Geometry Lesson Plans

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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A Response (Part I)

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

Loki wrote:

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

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

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

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

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

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

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

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

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

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

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The Innate Differences between Women and Math (Part 2)

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

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

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

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

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

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

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

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

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

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The Innate Differences between Women and Math (Part 1)

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

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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Descartes, Urban Planning, and Chaotic Systems

Motivated by shame, I’ve taken up Dan Aaron’s challenge to read more original work by giants who laid the foundations of my field. Being a hack mathematician with physicist tendencies, I’ve turned to RenĂ© Descartes—the man who married geometry and algebra to give us Cartesian coordinates, among other very important things—and his Discourse on Method. In the first part of the Discourse, he situates the reader a bit, explains his personal history, and how it led him to his approach to problem-solving. He tries very hard to be humble and sometimes he almost succeeds, but we know Descartes was a genius. He knew it, too. And even he can’t hide his high opinion of himself.

Descartes claims that the work of several authors is usually inferior in quality to that wrought by a single hand: it’s the old “too many cooks spoil the soup” theory. That’s how he justifies denouncing centuries of philosophers and their work: he had to do it start on his own from scratch; it’s impossible to sort through the ideas of others with any systematic clearity. But he stays out of the kitchen in his metaphor, instead traveling to the city:

Thus we notice that buildings conceived and completed by a single architect are usually more beautiful and better planned than those remodeled by several persons using ancient walls that had originally been built for quite other purposes. Similarly, those ancient towns which were originally nothing but hamlets, and in the course of time have become great cities [think London], are ordinarily very badly arranged compared to one of the symmetrical metropolitan districts which a city planner has laid out on an open plain according to his own designs [Brasilia, say: it looks like a bird or airplane]. It is true that when we consider their buildings one by one, there is often as much beauty in the first city as in the second, or even more; nevertheless, when we observe how they are arranged, here a large unit, there a small; and how the streets are crooked and uneven, one would rather supppose that chance and not the decisions of rational men had arranged them.

Now, Descartes’ philosophy aside, there’s a pretty interesting observation in there. Even though a very complicated system, such as a city, maybe be very orderly on small-scales, the over-all effect is painfully complex. I’ve declared more than a few times, proudly to visitors, that the streets of Boston were once the random paths of wandering cattle. So this Fathers’ Day, when my dad and sister came to visit my new digs, I had to remind them to look in all—not both—directions when crossing the street. “The streets around here are so wacky, that the cars going one way can be stopped at a red and still you can be hit in three other directions.” We all made it across, a little bit hurried but safe. Meanwhile, Manhattans live on a grid and seem to like it. Some have even claimed to prefer it.

A similar phenomenon exists in economics. A while back someone was awarded a Nobel Prize for showing that even though each person may act rationally, the market as a whole still might act irrationally. I don’t know much about how this works, but I can tell you about an analogue in math.

There are several competing definitions of chaos in mathematics, depending on just which subfield you subscribe to. One of the most tractable definitions comes from one-dimensional, real, discrete dynamics; that is, the study of iterated functions that live on the real number line. Simply, take a function, which is only a fancy word for a rule, which takes in a number (like 3) and spits out another (say 1.5) and repeat it over and over again. We say that a function is chaotic if it exhibits the following three behaviors:

  • The function has a dense set of periodic points. Periodic points are orderly; their trajectories are predictable. They start at one location. The function moves a periodic point to another point; another application of the function moves it again to another, and another, and another. Eventually, though, the point needs to wind up where it started. Think of periodic points as travelers on a multi-city, round-trip itinerary. A salesman might start in Boston, go to Denver, follow-up in Phoenix, stop short to family visit in D.C., but at the end of it all, he’s got to go home to Boston. In the language of discrete dynamics, our salesman is a periodic point. Dense is just a mathematician’s way of saying most. So, if you blindly pick out a number, it’ll most likely be a periodic point. And if it’s not, there are plenty of periodic points nearby. On a plane, not every passenger is needs to be a travelling salesman like the one in our example. But near each passenger there should be a few of them close by.
  • Notice that the definition of a chaotic map (or system or function—these terms all refer to the same thing) demands lots of order. Periodic points are simple. We know exactly where they go and exactly where they’ll end up: they travel in loops forever. However, chaos requires a little bit more.

  • The function displays sensitivity with respect to initial conditions. This requirement ensures that points which start out close don’t stay that way forever. You can think of functions which are sensitive to initial conditions as those maps which mix points up. Sensitive functions are not very tolerant of approximation. They hate playing horse-shoes, for example. And they’re very hard to plot on computers due to rounding errors. Even though we might be very accurate, a chaotic function will churn the points about so wildly that we cannot guarantee that anything we learn about one point will shed any insight on the whereabouts its neighbor.
  • So far, we require chaotic maps to be, on the one hand, very orderly—by way of a multitude of periodic points—, and simulatenously jumbled, on the other hand—in an indirect way, through its sensitivity. In a modern treatment, we could stop here. But to really drive things home, let’s add in a third requirement.

  • The function is topologically transitive. Topological transitivity is a mathematician’s way of saying that the function meanders. Pick any two points A and B. If the function is topologically transitive, then I can find another point as close to A as you want that eventually makes its way as close to B as you want.
  • One of the simplest examples of a chaotic maps is the so-called logistic map. It’s a quadratic (there are squares in there):

    xn+1= xn (1-xn)/2.

    Its continuous conterpart crops up in population dynamics as one of the simplest, foundational models. Some examples within the logistic equation’s domain include colonies of bacteria, blades of grass on a lawn, or frogs in a pond with no predators to eat them.

    So, as long ago as 1637, Descartes noticed, very carelessly, that order can breed chaos. Let this be a warning to those of you neat-freaks who work tirelessly to assure everything is in its place. Also, Descartes might argue that cities provide evidence against the precepts of Intelligent Design. I know that’s how I read it.

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Assessing Assessments

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.

Role Models and Welfare

On my way into Town last night, I turned on my favorite NPR affiliate WBUR to hear what was going on in the world. I caught the tail end of After Welfare, a radio documentary by the American RadioWorks on the 1996 federal welfare reform legislation which ceded funding to the states and some of its subsequent effects. The piece closed with a very interesting focus on marriage. Evidently, the bill Clinton signed into law has in it some very specific wording that promotes low-income marriages. The idea runs something like this: two low incomes can provide for a child better than one. In Oklahoma, just over two million dollars pay for one of the more radical programs to result from the shift to the states. It is called the Oklahoma Marriage Initiative.

Aimed at low-income expectant parents, couples volunteer to complete a 12-hour course during which they learn, review, and discuss what it takes to stay in a long-term relationship. I believe much of their time is devoted to ever important communication techniques. It’s hard to know what if any effect OMI and others programs like it will have. And we won’t know for years, but it’s worth trying, I suppose. Studies show that as a group single mothers hold some of the most conservative family values. They believe that being a mother is one of, if not the single most important thing a woman can do. They want a traditional, nuclear family, and the majority [in the study I can’t remember below] oppose abortion.

While you may not be suprised to learn that even poorer people don’t want to sabotage their own lives, many critics of the 1996 law were afraid that low income women would have more and more kids in order to up their monthly check from the state at the expense of tax-payers and their hypothetical children. Some ground-breaking research, which I can’t name off the top of my head, in which about 160 single, low-income mothers were interviewed, shows that these women didn’t get married not because they somehow lack morals and values—as others might suggest—but because they revere the institution of marriage as holy. They’re holding out for someone who can provide a stable, healthy environment for them and their kids. The only difference, it seems, between them and their middle- and upper-class counterparts is resources.

Professor Skip Gates of Harvard’s Afro-American studies department recently produced a several part PBS documentary on blacks in America. He found that many boys in impoverished areas grew up to do what their role models did: sell drugs and go to jail. But why? Because they didn’t know what else to do. Why go to school and learn things that might be useful years from now and make no money in the interim when you could sell some drugs and make a few thousand dollars in a few days? The problem of immediate gratification is ruining large portions of society. The sort of education we need here is of the utmost personal kind. It is important that children, as President Bush says, be exposed to as many possibilities as, uh, possible. If a parent tells a child that he can be whatever he wants to be when he grows up, the statement has very little empowering effect if the child can’t think of things to be.

So when I say that these women’s middle-class counterparts have more resources, I intend more than material means; I’m also talking about psychology and education.

If these women believe that motherhood is the highest form of success they can acheive, it’s no wonder that most low-income babies, while perhaps not planned, are purposefully not prevented. Among other things, we need to get more and different kinds role models and mentors to work especially within low-income populations.

Even when presented with alternatives, it’s easy to believe that you’re born into your part in society, that lots are cast. In America, parents reinforce this misconception all the time. When interviewed, American mothers will list innate ability as the single most important factor in determining a person’s long-term success. Chinese and Japanese mothers, on the other hand, choose effort and persistance. As a result, American children can easily believe that those things which come easy to them are the things that are meant for them, and the stuff that’s hard isn’t. Again, pretty unsuprisingly sociologists suspect that one reason kids join gangs is a thirst for immediate gratification. Gangs will get you where you want to be fast.

And that’s why good math education is so crucial. (I could see you waiting for it, so I won’t disappoint.) Math is the sort of subject that requires lots of forethought and whose reward is delayed gratification. Of course good mathematical training won’t cure all of society’s ills, but [because this post is already long I’ll keep this brief and end abruptly claiming wildly that] the psychology of mathematics couldn’t hurt.

Probability Follow-Up

After writing my last post, where I casually mentioned that probability and statistics are probably the single most important topics left out of the current standard math curriculum, I read the most recent entry at Cognitive Daily. They’re concerned that people don’t know probability, too. For an alarming example, they turn to a recent study on doctor-patient relationships.

In a course on noise and data analysis conducted in the astronomy department, we had to tackle problems like:

One test for a deadly, rare disease is 97% accurate. The disease is genetic and only about 1% of the population has it. The only treatment for this disease is expensive, but effective if you actually have the disease and potentially deadly if not you don’t. What would you do if your test results came back positive?

The correct answer, of course, is to take the test again. Even though the test is pretty accurate, because almost no one has the disease, the inaccuracy of the test will be responsible for most of the postive readings; false positives will far outnumber positives reported because disease is actually present. But because of the high accuracy of the test, the likelihood that you will get several false positives is low. The more times you test positive, the better the chances are you actually have the disease. And with the scenario I’ve presented, it’s worth your time to take the test again.

We also did astronomy in this class, too.

A Revelation.

Disclaimer: This post will be about as funny as the last and even more self-indulgent.

I can’t sleep. But moments ago I nearly nodded off. During that in-between-dream-and-wakeful-worlds time I suddenly simplified a solution to my blockmate Verena’s QFT homework. She had to verify some result about the Lorentz group — at least I thought I had. So I jumped up, ran to the computer, and wrote a technically inclined email to her about SL(2,C), a group of manifold importance to mathematicians and physicists alike. Satisfied, I went back to bed only to realize, again while nodding off, that I was wrong. What I wrote applied not to the special linear group but to the projective special linear group, PSL(2,C).

Neither of those emails contained the revelation mentioned in the title. No, it is this: if I think about this stuff to ease myself to sleep, maybe I do, indeed, want to be a mathematician when I grow up, at least for a little bit. Anyway, now you and I both have something to contemplate before bed.