long division and time

from James Milgram's talk:

In mathematics many skills must be developed for many years before they can be used effectively or before applications become available.

First of all, I claim that taking—even asking to take [long division] out of the curriculum—shows a profound ignorance of the subject of mathematics. The point is, in mathematics, many, many skills develop over an extended period of time and are not really fully exploited until perhaps 10, 12, or even 15 years after they've been introduced. Some skills begin to develop in the first or second grade and they do not come to fruition or see their major applications until maybe the second year of college. This happens a lot in mathematics and long division is one of the key examples.

• Students cannot understand why rational numbers are either terminating or (ultimately) repeating decimals without understanding long division.

• Long division is essential in learning to manipulate and factor polynomials.

• Polynomial manipulation and factoring are skills critical in calculus and linear algebra: partial fractions and canonical forms.
  

I regard the repeating decimal standard as relatively minor, but some people seem to think it is important. The next topic is critical and almost everyone thinks it's minor (Laughter). Long division is essential to learning to manipulate polynomials. Without it, you simply cannot factor polynomials.

So what, you ask? Again, this is a question that doesn't come up until the third year in college. At this point the skills that have come from long division through handling polynomials become essential to things like partial fraction decomposition which is important in calculus but finds its main applications in the study of systems of linear differential equations, particularly in using Laplace transforms, which is the critical construction in control theory. It is also essential in linear algebra for understanding eigenvalues, eigenvectors, and ultimately, all of canonical form theory -- the chief underpinning of optimization and design in engineering, economics, and other areas.

[snip]

What happens when you take long division out of the curriculum? Unfortunately, from personal and recent experience at Stanford, I can tell you exactly what happens. What I'm referring to here is the experience of my students in a differential equations class in the fall of 1998. The students in that course were the last students at Stanford taught using the Harvard calculus. And I had a very difficult time teaching them the usual content of the differential equations course because they could not handle basic polynomial manipulations. Consequently, it was impossible for us to get to the depth needed in both the subjects of Laplace transforms and eigenvalue methods required and expected by the engineering school.

But what made things worse was that the students knew full well what had happened to them and why, and in a sense they were desperate. They were off schedule in 4th and 3rd years, taking differential equations because they were having severe difficulties in their engineering courses. It was a disaster. Moreover, it was very difficult for them to fill in the gaps in their knowledge. It seems to take a considerable amount of time for the requisite skills to develop.