommodate the new problem.

I’ve discussed test design before, but for Loki’s benefit maybe I should quickly recapitulate. A Good Question should be able to distinguish between accidental correct answers due to rote memorization and intended correct answers resulting from mastery over the subject. Let’s build up a good problem from a bad one. When learning about prime factorization, teachers often introduce the concept of the least common multiple (LCM) and greatest common factor (GCF) of two numbers. Therefore, a natural question to ask on a test might be:

Standard Question. Find the least common multiple of 12 and 21.

In itself, there’s nothing especially bad with the Standard Question. It gets to the point, shows that the student has some computational understanding of what’s going on, and can reliably produce the answer to this type of question. In fact, a Good Question draws on the content of interest. If we’re interested in LCM, then this question is on its way to becoming a Good Question. But if the student taking the test has access to a TI-89 or other sophisticated calculator (as I did), then all he needs to do is to type LCM(12, 21) into the calculator. Surely, the use of technology is not something to be scoffed at. I’m using a computer to type up this paper, after all. I’m not about to propose everyone throw out their computer and write everything by hand. But if our aim is to teach kids something about the structure of numbers, then maybe a heavy dependence on technology gets in our way. We really need to come up with a Better Question, one that a calculator can’t do. Let’s try.

Better Question. Tricia says that you can find the least common multiple of two numbers by finding their product and dividing by their greatest common factor. Does Tricia’s method always work? Explain your answer.

Well, we’re getting there. Except now Loki might object, and rightfully, that this Better Question doesn’t readily test whether students can “ride the bike.” It asks them to identify the various parts. It even requires them to be able to build the bike. But it doesn’t ask them to ride it. So, maybe a Good Question does it all: it requires kids to build and ride their own bike. What more responsibility could we ask for then that?

Good Question. Tricia says that you can find the least common multiple of two numbers by finding their product and dividing by their greatest common factor. Does Tricia’s method always work? Explain your answer. Find the LCM of 12 and 21 in at least two different ways.

And notice that the Good Question requires students to calculate the LCM in at least two different ways—here we sort the lazy memorizers from the more dedicated kind. What makes the Good Question good, though, is that it asks the students to synthesize knowledge on the spot. That’s not a skill you can easily happen on by mistake. Sure, it’s a little bit harder to grade, but who cares; isn’t that the point of being a teacher?

As a test writer, I see myself in a very funny and useful position. Teachers have a habit of “teaching to the test.” So if I alter the way I write tests, it seems—at least in theory—that I accomplish real change in the way teachers prepare their students. Ideally, teachers would have enough mastery over their subject so that they could let students lead the learning themselves (as is done in the Math Circle run by the Kaplans at Northeastern and Harvard, or in schools which have adopted a curriculum tailored by Project SEED). In those classrooms, the shift in responsibility is more apparent—though perhaps no more real, since the teacher must keep a careful eye on the course of the class and give constant, mindful guidance. Perhaps this is more what Loki had in mind. I’m not sure; hopefully, he’ll elaborate. For now, I feel like I’m working on both the teacher and the student in a way that produces a broad effect on practice without having to sort through the politics of education policy.

In my next response, I’ll address the social component of learning more directly. Sorry guys, this post went in a different direction than I had initially intended. If I don’t use the words authoritarian and authoritative in my next post, please leave me an angry comment.

References

See, for example, Carroll, W. R., Rosenthal, T. L., & Brysh, C. G. Social transmission of grammatical parameters. Psychological Reports, 1971, 29, 1047–1050.

Rosenthal, T. L. & Zimmerman, B. J. Language and Verbal Behavoir: Social Learning of Synactic Constructions in Social Learning and Cognition, Academic Press: New York, 1978.

Technorati Tags: zpd, test design, assessment, math, education, psychology, teaching, zone of proximal development, vygotsky, social learning

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.

Technorati Tags:potential, education, calculus, curriculum, mathematics, competence, performance, mastery, motivation, ability, volume, community of learners, iq, intelligence

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?)

Technorati Tags:marx, vygotsky, sign, semiotics, education, innate differences, mathematics, science, women, memory, knowledge, representation, distributed cognition, symbol, understanding, language

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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Last night, I pointed out to my roommate DJ that a Democrat has finally smartened up and done the same. Massachusetts gubernatorial candidate Deval Patrick, whose website browser icon is funnily DP—I wonder if his marketing team are aware of this—, has used similarly effective however empty campaign slogans. The weakest of his tag lines claims that Patrick is No Ordinary Leader. Now this is good, sure, but it’s not great. It tries to exploit the constant dissatisfaction that most of us harbor against whatever we currently have (be it our government, job, or any other part of life). More than that, it presumes that ordinary is bad and that unsual is good. Just to keep us in line, I’d like to point out—and I know that I’m using an unfair extreme–that Hitler was No Ordinary Leader. I’m not going to argue with you now, so take it at face value when I say that Hitler was bad. A good leader, sure; a bad man, certainly. But like I said, Patrick’s got better ones.

Next in order of efficacy, I think, comes his invitation to join him. Together We Can his posters say. My sister’s boyfriend Andrew finds this one particularly stirring. Last night he told me, “It evokes a partnership between me, the common man, and the candidate for the leadership embodied in the State’s chief magistrate,” or something. “Also, this guy went to some farmers out west somewhere and told them, ‘I’m not a farmer. I don’t know about this stuff. Tell me what I should do to help you.’ He’s really thinking out of the box,” he went on to tell me. My roommate DJ nearly drowned in his own tears (of laughter) upon hearing this.

Andrew proves my point. Perhaps now I should make it.

Together We Can is genius simply because it promises nothing. Patrick’s team were very careful never to use punctuation after any of their slogans on any of their signs. Of course not. They’re fragments. You can’t put a period after a fragment, after all. Doing so might point out raise the attention of a lazy reader. Then he’d realize that you haven’t said anything at all. To Andrew I asked, “Together we can what?” Patrick doesn’t tell us. Instead, he lets our imaginations run wild. That’s right, I am going to help run this State. I am important. Wrong. This slogan is so compelling because it calls on the reader to finish the sentence according to his personal whims and then pretend that it’ll happen, that he’s effected the change, and it spares him the hassle of doing any, real work. People love to feel like they’ve contributed something useful; on the other hand, they hate to exert themselves. This slogan let’s you think you can have your cake and eat it, too. (I’ve never understood that saying.)

But undoubtedly the best slogan I’ve heard so far, Patrick saved for until after he won the primary. Now it’s showing up on bumper stickers. Patrick asks us to Believe Again. I can’t begin to explain how impressed I was when I read this slogan. I wanted to run up and shake him and cry and clap my hands uncontrollably. It’s really quite amazing. This slogan reaches the largest audience possible. Being the most devoid of content, it has the greatest reach. Believe Again entices the voter to conjure up the most romantic, idealized form of government possible. But it doesn’t stop there, the implications are unstoppable. It’s an easy jump from government to general quality of life. Improving one naturally improves the other, right? No matter what you believe in, Patrick does, too—at least according to this slogan. And shouldn’t you support someone who holds such a coincident and intimate commitment to those things you hold so dear? It’s hard to argue against him, because you’d have to argue against yourself. Imagine a leader who would allow you to Believe Again.

To test my claims that these are, indeed, worthy of the Republicans, DJ asked quite blankly, “Are you suggesting we Cut and Run?”

To which I answered, “It’ll take No Ordinary Leader.”

To which he countered, “But don’t you Support Our Troops?”

But then I hit him full-force with, “Together We Can. I want to Believe Again.”

It was over. The conversation left both of us stunned.

DJ then noted that we should write for the Colbert Report, or, maybe I should write for the Colbert Report, or, possibly, just to them, to let them know that someone else figured out how to play the word game.

What’s worth mentioning is that Patrick’s slogans are even more sinister than the Republican’s because they aren’t immediately negative. (No Ordinary Leader comes closest to being overtly aggressive, but is pretty sissy when flanked by Cut and Run and Support Our Troops. Notice, however, that Support Our Troops also makes the people who say it feel like they’ve really accomplished something even though they’ve taken no physical action.) Patrick’s tag lines get stuck in your ear, and, while there, make you feel better about him and about yourself. How empowering! I really can’t get over just how brilliant they are.

Moral: If don’t want people to disagree with you, don’t say anything that they can disagree with.

I’m telling you about the past—the past has nothing to do with today.

This is the dogma of the New Capitalism, and, coincidentally, the theme of a book by Richard Sennett I’ve mentioned before. With many industries looking towards consulting these days, many of us place our stock in potential rather than years of practice. This women’s belief is the end of craftsmanship.

I’m led to believe that before the dotcom boom of the early nineties—a time I know almost nothing about first-hand—employers hired and evaluated employees based on the history of their performance. With time and experience workers generally got better at their craft. Nowadays, however, there has been a shift from the past to the future. We hear lots of talk about so-called potential and adaptability. The idea is that the world is a rapidly changing place and those who cannot keep up are left behind. To me, this is an interesting departure from something that is at the very worst measurable to something that is at the very best ill-defined.

Society, even very conservative sects, believe that innovation and change are the same things as progress. Outwardly, such a tenet forces a meritocracy, and isn’t that the framers of the fledgling United States had in mind; aren’t we fully realizing Jefferson’s hope to establish a “natural aristocracy” founded not on the arbitrary forces of birth but by ability and good work? No, I don’t believe we are. [Nor do I necessarily think that we should. But to explain why might require another entry or two.]

We must question how we judge ability. We treat potential as if it were a fixed trait, born into us, and therefore just as arbitrary and unfair as family name. Growing up, I learned that the first grade teachers at my school had pooled together to bet which among us would be valedictorian. And I remember teachers and other adults saying of me that “he’s just not challenged enough.” To wit, nothing yet had tested me, forcing me to actualize my potential. Even as late as last week, my friend told me that I have more potential than he does. Somehow people are willing to overlook the past six months, during which I lived off my father and sister at home, fully unemployed and with little motivation to change. The reason why: potential.

But how does this conception of ability stand up in reality; should anyone get the job simply because he has potential? Let’s look at a specific case. Your goal is simply to identify the best piano player:

• Student 1 first sat down at a piano when she was 12 years old. Having never so much as plunked a single note on the beast before, she was able instantly to reproduce any theme, classical or contemporary, she heard perfectly. By 15, she was touring the country as guest soloist with more than a dozen symphony orchestras. She never had to practice once.
• Student 2 by contrast started playing when she was 4. She practiced constantly. By the time she finished high school and began college, she logged between four and six hours of practice daily. Student 2 studied music professionally and had several instructors who helped her to refine her talent and musical interpretation over the years. Eventually, she broke into the competitive circuit, and though not initially, was able to distinguish herself. Now she also tours and guest solos and boasts the same popularity and acclaim as Student 1.
• Student 3 is Student 1’s twin brother. By all accounts, he has the same capacity for virtuosity as his sister. In some cases, he can even play some of the most difficult passages on the piano with more ease and musical expression than his sister. Yet Student 3 does not practice his talent. Instead he chose to become a landscape designer. Today he manages fourteen professional golf courses and almost never listens to music, let alone plays the piano.

The question: who is the best piano player of the three described? The answer isn’t so straight-forward.

Potential alone, perhaps, isn’t good enough. Student 2 was able to equal Student 1 in success because she worked hard. Student 3 was not as successful a piano player as the other two because he didn’t work hard at it. And chances are no one will ask Student 3 to guest solo with an orchestra any time soon—despite his potential—because he lacks a good track record.

It is very hard, if not impossible, to measure potential because of this sticky business known as persistence. Sustained effort can and often does overcome the random distribution of powers and abilities. The son of a very rich man can die poor. The orphan children can grow up to be very rich. Be wary of tests which purport to predict ability. Tools like the IQ, which were designed merely as a diagnostic to assess the present—not the future—, have been misappropriated. The SAT, whose history begins as an officer exam for the US Army during World War II and has changed little since, is notoriously bad at guessing how students will fare in college. So bad is it, that they’ve changed the name from the Scholastic Aptitude Test to SAT. It’s no longer an acronym. The letters don’t mean anything, which reflects, I think, on just how much the test itself means.

Viewing ability as an innate, fixed trait can be extremely harmful. Girls outperform boys in math and science until about age 13. Perhaps in my next post I’ll explain some reasons why, and maybe respond to those infamous comments by former President Summers about women in science soon. For now, you can re-read what I’ve learned about praise.

And please, do not misread me. I am not advocating the end of testing. Far from it. But we should remember exactly what tests do under perfect situations: the most any test can do is to give an approximation of circumstances at the present. I’ll write a little more on testing for understanding soon, too.