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Topic: On Reform, Part 2
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Posts: 1,838
Registered: 12/6/04
On Reform, Part 2
Posted: Jun 23, 1997 11:43 AM
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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,

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

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.

--Lou Talman

(To be continued)

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