I thank you for mentioning this issue of the Notices from the AMS that contain these four replies to Dudley's article.
David A. Edwards from the University of Georgia says that relatively few positions even in science and engineering rarely use much mathematics beyond eighth-grade mathematics (whatever eighth grade mathematics is exactly, I'm not sure, and that seems to vary some from state to state anyway) and that they require technical degrees as merely filters. He also mentions Vivek Wadhwa's statement that America is producing more scientists and engineers than there are job openings. I take that he is referring to his article that I had found at http://www.businessweek.com/smallbiz/content/oct2007/sb20071025_827398.htm.
I'm no scientist or engineer, so I cannot say whether such claims are true or not. But I can say that our schools and even colleges are in such a mess these days that high school diplomas and even many college degrees are not worth more than the paper they are printed on. How else do you explain the countless floods of college students I see whose reading, writing, math, study, and even common sense skills are still stuck in second or third grade? It disturbs me to see all the discussions and assignments in class that appear as if they were written by the students' children rather than by the students themselves! I would conjecture that massive grade inflation troubles employers enough that many cannot trust that ones with only a high school diploma or college diploma truly have meaningful diplomas because of many who do manage to graduate without learning much of anything. I suppose that they can test the skills of those they might be interested in hiring, but I imagine that such testing is time consuming and expensive. I thought about that when I have thought about a job as a statisician, but I believe I would need a stronger background in statistics to qualify or at least to give myself a strong chance of getting such a job. But I then realized this dilemma if I choose to study some statistics on my own: How can I show that I have learned more statistics than what my degrees and transcripts show? I would not blame employers in the least bit if they did not believe me because anyone can make such claims. And it would take a lot of their and my time to show that I indeed did study on my own. So this thinking has told me that I am sure that employers want students with credentials but that there is tangible proof of such credentials and that the proof of such credentials is actually meaningful proof and not simply a fancy version of some scribbled note by someone saying that John Q. Smith really has these credentials and that I witnessed this myself.
I doubt these claims myself, but I don't work in science or engineering to know how to test this claim or to refute it. The best I can do for now is to ask some colleagues I know who work in science.
However, let us suppose Edwards' claim is true. Does this mean that students who stop with arithmetic are really competent enough to understand the mathematical side of science? Perhaps some are, but few would be. I myself would doubt this seriously because of the meaningless way that most elementary mathematics is taught and, on top of that, with the massive grade inflation these days so that many students can finish arithmetic with good grades but understand very little of it. And I would venture that most who work in education know that virtually all students finish arithmetic with good grades but don't understand much of it. Those who know what subject knowledge it takes to teach mathematics effectively generally realize and agree that teachers should know mathematics at a higher level than the level they will teach because the extra mathematics courses help them learn (at least they should, but that is not always the case) the mathematics they will teach much better than otherwise. A few other reasons are often given as well, but this one reason is an important one and pertinent to the discussion here. If any form of mathematics is a significant part of the job--whatever level that math may be, then the students should learn the mathematics and should learn it well. Furthermore, as Sherman Stein mentions in one of these replies, it is better to overprepare in mathematics than to underprepare in mathematics in case of changing career goals and also because further preparation in mathematics can help students understand better what mathematics they will use.
One of the most important goals of learning mathematics that is sorely missing from elementary math courses is teaching students how to study and learn mathematics for themselves and how to learn to mature mathematically. A major difficulty I see with students in mathematics and statistics courses is that students have little mathematical maturity and little idea of what it means to think mathematically.
And few of them understand symbolic reasoning, which makes it difficult for them to learn the reasoning behind mathematics. And that also makes it difficult for them to learn algebra. In fact, so many students I have seen have such little understanding of any form of symbolic reasoning that they have little idea of what it means to use a formula! Rather than stressing algebra as we do, why not emphasize more about the teaching of symbolic reasoning? Until students can learn to make sense of symbolic reasoning and learn to make sense of mathematical statements with letters in them, algebra and other symbolic mathematics will make little sense to them, and their learning will be wasted.
And we should emphasize logic and reasoning and conceptual understanding in arithmetic. Far too many students take arithmetic and "not get it." And far too many students end up thinking that mathematics is mechanical-- nothing but following recipes and plugging and chugging into formulas.