A Weblog

I have a number of summer resolutions. For one, I’d like to learn to play the piano. Again.

It’s not that I could ever really play in the first place. After a few years of self-imposed lessons, I could proudly state that I knew the first twenty bars to about a dozen songs. Whether I could actually play them was another story. Some of these selections were almost impressive, however. My teacher and I liked to hack away at four handed arrangements of things from time to time. He was a little better than me. But at least it explains why a copy of Beethoven’s symphonies (numbers one through five) and another of the Nutcracker Suite piano reduction loom behind me on the bookshelf. I figure if I live with them long enough eventually I’ll learn them.

Tonight after work, I settled down in front of a piano downstairs to practice for half-an-hour. I had made arrangements to set off several hundreds of dollars worth of fireworks with DJ and Bob from across the street, but the recent rains have left our sand flats a bit too muddy for Bob’s taste. This is probably all for the best, anyway.

Tonight DJ, Bob, and I realized that we have each, at some point, lived next to the other. When DJ moved from Brockton, his family took a house neighboring Bob’s. Then Bob moved and DJ moved again, this time down the street from me. That’s essentially how we met. Bob and I had met through soccer but were reacquainted through varisty cross country in high school. DJ also ran. Because he walked the same way home, we walked together. When polled, people will say that they are friends with their friends because they think similarly or share compatible beliefs; when friends are asked to take surveys independently of each, though, that is seldom the case. Instead it appears that friends usually share common interests or activities and not beliefs. Now that we don’t run anymore, it’s surprising that we’re still friends nearly eight years later. But of course, I haven’t completed the triangle.

While I was away at college, my dad and sister moved to the next town. And wouldn’t you know it, they found a place three houses down the street from Bob, his wife Trisha, and then only child Kyle. Luke, their second kid, is almost eighteen months old. We have each neighbored the other in pairs. The three of us and Kyle (and a few others) are going to see Superman at the Natick IMAX next Friday. But none of bears any consequence on my playing the piano.

Somehow sight reading seemed easier tonight than it did five or six years ago, whenever it was that I last tried to tame that beast with its eighty-eight keys. According to my piano teacher, there’s a direct line of tutelage between me and Chopin. Perhaps there’s something to that. I’ll let you know when my end-of-the-summer concert is.

I’ve been using growing up as an excuse to buy more and more expensive toys—that’s both more toys and more expensive toys, not just super expensive toys. My latest, emergency purchase was this sexy GPS/Palm device produced by Garmin, a respected name in GPS technology. Behold the iQue 3600.

It’s a fairly old technology by technology standards, and I would’ve prefered its updated, sleek cousin the M4, but I couldn’t stomach the price. After all, I can’t even afford the street price of the iQue 3600. Luckily, these things have been around long enough (read: a year) that they’ve shown up on ubid.com. Go quick and you’ll beat my winning bid by two dollars.

I totally would’ve rocked everyone else at those boy scout orienteering competitions if I had this bad boy instead of my crude but trusty compass.

Also, on the less expensive more toys front, I received A Primer of Algebraic D-modules in the mail today. “This book introduces D-modules and their applications avoiding all unnecessary over-sophistication.”

It’s like Christmas in July. In June!

My friend Jessica has asked me to install an air conditioner in her apartment sometime this week. In order to do this, however, we needed to get permission and a few tools from my friend, former boss, and Jessica’s building manager, Paul. Yesterday I stopped by his office after my morning swim. Miraculously, Jessica called at about the same time. Here’s how part of the phone conversation, at least the part that I could hear, went.

“Why, hello. Of course I know who Tutor Jessica is,” answered Paul. (Jessica isn’t some sort of perverse, self-involved creep. She lives in a dorm and acts as a resident tutor for the students in addition to her real-world, grown-up job.) After some obligatory chit-chat and his giving her a general hard time, Jessica was able to move him closer to her goal: the air conditioner.

Paul agreed to lend Jessica a ladder—so that I could access the wrought iron gate barring her window—and screwdriver—so that I could remove the gate and install the AC. Jessica, I believe, made a tactical error. In an attempt to be cute, she also asked for a hard hat, for my protection. Pandora herself had never opened such a box.

“Oh, yes. That’s very good. We’ll get Josh a hard hat. And we’ll get him construction boots, too.” He darted his glance over to me and smiled. “And how about a tool belt? We’ll have to get him a tool belt and a hammer,” he added.

I cut him off, “Paul, I’m not a stripper.”

Perhaps cognitive dissidence or an over-active sense of professionalism prevents my recalling what came next. I know I didn’t leave the office for another hour.

A week ago I promised a follow-up entry on testing. Here goes. We can divvy up tests into two broad categories: formative and summative. The first type, formative, is often the more useful of the two when discussing learning and understanding. It commonly goes by feedback or commenting. But the point is that it not only tells the student that he’s got it wrong, it points out why and how it could be changed so that it’s better. Think annotations on an English composition or history paper. This sort of test helps students to revise the way they’ve approached the subject and generally to improve their grasp of the material. Also, this sort of thing usually isn’t considered testing. It tends to happen during the draft rather than after the final project is complete, though it need not.

When I graded for a freshman honors, theoretical math course, it was not uncommon for me to mark up problem sets with comments like, “This is impossible. I see why you wrote this, but here’s why it doesn’t work…You’ve got the basic idea right, though.” Score: ten out of ten. Sometimes I’d go as low as seven, but you really had to push me there. The score wasn’t important, the reasoning was.

The second kind of assessment, summative, takes its name from the word summary. And as such, it usually signals the end of a unit, a chapter, a book, whatever. Once the lesson is done, summative testing quantifies student learning and spits out a grade. It’s dangerous for at least two different reasons.

Summative testing doesn’t help the student learn from his mistakes, at least not as easily. Say a second grader writes on his multi-digit addition exam that 112+37=482. Summative testing tells the student that he’s got it wrong. Formative testing identify the problem: he’s lining up the numbers in the wrong way. If he makes the same mistake consistently, formative assessment would address the root of the problem—he needs to review basic concepts about base ten number representations.

The second weakness of summative testing comes from the kinds of questions it asks and when it asks them. When kindergarteners enter school, most believe that the world is flat. This fact, after all, is confirmed by common experience. Once a teacher explains to his students that the world round and not flat, the student may accept this new fact—teachers are authorities, you know—but not in the way the teacher meant. If asked on a test, the kindergartener may successfully report that the earth is round, even though round to her might mean round like a pancake rather than like a ball. The trick, then, is to ask the right sort of question.

Too often summative testing (think SATs) requires a definitive right answer against which all other responses are considered wrong. Multiple-choice questions are especially bad, as the test taker may not know anything about the answer except that it has to be right in front of her. When the Princeton Review guarantees its instructors can raise test scores simply by teaching testing techniques, they mean it. There’s big money involved. And a lot of it comes simply from the format of the test.

So what should we have asked our kindergartener instead? Well, there’s nothing especially wrong with the first question. We can still ask what shape the earth is. But we need to supplement it with the sorts of questions that incorporate the student’s foundational knowledge: their conceptions, misconceptions, superstitions, and cultural beliefs—whatever, which they bring to the classroom before they ever enter it. So ask them to name another object that is the same shape as the earth and to draw it. It’s hard to hide a misunderstanding if you look for it in many different ways. For an older child, it might even be appropriate to ask her to design an experiment to confirm her answer. (How do we know that the earth is round like a ball short of going into space to see?)

Now that we know which kinds of questions to ask, we should think about when we ask them. Timing is crucial. So much summative assessment comes at the end of a chapter. This provides context. And the context may serve as a crutch, providing a sense of false understanding. If a calculus class has just finished a section on integration by parts, there’s a good chance that the questions on the test can all be solved by integration by parts. Many students dread cumulative final exams. They’re harder, if for nothing else, because the questions come out of context.

In that same freshman math class, a student came up to me during office hours after the midterm exam. She explained that there was one problem that was unlike any other that they had seen and that it was totally unfair and how could the professor do such a thing and how upset she was. After a short deep-breathing exercise and one and a half cups of cold water with a wedge of lemon, she was calm enough to identify the question. It was a three parter and went something like this:

(a) State the Rank-Nullity Theorem for linear operators.
(b) For a linear transformation A not the zero matrix from R³ to R³ such that A²=0, find a relationship between its image and kernel.
(c) What are the maximum possible dimensions of Image(A) and Ker(A)?

She was right. They had never discussed such a linear transformation (for the curious, such a creature is called nilpotent) in class before. They had, however, proved the Rank-Nullity Theorem. The problem above required students not only to have memorized the statement of the theorem but also understand what it meant enough to apply it to a slightly new situation.

I told the frazzled student that I recognized the question and thought it was very fair, and that’s why I had written it in the first place. She and I both were unmoved by the other.

The point is, it is possible to write good questions even in a summative testing environment.

Please excuse my unannounced blog hiatus—I started a new job and moved into a new place. Hopefully it won’t compromise my safety to mention that my days off are Thursday and Friday. Expect the follow-up posts I promised then. I hadn’t planned to write but dinner has moved me in unforeseeable ways.

Tonight I strolled down the street to my favorite Mexican, cheap-eats place for some pozole and enchiladas. Each summer I try to establish myself as a regular. Ordering the soup helps me stand out from the rest of the clientele. It also means I can sit down and eat right away while the rest of my meal bakes in the oven. About half way through the bowl my ears perk up and shoulders cringe forward.

“Um, quiero un más de queso,” I heard a shrill, female voice say. In high school my French language teacher, M. Labouiere used to say that bad French made his ears bleed. Not being fluent in Spanish myself, I can only guess how the man behind the counter’s ears felt. Even still, I shuddered. She continued. Whenever asked if she wanted to include some ingredient or other in her bean and cheese burrito extra grande, she had the gall to answer “sí” or “no” in a painful accent that mimicked, I guess, what Americans must think the Spanish sounds like. Since working at my new job, I’ve let on that I might understand more Spanish than I claim to and have been challenged thus. Just today Juan tried to get me to show off my [sadly lacking] skills when I butt into a conversation he and one of the workers were having in Spanish about the World Cup. Neither knew that Korea and Togo were matched in the morning game. My boss, Stephen, who spent a year in Quito while in college, thinks it shameful that my last name is Mexican and that I don’t speak Spanish. I may stick to Latin and Old English despite my so-called heritage. I was raised Boston Irish, after all.

But this shrill female, she didn’t know when to stop. Her trouble ordering clearly couldn’t've been her fault, so finally she checked, “Excuse me, are you speaking Spanish or Brazilian?” By now blood ceased to flow out of ears. Instead it was boiling within. I almost spoke up. It’s Portuguese, not Brazilian. Keep your sixteenth century, Catholic super powers straight.

Five minutes later, she was happy, burrito in hand. Her boyfriend didn’t speak much. He responded a simple and slightly embarrassed “sí,” I think for solidarity. The customer next in line asked for only a little bit of cheese on his quesadilla. I wonder it was surprising to learn that the man behind the counter was not only fluent in either Spanish or Brazilian but also fully conversant in American.

Now I know what my madrileña blockmate Verena means when she says that Americans speaking Spanish are among the world’s most obnoxious people. And it struck right here, in Cambridge, USA.

The reason why I ever noticed that depressing woman on the train was because of something she said that stuck with me:

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.