A Weblog

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.

Technorati Tags:stereotypes, frittata, williams-sonoma, cooking, frying pans, stainless steel, knowledge, nazis, torture, guantamino, philosophy, eggs

I’ve started doing what I like to do again. That’s right. I’ve decided that it’s been long enough. Why don’t I have a tag-driven, database-backed website up and running yet? I’ve been meaning to collect funny and meaningful quotes—Lord knows I run across so many good ones everday. But for some reason I have forced myself to sit down and set up the site. Rather than using some out-of-the-box, I’m going to take this as an opportunity to learn some skills. I’ve taken out three books that may or may not prove useful: two on data mining, one on interaction design. Because of the hype, I should probably get one that covers AJAX, too.

I’ll let you know where and when you can see what I’ve cooked up.

I consider this a living exercise in computer science and philosophy. I’m fascinated by these folksonomy things and what it means for information architecture. I think that these user-authored category systems are going to propel the semantic web in the immediate future. But then again, I’m not sure really I know what I’m talking about. Please don’t trust me. I need to do some more reading and thinking about distributed cognition. Please let me know about your insights on these subjects.

The bloodgood shed all its foliage. The core stem is green. I keep tricking myself into finding what are not new buds. Still, it looks healthy. And it’s beautiful, what with its sterile dignity. Maybe I’ll go purchase some clunky, black frames to match it.

Technorati Tags:late night musings, bloodgood, maple, folksonomy, technology, semantic web, tags, catagorization

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

Since I write these things for class, I’ve decided to post my disorganized ramblings on creativity and the creative problem solving. I’ll update with a new installment weekly (or thereabouts). You can always find the link to the right under Pages > Creativity Journals.

Since everyone has something to say about creativity, maybe some of you will comment. I’d love to know what the anonymous masses think.

Technorati Tags: creativity, creative process, creative problem solving, thinking, reasoning

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