Showing posts with label math. Show all posts
Showing posts with label math. Show all posts
Wednesday, February 10, 2010
Approximating Functions of Shapes
Every now and then, I end up writing about my research at a not-too-technical level. Here's my latest attempt:
The branch of mathematics known as topology is concerned with the study of shapes. Whereas shapes in geometry are fairly rigid objects, shapes in topology are much more flexible. If one flexible shape can be flexed and twisted and not-too-drastically mangled into another flexible shape, topology deems them to be the same. It becomes much more difficult, then, to tell if two shapes are different. The question of topology is, ``Given two flexible shapes, how can we tell if they are the same?''
In geometry, much time is spent studying shapes like triangles and quadrilaterals. If you draw one triangle on one sheet of paper, and another triangle on another sheet of paper, then there is an easy way to tell if those two triangles are the same. Namely, shift the papers around, possibly flip one or the other over, rotate them, and see if you can get the triangles to line up. If you can, then the triangles are ``the same'' (congruent). If you can't line up the triangles (and have put forth enough effort), then the triangles are not the same. Similarly, you could draw a square on one paper, a triangle on the other, and try to line those shapes up. Of course, you will never succeed, because the two shapes will always have a different number of sides.
In topology, we might consider the same scenario of triangles and squares, but instead of using ordinary paper, we would draw our shapes on a sheet of flexible rubber (an inflatable balloon, say). Indeed, topology is often referred to as ``rubber-sheet geometry.'' Now that the surface is flexible, by pulling it in the appropriate directions one can get the square to look like a triangle. Or either shape to look like a circle. To a topologist, this means that all of these shapes are ``the same'' (homeomorphic). The topologist still has some rules in place, though. You are not allowed to cut holes in the balloon, or glue parts of the balloon together.
With all of the additional flexibility, we have a harder time telling two shapes apart. We can no longer rely on the number of sides, or the lengths of sides. Other methods must be found.
It turns out that a good way to get a sense of how similar two shapes are is to understand how the shapes are related to yet another shape. Suppose we have two flexible shapes, called Shape 1 and Shape 2. One might try to obtain information about Shape 1 in isolation, and compare that to information obtained about Shape 2 in isolation. However, as the shapes are flexible, there's only so much we can tell by looking at each shape independently. Instead, we introduce another flexible shape, Shape 3. Generally Shape 3 is a shape that has been much studied, like a circle. Now we ask, ``How many ways can Shape 3 be put into Shape 1?'' and, ``How many ways can Shape 3 be put into Shape 2?'' We call a single way of putting Shape 3 into Shape 1, if it is reasonably well-behaved, an ``embedding.'' If the embeddings of Shape 3 into either of the two original Shapes are similar, we might decide that the two shapes themselves are fairly similar. We can then replace Shape 3 with Shape 4, repeat the question, and get an even finer appreciation for how similar Shapes 1 and 2 are.
The entire collection of embeddings of Shape 3 into Shape 1 can all be gathered up and reasonably interpreted as making up yet another shape. However, it can be a fairly difficult shape to get a good sense of. Instead, one might ask if there is a way to approximate this shape. Perhaps an initial approximation can be attained, and then that approximation can be refined, and refined further, giving a better and better understanding of the complicated shape we originally sought. Using geometry as an example, one might approximate a circle by a square, and then refine the approximation to an octagon, and carry on the process of adding more sides to the approximating figure. This was the method used, historically, for calculating the area of a circle.
But the idea of approximating the shape of the collection of embeddings can be further generalized. If we have a strong handle on a particular shape, say S, then our first step in understanding any new shape that comes along might be to find all the ways S can be put into the new shape. We have, then, described a function. Given a shape, X, we can ask for the shape of ways that S can be embedded in X. We might denote this function by E(X), and call it the embedding function for S. Now instead of understanding a particular value, E(X), of this function, we might ask about understanding the function E as a whole. While the function itself is likely complicated, we can try to approximate it, as before.
In my research, I study a family of embeddings functions. The functions are based on the ``Euclidean spaces'' of various dimensions: a point, a line, a plane, three-dimensional space, and on to higher dimensions. From a topologist's perspective, these are the easiest shapes there are, but they can be used to build up nearly all of the other shapes worth studying, so are of fundamental importance. Instead of just thinking about one such space as the basis for my function, I might make a collection of them; say, 3 lines and 2 planes. I only ask about E(X) when X is, itself, a single Euclidean space.
It might seem that by restricting my study to particularly nice shapes S and X, and particularly nice ways to put S into X, I have eliminated any hope of doing anything useful. However, this is not the case. An important class of shapes, called manifolds, are built up from Euclidean spaces by gluing them together. By obtaining a particularly nice description of embedding functions using my restricted shapes S, there are known ways to assemble them into the embedding functions for more general manifolds.
Previous work has found nice ways to describe embedding functions for some of the cases I am considering. If the base space, S, is a finite collection of points, then spaces of embeddings are what are known as ``Configuration Spaces,'' and these detect one type of behavior for embeddings. Alternatively, if S has just a single space, but of any dimension, the spaces of embeddings are detecting a different sort of behavior. Nice descriptions for the embedding functions in both of these cases are known. It has been my task to unify the two descriptions, allowing for an understanding of embedding functions when the known space, S, is a collection of any number of Euclidean spaces.
Update 20100225: There's a somewhat different version of this on my other blog.
The branch of mathematics known as topology is concerned with the study of shapes. Whereas shapes in geometry are fairly rigid objects, shapes in topology are much more flexible. If one flexible shape can be flexed and twisted and not-too-drastically mangled into another flexible shape, topology deems them to be the same. It becomes much more difficult, then, to tell if two shapes are different. The question of topology is, ``Given two flexible shapes, how can we tell if they are the same?''
In geometry, much time is spent studying shapes like triangles and quadrilaterals. If you draw one triangle on one sheet of paper, and another triangle on another sheet of paper, then there is an easy way to tell if those two triangles are the same. Namely, shift the papers around, possibly flip one or the other over, rotate them, and see if you can get the triangles to line up. If you can, then the triangles are ``the same'' (congruent). If you can't line up the triangles (and have put forth enough effort), then the triangles are not the same. Similarly, you could draw a square on one paper, a triangle on the other, and try to line those shapes up. Of course, you will never succeed, because the two shapes will always have a different number of sides.
In topology, we might consider the same scenario of triangles and squares, but instead of using ordinary paper, we would draw our shapes on a sheet of flexible rubber (an inflatable balloon, say). Indeed, topology is often referred to as ``rubber-sheet geometry.'' Now that the surface is flexible, by pulling it in the appropriate directions one can get the square to look like a triangle. Or either shape to look like a circle. To a topologist, this means that all of these shapes are ``the same'' (homeomorphic). The topologist still has some rules in place, though. You are not allowed to cut holes in the balloon, or glue parts of the balloon together.
With all of the additional flexibility, we have a harder time telling two shapes apart. We can no longer rely on the number of sides, or the lengths of sides. Other methods must be found.
It turns out that a good way to get a sense of how similar two shapes are is to understand how the shapes are related to yet another shape. Suppose we have two flexible shapes, called Shape 1 and Shape 2. One might try to obtain information about Shape 1 in isolation, and compare that to information obtained about Shape 2 in isolation. However, as the shapes are flexible, there's only so much we can tell by looking at each shape independently. Instead, we introduce another flexible shape, Shape 3. Generally Shape 3 is a shape that has been much studied, like a circle. Now we ask, ``How many ways can Shape 3 be put into Shape 1?'' and, ``How many ways can Shape 3 be put into Shape 2?'' We call a single way of putting Shape 3 into Shape 1, if it is reasonably well-behaved, an ``embedding.'' If the embeddings of Shape 3 into either of the two original Shapes are similar, we might decide that the two shapes themselves are fairly similar. We can then replace Shape 3 with Shape 4, repeat the question, and get an even finer appreciation for how similar Shapes 1 and 2 are.
The entire collection of embeddings of Shape 3 into Shape 1 can all be gathered up and reasonably interpreted as making up yet another shape. However, it can be a fairly difficult shape to get a good sense of. Instead, one might ask if there is a way to approximate this shape. Perhaps an initial approximation can be attained, and then that approximation can be refined, and refined further, giving a better and better understanding of the complicated shape we originally sought. Using geometry as an example, one might approximate a circle by a square, and then refine the approximation to an octagon, and carry on the process of adding more sides to the approximating figure. This was the method used, historically, for calculating the area of a circle.
But the idea of approximating the shape of the collection of embeddings can be further generalized. If we have a strong handle on a particular shape, say S, then our first step in understanding any new shape that comes along might be to find all the ways S can be put into the new shape. We have, then, described a function. Given a shape, X, we can ask for the shape of ways that S can be embedded in X. We might denote this function by E(X), and call it the embedding function for S. Now instead of understanding a particular value, E(X), of this function, we might ask about understanding the function E as a whole. While the function itself is likely complicated, we can try to approximate it, as before.
In my research, I study a family of embeddings functions. The functions are based on the ``Euclidean spaces'' of various dimensions: a point, a line, a plane, three-dimensional space, and on to higher dimensions. From a topologist's perspective, these are the easiest shapes there are, but they can be used to build up nearly all of the other shapes worth studying, so are of fundamental importance. Instead of just thinking about one such space as the basis for my function, I might make a collection of them; say, 3 lines and 2 planes. I only ask about E(X) when X is, itself, a single Euclidean space.
It might seem that by restricting my study to particularly nice shapes S and X, and particularly nice ways to put S into X, I have eliminated any hope of doing anything useful. However, this is not the case. An important class of shapes, called manifolds, are built up from Euclidean spaces by gluing them together. By obtaining a particularly nice description of embedding functions using my restricted shapes S, there are known ways to assemble them into the embedding functions for more general manifolds.
Previous work has found nice ways to describe embedding functions for some of the cases I am considering. If the base space, S, is a finite collection of points, then spaces of embeddings are what are known as ``Configuration Spaces,'' and these detect one type of behavior for embeddings. Alternatively, if S has just a single space, but of any dimension, the spaces of embeddings are detecting a different sort of behavior. Nice descriptions for the embedding functions in both of these cases are known. It has been my task to unify the two descriptions, allowing for an understanding of embedding functions when the known space, S, is a collection of any number of Euclidean spaces.
Update 20100225: There's a somewhat different version of this on my other blog.
Friday, December 11, 2009
Changing Calculus
Apparently toward the end of every semester I decide to write a post about changing calculus. At least, this is my second go at it. But today's attempt is more a follow-up to my most recent post, about the role of higher education.
My initial motivation in asking the questions in my last post was to provide some structure for what a calculus class should look like. Is the role of calculus really to make sure that all freshman know how to compute integrals by hand, and can use this to find the volume and surface area of a solid of revolution? That rising second years can go through mindless algebraic manipulations to arrive at an answer to a question that was asked without any context or applicability? Moreover, that they need to be able to do so by hand? Quick, find the antiderivative of $\sqrt{3-2x-x^2},ドル no computers. Who gives a crap? How many times will that function, or one like it, actually show up in the lifetime of my student? I get paid to do math, and the only time I've ever cared about the answers to these algebraic questions was when I was taking or teaching a calculus course.
Looking back, I'm a little ashamed that the problem above was actually one I assigned as homework for my students this semester. What good does it do to have them work this integral by hand? Surely the answer depends on the students, right? My students going away from the sciences have gotten nothing out of this assignment. My students staying in the sciences have spent another... 5 minutes? 30?... practicing their algebra skills. Hurray‽
Look, I'm not saying algebra isn't useful. But I really do think it is over-emphasized in calculus courses (the ones I've encountered). My calc II course covered: techniques of integration, improper integrals, arc length and surface area of revolution, parametric curves, polar curves, multiple integrals, and infinite sequences and series. Lots of these are wonderfully fascinating and fun topics. Parametric and polar curves force students to start thinking about curves differently, to start re-interpreting functions and points in the plane. Infinite sequences and series are so unlike anything that's ever come before that they can't help but be interesting. Add up infinitely many things that get infinitely small, and produce an infinite value? Or a finite one? How the hell does that work? What does it even mean?
If not a single one of my students can do the integral mentioned above, I'm not sure that I'd feel bad about my semester. If not a significant portion of my students could tell me what was new about a parametric or polar curve, I think I'd be fairly upset (admittedly, now I'm a little afraid to ask).
My problem with my course is that algebraic manipulations are essentially given precedence over conceptual understanding. This is done by making exams that test algebraic nonsense, forcing instructors to make sure their students can do the algebraic nonsense, taking up valuable time that could be better used in other ways. With less time spent on algebra, more time could be spend on concepts, and more time could be spent on improving student's writing. The time spent on concepts makes it a more mathematically interesting and, I'd argue, worthwhile course. Whatever negative impact on my students is brought on by slightly less algebra practice will be more than compensated for by having them spend time thinking about ideas. More time spent teaching students how to write mathematics will force them to think more precisely about what they are saying and how to say it. How could this not be advantageous to students, whether they will be taking more math or not? Unless, of course, thinking about these ideas and how to write them is not aligned with the goals of my university. If the role of higher education isn't something that thinks this would be better, then I'm getting off this boat before it sinks.
So why not go for it? Spend drastically less time on techniques of integration and series convergence tests. Force students to turn in well-written assignments. Spend class time talking about what well-written math is. Spend time letting students play with computers to get a feel for parametric and polar curves. Computers will be faster and more accurate in plotting curves than any student ever will be. Give students time to plot lots of curves, changing parameters to see the effect on the graph. Make students look up resources online to learn how to find arc lengths or surface areas, instead of just telling them the formula (or even deriving it) and having them memorize it. Spend class time talking about how to read the resources they find, and how to evaluate them for quality.
My advisor pointed out, during our conversation mention in my last post, that there are some issues with trying new things. At the lowest level: who gets to teach such a course, and where does their funding come from? But I think better concerns that he mentioned are two that bring me back to some of my earlier questions.
These questions brought me back, full circle, to my initial desire for some formal guidelines for courses. I think it would be hasty to change all of our calculus courses around right away. I think a better solution involves some experimenting. Make a new course, make grad students apply to teach it, arguing for what they want to do differently, and require the instructor to report back on how it went. I think having formal guidelines for courses would aid the experimentation, because it gives an objective measure for comparing a newly designed course with a more traditional one.
I know that much of what I'm saying is oriented toward the way things are set up in my department. How is your department set up differently, and what does that mean about how calculus courses are structured?
Thoughts? Objections? (non-spammy) Links?
My initial motivation in asking the questions in my last post was to provide some structure for what a calculus class should look like. Is the role of calculus really to make sure that all freshman know how to compute integrals by hand, and can use this to find the volume and surface area of a solid of revolution? That rising second years can go through mindless algebraic manipulations to arrive at an answer to a question that was asked without any context or applicability? Moreover, that they need to be able to do so by hand? Quick, find the antiderivative of $\sqrt{3-2x-x^2},ドル no computers. Who gives a crap? How many times will that function, or one like it, actually show up in the lifetime of my student? I get paid to do math, and the only time I've ever cared about the answers to these algebraic questions was when I was taking or teaching a calculus course.
Looking back, I'm a little ashamed that the problem above was actually one I assigned as homework for my students this semester. What good does it do to have them work this integral by hand? Surely the answer depends on the students, right? My students going away from the sciences have gotten nothing out of this assignment. My students staying in the sciences have spent another... 5 minutes? 30?... practicing their algebra skills. Hurray‽
Look, I'm not saying algebra isn't useful. But I really do think it is over-emphasized in calculus courses (the ones I've encountered). My calc II course covered: techniques of integration, improper integrals, arc length and surface area of revolution, parametric curves, polar curves, multiple integrals, and infinite sequences and series. Lots of these are wonderfully fascinating and fun topics. Parametric and polar curves force students to start thinking about curves differently, to start re-interpreting functions and points in the plane. Infinite sequences and series are so unlike anything that's ever come before that they can't help but be interesting. Add up infinitely many things that get infinitely small, and produce an infinite value? Or a finite one? How the hell does that work? What does it even mean?
If not a single one of my students can do the integral mentioned above, I'm not sure that I'd feel bad about my semester. If not a significant portion of my students could tell me what was new about a parametric or polar curve, I think I'd be fairly upset (admittedly, now I'm a little afraid to ask).
My problem with my course is that algebraic manipulations are essentially given precedence over conceptual understanding. This is done by making exams that test algebraic nonsense, forcing instructors to make sure their students can do the algebraic nonsense, taking up valuable time that could be better used in other ways. With less time spent on algebra, more time could be spend on concepts, and more time could be spent on improving student's writing. The time spent on concepts makes it a more mathematically interesting and, I'd argue, worthwhile course. Whatever negative impact on my students is brought on by slightly less algebra practice will be more than compensated for by having them spend time thinking about ideas. More time spent teaching students how to write mathematics will force them to think more precisely about what they are saying and how to say it. How could this not be advantageous to students, whether they will be taking more math or not? Unless, of course, thinking about these ideas and how to write them is not aligned with the goals of my university. If the role of higher education isn't something that thinks this would be better, then I'm getting off this boat before it sinks.
So why not go for it? Spend drastically less time on techniques of integration and series convergence tests. Force students to turn in well-written assignments. Spend class time talking about what well-written math is. Spend time letting students play with computers to get a feel for parametric and polar curves. Computers will be faster and more accurate in plotting curves than any student ever will be. Give students time to plot lots of curves, changing parameters to see the effect on the graph. Make students look up resources online to learn how to find arc lengths or surface areas, instead of just telling them the formula (or even deriving it) and having them memorize it. Spend class time talking about how to read the resources they find, and how to evaluate them for quality.
My advisor pointed out, during our conversation mention in my last post, that there are some issues with trying new things. At the lowest level: who gets to teach such a course, and where does their funding come from? But I think better concerns that he mentioned are two that bring me back to some of my earlier questions.
- It is hard to test conceptual understanding. It is much easier to test if a student can do some algebra. With several hundred students going through calculus every semester, this is a practical concern. How does a university handle scale?
- Institutional inertia would have to be overcome. Change is hard. We do this because this is how we've done this for a while now,
(削除) and it is clearly the best way to do things (削除ここまで)(削除) and it's going ok (削除ここまで)(削除) and nobody has complained (削除ここまで).
These questions brought me back, full circle, to my initial desire for some formal guidelines for courses. I think it would be hasty to change all of our calculus courses around right away. I think a better solution involves some experimenting. Make a new course, make grad students apply to teach it, arguing for what they want to do differently, and require the instructor to report back on how it went. I think having formal guidelines for courses would aid the experimentation, because it gives an objective measure for comparing a newly designed course with a more traditional one.
I know that much of what I'm saying is oriented toward the way things are set up in my department. How is your department set up differently, and what does that mean about how calculus courses are structured?
Thoughts? Objections? (non-spammy) Links?
Wednesday, November 18, 2009
Grade Eachother?
I started doing daily homework assignments for my calculus class when we got to series convergence tests. The idea is that they will read the section in the book and work some problems to turn in, before we talk about it in class. It isn't going well.
Mostly students are "getting the right answer," but the write-ups are fairly bad (mostly making me feel like understanding is pretty low). I spent about 20-30 minutes in class Monday telling them about the things they were doing that are driving me crazy.
We've got three more of these daily assignments due over the next week. I'm thinking about doing the following: During class, have the students gather themselves up in pairs and have each student write their name on their partner's paper. The students will spend time looking for mistakes on their partner's paper. And I'll tell them that whatever points I take off of a paper will also be taken off of the partner's paper.
Surely somebody out there has done something like this before. How does it go? Does it help? What do I need to watch out for? I'm a little worried that "correcting" won't happen as much as "copy down what is hopefully a better answer, without understanding it".
Mostly students are "getting the right answer," but the write-ups are fairly bad (mostly making me feel like understanding is pretty low). I spent about 20-30 minutes in class Monday telling them about the things they were doing that are driving me crazy.
We've got three more of these daily assignments due over the next week. I'm thinking about doing the following: During class, have the students gather themselves up in pairs and have each student write their name on their partner's paper. The students will spend time looking for mistakes on their partner's paper. And I'll tell them that whatever points I take off of a paper will also be taken off of the partner's paper.
Surely somebody out there has done something like this before. How does it go? Does it help? What do I need to watch out for? I'm a little worried that "correcting" won't happen as much as "copy down what is hopefully a better answer, without understanding it".
Tuesday, November 10, 2009
Solid
If you are still following this blog, after two months with no posts followed by a rambling personal post, then I owe you something. The best I have is the following...
A few weeks ago in a seminar, I was sitting behind a professor and noticed that his t-shirt had a pretty cool picture of a solid shape on it. I drew the picture, and decided I would love to have a 3-d version. My original (and, now, long-term) goal was to make the solid represented in the picture by a piece of wood. Wandering around the craft store, I decided foam was a bit more feasible for me right now. So anyway... I'll just give you a link to the album I put all of the pictures in (click the picture for the album):
I also made a little video of me spinning the thing, to give an idea what it looks like from more angles.
[埋込みオブジェクト:http://www.youtube.com/v/Rynlkh4rIu4&hl=en&fs=1&]
This was my first attempt at this project, and my first attempt carving something out of foam. I'm honestly fairly pleased with the result. Certainly I see room for improvement though, and plan on trying again soonish. I've got two more blocks of foam still...
A few weeks ago in a seminar, I was sitting behind a professor and noticed that his t-shirt had a pretty cool picture of a solid shape on it. I drew the picture, and decided I would love to have a 3-d version. My original (and, now, long-term) goal was to make the solid represented in the picture by a piece of wood. Wandering around the craft store, I decided foam was a bit more feasible for me right now. So anyway... I'll just give you a link to the album I put all of the pictures in (click the picture for the album):
I also made a little video of me spinning the thing, to give an idea what it looks like from more angles.
[埋込みオブジェクト:http://www.youtube.com/v/Rynlkh4rIu4&hl=en&fs=1&]
This was my first attempt at this project, and my first attempt carving something out of foam. I'm honestly fairly pleased with the result. Certainly I see room for improvement though, and plan on trying again soonish. I've got two more blocks of foam still...
Wednesday, September 9, 2009
Ask Again
I've been going into my calculus classes to teach with fewer and fewer notes this semester. I've got a few examples in mind, but that's about it. And I really think it is going well. I think it helps that I started this way on the first day. I also seem to have a pretty good class - I've certainly been pleased so far. After today's class, and some others recently, I'm starting to think that posing the same question to my class a few times, either asking for a different solution, or just a differently-stated solution, is a good habit.
Last week we were talking about improper integrals in class. After some introductory discussion, I asked them to tell me which of the integrals, $\int_1^{\infty} \frac{e^{\sqrt{x}}}{\sqrt{x}}\ dx$ or $\int_1^{\infty} \frac{1}{\sqrt{x}e^{\sqrt{x}}\ dx$ could possibly converge, and why. The first answer I got was to actually compute the first integral and notice that it diverges, so that couldn't be it. A good answer, certainly, with an opportunity to talk about $u$-substitution with improper integrals. And then I asked again, for another reason. A student brought up the integral $\int_1^{\infty} \frac{1}{\sqrt{x}}\ dx,ドル which we already knew diverged, and this gave us the chance to talk about the comparison test. I think at this point we about ran out of time, but I know somewhere in there we also had a discussion about functions that approach 0 not necessarily having a convergent improper integral to infinity. We didn't get to L'Hospital's rule (hopefully a reminder, this being calc 2) that day, but did eventually, with this same example.
Today I was talking about parametric curves. I asked them to take a few minutes to try to sketch the parametric curve $(t^2-1,t^3),ドル and then had students come to the board to draw what they had. I was happy to get two different answers - something like a sideways parabola, and then the correct graph, with a cusp. I asked them how we could tell which could possibly be the correct graph, because they both went through the first few easy points, (0,-1), (-1,0), and (0,1). A student pointed out that $y$ was changing faster than $x,ドル so it shouldn't be the sideways parabola. I didn't want to talk about concavity just yet, but we did get to it eventually.
Then I asked basically the same question, drawing three different curves that weren't the sideways parabola. One had a cusp at (-1,0) where the derivative approached 0 from both sides, another had a cusp where the derivative approached +/-1 (or so), and the third had a vertical asymptote at (-1,0), almost like the graph of $e^{-x^2}$ flipped sideways. I asked which of these it could possibly be, several times. I was delighted to get lots of answers, and continue to pressure the students to reformulate their answers to try to be more precise. I like to think that I waited through the quiet moments while students were working on more answers, or reformulating old answers. I like to think that's what they were doing, instead of just waiting for me to tell them the answer.
I love the feeling that nobody in the class is talking (me especially) because everybody is thinking hard about the same question. I hope that was actually the case with the quiet moments today. I think this process of asking the same question a few ways, or asking for students to answer it a few ways, and me harassing them about their answers, is a good process. We're not just going through "routine" calculations, and I like that. I also hope that I can continue to get lucky with examples that I can do this with.
Last week we were talking about improper integrals in class. After some introductory discussion, I asked them to tell me which of the integrals, $\int_1^{\infty} \frac{e^{\sqrt{x}}}{\sqrt{x}}\ dx$ or $\int_1^{\infty} \frac{1}{\sqrt{x}e^{\sqrt{x}}\ dx$ could possibly converge, and why. The first answer I got was to actually compute the first integral and notice that it diverges, so that couldn't be it. A good answer, certainly, with an opportunity to talk about $u$-substitution with improper integrals. And then I asked again, for another reason. A student brought up the integral $\int_1^{\infty} \frac{1}{\sqrt{x}}\ dx,ドル which we already knew diverged, and this gave us the chance to talk about the comparison test. I think at this point we about ran out of time, but I know somewhere in there we also had a discussion about functions that approach 0 not necessarily having a convergent improper integral to infinity. We didn't get to L'Hospital's rule (hopefully a reminder, this being calc 2) that day, but did eventually, with this same example.
Today I was talking about parametric curves. I asked them to take a few minutes to try to sketch the parametric curve $(t^2-1,t^3),ドル and then had students come to the board to draw what they had. I was happy to get two different answers - something like a sideways parabola, and then the correct graph, with a cusp. I asked them how we could tell which could possibly be the correct graph, because they both went through the first few easy points, (0,-1), (-1,0), and (0,1). A student pointed out that $y$ was changing faster than $x,ドル so it shouldn't be the sideways parabola. I didn't want to talk about concavity just yet, but we did get to it eventually.
Then I asked basically the same question, drawing three different curves that weren't the sideways parabola. One had a cusp at (-1,0) where the derivative approached 0 from both sides, another had a cusp where the derivative approached +/-1 (or so), and the third had a vertical asymptote at (-1,0), almost like the graph of $e^{-x^2}$ flipped sideways. I asked which of these it could possibly be, several times. I was delighted to get lots of answers, and continue to pressure the students to reformulate their answers to try to be more precise. I like to think that I waited through the quiet moments while students were working on more answers, or reformulating old answers. I like to think that's what they were doing, instead of just waiting for me to tell them the answer.
I love the feeling that nobody in the class is talking (me especially) because everybody is thinking hard about the same question. I hope that was actually the case with the quiet moments today. I think this process of asking the same question a few ways, or asking for students to answer it a few ways, and me harassing them about their answers, is a good process. We're not just going through "routine" calculations, and I like that. I also hope that I can continue to get lucky with examples that I can do this with.
Monday, September 7, 2009
Parametric Explorer
If you've been following my twitter stream the past few days, you may have noticed that I've been mentioning (and shamefully linking to) a webpage I've been working on to explore parametric curves. Today I decided to make another round of improvements and it was going well, so I thought I'd share here as well. I like to think it is something that calculus teachers might find useful.
Anyway, the page is sort of introduced on my personal website, and the actual "play around with this" page is here. I've only gotten it to work in Firefox 3.5 and Google Chrome. That's enough for me to play with it, so I don't intend to do much more in terms of browser compatibility work on it.
The idea is that your mouse coordinates describe a parametric curve as you move around the screen. The webpage then also graphs the individual curves $x(t)$ and $y(t)$.
There's certainly room for improvement, but I'm happy enough with it as it is to not worry about it for now.
Anyway, the page is sort of introduced on my personal website, and the actual "play around with this" page is here. I've only gotten it to work in Firefox 3.5 and Google Chrome. That's enough for me to play with it, so I don't intend to do much more in terms of browser compatibility work on it.
The idea is that your mouse coordinates describe a parametric curve as you move around the screen. The webpage then also graphs the individual curves $x(t)$ and $y(t)$.
There's certainly room for improvement, but I'm happy enough with it as it is to not worry about it for now.
Wednesday, July 1, 2009
Summer Calc II, Second Half
My summer calculus class is almost over. Not that I don't love teaching, or calculus, or that I don't like my students, but I won't mind when it's over. I've got plenty of other things I should be (and shouldn't be :)) working on this summer. But anyway. Our final exam is tomorrow.
Hopefully it will go better than the first midterm. Admittedly, the first midterm was after only 8 class periods, and had content from 4 different chapters. The exam I wrote is one I was pretty pleased with. Other instructors told me that it was conceptual, which I take as a complement. The average, though, was sadly low. Our final, tomorrow, only covers series. I don't think it's quite as interesting as the first, but other instructors seem to think it is still fair. So we'll see.
I approached series a little bit differently than I have in the past. The last few semesters, I've followed the outline of the book, starting with sequences, then series, on to convergence tests, power series, and wrapping up with Taylor series. I thought this semester I'd try to motivate the discussion of series a little differently. Instead of just "I'd really like to add up a bunch of numbers", I began with Taylor polynomials because "I'd really like to approximate a function". This felt like a good fit with the earlier material, when we found arc lengths (and similar things) by approximating the calculation with an easy one and taking a limit (and calling it an integral). So we're going to approximate a function by an "easy" one (polynomial) and then take a limit.
After that goal is set, it's easy enough to say that "good approximation" means "the derivatives match" (at the point in question), and derive the formula for the coefficients of the Taylor polynomial. Then I pointed out that we could do the process as long as we want and make polynomials of degrees as large as we wanted, and pointed out that this would end up looking very much like taking a limit.
Next I talked about power series in general, power series being what we get "in the limit" of our Taylor polynomial calculations. I talked about differentiating, integrating, and substituting into power series, to try to give some indication that they are useful and easy to work with. I'm not sure this part went over particularly well. It might have been better to do this later, with radius and interval of convergence (see below).
So once we have power series, I pointed out that evaluating a power series at a point meant you had to sum infinitely many values. But that we basically knew how, since we started with Taylor polynomials and took the limit. To sum infinitely many values, you just take a limit of partial sums. So I talked a little about general sequences and series at this point (and also spend an hour talking about fun examples - continued fractions, 3n+1 problem, koch snowflake, cantor set,...).
Then we had a whole day (2 hours of class, +/-) where I just talked about all of the convergence tests, followed by a day for them to work on convergence tests in class (work on homework, or just try other problems). I think that for a summer class, with the odd schedules and times, this almost works. During a normal semester, though, I wouldn't do things this way. I'd probably try to get the students to learn the problems from the book themselves. In fact, I did try this last semester, but that's a separate story.
My philosophy with the convergence tests (and most other topics) is that there isn't much point in my doing more than one or two examples at the board. Math is always easier when you watch somebody else do it. The only way to get comfortable with the convergence tests is to work a whole bunch of problems for yourself. Hopefully the longer-than-average assignment I gave my class helped them gain that comfort.
Once we have convergence tests, we can return to power series, and ask "for which x is this power series defined?" I like to think that this helped tie things together from our initial discussion of power series. I'm not convinced that's the case, but I'm also not sure how to tell (ask the students?). I think this maybe would be a better time for the discussion about how to manipulate power series than early on. I'll probably try it this way next time (this fall, for my fourth straight calc II class).
That covers all of the content for the series chapter. So then I spent a full class period talking about all sorts of fun and exciting uses of series. Today in class I gave them time to review (gave them a copy of last semester's exam, essentially), after answering whatever review questions they came in with.
I'm hoping this exam goes well. And that after it's all over, I have a productive summer.
Hopefully it will go better than the first midterm. Admittedly, the first midterm was after only 8 class periods, and had content from 4 different chapters. The exam I wrote is one I was pretty pleased with. Other instructors told me that it was conceptual, which I take as a complement. The average, though, was sadly low. Our final, tomorrow, only covers series. I don't think it's quite as interesting as the first, but other instructors seem to think it is still fair. So we'll see.
I approached series a little bit differently than I have in the past. The last few semesters, I've followed the outline of the book, starting with sequences, then series, on to convergence tests, power series, and wrapping up with Taylor series. I thought this semester I'd try to motivate the discussion of series a little differently. Instead of just "I'd really like to add up a bunch of numbers", I began with Taylor polynomials because "I'd really like to approximate a function". This felt like a good fit with the earlier material, when we found arc lengths (and similar things) by approximating the calculation with an easy one and taking a limit (and calling it an integral). So we're going to approximate a function by an "easy" one (polynomial) and then take a limit.
After that goal is set, it's easy enough to say that "good approximation" means "the derivatives match" (at the point in question), and derive the formula for the coefficients of the Taylor polynomial. Then I pointed out that we could do the process as long as we want and make polynomials of degrees as large as we wanted, and pointed out that this would end up looking very much like taking a limit.
Next I talked about power series in general, power series being what we get "in the limit" of our Taylor polynomial calculations. I talked about differentiating, integrating, and substituting into power series, to try to give some indication that they are useful and easy to work with. I'm not sure this part went over particularly well. It might have been better to do this later, with radius and interval of convergence (see below).
So once we have power series, I pointed out that evaluating a power series at a point meant you had to sum infinitely many values. But that we basically knew how, since we started with Taylor polynomials and took the limit. To sum infinitely many values, you just take a limit of partial sums. So I talked a little about general sequences and series at this point (and also spend an hour talking about fun examples - continued fractions, 3n+1 problem, koch snowflake, cantor set,...).
Then we had a whole day (2 hours of class, +/-) where I just talked about all of the convergence tests, followed by a day for them to work on convergence tests in class (work on homework, or just try other problems). I think that for a summer class, with the odd schedules and times, this almost works. During a normal semester, though, I wouldn't do things this way. I'd probably try to get the students to learn the problems from the book themselves. In fact, I did try this last semester, but that's a separate story.
My philosophy with the convergence tests (and most other topics) is that there isn't much point in my doing more than one or two examples at the board. Math is always easier when you watch somebody else do it. The only way to get comfortable with the convergence tests is to work a whole bunch of problems for yourself. Hopefully the longer-than-average assignment I gave my class helped them gain that comfort.
Once we have convergence tests, we can return to power series, and ask "for which x is this power series defined?" I like to think that this helped tie things together from our initial discussion of power series. I'm not convinced that's the case, but I'm also not sure how to tell (ask the students?). I think this maybe would be a better time for the discussion about how to manipulate power series than early on. I'll probably try it this way next time (this fall, for my fourth straight calc II class).
That covers all of the content for the series chapter. So then I spent a full class period talking about all sorts of fun and exciting uses of series. Today in class I gave them time to review (gave them a copy of last semester's exam, essentially), after answering whatever review questions they came in with.
I'm hoping this exam goes well. And that after it's all over, I have a productive summer.
Monday, June 15, 2009
Summer Calc II, Week 1
Last Tuesday I started teaching my summer Calc II course. We meet for 2 hours every day (M-F), and then the students also have a 45 minute discussion section with our TA. I decided to set up the course a little differently from how I've done Calc II the last few times, and thought I might try to describe some of it here.
The, say, "standard" way this course would go is to start with techniques of integration, do improper integrals, arc length and surface area of revolution, parametric curves, polar curves, iterated integrals, and then series. Last semester, we had one exam on techniques of integration and arc length and surface area, one on parametric and polar curves and iterated integrals, and one on series.
When I was setting up my course for the summer, I looked at how many days to spend on each topic, and how to try to schedule things best to naturally have exams and things. I decided to shoot for having all of the topics besides series as a single exam, around halfway through the course, and then have series be their own exam. We're just about to the first exam (it'll be Friday), and it looks like we can meet that schedule (with in class review time before the exam even!).
I was able to condense the material down in a few ways. First, the arc length and surface area calculations we do with usual $y=f(x)$ curves, we later redo with parametric curves (in the "standard" class). I decided to start with doing them as parametric and polar curves, and then make a few comments about how to convert a usual $y=f(x)$ curve to a parametric curve. [While I'm talking about surface area... I know that when you rotate around whichever axis, you have (sometimes) two choices for how to set up the integral. You could do it dy or dx. Probably I just haven't dug into enough examples, but is there one when the integral is "doable by hand" one way, but not "doable by hand" the other way, even though both can be set up?]
The other way I crunched things together was to leave more of the responsibility to the students to learn the techniques of integration on their own, outside of class, for homework. I made a few comments here or there, but largely it's been up to them. In my mind, the only real techniques of integration are u-substitution and integration by parts anyway, both of which I did talk about in class. The others, in the text, are trig integrals ("apply lots of trig identities"), trig substitution ("memorize a chart of what substitution to make when"), and partial fractions (I've got no beef with partial fractions). It wouldn't matter if I spent all day for a few class periods working integrals, the students would leave feeling like they weren't too bad, and would get stuck on them at home. So I cut out more of the "they aren't that bad" feeling, and mostly threw the students in to get stuck. My hope was that I'd see lots of kids in office hours, which has not (yet) been the case.
My other thoughts, along the lines of techniques of integration, are concerned with how useful they actually are. This general question is something me and every other math teacher with a blog have brought up before, especially recently with Wolfram Alpha doing integrals (and showing steps!). What is it we should actually be teaching these kids? I'm not convinced I've ever worked a trig-substitution integral outside of calculus class, but would love to hear about it if anybody out there had. Ok, sure, working them gives you lots of good practice with algebraic rules, and so I can now manipulate symbols like it's my job. But I've been trying to set up my class to sort of tend away from this. I really want to spend more time on, and get my students to answer more questions about, how to set up integrals to calculate things you might want to calculate, or what a given integral calculates (in a picture). I also try, when we work an integral, to talk them through any sort of consistency checking I would do for the answer (is it positive or negative? should it be? is it too big, or too small?). I like to think that I am giving them practice taking a problem and converting it to something a computer can easily do, and then thinking about if the answer is reasonable.
Part of my other goal in having them learn techniques of integration with somewhat less guidance, was to give them experience learning from the textbook directly. Probably I want to do this because it is how the calc class I took as an undergrad was set up.
I'm not convinced, based on homework grades, that things are going terribly well so far. I've asked them to give me anonymous feedback about any changes they want to have made, and tried to stress that I want them to ask questions and come to office hours. I've not gotten too much back in the way of feedback yet. The one, really, so far was that I should work harder problems in class. I think I get this every semester, so probably it is something I should seriously look into. But most of me thinks: they'll always look easier in class, you have to get stuck on them yourself. Math is not easy. I don't know, perhaps that's my laziness shining through. But at the same time, I can't really start with hard examples in class, because they won't make sense right away, and there's only so much time during class.
At some point I was preparing lectures and getting frustrated by the examples I had, and with trying to find new ones. (Is there a place online where teachers keep their favorite examples of all sorts of problems? Some sort of examples repository or something?) I came up with an idea I'm still mulling over for how to teach my classes. My thought was maybe I'll start each day with a little discussion about whatever new concepts there are for the day, at sort of the theoretical level. And then I'll just grab the first couple (or just one) odd problems from the text, and work through them. And then the rest of class time will be students working through as many more of the problems as we have time for. They can work in groups if they want, but don't have to. This gives them the chance to get stuck on problems, but quickly start asking questions to get over initial hurdles. They'll have their textbooks with them, to dig through examples, and will also have other students (and me) to ask when they're really stuck. Anybody have any thoughts on this setup?
The, say, "standard" way this course would go is to start with techniques of integration, do improper integrals, arc length and surface area of revolution, parametric curves, polar curves, iterated integrals, and then series. Last semester, we had one exam on techniques of integration and arc length and surface area, one on parametric and polar curves and iterated integrals, and one on series.
When I was setting up my course for the summer, I looked at how many days to spend on each topic, and how to try to schedule things best to naturally have exams and things. I decided to shoot for having all of the topics besides series as a single exam, around halfway through the course, and then have series be their own exam. We're just about to the first exam (it'll be Friday), and it looks like we can meet that schedule (with in class review time before the exam even!).
I was able to condense the material down in a few ways. First, the arc length and surface area calculations we do with usual $y=f(x)$ curves, we later redo with parametric curves (in the "standard" class). I decided to start with doing them as parametric and polar curves, and then make a few comments about how to convert a usual $y=f(x)$ curve to a parametric curve. [While I'm talking about surface area... I know that when you rotate around whichever axis, you have (sometimes) two choices for how to set up the integral. You could do it dy or dx. Probably I just haven't dug into enough examples, but is there one when the integral is "doable by hand" one way, but not "doable by hand" the other way, even though both can be set up?]
The other way I crunched things together was to leave more of the responsibility to the students to learn the techniques of integration on their own, outside of class, for homework. I made a few comments here or there, but largely it's been up to them. In my mind, the only real techniques of integration are u-substitution and integration by parts anyway, both of which I did talk about in class. The others, in the text, are trig integrals ("apply lots of trig identities"), trig substitution ("memorize a chart of what substitution to make when"), and partial fractions (I've got no beef with partial fractions). It wouldn't matter if I spent all day for a few class periods working integrals, the students would leave feeling like they weren't too bad, and would get stuck on them at home. So I cut out more of the "they aren't that bad" feeling, and mostly threw the students in to get stuck. My hope was that I'd see lots of kids in office hours, which has not (yet) been the case.
My other thoughts, along the lines of techniques of integration, are concerned with how useful they actually are. This general question is something me and every other math teacher with a blog have brought up before, especially recently with Wolfram Alpha doing integrals (and showing steps!). What is it we should actually be teaching these kids? I'm not convinced I've ever worked a trig-substitution integral outside of calculus class, but would love to hear about it if anybody out there had. Ok, sure, working them gives you lots of good practice with algebraic rules, and so I can now manipulate symbols like it's my job. But I've been trying to set up my class to sort of tend away from this. I really want to spend more time on, and get my students to answer more questions about, how to set up integrals to calculate things you might want to calculate, or what a given integral calculates (in a picture). I also try, when we work an integral, to talk them through any sort of consistency checking I would do for the answer (is it positive or negative? should it be? is it too big, or too small?). I like to think that I am giving them practice taking a problem and converting it to something a computer can easily do, and then thinking about if the answer is reasonable.
Part of my other goal in having them learn techniques of integration with somewhat less guidance, was to give them experience learning from the textbook directly. Probably I want to do this because it is how the calc class I took as an undergrad was set up.
I'm not convinced, based on homework grades, that things are going terribly well so far. I've asked them to give me anonymous feedback about any changes they want to have made, and tried to stress that I want them to ask questions and come to office hours. I've not gotten too much back in the way of feedback yet. The one, really, so far was that I should work harder problems in class. I think I get this every semester, so probably it is something I should seriously look into. But most of me thinks: they'll always look easier in class, you have to get stuck on them yourself. Math is not easy. I don't know, perhaps that's my laziness shining through. But at the same time, I can't really start with hard examples in class, because they won't make sense right away, and there's only so much time during class.
At some point I was preparing lectures and getting frustrated by the examples I had, and with trying to find new ones. (Is there a place online where teachers keep their favorite examples of all sorts of problems? Some sort of examples repository or something?) I came up with an idea I'm still mulling over for how to teach my classes. My thought was maybe I'll start each day with a little discussion about whatever new concepts there are for the day, at sort of the theoretical level. And then I'll just grab the first couple (or just one) odd problems from the text, and work through them. And then the rest of class time will be students working through as many more of the problems as we have time for. They can work in groups if they want, but don't have to. This gives them the chance to get stuck on problems, but quickly start asking questions to get over initial hurdles. They'll have their textbooks with them, to dig through examples, and will also have other students (and me) to ask when they're really stuck. Anybody have any thoughts on this setup?
Saturday, May 30, 2009
Walpha Wiki
Just in case you missed it, I thought I'd share the link for the Walpha Wiki. The same evening that I posted asking if anybody had a wiki going, Derek Bruff started this one. I've not contributed as much as I want to have yet, but still intend to. Won't you help?
The blog also includes discussion about the impact of W|alpha on mathematics education, a topic that's been on my mind recently.
The blog also includes discussion about the impact of W|alpha on mathematics education, a topic that's been on my mind recently.
Tuesday, May 26, 2009
A New Kind of Wiki
Well, not really... I'll just explain.
So, when Wolfram|Alpha (referred to as w|a below because I'm lazy) came out, I, like many of you, was pretty excited to play with it. I was primarily interested in its use as a free, online, computer algebra system (CAS). So when I tested it, I gave it the sorts of questions that I give my calculus students (in fact, I essentially tested it with exams I've given students). In many areas it was obvious what to do, in some areas I could mess around and get a reasonable answer, and in a remaining few areas, w|a seemed to come up lacking.
I thought it would be great to have a resource telling how to input questions you might typically ask a CAS, since apparently entering straight-up Mathematica code doesn't always work (I guess Wolfram still wants to sell copies of Mathematica). One of my early thoughts was that I should make one. And then I thought, surely somebody else has already done so. In fact, the folks at w|a probably already have some nice documentation online. I made a note to look into it, and thought it funny that I was hoping to find documentation for such an online system.
Not long after that, and before I did any more playing with things, Maria Anderson, @busynessgirl on Twitter, posted a tweet: "I am toying with the idea of taking a standard algebra TOC and putting up a webpage that shows which topics W|A can do." A fantastic idea (which she quickly refined: webpage -> wiki). Extend it to calculus, and I'm there. And show not just what it can do, but what it can't do, what it does wrong (or oddly), and ways to make it do what it can do.
I think such a thing should come into being. Perhaps it already has, and I missed it? Or perhaps there is some nice documentation for w|a that I've not yet found? If either of these is the case, could somebody point me to it?
If there is no such thing yet, I say it's time to make one. I'm getting antsy. In the comments below, if you want such a wiki to exist, would you please leave some helpful feedback? I'm particularly interested in: (1) What (free, hosted) wiki software would you suggest or suggest avoiding? I think right now I'm leaning toward wikispaces, though I've not looked into things a whole lot. (2) What should it be called? (3) Any other comments or suggestions you have.
To get things rolling, I'll say that this coming Saturday (May 30), if no links are provided to an existing webpage, I'll start a wiki somewhere that seems to fit the consensus of the comments (I hope there are comments, and they have a consensus). I'll then let you know where it is.
Update 20090526: Derek Bruff left a comment that he was starting one, and posted the link http://walphawiki.wikidot.com/calculus-i via twitter. Looks promising!
So, when Wolfram|Alpha (referred to as w|a below because I'm lazy) came out, I, like many of you, was pretty excited to play with it. I was primarily interested in its use as a free, online, computer algebra system (CAS). So when I tested it, I gave it the sorts of questions that I give my calculus students (in fact, I essentially tested it with exams I've given students). In many areas it was obvious what to do, in some areas I could mess around and get a reasonable answer, and in a remaining few areas, w|a seemed to come up lacking.
I thought it would be great to have a resource telling how to input questions you might typically ask a CAS, since apparently entering straight-up Mathematica code doesn't always work (I guess Wolfram still wants to sell copies of Mathematica). One of my early thoughts was that I should make one. And then I thought, surely somebody else has already done so. In fact, the folks at w|a probably already have some nice documentation online. I made a note to look into it, and thought it funny that I was hoping to find documentation for such an online system.
Not long after that, and before I did any more playing with things, Maria Anderson, @busynessgirl on Twitter, posted a tweet: "I am toying with the idea of taking a standard algebra TOC and putting up a webpage that shows which topics W|A can do." A fantastic idea (which she quickly refined: webpage -> wiki). Extend it to calculus, and I'm there. And show not just what it can do, but what it can't do, what it does wrong (or oddly), and ways to make it do what it can do.
I think such a thing should come into being. Perhaps it already has, and I missed it? Or perhaps there is some nice documentation for w|a that I've not yet found? If either of these is the case, could somebody point me to it?
If there is no such thing yet, I say it's time to make one. I'm getting antsy. In the comments below, if you want such a wiki to exist, would you please leave some helpful feedback? I'm particularly interested in: (1) What (free, hosted) wiki software would you suggest or suggest avoiding? I think right now I'm leaning toward wikispaces, though I've not looked into things a whole lot. (2) What should it be called? (3) Any other comments or suggestions you have.
To get things rolling, I'll say that this coming Saturday (May 30), if no links are provided to an existing webpage, I'll start a wiki somewhere that seems to fit the consensus of the comments (I hope there are comments, and they have a consensus). I'll then let you know where it is.
Update 20090526: Derek Bruff left a comment that he was starting one, and posted the link http://walphawiki.wikidot.com/calculus-i via twitter. Looks promising!
Thursday, May 7, 2009
Changing Calculus
Calculus, at least derivatives, are the (a?) study of rates of change. What I've been wondering recently is how instructors are thinking about change - in their curricula.
I know we've had calculators for quite some time that can do lots of the work we assign our kids. There has always been a price barrier for students using them though. I'm thinking Wolfram Alpha is about to change that (when it goes live later this month).
There has always (well, for quite some time, anyway) been integrals.wolfram.com, which will compute integrals (a big part of a calc 2 course). However, no indication is given there about how to obtain the solution. According to the ReadWriteWeb account of Wolfram Alpha, you can ask it to do an integral, and also ask to see the steps in the computation.
I think this is just one sign, of many, that calculus class will be changing. Sure, technology has been around (behind a price barrier) that will give students answers. Teachers could typically rely on "Show All Work" to hopefully get their students to not bother with the calculators. But now, perhaps, "Show All Work" is also done by the machines, and now it's free. How should I be changing the setup of my calculus class to accommodate this shift?
It seems to me that my classes should start spending less time going through the algebra and "doing integrals" (though not completely removing this from the syllabus), and spend more time finding ways to use them to solve problems. Perhaps try to work some more theory into things, besides just "Oh, look, with functions that look like blah, a substitution blah makes them easier to integrate". I need to figure out how to shift my classes from "do the algebra to work out this computation" to "set up a computation that will determine the answer to this 'interesting' question".
Wolfram Alpha, which has brought this issue up most pressingly (in my mind), might also be a useful tool in shifting how my calculus courses are set up. By the looks of things, Wolfram Alpha has access to lots and lots of data, and can do lots and lots of interesting computation with it. So perhaps it will be a great way to find and create new problems, and give students interesting opportunities to find solutions. Of course, it's too soon to say, because the service isn't up yet. But it will be soon.
So, have people already started making these changes, and I'm just behind in my teaching (as it the rest of my school)? If so, how do I get to where you are? What should I be doing? What are the "interesting" problems I should have my students thinking about, instead of the interesting (in terms of symbol pushing) problems they currently do? Perhaps the tools that I'm just starting to see available for free in Wolfram Alpha are already around (anybody have some links for us)? Or is this all a non-issue, because doing 10 steps of algebra in each of 10 problems, each with a different algebra trick, is what we want our students to be able to do after they're through a calculus class (because in the "real world" (which I'm assuming is out there) they'll have to do everything by hand, no computers)?
I know the technology in math classes debate is not a new one. But I think it is getting more pressing. Maybe I've just been reading too much online/tech news.
I also know this is not the only question that should go into changing courses (if a change is going to happen). What is the goal of a calculus course? How does it fit into the entire mathematics curriculum? And what are the answers to these questions in terms of students going into mathematics, versus science, versus the arts? What actual calculus (and other math) should they be getting out of my class? What other things should they be getting out of my class (how to read a math text? how to present a mathematical solution? how to write one?)? What other questions am I supposed to be asking?
Apparently giving a final exam today is making me philosophical.
I know we've had calculators for quite some time that can do lots of the work we assign our kids. There has always been a price barrier for students using them though. I'm thinking Wolfram Alpha is about to change that (when it goes live later this month).
There has always (well, for quite some time, anyway) been integrals.wolfram.com, which will compute integrals (a big part of a calc 2 course). However, no indication is given there about how to obtain the solution. According to the ReadWriteWeb account of Wolfram Alpha, you can ask it to do an integral, and also ask to see the steps in the computation.
I think this is just one sign, of many, that calculus class will be changing. Sure, technology has been around (behind a price barrier) that will give students answers. Teachers could typically rely on "Show All Work" to hopefully get their students to not bother with the calculators. But now, perhaps, "Show All Work" is also done by the machines, and now it's free. How should I be changing the setup of my calculus class to accommodate this shift?
It seems to me that my classes should start spending less time going through the algebra and "doing integrals" (though not completely removing this from the syllabus), and spend more time finding ways to use them to solve problems. Perhaps try to work some more theory into things, besides just "Oh, look, with functions that look like blah, a substitution blah makes them easier to integrate". I need to figure out how to shift my classes from "do the algebra to work out this computation" to "set up a computation that will determine the answer to this 'interesting' question".
Wolfram Alpha, which has brought this issue up most pressingly (in my mind), might also be a useful tool in shifting how my calculus courses are set up. By the looks of things, Wolfram Alpha has access to lots and lots of data, and can do lots and lots of interesting computation with it. So perhaps it will be a great way to find and create new problems, and give students interesting opportunities to find solutions. Of course, it's too soon to say, because the service isn't up yet. But it will be soon.
So, have people already started making these changes, and I'm just behind in my teaching (as it the rest of my school)? If so, how do I get to where you are? What should I be doing? What are the "interesting" problems I should have my students thinking about, instead of the interesting (in terms of symbol pushing) problems they currently do? Perhaps the tools that I'm just starting to see available for free in Wolfram Alpha are already around (anybody have some links for us)? Or is this all a non-issue, because doing 10 steps of algebra in each of 10 problems, each with a different algebra trick, is what we want our students to be able to do after they're through a calculus class (because in the "real world" (which I'm assuming is out there) they'll have to do everything by hand, no computers)?
I know the technology in math classes debate is not a new one. But I think it is getting more pressing. Maybe I've just been reading too much online/tech news.
I also know this is not the only question that should go into changing courses (if a change is going to happen). What is the goal of a calculus course? How does it fit into the entire mathematics curriculum? And what are the answers to these questions in terms of students going into mathematics, versus science, versus the arts? What actual calculus (and other math) should they be getting out of my class? What other things should they be getting out of my class (how to read a math text? how to present a mathematical solution? how to write one?)? What other questions am I supposed to be asking?
Apparently giving a final exam today is making me philosophical.
Sunday, March 29, 2009
Hyperbolic Space
I recently ran across, via this post on Division by Zero, a way to make hyperbolic space from paper, a project that I couldn't resist. In fact, it claims to be a hyperbolic soccer ball. Digging through Reader to find the link again, I found this other recent post about shapes relating to soccer balls, so thought I'd share it as well.
Anyway, the idea is that to typically make a soccer ball, you place a ring of hexagons around a pentagon, and iterate. The pentagons introduce some positive curvature in the process (hexagons alone have 0 curvature - they tile the plane), and you end up with something fairly spherical. If you place hexagons around a central heptagon though, you get negative curvature.
The directions to actually make one yourself are at The Institute for Figuring, and are available here (pdf). Following the basic instructions, I ended up with the following:
Which my cats only took fleeting interest in:
The directions suggested that you could continue adding more rings of hexagons, with heptagons in appropriate locations, and extend the model. I was curious to see what would happen when I did, so I printed out six more of the base sheets (3 of which gave the starting model, above). It's a bit of a monster:
Anyway, the idea is that to typically make a soccer ball, you place a ring of hexagons around a pentagon, and iterate. The pentagons introduce some positive curvature in the process (hexagons alone have 0 curvature - they tile the plane), and you end up with something fairly spherical. If you place hexagons around a central heptagon though, you get negative curvature.
The directions to actually make one yourself are at The Institute for Figuring, and are available here (pdf). Following the basic instructions, I ended up with the following:
Which my cats only took fleeting interest in:
The directions suggested that you could continue adding more rings of hexagons, with heptagons in appropriate locations, and extend the model. I was curious to see what would happen when I did, so I printed out six more of the base sheets (3 of which gave the starting model, above). It's a bit of a monster:
Tuesday, March 17, 2009
Semi-Sierpinski St. Paddy's Sugar Cookies
Today I made some sugar cookies, with hopes of them coming out something close to the awesome Sierpinski Cookies I found here. This was my second time trying. The first time I used a package of sugar cookie mix, and it didn't work out for me at all. This time, I followed the recipe linked to from the above page and made my dough from scratch. They just about came out reasonable:
In that picture I have a few levels of iteration of the general idea. You'll notice, though, that at the highest iteration, the cookies aren't quite right. There is supposed to be one big green square in the middle, and 8 little green squares around the outside. Let's have another look at how one actually came out:
Where's the missing green square? I honestly have no idea. But all of my cookies are missing at least one, and sometimes two, of the green squares.
Ah well, they still taste good. And they came out well enough for me to maybe try again sometime.
In that picture I have a few levels of iteration of the general idea. You'll notice, though, that at the highest iteration, the cookies aren't quite right. There is supposed to be one big green square in the middle, and 8 little green squares around the outside. Let's have another look at how one actually came out:
Where's the missing green square? I honestly have no idea. But all of my cookies are missing at least one, and sometimes two, of the green squares.
Ah well, they still taste good. And they came out well enough for me to maybe try again sometime.
Thursday, March 12, 2009
Math Blogroll in OPML
I don't spend a lot of time visiting the actual pages for many of the blogs I follow, since I get all (or at least, most, and the main portion) of the content from their rss/atom feeds. Recently though, one of the feeds I had indicated that the author was quitting. For whatever reason (perhaps because it mentioned having the world's best math blogroll), I was inspired to visit the actual page, instead of just removing the feed from my list (or doing nothing).
The feed was from Vlorbik on Math Ed. Upon visiting the page, I found that Vlorbik kept a pretty substantial blogroll of math blogs. Liking to be in the know, I figured I might subscribe to some. Of course, I already probably do subscribe to some (and author some :)), but there are surely plenty there that I don't subscribe to. And perhaps some of them are ones I would like to follow. But I didn't want to click each link, load each page, find it's feed, and add it to Google Reader. I'm pretty lazy, and my computer would slow down a bit and frustrate me.
This evening, though, I decided to see if I could write a script to grab the rss/atom feeds for any or all of the linked pages in the blogroll. I had a great time doing so. Remembering some fun pattern matching variables in perl(like \$' and \$& (the \ there only because of how I'm doing LaTeX in Blogger)), and using curl to grab the pages... good times. Then some reformatting of appropriate strings, and out pops an OPML file. Handy, because that's what Google Reader expects if you want to import a bunch of feeds. I've played with similar things before.
Anyway, the long and short of it is, I thought perhaps other people might find this OPML file helpful. Blogger won't let me upload anything besides pictures (where's my damn GDrive?), so the file is currently (as of this writing) on my UVA personal page, here. If you'd like to blindly add these feeds to your feed reader, and then trim them down individually based on content or whatever, I encourage you to do so. The only reader I've used is Google's, so I'll give some instructions for that.
The first step is to download my OPML file, and save it somewhere convenient (you only need it temporarily on your computer). In Reader, at the bottom of the left-hand pane is the 'Manage Subscriptions' link. Click on that, and then the 'Import/Export' link at the top of the settings page that pops up. In the file upload form where it says 'Select an OPML file to upload', pick the file out from wherever you saved it, and then click 'Upload'. Wait patiently as Google imports the new feeds (it really doesn't take that long, though it might take longer for news items to start flowing in). It'll send you back to the main settings page, so click 'Back to Google Reader' to start reading. You'll notice that the feeds all show up in a folder in your subscriptions panel, called 'vlorbik' (if you already have such a folder, you might modify my OPML file before upload... I should have told you that earlier). If you already subscribed to one of the feeds, it won't mess anything up, and they won't show up as duplicates in your news stream. Of course, when making this file I grabbed the atom files, where available, so if you are subscribed to the rss feed (as I am, in many cases), then you will have duplicates. But whatever, I'll let you sort out your own subscription list.
So, with this success, I feel like perhaps I should visit actual pages (instead of just watching the news stream go by in reader) more often. Perhaps find some other blogrolls?
Anyway, enjoy. Sorry, Vlorbik, that I only started getting to know you on your way out.
The feed was from Vlorbik on Math Ed. Upon visiting the page, I found that Vlorbik kept a pretty substantial blogroll of math blogs. Liking to be in the know, I figured I might subscribe to some. Of course, I already probably do subscribe to some (and author some :)), but there are surely plenty there that I don't subscribe to. And perhaps some of them are ones I would like to follow. But I didn't want to click each link, load each page, find it's feed, and add it to Google Reader. I'm pretty lazy, and my computer would slow down a bit and frustrate me.
This evening, though, I decided to see if I could write a script to grab the rss/atom feeds for any or all of the linked pages in the blogroll. I had a great time doing so. Remembering some fun pattern matching variables in perl(like \$' and \$& (the \ there only because of how I'm doing LaTeX in Blogger)), and using curl to grab the pages... good times. Then some reformatting of appropriate strings, and out pops an OPML file. Handy, because that's what Google Reader expects if you want to import a bunch of feeds. I've played with similar things before.
Anyway, the long and short of it is, I thought perhaps other people might find this OPML file helpful. Blogger won't let me upload anything besides pictures (where's my damn GDrive?), so the file is currently (as of this writing) on my UVA personal page, here. If you'd like to blindly add these feeds to your feed reader, and then trim them down individually based on content or whatever, I encourage you to do so. The only reader I've used is Google's, so I'll give some instructions for that.
The first step is to download my OPML file, and save it somewhere convenient (you only need it temporarily on your computer). In Reader, at the bottom of the left-hand pane is the 'Manage Subscriptions' link. Click on that, and then the 'Import/Export' link at the top of the settings page that pops up. In the file upload form where it says 'Select an OPML file to upload', pick the file out from wherever you saved it, and then click 'Upload'. Wait patiently as Google imports the new feeds (it really doesn't take that long, though it might take longer for news items to start flowing in). It'll send you back to the main settings page, so click 'Back to Google Reader' to start reading. You'll notice that the feeds all show up in a folder in your subscriptions panel, called 'vlorbik' (if you already have such a folder, you might modify my OPML file before upload... I should have told you that earlier). If you already subscribed to one of the feeds, it won't mess anything up, and they won't show up as duplicates in your news stream. Of course, when making this file I grabbed the atom files, where available, so if you are subscribed to the rss feed (as I am, in many cases), then you will have duplicates. But whatever, I'll let you sort out your own subscription list.
So, with this success, I feel like perhaps I should visit actual pages (instead of just watching the news stream go by in reader) more often. Perhaps find some other blogrolls?
Anyway, enjoy. Sorry, Vlorbik, that I only started getting to know you on your way out.
Monday, March 9, 2009
Finding Mistakes
Of all the questions I get in the office hours for my calculus classes, the most frequent are probably from students who have worked through a problem and gotten the wrong answer, but can't find their mistake. I sit down with these students and go through each line of their work, ideally getting them to explain each of their steps to me. Sometimes, students are able to spot their own errors when we do this. Frequently, though, they can't.
While it's generally not terribly difficult for me to find errors, it's a skill I believe I've developed after several years in my math classes. It's a skill I'd like for my students to develop. Recently, I struck on an idea for how to run class that might help students find mistakes.
Our calculus classes are accompanied by an additional class period, called the fourth hour or discussion section. Mostly what happens during this time is that students ask questions from the homework, and the TA works them. Or, at least, gives some hints for how to work them. Sometimes the TA for the discussion section is just the instructor, sometimes it is another graduate student. While students certainly appreciate the chance to ask these questions and get answers to their homework, this setup has always frustrated me.
Part of the problem with this setup is the partition of the class into students who have started problems but gotten stuck or made a mistake, and students who have not started problem. The students who have not started are waiting for as many answers in the discussion as possible before doing whatever few remaining problems there are on their own. My hope is that these students do poorly on the exam, if I'm honest. The other students, the ones who ask the questions, because they have started their work, are also not gaining much from most of the time spent answering their question. This is because their mistake shows up, or they got stuck, mid-way through the problem, so all the time used in class getting to that point of the solution isn't much help. However, starting mid-way through the problem won't work, since most of the rest of the class will be lost.
I think a better plan would be to have students bring in their work, and spend most, if not all, of discussion sections working on finding mistakes. I'm trying to think about implementation details for this, and thought I'd see about getting some feedback here. I envision students writing each problem that they worked on but didn't get correct on a separate sheet of paper, and bringing those to discussion sections. Then, during class, the papers would all be gathered up and redistributed to all the students. Depending on how many papers there are, students might break into groups to tackle a paper, or perhaps collections of papers (or they could work alone). With all of those eyes, bugs are, famously, shallow. Groups would make notes on the paper about errors, or tips on how to proceed, and then papers would be returned to their owner. If, after this time, nobody can find a mistake on a paper, or everybody is stuck at some point in the same problem, the TA can talk through the problem with the class.
So, a few questions.
While it's generally not terribly difficult for me to find errors, it's a skill I believe I've developed after several years in my math classes. It's a skill I'd like for my students to develop. Recently, I struck on an idea for how to run class that might help students find mistakes.
Our calculus classes are accompanied by an additional class period, called the fourth hour or discussion section. Mostly what happens during this time is that students ask questions from the homework, and the TA works them. Or, at least, gives some hints for how to work them. Sometimes the TA for the discussion section is just the instructor, sometimes it is another graduate student. While students certainly appreciate the chance to ask these questions and get answers to their homework, this setup has always frustrated me.
Part of the problem with this setup is the partition of the class into students who have started problems but gotten stuck or made a mistake, and students who have not started problem. The students who have not started are waiting for as many answers in the discussion as possible before doing whatever few remaining problems there are on their own. My hope is that these students do poorly on the exam, if I'm honest. The other students, the ones who ask the questions, because they have started their work, are also not gaining much from most of the time spent answering their question. This is because their mistake shows up, or they got stuck, mid-way through the problem, so all the time used in class getting to that point of the solution isn't much help. However, starting mid-way through the problem won't work, since most of the rest of the class will be lost.
I think a better plan would be to have students bring in their work, and spend most, if not all, of discussion sections working on finding mistakes. I'm trying to think about implementation details for this, and thought I'd see about getting some feedback here. I envision students writing each problem that they worked on but didn't get correct on a separate sheet of paper, and bringing those to discussion sections. Then, during class, the papers would all be gathered up and redistributed to all the students. Depending on how many papers there are, students might break into groups to tackle a paper, or perhaps collections of papers (or they could work alone). With all of those eyes, bugs are, famously, shallow. Groups would make notes on the paper about errors, or tips on how to proceed, and then papers would be returned to their owner. If, after this time, nobody can find a mistake on a paper, or everybody is stuck at some point in the same problem, the TA can talk through the problem with the class.
So, a few questions.
- Do I have students write their names on the papers, so that it's easy to get them returned? Or does this violate some sort of anonymity that should be preserved? Would writing some fixed number on the paper be better? Or perhaps initials? Maybe have students write something on their papers that they can identify, but other students wont, and then when passing papers back, just hand back all the papers at once, to be passed around so students can grab their own?
- What about students who haven't started the assignment? Or completed it successfully? Should I set things up so the assignment is due very shortly after the discussion section, encouraging students to have looked at it before-hand? Or will this lead to more students in office hours, avoiding postponing getting their problems fixed, and thus defeating the purpose of the group mistake-finding exercise? Should I have students who have already finished pick a problem and write up a fake solution, artificially introducing an error, to give somebody (whoever ends up with their paper) a chance to try to find the mistake?
- What could go wrong with this setup? What policies should be put in place? What do I need to be careful of, or think more about?
Wednesday, March 4, 2009
Mm, Donuts
As a topologist (or, perhaps more accurately - somebody who has taken primarily topology courses recently), I see tori (donuts) all the time. And people always talk about a meridian circle on a torus, versus latitudes. I generally have a hard time remembering which is which. But recently I decided a fun way to remember it by thinking about donuts.
Your first bite into a donut, unless you're doing something wrong, is essentially along a meridian. That works out well for remembering things, because your first bite should also be accompanied by an 'mmmm', and 'mmmm' starts 'mmmmeridian'.
Your first bite into a donut, unless you're doing something wrong, is essentially along a meridian. That works out well for remembering things, because your first bite should also be accompanied by an 'mmmm', and 'mmmm' starts 'mmmmeridian'.
Monday, January 19, 2009
Borromean Rings
The Borromean rings are quite possibly my favorite link. It's 3 circles in space, no two of which are linked, but together, they can't be unlinked. Fantastic. After completing the recent Menger sponge project, I still had lots of cards leftover, and these rings came to mind pretty quickly. So, several more hours of folding later, and I've got a decent model:
Click the picture (or the upcoming text link) for the Picasa gallery, with 2 more pictures (whoo! exciting!).
I'll probably not do too many more projects like this for a while. Of course, that'd probably change quickly if I found a nice model for the trefoil knot (as I've been hoping to stumble across for a while).
Click the picture (or the upcoming text link) for the Picasa gallery, with 2 more pictures (whoo! exciting!).
I'll probably not do too many more projects like this for a while. Of course, that'd probably change quickly if I found a nice model for the trefoil knot (as I've been hoping to stumble across for a while).
Saturday, January 17, 2009
Integration Techniques Discovery Project
I decided to try something new this semester, in terms of how I teach techniques of integration (trig integrals, trig substitution, and partial fractions). Last semester I just lectured through it, presenting as many examples as time permitted. We got through it, but it seemed like it could have been improved.
This semester, I decided to make a project out of it. I decided I would teach u-substitution and integration by parts (which, in my mind, are the only "real" techniques of integration), but let my students learn the other techniques on their own. To support them, I gave them a list of 40 integrals, and solutions for each integral. I specifically told them I didn't them want to look in their book for this part of the class, only look at the worked examples. The examples were, essentially, all the integrals from the appropriate sections of the text, as well as whatever other examples I had found last year to use. Additionally, I gave them a list of 50 integrals (I have about 35 students in my class), without solutions. The idea is that they will look at the unsolved integrals and find one that looks interesting or familiar (or just pick one at random). Then they will consult the solved integrals, looking for integrals that seem similar, and analyzing their solutions. After studying the given solutions, they try to apply similar tricks (substitutions, ways of re-writing) to the integral they chose to solve. Once they have solved an integral, the will present it to the class, and submit a writeup (which will be posted online to share with the class).
I decided it wasn't particularly efficient to print out copies of the handouts I had made (page of worked integrals, page of unworked integrals, 13 page packet of solutions). Instead, I posted each as a pdf on our course page (UVA uses some tailored version of 'Sakai', by my understanding). To make things even more useful, I also made a webpage for the worked integrals. The base page just has the list of integrals, without solutions, and then clicking on an integral reveals its solution. This means that when a student is looking for integrals similar to the integral they have chosen, they can see the whole list of integrals at one time. At some point I thought it would be nice if each student had their own webpage for this, or something, so that they could rearrange the integrals and group them to their heart's content. But I decided to leave that up to them.
My original thought for this project was that I would simply start every class by asking 'Who has an integral to present?'. Whoever raised his or her hand first got to present. I figured this would encourage students to present as soon as they were ready, to avoid getting their problem stolen. It also meant students would start this project early, to try to get an easy problem. However, before the class started I decided to make a wiki page for all of the unworked integrals. When a student was ready to present a problem, they would go to the wiki and move that problem to a 'Claimed Integrals' section, and put their name next to it. This also allows me to see who is ready to present, before class starts, so that I can chose the order to have students present, and hopefully do a bunch of similar problems at the same time.
After a student presents their solution in class, while they are still at the board, I like to ask them little challenging questions. "What if that exponent were a 3 instead of a 2, would your solution still work?" "Can you do this as a definite integral from 0 to 1?" I also encourage the class to ask questions, though mostly (so far) they're pretty quiet.
We've had 2 days worth of presentations so far, and will need to have several more to give everybody time to present. Things seem to be going well. I was quite pleased that there were already more than 10 integrals claimed after the first day of class. The presentations so far have been very good, which I'm glad about. I think starting off with good presentations will give the students a good frame of reference. I've also gotten some phenomenal writeups, in which students carefully explained all of their steps - instead of just writing down their solution.
I'm pleased with this project so far, and have noticed a few things to change if/when I decide to try it again. I really want a student to claim an integral on the wiki only after they are ready to present a problem. I think in the future I will make sure to tell students that they will lose a point if they are asked to present their integral, but are not ready to do so (including not being present!). I may also have students, as a first assignment, go through the worked integrals and gather them up in to groups of integrals that look similar.
Though each student is only working one of these problems, I will have separate assignments (normal homework assignments) to make sure everybody does some of each type of integral. For example, now that we've seen most of the trig integrals in class, they have a homework assignment due on these sorts of problems.
My only real concern about this project occurred sometime after the first day of presentations. I'm worried that strong students will look at the list sooner (or be ready to claim an integral sooner, if nothing else), and will end up working easier problems. This is a two-fold issue, because it means the stronger students aren't being challenged enough, and that weaker students are potentially getting stuck with much harder problems. Perhaps I'll have a better sense if this is the case after a few more days of presentations. I'm not really sure how to avoid this though, if it is the case.
This semester, I decided to make a project out of it. I decided I would teach u-substitution and integration by parts (which, in my mind, are the only "real" techniques of integration), but let my students learn the other techniques on their own. To support them, I gave them a list of 40 integrals, and solutions for each integral. I specifically told them I didn't them want to look in their book for this part of the class, only look at the worked examples. The examples were, essentially, all the integrals from the appropriate sections of the text, as well as whatever other examples I had found last year to use. Additionally, I gave them a list of 50 integrals (I have about 35 students in my class), without solutions. The idea is that they will look at the unsolved integrals and find one that looks interesting or familiar (or just pick one at random). Then they will consult the solved integrals, looking for integrals that seem similar, and analyzing their solutions. After studying the given solutions, they try to apply similar tricks (substitutions, ways of re-writing) to the integral they chose to solve. Once they have solved an integral, the will present it to the class, and submit a writeup (which will be posted online to share with the class).
I decided it wasn't particularly efficient to print out copies of the handouts I had made (page of worked integrals, page of unworked integrals, 13 page packet of solutions). Instead, I posted each as a pdf on our course page (UVA uses some tailored version of 'Sakai', by my understanding). To make things even more useful, I also made a webpage for the worked integrals. The base page just has the list of integrals, without solutions, and then clicking on an integral reveals its solution. This means that when a student is looking for integrals similar to the integral they have chosen, they can see the whole list of integrals at one time. At some point I thought it would be nice if each student had their own webpage for this, or something, so that they could rearrange the integrals and group them to their heart's content. But I decided to leave that up to them.
My original thought for this project was that I would simply start every class by asking 'Who has an integral to present?'. Whoever raised his or her hand first got to present. I figured this would encourage students to present as soon as they were ready, to avoid getting their problem stolen. It also meant students would start this project early, to try to get an easy problem. However, before the class started I decided to make a wiki page for all of the unworked integrals. When a student was ready to present a problem, they would go to the wiki and move that problem to a 'Claimed Integrals' section, and put their name next to it. This also allows me to see who is ready to present, before class starts, so that I can chose the order to have students present, and hopefully do a bunch of similar problems at the same time.
After a student presents their solution in class, while they are still at the board, I like to ask them little challenging questions. "What if that exponent were a 3 instead of a 2, would your solution still work?" "Can you do this as a definite integral from 0 to 1?" I also encourage the class to ask questions, though mostly (so far) they're pretty quiet.
We've had 2 days worth of presentations so far, and will need to have several more to give everybody time to present. Things seem to be going well. I was quite pleased that there were already more than 10 integrals claimed after the first day of class. The presentations so far have been very good, which I'm glad about. I think starting off with good presentations will give the students a good frame of reference. I've also gotten some phenomenal writeups, in which students carefully explained all of their steps - instead of just writing down their solution.
I'm pleased with this project so far, and have noticed a few things to change if/when I decide to try it again. I really want a student to claim an integral on the wiki only after they are ready to present a problem. I think in the future I will make sure to tell students that they will lose a point if they are asked to present their integral, but are not ready to do so (including not being present!). I may also have students, as a first assignment, go through the worked integrals and gather them up in to groups of integrals that look similar.
Though each student is only working one of these problems, I will have separate assignments (normal homework assignments) to make sure everybody does some of each type of integral. For example, now that we've seen most of the trig integrals in class, they have a homework assignment due on these sorts of problems.
My only real concern about this project occurred sometime after the first day of presentations. I'm worried that strong students will look at the list sooner (or be ready to claim an integral sooner, if nothing else), and will end up working easier problems. This is a two-fold issue, because it means the stronger students aren't being challenged enough, and that weaker students are potentially getting stuck with much harder problems. Perhaps I'll have a better sense if this is the case after a few more days of presentations. I'm not really sure how to avoid this though, if it is the case.
Wednesday, December 31, 2008
Some Parametric Curves
If you aren't following my other blog, you just missed two fun posts trying to find some parametric curves. I'll try not to spam this blog too much with links to my other one. But I enjoyed these curves, and thought you might too.
Happy New Year!
Happy New Year!
Monday, December 29, 2008
Teaching Calc 2
Last semester I had the opportunity to teach Calc 2 here at the University of Virginia. And I get to this coming semester as well. While my course evaluations from last semester went well, there's still plenty of things I want to change. I thought I'd post some of my thoughts here, in hopes of maybe organizing them and, more importantly, getting some feedback.
First off, some parameters. I'm teaching out of Stewart's Calculus book, chapters 7 (Techniques of Integration), 8 (Applications - Arc Length and Surface Area), 10 (Parametric and Polar), 15 (Multiple Integrals), and then 11 (Series (Hurray!)). Yes, we skip the chapter on differential equations (in addition to some scattered sections), and chapter 15 might typically be considered Calc 3 material (that is, after all, where it is in the book). But the course I'm teaching is 'coordinated', meaning that there are several sections of it, and we all have common exams. This also means I'm to cover the above chapters in the order I list them in (or, I suppose, I could switch 7 and 8, or 10 and 15). I get to teach 3 times a week, for an hour each time. Finally, I have a TA who will be running an additional 'discussion section' (alternatively called a '4th hour'). Oh, and I expect to have approximately 40 students.
One of my concerns is finding how to best use my TA. The first (and only) time I had a TA, she got the usual duty (for calc classes here, anyway) of answering whatever questions the students had ("do this problem please") and giving a shortly weekly quiz. I didn't really like the plan, but had no idea what else to do. I still don't like the plan, and don't yet have much of a better plan on what to do. Besides perhaps not having the quiz during the discussion section itself, to save time (which I heard was a complaint among other classes last semester). When I have run my own discussion sections in the past, I haven't given a quiz at all.
So, some thoughts:
I'd also like to incorporate technology more. The classroom I'm teaching in has a computer and projector. I'm pretty sure (though I'll have to check) that the computers on campus (in labs and so forth) have Maple and/or Mathematica. I feel like this will help immensely in visualizing parametric and polar curves, and multiple integrals. Plus it gives me more fun things to play with :)
Oh, and I was also debating about making videos of me working through problems, and post them on youtube. I'm not exactly sure why (besides giving me an excuse to play with such things more). My students last semester did ask that I complete more problems during class, but I don't really feel like there's enough time. Plus, I'd rather have them finish up the problems. I like to only work the new parts of a problem, and let them finish up the algebra (or integrals using techniques from older sections). I certainly could write up the solutions (I do love LaTeX), but I wonder if the video approach might be more appreciated by students? Less intimidating, perhaps? Maybe I should have them make the videos? Or allow more freedom - let them make presentations posted online somewhere? Or not have them do any such thing, to avoid copyright worries and things?
First off, some parameters. I'm teaching out of Stewart's Calculus book, chapters 7 (Techniques of Integration), 8 (Applications - Arc Length and Surface Area), 10 (Parametric and Polar), 15 (Multiple Integrals), and then 11 (Series (Hurray!)). Yes, we skip the chapter on differential equations (in addition to some scattered sections), and chapter 15 might typically be considered Calc 3 material (that is, after all, where it is in the book). But the course I'm teaching is 'coordinated', meaning that there are several sections of it, and we all have common exams. This also means I'm to cover the above chapters in the order I list them in (or, I suppose, I could switch 7 and 8, or 10 and 15). I get to teach 3 times a week, for an hour each time. Finally, I have a TA who will be running an additional 'discussion section' (alternatively called a '4th hour'). Oh, and I expect to have approximately 40 students.
One of my concerns is finding how to best use my TA. The first (and only) time I had a TA, she got the usual duty (for calc classes here, anyway) of answering whatever questions the students had ("do this problem please") and giving a shortly weekly quiz. I didn't really like the plan, but had no idea what else to do. I still don't like the plan, and don't yet have much of a better plan on what to do. Besides perhaps not having the quiz during the discussion section itself, to save time (which I heard was a complaint among other classes last semester). When I have run my own discussion sections in the past, I haven't given a quiz at all.
So, some thoughts:
- Chapter 7, Techniques of Integration, is concerned with learning four techniques (and when they are mostly likely usefully applied). One thought I had was to break the class into four groups, and have each group teach one section (because you learn best when you teach). Of course, I'd meet with the groups and make suggestions and corrections and oversee to make sure things went mostly ok. Another idea I had, and might be leaning towards, is a bit more complicated. I thought I might write up a bunch of worked examples (some from the text, some solved problems, some examples from other texts) to distribute to the class along with a collection of unworked problems. I would not permit them to look at their book, at all, during this part of the class. Instead, the idea would be that they should look at the worked examples, try to find worked examples that look similar to the unworked problems, and learn their own way through the technique. I rather like this idea of 'pattern finding', which I like to think is part of what mathematics 'is' - as opposed to 'learn these mechanical procedures and repeat'. I may also structure the class so that each student presents a problem to the class. This might get them to start early so they can have more freedom in which problem they do (I'm thinking just ask for volunteers during each class, they get to work any problem that hasn't yet been done). It might also occupy some of the time in the first few discussion sections, which would be convenient.
- In Chapter 10, on parametric and polar equations, I thought I might give them a project: find parametric equations (likely piecewise, discontinuous) that will draw their initials. I'd probably allow piecewise linear functions, but would certainly encourage (5 points of the project for non-linearity?) something a bit smoother. This makes me wonder if it would be a bonus, extra credit project, or count as a weekly homework, or if I should also find some sort of similar project from each chapter (or for each of the exams?) so the course has somewhat more regularity (1 random project seems odd).
- I'm almost certain, though I distrust my memory, that in my calculus class as an undergraduate, we were given an assignment on a section before we ever talked about it in class. We were expected to go home, read the section, go to office hours if necessary (or work in groups), and complete problems from the section before the next class. I loved it. I have friends here in grad school that say they would have hated it. But I was thinking I might structure one chapter this way, perhaps chapter 8 (arc length and surface area). I recall last semester noticing that chapter 15 wasn't the most friendly read, but I wonder if chapter 8 might be better for that. I think if I do this I might pick the first couple of problems from each section and have them due at the beginning of class. These first few problems are typically pretty easy.
I'd also like to incorporate technology more. The classroom I'm teaching in has a computer and projector. I'm pretty sure (though I'll have to check) that the computers on campus (in labs and so forth) have Maple and/or Mathematica. I feel like this will help immensely in visualizing parametric and polar curves, and multiple integrals. Plus it gives me more fun things to play with :)
Oh, and I was also debating about making videos of me working through problems, and post them on youtube. I'm not exactly sure why (besides giving me an excuse to play with such things more). My students last semester did ask that I complete more problems during class, but I don't really feel like there's enough time. Plus, I'd rather have them finish up the problems. I like to only work the new parts of a problem, and let them finish up the algebra (or integrals using techniques from older sections). I certainly could write up the solutions (I do love LaTeX), but I wonder if the video approach might be more appreciated by students? Less intimidating, perhaps? Maybe I should have them make the videos? Or allow more freedom - let them make presentations posted online somewhere? Or not have them do any such thing, to avoid copyright worries and things?
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