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## Plagiarism in high school

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About two dozen seniors at Hamilton Southeastern High School in the affluent northern suburbs of Indianapolis have been caught plagiarizing in a dual-enrollment college course, thanks to turnitin.com. Full story with video here, and there’s an official statement from the HSE superintendent on this issue here (.DOC, 20KB).

This would be an ordinary, though disappointing, story about students getting caught cheating if it weren’t for some head-scratchers here. First, this bit from the superintendent’s statement:

We took immediate action because the end of the school year was rapidly approaching. Several students were in danger of not graduating on time. We found a teacher who was willing to step up and administer a complete but highly accelerated online version of a class that would replace the credit that was lost due to cheating. Each student who wishes to graduate on time and participate in commencement now has the opportunity to do so. [my emphasis]

It’s troublesome that the superintendent chooses to describe the teacher as “stepping up” to deliver an online makeup course. “Stepping up” is what you call it when there’s something that needs to be done and somebody agrees to get it done. But it seems to me that the school system here owes these students absolutely nothing. HSE, in conjunction with Indiana University, offered a legitimate college course with clearly-defined parameters for academic performance, and HSE did a particularly thorough job describing the boundaries of academic honesty. The students chose to violate that contract and cheat. The school system is therefore not obliged to offer an online makeup course, or indeed to offer anything to these students at all. To imply that HSE does owe the students a path to graduate on time is like saying that if someone gets caught shoplifting, the grocery store owes it to the shoplifter to find a way to help him buy his groceries.

Also, what is the teacher who “stepped up” being paid to run this online course? If the teacher is being paid from public school coffers for this, and if I lived in Hamilton County, I would have a big problem with my tax money being spent to offer online courses to students guilty of cheating just so they can graduate on time — especially when public school money is historically scarce right now. Let the students find their own way to graduate. It’s not like they were barred from graduating on time, fair and square, in the first place. Let the residents’ school money go to help the students who are working hard and doing things the right way instead. (If the teacher’s doing it for free, then other questions arise.) This is the way we’d do it in college, and this is a college course, right?

HSE might think it’s doing right by the students in “allowing each student to work his or her way back toward the proper path so they can graduate on time, continue their educations [sic] and understand the benefits of making good choices” (quote from the superintendent’s statement). But isn’t this really illustrating the benefits of making bad choices — as in, go ahead and cheat, because the school will find a way to let you graduate on time anyway? Other than potentially not getting into IU, what consequences are these students having to face, exactly, other than sacrificing a bit of their summer to retake a course at taxpayer expense? (By the way, if this course is dual-credit, whose rules about academic dishonesty are supposed to be followed? IU’s appear to be more strict that Hamilton Southeastern’s.)

This bit from a fellow student is equally disturbing:

“If you’re going to do something dishonorable, there’s going to be consequences for it,” said [a fellow student, not part of the plagiarizing group]. But she says she sympathizes with her friends who were caught cheating. She claims students have been cheating for years, but this is the first year teachers have used the software system that gives them the ability to easily catch cheaters. She believes this incident likely serves as a lesson for students for years to come.

So, it’s about the consequences, not so much the act itself. The sympathy didn’t show up until turnitin.com caught them. Until we stop “sympathizing” with plagiarists and start treating plagiarism on the same level as lying and stealing — which it is both — this problem isn’t going to go away.

What’s your take on all this? Is HSE acting honorably or just enabling future plagiarism? What’s the best way to punish teenage plagiarists on the one hand but really help them make better choices on the other?

## The semester in review

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I’ve made it to the end of another semester. Classes ended on Friday, and we have final exams this coming week. It’s been a long and full semester, as you can see by the relative lack of posting going on here since around October. How did things go?

Well, first of all I had a record course load this time around — four different courses, one of which was the MATLAB course that was brand new and outside my main discipline; plus an independent study that was more like an undergraduate research project, and so it required almost as much prep time from me as a regular course.

The Functions and Models class (formerly known as Pre-calculus) has been one of my favorites to teach here, and this class was no exception. We do precalculus a bit differently here, focusing on using functions as data modeling tools, so the main meat of the course is simply looking at data and asking, Are the data linear? If not, are they best fit by a logarithmic, exponential, or power function? Or a polynomial? And what should be the degree of that polynomial? And so on. I enjoy this class because it’s primed for the kind of studio teaching that I’ve come to enjoy. I just bring in some data I’ve found, or which the students have collected, and we play with the data. And these are mainly students who, by virtue of having placed below calculus on our placement exam, have been used to a dry, lecture-oriented math environment, and it’s very cool to see them light up and have fun with math for a change. It was a small class (seven students) and we had fun and learned a lot.

The Calculus class was challenging, as you can tell from my boxplots posts (first post, second post). The grades in the class were nowhere near where I wanted them to be, nor for the students (I hope). I think every instructor is going to have a class every now and then where this happens, and the challenge is to find the lesson to learn and then learn them. If you read those two boxplots posts, you can see some of the lessons and information that I’ve gleaned, and in the fall when I teach two sections of this course there could be some significant changes with respect to getting more active work into the class and more passive work outside the class.

Linear Algebra was a delight. This year we increased the credit load of this class from three hours to four, and the extra hour a week has really transformed what we can do with the course. I had a big class of 15 students (that’s big for us), many of whom are as sharp as you’ll find among undergraduates, and all of whom possess a keen sense of humor and a strong work ethic that makes learning a difficult subject quite doable. I’ll be posting later about their application projects and poster session, which were both terrific.

Computer Tools for Problem Solving (aka the MATLAB course) was a tale of two halves of the semester. The first half of the semester was quite a struggle — against a relatively low comfort level around technology with the students and against the students’ expectations for my teaching. But I tried to listen to the students, giving them weekly questionnaires about how the class is going, and engaging in an ongoing dialogue about what we could be doing better. We made some changes to the course on the fly that didn’t dumb the course down but which made the learning objectives and expectations a lot clearer, and they responded extremely well. By the end of the course, I daresay they were having fun with MATLAB. And more importantly, I was receiving reports from my colleagues that those students were using MATLAB spontaneously to do tasks in those courses. That was the main goal of the course for me — get students to the point where they are comfortable and fluent enough with MATLAB that they’ll pull it up and use it effectively without being told to do so. There are some changes I need to make to next year’s offering of the course, but I’m glad to see that the students were able to come out of the course doing what I wanted them to do.

The independent study on finite fields and applications was quite a trip. Andrew Newman, the young man doing the study with me, is one of the brightest young mathematicians with whom I’ve worked in my whole career, and he took on the project with both hands from the very beginning. The idea was to read through parts of Mullen and Mummert to get basic background in finite field theory; then narrow down his reading to a particular application; then dive in deep to that application. Washington’s book on elliptic curves ended up being the primary text, though, and Andrew ended up studying elliptic curve cryptography and the Diffie-Hellman decision problem. Every independent study has a creative project requirement attached, and his was to implement the decision problem in Sage. He’s currently writing up a paper on his research and we hope to get it published in Mathematics Exchange. (Disclaimer: I’m on the editorial board of Math Exchange.) In the middle of the semester, Andrew found out that he’d been accepted into the summer REU on mathematical cryptology at Northern Kentucky University/University of Cincinnati, and he’ll be heading out there in a few weeks to study (probably) multivariate public-key systems for the summer. I’m extremely proud of Andrew and what he’s been able to do this semester — he certainly knows a lot more about finite fields and elliptic curve crypto than I do now.

In between all the teaching, here are some other things I was able to do:

• Went to the ICTCM in Chicago and presented a couple of papers. Here’s the Prezi for the MATLAB course presentation. Both of those papers are currently being written up for publication in the conference proceedings.
• Helped with hosting the Indiana MAA spring meetings at our place, and I finished up my three-year term as Student Activities Coordinator by putting together this year’s Indiana College Mathematics Competition.
• Did a little consulting work, which I can’t really talk about thanks to the NDA I signed.
• I got a new Macbook Pro thanks to my college’s generous technology grant system. Of course Apple refreshed the Macbook Pro lineup mere weeks later, but them’s the breaks.
• I’m sure there’s more, but I’ve got finals on the brain right now.

In another post I’ll talk about what’s coming up for me this summer and look ahead to the fall.

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## Must the tenure process really be like this?

Like a lot of people in higher ed, I’ve been following Friday’s deadly shooting at the University of Alabama-Hunstville. (Click the link for background in case you missed the story. I have no idea how much press it is or is not getting in the national mainstream media.) It’s known that Amy Bishop, the UAH biology professor being charged with the shooting, was denied tenure in April and had made an unsuccessful appeal regarding her tenure denial. It’s not clear that the shooting was related to the tenure situation, but the speculation — especially in the article at the second link — is that there’s a connection.

What is clear, at least from my perspective as a professor and as somebody in the fourth year of a five-year appointment to my college’s Promotion and Tenure Committee, is that something is really badly wrong with UAH’s tenure system, and perhaps with tenure as a concept. Listen to this description of Prof. Bishop’s situation from William Setzer, chemistry department chair at UAH:

As for why she had been turned down for tenure, Mr. Setzer said he had heard that her publication record was thin and that she hadn’t secured enough grants. Also, there were concerns about her personality, he said. In meetings, Mr. Setzer remembered, she would go off on “bizarre” rambles about topics not related to tasks at hand — “left-field kind of stuff,” he said. […]

While there were those who supported her tenure and promotion, Mr. Setzer said, he didn’t believe she had any friends in the department.

There was no doubt, however, about her intelligence or pedigree. “She’s pretty smart,” said Mr. Setzer. “That was not a question. There might have been some question about how good of a [principal investigator] and mentor she was. Yeah, she knows her stuff, and she’s a good technical person, but as far as being the boss and running the lab, that was kind of the question.”

Mr. Setzer might not be giving an accurate description of how people get tenure at UAH, but is this really what tenure is all about? Publication records? Grants? Personalities? Whether or not you have enough friends, or the right friends?  UAH does state up-front that it is a research-intensive university, but where is teaching in all of this?

Now look at the depressing remarks of  Cary Nelson, president of the American Association of University Professors, about the realities (?) of tenure:

“The most likely result of being denied tenure in this nonexistent job market is that you will not be able to continue teaching,” said [Nelson]. “You probably can’t get another job.”

Nelson, who teaches at the University of Illinois at Urbana-Champaign, said the review is a period of great stress for even the most likely candidates. They feel judged. Denial can lead to isolation.

“If you have underlying problems,” Nelson said, “then there’s a good chance that they will surface during the tenure process because you are under so much stress.”

We do not know, yet, just how much Prof. Bishop’s tenure denial contributed to her actions, which (I stress) are not justifiable under any circumstance. But honestly — if this is what getting tenure is like at your school, then your school is doing it wrong.

The tenure process can, and should, be an open and transparent process whereby junior faculty are guided in their professional development by senior faculty with a view towards making positive contributions to their institution for 30 or 40 years or more. Done right, tenure can be a transformative and powerful experience for faculty, institutions, and students alike. Done as it is described above, though, it is bound to be petty, political, focused on all the wrong things, and producing professionally unbalanced faculty who have merely learned to play the game properly. No wonder so many schools are considering dropping tenure. But isn’t there some middle ground where tenure can be redeemed?

Filed under Education, Higher ed, Life in academia, Tenure

## A simple idea for publishers to help students (and themselves)

Image by Getty Images via Daylife

I’m doing some research, if you can call it that, right now that involves looking at past editions of popular and/or influential calculus books to track the evolution of how certain concepts are developed and presented. I’ll have a lot to say on this if I ever get anywhere with it. But in the course of reading, I have been struck with how little some books change over the course of several editions. For example, the classic Stewart text has retained the exact wording and presentation in its section on concavity in every edition since the first, which was released in the mid-80’s. There’s nothing wrong with sticking with a particular way of doing things, if it works; but you have to ask yourself, does it really work? And if so, why are we now on the sixth edition of the book? I know that books need refreshing from time to time, but five times in 15 years?

Anyhow, it occurred to me that there’s something really simple that textbook companies could do that would both help out students who have a hard time affording textbooks (which is a lot of students) and give themselves an incentive not to update book editions for merely superficial reasons. That simple thing is: When a textbook undergoes a change in edition, post the old edition to the web as a free download. That could be a plain PDF, or it could be a  Kindle or iBooks version. Whatever the format, make it free, and make it easy to get.

This would be a win-win-win for publishers, authors, and students:

• By charging the regular full price for the “premium” (= most up-to-date) edition of the book, the publisher wouldn’t experience any big changes in its revenue stream, provided (and this is a big “if”) the premium edition provides significant additional value over the old edition. In other words, as long as the new edition is really new, it would cost the publisher nothing to give the old version away.
• But if the premium edition is just a superficial update of the old one, it will cost the publisher big money. So publishers would have significant incentive not to update editions for no good reason, thereby costing consumers (students) money they didn’t really need to spend (and may not have had in the first place).
• All the add-ons like CD-ROMs, websites, and other items that often get bundled with textbooks would only be bundled with the premium edition. That would provide additional incentive for those who can afford to pay for the premium edition to do so. (It would also provide a litmus test for exactly how much value those add-ons really add to the book.)
• It’s a lot easier to download a PDF of a deprecated version of a book, free and legally, then to try your luck with the various torrent sites or what-have-you to get the newest edition. Therefore, pirated versions of the textbook would be less desirable, benefitting both publishers and authors.
• Schools with limited budgets (including homeschooling families) could simply agree not to use the premium version and go with the free, deprecated version instead. This would always be the case if the cost of the new edition outweighs the benefits of adopting it — which again, puts pressure on the publishers not to update editions unless there are really good reasons to do so and the differences between editions are really significant.
• The above point also holds in a big, big way for schools in developing countries or in poverty-stricken areas in this country.
• Individual students could also choose to use the old edition, and presumably accept responsibility for the differences in edition, even if their schools use the premium edition. Those who teach college know that many students do this now already, except the old editions aren’t free (unless someone gives the book to them).
• All this provides publishers and authors to take the moral high road while still preserving their means of making money and doing good business.

Some individual authors have already done this: the legendary Gil Strang and his calculus book, Thomas Judson and his abstract algebra book (which I used last semester and really liked), Fred Goodman and his algebra book. These books were all formerly published by major houses at considerable cost, but were either dropped or deprecated, and the authors made them free.

How about some of the major book publishers stepping up and doing the same?

## Piecewise-linear calculus part 2: Getting to smoothness

Image via Wikipedia

This is the second post (here’s the first one) about an approach to introducing the derivative to calculus students that is counter to what I’ve seen in textbooks and other traditional treatments of the subject. As I wrote in the first post, in the typical first contact with the derivative, students are given a smooth curve and asked to find the slope of a tangent line to this curve at a point. But I argued that it would be more helpful to students’ understanding of the derivative to start with a simpler case first, namely to use only piecewise-linear functions at the beginning. This way, as we saw, we can develop some important core ideas about the derivative without resorting to anything more than pictures and an occasional slope calculation.

But now, we need to deal with the main problem: What happens if we do have a smooth curve, not a straight line or piecewise-linear graph, and we want to answer the same kinds of questions as we posed in the first post? Again, here’s how this might play out in a classroom setting.

Let’s go back to Charlie from the example in the previous post, who travels 100 meters over a 120-second time span to the cafeteria according to this graph:

The piecewise-linearity of this graph makes it easy to calculate Charlie’s velocity at (almost…) any point. But there’s a problem. Can a human being possibly change velocities, as Charlie does at t = 60 and t = 90, without slowing down first? That is clearly not in line with the laws of physics unless you have no mass. So, although the piecewise-linear graph can be a pretty good approximation to real life, in real life no person would ever move like this. Instead, Charlie’s motion is probably more like this:

Charlie’s story as told by this graph is basically the same as before. But now the curve is smoothed out where Charlie changes direction to account for the physical realities of motion. Now let’s ask the same kind of question as before: How fast was Charlie going at, say, 30 seconds?

I like just to give this problem to the students to see what they can make of it. We’ve done instantaneous rates several times by this point, but all for piecewise-linear functions. That was easy; how can you adapt this method to a function that is not linear? Students who come up with any sort of idea at all usually come up with the right one: Somehow approximate the curve with a straight line at t = 30 and then measure that line’s slope.

Some students do this by arguing that the graph from t=0 to t=60  is essentially linear already; that tiny bit of curvature we see is so small it can be neglected, so just find the slope of the “line” from 0 to 60 using the origin (through which the graph clearly passes) and either (30, 50) or (60, 80). Other students will draw the tangent line to the graph at t = 30 — without ever having been told what a tangent line is or having seen one — and measure its slope. The first approach, of course, is using a secant line, the second one a tangent line.

Both of these approaches are quite natural and also pretty accurate in this case. But eventually we want students to understand that the best approach is to create not a picture but a process whereby we can get an approximate slope to any degree of accuracy we like — and eventually define . The usual way to do this is in the calculus books — fix the point of tangency (e.g. t = 30) and select a movable second point (a, y(a)); calculate the slope of the secant line; repeat until the differences in the secant slopes become negligible. The result is the slope of the tangent line. There’s nothing wrong with that, but here’s another approach that retains the piecewise-linear flavor of the initial encounter.

We don’t (yet) know exactly what it means when we talk about the “slope” of a curve. So let’s take a step backwards. Suppose we broke Charlie’s distance graph into a number of straight pieces by picking a bunch of points on the curve and connecting the dots, like so:

(Here the dots are plotted at t = 0, 30, 60, … , 120.) Voíla — we have piecewise-linearized the graph! Now, if there is a single line segment that contains t=30, just locate it and find its slope. This requires approximation, but that’s the price we pay. (On the other hand, if we had a formula, we wouldn’t need to approximate; that’s a seperate calculation and in the spirit of keeping things relatively algebra-free here, we won’t go into that.)

But since two pieces of data are often better than one, a potentially even better approach is to plot a bunch of dots and make t = 30 one of them, as we have done above. This will create a line segment before t=30 and a line segment after t=30. Then we can estimate the two slopes and average the result.

Question: How accurate is this, and can we make it more accurate? Intuitively, as long as the function is relatively well-behaved at t = 30, the more dots we plot on the graph, the better accuracy we get. So go back through and (say) double the number of dots you plot and repeat. This sounds like a lot of busy work until you realize you only need three dots: one at t=30, another just before t=30, and another just after t=30. For simplicity, make the two “outside” dots the same distance from t=30, say 0.1 units away. Find the slope from t=29.9 to t=30 and then from t=30 to t = 31.1; average the results; and that’s a better approximation. Reduce the size of the offset if you want even more accuracy. And if you want a clear idea of what the “slope” of a curve at a point is, reduce the offset size repeatedly and see what the average slopes approach.

All we’re doing here is reformulating the standard method of getting the derivative. If we let $h$ represent the offset described above, and if $y = f(t)$ is the function of interest, then the “slope just before t=30” is

$\displaystyle{\frac{y(30-h) - y(30)}{30 - h - 30} =- \frac{y(30- h) - y(30)}{h}}$

and similarly, the “slope just after t=30” is

$\displaystyle{\frac{y(30+h) - y(30)}{30 + h - 30} = \frac{y(30+ h) - y(30)}{h}}$

the average of these two is

$\displaystyle{\frac{y(30+h) - y(30-h)}{2h}}$

and this is known as the symmetric difference quotient, a standard means of calculating numerical derivatives and perhaps the best choice for differentiating functions that are given as tables of data. What we are doing by “shrinking the offset” is merely letting $h \rightarrow 0$. So ultimately we are setting up the definition of the derivative at t=30 to be:

$\displaystyle{y'(30) = \lim_{h \to 0} \frac{y(30+h) - y(30-h)}{2h}}$

Of course this produces the same derivative values as the usual limit-based definitions of the derivative. What makes this possibly preferable to the usual formulas, though, is that it arises out of the piecewise-linear approach; and it applies itself very well to functions given as tables of data (if you knock out the limit). The method of going through a smooth curve, putting a bunch of equally-spaced dots on it, and then connecting the dots is also precisely how the formula for arc length is developed when students get around to applications of the integral. So this approach also provides a bit of unification between differential and integral calculus.

But integration, and how the piecewise-linear approach might be useful there, is the subject of the next post.

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Filed under Calculus, Math, Teaching

## Wolfram|Alpha as a self-verification tool

Last week, I wrote about structuring class time to get students to self-verify their work. This means using tools, experiences, other people, and their own intelligence to gauge the validity of a solution or answer without uncritical reference an external authority — and being deliberate about it while teaching, resisting the urge to answer the many “Is this right?” questions that students will ask.

Among the many tools available to students for this purpose is Wolfram|Alpha, which has been blogged about extensively. (See also my YouTube video, “Wolfram|Alpha for Calculus Students”.) W|A’s ability to accept natural-language queries for calculations and other information and produce multiple representations of all information it has that is related to the query — and the fact that it’s free and readily accessible on the web — makes it perhaps the most powerful self-verification tool ever invented.

For example, suppose a student were trying to calculate the derivative of $y = \frac{e^x}{x^2 + 1}$. Students might forget the Quotient Rule and instead try to take the derivative of both top and bottom of the fraction, giving:

$y' = \frac{e^x}{2x}$

Then, if they’re conscientious students, they’ll ask “Is this right?” What I suggest is: What does Wolfram|Alpha say? If we type in derivative of e^x/(x^2+1) into W|A, we get:

The derivative W|A gets is clearly nowhere near the derivative we got,  so one of us is wrong… and it’s probably not W|A. Even if we got the initial derivative right in an unsimplified form, the probability of a simplification error is pretty high here thanks to all the algebra; we can check our work in different ways by looking at the alternate form and at the graphs. (Is my derivative always nonnegative? Does it have a root at 0? If I graph my result on a calculator or Winplot, does it look like the plot W|A is giving me? And so on.)

But how is this better than just having a very sophisticated “back of the book”, another authority figure whose correctness we don’t question and whose answers we use as the norm? The answer lies in the  “Show steps” link at the right corner of the result. Click on it, and we get the sort of disclosure that oracles, including backs of books, don’t usually provide:

Every step is generated in complete detail. Some of the details have to be parsed out (especially the first line about using the Quotient Rule), but nothing is hidden. This makes W|A much more like an interactive solutions manual than just the back of the book, and the ability given to the student to verify the correctness of the computer-generated solution is what makes W|A much more than just an oracle whose results we take on faith.

Using W|A as a self-checking tool also trains students to think in the right sort of way about reading — and preparing — mathematical solutions. Namely, the solution consists of a chain of steps, each of which is verifiable and, above all, simple. “Differentiate the sum term by term”; “The derivative of 1 is zero”. When students use W|A to check a solution, they can sit down with that solution and then go line by line, asking themselves (or having me ask them) “Do you understand THIS step? Do you understand THE NEXT step?” and so on. They begin to see that mathematical solutions may be complex when taken in totality but are ultimately made of simple things when taken down to the atomic level.

The very fact that solutions even have an “atomic level” and consist of irreducible simple steps chained together in a logical flow is a profound idea for a lot of students, and if they learn this and forget all their calculus, I’ll still feel like they had a successful experience in my class. For this reason alone teachers everywhere — particularly at the high school level, where mechanical fluency is perhaps more prominent than at the college level — ought to be making W|A a fixture of their instructional strategies.

## Friday Random 10: 1/1/2009

No re-start of this blog would be complete without a return to the Friday Random 10 feature, where I pull off 10 random songs in a row from the iPod and do some kind of video focus on one song or artist that shows up. Here you go:

1. “Black Friday” (Steely Dan, Katy Lied)
2. “Broken” (Jack Johnson, Sing-a-Longs and Lullabies (Curious George soundtrack))
3. “Hammer to Fall” (Queen, Classic Queen)
4. “The Dancing Flowers” (The Wiggles, Whoo Hoo Wiggly Gremlins)
5. “Work in Progress” (Alan Jackson, Drive)
6. “Let Everything That Has Breath” (Phillips, Craig, and Dean, Let My Words Be Few)
7. “Spanish Fantasy” (Phil Keaggy, Acoustic Sketches)
8. “Can You (Point Your Fingers and Do the Twist)” (The Wiggles, Here Comes the Big Red Car)
9. “Partita #3 (iv)” (Paul Galbraith, Bach: The Sonatas and Partitas)
10. “The Calling” (Yes, Talk)

If by some accident you have never heard of Phil Keaggy (#7), here’s a video that gives an idea why he’s all over my music library. This is “Addison’s Walk” from Beyond Nature, which was a staple of my graduate school-days listening diet.

That’s just ridiculous.

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Filed under Friday Random 10, Music, Weekly features