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This week (and last) in screencasting: Functions!

So we started  back to classes this past week, and getting ready has demanded much of my time and blogging capabilities. But I did get some new screencasts done. I finished the series of screencasts I was making for our calculus students to prepare for Mastery Exams, a series of short untimed quizzes over precalculus material that students have to pass with a 100% score. But then I turned around and did some more for my two sections of calculus on functions. There were three of them. The first one covers what a function is, and how we can work with them as formulas:

The second one continues with functions as graphs, tables, and verbal descriptions:

And this third one is all on domain and range:

The reason I made these was because we were doing the first section of the Stewart calculus book in one day of class. If you know this book, you realize this is impossible because there is an enormous amount of stuff crammed into this one section. Two items covered in that section are how to calculate and reduce the difference quotient \frac{f(a+h) - f(a)}{h} and doing word problems. Each of these topics alone can cover multiple class meetings, since many students are historically rusty or just plain bad at manipulating formulas correctly and suffer instantaneous brain-lock when put into the presence of a word problem. So, my thought was to go all Eric Mazur on them and farm out the material that is most likely to be easy review for them as an outside “reading” assignment, and spend the time in class on the stuff that on which they were most likely to need serious help.

Our first class was last Tuesday and the second class wasn’t until Thursday, so I assigned the three videos and three related exercises from the Stewart book for Thursday, along with instructions to email questions on any of this, or post to our Moodle discussion board. I made up some clicker questions that we used to assess their grasp of the material in these videos, and guess what? Many students didn’t have any problems at all with this material, and those who did got their issues straightened out through discussions with other students as part of the clicker activity.

They’ll be assessed in 2 or 3 other ways on this stuff this week to make sure they really have the material down and are not just being shy about not having it. But it looks like using screencasts to motivate student contact with the material outside of class worked fine, at least as effectively as me lecturing over it. And we had more time for the hard stuff that I wouldn’t expect students to be able to handle, not all of them anyway.

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Filed under Calculus, Education, Educational technology, Math, Peer instruction, Screencasts, Teaching

Free textbooks: It can be done

The last time I taught abstract algebra, I used no textbook but rather my own homemade notes. That went reasonably well, but in doing initial preps for teaching the course again this coming fall I realized my notes needed a serious overhaul; and since I’m playing stay-at-home dad to three kids under 6 this summer, this is looking more like a sabbatical project than something I can get done before August. So last month I set about auditioning textbooks.

I looked at the usual suspects — the excellent book by Joe Gallian which I’ve used before and really liked, Hungerford’s undergraduate text*, Rotman — but in the end,  I went with Abstract Algebra: Theory and Applications by Tom Judson. I would say it’s comparable to Gallian, with a little more flexibility in the topic sequencing and a greater, more integrated treatment of applications to coding theory and cryptography. (This last was something I was really looking for.) There’s even a free companion to the book which incorporates Sage, which I am sorely tempted to use as well because learning Sage has been a pet project of mine.

But what’s really different about this book is that it’s free, licensed under the GNU Free Documentation License. I am having the bookstore prepare print copies for the students — I asked the students if they wanted a print version in addition to the free PDF’s online, and they said “yes” — which the bookstore will sell for a whopping $16.95, just enough to cover the costs of copying and 3-hole punching the 400+ pages of the book. I’m happy because I found a book that really fits my needs; the students are happy because they get a good book too, for a tremendous bang-to-buck ratio.

In the long and contentious comment thread for my post about James Stewart’s new $24M mansion, I suggested that Stewart should consider topping off his impressive (and apparently lucrative) teaching and writing career by making his Calculus book freely available online for anybody who wants it. That suggestion was met with shocked incredulity: “If you had any idea how much work it was to write and maintain a textbook, you’d never consider making it free.” Well, I’m happy to report that hard work and good writing need not necessarily be mutually exclusive with giving it away.

In fact, as more well-written textbooks appear for free online — and there were even more free abstract algebra e-books I did not end up selecting — the commercial market might find itself in trouble.

* Actually, I requested the Hungerford algebra book, complete with a crystal-clear note that I needed to have it in hand by April 10 in order to be able to adopt it in time for our bookstore. To this date I have not received it. Another problem with commercial textbooks: the distribution model for review copies is dreadful. I’m always receiving multiple copies of books I neither need nor am interested in, and not getting the books I do need and am interested in.

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Why do we overcomplicate calculus like this?

A labelled tangent to a curve. Created to repl...
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In the Stewart calculus text, which we use here, the first chapter is essentially a precalculus review. The second chapter opens up with a treatment of tangent lines and velocities, with the idea of secant line slopes converging to tangent line slopes and average velocities converging to instantaneous velocities taking center stage.

Calculating average velocity is just a matter of identifying two time values and two position values and then performing two subtractions and a division. It is not complicated. Doing this several times for shorter and shorter time periods is also not complicated, and then using the results to guess the instantaneous velocity is a little complicated but not that bad once you understand the (essentially qualitative, not quantitative) idea behind shrinking the length of the interval to get an instantaneous value out of a sequence of averages.

So I nearly hit the roof when a student came in this morning needing help understanding the Student Solutions Manual for the Stewart text on a problem where you had to find the average velocity of a moving object from 2 seconds to 2.5 seconds. A formula for position is given, y = s(t). The simple way to do this — the way that works, does not dumb the process down, and yet makes it understandable to the broadest possible audience and therefore sets  up general understanding of the more complicated idea of derivative calculations later — is to calculate s(2.5), calculate s(2), and then calculate \frac{s(2.5)-s(2)}{2.5 - 2}. Fifth-graders do this.

Instead, the Student Solution Manual does it like this:

  • Let h represent some positive number.
  • Calculate and fully simply the expression \frac{s(2+h)-s(2)}{h}.
  • Plug in h = 0.5.

This is crazy, absurd, and downright dangerous. It’s as if Stewart, and the person who wrote the manual, really believe that calculus is made up of algebra, and students who are in calculus are uniformly comfortable and skilled with algebra to the point that their way is just as transparent and simple as calculating distance divided by time — as if the algebraic work that ensues when you perform step (2) above were as natural as the concept of velocity itself and students spoke algebra like a first or second language.

Yes, the book’s approach works — and it closely mirrors what’s going to happen later when we want to get an exact value of the instantaneous velocity by letting h \rightarrow 0. But that’s not what students are doing right now. What students are doing is trying to understand the concept of average velocity. It’s not complicated. The complications should come, if at all, on the back end of the subject — where we are trying to make the concept of instantaneous velocity precise through limit calculations — but not on the front end when students are just trying to figure out what’s going on.

In the middle of typing this post out, another student came in, equally confused about the exact same problem. I told him to close his solutions manual. I asked him: What’s the definition of average velocity? He thought about it, and then gave me the right definition. “OK, then,” I said, “How would you get the average velocity from t=2 to t=2.5 here?” And he gave me an exactly right description of the process. The relief on his face was palpable. He understood this concept but the student solutions manual made it appear that he didn’t! How bad is it when you need a manual for the student manual?

Calculus is a really simple subject when you get to its core. I wish the book treated it that way.

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