Category Archives: Education

Geometry Lesson Plans

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

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

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

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

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

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

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

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Testing Responsibility

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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.

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Creativity Journals

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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.

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A Response (Part I)

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

Loki wrote:

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

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

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

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

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

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

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

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

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

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

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The Innate Differences between Women and Math (Part 2)

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

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

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

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

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

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

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

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

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

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The Innate Differences between Women and Math (Part 1)

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

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

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

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

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

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

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

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

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

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Hurting Children

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

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

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

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

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

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

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

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

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

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

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

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

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Grading Woes

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Earlier this week my friend Emma the Grecian Sailor lamented that grading for her fluid mechanics class takes up too much of her time—there are only precious few good sailing days left this season, you know. And she’s right. How does one give a responsible, formative assessment fast enough to have time left over to bat around on the Charles? This question is as old as grading itself. Likewise, Verena has complained that Physics 15b has taken over her life. So, to my grad school buddies, I have an illustrative anecdote and suggestion: Like a magician would, confuse your students—divert their attention.

A few years ago, I helped teach what has since become a notoriously large freshman honors math course: Math 23ab tries to train those heretofore uninitiated in mathematics to think, write, and speak like a mathematician. We try to sneak in a little multivariable calculus and linear algebra along the way. (I threw in a little geometry and physics, too, when I could.) But being high-performing over-acheivers, these kids were super-aware of their grades. When one of the teaching staff gave one student a full zero out of ten points on her set, she objected and cried—in class! We had an emergency staff meeting to discuss the matter. The grading, it seemed, had to change.

Isadora and I complained that the last set had taken us each more than twenty hours to grade. (And for those of you economists and computer scientists out there, we weren’t doubling up on the work. Each course assistant graded exactly one problem from the set each week. If the problems were shorter, sometimes two.) As is usually the case in any collaborative venture, we couldn’t come to a consensus. We did have some wonderful lunches together at the faculty club, though.

To address the problem personally, I devised the following tactics:

  • Never grade out of a small, round number. Tens, twentys, and hundreds are strictly out. Kids can figure out their percentage right pretty easily if you do. And that means they’ll protest their grades more often. You don’t want them to pass their sets back in once you’ve passed them out. It’s just no fun.
  • Use outrageously large, unround numbers instead. Typically, I’d increase the worth of each question the further we were into the semester. Say, for example, a question might be work 268 points in September; 94760 in January. When the point values climb, students are less likely to care if you took 79 points off for something. Also, they almost never divide out to find their percentages anymore. Who cares what 6432/7356 is? Those are mean-looking numbers, after all!
  • Vary point values throughout a single set. If I had more than one question to grade, or one multi-part question, I’d mix it up. Part A might be worth 2305 points, whereas Part B was out of 7342. It shifts the focus off the grades, and off of you.
  • Don’t worry about strict consistancy. Setting up and maintaining a rubric is hard work, especially if you change your mind thirty-three sets into a hundred. Because the point values are so wacky, the students will assume that your grading schema is complicated and often won’t challenge or compare grades. It risks entering the rigamarole of your mind. That said, try hard to be fair.
  • Write encouraging remarks on their work. My favorites were wizard, way-to-go, good effort, and ingenious. Always follow your comments up with an exclamation mark.
  • Never grade in red. It puts them in a bad place, psychologically. And by bad, I mean nervous and contemptuous. I prefered orange Crayola markers. If I felt especially fiesty, I’d use purple.
  • Give explanatory feedback when appropriate. One former student reminded me that I had once graded her set, “This is impossible. [short explanation] See solution set. 10/10.” To be fair to me, she had it conceptually correct, except for, of course, the part that was impossible, which I circled in orange.
  • Write clear solution sets and post them in a timely manner. Another student, who had turned her set in late, told me that she used my solution set for help. I asked if she cited it. She had. My answers looked right to me. And so she got full-credit.

Hopefully, these techniques will redirect the students’ attention from their grades (product) onto their arguments (process). The idea is to retrain them, many of whom hold strongly developed outcome-orientations, to care about how they got the answer, and not merely that they got the answer. (Last night Michelle told me that a biologist told her that you can train just about anything larger than an amoeba, and that includes people. Of course people in social learning theorists have been saying that for years.) [Check out an old post for more.] And with any luck, it’ll make your life as a grader a little more comfortable.

Believe Again

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Yes, yes. We’ve all heard that the pen is mightier than the sword. Somehow it’s easy to forget, though, just how powerful those silly little words can be. The Republicans seem to know. They’ve sent out now ubiquitous catch-phrases—who doesn’t know to Support Our Troops?—to rally Americans to their causes without actually giving any cause to do so. These slogans are short, to the point, and entirely devoid of content. And still they have proven to be incredibly powerful. Remember when Colbert talked Geoffrey Nunberg, linguist and author of Talking Right: How Conservatives Turned Liberalism into a Tax-Raising, Latte-Drinking, Sushi-Eating, Volvo-Driving, New York Times-Reading, Body-Piercing, Hollywood-Loving, Left-Wing Freak Show, into the ground with only three carefully crafted phrases? (If not, search through the archive tapes for the show originally aired August 21, 2006. Comedy Central has clips: Part 1, Part 2, Part 3.)

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.

Book Reviews

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Since I started a new job and classes at just about the exact same time, I haven’t had much time to sit down and write. Because my boss took the day off, I can put a whole day’s worth of work into this thing—the problem is, though, I’m out of the blog zone and I’m not sure how to get back in. When DJ and I play tennis, the one who lost the point has to sprint around the court. At first, the loser of a single point continued to lose several points. Running made it hard to concentrate on the game. But as we played on, we got better. The interpoint sprints actually honed our mental and physical stamina. I don’t know of an analogous blogging exercise.

So, without any prepared material, I write on. Hoping that you’ll keep reading. And while I haven’t been writing lately, I have been reading. In the past week I’ve started four books. You’ll see that three of them fall into an obvious theme. Maybe you can guess it by the first’s title:

Social Learning and Cognition, by Rosenthal and Zimmerman, was written in 1978. I don’t know how much of the book is still current, but what they say seems to make sense. Like the other books I’ll mention, I haven’t made it very far: I’m only in the first chapter. To be fair, this one only has three, individually long chapters. Basically, social learning combines information processing—which came about once people began that quest for artificial intelligence—with a behavioral twist. As far as I can tell, this sort of thing has been applied mostly to criminology. What I’m reading smacks of Vygotsky, who, due to political barriers, never made it big in the West. Bandura, the guy who sort pioneered this sort of thing, applied his work most closely to violent behavior. Hence the trajectory towards criminology. However, just about anyone—folks in public service, education, corporate training, and community building at large—should know about this stuff. We learn from each other all the time.

The second book I’m reading for class. In fact, I found Social Learning only because it was near Uncommon Genius on the shelf. The author, Shekerjian, tries to figure out what creativity is through interviews with forty of MacArthur fellows: those men and women given a cool half mil from what has been popularly dubbed the “genius award.” Sadly, she didn’t interview the two MacArthur fellows I know. To be fair, Zaldarriaga, the guy who co-taught a course on cosmology I took last year, neither knows me nor had received the award before the time of this book’s publication. It’s an enjoyable read. Don’t expect any research, though. Sheekerjian warns you from the outset that her book isn’t rigorous investigation of creative thought. The anecdotes are apt and her writing is smooth. Her analysis falls into same linear model of thought as much of the research literature on creativity, though. Pick it up if you have a short flight and you’re bored.

The last book hasn’t been published yet. The Emotion Machine is Marvin Minsky’s soon to be released follow up to Society of Mind. If you can’t wait until November to read it, you can find a draft online on his personal website, though I’m not sure for how much longer. Minsky, who you can tell is a trained mathematician by his style, gives a very easy introduction into the basics of artificial intelligence, which, by the way, doesn’t preclude its telling us something about human intelligence along the way. Minsky writes in a semi-dialogue form, injecting objections and commentary by invented philosopher, student, and citizen characters. They move along the discussion in an informal way which hides its unusual directness. If you had to choose among these three, you should probably choose this one.

The fourth book, which I only started about an hour ago, is Sakurai’s Modern Quantum Mechanics. I started this one based on a recommendation of one of my college roommates, who’s gone on to do her PhD in physics. She says that everyone agrees, Sakurai’s exposition is wonderfully clear though advanced. I guess they use this in a graduate-level QM course either at Harvard or MIT. (She cross-registers a lot.) Being an undergraduate mathematician, I missed out on quantum mechanics. As a high schooler, I thought that it would be my ultimate achievement. In the meantime, my dad has started spending a lot of money on some heinous line of products based on the study of so-called biophotons. This ever-authoritative Wikipedia entry sums it up nicely:

The field of biophoton related study also appears to have recently become rife with new age, complementary and alternative medicine, and quantum mysticism claims from those wishing to exploit such clams [sic] for financial benefit. Numerous claims are even made that by “harnessing the energy of biophotons” that supposed natural cures for cancer are guaranteed. Mainstream medicine and science strongly reject these claims as outright fraud and a dangerous diversion from actual medical treatment for someone who is suffering from such disease.

I figure if I can master the basics of QM, maybe I can have my dad ask questions that will confuse his prophets—because that’s what this has become. He defends these guys as if they were his gods.—and demonstrate that they’re just out to get his money. Even if that’s not your aim, you should read Sakurai, especially if you have a strong background in linear algebra (including Fourier analysis).