Homework Helper

I've been a little bit frustrated with the way the discussion session for my calculus class has been going recently. That time is set aside as a time for students to ask whatever questions they have, without me lecturing on any new content. Of course, generally the questions they have are 'can you do this homework problem?' or 'I got stuck on this problem, can we go through it?'. Generally, those are fine questions that I'm happy to answer. However, I'm getting the impression that many of the students have not yet looked at the assignment yet, and are just waiting for me to do half of it for them. Of course, this'll come back to bite them on the exam, but it's fairly frustrating all around. So I've been trying to decide what to do about it.

It occurred to me today that even just answering those questions asked by the students who have looked at the assignment isn't very efficient. If they've already started, but gotten stuck, it'd be fairly quick for me to sit down with them individually, find their error, and send them on their merry way. Even there, though, that's not what I should be doing. It's easy for me to spot errors, generally. Especially when I've already looked at the problem with several other students. But it would be hugely valuable for students to be able to find their own mistakes. It can be maddening trying to find your own mistakes, of course, but it's an important skill to have.

A good way to practice finding mistakes, even if they get all of their own problems correct, would be to help identify mistakes in other people's work. Of course, this process can be ironed out a little online. I am envisioning a system where students can go and enter the work they have on a problem, up to the point where they got stuck. Then other students could go and try to find errors in people's work. This way people that get stuck can get help whenever it's convenient for them (as opposed to waiting for office hours or something), and students can practice finding errors in work.

It seems there should be some sort of credits system involved. At the beginning of the semester, students have, say... 3 credits, or 5 or something. A credit gives you permission to ask a question. To earn credits, you submit a bug report on another person's question. Perhaps a bug report just identifies what line the error occurs on, without identifying the error. And I guess answers would need to be verified before credit is added to the person who submitted the answer. Perhaps the person asking the question verifies it?

That's about as far as I've taken the idea today. Clearly there'd have to be an easy way to enter work, perhaps with some sort of graphical formula editor. Also probably some anonymity, so you can see questions, but not who submitted them (nor who answered them?). It also seems like what might happen is that the people who have the most questions might have a hard time spotting other people's mistakes in order to earn credits to ask more questions. So perhaps there's a way to account for that. Something like... if you earn lower than an N on the exam, each point less than that gets you a free credit?

Anyway, that's a day's thought on the idea. What do you all think? Do you know of a system that does something like this already? Could something like the above idea be worthwhile and helpful? How could it fail? Where does it need improvement? What additional policies might you use?

GEER

Today the annual Great Eastern Endurance Run was held at Sherando Lake. In fact, it is still being held. With the 6am start time, and a 19 hour time limit for the 100k group, there are likely plenty of runners still out there. I only ran the 50k, so I (thankfully) have been done for a little while.

I've been signed up for this run since sometime in the summer, and it has been my primary training goal since then. What is surprising, to me anyway, is that I don't think I ran too many runs in all that time that were longer than 20 miles. I think no more than 5. I have, though, run every single day for the last 6 weeks (as long as my 2 mile run yesterday counts). I'd like to continue this streak, and run a little tomorrow, but I'm going to see how I feel.

This last week I've been eating lots of pasta, and the last two days several bagels. Yesterday I drank a whole lot of water, and some Gatorade. Also in this past week I watched 'Running on the Sun', a documentary about the Badwater 135 mile ulta-marathon. I think it was a good move (and movie), since it made me much less nervous about my own run. My run seemed entirely reasonable next to those guys.

All the same, last night I was reasonably nervous. It didn't help that it had been raining for two days, and the forecast called for more today. I also wasn't looking forward to the waking up at 3:30 part of the plan. Luckily I got to bed pretty early, and my cats actually let me sleep. At some point a thunderstorm woke me up, so I got to worry about running in one of those. The primary concern there being that the GEER run is out in, and on top of, some of the mountains near the Blue Ridge Parkway. On top of a mountain doesn't sound like the ideal place to be in a thunderstorm.

Waking up early gave me time for a bagel and a banana, and some water. A little after 4 I met with another runner (doing the half marathon) that I was carpooling with, and then we picked up a volunteer for the run on our way out of town. Around 5 we were down at the park, still earlier than many people, which meant we got a good parking space, and getting our packets wasn't too much of a hastle. A couple of trips to the bathroom, a pre-race briefing, and it was go time.

Of course, it was still pretty dark out. Most of the runners had headlamps, and I was running just carrying a flashlight. The first mile or so was on the road, so it wasn't too much of a problem. The next stretch was up a pretty steep trail, but even so, the light wasn't too much of an issue. I was running, at the time, with the one other guy I knew, and we were somewhere in the top 10. Which was a good plan, so we didn't have to worry too much about getting stuck behind people on the relatively narrow trail. Eventually the sun starting coming out, and we were on top of a ridge, running in the fog or clouds or whatever. It was quite nice.

The rest stops all had lots of goodies. I went through handfuls of gummi bears, as well as a few oreos and pretzels, and even a quartered boiled potato. The stops also had bananas, but also other candies, mini chocolate bars, and even twinkies at one point. I had brought along some Clif Shot Blocks, courtesy of my carpooling friend, 3 packets of Gu, and a powerbar. The Shot Blocks and Gu were a huge help, and I never needed the powerbar.

By mile 14 I had moved up to somewhere around 5th place. The way the course is setup, there are two places where you run out and then turn around, and run back. This gives you a good chance to see how many people are ahead of you (if you are keeping track), and gives some indication how far back you are. I was surprised that the first place runner didn't seem much further ahead.

Around mile 20 I took 4th place, and within another 2 miles had passed two more runners on a bit of an uphill. I was quite surprised by all of this, but it made me pretty excited. Surely the top 3 finishers get something cool! Of course, that left me looking over my shoulder for the remaining several miles. Generally in the runs I do I start off far enough back that I spend the entire time passing people, which is a great feeling. Running in the front, all you can do is get passed. That's not as fun.

All the same, I decided I was going to hold on to second place. That (along with all the gummi bears and other goodies) kept me pushing the remaining miles away. My legs had started hurting a bit, and I was worried that my right quad was going to cramp up. It was not appreciating uphills, but luckily the last 7 or so miles are primarily downhill or flat.

I had decided not to run with a watch. This was my first run at this distance, I just wanted to go out and enjoy myself and not stress about my time. At some point about half way through I caught a guy looking at his watch, and got the time from him. I felt a little bad interrupting his run, but it seemed to work out ok. When I'd been thinking about the run, I had hoped to finish in 6 hours. With the weather, I thought 7 might be more reasonable. In the end, I came in just under 5 hours, holding on to second place (just about 4 minutes behind first). Which felt amazing. I'm still pretty excited about it. I got not only a pretty cool jacket (even though it's got a big logo on the back), but also an additional running t-shirt (all runners got 1 for the event).

It's been about 7 hours since I finished running. My legs can definitely tell that they did something this morning, but I feel pretty good. This week one of my professors was talking about how when he goes running he goes for short, hard runs. He wants to get tired as quickly as possible. I told him I'd much rather go and run for a few hours and get tired that way. Today I realized that perhaps part of that is that with a nice long run, you still feel like you did something hours, or even days later. Sure, a half hour hard run will make you tired, but it's not the same tired. Either way though, as long as you enjoy it, that's what counts. Right?

Flatland, the Game

I was recently directed to the game here, and thought I'd forward it on to you, devoted reader. It's pretty entertaining, and a great way to think about dimensions. If you haven't read Flatland (wiki), you should probably get on that. After that, you can reward yourself by playing the game.

What happens in the game is two dimensional shapes (letters, numbers, etc) get sliced up into horizontal lines, and all you see is a movie of those horizontal lines. It is then your job to figure out what shape the came from. This is very much like the Sphere passing through the plane in Flatland.

Update 9/24: I was just reminded that Google recently did something where you can embed books. Flatland just happens to be one of them:


Hmm, I seem to not have it right. Suggestions anybody?

News in Class

I was happy this week about the two new Mersenne primes being found/verified (mathworld). It gave me something outside or the usual calculus material to talk to my class about. Don't get me wrong, the calc is, of course, fascinating. But when you're in the midst of it, day in and day out, a change is welcome, I expect. I feel the same way about my research, incidentally. Looking at it on the broad scale, it's all interesting stuff. In the daily work, it's terribly frustrating. But I digress (seems to be what I do).

So to start class off Friday I mentioned the news. I asked if anybody had heard about it, and can't say I was particularly surprised that nobody had. I talked briefly about GIMPS and about the importance of primes in modern cryptography. I repeated the statements I'd seen in various blogs about the number of digits being vastly larger than the estimated number of atoms in the universe. The students seemed to be enjoying it, for which I was glad.

At some point, one of my students asked "What do you want to do when you grow up?" That may not be an exact quote, but that was the question. This still strikes me as fairly random for the conversation. I guess he was looking at the math I was talking about, perhaps wondering why anybody would care, and if that was the sort of thing I hoped to work on when I was done here. I had to answer honestly that I didn't quite know what I wanted to do, though I do enjoy teaching.

I'd like to continue bringing in math news items for my calc class. I feel fairly well connected online, so that if something I could share were to happen, I'd hear about it. But I'm trying to remember what the last similar sort of math news I could share would be, and it doesn't seem like the sort of thing that happens often. Anybody have any recommendations where I should be looking?

Techniques of Integration

This semester I'm teaching Calc II from Stewart's text, Calculus. We start in chapter 7, which is about techniques of integration. One of my students made a very good observation early on in the process. We had already covered u-substitution and integration by parts, and were beginning trig integrals (integrating functions that are mostly build up from trig functions, products of sines and cosines and such). He realized that the 'technique' of the section was not, exactly, a new way to integrate. It was really just a way to re-write the integrand so that you could either apply u-substitution or parts. After pointing this out in class, I realized that he was correct, and that for our purposes, there really are only two techniques of integration - u-substitution and integration by parts. Everything past that is algebraic manipulations to change how the integrand 'looks'. That isn't to say the rest of it is easy. There are plenty of strange little tricks that pop up. Odd substitutions, or strange ways to split up the function.

I've made a point of trying to convince my class that they have to go home and do lots of integrals on their own. It won't matter how many examples I do in class (where the time is limited anyway, so we can't do too many). It is always easy to watch somebody doing math and think you understand, but then get completely stuck when you sit down to do similar problems yourself. Certainly there is value in doing examples in class. For example, trig substitutions are not something most people would be likely to come up with on their own. But after an example or two, the idea is something that can be readily used, and should be struggled with individually.

What I've been doing to present examples in class is to have the class guide me, as much as possible, through a problem. I put an integral up on the board and ask for ideas. We might go through a couple of failed u-substitutions or parts attempts before striking on the right way to proceed. But that's all part of the process, and it's good to see what happens when things don't work. After we get going, I try to get the students to tell me about each step. I hope that this keeps the class involved and focused on the problem. This means the examples take more time than if I was just presenting a quick path to the answer. So I'm, perhaps, not getting to as many examples as I could be, but I think I'm happy with how it's going so far.

We've still got partial fractions to go in the chapter. I think I'm happy with how the class is going so far, and hope that feeling continues.

Back to School

The semester, here at UVA, started this week. In fact, I kicked it off with an 8am calculus class on the first day. It's been a busy week, largely due to my failure this summer to use my time wisely, but it's been fun.

Last semester I didn't teach. To compensate, I worked as a tutor for athletes in their tutoring center. It was ok, and certainly had it's moments, but it's nothing like having your own class that you meet with several times a week. I was nervous before the semester that I wouldn't have my old enthusiasm for teaching. It's always been what's gotten me through grad school (both the teaching now, and the hope to teach when I'm done). This summer, though, I was frustrated and couldn't motivate myself to think about the semester. It didn't seem like a good sign, but it's just a few days in, and I'm feeling better already.

I've been reminded of the fun of starting the fall semester, with mostly first year (we don't call them freshmen here, I don't know why) students in my class. This is my 5th year in grad school, and the first class of my own that I taught was 4 years ago. That means most of that class will be graduating this year (I hope). I believe that today, walking around on campus, I passed one of the students from that first semester, but they made no indication that they recognized me. That'll happen, I guess. They probably forgot most of the math even quicker.

I always want my class to go well (who doesn't?). I want to show them interesting things, and the things they'll need to know. Of course, in calc classes (any math classes), there is no shortage of interesting things. It's harder to figure out what it is they need to know. Since my class is one of several coordinated calculus classes, I at least have a pretty good idea what will be on exams. Certainly that's something they'll want to know. Whether it's the "right" content for the exam or not is harder to tell. Do we teach something because it's content a later class depends on? Or do we teach it just out of tradition? Is there a national standard base-level curriculum at the college level that we're trying to follow? Harder still is determining if there is a better way to structure a class than having it be largely exam (and homework set) based. Perhaps there's a nice way to do the class that's project based? And just how much technology should we be using? The old "calculators question".

The start of the semester has it's frustrations. We use the online homework system webwork in my class, and I really like it. Of course, every time I use it, I think about other ways to have it set up. Since it's open source, I should probably try to contribute some of these ideas in code. Not right now though. For now, I've got the usual issue of making sure everybody can log in. Some changes were made to our setup here, but there still seem to be some login issues. My class is also set up to use an online waitlist for enrollment. Which means I'm getting lots of emails from students asking to get in, and if they have a chance to get in off the waitlist. And since they've now missed two class periods... it's a hastle.

It'll be good when the administrivia settles down, and we can concentrate on the math. We're doing improper integrals tomorrow (and probably a little into Monday), and then spending several class periods covering the rest of the chapter on integration techniques. I've got an idea I'm excited about for teaching it, but I think I'll leave that for a later post, when I can tell you how it went. For now, the Mythbusters are busting some lunar landing myths.

TN Roadtrip, Day 2 (13 Aug, 2008)

I had thought about getting a pretty early start to the morning to go for a run at an area of the park known as Cades Cove. There is a scenic 11 mile road there, with lots of places to stop and look at things, like historic log cabins. All of the roads in the park (that I was on) had lots of places to stop and look at the scenery, and probably all of the stops are worthwhile, if you've got lots of time. Anyway, the reason I had thought about doing this run was that the road is closed Wednesday (which it was) and Saturday mornings until 10am, to allow cyclists and pedestrians to have the road to themselves. However, when I talked to somebody at the information desk the day before, he mentioned that I was very likely to see bear if I did this. Seeing bear driving is one thing, but running is a whole new matter. Especially because one of the first things they tell you, for if you encounter a bear, is not to run. Besides his warning, at dinner the night before I had seen on the news that there had been a bear attack on a human that same day. It was the first attack since 2000, and billed as quite rare, but I was still a bit un-nerved. I also saw in the news that there was a local school that wasn't starting on time because they couldn't get their budget worked out. I was pretty surprised by this, but when I mentioned it to a friend, he said this was happening in several places. He said some of the schools are going to 4-day weeks to save some costs.

So I decided not to go on the run, planning instead on hiking a trail loop starting at Cades Cove. Cades Cove is less than 30 miles from the Sugarlands Visitor's Center, at the entrance to the park, and it takes about an hour to drive. I figured I could head out and do the hike, then drive the popular road loop. One of the first things I did when I started the hike was to pick up a walking stick. The main reason was that I didn't want to have to be finding sticks to defend myself with from a bear as it attacked me. I wanted to have one ready to go, just in case. Probably silly, but there you have it.

Perhaps a mile or so in on the trail, I came to a clearing with this log cabin:
While still basically in sight of the cabin, along the trail, I saw 2 dear in the woods. I was glad I was doing the hike, instead of one of the many people on the road who didn't get to see the deer. Sure, a cabin or two is probably interesting to see, but deer are more fun. They didn't seem too upset with me, and let me pass without running away. The trail continued slightly uphill for a short while, at which point I encountered my first bear of the day. It was right there on the trail, and actually I kinda came around a corner and saw it, so got closer than I was happy with. After backing off, and getting my picture,
I waited around and listened to the bear wander off slowly into the woods. When it sounded far enough away, I continued on along the trail.

From that point on, I was pretty nervous hiking. Every sound in the woods gave me pause. Not long (15 minutes, maybe) after my first bear encounter, I was heading up a larger hill, and heard something off to my left. I couldn't see anything, but it wasn't far off the trail. So I backed up, and tried making a little bit of noise (coughing, tossing a stone at a larger rock on the trail). I wasn't trying to scare whatever it was, I just wanted it to know I was there, so I didn't surprise it. Bear Grylls [wiki] taught me to make my presence known. Seems to me a frightened animal is more dangerous, and I didn't want to encounter any of that. So anyway, I stood there for a while, not seeing anything, and not hearing it wander off. So I tried heading up the trail again, keeping a keen eye out. Before I got too close, I saw that it was, indeed, another bear. They tell you, in addition to not running, that if a bear attacks you are supposed to make yourself look bigger, by holding up your arms and standing on a rock or tree stump. Being downhill, this knowledge made me fairly uncomfortable. I backed off again from the bear (couldn't get a good picture this time), and decided that perhaps I'd try my luck along another trail (one that I was actually more interested in, since it was in higher mountains).

On my way back down the trail, right where I had seen the first bear, I heard something that sounded large on the side of the trail again. This time the sound was coming up, toward the trail, from a little valley, so I couldn't see anything. I started moving faster, keeping myself pointed toward the noise, holding my arms up, and talking at the noise. I never did see anything, happily. Shortly thereafter, I passed a few hikers heading back in that direction, and told them to keep an eye out. They seemed eager to see a bear, so I hope they got the opportunity. I, on the other hand, was fed up with seeing bears, and was glad not to have any more encounters. However, the rest of the day I was nervous, and on high alert.

The next main portion of my trip was a long hike along the Bull Head and Rainbow Falls trails (in that order), and was probably the highlight of my Smoky Mountains touring. This is a 12+ mile loop, and there's an additional half-mile spur you can do near the top to bring you to some lodges near the peak of Mt. Le Conte, bringing the total to 13+ miles. This made me feel a little better about not running in the morning, since I'd still be getting a workout. The hike took me just more than 5 hours, but I was kinda pushing myself. The first 2 hours were nearly all uphill, which I simply loved (modulo the being nervous about bears bit). I saw only 4 other hikers this entire time, in 2 groups of 2. It's wonderful how much elevation you can get walking uphill for 2 hours. I was also amazed by how much the vegetation changed along my walk. The trip back down Rainbow Falls trail bring you past, get this - Rainbow Falls. Amazing huh? The falls themselves weren't entirely impressive, though I expect that at wetter times of the year this might not be the case. Even so, it's a beautiful area.
The falls are about 2.5 miles from the nearby parking area, so this section of trail is a little bit busier. One group I caught up to mentioned a mama bear with cubs that I'd apparently just missed on my way down the trail. The bears were apparently just on the other side of the creek from the trail. Probably the cubs were cute, but mama bears are especially dangerous, so I'm not sure how disappointed I am that I missed these guys.

After the hike, I followed the Roaring Fork Motor Nature Trail, which is a single-lane, one-way road. I think if you buy the \$1 visitor's guide, it has some information that accompanies the various numbered signs you'll see along the way. It was a quite nice loop, and fun to drive, with lots of nice scenery.
This picture was taken from a stop about a mile from the start of the loop. I almost wished I'd just stayed there the remaining time until sunset. I bet it was amazing. I'd recommend this loop to anybody who was in the area. It doesn't take too long to drive (I think it's only about 6 miles), and is quite pretty.

So there you have it. The Smoky Mountains Park is wonderful, and I highly recommend it. Stop in the visitor's center on your way in and buy some maps to both support the park and help you decide what to do with your time. If you plan on hiking alone, some pepper spray might also help you feel more comfortable about bear encounters. I found the webpage also to be helpful, with lots of good information about planning your trip. The trails I was on were all well marked, and very well maintained. There were signs at all the trail junctions indicating which trail was which, and also indicating distances to the next trail. If you are driving the park, pick any (better: several) of the little stop-offs to get out of your car and wander around. I've posted a few more of my (favorite) pictures over on flickr, if you are interested.