## Wednesday, December 23, 2009

### Desktop OAuth/Python Question

This post is slightly more tech-heavy than most of my recent posts. If that's not your thing, feel free to move on now.

[Update 20091223: NEVERMIND. I got it sorted. The url you re-direct a user to with OAuth doesn't need extra OAuth headers. So you really could just use webbrowser.open(). My bad.]

Suppose I want to interact with the twitter API via some python code running on my personal computer. Suppose, for grins, that instead of using the Basic Authentication I'd rather try OAuth (even though it's all running locally...). Part of the flow of OAuth authentication is that my script is supposed to direct the user to an address at twitter (oauth/authorize), with some OAuth specification headers in the HTTP request for that address (I hope I'm saying things within a few shades of correctly). Well, python provides the webbrowser module which should open up a url in the users browser of choice. And it does, pretty easily (based on my 1-test sample). The problem is that for the OAuth dealings, I'm supposed to pass additional HTTP headers, and I can't figure out how to do that with the webbrowser module. I tried creating a Request object, from the urllib2 module. If I were just making a url request using this library, I could make the Request object, with the extra headers, and things would go fine. But the webbrowser.open() method seems to want its url parameter to be a string, not a Request object.

So... how am I supposed to do this? Or am I not supposed to do this?

Am I supposed to use some other existing python based browser? How is the user supposed to feel like I'm not still in the middle of the authentication process? I mean, if my script displays a webpage using some graphical widget, and waits for the user's input, then I could just be grabbing their username and password while they log in to twitter, no? The point of having the user go to twitter and get a pin is that the user then tells me that pin. I don't put myself in the middle and grab the pin (or their username/password) somehow.

Does any of this make sense? Can somebody point me at a solution? Existing code that solves this problem?

## 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 $\sqrt{3-2x-x^2}$, 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.
1. 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?
2. 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, and it is clearly the best way to do things and it's going ok and nobody has complained.
So let me ask again, what is the role of the university, and how will the university adapt to the changing world? Is the role to grant degrees, and so is the role of my lowly calculus class to funnel students on to further classes until we can get them out of here with their paper in hand? What market are universities in? If we know, then we can see if our institutional inertia is good or bad. If we don't know, we should probably get on that.

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?

### Higher Ed

I've been wrestling recently with identifying the role of higher education. I'm not much of a wrestler, physically or intellectually, so I thought I'd see what I could put into words and have you folks comment on. Thoughts of yours, or links to thoughts of others are much appreciated.

What brought on my recent concerns, or possibly made the pot boil over, as it were, was some frustration in making a final exam for my calculus course. The course I teach is one section among several, and we all have common exams. When we went to make the exam this semester, there was some disagreement about what should be tested. The lack of a prescribed vision for the course (at least, one I was aware of) started to upset me. And then I decided that such a vision should be accompanied by a broader vision of the math courses offered by the department. It should include formal indication of how the courses fit together, what is assumed of students coming in to each course, and what is assumed of students passing a course (I'm thinking something more formal and precise than the short list of pre-reqs for a course). And then I wondered where this vision came from, if it was internal to the department or how much say other departments had in what they wanted math classes to require. I'm not expecting the English department to care at all what is covered in Calc I or II. But perhaps Economics has some input for us? Or other departments? And then I wondered what the point of any of it was. What is the role of higher education, and how does my Calc I or II class fit into that?

Clearly things were getting out of hand.

Luckily, my thesis advisor turns out to have the title of Director of Undergraduate Affairs. Furthermore, he was happy to talk to me about these things, and brought a welcome level-headedness up against my relative insanity. Of course, I'm sure he won't mind much when I get back to research... :)

He pointed out that probably having formal guidelines for all courses is a bit much - that professors teaching upper level courses should be granted plenty of flexibility. I can't really argue with that. I have vast respect for all of my professors. But this flexibility doesn't really help in getting a vision for Calc I or II together. And if the courses are coordinated among several sections, there should be some vision backing it, right?

Part of my initial inquiry was if there was a curriculum review that happened periodically, and how long ago the most recent was. My guess was that if there was such a thing, it happened before nearly all of my students had personal laptops (or possibly even personal computers in their dorm room). I know I'm a bit of a nerd, but I really think it is important to adapt to the changes brought about by each student having their own computer (that they maybe can carry in their pocket!) connected to the internet. The explosion of infinite goods and real-time global communications, for "everybody" (yes, I know I live in a privileged region, or whatever is the PC way to say it. I'm sorry?), has been changing, is changing, and will continue to change... well... nearly everything. When the fundamentals of economics and communication change, what isn't impacted? [Side note: I know very little about economics, I'm probably saying things incorrectly]

Bringing things back down closer (somewhat) to the level of a graduate student who knows basically nothing about anything useful... how does the internet change higher ed? What happens when lectures become an infinite good, available to be watched freely online? What happens when the content that makes up our textbooks is available in a multitude of places online? What happens if students have direct access to experts in any field at campuses on the other side of the globe? Gah, I'm getting out of hand again.

Two recent posts online have been on my mind a bit as I think about this question:
1. A post on ars technica about the difference between the movie industry and the music industry. For some reason this article made me realize that I probably wasn't thinking much about what was distinct about higher ed, as opposed to the entertainment or news industries, and what that meant about the impact of technology.
2. A video on techdirt explaining the innovator's dilemma. This video makes me think that in order to adapt, universities should be asking what market they are in. I've been phrasing the question in terms of what role they play, but I think maybe they're pretty similar questions. Are we in the horse-and-buggy market, following the example in the video, or the transportation market? Are we in the market of churning out degrees? Are we in the certification market? Is that our primary goal and purpose?
I truly hope that I'm not saying anything surprising to people involved in higher education (or other fields!), or asking questions you've not heard before. I really do hope that I'm late to this conversation - that smart, dedicated people in the right places are asking these questions. Moreover, that they are coming up with some answers. I hope that I am just not seeing this conversation at my university, or at others, because I am not involved enough somewhere.

Anybody have some thoughts or links for me?

This post is somewhat of a part 1 in a series. The second focuses more on teaching calculus.

## Monday, November 23, 2009

### Phones

Today I made a few changes to the way I intend to interact with phones, and I thought I'd see if any folks out there had any advice for me about these changes.

I switched to a pay-per-minute plan for my mobile phone today, and also signed up for Google Voice (thanks to an invite from a twitter follower), with a new Google number. My mobile plan is now 1 dollar every day I make or receive (actually answer) a call, plus 10 cents for each minute of those calls.

So here's what I'm planning on doing: I'll tell people my Google number. They'll call that, and it'll call my phone, and I'll not (in general) answer it, but I'll notice that somebody called because my phone will ring. Then I can wait for them to leave a voicemail, and check it online (assuming I'm near a computer (which I probably am)). If it's important (and the person isn't online where I can just chat with them) I can then call them back.

Perhaps this is selfish and a hastle for the people that call me. But there simply aren't that many people that call me, and most of them (Mom, e.g.) probably won't mind or even notice the slight hastle. And if I know somebody will be calling (a friend arriving somewhere, say), I don't have to let the call go to voicemail. And I can tell folks that if they'd really like to reach me, now and on the phone, they can just call twice or something.

I thought I'd see if anybody out there does, or has done, or has had a friend do, something similar. If you've got any comments of feedback about my new scheme for being a cheap-ass hermit, please leave one below. Don't bother calling me :)

That's basically the end of this post. But my original version of the post started with the story behind making these changes, and it's all written out, so I thought I'd include it anyway. If you don't care, then go find something else to do, I won't be offended (I won't even notice, but even if I did, I wouldn't be offended). If you're bored... here's the story:

I think I've never really been a fan of talking to people on the phone. Certainly not recently. I'd almost invariably rather talk to somebody via email or instant message. But I've got a phone, because apparently you've gotta have one. And, I'll concede, they have their uses.

When I started grad school, I got my own mobile phone (look at me all grown up), and an individual phone plan. I've been on the cheapest mobile phone plan since then (5+ years now). I have never gotten particularly close to using anything like my allotted minutes. My most recent bill claims I used 6 minutes total. Not exactly something I was happy paying 50 dollars a month for, but I was too lazy to look into many other options.

The other day I accidentally left my phone in my pants pocket when I put those pants through a cycle in the laundry machine. It was an old phone that I'd gotten for free for signing up for whatever plan anyway, so I can't say I was too upset when it came out of the laundry and wouldn't turn on. I was mostly amused, and glad it didn't do any damage to the laundry machine (water and electronics being what they are... I figured).

After letting it sit for a few days on the hope it might sort itself out, with no luck, I decided it was time for something new. What I probably most wanted to do was get a Droid. This would mean paying more for my phone service, but it'd be paying for something I'd almost certainly use. Not the voice service, so much, but certainly data. However, I listened to the voice of reason (for now :)) and decided instead to switch to a pay-per-minute plan with AT&T (my current provider, inertia being what it is). I went and got a new phone (the cheapest), and am now (after some slight hastle with the SIM card, and having lost all of my contacts) on a plan where I pay 1 dollar every day that I make or receive a call plus 10 cents per minute that those calls take. While I certainly can see that some months will be more expensive than others (due to travel or getting stuck on hold doing something stupid), I have a hard time believing any will be more than the 50 dollars a month I was paying. That's something like 6 minutes of calls every day, or a couple of hour long calls scattered through the month. I'm not sure there's anybody I want to spend an hour on the phone with.

None of that is particularly interesting or exciting, I suppose. It's new for me, so a little exciting, but I can't see why you'd care. What's more exciting is Google Voice. Thanks to an invite from a twitter follower, I now have a Google number. In fact, it's HAM-BLET (in an area code that isn't where I am), which is a little fun (ANIDIOT wasn't available). And I guess the idea is I have the number... indefinitely. And then I can add any of my usual phone numbers (the one I've had for a while now, e.g.) to my Voice account, and whenever somebody calls my Google number, I see the call on my normal phone. And if I decide to not answer, Google will record the voicemail for me and send me an email (or text message, but I turned that off) with a transcript, or I can even listen to it online. That's probably the most exciting thing I've seen in... rather a while. Pressing some buttons online and having my phone ring is pretty magical.

## Wednesday, November 18, 2009

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".

## 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.

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...

## Monday, November 9, 2009

### Still Right Here

This post was inspired by my lack of having posted anything else here in a while (2 months, to the day, apparently). Of course, if you've got nothing to say, which I don't, then not posting isn't a bad thing. Another inspiration for this post is that when I sat down to do "real work" (research, toward my Ph.D.) today, I found that I was on the last page of yet another notebook. Seems like a good time for reflection. I wish I could say that I felt like this last notebook had useful ideas. Or that I felt that way about any of the previous notebooks. I guess if it were true I might be out of here by now, or on my way out in the spring.

This post is fairly personal. The only thing you'll learn about, from reading it, is me. And I'm not a particularly interesting subject, I promise. Go find something else to do, there's plenty out there.

For some reason this academic year has been a huge source of confusion and frustration for me. I'm now in my 6th year of graduate school (for math, in case you forgot that part). For the past 5 years, I knew why I was here: I wanted to be a math professor. Maybe I still do, but I'm not so sure any more. My thesis advisor says he thinks I'd make a good prof at some small school, which was always the goal. And yet, I have a hard time convincing myself that this is still what I want. It's sort of an odd feeling to have your main plan in life for... a decade?... not really matter to you any more. Or, to maybe not matter.

I only have some vague idea(s) why I don't care as much about being a professor as I used to. I'll see what I can put into words, as much (more) for my benefit as yours (hopefully you stopped reading around the end of the second paragraph).
• Research sucks. Or I suck at research. Or... something like that. I've been reading math books for fun since high school (after I read all the books on sharks at the local library, and then decided I wasn't one for the water). I still do. And I love reading about math. But perhaps "doing math" is not something I care much for. I know, I know... math isn't a spectator sport, and... if you aren't reading with a pencil and paper and trying to guess what comes next, you're doing it wrong. But F that. I love reading math, the way I read math. If the way I read math means I'm not a "real mathematician", then maybe I shouldn't be here anyway, or shouldn't be teaching the next batch of math students.
• How am I supposed to be a professor, and tell my students how to learn math, if I don't do it myself? How can I tell my students to go home and work more problems (a habit I never had), and bang their heads against problems for a while (when, every time I sit down to do research I find something else to do as quickly as possible)?
• Sure, I can tell my students about math. I can tell them definitions and theorems and how to work problems, and maybe even tie it all together in some meaningful way. But lots of people have already done that, and their work exists in textbooks and, increasingly, online.
• Speaking of online, I feel like (and I know I'm not the only one) higher ed. (and probably other ed., and plenty more) is going to be going through a bit of an upheaval in the near future. I'm not sure I see how small, private, liberal arts colleges (like the one I went to, and always envisioned myself teaching at) are in a sustainable position currently. They are too expensive, and for what? I believe that many people are going to start recognizing that the diploma you get from such an institution isn't as valuable as, say, an impressive online resume, which is now something anybody can create with little effort (besides the "doing things that go on the resume" part). People can show everything they are capable of online, for everybody to see. What good is another diploma in relation to that? (I know that a diploma is still good... I'm not going to argue any of the things I say here)
• And also, while I'm on the subject of "online"... the experts are out there posting work online. Awesome teachers are posting full lesson plans, and all sorts of incredible resources. And my students could get to it as easily as I can. What extra value do I bring to the table? A convenient face to bounce ideas off of, to ask questions to (before thinking about the problem long enough alone)? Scheduled hours when I'll be around? I think there is a place for web collaboration tools in education, and I'm not sure how I complete with the sorts of individuals that my students have access to online.
• And a final thought: even going in to grad school, I was making a choice between grad school for math, or grad school for computer science. By the time senior year rolled around I was decided on math. I know at least one of my closest friends at the time was surprised. I sort of wish I had talked to my advisor and my CS professors a bit more about my decision, before making it. These days, I feel like I was probably wrong. I can spend all day online reading about computer/tech/programming stuff. I'll work on Project Euler problems, happily, until I solve them - in contrast to research, which I have a very hard time convincing myself to spend even an hour a day "doing". I don't know, maybe this is just a case of "the grass is always greener". Also, Project Euler problems, from what I've seen, aren't meant to be long, whereas math Ph.D. problems are sort of meant to take a little while. But what gets me excited are the projects I want to work on as a programmer, not as a mathematician.

So, anyway. I don't really know where I am. I don't really know where I am going. I'm apparently in not too much of a hurry to find out. I've killed another hour that I should have spent on research.

And, yes, the title of this post is a nod to Tool.

## Wednesday, September 9, 2009

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 $\int_1^{\infty} \frac{1}{\sqrt{x}}\ dx$, 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 $(t^2-1,t^3)$, 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 $x$, 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.

## Tuesday, August 25, 2009

### First Day

Today was the start of yet another semester at the University of Virginia. The last few days I've been depressed and frustrated and probably a whole host of other negative things, but I think "the routine" starting back up again is helping me out of it. Nice to know that I still love teaching. It also helps that I have friends that treat me far better than I deserve, but that's a story for another day, perhaps.

I had my first class this morning. In fact, I had an 8am class, so I was one of several instructors who taught students their first college class ever. Kinda a fun thought. I hope I didn't screw it up too bad.

I'm teaching Math 132 again, which is Calculus II for math/science folks. I enjoy the course. Which is nice, since this is my fourth time in a row teaching it.

Since I've taught this course several times recently, I'm pretty comfortable with how the material flows, and how I want to make it flow. That, coupled with whatever nonsense I was going through this last weekend, means I hardly prepared for today at all. I had in my mind that I was going to walk in and ask students to tell me about integrals. Just whatever they knew, completely open ended. About 20 minutes before class started I thought maybe it'd be good to review a couple basic derivative rules too - power, e^x, sin, cos. We also talked about some trig identities, since students brought it up.

It's always fun to ask a group of 30-40 students what an integral is. You're bound to get many answers, and I've always been pleased that many of them were ones I had hoped for. It's also interesting to hear ones you weren't expecting. Today we talked through antiderivatives being derivatives "the other way". When asked what antiderivatives were, there was a lovely pause after I fained ignorance about what "the other way" meant ("write it right to left? bottom to top?"). I could feel the class thinking about another way to say this. We got around to it in time. Then we moved on to definite integrals as areas under the curve (but, really, between the curve and the axis, and really the signed version of that). And how they are related to Riemann sums, and how that was different from just a finite sum of rectangles (and did rectangles have to be below the curve? all the time?).

It was a great discussion. I had a wonderful time, and the students seemed to be reasonably content as well. I was shocked to hear what sounded like an entire classroom full of students tell me the derivatives of several functions, basically in unison. At 8:15 in the morning. I hope they can maintain this energy throughout the semester.

I also hope that the course setup can be like this many days, where it's not so much a lecture as a group discussion, with them leading the way. We'll see how that goes. I'm not expecting to try it all of the time. I'm sure that carefully planned examples will be a necessity before too long. I get kinda frustrated thinking about which examples to work. No matter what I do, the problems will always be easier when the students watch me do them, or we work through it together in class. They have to try problems themselves. They have to get stuck. It's the pain about math.

We meet again tomorrow, though, thankfully, not nearly as early. Tomorrow we start techniques of integration, kicking things off with "undoing the chain rule" (u-substitution), and perhaps hinting at "undoing the product rule" (parts), depending on timing. After that, I'm making them learn the other techniques on their own. The other techniques frustrate me. They're not techniques of integration - they are algebraic manipulations to try that come up in lots of integral problems (the problems that show up in calc textbooks anyway). And I would have no idea how to respond if a student asked why they are useful. They're doable by machine (wolframalpha.com, we've been through this before). My hope with making them learn this by themselves, from the textbook (and I encouraged them to find things online, and share what they found with the class), is that the exercise becomes "learn how to learn math by yourself" (well, starting off by yourself, and finding help where you can) instead of "learn these particular tricks". In the long run, that's a more useful thing anyway. In the short run, it might cause some issues, since the midterm will have a hard time testing if they've learned how to learn math by themselves. After they've turned in their first assignment on these techniques, I'll do some review in class, and then give them some more practice problems before the exam.

## Saturday, August 15, 2009

### Carter Mountain Orchard

Today I decided to go visit the Carter Mountain Orchard.

I'd never been, but enjoyed it, and recommend it if you're around Charlottesville. Especially if you want fruit. But you could also just go wander a little, it's got some nice views, as you might expect for the top of (what classifies around here as) a mountain.
The white building in the middle of that picture is the UVA Hospital, or so.

After wandering around the little country store they've got, I decided I'd go pick some fruit. It's enjoyable to just walk around between fruit trees.

I was surprised to notice that not all types of apple grow the same. I have this image in my head of apple trees with all of the apples isolated, hanging from little stems. Apparently this isn't always the case (or perhaps today was just early in the season?). "Golden Ginger" (I think) seem to grow in pairs, right next to the branch:
while these ones (possibly Red Delicious) grow more in clusters:

After a small sampling of apples, I wandered over to the peach section. The first several trees I saw had no obvious peaches, and I worried that I'd missed the season somehow. As I walked, though, the trees had more and more peaches. Some were a bit high for me (I saw people with little ladders, I wonder if you can rent them), but there were still plenty of delicious looking peaches in arms length.
Many of these peaches are still ripening, so there's still time left this fall to get delicious peaches.

The peaches are on the opposite side of the mountain from cville, with more nice views:

I kinda liked this tree:

On my way out, I stopped in the store again and picked up, among other things, some donuts. I'm always a sucker for donuts, and these looked delicious. I'm happy to report that they did not disappoint. If you don't have a particularly sweet tooth, though, you may want to share one with somebody.

## Sunday, July 19, 2009

### Handling Personal Issues

I never have any idea what to do with students who are going through some sort of difficulty outside of the classroom. I understand that bad things happen all of the time, and there's never a good time for it. But at the same time, I don't see how it helps for a student to tell me what they are going through, even just in broad terms. They don't have to tell me if there is some family issue, or relationships, or health, or... anything. There are outside circumstances, I understand, and that's all I really need to know about.

But I don't know how to actually interact with these students when they come to my office and want to tell me what is going on. Most of me wants to stop them from talking about what, in any general or specific terms, is going on. There's nothing I can do about it. But I don't want to come across as not caring. Of course I'm sorry that they are going through whatever difficulty it is.

## 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.

## 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?

## 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.

## 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!

## Thursday, May 21, 2009

### Octagons

I've been at the University of Virginia for 5 years now. I've walked on campus, then, many hundreds of times. It's a pretty campus. Since my most recent move, I've adjusted my walking path to the office. Now I typically walk along some of the "garden" paths, just off of "The Lawn".

In case you're curious about those wavy walls, that was Mr. Jefferson's idea. And we couldn't be more proud.

But in all of my walks, up until just recently, I never noticed anything interesting about those little posts lining the driveway in the picture above. Then, from above:

Octagons!

## Wednesday, May 20, 2009

### The Education and Music Industries

Recently, a friend of mine pointed me at the article, "Psst! Need the Answer to No. 7? Click Here.", from the New York Times. I found it an interesting article, and wondered how many of my students were using online services to look at former students' notes, or for solutions. Every semester, I put a question on my course evaluations asking what outside resources students used. This is mostly to help point me at new and helpful sources. However, I've yet to get a direct reference (besides "Google"). It's always frustrating.

But anyway, after reading the article, and observing, as my friend pointed out, that all of the solutions for the calculus text that we use (Stewart) are on cramster.com, I was struck by a sort of comparison with the music industry. A decade ago (and still) the music industry was put into turmoil because it became easy to put music online, accessibly and freely. Whole albums and individual songs were there for the taking. The parallel to seeing whole solution manuals, and solutions to individual solutions, struck me.

I like to think there is an important difference. Downloading songs online, without paying for them, didn't hurt the consumer (directly - now we've gotta deal with all sorts of crap, but that's not my point). Besides, I suppose some might argue, a sort of moral degradation or something. With solutions being freely accessible online, it seems like only the consumer (student) is being hurt (to me, at first glance, anyway).

Of course, they're only being hurt if they aren't appropriately using the solutions. If they take the easy route on homeworks, and copy a few solutions, they'll likely have trouble at test time (if they don't, more power to them - or not). I don't think there's any argument that these resources can be used to improve the learning experience, instead of cheat it.

Should I spend some of my class time teaching students how to effectively use solutions manuals? Does anybody point their students at solutions manuals (or similar things online)? Do you incorporate them into your teaching? Or does it seem a non-issue, because students don't use them? Are students more likely to come to office hours to get help?

It occurs to me that while cramster might have a certain convenience, the solution manual for our textbook is available for short-term (something like 3 hours I think) checkout at the library.

## Wednesday, May 13, 2009

### It's!

Leonhard Euler's Flying Circus!

That will be the home of the little Python/Sage learning group I mentioned recently.

Now, let's see if this thing gets off the ground...

## 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.

## Saturday, April 25, 2009

### Long Exams

I was recently struck by the idea of writing an exam that was purposefully too long. Basically, the instructor just writes down whatever problems they come up with, related to the subject at hand, and give the complete list of questions to the students. Tell the students they aren't expected to get through everything, that they should look through the list and attack problems they know how to do. And if students finish all the problems they know how to do before time is up, they just keep trying as many problems as possibly. Students have whatever fixed amount of time to get through as many problems as they possibly can.

I was wondering if anybody out there had tried this, or what people's thoughts were about this idea. It seems like setting up an exam this (kinda lazy) way, the instructor can easily see what topics students are comfortable with. I suppose this is possibly the case for more traditionally designed exams. Perhaps we should ask students to rate how confident they are about their answers on exams?

The first question about this method probably is about grading. I'd say each problem is assigned a point value, before being distributed to students. After the exam, the instructor grades whatever work students turn in, and this gives them a point distribution for the class. The highest grade gets an A (presumably), and then you work out what to do with lower grades, perhaps based on the distribution that arises. Perhaps you set a level of minimum points you're going to allow for a... C say, to make sure the students don't conspire to all just do one problem and all get the same grade?

To give credit where credit is due I should perhaps describe the circumstances that led me to the idea. There are 5 sections of the course I am teaching this semester (Calc 2), and we have coordinated exams. This means the 5 instructors all meet sometime before the exam, and write a common exam. The setup we have adopted this semester is to split up the sections that are on the exam, and have each instructor write questions from the sections they are assigned (and whatever other fun questions they want). Then when we meet we have 5 pages of questions, one from each instructor. We meet and decide which problems to keep, and then the exam gets written.

At this last meeting we had, Katherine, one of the other instructors, joked that we should just make copies of the sheets we were looking at, and hand the set to our students as their exam. Naturally, this was the inspiration for the idea asked about above.

So... thoughts? Anybody tried it?

## Monday, April 20, 2009

### Learning Group Name?

I recently ran across projecteuler.net, a collection of intriguing programming puzzles with a mathematical bent. Before long, I had decided that it would be fun to get a group of my fellow UVA math grad students together to work through these problems. I've wanted to learn python for a while now, and thought perhaps others might as well, and that using the problems from projecteuler.net would be fun. So I sent out an email and have gotten several others who would like to join me, which is pretty encouraging. We're going to start after the semester ends, which is just a few weeks away.

When I was thinking about the group initially, I thought perhaps we'd organize some meeting time and talk about our code. But then do we print out our code and pass it around? Write it on the chalkboard? Perhaps bring in thumb drives and a laptop and projector and present out code to each other? And then I thought maybe just putting all of the code up on a group-run blog would be the best idea. We don't have to worry about organizing meetings, people can look at anything on their own time, and, for what it's worth, our work would be out in the wild for anybody to see.

My question is... can you think of a clever name for such a blog (one that you'd happily let us use)? My first thought, something like "Let's Learn Python", or so, isn't hugely fun. I feel like there's potential using π instead of "py" in "python"...

Once we get going, you can expect to see a link.

Update 20090513: Here it is: Leonhard Euler's Flying Circus!

## 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:

## Tuesday, March 17, 2009

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.

## 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.

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.
1. 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?
2. 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?
3. 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?
Thanks for any feedback.

## 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'.

## Saturday, February 28, 2009

### Acronym

A
Big
Cod
Descended
Energetically
From
George's
House
In
Jupiter.
Kevin
Left
My
Never
Open
Popsicle
Quarry
Running
Stupidly
Toward
Undiscovered
Volcanoes
Where
Xylophones
Yodel
Zanily

Ah, the senselessness of game nights.

## Friday, February 13, 2009

### Layout Update

I realized that I could put a feed widget in the sidebar here, containing the feed for the math fork of this blog over at wordpress. Of course, I could also just put a friendfeed widget there, and replace my reader share and twitter feed. But I haven't decided to do that yet.

So I'll stop mentioning, here, when I write a post over at my wordpress site. If you aren't yet subscribed to that blog, and aren't planning on it, I promise I'll stop mentioning it now. I guess that means I have to come up with more non-math things to write about here...

## 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).

## 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.

## Wednesday, January 7, 2009

### Cumulative Final

In less than a week my Calculus 2 class will be starting up. As you may have noticed, I've been thinking about ways to run this class differently than I ran it last time. One of the things that has come up in my thinking is how to structure exams. Last semester we had 3 midterm exams and a cumulative final. What I hope to try this semester is two midterms and a non-cumulative final.

First, some background from last semester. Our first midterm covered 2 chapters, the second midterm covered 2 more chapters, and the third covered all but two sections of a single chapter (the chapter on sequences and series). The remaining two sections of that chapter were untested going in to the final. That insured a couple of problems from those sections on the final, as well as all the other material.

What I think I'd rather do is have 2 midterms, covering the same material as the first two midterms last semester, and then have our final exam cover that whole last chapter. Each of these exams will be given the same weight for the overall grade, instead of having a somewhat more heavily weighted final exam.

I think to explain why I like this idea, I need to mention the content that we cover in somewhat more detail. The first midterm covers techniques of integration and a few applications (arc length and surface area for surfaces of revolution). The next exam covers parametric and polar curves (derivatives, areas, and arc lengths) and iterated integrals. The third midterm, last semester, covered sequences and series (definitions and convergence tests) leaving Taylor series for the final.

So, why do I not want a cumulative final? Looking at the 'old material' that students would need to go back and learn, I see a lot of formula memorization. "Make this non-obvious trig substitution when an integral involves...", "to break up a function using partial fractions, do this strange procedure", "the formula for the (arc length, surface area) of a ((polar) function, parametric curve) is ...". These are things I want my kids to know about at the end of the semester. But if they forget formulas, or the appropriate substitutions, I'm perfectly ok with that. They should be allowed, after this semester, to look up all of these things in a book. I expect that the techniques of integration won't be used by hand by students again (unless they end up teaching calc), because they will be permitted to use computers or tables of integrals in the future. And if they are needed, that's what the textbook is for.

Ask any former calc student for a technique of integration. If you're very lucky they'll remember that sometimes something like a trig substitution is useful. Ask them how to find the arc length of a parametric curve. I'd guess you'll see a lot of blank stares. Same goes for polar areas. Why should I test my students twice on material I'm happy to let them forget details of, after demonstrating competence at least once? My vote is for non-cumulative finals.

Thoughts? Why am I wrong?