This semester I have the privilege/responsibility of teaching a class whose students, all very intelligent and highly motivated, are trying to get a jump start on higher level mathematics. The course is advertised as Multivariable Calculus "honors" course. I hesitate to even call it a course in multivariable calculus, as we won't cover any of that material until the second half of the course. The first half of the course will be laying a very theoretical groundwork for what students in other classes are already learning. Our class will have the advantage of seeing these concepts the "right" way--understanding the derivative in terms of linear functions, understanding functions in a rigorous way, and understanding vectors and vector spaces abstractly.
This week I am focusing on function theory and set theory. That means proofs--and lots of them. One of my students in office hours was complaining as much. (My office hours, by the way, had to be held in the common room in the math building, because apparently my office must now be under construction until an unspecified time. There was no notice of this occurrence before it started, and those of us who occupy said office still have no idea when it will end. Also, today I went in my office to see if it was finished, and I noticed that someone had messed with my Rubick's cube! All of the stickers had been taken off! Who messed with my stuff, anyway? But I digress...)
So, proofs... Yes, entering the world of mathematical rigor is not easy for most students. I have to be honest, it was pretty easy for me when I first started seeing proofs in college. Part of that was that I had a proof-based course in high school through the EPGY. Another part of it is that I've always been a naturally skeptical person, and very particular about the meaning of statements. For instance, once in fourth grade I actually battled my entire class, including my teacher, over the statement, "A square is a rectangle, but a rectangle is not a square." Obviously, the correct statement would be, "Not all rectangles are squares," but the subtlety was lost on my peers, and, yes, even my teacher, as far as I can tell to this day.
Proof writing is a beautiful and delicate art. The truly great mathematical writers are the ones who get beyond just the brute calculations and really explain what's going on. Often people ask me how one actually does research in mathematics. The answer is the same as in any other field; you come up with a new idea and you present it. This is done on a scale from acceptable to great. Acceptable mathematics presents true statements and sufficient details to prove those statements. Great mathematics presents not only true statements, but powerful ones; and it doesn't just prove the results, it also expounds the key ideas that make the results true. In other words, great mathematics is great writing, just as in any other field.
Showing posts with label teaching. Show all posts
Showing posts with label teaching. Show all posts
Monday, September 5, 2011
Tuesday, May 10, 2011
Grading on a curve
Overheard from a professor in the math building: "What does this mean, 'grade on a curve?' What are they saying?"
I like math professors. Especially foreign ones.
I like math professors. Especially foreign ones.
Friday, February 11, 2011
Abstract thought in applications
Sitting here in Para Coffee grading my students' first exam of the semester. It's not going so well, or at least not as well as I'd hoped. I feel like the last time I taught this course my students were a little more prepared for calculus. I guess what I have to remember is that this is spring term--a chance for those who really hate math to get this class out of the way. Oh boy.
As I grade, I'm realizing how silly it is when we break things down into "abstract" and "concrete." In math, solving concrete problems takes abstract thought--that's the whole point of even learning this stuff. You need to build something with a given amount of material and optimize the amount of space it contains. You need to maximize the speed of your vehicle. Whatever. The point is that if you just attacked the problem with no abstract thought whatsoever, you'd be stuck. If you can't imagine the problem as anything other than what it looks like on the surface, you can't solve it.
The real distinction, in my opinion, is between abstract and habitual. Students hate word problems. Why? Because they don't always look the same as every other problem they've ever done. They actually have to approach the problem on its own terms. That's frustrating, and students commonly express the opinion that their teacher "never gave us anything like this before." Students can do habitual assignments pretty well. It's much harder to get them to actually solve a problem. That's why they'd prefer to just take a bunch of meaningless limits and crank out a number or a "DNE" (does not exist) than solve real problems that someone in real life might actually ask. I worry about the implications for their future occupations. Will they be stuck just doing the same task over and over again, simply because that's easier? I suppose there's nothing wrong with that. It's just that you'd think a UVA student might imagine something more. I don't know.
The problem is that students have a hard time seeing the heart of a problem. They don't want to sit down and think about principles. It's so easy to treat everything mechanically. Yet it's so much easier to actually solve a problem by first figuring out what in the world they're looking for. And yes, that means abstractly. For instance, if you're looking for area, write down area = length times width. This is a thoroughly abstract statement, but it gives you a guide to solving the problem. You can't approach a concrete problem and solve it without identifying the abstract principles governing it. Indeed, one's knowledge of the abstract is tested best in application questions.
That's it for now--just a short rant on the difficulties of teaching math. I admit I can't help like feeling that my students' failure is my failure, even though that's absurd. I'm just some grad student the school throws up in front of these kids for less than three hours a week. Most of what these students need to know, there's no way I can teach them in three weeks before their first exam.
As I grade, I'm realizing how silly it is when we break things down into "abstract" and "concrete." In math, solving concrete problems takes abstract thought--that's the whole point of even learning this stuff. You need to build something with a given amount of material and optimize the amount of space it contains. You need to maximize the speed of your vehicle. Whatever. The point is that if you just attacked the problem with no abstract thought whatsoever, you'd be stuck. If you can't imagine the problem as anything other than what it looks like on the surface, you can't solve it.
The real distinction, in my opinion, is between abstract and habitual. Students hate word problems. Why? Because they don't always look the same as every other problem they've ever done. They actually have to approach the problem on its own terms. That's frustrating, and students commonly express the opinion that their teacher "never gave us anything like this before." Students can do habitual assignments pretty well. It's much harder to get them to actually solve a problem. That's why they'd prefer to just take a bunch of meaningless limits and crank out a number or a "DNE" (does not exist) than solve real problems that someone in real life might actually ask. I worry about the implications for their future occupations. Will they be stuck just doing the same task over and over again, simply because that's easier? I suppose there's nothing wrong with that. It's just that you'd think a UVA student might imagine something more. I don't know.
The problem is that students have a hard time seeing the heart of a problem. They don't want to sit down and think about principles. It's so easy to treat everything mechanically. Yet it's so much easier to actually solve a problem by first figuring out what in the world they're looking for. And yes, that means abstractly. For instance, if you're looking for area, write down area = length times width. This is a thoroughly abstract statement, but it gives you a guide to solving the problem. You can't approach a concrete problem and solve it without identifying the abstract principles governing it. Indeed, one's knowledge of the abstract is tested best in application questions.
That's it for now--just a short rant on the difficulties of teaching math. I admit I can't help like feeling that my students' failure is my failure, even though that's absurd. I'm just some grad student the school throws up in front of these kids for less than three hours a week. Most of what these students need to know, there's no way I can teach them in three weeks before their first exam.
Monday, October 25, 2010
Concerning philosophy and one year olds
rationalism -noun
1. the principle or habit of accepting reason as the supreme authority in matters of opinion, belief, or conduct.
The life of a graduate student, in my experience, is typically spent around peers, around professors, or in solitude. Marriage is common enough in graduate school, but I would venture to guess that the majority of us aren't married. Parents make up a much smaller percentage of graduate students. In this environment it is taken as a truism that having children hinders progress in our studies. (I have heard stunning exceptions to this. One student in religious studies once told me that his advisor recommends having children as a graduate student--then suddenly you will be unable to waste time.)
Well, you'll get no argument from me on the practical implications of having children--they take a lot of work. What I'm interested in is the effect of simply not being around children, of being immersed in a world of professionals. Of course we could go on all day about how this affects our priorities, what it does to our personal development, and so on. But I'm really interested in the intellectual side of it. How does being absent from children all the time affect the way we think, about ourselves and about our beliefs?
Wednesday, October 6, 2010
What's wrong with math education...
see more funny videos
Another sign that our math education is based on mindless application of formulas rather than common sense...
P.S. The teacher is wrong here, not the student.
Saturday, September 18, 2010
Training skeptics
This semester my teaching experience has been much different than last year. Whereas last year I taught a "throw-away" calculus course designed to fulfill a requirement for certain majors (but not designed to prepare students for any mathematics beyond calculus), this semester I'm teaching an honors course in multivariable calculus specifically designed to train students for higher level mathematics. Hence one of the essential parts of the course is teaching students to write proofs.
There is a certain level at which young mathematicians regard "proofs" almost as a specific area of math. They've learned algebra, trigonometry, calculus, and now they're learning "proofs." If they continue on in mathematics, they can't continue to think this way for long. Proof is not a part of mathematics. Proof is mathematics.
On the list of ways in which we teach mathematics badly in American schools, I would certainly add that we teach students that writing is not a part of mathematics. Writing is part of everything. If you can't write it, you can't communicate it. Perhaps part of the problem is that most people (even those who teach mathematics in high school and below) don't even realize mathematics needs to be communicated. Why would you need to convince someone that something is true in mathematics? Hasn't it all been figured out already? I kid you not--I would not be surprised if half the people I meet don't realize there's still being research done in mathematics. As in there's still math we don't know!
Since we teach students to do math without actually writing sentences, we also implicitly teach them that math is nothing more than a series of arcane symbolic manipulations that magically results in an answer. Many students will simply never grasp the idea that mathematics is really a set of questions that real people have asked out of genuine curiosity, and have answered purely through deductive reasoning. No magic necessary.
My students are a little sharper than that, but many of them still have a rough transition to make. They know how to get the answer, but they're totally new to actually writing mathematics. Thus if they were asked to prove that if x is a number other than 1, then 1 + x + x^2 + ... + x^n = (x^{n+1} - 1)/(x-1), they might give me an argument that looks like this:
writing a check mark beside the last line. It seems like a legitimate way to argue, because the symbolic manipulations are exactly what they've been trained to do all through their previous math education, particularly when they're trying to solve equations. And the symbolic manipulations generate something that's true! So it must be right.
There is a sense in which this argument is correct. If the student were to indicate that each line is logically equivalent to the line before it, then I suppose the argument would work. But it still wouldn't feel like good style. Everyone who is trained in higher mathematics (or in logic) understands this basic fact about a good argument: you don't start with what you're trying to prove. You start with your hypothesis, then argue step by step to the conclusion.
A reasonable argument of the proposition I just mentioned might simply reverse the lines of the poorer argument I gave, but that would be rather clunky. A good argument would actually use words! It would be much easier and clearer to simply write the following:
Here's what I tell my students: try to write as if you're trying to convince the most skeptical person in the world. Every line you write has to be 100% convincing. Starting with what you're trying to prove will never satisfy a skeptic, because a skeptic knows that you can get anything to be true if you just start by assuming it to be true. For instance, let's say I want to prove that -1 = 1. Well, my argument would be rather short and sweet:
Since what I got in the end is clearly true, the argument must work, right? Well, of course not, because -1 does not equal 1. But that's basically how students will argue when they're first starting to write proofs in mathematics. They've never had to convince a skeptic before. All they had to do was convince their teachers, who really just wanted to see a bunch of symbolic manipulations that magically resulted in an answer. In other words, all they had to do before was computations. Now they actually have to write arguments.
I have a great deal of uneasiness about the whole process of teaching good mathematical writing. I can tell them all day what's wrong with their proofs, and they might write little notes to themselves to try to figure out what I want to see on their homework. But that's the last thing I want them to learn. I don't want them to be able to convince me. I want them to be able to convince anyone who understands the symbols being used. There's no precise way to say this, but what I'm really going for is that they would be able to convince reason itself. Somehow they have to transcend the personal motives of finding acceptance from their teacher and getting good grades. They have to develop an innate desire to critique their own arguments, and write something about which they can confidently say, This is simply irrefutable.
Of course, mathematics is never really exactly like that in the real world of research. Mistakes are made. In the rush to get results published, sometimes mathematicians have indeed overlooked important details. But members of the mathematical community are constantly attempting to hold each other to that rigorous standard of irrefutable proof. This principle is emulated in all the sciences, yet mathematics has the privilege of working by pure logic. There is no "methods" section of a research paper in mathematics.
What an interesting sort of community this is, though. We're not a community of experimentalists, who argue over the analysis of data. We're not a community of theologians or philosophers, who argue over the meanings of words or doctrines. In fact, arguments between mathematicians are rarely over the actual content of what they study. We can argue over what the best method of proof is (though such arguments can be absurd), or we can argue over what we think the solution to a problem will be (though such arguments are rendered somewhat meaningless once the problem is actually solved). (There are also plenty of arguments between mathematicians mainly concerning their own egos; in that our community is not at all unique.)
If all the theologians or philosophers in the world were asked to reach a conclusive solution to a problem in one of these disciplines, they would only succeed in dividing into several camps. Yet do the same thing for all the mathematicians in the world, and all of them, working entirely independently from one another, can still reach the same solution. There will be no disagreement on the final result. Of all the other academic disciplines, the scientific community is most similar to this; yet the scientific community routinely has to correct past theories.
Does that make the mathematics community somehow ideal, which the whole world ought to emulate? Not at all. Mathematics progresses through skepticism; society in general does not. Part of skepticism is the ability to say, "I don't know." In the case of mathematics, we can say that for quite a long time--many famous problems have taken centuries to solve. It would be absurd to advocate the kind of rigor that mathematics demands as a way to unite all people. There are plenty of decisions in life that need to be made now. There are plenty of questions of profound importance which demand incomplete answers. I believe it was an insight of Friedrich Hayek that if we only accepted what we could justify on purely rational grounds, no human institutions could exist.
In short, mathematics is a luxury sport (like everything academic). That's not to say one must come from a wealthy family to do it. But it is certainly something that society could never have developed if it had never moved beyond the point of just doing enough to survive. More than that, it fundamentally arises out of a certain kind of dissatisfaction with mere acquaintance with the world around us. Mathematics can be characterized by a strange and relentless longing to see the inherent logical connections between ideas.
That is basically what I'm trying to train my students into. It should not be surprising if some of them never get it. If this innate desire is in you, you don't really need your teacher to justify it. On the other hand, if this desire isn't in you, then I wonder if you can ever really understand more than what it is that your teacher wants to see on your homework.
The point is that if a student wants to really do mathematics, he has to be skeptical of himself, first of all. He has to understand that it doesn't really matter whether or not he has the basic gist of a problem. He has to learn to expect a kind of precision of himself that can only be based on a relentless desire to see clearly the logical connections between ideas. If one is genuine about this, it really demands a great deal of humility. It demands the ability to say, I don't know until I have truly seen. And when one has seen clearly, then one has to humbly accept that there is no denying what one has seen. Yet while this is a humbling experience, it is simultaneously empowering, as it grants the ability to prove irrefutable claims.
Even as I teach students this, I am still learning it myself. It is an endless process. Yet I have to believe there is some inherent value in it. On the other hand, whether or not mathematics has value is hardly a mathematical question.
There is a certain level at which young mathematicians regard "proofs" almost as a specific area of math. They've learned algebra, trigonometry, calculus, and now they're learning "proofs." If they continue on in mathematics, they can't continue to think this way for long. Proof is not a part of mathematics. Proof is mathematics.
On the list of ways in which we teach mathematics badly in American schools, I would certainly add that we teach students that writing is not a part of mathematics. Writing is part of everything. If you can't write it, you can't communicate it. Perhaps part of the problem is that most people (even those who teach mathematics in high school and below) don't even realize mathematics needs to be communicated. Why would you need to convince someone that something is true in mathematics? Hasn't it all been figured out already? I kid you not--I would not be surprised if half the people I meet don't realize there's still being research done in mathematics. As in there's still math we don't know!
Since we teach students to do math without actually writing sentences, we also implicitly teach them that math is nothing more than a series of arcane symbolic manipulations that magically results in an answer. Many students will simply never grasp the idea that mathematics is really a set of questions that real people have asked out of genuine curiosity, and have answered purely through deductive reasoning. No magic necessary.
My students are a little sharper than that, but many of them still have a rough transition to make. They know how to get the answer, but they're totally new to actually writing mathematics. Thus if they were asked to prove that if x is a number other than 1, then 1 + x + x^2 + ... + x^n = (x^{n+1} - 1)/(x-1), they might give me an argument that looks like this:
1 + x + x^2 + ... + x^n = (x^{n+1} - 1)/(x-1)
(x-1)*(1 + x + x^2 + ... + x^n) = (x-1)*(x^{n+1} - 1)/(x-1)
(x + x^2 + ... + x^{n+1}) - (1 + x + x^2 + ... + x^n) = x^{n+1} - 1
x^{n+1} - 1 = x^{n+1} - 1
There is a sense in which this argument is correct. If the student were to indicate that each line is logically equivalent to the line before it, then I suppose the argument would work. But it still wouldn't feel like good style. Everyone who is trained in higher mathematics (or in logic) understands this basic fact about a good argument: you don't start with what you're trying to prove. You start with your hypothesis, then argue step by step to the conclusion.
A reasonable argument of the proposition I just mentioned might simply reverse the lines of the poorer argument I gave, but that would be rather clunky. A good argument would actually use words! It would be much easier and clearer to simply write the following:
Observe that (x-1)*(1 + x + x^2 + ... + x^n) = (x + x^2 + ... + x^{n+1}) - (1 + x + x^2 + ... + x^n) = x^{n+1} - 1. Now divide both sides of the equation by (x-1) to obtain the desired conclusion. QED
Here's what I tell my students: try to write as if you're trying to convince the most skeptical person in the world. Every line you write has to be 100% convincing. Starting with what you're trying to prove will never satisfy a skeptic, because a skeptic knows that you can get anything to be true if you just start by assuming it to be true. For instance, let's say I want to prove that -1 = 1. Well, my argument would be rather short and sweet:
-1 = 1
(-1)*(-1) = 1*1
1 = 1 (check mark!)
I have a great deal of uneasiness about the whole process of teaching good mathematical writing. I can tell them all day what's wrong with their proofs, and they might write little notes to themselves to try to figure out what I want to see on their homework. But that's the last thing I want them to learn. I don't want them to be able to convince me. I want them to be able to convince anyone who understands the symbols being used. There's no precise way to say this, but what I'm really going for is that they would be able to convince reason itself. Somehow they have to transcend the personal motives of finding acceptance from their teacher and getting good grades. They have to develop an innate desire to critique their own arguments, and write something about which they can confidently say, This is simply irrefutable.
Of course, mathematics is never really exactly like that in the real world of research. Mistakes are made. In the rush to get results published, sometimes mathematicians have indeed overlooked important details. But members of the mathematical community are constantly attempting to hold each other to that rigorous standard of irrefutable proof. This principle is emulated in all the sciences, yet mathematics has the privilege of working by pure logic. There is no "methods" section of a research paper in mathematics.
What an interesting sort of community this is, though. We're not a community of experimentalists, who argue over the analysis of data. We're not a community of theologians or philosophers, who argue over the meanings of words or doctrines. In fact, arguments between mathematicians are rarely over the actual content of what they study. We can argue over what the best method of proof is (though such arguments can be absurd), or we can argue over what we think the solution to a problem will be (though such arguments are rendered somewhat meaningless once the problem is actually solved). (There are also plenty of arguments between mathematicians mainly concerning their own egos; in that our community is not at all unique.)
If all the theologians or philosophers in the world were asked to reach a conclusive solution to a problem in one of these disciplines, they would only succeed in dividing into several camps. Yet do the same thing for all the mathematicians in the world, and all of them, working entirely independently from one another, can still reach the same solution. There will be no disagreement on the final result. Of all the other academic disciplines, the scientific community is most similar to this; yet the scientific community routinely has to correct past theories.
Does that make the mathematics community somehow ideal, which the whole world ought to emulate? Not at all. Mathematics progresses through skepticism; society in general does not. Part of skepticism is the ability to say, "I don't know." In the case of mathematics, we can say that for quite a long time--many famous problems have taken centuries to solve. It would be absurd to advocate the kind of rigor that mathematics demands as a way to unite all people. There are plenty of decisions in life that need to be made now. There are plenty of questions of profound importance which demand incomplete answers. I believe it was an insight of Friedrich Hayek that if we only accepted what we could justify on purely rational grounds, no human institutions could exist.
In short, mathematics is a luxury sport (like everything academic). That's not to say one must come from a wealthy family to do it. But it is certainly something that society could never have developed if it had never moved beyond the point of just doing enough to survive. More than that, it fundamentally arises out of a certain kind of dissatisfaction with mere acquaintance with the world around us. Mathematics can be characterized by a strange and relentless longing to see the inherent logical connections between ideas.
That is basically what I'm trying to train my students into. It should not be surprising if some of them never get it. If this innate desire is in you, you don't really need your teacher to justify it. On the other hand, if this desire isn't in you, then I wonder if you can ever really understand more than what it is that your teacher wants to see on your homework.
The point is that if a student wants to really do mathematics, he has to be skeptical of himself, first of all. He has to understand that it doesn't really matter whether or not he has the basic gist of a problem. He has to learn to expect a kind of precision of himself that can only be based on a relentless desire to see clearly the logical connections between ideas. If one is genuine about this, it really demands a great deal of humility. It demands the ability to say, I don't know until I have truly seen. And when one has seen clearly, then one has to humbly accept that there is no denying what one has seen. Yet while this is a humbling experience, it is simultaneously empowering, as it grants the ability to prove irrefutable claims.
Even as I teach students this, I am still learning it myself. It is an endless process. Yet I have to believe there is some inherent value in it. On the other hand, whether or not mathematics has value is hardly a mathematical question.
Tuesday, September 7, 2010
It can't be true!
Yesterday I had the joy of sharing Cantor's famous diagonal argument, proving that the real numbers form an uncountable set. This semester, instead of teaching my own class, I get to be a TA for an honors Calculus III course, which goes through many advanced topics to prepare students for higher level mathematics. As a result, I get to share moments like these, in which students go from being calculators to being real mathematicians.
Mathematics finally means something to a student when he encounters a proof of something that he can't believe. I had at least one student express his inability to accept the diagonal argument after I gave it. Of course, the proof is irrefutable--it is logically sound, and despite the large number of attempts made every year by crank mathematicians, it can never be overturned.
That's part of the sheer beauty of mathematics. It shows how pure logic can yield surprising results. Why are we surprised by things that are logically irrefutable? This is not a mathematical question, but a question about the human spirit; yet in some respects it is one which only the mathematician can encounter. No one else can experience quite the same thrill, or frustration, at coming face to face with those results of pure logic that seem to break down every sense of intuition you ever had.
Why does Cantor's proof frustrate students so?
The result itself frustrates people in general, at least in my experience. When you explain that the real numbers can never be counted, even if you counted them for all eternity, even if you go all the way to infinity, they do not believe you. Or they don't understand what you even mean. The young student of mathematics, on the other hand, can see and accept each line of the proof; yet much like any other person, he is intuitively troubled by the result itself. This manifests itself in the form of complaints about the proof; yet what it always seems to come down to is that the result is incomprehensible on any intuitive level.
Why is this? I don't really think that it is because the logical steps are hard to follow. Indeed, I think that even a person relatively untrained in mathematics can understand each step in the proof. What I think people can't really and truly deal with are the starting assumptions themselves. To embrace those assumptions takes imagination.
What is infinity, in the first place? To the extent that it touches upon our every day experience, infinity can actually be something quite small. The national debt, for instance, is for all practical purposes, infinite. No single human being can fathom having control of that much money. If I had $13 trillion in my bank account, I would simply never run out of money. I could spend $5000 every second for the next 80 years and still not run out of money.
Infinity, then, is a guarantee: there's always something left. The truth is, though, 13 trillion can be a very small number of some things. Avogadro's constant is somewhere on the order of a trillion trillions; yet this ridiculously large number is equal to the number of molecules in a measly 32 grams of oxygen. Numbers in this universe get so ridiculously large that our minds stop distinguishing between them all. (This has unfortunate political consequences, as people have yet to truly comprehend how astronomical their own governments' spending really is.)
So then, one who is willing to stretch out with the imagination can make this assumption: for every number n, there is always a number n + 1. You can always go, as Christopher Guest on Spinal Tap might put it, "one louder." We have formalized the guarantee of infinity. We have assumed the existence of an infinite set of numbers. One, two, three, four, five, ... ten, ..., twenty, ... thirty, ... one hundred, ... one thousand, ..., one million, ..., one trillion, ... why should it ever end?
But that isn't the only kind of guarantee one seeks in this world of computation. We also seek the guarantee of being really close to something. For instance, there is this ugly business of finding the circumference of a circle. A circle, of all things! A beautiful, simple shape--nothing could be simpler, really. And yet, the ratio between its circumference and its diameter is such a monster of a number that we are forced to give it a Greek name--"pi"--and leave it at that.
Or are we? We all heard in grade school that pi = 3.14... People sometimes wonder if I have all the digits of pi memorized, although some more sophisticated folks know that you can't memorize all of them but still wonder how many I know. Well I know up to about eight, I think: 3.1415926... And I've heard school children will sometimes have contests to see how many digits of pi they can memorize. Yet no matter how many digits they memorize, they will never have actually gotten pi. Never. Not in a million billion trillion years--not ever.
But they are getting closer and closer. How much closer is rather easy to quantify. If I guess pi = 3.14, then I am within one-hundredth of pi--that is, pi is between 3.14 and 3.15. If I guess pi = 3.1415926, then I am within one 10 millionth of pi. That's pretty close. Never exactly right, but closer and closer. So that's my guarantee. I can always get closer.
There are plenty of other common ratios in the real world which have no exact finite decimal representation--in fact, no repeating pattern ever emerges in their decimal expansions. The square root of 2 is one of them--this is simply the length across the diagonal of a square. The square roots of most numbers are the same way, and these can all be represented using common, everyday shapes. In each case, we do have a guarantee, as we did with pi. We can get closer and closer by taking more and more decimal places. That is, we can take the ratio between two ordinary natural numbers and be as close as we want to be to a weird number like pi.
Now what if I imagine that there is a number at the end of every conceivable decimal expansion? I say to myself, "Look, I can just start typing numbers after a decimal point, and there's no stopping me." In fact, just like at me as I cough up numbers right now:
.011923985461928384093745601092983562390523984691348612340213901098234908123984...
Try it! It's kind of therapeutic, actually.
Now I stretch out my imagination and put my faith in a new assumption: that this process of picking new digits can continue on forever and ever, and that no matter how the digits are picked at each step, the result can rightfully be called a number. I don't know what that number is, but I know that I'm getting closer and closer to it with every arbitrary choice of a new digit. Just as it took imagination to take on the assumption that there is always one more number, so it also takes imagination to embrace this new assumption. It means formalizing a guarantee.
It is, in a sense, an act of faith. Implicitly it means trusting that guarantee to have some sort of meaning. Otherwise, what would be the point of studying the logical results of that guarantee?
If you've gone with me this far, if you have enough faith to believe that after every number n there's always n+1 and that any decimal that can be continued on forever should be considered a number, then congratulations! You have, more or less, just embraced what mathematicians call the real numbers. If not, don't worry. Chances are your world can do without such big numbers. Even the national debt is probably higher than you'll ever need to imagine.
Young students of mathematics have generally been unwittingly indoctrinated into having faith in the real numbers. Duh, there's always a bigger number, and of course every decimal is a number. Why would we have to "imagine" that? Why, indeed! The ancient Greeks didn't even believe in zero.
Our educational system teaches these principles from a very young age. Enter pi in on your calculator. See? A decimal comes up! And those digits can keep going and going... Thus the youth are catechized into the traditions of their elders, unaware that without imagination, none of the structure in their mathematical universe could ever have arisen.
No wonder young mathematicians are so shaken by Cantor's proof! Indoctrinated into the assumptions which the proof begins with, these young minds are totally unprepared to handle the consequences of those assumptions. For as soon as the mind willingly submits to the two assumptions I have fleshed out here, then simple logic reveals a truth that is so staggering that it has drawn downright hostility from philosophers and logicians ever since Cantor first made his argument.
The claim is simple. Take all of those counting numbers: 1, 2, 3, 4, 5, 6, 7, ...
And I mean all of them, all infinitely many of them, because you know there is always one more! And now to each one of those counting numbers, assign some decimal, like this:
Here's what happens: you'll never get all of the decimals!
Never! Never ever! No matter how cleverly you chose which decimals to match to each of your counting numbers! Even though you always have one more counting number--that is, even though there are infinitely many counting numbers--you still don't have enough. The decimals are to the counting numbers what the national debt is to your savings account.
And the argument is quite simple: just read down the diagonal of that list you just made. Take the first digit of the first number, change it to another digit, and write it down. Take the second digit of the second number, change it, and it write it down next to the first one you just wrote. Do the same with the third, and the fourth, and so on. You'll get a new decimal. Mine would start to look like this:
Now, is this new number in your list? No! It can't be. It's not the same as your first number, because the first digit is different. It's not the same as your second number, because the second digit is different. It's not the same as your third number, because the third digit is different. And so on, even for every single counting number.
That's Cantor's proof. As shocking as the result is, the proof is nothing more than simple logic. Yet logic has to build on certain assumptions, and it's really those assumptions that set up this amazing result. It must have been that Cantor was the first person to fully buy into all of those assumptions. He was a mathematician of true faith, and true imagination.
And also true bravery. When I say that Cantor fully embraced those assumptions about numbers, I mean that he was even willing to embrace the logical consequences of those assumptions. That is faith. And without it, there can be no progress.
I'm allowed to say such things, because it's my blog. But I seriously wonder, how many assumptions do we take for granted, yet without being willing to accept their logical consequences? It is often only when someone shows you what the logical consequences are that you're able to see what the assumptions actually mean.
This, to me, is what's so liberating about mathematics. Whatever faith you have in your assumptions will be thrown to the fire to be tested. You must seek out the logical consequences of whatever you start with. And if you are able to overcome your initial fear, you might just find that the universe is a much grander, more majestic, and more mysterious place than you had ever imagined.
Mathematics finally means something to a student when he encounters a proof of something that he can't believe. I had at least one student express his inability to accept the diagonal argument after I gave it. Of course, the proof is irrefutable--it is logically sound, and despite the large number of attempts made every year by crank mathematicians, it can never be overturned.
That's part of the sheer beauty of mathematics. It shows how pure logic can yield surprising results. Why are we surprised by things that are logically irrefutable? This is not a mathematical question, but a question about the human spirit; yet in some respects it is one which only the mathematician can encounter. No one else can experience quite the same thrill, or frustration, at coming face to face with those results of pure logic that seem to break down every sense of intuition you ever had.
Why does Cantor's proof frustrate students so?
The result itself frustrates people in general, at least in my experience. When you explain that the real numbers can never be counted, even if you counted them for all eternity, even if you go all the way to infinity, they do not believe you. Or they don't understand what you even mean. The young student of mathematics, on the other hand, can see and accept each line of the proof; yet much like any other person, he is intuitively troubled by the result itself. This manifests itself in the form of complaints about the proof; yet what it always seems to come down to is that the result is incomprehensible on any intuitive level.
Why is this? I don't really think that it is because the logical steps are hard to follow. Indeed, I think that even a person relatively untrained in mathematics can understand each step in the proof. What I think people can't really and truly deal with are the starting assumptions themselves. To embrace those assumptions takes imagination.
What is infinity, in the first place? To the extent that it touches upon our every day experience, infinity can actually be something quite small. The national debt, for instance, is for all practical purposes, infinite. No single human being can fathom having control of that much money. If I had $13 trillion in my bank account, I would simply never run out of money. I could spend $5000 every second for the next 80 years and still not run out of money.
Infinity, then, is a guarantee: there's always something left. The truth is, though, 13 trillion can be a very small number of some things. Avogadro's constant is somewhere on the order of a trillion trillions; yet this ridiculously large number is equal to the number of molecules in a measly 32 grams of oxygen. Numbers in this universe get so ridiculously large that our minds stop distinguishing between them all. (This has unfortunate political consequences, as people have yet to truly comprehend how astronomical their own governments' spending really is.)
So then, one who is willing to stretch out with the imagination can make this assumption: for every number n, there is always a number n + 1. You can always go, as Christopher Guest on Spinal Tap might put it, "one louder." We have formalized the guarantee of infinity. We have assumed the existence of an infinite set of numbers. One, two, three, four, five, ... ten, ..., twenty, ... thirty, ... one hundred, ... one thousand, ..., one million, ..., one trillion, ... why should it ever end?
But that isn't the only kind of guarantee one seeks in this world of computation. We also seek the guarantee of being really close to something. For instance, there is this ugly business of finding the circumference of a circle. A circle, of all things! A beautiful, simple shape--nothing could be simpler, really. And yet, the ratio between its circumference and its diameter is such a monster of a number that we are forced to give it a Greek name--"pi"--and leave it at that.
Or are we? We all heard in grade school that pi = 3.14... People sometimes wonder if I have all the digits of pi memorized, although some more sophisticated folks know that you can't memorize all of them but still wonder how many I know. Well I know up to about eight, I think: 3.1415926... And I've heard school children will sometimes have contests to see how many digits of pi they can memorize. Yet no matter how many digits they memorize, they will never have actually gotten pi. Never. Not in a million billion trillion years--not ever.
But they are getting closer and closer. How much closer is rather easy to quantify. If I guess pi = 3.14, then I am within one-hundredth of pi--that is, pi is between 3.14 and 3.15. If I guess pi = 3.1415926, then I am within one 10 millionth of pi. That's pretty close. Never exactly right, but closer and closer. So that's my guarantee. I can always get closer.
There are plenty of other common ratios in the real world which have no exact finite decimal representation--in fact, no repeating pattern ever emerges in their decimal expansions. The square root of 2 is one of them--this is simply the length across the diagonal of a square. The square roots of most numbers are the same way, and these can all be represented using common, everyday shapes. In each case, we do have a guarantee, as we did with pi. We can get closer and closer by taking more and more decimal places. That is, we can take the ratio between two ordinary natural numbers and be as close as we want to be to a weird number like pi.
Now what if I imagine that there is a number at the end of every conceivable decimal expansion? I say to myself, "Look, I can just start typing numbers after a decimal point, and there's no stopping me." In fact, just like at me as I cough up numbers right now:
.011923985461928384093745601092983562390523984691348612340213901098234908123984...
Try it! It's kind of therapeutic, actually.
Now I stretch out my imagination and put my faith in a new assumption: that this process of picking new digits can continue on forever and ever, and that no matter how the digits are picked at each step, the result can rightfully be called a number. I don't know what that number is, but I know that I'm getting closer and closer to it with every arbitrary choice of a new digit. Just as it took imagination to take on the assumption that there is always one more number, so it also takes imagination to embrace this new assumption. It means formalizing a guarantee.
It is, in a sense, an act of faith. Implicitly it means trusting that guarantee to have some sort of meaning. Otherwise, what would be the point of studying the logical results of that guarantee?
If you've gone with me this far, if you have enough faith to believe that after every number n there's always n+1 and that any decimal that can be continued on forever should be considered a number, then congratulations! You have, more or less, just embraced what mathematicians call the real numbers. If not, don't worry. Chances are your world can do without such big numbers. Even the national debt is probably higher than you'll ever need to imagine.
Young students of mathematics have generally been unwittingly indoctrinated into having faith in the real numbers. Duh, there's always a bigger number, and of course every decimal is a number. Why would we have to "imagine" that? Why, indeed! The ancient Greeks didn't even believe in zero.
Our educational system teaches these principles from a very young age. Enter pi in on your calculator. See? A decimal comes up! And those digits can keep going and going... Thus the youth are catechized into the traditions of their elders, unaware that without imagination, none of the structure in their mathematical universe could ever have arisen.
No wonder young mathematicians are so shaken by Cantor's proof! Indoctrinated into the assumptions which the proof begins with, these young minds are totally unprepared to handle the consequences of those assumptions. For as soon as the mind willingly submits to the two assumptions I have fleshed out here, then simple logic reveals a truth that is so staggering that it has drawn downright hostility from philosophers and logicians ever since Cantor first made his argument.
The claim is simple. Take all of those counting numbers: 1, 2, 3, 4, 5, 6, 7, ...
And I mean all of them, all infinitely many of them, because you know there is always one more! And now to each one of those counting numbers, assign some decimal, like this:
1: 0.123846130865861902034109826394384...and so on. Except, don't just do it how I just did it--do it however you want! Be completely arbitrary!
2: 0.988349102039136358923403948190234...
3: 0.238463985658654865435864238653428...
4: 0.843565643854843643874398340954893...
5: 0.458430938230483240893291293048239...
6: 0.789234923902309238942394823094872...
7: 0.097138428094217089340873078320683...
...
Here's what happens: you'll never get all of the decimals!
Never! Never ever! No matter how cleverly you chose which decimals to match to each of your counting numbers! Even though you always have one more counting number--that is, even though there are infinitely many counting numbers--you still don't have enough. The decimals are to the counting numbers what the national debt is to your savings account.
And the argument is quite simple: just read down the diagonal of that list you just made. Take the first digit of the first number, change it to another digit, and write it down. Take the second digit of the second number, change it, and it write it down next to the first one you just wrote. Do the same with the third, and the fourth, and so on. You'll get a new decimal. Mine would start to look like this:
0.2996455...All I did was add 1 to each of the "diagonal elements" of my list. You can't add 1 to a 9, but you can just change 9 to 8, and the same idea holds.
Now, is this new number in your list? No! It can't be. It's not the same as your first number, because the first digit is different. It's not the same as your second number, because the second digit is different. It's not the same as your third number, because the third digit is different. And so on, even for every single counting number.
That's Cantor's proof. As shocking as the result is, the proof is nothing more than simple logic. Yet logic has to build on certain assumptions, and it's really those assumptions that set up this amazing result. It must have been that Cantor was the first person to fully buy into all of those assumptions. He was a mathematician of true faith, and true imagination.
And also true bravery. When I say that Cantor fully embraced those assumptions about numbers, I mean that he was even willing to embrace the logical consequences of those assumptions. That is faith. And without it, there can be no progress.
I'm allowed to say such things, because it's my blog. But I seriously wonder, how many assumptions do we take for granted, yet without being willing to accept their logical consequences? It is often only when someone shows you what the logical consequences are that you're able to see what the assumptions actually mean.
This, to me, is what's so liberating about mathematics. Whatever faith you have in your assumptions will be thrown to the fire to be tested. You must seek out the logical consequences of whatever you start with. And if you are able to overcome your initial fear, you might just find that the universe is a much grander, more majestic, and more mysterious place than you had ever imagined.
Thursday, November 12, 2009
Students
Next week my students have their last exam of the semester before the Final Exam. Our Applied Calculus I course at UVA is brilliantly structured so that the first two exams are before the Withdrawal Deadline, after which they are stuck in the class for the rest of the semester! So now they're stuck taking the hardest of the three exams, and there's no way out of it. Muahahahahaha...
I am having more and more people come to office hours lately. It's almost like they're just finding out I can talk to them outside of class! Some students find this out right from the beginning of the course, and they never let go. Others are shy, and couldn't possibly imagine taking any of my precious time outside of class. Both of these traits can be detrimental--the first because it means they're not thinking for themselves, and the second because it means they're not getting help.
On the other hand, my honest assessment is that there's basically no correlation between how often a student comes to office hours and how well they do in the class. Some students don't come to office hours because they don't need to, others don't come even though they should. Some students come to office hours because they're really motivated and want to get an A or an A+, while others come to office hours because they're really not getting it.
But the main thing I realized this week is that I'm actually getting personally attached. It happened when some students ask me whether I'd be teaching Applied Calculus II next semester, so they could sign up for my section. I had to say I honestly don't know. And then it hit me. I actually would like to see these students again. I kind of like teaching them.
It was so easy back when the department sent out the class request list to write down that I just wanted to teach the same course again--that way I don't have to write new lecture notes, I don't have to work so hard to develop a course curriculum, and I can focus on my research. But now I'm realizing, oh snap, I actually do care about something more than my own goals; I actually care about my students.
Since the department has yet to assign grad students to teaching sections, I technically don't know yet whether or not I might actually get to see some of my students next semester; but the reality is, I probably won't. It's weird just being a grad student and teaching. You go into it feeling like it's mostly just a job--like I was telling one of my students the other day, we all do it to pay the bills. What I didn't count on exactly was that it might actually mean more to me than that. I guess I didn't factor that into my equation.
And I thought I was good at math.
I am having more and more people come to office hours lately. It's almost like they're just finding out I can talk to them outside of class! Some students find this out right from the beginning of the course, and they never let go. Others are shy, and couldn't possibly imagine taking any of my precious time outside of class. Both of these traits can be detrimental--the first because it means they're not thinking for themselves, and the second because it means they're not getting help.
On the other hand, my honest assessment is that there's basically no correlation between how often a student comes to office hours and how well they do in the class. Some students don't come to office hours because they don't need to, others don't come even though they should. Some students come to office hours because they're really motivated and want to get an A or an A+, while others come to office hours because they're really not getting it.
But the main thing I realized this week is that I'm actually getting personally attached. It happened when some students ask me whether I'd be teaching Applied Calculus II next semester, so they could sign up for my section. I had to say I honestly don't know. And then it hit me. I actually would like to see these students again. I kind of like teaching them.
It was so easy back when the department sent out the class request list to write down that I just wanted to teach the same course again--that way I don't have to write new lecture notes, I don't have to work so hard to develop a course curriculum, and I can focus on my research. But now I'm realizing, oh snap, I actually do care about something more than my own goals; I actually care about my students.
Since the department has yet to assign grad students to teaching sections, I technically don't know yet whether or not I might actually get to see some of my students next semester; but the reality is, I probably won't. It's weird just being a grad student and teaching. You go into it feeling like it's mostly just a job--like I was telling one of my students the other day, we all do it to pay the bills. What I didn't count on exactly was that it might actually mean more to me than that. I guess I didn't factor that into my equation.
And I thought I was good at math.
Friday, September 25, 2009
Grading
This week has basically been consumed by teaching. I teach a section of Applied Calculus at UVA--it's my first year ever teaching a class. It's quite an experiment, really. Let's take a bunch of grad students, who have been admitted to the PhD program entirely because of academic merit in mathematics and not whatsoever on teaching merit, and throw them in front of some college freshmen (oh, I'm sorry, "first-years," as we say at UVA). Surely some learning will take place.
Tuesday was my students' first of three mid-term exams. I felt like it went well. There's nothing quite like grading an exam. Even in mathematics, it's highly subjective. The only time it isn't subjective is when the answer is entirely correct. I guess that does make it different from, say, grading a paper in philosophy. Is there ever an entirely correct answer in philosophy? This in itself is a philosophical question, of course... but the practical answer is, "no."
Subjectivity naturally comes into play, because for any given problem, a significant percentage of my students will have written something not 100% correct, and then I'm faced with the tricky problem of trying to measure how much understanding the answer communicates. How do you quantify understanding? Sometimes the whole process seems absurd.
I'm conditioned to expect a certain grade distribution, and I can kind of use this expectation to judge the validity of the grades I have come up with for my students. The truth is, though, I desperately try to find things that I can give credit for. Any signs of intelligent thought are rewarded with points.
Mathematics could be the most easily self-regulated discipline in the world. Every mathematician is highly critical by nature. One simply cannot succeed in mathematics without having this irrepressible urge to spot every important detail in a problem and verify it. Thus, if you tell any mathematician to grade a collection of calculus exams, his first inclination is simply to rip every answer to pieces.
Yet that same critical impulse generally causes me to turn inward and ask myself, what purpose would it serve to be hyper-critical of responses on a calculus exam? That combined with a genuine caring for my students as people causes me to try to find the best in every solution.
I never have to worry that I'm being overly generous. It is simply not physically possible for a mathematician to go above a certain threshold in giving credit for only partially correct solutions. The mistakes that appear in answers students give can range from humorous to painful, but in any case I just can't ignore them. No matter how much credit I want to give a student, I am forever bound by a mysterious force beyond my power to be more honest in my grading than perhaps my students would like me to be.
It's pretty brilliant, isn't it? The department never has to look over my shoulder to see if I'm actually being an ethical grader. They know that I am already constrained by a power far greater than any threats that bureaucracy might employ to keep me on track.
All of this is not to say that all of us grad students grade our students the same way. Sometimes we don't agree on what exactly the students are expected to understand about a problem. Math has this attractive quality to it because it is so objective, but setting learning goals for students (and then measuring success) is a more or less entirely subjective matter.
I could make it entirely objective, of course--each answer would be either right or wrong, no in between. But this probably would be the most unethical way to grade, because it would completely rob students of any way to communicate their ability to develop some sort of problem solving strategy, which is actually what we're trying to teach. Math, as it turns out, is not really about the answer.
When it comes right down to it, every academic discipline, math and the hard sciences included, are mostly about creating a dialog about something we wish to understand better. The reason we grade our students is to try to get them to make their ideas clearer, not just "right." There's something deeply personal about the whole thing, actually.
What's sad is how so many students go on asking what's going to be good enough--good enough for you, the teacher, to find them acceptable enough to label with an 'A' so they can go on and accomplish their own goals quite apart from anything you've told them. If only they realized that they'd really earn their 'A' if only they opened themselves up to actually communicating ideas.
I suppose it's really just the difference between being a consumer and thinking of your teacher as a supplier of a good, and being a person and thinking of your teacher as another person. If you're just a consumer, you can certainly get your 'A,' but only because our culture encourages grade inflation (education is subject to market forces, after all). If you really want to understand the material, you have to move from being a consumer to being a real person desiring real communication.
I understand why many students can't move this direction. I'm actually quite okay with most students in my class not caring about math. I'll do my best to be a supplier of a good--maybe if they work hard, I can even supply 'A's for them. There is, after all, a certain level of dignity in that kind of transaction.
But being a teacher, while also being a student, has caused me to reflect that what's really behind all those commands to "show your work" is a desire for something more personal. I wonder if my students will ever get that.
Tuesday was my students' first of three mid-term exams. I felt like it went well. There's nothing quite like grading an exam. Even in mathematics, it's highly subjective. The only time it isn't subjective is when the answer is entirely correct. I guess that does make it different from, say, grading a paper in philosophy. Is there ever an entirely correct answer in philosophy? This in itself is a philosophical question, of course... but the practical answer is, "no."
Subjectivity naturally comes into play, because for any given problem, a significant percentage of my students will have written something not 100% correct, and then I'm faced with the tricky problem of trying to measure how much understanding the answer communicates. How do you quantify understanding? Sometimes the whole process seems absurd.
I'm conditioned to expect a certain grade distribution, and I can kind of use this expectation to judge the validity of the grades I have come up with for my students. The truth is, though, I desperately try to find things that I can give credit for. Any signs of intelligent thought are rewarded with points.
Mathematics could be the most easily self-regulated discipline in the world. Every mathematician is highly critical by nature. One simply cannot succeed in mathematics without having this irrepressible urge to spot every important detail in a problem and verify it. Thus, if you tell any mathematician to grade a collection of calculus exams, his first inclination is simply to rip every answer to pieces.
Yet that same critical impulse generally causes me to turn inward and ask myself, what purpose would it serve to be hyper-critical of responses on a calculus exam? That combined with a genuine caring for my students as people causes me to try to find the best in every solution.
I never have to worry that I'm being overly generous. It is simply not physically possible for a mathematician to go above a certain threshold in giving credit for only partially correct solutions. The mistakes that appear in answers students give can range from humorous to painful, but in any case I just can't ignore them. No matter how much credit I want to give a student, I am forever bound by a mysterious force beyond my power to be more honest in my grading than perhaps my students would like me to be.
It's pretty brilliant, isn't it? The department never has to look over my shoulder to see if I'm actually being an ethical grader. They know that I am already constrained by a power far greater than any threats that bureaucracy might employ to keep me on track.
All of this is not to say that all of us grad students grade our students the same way. Sometimes we don't agree on what exactly the students are expected to understand about a problem. Math has this attractive quality to it because it is so objective, but setting learning goals for students (and then measuring success) is a more or less entirely subjective matter.
I could make it entirely objective, of course--each answer would be either right or wrong, no in between. But this probably would be the most unethical way to grade, because it would completely rob students of any way to communicate their ability to develop some sort of problem solving strategy, which is actually what we're trying to teach. Math, as it turns out, is not really about the answer.
When it comes right down to it, every academic discipline, math and the hard sciences included, are mostly about creating a dialog about something we wish to understand better. The reason we grade our students is to try to get them to make their ideas clearer, not just "right." There's something deeply personal about the whole thing, actually.
What's sad is how so many students go on asking what's going to be good enough--good enough for you, the teacher, to find them acceptable enough to label with an 'A' so they can go on and accomplish their own goals quite apart from anything you've told them. If only they realized that they'd really earn their 'A' if only they opened themselves up to actually communicating ideas.
I suppose it's really just the difference between being a consumer and thinking of your teacher as a supplier of a good, and being a person and thinking of your teacher as another person. If you're just a consumer, you can certainly get your 'A,' but only because our culture encourages grade inflation (education is subject to market forces, after all). If you really want to understand the material, you have to move from being a consumer to being a real person desiring real communication.
I understand why many students can't move this direction. I'm actually quite okay with most students in my class not caring about math. I'll do my best to be a supplier of a good--maybe if they work hard, I can even supply 'A's for them. There is, after all, a certain level of dignity in that kind of transaction.
But being a teacher, while also being a student, has caused me to reflect that what's really behind all those commands to "show your work" is a desire for something more personal. I wonder if my students will ever get that.
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