On Thu, 12 Dec 2013 12:51:21 -0700, Robert Hansen <email@example.com> wrote:
...what does “moment of inertia” have to do with it?
That is indeed the question. At the level the kids are working, "moment of inertia" has absolutely nothing to do with it, and it is dishonest to suggest that it is otherwise. The kids are learning to substitute numbers for the symbols found in a formula. In spite of the "moment of inertia" label, this formula is meaningless to them. What is the point of pretending otherwise? Be honest: Give them a meaningless formula, tell them it's meaningless, and ask them to make the substitutions.
Can you create as involved an expression about something they would know about?
Why would I try? There's nothing wrong with involved expressions in general, or with this one in particular. But the issue here isn't how involved the expression is, or even where it comes from. It's the pretense that the student is doing something meaningful---when that is anything but the case for the student. When we make such misrepresentations, we teach students things we don't want them to learn---as you confessed below.
Granted, this is an algebra 2 problem. I don’t know why Dan is commenting on an algebra problem at all, at any level, he doesn't teach it in his blog. But he did comment on it. I look at this problem in the context of the level it is meant to be. It would be a horrible arithmetic problem.
Exactly: Look at the problem "in the context of the level it is meant to be." That context includes no understanding of where those complicated formulae come from or how. Why pretend otherwise by giving the formulae meaningless labels and suggesting that the kids are doing something real? Every one of them who thinks at all knows damn good and well that they are not. I'm not objecting to the substitution part of the problem; I'm objecting to the cloud of misrepresentation that surrounds it.
No, it shouldn’t be mysterious.
Then what is the point of making it so?
It appears to be mysterious to Dan and his group as well.
What is mysterious to Dan and his group is something very different. They understand only that there is an issue; they haven't identified it.
But the stimulus is there. We went right to it. It’s that mess of letters and numbers off to the left. That is the challenge, to train the student to see the math.
But here, the real mathematics is beyond the level at which the students are working. It lies in knowing what moments of inertia are, what they mean, and how the formula that confronts them is developed. That's the irremediable flaw in presenting the problem as "real engineering mathematics" when it is really nothing more than a poorly camouflaged exercise in making simple substitutions.
Do you really want the majority of students to think that what engineers do consists of sitting around and spending their days making simple substitutions in complicated formulae they don't understand?
Students figure out very soon that mathematics is not just a collection or rules to be memorized. Well, differential equations is, but the rest of it isn’t.:) You know how they figure this out? Cause they fail the damn tests!
A few do this. The vast majority write mathematics off as devoid of intellectual content.
Assuming of course that the tests are of sufficient quality. Our challenge is to make them think and not about something entirely else, but about this. About all the things around the collection of rules to be memorized and that make the collection of rules work in the first place. The more you do that, then the more they will rely on sense and instinct and less on memorization. But you still have to have substance to do this and technical problems like this are a mainstay in that exercise, for several reasons. Primary of which is that in applied math these are the problems algebra is used for. Secondly, you just can’t find a better source of technical problems than nature itself.
Are you suggesting that *this* is a technical problem? That's funny! Especially because problems whose goal is learning to substitute numbers for letters are problems that the kids you call "mathy" shouldn't need to do more than two or three of!
Sure, the subject of any problem with this many terms is going to be unknown to the students, but that doesn’t hurt the exercise when your focus is on the math and not the subject.
That's the trouble. It doesn't hurt the exercise--it hurts the vast majority of students.
The real solution is to inch the students into technical problems in subjects the student doesn’t know.
The real solution is to inch the students into the mathematics that the students don't know. The time for what you are now calling "technical problems" is when they are studying the science itself, using appropriate mathematics that they have already learned or are learning at the same time.
This is the type of stuff I was expecting from Stanford edu. Instead they sent us Dan.
Not being one, I don't know if engineers approach mathematics this way or not. But I've had engineering students in my classes, and I have my suspicions. Bob's suggestion that the problem in question is a good one tends to confirm those suspicions.
It is applied mathematics. That is why I majored in physics and not engineering. Not because the math was better, it was pretty much the same. But at least the physics was better.
Could that just possibly be because there is no excuse for pretending that we are doing physics in a mathematics course, when the real physics is beyond the students' mathematical capacity? And yet, your teachers evidently made the mistake of trying to carry out that pretense. And in so doing, they taught you something about mathematics that is incorrect.
What is bad is that students are not born with a complete understanding of the principles of everything, except math, so that we can then show them how to apply math to everything. It is not feasible, effective nor pedagogically sound to design an algebra course around the theory of the physical world an 8th or 9th grader has. That’s one of the challenges of being an algebra teacher.
Just so. (Except that students aren't born with a complete understanding of the principles of mathematics, either.) And if it is neither feasible, nor effective, nor pedagogically sound "to design an algebra course around the theory of the physical world an 8th or 9th grader has," then why pretend to have done so?
The point is this: Students know when we are lying to them, and they resent it. And regardless of whether or not engineers work problems like this, the suggestion that this problem should mean something to students of elementary algebra because "It's an application" is a lie. It's an exercise (nothing wrong with that!) dressed up in Sunday-go-to-meeting-clothes in a dishonest attempt to convince students otherwise (and there's plenty wrong with that!).
Department of Mathematical & Computer Sciences
Metropolitan State University of Denver