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).

## Monday, January 19, 2009

## Saturday, January 17, 2009

### Integration Techniques Discovery Project

I decided to try something new this semester, in terms of how I teach techniques of integration (trig integrals, trig substitution, and partial fractions). Last semester I just lectured through it, presenting as many examples as time permitted. We got through it, but it seemed like it could have been improved.

This semester, I decided to make a project out of it. I decided I would teach u-substitution and integration by parts (which, in my mind, are the only "real" techniques of integration), but let my students learn the other techniques on their own. To support them, I gave them a list of 40 integrals, and solutions for each integral. I specifically told them I didn't them want to look in their book for this part of the class, only look at the worked examples. The examples were, essentially, all the integrals from the appropriate sections of the text, as well as whatever other examples I had found last year to use. Additionally, I gave them a list of 50 integrals (I have about 35 students in my class), without solutions. The idea is that they will look at the unsolved integrals and find one that looks interesting or familiar (or just pick one at random). Then they will consult the solved integrals, looking for integrals that seem similar, and analyzing their solutions. After studying the given solutions, they try to apply similar tricks (substitutions, ways of re-writing) to the integral they chose to solve. Once they have solved an integral, the will present it to the class, and submit a writeup (which will be posted online to share with the class).

I decided it wasn't particularly efficient to print out copies of the handouts I had made (page of worked integrals, page of unworked integrals, 13 page packet of solutions). Instead, I posted each as a pdf on our course page (UVA uses some tailored version of 'Sakai', by my understanding). To make things even more useful, I also made a webpage for the worked integrals. The base page just has the list of integrals, without solutions, and then clicking on an integral reveals its solution. This means that when a student is looking for integrals similar to the integral they have chosen, they can see the whole list of integrals at one time. At some point I thought it would be nice if each student had their own webpage for this, or something, so that they could rearrange the integrals and group them to their heart's content. But I decided to leave that up to them.

My original thought for this project was that I would simply start every class by asking 'Who has an integral to present?'. Whoever raised his or her hand first got to present. I figured this would encourage students to present as soon as they were ready, to avoid getting their problem stolen. It also meant students would start this project early, to try to get an easy problem. However, before the class started I decided to make a wiki page for all of the unworked integrals. When a student was ready to present a problem, they would go to the wiki and move that problem to a 'Claimed Integrals' section, and put their name next to it. This also allows me to see who is ready to present, before class starts, so that I can chose the order to have students present, and hopefully do a bunch of similar problems at the same time.

After a student presents their solution in class, while they are still at the board, I like to ask them little challenging questions. "What if that exponent were a 3 instead of a 2, would your solution still work?" "Can you do this as a definite integral from 0 to 1?" I also encourage the class to ask questions, though mostly (so far) they're pretty quiet.

We've had 2 days worth of presentations so far, and will need to have several more to give everybody time to present. Things seem to be going well. I was quite pleased that there were already more than 10 integrals claimed after the first day of class. The presentations so far have been very good, which I'm glad about. I think starting off with good presentations will give the students a good frame of reference. I've also gotten some phenomenal writeups, in which students carefully explained all of their steps - instead of just writing down their solution.

I'm pleased with this project so far, and have noticed a few things to change if/when I decide to try it again. I really want a student to claim an integral on the wiki only after they are ready to present a problem. I think in the future I will make sure to tell students that they will lose a point if they are asked to present their integral, but are not ready to do so (including not being present!). I may also have students, as a first assignment, go through the worked integrals and gather them up in to groups of integrals that look similar.

Though each student is only working one of these problems, I will have separate assignments (normal homework assignments) to make sure everybody does some of each type of integral. For example, now that we've seen most of the trig integrals in class, they have a homework assignment due on these sorts of problems.

My only real concern about this project occurred sometime after the first day of presentations. I'm worried that strong students will look at the list sooner (or be ready to claim an integral sooner, if nothing else), and will end up working easier problems. This is a two-fold issue, because it means the stronger students aren't being challenged enough, and that weaker students are potentially getting stuck with much harder problems. Perhaps I'll have a better sense if this is the case after a few more days of presentations. I'm not really sure how to avoid this though, if it is the case.

This semester, I decided to make a project out of it. I decided I would teach u-substitution and integration by parts (which, in my mind, are the only "real" techniques of integration), but let my students learn the other techniques on their own. To support them, I gave them a list of 40 integrals, and solutions for each integral. I specifically told them I didn't them want to look in their book for this part of the class, only look at the worked examples. The examples were, essentially, all the integrals from the appropriate sections of the text, as well as whatever other examples I had found last year to use. Additionally, I gave them a list of 50 integrals (I have about 35 students in my class), without solutions. The idea is that they will look at the unsolved integrals and find one that looks interesting or familiar (or just pick one at random). Then they will consult the solved integrals, looking for integrals that seem similar, and analyzing their solutions. After studying the given solutions, they try to apply similar tricks (substitutions, ways of re-writing) to the integral they chose to solve. Once they have solved an integral, the will present it to the class, and submit a writeup (which will be posted online to share with the class).

I decided it wasn't particularly efficient to print out copies of the handouts I had made (page of worked integrals, page of unworked integrals, 13 page packet of solutions). Instead, I posted each as a pdf on our course page (UVA uses some tailored version of 'Sakai', by my understanding). To make things even more useful, I also made a webpage for the worked integrals. The base page just has the list of integrals, without solutions, and then clicking on an integral reveals its solution. This means that when a student is looking for integrals similar to the integral they have chosen, they can see the whole list of integrals at one time. At some point I thought it would be nice if each student had their own webpage for this, or something, so that they could rearrange the integrals and group them to their heart's content. But I decided to leave that up to them.

My original thought for this project was that I would simply start every class by asking 'Who has an integral to present?'. Whoever raised his or her hand first got to present. I figured this would encourage students to present as soon as they were ready, to avoid getting their problem stolen. It also meant students would start this project early, to try to get an easy problem. However, before the class started I decided to make a wiki page for all of the unworked integrals. When a student was ready to present a problem, they would go to the wiki and move that problem to a 'Claimed Integrals' section, and put their name next to it. This also allows me to see who is ready to present, before class starts, so that I can chose the order to have students present, and hopefully do a bunch of similar problems at the same time.

After a student presents their solution in class, while they are still at the board, I like to ask them little challenging questions. "What if that exponent were a 3 instead of a 2, would your solution still work?" "Can you do this as a definite integral from 0 to 1?" I also encourage the class to ask questions, though mostly (so far) they're pretty quiet.

We've had 2 days worth of presentations so far, and will need to have several more to give everybody time to present. Things seem to be going well. I was quite pleased that there were already more than 10 integrals claimed after the first day of class. The presentations so far have been very good, which I'm glad about. I think starting off with good presentations will give the students a good frame of reference. I've also gotten some phenomenal writeups, in which students carefully explained all of their steps - instead of just writing down their solution.

I'm pleased with this project so far, and have noticed a few things to change if/when I decide to try it again. I really want a student to claim an integral on the wiki only after they are ready to present a problem. I think in the future I will make sure to tell students that they will lose a point if they are asked to present their integral, but are not ready to do so (including not being present!). I may also have students, as a first assignment, go through the worked integrals and gather them up in to groups of integrals that look similar.

Though each student is only working one of these problems, I will have separate assignments (normal homework assignments) to make sure everybody does some of each type of integral. For example, now that we've seen most of the trig integrals in class, they have a homework assignment due on these sorts of problems.

My only real concern about this project occurred sometime after the first day of presentations. I'm worried that strong students will look at the list sooner (or be ready to claim an integral sooner, if nothing else), and will end up working easier problems. This is a two-fold issue, because it means the stronger students aren't being challenged enough, and that weaker students are potentially getting stuck with much harder problems. Perhaps I'll have a better sense if this is the case after a few more days of presentations. I'm not really sure how to avoid this though, if it is the case.

## Wednesday, 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?

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?

## Saturday, January 3, 2009

### Menger Sponge!

My roommate, Chris, and I just completed our own level 2 Menger sponge origami project, built from 3456 business cards in something around 31 total working hours spread over 4 days (click picture for more pictures):

In all honesty, they weren't exactly business cards. A shortish chain of relations gets Chris and I to a guy with access to card stock and an industrial paper cutter, or so. For Christmas we got 4000 cards cut. They're business card size, and I think approximately the same weight. But then, I don't interact too much with business cards, so I could be mistaken.

Anyway, I should stop rambling. This is not our first large-scale origami project. A few summers ago, Chris and I, along with a then-first-year graduate student (Sean) in the department, built a modular torus:

It was a fun project, but didn't hold up under it's own weight. And since we couldn't find a place to put it, it sadly didn't last long. What's more sad about that is that we didn't even light it on fire, which would have looked simply amazing, I reckon. But, again, I digress.

So, this big sponge. It was most recently inspired by Thomas Hull's Project Origami book, but it's possible I'd heard about it before reading said book. On my sister site, I posted my calculations for finding the formula for the number of cards needed, so you can check that out, if you are interested. And over at picasa, I made an album of some of the pictures I took, which is probably more interesting. I'd also like to direct you to this picture, of somebody else's project, because they also made an 'anti-sponge', which looks awesome (like I needed another project).

In the process of building this sponge, I decided this (well, perhaps the smaller level 1 sponge) might just be the best modular origami project for somebody just starting to do modular origami. The pieces are very easy to fold (just 2 creases per unit), and pretty forgiving (if you're folds aren't quite as accurate as they could be, it should still go together pretty well). The pieces are fairly easy to put together (though, of course, it takes a little practice) - in particular because the whole thing won't fall apart if you unfold part of one piece in order to stick another piece somewhere. The final result is very nice and quite sturdy. So, if you are just starting modular origami, you might consider making this model. I recommend Hull's book, but you could probably find resources online for instructions putting these together. If you're looking for other modular origami books, my favorite is probably Tomoko Fuse's 'Unit Origami'.

I know the lighting is pretty poor in this picture, but I like it anyway:

I can't wait to take this thing outside and balance it on my chin :)

In all honesty, they weren't exactly business cards. A shortish chain of relations gets Chris and I to a guy with access to card stock and an industrial paper cutter, or so. For Christmas we got 4000 cards cut. They're business card size, and I think approximately the same weight. But then, I don't interact too much with business cards, so I could be mistaken.

Anyway, I should stop rambling. This is not our first large-scale origami project. A few summers ago, Chris and I, along with a then-first-year graduate student (Sean) in the department, built a modular torus:

It was a fun project, but didn't hold up under it's own weight. And since we couldn't find a place to put it, it sadly didn't last long. What's more sad about that is that we didn't even light it on fire, which would have looked simply amazing, I reckon. But, again, I digress.

So, this big sponge. It was most recently inspired by Thomas Hull's Project Origami book, but it's possible I'd heard about it before reading said book. On my sister site, I posted my calculations for finding the formula for the number of cards needed, so you can check that out, if you are interested. And over at picasa, I made an album of some of the pictures I took, which is probably more interesting. I'd also like to direct you to this picture, of somebody else's project, because they also made an 'anti-sponge', which looks awesome (like I needed another project).

In the process of building this sponge, I decided this (well, perhaps the smaller level 1 sponge) might just be the best modular origami project for somebody just starting to do modular origami. The pieces are very easy to fold (just 2 creases per unit), and pretty forgiving (if you're folds aren't quite as accurate as they could be, it should still go together pretty well). The pieces are fairly easy to put together (though, of course, it takes a little practice) - in particular because the whole thing won't fall apart if you unfold part of one piece in order to stick another piece somewhere. The final result is very nice and quite sturdy. So, if you are just starting modular origami, you might consider making this model. I recommend Hull's book, but you could probably find resources online for instructions putting these together. If you're looking for other modular origami books, my favorite is probably Tomoko Fuse's 'Unit Origami'.

I know the lighting is pretty poor in this picture, but I like it anyway:

I can't wait to take this thing outside and balance it on my chin :)

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