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.
Sunday, July 19, 2009
Wednesday, July 1, 2009
Summer Calc II, Second Half
My summer calculus class is almost over. Not that I don't love teaching, or calculus, or that I don't like my students, but I won't mind when it's over. I've got plenty of other things I should be (and shouldn't be :)) working on this summer. But anyway. Our final exam is tomorrow.
Hopefully it will go better than the first midterm. Admittedly, the first midterm was after only 8 class periods, and had content from 4 different chapters. The exam I wrote is one I was pretty pleased with. Other instructors told me that it was conceptual, which I take as a complement. The average, though, was sadly low. Our final, tomorrow, only covers series. I don't think it's quite as interesting as the first, but other instructors seem to think it is still fair. So we'll see.
I approached series a little bit differently than I have in the past. The last few semesters, I've followed the outline of the book, starting with sequences, then series, on to convergence tests, power series, and wrapping up with Taylor series. I thought this semester I'd try to motivate the discussion of series a little differently. Instead of just "I'd really like to add up a bunch of numbers", I began with Taylor polynomials because "I'd really like to approximate a function". This felt like a good fit with the earlier material, when we found arc lengths (and similar things) by approximating the calculation with an easy one and taking a limit (and calling it an integral). So we're going to approximate a function by an "easy" one (polynomial) and then take a limit.
After that goal is set, it's easy enough to say that "good approximation" means "the derivatives match" (at the point in question), and derive the formula for the coefficients of the Taylor polynomial. Then I pointed out that we could do the process as long as we want and make polynomials of degrees as large as we wanted, and pointed out that this would end up looking very much like taking a limit.
Next I talked about power series in general, power series being what we get "in the limit" of our Taylor polynomial calculations. I talked about differentiating, integrating, and substituting into power series, to try to give some indication that they are useful and easy to work with. I'm not sure this part went over particularly well. It might have been better to do this later, with radius and interval of convergence (see below).
So once we have power series, I pointed out that evaluating a power series at a point meant you had to sum infinitely many values. But that we basically knew how, since we started with Taylor polynomials and took the limit. To sum infinitely many values, you just take a limit of partial sums. So I talked a little about general sequences and series at this point (and also spend an hour talking about fun examples - continued fractions, 3n+1 problem, koch snowflake, cantor set,...).
Then we had a whole day (2 hours of class, +/-) where I just talked about all of the convergence tests, followed by a day for them to work on convergence tests in class (work on homework, or just try other problems). I think that for a summer class, with the odd schedules and times, this almost works. During a normal semester, though, I wouldn't do things this way. I'd probably try to get the students to learn the problems from the book themselves. In fact, I did try this last semester, but that's a separate story.
My philosophy with the convergence tests (and most other topics) is that there isn't much point in my doing more than one or two examples at the board. Math is always easier when you watch somebody else do it. The only way to get comfortable with the convergence tests is to work a whole bunch of problems for yourself. Hopefully the longer-than-average assignment I gave my class helped them gain that comfort.
Once we have convergence tests, we can return to power series, and ask "for which x is this power series defined?" I like to think that this helped tie things together from our initial discussion of power series. I'm not convinced that's the case, but I'm also not sure how to tell (ask the students?). I think this maybe would be a better time for the discussion about how to manipulate power series than early on. I'll probably try it this way next time (this fall, for my fourth straight calc II class).
That covers all of the content for the series chapter. So then I spent a full class period talking about all sorts of fun and exciting uses of series. Today in class I gave them time to review (gave them a copy of last semester's exam, essentially), after answering whatever review questions they came in with.
I'm hoping this exam goes well. And that after it's all over, I have a productive summer.
Hopefully it will go better than the first midterm. Admittedly, the first midterm was after only 8 class periods, and had content from 4 different chapters. The exam I wrote is one I was pretty pleased with. Other instructors told me that it was conceptual, which I take as a complement. The average, though, was sadly low. Our final, tomorrow, only covers series. I don't think it's quite as interesting as the first, but other instructors seem to think it is still fair. So we'll see.
I approached series a little bit differently than I have in the past. The last few semesters, I've followed the outline of the book, starting with sequences, then series, on to convergence tests, power series, and wrapping up with Taylor series. I thought this semester I'd try to motivate the discussion of series a little differently. Instead of just "I'd really like to add up a bunch of numbers", I began with Taylor polynomials because "I'd really like to approximate a function". This felt like a good fit with the earlier material, when we found arc lengths (and similar things) by approximating the calculation with an easy one and taking a limit (and calling it an integral). So we're going to approximate a function by an "easy" one (polynomial) and then take a limit.
After that goal is set, it's easy enough to say that "good approximation" means "the derivatives match" (at the point in question), and derive the formula for the coefficients of the Taylor polynomial. Then I pointed out that we could do the process as long as we want and make polynomials of degrees as large as we wanted, and pointed out that this would end up looking very much like taking a limit.
Next I talked about power series in general, power series being what we get "in the limit" of our Taylor polynomial calculations. I talked about differentiating, integrating, and substituting into power series, to try to give some indication that they are useful and easy to work with. I'm not sure this part went over particularly well. It might have been better to do this later, with radius and interval of convergence (see below).
So once we have power series, I pointed out that evaluating a power series at a point meant you had to sum infinitely many values. But that we basically knew how, since we started with Taylor polynomials and took the limit. To sum infinitely many values, you just take a limit of partial sums. So I talked a little about general sequences and series at this point (and also spend an hour talking about fun examples - continued fractions, 3n+1 problem, koch snowflake, cantor set,...).
Then we had a whole day (2 hours of class, +/-) where I just talked about all of the convergence tests, followed by a day for them to work on convergence tests in class (work on homework, or just try other problems). I think that for a summer class, with the odd schedules and times, this almost works. During a normal semester, though, I wouldn't do things this way. I'd probably try to get the students to learn the problems from the book themselves. In fact, I did try this last semester, but that's a separate story.
My philosophy with the convergence tests (and most other topics) is that there isn't much point in my doing more than one or two examples at the board. Math is always easier when you watch somebody else do it. The only way to get comfortable with the convergence tests is to work a whole bunch of problems for yourself. Hopefully the longer-than-average assignment I gave my class helped them gain that comfort.
Once we have convergence tests, we can return to power series, and ask "for which x is this power series defined?" I like to think that this helped tie things together from our initial discussion of power series. I'm not convinced that's the case, but I'm also not sure how to tell (ask the students?). I think this maybe would be a better time for the discussion about how to manipulate power series than early on. I'll probably try it this way next time (this fall, for my fourth straight calc II class).
That covers all of the content for the series chapter. So then I spent a full class period talking about all sorts of fun and exciting uses of series. Today in class I gave them time to review (gave them a copy of last semester's exam, essentially), after answering whatever review questions they came in with.
I'm hoping this exam goes well. And that after it's all over, I have a productive summer.
Monday, June 15, 2009
Summer Calc II, Week 1
Last Tuesday I started teaching my summer Calc II course. We meet for 2 hours every day (M-F), and then the students also have a 45 minute discussion section with our TA. I decided to set up the course a little differently from how I've done Calc II the last few times, and thought I might try to describe some of it here.
The, say, "standard" way this course would go is to start with techniques of integration, do improper integrals, arc length and surface area of revolution, parametric curves, polar curves, iterated integrals, and then series. Last semester, we had one exam on techniques of integration and arc length and surface area, one on parametric and polar curves and iterated integrals, and one on series.
When I was setting up my course for the summer, I looked at how many days to spend on each topic, and how to try to schedule things best to naturally have exams and things. I decided to shoot for having all of the topics besides series as a single exam, around halfway through the course, and then have series be their own exam. We're just about to the first exam (it'll be Friday), and it looks like we can meet that schedule (with in class review time before the exam even!).
I was able to condense the material down in a few ways. First, the arc length and surface area calculations we do with usual $y=f(x)$ curves, we later redo with parametric curves (in the "standard" class). I decided to start with doing them as parametric and polar curves, and then make a few comments about how to convert a usual $y=f(x)$ curve to a parametric curve. [While I'm talking about surface area... I know that when you rotate around whichever axis, you have (sometimes) two choices for how to set up the integral. You could do it dy or dx. Probably I just haven't dug into enough examples, but is there one when the integral is "doable by hand" one way, but not "doable by hand" the other way, even though both can be set up?]
The other way I crunched things together was to leave more of the responsibility to the students to learn the techniques of integration on their own, outside of class, for homework. I made a few comments here or there, but largely it's been up to them. In my mind, the only real techniques of integration are u-substitution and integration by parts anyway, both of which I did talk about in class. The others, in the text, are trig integrals ("apply lots of trig identities"), trig substitution ("memorize a chart of what substitution to make when"), and partial fractions (I've got no beef with partial fractions). It wouldn't matter if I spent all day for a few class periods working integrals, the students would leave feeling like they weren't too bad, and would get stuck on them at home. So I cut out more of the "they aren't that bad" feeling, and mostly threw the students in to get stuck. My hope was that I'd see lots of kids in office hours, which has not (yet) been the case.
My other thoughts, along the lines of techniques of integration, are concerned with how useful they actually are. This general question is something me and every other math teacher with a blog have brought up before, especially recently with Wolfram Alpha doing integrals (and showing steps!). What is it we should actually be teaching these kids? I'm not convinced I've ever worked a trig-substitution integral outside of calculus class, but would love to hear about it if anybody out there had. Ok, sure, working them gives you lots of good practice with algebraic rules, and so I can now manipulate symbols like it's my job. But I've been trying to set up my class to sort of tend away from this. I really want to spend more time on, and get my students to answer more questions about, how to set up integrals to calculate things you might want to calculate, or what a given integral calculates (in a picture). I also try, when we work an integral, to talk them through any sort of consistency checking I would do for the answer (is it positive or negative? should it be? is it too big, or too small?). I like to think that I am giving them practice taking a problem and converting it to something a computer can easily do, and then thinking about if the answer is reasonable.
Part of my other goal in having them learn techniques of integration with somewhat less guidance, was to give them experience learning from the textbook directly. Probably I want to do this because it is how the calc class I took as an undergrad was set up.
I'm not convinced, based on homework grades, that things are going terribly well so far. I've asked them to give me anonymous feedback about any changes they want to have made, and tried to stress that I want them to ask questions and come to office hours. I've not gotten too much back in the way of feedback yet. The one, really, so far was that I should work harder problems in class. I think I get this every semester, so probably it is something I should seriously look into. But most of me thinks: they'll always look easier in class, you have to get stuck on them yourself. Math is not easy. I don't know, perhaps that's my laziness shining through. But at the same time, I can't really start with hard examples in class, because they won't make sense right away, and there's only so much time during class.
At some point I was preparing lectures and getting frustrated by the examples I had, and with trying to find new ones. (Is there a place online where teachers keep their favorite examples of all sorts of problems? Some sort of examples repository or something?) I came up with an idea I'm still mulling over for how to teach my classes. My thought was maybe I'll start each day with a little discussion about whatever new concepts there are for the day, at sort of the theoretical level. And then I'll just grab the first couple (or just one) odd problems from the text, and work through them. And then the rest of class time will be students working through as many more of the problems as we have time for. They can work in groups if they want, but don't have to. This gives them the chance to get stuck on problems, but quickly start asking questions to get over initial hurdles. They'll have their textbooks with them, to dig through examples, and will also have other students (and me) to ask when they're really stuck. Anybody have any thoughts on this setup?
The, say, "standard" way this course would go is to start with techniques of integration, do improper integrals, arc length and surface area of revolution, parametric curves, polar curves, iterated integrals, and then series. Last semester, we had one exam on techniques of integration and arc length and surface area, one on parametric and polar curves and iterated integrals, and one on series.
When I was setting up my course for the summer, I looked at how many days to spend on each topic, and how to try to schedule things best to naturally have exams and things. I decided to shoot for having all of the topics besides series as a single exam, around halfway through the course, and then have series be their own exam. We're just about to the first exam (it'll be Friday), and it looks like we can meet that schedule (with in class review time before the exam even!).
I was able to condense the material down in a few ways. First, the arc length and surface area calculations we do with usual $y=f(x)$ curves, we later redo with parametric curves (in the "standard" class). I decided to start with doing them as parametric and polar curves, and then make a few comments about how to convert a usual $y=f(x)$ curve to a parametric curve. [While I'm talking about surface area... I know that when you rotate around whichever axis, you have (sometimes) two choices for how to set up the integral. You could do it dy or dx. Probably I just haven't dug into enough examples, but is there one when the integral is "doable by hand" one way, but not "doable by hand" the other way, even though both can be set up?]
The other way I crunched things together was to leave more of the responsibility to the students to learn the techniques of integration on their own, outside of class, for homework. I made a few comments here or there, but largely it's been up to them. In my mind, the only real techniques of integration are u-substitution and integration by parts anyway, both of which I did talk about in class. The others, in the text, are trig integrals ("apply lots of trig identities"), trig substitution ("memorize a chart of what substitution to make when"), and partial fractions (I've got no beef with partial fractions). It wouldn't matter if I spent all day for a few class periods working integrals, the students would leave feeling like they weren't too bad, and would get stuck on them at home. So I cut out more of the "they aren't that bad" feeling, and mostly threw the students in to get stuck. My hope was that I'd see lots of kids in office hours, which has not (yet) been the case.
My other thoughts, along the lines of techniques of integration, are concerned with how useful they actually are. This general question is something me and every other math teacher with a blog have brought up before, especially recently with Wolfram Alpha doing integrals (and showing steps!). What is it we should actually be teaching these kids? I'm not convinced I've ever worked a trig-substitution integral outside of calculus class, but would love to hear about it if anybody out there had. Ok, sure, working them gives you lots of good practice with algebraic rules, and so I can now manipulate symbols like it's my job. But I've been trying to set up my class to sort of tend away from this. I really want to spend more time on, and get my students to answer more questions about, how to set up integrals to calculate things you might want to calculate, or what a given integral calculates (in a picture). I also try, when we work an integral, to talk them through any sort of consistency checking I would do for the answer (is it positive or negative? should it be? is it too big, or too small?). I like to think that I am giving them practice taking a problem and converting it to something a computer can easily do, and then thinking about if the answer is reasonable.
Part of my other goal in having them learn techniques of integration with somewhat less guidance, was to give them experience learning from the textbook directly. Probably I want to do this because it is how the calc class I took as an undergrad was set up.
I'm not convinced, based on homework grades, that things are going terribly well so far. I've asked them to give me anonymous feedback about any changes they want to have made, and tried to stress that I want them to ask questions and come to office hours. I've not gotten too much back in the way of feedback yet. The one, really, so far was that I should work harder problems in class. I think I get this every semester, so probably it is something I should seriously look into. But most of me thinks: they'll always look easier in class, you have to get stuck on them yourself. Math is not easy. I don't know, perhaps that's my laziness shining through. But at the same time, I can't really start with hard examples in class, because they won't make sense right away, and there's only so much time during class.
At some point I was preparing lectures and getting frustrated by the examples I had, and with trying to find new ones. (Is there a place online where teachers keep their favorite examples of all sorts of problems? Some sort of examples repository or something?) I came up with an idea I'm still mulling over for how to teach my classes. My thought was maybe I'll start each day with a little discussion about whatever new concepts there are for the day, at sort of the theoretical level. And then I'll just grab the first couple (or just one) odd problems from the text, and work through them. And then the rest of class time will be students working through as many more of the problems as we have time for. They can work in groups if they want, but don't have to. This gives them the chance to get stuck on problems, but quickly start asking questions to get over initial hurdles. They'll have their textbooks with them, to dig through examples, and will also have other students (and me) to ask when they're really stuck. Anybody have any thoughts on this setup?
Saturday, May 30, 2009
Walpha Wiki
Just in case you missed it, I thought I'd share the link for the Walpha Wiki. The same evening that I posted asking if anybody had a wiki going, Derek Bruff started this one. I've not contributed as much as I want to have yet, but still intend to. Won't you help?
The blog also includes discussion about the impact of W|alpha on mathematics education, a topic that's been on my mind recently.
The blog also includes discussion about the impact of W|alpha on mathematics education, a topic that's been on my mind recently.
Tuesday, May 26, 2009
A New Kind of Wiki
Well, not really... I'll just explain.
So, when Wolfram|Alpha (referred to as w|a below because I'm lazy) came out, I, like many of you, was pretty excited to play with it. I was primarily interested in its use as a free, online, computer algebra system (CAS). So when I tested it, I gave it the sorts of questions that I give my calculus students (in fact, I essentially tested it with exams I've given students). In many areas it was obvious what to do, in some areas I could mess around and get a reasonable answer, and in a remaining few areas, w|a seemed to come up lacking.
I thought it would be great to have a resource telling how to input questions you might typically ask a CAS, since apparently entering straight-up Mathematica code doesn't always work (I guess Wolfram still wants to sell copies of Mathematica). One of my early thoughts was that I should make one. And then I thought, surely somebody else has already done so. In fact, the folks at w|a probably already have some nice documentation online. I made a note to look into it, and thought it funny that I was hoping to find documentation for such an online system.
Not long after that, and before I did any more playing with things, Maria Anderson, @busynessgirl on Twitter, posted a tweet: "I am toying with the idea of taking a standard algebra TOC and putting up a webpage that shows which topics W|A can do." A fantastic idea (which she quickly refined: webpage -> wiki). Extend it to calculus, and I'm there. And show not just what it can do, but what it can't do, what it does wrong (or oddly), and ways to make it do what it can do.
I think such a thing should come into being. Perhaps it already has, and I missed it? Or perhaps there is some nice documentation for w|a that I've not yet found? If either of these is the case, could somebody point me to it?
If there is no such thing yet, I say it's time to make one. I'm getting antsy. In the comments below, if you want such a wiki to exist, would you please leave some helpful feedback? I'm particularly interested in: (1) What (free, hosted) wiki software would you suggest or suggest avoiding? I think right now I'm leaning toward wikispaces, though I've not looked into things a whole lot. (2) What should it be called? (3) Any other comments or suggestions you have.
To get things rolling, I'll say that this coming Saturday (May 30), if no links are provided to an existing webpage, I'll start a wiki somewhere that seems to fit the consensus of the comments (I hope there are comments, and they have a consensus). I'll then let you know where it is.
Update 20090526: Derek Bruff left a comment that he was starting one, and posted the link http://walphawiki.wikidot.com/calculus-i via twitter. Looks promising!
So, when Wolfram|Alpha (referred to as w|a below because I'm lazy) came out, I, like many of you, was pretty excited to play with it. I was primarily interested in its use as a free, online, computer algebra system (CAS). So when I tested it, I gave it the sorts of questions that I give my calculus students (in fact, I essentially tested it with exams I've given students). In many areas it was obvious what to do, in some areas I could mess around and get a reasonable answer, and in a remaining few areas, w|a seemed to come up lacking.
I thought it would be great to have a resource telling how to input questions you might typically ask a CAS, since apparently entering straight-up Mathematica code doesn't always work (I guess Wolfram still wants to sell copies of Mathematica). One of my early thoughts was that I should make one. And then I thought, surely somebody else has already done so. In fact, the folks at w|a probably already have some nice documentation online. I made a note to look into it, and thought it funny that I was hoping to find documentation for such an online system.
Not long after that, and before I did any more playing with things, Maria Anderson, @busynessgirl on Twitter, posted a tweet: "I am toying with the idea of taking a standard algebra TOC and putting up a webpage that shows which topics W|A can do." A fantastic idea (which she quickly refined: webpage -> wiki). Extend it to calculus, and I'm there. And show not just what it can do, but what it can't do, what it does wrong (or oddly), and ways to make it do what it can do.
I think such a thing should come into being. Perhaps it already has, and I missed it? Or perhaps there is some nice documentation for w|a that I've not yet found? If either of these is the case, could somebody point me to it?
If there is no such thing yet, I say it's time to make one. I'm getting antsy. In the comments below, if you want such a wiki to exist, would you please leave some helpful feedback? I'm particularly interested in: (1) What (free, hosted) wiki software would you suggest or suggest avoiding? I think right now I'm leaning toward wikispaces, though I've not looked into things a whole lot. (2) What should it be called? (3) Any other comments or suggestions you have.
To get things rolling, I'll say that this coming Saturday (May 30), if no links are provided to an existing webpage, I'll start a wiki somewhere that seems to fit the consensus of the comments (I hope there are comments, and they have a consensus). I'll then let you know where it is.
Update 20090526: Derek Bruff left a comment that he was starting one, and posted the link http://walphawiki.wikidot.com/calculus-i via twitter. Looks promising!
Thursday, May 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!
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.
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.
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