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.
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4 comments:
Neat project. I really need to come up with somewhere to start keeping a list of neat teaching ideas that I see other people doing. I've always hated teaching trig substitution, so anything to move some of the responsibility for learning onto the students ought to be good. Is your class all lecture or lecture/recitation? If you've got a lecture/recitation format, are the presentations in lecture or recitation?
Dealing with the better students presenting first, and therefore taking all the easy ones, are you willing to acknowledge that some are more difficult in advance? If so, what about giving extra credit for harder problems that are ready to present early? Maybe a decreasing scale of how many extra credit points they can get based on how long the question's been out there. That might provide just enough incentive for the better students to go for the harder problems. They'd probably solve the easy ones first but realize that they can get more from doing the harder ones and so aim to present them. Since it'd be early in the term, hopefully the weaker students wouldn't think they should aim for extra credit right away and then not actually solve the ones they should be able to.
Thanks @Mitch. The class has a "4th hour" or "discussion section", in addition to 3 hours of class each week. I've been using both class and discussion time to do presentations, to make sure we get through them all.
I think I like your idea about extra credit, possibly on a scale decreasing with time, for harder problems. I've always refrained from any sort of extra credit before, but it seems like you might be on to something here. I wonder how implementing such an idea in this project would affect the order students present their integrals in. Somebody could claim a hard integral with lots of extra credit, but then having them present on the second day of class isn't likely to help many other students see the methods involved. But if they don't present right away, they aren't getting all the extra credit for the problem? Perhaps I have some sort of check, outside of class. When a student is ready to claim an integral they come to me, I make sure they can do the problem (and they get whatever extra credit at that point), and then they present whenever it seems appropriate?
Yeah, you'll probably want to have them claim problems but then have to submit verification that they've got it figured out in order to get the credit while they wait for the time when you think it's right for a presentation to happen. You could have them come by your office (possibly very time-consuming depending on class size) or turn in the write-up right away. If you've got a TA or grader, he/she could help verify claims to make it easier on you.
This is sounding pretty feasible. I'll have to revisit it next time I try this approach.
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