Keep the Internet Open: Update

Last week, the Senate voted on SJ Res. 6, that piece of legislation that stated plainly: Congress disapproves of openness on the internet. That is, that companies should be able to block your traffic on the internet unless you paid for it. Fortunately, the resolution was rejected (by a slim margin of four votes). See the tally here.

Imagine how this sort of deregulation would work in the phone industry. Phone companies could monitor who were calling, listen in to what you were talking about, and then decide how much to charge you accordingly. Would you want the phone companies listening in to your phone conversations and then deciding how much to charge you based on what you said? “Oh, he’s calling his sister again—I bet he’d pay more to talk to his family.” “Her doctor is about to say something important, that phone call’ll cost extra.” Eek.

The FCC has been charged with the responsibility to make sure the internet remains open, transparent, and is free from blocking and unreasonable discrimination. According to the FCC, if it’s legal, you can do it and the internet providers should respect that. Since this policy seems like a good idea to me as it promotes technological innovation and rightful consumer protection, I was shocked that Scott Brown voted to dismantle openness on the internet. I’ve written to his office for an explanation for his vote. I’ll let you know the reasons if they respond.

I’m thankful to Kerry for supporting net neutrality. And I’m surprised and pleased to see that the White House had the backs of the American people, even if a near majority of the Senate didn’t. In an official statement, the President openly opposed SJ Res. 6 because he is in favor of job creation. And this resolution would have stifled technological innovation. Good work.

As an aside: in the end Olympia Snowe, Susan Collins, and Kelly Ayotte did vote for big business. If anyone can tell me how this resolution would have helped families, consumers, the poor, and/or children in Massachusetts, Maine, or New Hampshire, I’d like to know!

Keep the Internet Open

The internet is this country’s greatest, most used, and largest public library. Nearly a year ago, the FCC adopted FC 10-201 to keep the doors of the internet open to American citizens. In this regulation, the FCC cites evidence that broadband providers had been covertly blocking or degrading Internet traffic, and that cable companies have financial incentive and ability to shut things down even more.

Here’s a summary of values stated in the report:

PRESERVING THE FREE AND OPEN INTERNET

1. Today the Commission takes an important step to preserve the Internet as an open platform for innovation, investment, job creation, economic growth, competition, and free expression. To provide greater clarity and certainty regarding the continued freedom and openness of the Internet, we adopt three basic rules that are grounded in broadly accepted Internet norms, as well as our own prior decisions:

i. Transparency.
Fixed and mobile broadband providers must disclose the network
management practices, performance characteristics, and terms and conditions of their
broadband services;
ii. No blocking.
Fixed broadband providers may not block lawful content, applications, services, or non-harmful devices; mobile broadband providers may not block lawful websites, or block applications that compete with their voice or video telephony services; and
iii. No unreasonable discrimination.
Fixed broadband providers may not unreasonably discriminate in transmitting lawful network traffic.

We believe these rules, applied with the complementary principle of reasonable network management, will empower and protect consumers and innovators while helping ensure that the Internet continues to flourish, with robust private investment and rapid innovation at both the core and the edge of the network. This is consistent with the National Broadband Plan goal of broadband access that is ubiquitous and fast, promoting the global competitiveness of the United States.

But NO! Senator Kay Hutichson (R-TX) introduced S.J. Resolution 6 to the Senate floor to strike down openness and transparency on the internet. The resolution states in plain English that Congress is against openness on the internet! If passed, this resolution will make the FCC unable to ensure the doors of the world’s greatest public library stay open to the public. This resolution is job-killing, innovation-stalling, and education-thwarting.

The resolution is so short, I’ll post its contents in their entirety for you to see for yourself:

Disapproving the rule submitted by the Federal Communications Commission with respect to regulating the Internet and broadband industry practices.

Resolved by the Senate and House of Representatives of the United States of America in Congress assembled, That Congress disapproves the rule submitted by the Federal Communications Commission relating to the matter of preserving the open Internet and broadband industry practices (Report and Order FCC 10-201, adopted by the Commission on December 21, 2010), and such rule shall have no force or effect.

Today I called Scott Brown’s office in DC to tell him to vote NO on S.J. Res. 6. The number is (202) 224-4543. Do you use the internet? Do you plan to use it in the future? If so, then please call your senators to tell them to vote for American innovation and against S.J. Res. 6.

p.s. — If you live in Maine, ask Olympia Snowe (202) 224-5344 and Susan Collins (202) 224-2523 how this resolution will create jobs in Maine. If you live in New Hampshire, ask Kelly Ayotte (202) 224-3324 whether this resolution will help New Hampshire families make ends meet. They must believe it will. They co-sponsored the resolution, after all.

Race is Different than Money

This winter I’m taking a course on urban education. Our first topic: segregation and desegregation in schools.

Firstly, what do we mean by segregation? As a working definition, I’ll offer that segregation is the spatial pattern of people across some attribute. So we could talk about segregation by race, by income, or by favorite ice cream flavor. Once we pick something to measure against, we find that every city is segregated according to this definition. What matters is in what way the segregation manifests and the consequences on the populace the pattern has. Segregation patterns can be uniform, with all groups distributed more or less evenly within a region, or clustered. Likewise, we could also calculate the extent to which subpopulations are isolated from each other—which also gives a rough estimation of how often members of one group is likely to run into someone outside of their group. I think when we talk about ‘segregated’ groups, we typically mean highly clustered populations that are isolated from the other groups in the city.

I don’t think that clustered, isolated groups are necessarily bad on their own. I love visiting the North End and Chinatown. Because they’re both T-accessible, it’s easy for me to get there. (Though, both neighborhoods have had rough pasts.) And Harvard Square is the nicest place I’ve ever lived. Score one for segregation!

Moral judgments aside, self-selection can have a big influence on patterns of segregation, at least it can in models. The positive feedback loops reinforce small, individual choice to generate large-scale patterning. Schelling’s model of segregation is a classic, good first example of what I mean. In this model individuals exhibit only a slight preference to have neighbors that are similar to them. The individuals in this model are not racist. (Or maybe they are. I don’t have a good functional definition of racism yet.) When individuals find themselves in a neighborhood that is too unlike themselves, they move somewhere else at random, possibly to a neighborhood more dissimilar from themselves than the last. Even with this mild, partially blind behavior, a totally segregated structure emerges.

In more relaxed models that completely ignore race, even more realistic patterns of segregation form. In this class of model, individuals simply choose to live in the nicest area they can afford. As if by magic, isolated poor and rich neighborhoods form. Depending on the details of the model, wealthy suburbs appear spontaneously. If we use socioeconomic status as a proxy for race, it’s the same old story. Except this time, we have a systems-level mechanism that generates isolated, poor communities that lack the power to advocate for equitable resources and very rich communities with disproportionately high share of public goods insulated by a buffer of middle class individuals. Race was not the cause; money was.

When was ask whether it’s morally justified for a white family to send their kid to a predominantly white school, I think it’s important to know what about the school is so attractive. Do all parents value differentiated cultural and social understanding across many kinds of experience? Are they likely to value it more than a pretty campus or reputation of success by its graduates? Sure, in some cases the choice may be motivated largely by racism. But I’d expect that in many cases, it’s mostly a matter of ensuring access to the most and best resources possible for their child. It just so happens that low-resource groups aggregate, even in the absence of race.

I believe that diversity (of background, experience, perspective, and the like) is important in schools because, as has been mentioned a few times by others, students learn how to navigate social situations outside of school from the people they meet in school. But when we talk about diversity, do we really mean racial diversity? As an example, imagine that an elite, wealthy, mostly white college in the Northeast has recently been chastised for admitting a student body that is not sufficient diverse. Consequently, the school begins recruiting wealthy black students from Africa, some of whom attended the same boarding schools as students already enrolled in the college. In time, the student body comes to be half white, half black with an even mix in all classes and housing situations. In what sense, if any, has the college increased diversity on campus? Do you think the college has produced the diversity they were previously lacking?

While I think that racial segregation is a problem, I don’t think race is necessarily the capital-C cause. In a world without racism, economic segregation will still exist. But I’m willing to bet that in a world with no financial disparity, a lot of the troubles we associate with racism would evaporate. And so, I think race will play a secondary part in the solution to segregation. In fact, I think that race may even obscure the issue of access to equitable education for all. (I’m not sure if that’s what we’re really trying to achieve, but I think it’s a good start.) Instead, I believe that the struggle of the American education system is one of power and status. As such, I think we should talk about resource allocation (including strategies that move students to resources as well as bringing more resources to students), causes and effects of socioeconomic segregation, and cultural and pedagogical practices that systematically discourage/motivate students to learn the skills required to become an informed and capable citizens.

Critical Thinking Journals/Skills and Dispositions

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

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

Journal 2 Journal 2: Skills and Dispositions

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

Let me know what you think.

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

Technorati Tags:, , , , , , , , , , , ,

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.

Technorati Tags:, , , , , , , , , , , , , ,

Oh, the French

I didn’t write anything explicitly to welcome the new year. I suppose that that’s partly because I was trying to resist the reality of it all. It looks like I’m not alone, either. The French up in Nantes, however, took a more direct approach. Good thing the BBC was there to cover it.

Technorati Tags:, , , , , , , ,

Testing Responsibility

You may remember that reader Loki on the run wrote:

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: , , , , , , , , ,

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.

Technorati Tags:, , , , , , , , , , , , ,

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?)

Technorati Tags:, , , , , , , , , , , , , , ,