Students have been too weak at algebra since before I started teaching more than three decades ago. Nevertheless, today's students are remarkably weaker than those of the Sixties. Three years ago, I had a class of 15 first semester calculus students *none* of whom had a clue about how to deal with the difference quotient associated with finding the derivative of the cubing function. And these were the 15 out of perhaps 25 or 30 who had stayed in the class after it had become clear that I would not allow them to decline to engage the material, as they clearly had done in previous courses.
I would like to suggest that it is the latter students (some of whom had quite good technical skills, as evidenced by their scores on our placement test and their grades in prerequisite courses) who have poor study skills. They are the ones who think they should be able to assimilate mathematics without investing time and effort.
Someone in this discussion (perhaps in a private message to me, which I've lost) noted that the tension that results from failing large numbers of students in calculus is worth it to keep standards high. It was a remarkably vague statement: What is a "large number" and what is "high"? We can not consistently flunk 80% to 90% of our calculus students--regardless of the level of algebra skill they bring to our calculus courses. Once the kids are in calculus, we have to deal with them as they are. And, as several posts to this list have pointed out, the cost of keeping them out can be quite high, also.
In this connection, Wayne Bishop wrote:
> Ill prepared arithmetic students have been flunking algebra in > droves throughout my lifetime and long before and ill prepared algebra and > trig students have been flunking calculus in droves as well.
From this he concludes that it must always be so. And perhaps he's right. On the other hand, it's safe to guess that, in 1800, nobody had ever travelled from New York to San Francisco in under a month. People of that year might well have predicted, on the basis of their experience that it had never been done, that it could not be done. Technology, in the form of the railroad, changed the picture entirely, so that old experience no longer held predictive value.
It used to be the case that students flunked trigonometry because they hadn't mastered calculation with common logarithms--especially those who also hadn't mastered long division. Because of technology, that no longer happens; I know of nobody who regrets it.
I am quite willing to let a machine handle the mechanical details of algebra, just as I am willing to let a machine do my long divisions and calculate my logarithms--when I need logarithms, which is not as often as it used to be. A computer will do the mechanical part of partial fractions decomposition just fine, and that's OK with me, too. Students ought, nevertheless, to know what a partial fractions decomposition is and what form to expect the p.f. decomposition of a given rational function to have. That might be half a lecture's worth or half an hour's reading. (Yes--this reformer still lectures. Sometimes.)
The question of related rates is somewhat different. I've included this topic under the Algebra heading because I think that the difficulty students have here is more with algebra than with calculus. Once they have the right equations, the majority know what they must do and how to do it. The difficulty isn't with manipulation, but with translation from English to algebra. I'm convinced that there are deep cognitive issues involved here; most students simply don't seem to be able to make the right kinds of connections. I don't think that drill of the "simplify these; rationalize those, etc." sort will cure this kind of difficulty. Nor will machines do that kind of translation. This is a thorny problem that I don't have a solution for. I do ask students to do some simple related rates problems.