Monday, February 27, 2006


Well, for those who were wondering, the pile is gone now. (See below.) It took about a week, but someone hauled it off.

But the existence of the pile launched an interesting debate with the fellow I carpool with, about what I will call "classifications of rudeness", for want of a better term. I was upset at what I perceived as a case of generic public rudeness which I find too common. I see the reasoning as "We don't need anything from him anymore, so there's no point in being nice. Just dump it in his yard and let it be his problem." (I'm going to make what I think is an entirely reasonable assumption: there is no fair reason for someone to dump junk in my yard for a week without even mentioning it to me.)

My friend responded that it's not unlike current politics: "Screw the women, children, gays, minorities, and poor people--I want a tax cut!" I mostly agree with him here, but I do disagree on a minor point: I don't think most people thinking "I want a tax cut!" actually realize that the money comes from somewhere, or that anyone is actually hurt by whatever politics they endorse. It's too abstract in their minds. Tax cuts are just magic, and don't affect anyone else. In that sense, I think it's more like the trash you see dumped by the side of a highway somewhere. I get the sense that the people dumping there don't actually think about the fact that, yes, someone has to clean up this mess eventually. As a semi-public area, I think they see it as belonging to no one at all. If you actually asked these people what would happen to it, they would eventually acknowledge that someone else would indeed have to fix the problem, and I generously assume that at least a few people would be abashed and pick their stuff back up again when they realized what they had done. (Maybe not many, but at least a few. I also think at least a few dumpers would feel guilty if they realized that someone actually has to clean up their mess, even if they still dumped it.)

In the case of dumping stuff on your neighbor's lawn, I think no one can possibly be unaware that they are creating a problem for someone else. (Heck, in that case you can point your finger and say "I'm passing my problem on to him .") Both are good examples of a lack of empathy, which I think is a HUGE social problem, but I do think there is a (small) distinction.

My friend thinks I give too much credit (or too little, depending on how you view it). He thinks people dumping stuff by the highway are perfectly aware that someone else has to clean up the mess, and they just don't care. I don't doubt that there are people who are aware and still dump, but I think for most people it's just magic. Somehow the stuff just disappears without anyone having to do anything. (They wouldn't say that if someone asked, but I think it's the way some people think.)

I'm certainly not arguing that there's no connection between dumping stuff in a neighbor's yard and dumping stuff by the highway. I'm just arguing a slight distinction in how some people decide to do it.

The whole discussion however reminds me of something I saw during the Republican national convention in 1996. I was watching a bit one morning when Newt Gingrich was speaking to a group of college Republicans. He joked that "Conservatism starts when you get your first paycheck from your first job and you see how much money got taken out of it." (Very rough paraphrase; I have no idea the actual wording.) I remember responding immediately: "Yes, Newt. And Liberalism starts when you finally get your nose out of your own paycheck and notice your neighbor's kids are starving." Of course there are intermediate stages. For example, conservatism gets moderated a bit the first time you hit a massive pothole which causes a few thousand dollars damage to your BMW and decide that maybe some government spending on things like roads would be OK after all. But I think Newt characterized his movement perfectly: "Me, me, me." Maybe I should just dump stuff I don't want by the side of the highway. And on an unrelated topic, why do we have to pay so much tax money to road crews?

Thursday, February 16, 2006

Student Evaluations

In my student evaluations from a class last semester I found the following two comments back-to-back:
He makes class more fun through his voice expressions and overall demeanor.
His monotone voice created such a doldrom environment that made me want to cease to exist.
Without having to get into a question of whether I'm an exciting speaker or not, could these two students have been sitting in the same classroom? That said, a thank you to the first student. If I hadn't seen that comment right before I found I had the power to make people want to cease to exist, I would have been depressed for a week. Now I just feel disappointed I couldn't talk to the second student for longer.

Wednesday, February 15, 2006

Calculators, II (The Social Rant)

Hey, welcome back to all the Not-Calculus-Dweebs who skipped the previous post. :-) Fair warning: I'm probably going to sneak some math in on you in this post anyway.

In working in a committee on a question for a calculus test, we ran into some difficulties in writing a question that would not be made completely pointless by the students' calculators. It led to an interesting argument between me and someone else over the use of calculators in math classes. Her position was that students should never use calculators in any math class. (Her words; I'm really not exaggerating here.)

I know this sentiment (or something similar) is not uncommon, but I've never understood it. I didn't see the point when I was a student of spending lots of time in long, tedious computations that the machine could do both faster and more accurately, and I don't see it now. I was slow on my times tables when I was in grade school, and hated math. Who wanted to do something that involved memorizing endless lists of facts? (I was particularly perplexed by the insistence that some people had on memorizing the times tables up to 12. If you knew up to 9, you could do all the rest by hand anyway; why bother?) Through grade school and junior high and even the first year of high school, I pretty much hated math. Math was boring. Math consisted of memorizing tables, finding common denominators, memorizing and practicing long division algorithms, and a bunch of other really boring things that amounted to spending 20 minutes of hard work with pencil and paper to tell you what a calculator could have told you in 20 seconds anyway.

The first time I realized I had some interest in math was in my geometry class in high school, where we started doing proofs. Proofs were interesting. This involved thinking and being creative, as well as being precise and logical.

Do I occasionally shake my head at my students inability to add, subtract, multiply, or divide without a calculator? Yes, I do indeed. (I had a student yesterday with the unreduced fraction 9/18.) Do I think this is a major problem? No, not really. If it's a common problem someone faces, they will learn how to deal with it through practice. If it's something they rarely have to deal with, I doubt it makes much difference.

As an aside, a certain amount of calculator dependency is different from a basic lack of understanding. I am concerned by students who cannot simplify things like (2)(7/2). A friend of mine had a student complain during an exam that "You said we wouldn't need a calculator!" The student was stuck because their answer had come out to the square root of 0^2. These are not just computational weakness, they are fundamental misunderstanding of how the operations work.

One point we debated over was whether students needed to memorize certain values of the trig functions, like that sin(pi/4) = 1/sqrt(2). I'm not sure that it matters much most of the time whether they say sin(pi/4) is 1/sqrt(2) or approximately 0.7071. She says that it's important to emphasize exactness. If that's true, I want to know what sin(1) is. Exactly. The point being that we like to talk about a very tiny subset of possible angles which we can reason out another expression for. I certainly feel like my students should at least be able to reason out these common angles, but don't see a point in memorizing them. (Of course, I admit I'm a total hypocrite sometimes; I usually feel fairly exasperated when they don't know what sin(0) or cos(0) are.)

So I'm basically not a big fan of lots of memorizing in mathematics, even when it's things I know, but I'm still conflicted sometimes. To what extent am I expecting my students to memorize things just because I know them?

I often think of a story my Math Ed professor in college told. He explained that when typewriters and pencils were becoming commonplace and available to all, the naysayers lamented loudly that there would be a vast decline in the quality of people's handwriting. People would no longer devote time and effort to perfecting their handwriting, and most people would end up with very poor script, perhaps all but illegible. And, he says, they were right. Handwriting did decline, and most people have horrendous penmanship when compared to people of 50 or 100 years ago. We are almost completely dependent on aids like typewriters (now computers) to communicate clearly with any speed. Sure, if absolutely necessary, we can still write by hand--slowly and laboriously, and probably still not as neatly as generations before. But the world didn't collapse; civilization didn't come to an end. People use crutches, but the crutches aren't going anywhere, so how much harm was really done? Now it seems silly to worry about this, and maybe in another 50 years it will seem silly to worry about whether someone uses a calculator for most of their basic math.

Calculators, I (The Technical Post)

Gentle readers: This post contains math. If you are a Not-Math-Dweeb you may not find it at all interesting and may want to just skip to "Calculators, II" above. If you are (like me) a Math-Dweeb of some flavor, you may find this interesting. I've been tinkering with these two posts since last semester when the issue came up and finally decided to just post.

I've was on a committee last semester (yes, more meetings) to design a calculus test. We'd run into some difficulty on asking about using a linearization to estimate a function's value. On a previous test, we had a question similar to this:
Use a linearization to estimate the value of ln(1.1).
I had a student who responded with something like this:
L(x) = ln(1.1) + (1/1.1)(1.1-x) (approximately equals) 0.0953 + (1/1.1)(1.1-x),
so ln(1.1) (approximately equals) 0.0953 + 0 = 0.0953.
(I've probably been somewhat more clear than the student has, but you get the idea.) How do I grade this? The student has in fact used a linearization to estimate the value of ln(1.1)--they just chose to use the linearization of ln(x) at the value x = 1.1.

In the narrow view, the student has answered the question correctly and deserves full credit. (Heck, they even know how to linearize a function, which is essentially what the question is asking.) In the broad view however, they are completely clueless as to why we might use a linearization. In this sense, they have no idea what is going on in the problem, and should probably get very little credit. (I don't remember anymore what I eventually decided when I graded this.)

Of course, they were only able to do this because they had a calculator that could give a decimal approximation to ln(1.1). This makes writing tests in a calculator age all the more challenging: How do you write a question like this in a way so as to circumvent such an answer?

There are ways to get around it, but most are unsatisfactory. Here's some of the initial options we discussed:
  1. Tell them where to linearize the function. (Downside: They don't have to make an intelligent decision about where would be a good place to linearize, which is part of what we would like to test.)
  2. Give them information about f(1) and f '(1) for some unknown function, and ask them to estimate this. (Same downside as #1.)
  3. Tell them to make the linearization at a point where ln(x) is a whole number. (Downside: This still seems to give away a lot, really.)
  4. Tell them they have to do this based entirely on values for ln(x) which they already know without use of a calculator. (Downside: Someone who is a Wise Ass (but who may or may not be Tall) will say they just "happened to know" that ln(1.1) is about .0953.)
Not very good choices, really. But I came up with two new ones while I was starting this post, which I like:
  1. Give them a table of values for an unknown function f and f ', and ask them to use a (wisely chosen) linearization to estimate the value of f at some intermediate point.
  2. Have them come up with the linearization for a function based at two different points, then ask them to choose an appropriate one of the two to use for estimating the desired value.
See, and you thought writing these posts wasn't accomplishing anything for me? (We ultimately went with the second choice.)

So I've solved the intermediate problem, and now I have a deeper problem to work with: Why do I care if they can do without a calculator what the calculator can do for them?

The classical problems in most calculus texts covering linearization tend to be problems like estimating the values of ln(1.1) or the square root of 10. While it's relatively cool that some simple calculus techniques and a back-of-the-envelope calculation can give you such estimates, this is not really what most people do. There are certainly many reasons to use linearization other than these kinds of estimates. I discuss with my class the linearization of the motion of a pendulum, for example (although they are not ready to understand the details of the differential equation). There's a lot of physics based around simple linearizations like sin(x) (approx equals) x or ln(x+1) (approx equals) x-1. Also, it's the beginnings of the theory for how the calculators and computers know the value of ln(x) or sin(x). (Admittedly, we are talking baby steps here, but you have to start somewhere.)

I think there might be value in asking the students to come up with and use a linearization in the simple way we do, even if their calculator can find the value more quickly and accurately. It's a good check to see if they understand both how to find a linearization, and what it means--namely, that it is a simplification of a complicated function. So maybe it's an unrealistic problem which is simultaneously a good test question.

Tuesday, February 14, 2006

1984 took a little longer than expected

I ran across the following three articles within about two minutes of each other, and the headline above is the first thing that came to mind. (All three came to me via "Wren's Nest Spirit News", which generally includes a lot of interesting articles. I'm quoting a small subset of what she quoted.)

"E-Tracking Through Your Cell Phone" (from CNet)
You may already know this, but your cell phone happens to be a miniature tracking device that can be used to monitor your location from afar. [. . .]

But the FBI and the U.S. Department of Justice have seized on the ability to locate a cellular customer and are using it to track Americans' whereabouts surreptitiously--even when there's no evidence of wrongdoing.
(The government says "Trust us; we'll only do it to bad people. We promise.")

Next, we have
"US Group Implants Electronic Tags In Workers": (From the Financial Times)
An Ohio company has embedded silicon chips in two of its employees - the first known case in which US workers have been 'tagged' electronically as a way of identifying them.
(The company explains there's nothing wrong with this--why, it's just like an ID card. A permanent, surgically implanted ID card that lets people track you surreptitiously.)

Finally, after both of these is the following story, which may explain why I don't feel an overwhelming sense of trust about the first two stories:
"Mere Mention Of Violent Talk Brings Silence" (From The Albuquerque Tribune)
I would have told Laura Berg that, until she came along, I hadn't heard the word sedition since journalism school. [. . .]

I would have asked her if the guys who took possession of her work computer not only peeped all around it, but maybe left a little something inside it. In a memo, her lawyers say, the VA said it suspected her of committing an act of sedition. [...]

I would have said all of that and more, but these days, Berg isn't talking. Not after her employer of 15 years, the Veterans Affairs Hospital, took offense at a letter to the editor in which she sharply criticized the Bush administration.
That's all I have to say about that for the moment.

Tuesday, February 07, 2006

Reasons No. 283, 186, and 119 I don't like large classes

Reason No. 283: Inflexible scheduling. I have two students this semester who are in my large class who have a conflict between class time and their work schedule. (The students have to work to pay their tuition.) Of course, there are only two sections of this class now, and they meet at the same time every semester. There are no other sections because the two sections can hold collectively nearly 500 students. I have a similar problem with students on the swim team who get out of practice across campus 15 minutes before my class starts and have to dry, change, and run to class late. (The only other section of the course meets before mine, during their practice.) The class is required for both groups of students.

Reason No. 186: accommodating individual students is prohibitive. In a small class, I'm often fairly understanding about late homework or special circumstances, unless I have a reason not to be. One student missed an online quiz? I'll take care of it manually. Forgot the date of the test? I'll e-mail it to you. In a class of 350, what was a couple of students out of thirty-five has now become twenty students missing deadlines and e-mailing me asking about quiz dates that are already published in the course policies. (Or twenty cell phones going off in class throughout the semester, but that may really be reason no. 187.) So usually I just build some flexibility into deadlines from the start and offer some blanket extensions or opportunities to make some things up. And lots and lots of online stuff that I don't have to grade by hand.

Reason No. 119: I don't know my students. I get occasional requests from students for letters of recommendation for things like tutoring or RA jobs, or for scholarships. I rarely have much I can say about my students, since I barely know most of them. Usually I can't write much more than "According to my records, came to class, turned in all homework, got a B." Wow, that will take them far. Why are they asking me? All their classes are large now, and I'm one of the few teachers who makes an effort to find out anything about my students in large sections. So I have to try to avoid writing things like
Pat turned in all the homework and is quite tall. I think Pat's hair is brown, and I am 99 percent certain Pat is male. But don't push me on that one.

Wednesday, February 01, 2006

What I've Learned from my Teaching Job

When I was in college, I postulated that the dining hall tables had infinite capacity:
Theorem: Dining hall tables have no finite bound on the number of people sitting at them.
Proof: Suppose that n were a bound on the number of people sitting at a table, and suppose that we have n people sitting at that table. If an n+1st friend comes along, everyone will rearrange their stuff so as to make room for this person. Thus there can be no such upper bound.
It turns out that apparently the same thing applies to teaching. There is no upper bound on the amount of material I can "cover" in a fixed class period. In the job I've worked for years, we have rigid schedules for most lower level courses: if it's Wednesday, it must be the chain rule. In most cases, even the homework problems are specified by the department in advance. The first time I taught (ever), I did not reach the end of the syllabus at the end of the semester. (No one else I talked to did either; the course was ridiculously overfull, in a topic-a-day, cobbled-together kind of way.)

My problem? I was worried about whether students understood what I was teaching. Silly me. After enough departmental pressure, I learned to correct the problem. Instead of teaching, I had learned to, like a cat in a little box, "cover material." After I few semesters I could be right on time with the schedule. If needed, cut down on examples, cut out theory, stick to material that was bound to be on the final exam (if there was a common exam), and forge ahead at all costs. (Of course, was still trying to do this with a minimum of critically lost students. I can't completely change my spots.)

Admittedly, I never quite achieved the mastery of a graduate professor of mine, who covered some sections in the book "by fiat." In other words, he covered it because he said he covered it, and it was up to us to fill in what we needed to know from the text. Someday I plan to use this.

Nonetheless, I've certainly learned from my teaching job how to stick to a schedule no matter what, and finish in one day whatever appears on the schedule for that day. Which brings me to tonight. I was working off another schedule someone else designed, and found that what used to be two relatively full days had become one class period. And yet, I finished essentially on time, and with a minimum of lost students. Somehow I've covered the twice the material in the same time. Which make me wonder if there is an upper bound to the amount of material that can be covered in one class period.

Of course, all this means that I've also learned two other things:
  1. I like much better classes in which I set my own schedule and don't work off someone else's; and
  2. I really want a new job.